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

Author:
Baha Abulnaga

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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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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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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.

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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

<12500

<1

<0.0004 1.14

Mud

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TABLE 1-7 Testing Parameters on Soils to Determine Dredgeability, Suitability for Placer Mining, or Slurry Preparation Type of soil

Testing parameters

Sand

Density Water content Specific gravity of grains Grain size Water permeability Frictional properties Lime content Organic content

Silt

Density Water content Water permeability Shear strength or sliding resistance Plasticity Lime content Organic content

Clay

Density Water content Sliding resistance Consistency ranges (plasticity) Organic content

Peat

Same parameters as clay

Gravel

Same parameters as sand

The difference between the liquid and plastic limits is defined as the plasticity index: PI = LL – PL

(1-3)

1-2 SLURRY FLOWS A slurry mixture is essentially a mixture of a carrying fluid and solid particles held in suspension. The most commonly used fluid is water, but over the years, attempts have been made to use crude oils with milled coal, and even air in pneumatic conveying. The flow of slurry in a pipeline is much different from the flow of a single-phase liquid. Theoretically, a single-phase liquid of low absolute (or dynamic) viscosity can be allowed to flow at slow speeds from a laminar flow to a turbulent flow. However, a twophase mixture, such as slurry, must overcome a deposition critical velocity or a viscous transition critical velocity. The analogy can be made here in terms of an airplane: if the speed drops excessively, the airplane stalls and stops flying. If the slurry’s speed of flow is not sufficiently high, the particles will not be maintained in suspension. On the other hand, in the case of highly viscous mixtures, if the shear rate in the pipeline is excessively low, the mixture will be too viscous and will resist flow. Sections 1-2-1 and 1-2-2 define the two basic slurry flows.

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1-2-1 Homogeneous Flows Solids are uniformly distributed throughout the liquid carrier. An example of homogeneous flows is copper concentrate slurry after undergoing a process of grinding and thickening. Particles are then very fine and the mixture is at a high concentration (50–60% by weight). As the concentration of particles is increased (beyond 40% by weight for many slurries), the mixture becomes more viscous and develops non-Newtonian properties. Apart from rich concentrate slurries, drilling mud, sewage sludge, and fine limestone (cement kiln feed slurry) behave as homogeneous flows. Typical particle sizes for homogeneous mixtures are smaller than 40 m to 70 m (325–200 mesh), depending on the density of the solids. The presence of clays in certain circuits must not be ignored. If clay is not separated, the slurry can be quite viscous. Pumps and pipes must be sized properly to handle the resultant absolute (or dynamic) viscosity. Certain mines in Peru contain material called soft high clay, which can increase the absolute (or dynamic) viscosity of the slurry up to 400 mPa at weight concentrations in excess of 45%. Dilution to lower concentration and changes to recycling load are solutions to such a problem.

1-2-2 Heterogeneous Flows In a heterogeneous flow, solids are not uniformly mixed in the horizontal plane. A gradient of concentration exists in the vertical plane. Dunes or a sliding bed may form in the pipe, with the heavier particles at the bottom and the lighter ones in suspension, particularly at the critical deposition velocity. The different phases retain their properties and the largest particles do not necessarily cause the biggest problems; it really depends on the ratio that they are mixed with finer particles. Heterogeneous slurries are encountered in many placer mining, phosphate rock mining, and dredging applications. Concentration of particles remains low, typically less than 25% by weight in many dredging applications and below 35% by weight in many tailing disposal applications. Heterogeneous flows require a minimum carrier velocity. In some tailing applications of the Taconite mines of Minnesota, the typical deposition velocity is in excess of 3.4–4 m/s (11–13 ft/s). Nature being complex, flows are encountered that have the characteristics of heterogeneous or homogeneous flows. The concept of pseudohomogeneous flows is also used when a large fraction of particles are fine but there remain a sufficient fraction of coarse particles that may deposit as the flow speed is reduced below a minimum value.

1-2-3 Intermediate Flow Regimes Intermediate regimes occur when some of the particles are homogeneously distributed and others are heterogeneously distributed. Intermediate regime flows include tailings from mineral processing plants and a wide range of industrial slurries.

1-2-4 Flows of Emulsions Strictly speaking, an emulsion is not slurry. An emulsion is a mixture of two phases at certain temperatures resulting in an essentially homogeneous flow. An example of an emulsion is a mixture of bitumen at 70% by volume with water at 30% by volume. If sur-

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factants are used, the bitumen remains well mixed with water in a certain temperature range. Emulsions can become unstable under certain high shear rates or through very tight clearances of pumps according to Nunez et al. (1996). In the 1990s, PDVSA-BITOR constructed a 300 km (188 mi) pipeline in Venezuela with a diameter of 660–915 mm (26–36 in) for transporting their ORIMULSION fuel, a mixture of highly concentrated bitumen and water. The fuel is a substitute for coal in thermal plants. Emulsions do not encounter the deposition velocity of slurries, but as flows become unstable, fine droplets of heavy oils or bitumen may coalesce into larger ones, causing changes of flow.

1-2-5 Flows of Emulsions—Slurry Mixtures A mixture of an emulsion and solids such as fine coal could be used to produce a fluid with a high calorific value. Coal must be very fine to burn readily in combustion furnaces. The flow is similar to a homogeneous flow with a high absolute (or dynamic) viscosity component.

1-3 SINKING VELOCITY OF PARTICLES AND CRITICAL VELOCITY OF FLOW Various parameters of speed determine whether a mixture may separate or continue to flow. In fact, the designer of a thickener or a mixer is often more interested in the sinking velocity of particles. On the other hand, the designer of a pipeline has to pay attention to the critical velocity of flow, settling speed, and whether the flow is vertical or horizontal, particularly in the case of heterogeneous flows.

1-3-1 Sinking or Terminal Velocity of Particles This is the minimum speed needed to maintain particles in suspension, particularly in a process of mixing or thickening. This velocity is not identical with the critical velocity of flow, and should not be confused with it. Table 1-8 presents examples of sinking velocity of various soils. The designer of a mixing system or a thickener is encouraged to conduct lab tests, since clays may be mixed with sands in some areas, or the soil may be stratified, with layers of different materials.

1-3-2 Critical Velocity of Flows In Chapters 3, 4, and 5 the mechanics of solid suspensions are described in detail. An important parameter to introduce in this chapter is the critical velocity of a slurry flow. Figure 1-8 plots the pressure loss per unit length on the y-axis, versus the velocity V of a slurry flow on the x-axis. Five points are shown for flow at a constant volume concentration. For this slurry of moderate viscosity, the flow is stationary and the solids clog the pipeline below point 1. There is insufficient speed to move the particles. As the flow is accelerated, the speed reaches point 1, which is called the deposition critical velocity VD, or minimum speed to start the flow. Between points 1 and 2, the bed builds up, dunes form, and the different phases are well separated. Between points 2 and 3 the flow is streaking but

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TABLE 1-8 Sinking Velocity of Soil Particles (after Sulzer Pumps, 1998, with permission of Elsevier) Particle diameter, Mesh size, micrometers US fine

Pressure drop per unit of length

0.2 0.6 1 2 5 6 20 50 60 100 200 250 300 500 600 1000 2000

Soil grain size identification Sinking velocity, m/s

Sinking velocity, ft/s

3 × 10–8 2.8 × 10–7 7 × 10–6 9.2 × 10–6 17 × 10–6 25 × 10–6 28 × 10–5 17 × 10–4 25 × 10–4 0.07 0.021 0.026 0.032 0.053 0.063 0.10 0.17

270 230 150 70 60 35 30 18 10

Grain size by ASTM

Fine clay Coarse clay

Silt

Fine silt

Fine sand

Coarse silt Intermediate silt to sand Fine sand Medium coarse sand

Coarse sand Coarse sand Very coarse sand

slurry

4-5 Pseudohomogeneous

5

3-4 Jumping and rolling

4

2-3 Streaking

3 1

1-2 Bedding

2

wa

Grain size, international

Clay

ter

y/D concentration

Below 1 Stationary and clogging

Speed of flow FIGURE 1-8 Pressure drop versus velocity for water and for a slurry mixture. (After Sulzer Pumps, 1998, with permission of Elsevier.)

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momentum is building up. Between points 3 and 4, there is sufficient speed to cause jumping and rolling of the coarse particles. Above point 4, the speed is sufficiently high to allow a pseudohomogeneous flow in which the fine particles act as a carrier for the coarse particles. These stages are extensively reviewed in Chapter 4. When absolute (or dynamic) viscosity is an important factor, such as in clayish slurries or homogeneous flows, another parameter, the viscous transition critical velocity VT must be determined. There are two regimes for flow of homogeneous mixtures. Flow at speeds less than VT is associated with laminar flows, whereas flows above VT are characteristic of turbulent flows. Flow in the laminar regime is often characterized by a friction loss factor, which is 64/Re for the Darcy factor or 16/Re for the Fanning factor (this topic will be discussed in more details in Chapter 2). As a result, the losses in the laminar regime appear to be a linear function of speed, whereas in the turbulent regime they are proportional to the square of the speed. As we will see in Chapter 5, researchers have struggled with special definitions of a modified Reynolds number for non-Newtonian flows. Emulsions have been pumped over long distances in laminar flow. Nunez et al. (1996) demonstrates the existence of certain effects similar to comminution, i.e., breakup of large droplets into finer ones, as well as coalescence of small particles into larger ones under different flow regimes, shear rates, and constraints. A question often asked is what the relationship between the sinking speed (as per Table 1-8) and the deposition of critical velocity VD? This question comes up when the rheology laboratory produces the results of thickening tests, and when there is not enough money or time to conduct proper slurry loop tests. This point will be examined in Chapter 4 and various approaches have been adopted over the years, from the simplest assumption that the critical deposition velocity is 17 times as large as the terminal or sinking velocity, to more complex mathematical formulae. Often in a lab test, the coarse particles deposit rapidly while the fine particles are still in suspension. Proper pump tests are often recommended, particularly when a multimillion dollar pipeline is being designed. Tests can be conducted at a number of universities, provincial and state research centers, or with the help of manufacturers of slurry pumps. Examples include the Saskatchewan Science Research Center in Canada; the GIW research lab of KSB Pumps (USA), described by Wilson et al. (1992); the Slurry Research Lab of Mazdak International Inc. (U.S.A.); Texas A&M University (U.S.A.); and Melbourne University (Australia), among others.

1-4 DENSITY OF A SLURRY MIXTURE The density of a slurry mixture is a function of 앫 The density of the carrier fluid 앫 The density of the solid particles 앫 The concentration by volume of the solid phase The density of the solid particles is determined carefully by various experimental methods. Fine particles tend to entrap air, which the lab technician must remove by proper agitation or by adding a small quantity of wetting agent. Some materials exhibit a change of packing abilities and therefore density as a function of particle size. If the solids are to be passed through a comminution process, a SAG (semiautogeneous), or ball mill, they can occupy more volume per unit mass as they be-

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come finer. The slurry engineer is therefore encouraged to measure the solid’s density at the proposed size of particles to be transported in slurry form. Certain errors can occur in evaluating the density of solids with heterogeneous mixtures. If the heavier slurry particles settle out and a sample is taken, it may reflect a greater density of finer particles. Due to these possible sources of error, the engineer is encouraged to measure the density of the slurry mixture after proper mixing, and to use the data on concentration by weight or by volume to work back to the density of the solids. The density of a slurry mixture is expressed as 100 m = ᎏᎏᎏ Cw/s + (100 – Cw)/L

(1-4)

where Cw = concentration by weight m = density of the mixture phase l = density of the liquid phase s = density of the solid phase Engineers use the term concentration by weight, as it is easier to convert back into the total tonnage of solids to be transported through a pipeline or across an extractive metallurgy plant. However, the characteristics of the mixture, the mechanics of flow, and the resultant physical properties are more related to the concentration by volume. The concentration by volume of solids in a mixture is expressed as Cwm 100 Cw/s Cv = ᎏ = ᎏᎏᎏ s Cw/s + (100 – Cw)/L

(1-5)

The concentration by weight of solids in a mixture is expressed as Cvs Cvs Cv = ᎏ = ᎏᎏ m Cvs + (100 – Cv)

(1-6)

Example 1-A A pipeline is designed to transport 140 metric ton (308,000 lb) of sand per hour. The specific gravity of the sand particles is 2.65 (or the density is 2650 kg/m3). The concentration by weight is 30%. Determine the density of the mixture if the carrier fluid is water and determine the resultant flow rate. 앫 Weight of sand in the mixture over a period of 1 hr is 140 metric ton (308,000 lb) 앫 Weight of equivalent volume of water equivalent to the sand content is 140,000 kg/2,650 kg/m3 = 52,800 앫 Weight of water in the slurry mixture at a concentration of 30% by weight = 140,000 kg (100 – 30)/30 or 326,600 kg 앫 Total weight of slurry mixture transported in 1 hr is the sum of the weight of water and sand or 140,000 + 326,600 = 466,600 kg/hr 앫 Total weight of equivalent volume of water is 326,000 + 52,800 = 378,800 kg or 378 m3 of liquid, since density of water is 1000 kg/m3 The density of the slurry mixture is therefore 466,600 kg/378 m3 = 1,230 kg/m3. Alternatively, the specific gravity of the slurry compared to water is 1.23. The flow rate, being the volume per unit of time, is equivalent to 378 m3/hr.

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1-5 DYNAMIC VISCOSITY OF A NEWTONIAN SLURRY MIXTURE Although density is essentially a static property, absolute (or dynamic) viscosity is a dynamic property and tends to reduce in magnitude as the shear rate in a pipeline increases. Thus, engineers have had to define different forms of viscosity over the years, everything from dynamic viscosity, to kinematic viscosity, to effective pipeline viscosity. The effective pipeline viscosity will be discussed in detail in Chapters 3, 4, and 5. In this chapter, the reader is introduced to basic concepts of the mixture of slurry in a stationary state. This is effectively what the pump, or a mixer, might see at the start-up of a plant. As is often the case, when the driver cannot deliver enough torque to overcome the absolute (or dynamic) viscosity, the operator is forced to dilute the slurry mixture. Plasticity as defined in Section 1-1-4 is an important parameter in determining overall absolute (or dynamic) viscosity of a mixture of clay and water. There are, however, numerous soils in nature, such as sand and water or gravel and water, in which the solids contribute little to the overall absolute (or dynamic) viscosity, except in terms of their concentration by volume.

1-5-1 Absolute (or Dynamic) Viscosity of Mixtures with Volume Concentration Smaller Than 1% For such solid–liquid mixtures in diluted form, Einstein developed the following formula for a linear relationship between absolute (or dynamic) viscosity and volume concentration:

m ᎏ = 1 + 2.5 L

(1-7)

where m = absolute (or dynamic) viscosity of the slurry mixture L = absolute (or dynamic) viscosity of the carrying liquid This is a very simple equation that is based on the following assumptions: 앫 Particles are fairly rigid 앫 The mixture is fairly dilute and there is no interaction between the particles Such a flow is not encountered, except in laminar regimes of very dilute concentrations (below a volume concentration of 1%).

1-5-2 Absolute (or Dynamic) Viscosity of Mixtures with Solids with Volume Concentration Smaller than 20% Thomas (1965) took the equation of Einstein further by calculating for higher volumetric concentrations of Newtonian mixtures:

m ᎏ = 1 + K1 + K22 + K33 + K44 + . . . L where K1, K2, K3, and K4 are constants

(1-8)

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K1 is the Einstein constant of 2.5 (from Equation 1-7), and K2 has been found to be in the range of 10.05–14.1 according to Guth and Simha (1936). It is difficult to extrapolate the higher terms K3 and K4 in Equation 1-8. They are ignored with volumetric concentrations smaller than 20%.

1-5-3 Absolute (or Dynamic) Viscosity of Mixtures with High Volume Concentration of Solids For higher concentrations, Thomas (1965) proposed the following equation with an exponential function:

m ᎏ = 1 + K1 + K22 + A exp(B) L

(1-9)

where K2 = 10.05 A = 0.00273 B = 16.6 Figure 1-9 is based on Equation 1-9 and is widely accepted in the slurry industry for heterogeneous mixtures of a Newtonian rheology.

1-6 SPECIFIC HEAT Thomas (1960) derived an equation for the specific heat of a mixture as a function of the specific heat of the liquid and solid phases: CpmCws + CpmCwL Cpm= ᎏᎏ 100

(1-10)

1-7 THERMAL CONDUCTIVITY AND HEAT TRANSFER Thermal conductivity is difficult to measure, as solids may settle during the test. Sometimes it is recommended to apply a small quantity of gel to maintain the solids in suspension. Orr and Dalla Valle (1954) derived the following equation for the thermal conductivity of slurry mixtures: 2kl + ks – 2(kl – ks) km = kl ᎏᎏᎏ 2kl + ks – 2(kl + ks)

冢

where k = thermal conductivity and subscripts l = liquid m = mixture s = solids.

冣

(1-11)

L m 70 60 50 40 30 20 10 0

0

10

20

30

40

50

60

70

80

90

CV Volumetric concentration of solids

10 12 14 16 18 20 22 25 27 29 31 33 35 37 39 42 43 44

CV [%] 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

m

L

8.950 9.932 11.07 12.40 13.94 15.75 17.86 20.33 23.22 26.62 30.61 35.29 40.80 47.28 54.91 63.89 74.47 86.94 101.63 118.95 139.4

FIGURE 1-9 Ratio of viscosity of mixture versus viscosity of carrier in accordance with the Thomas equation for coarse slurries.

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viscosity of carrier liquid

80

L

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3 5 7

90

m 1.029 1.089 1.156 1.233 1.365 1.465 1.575 1.696 1.83 1.978 2.142 2.426 2.649 2.907 3.210 3.573 4.017 4.570 5.273 6.734 7.37 8.103

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Ratio of viscosity of slurry mixture vs.

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100

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Metzner et al. (1959, 1960) published articles on heat transfer for slurries. The applications for heat transfer problems have been confined to the nuclear industry, the processing of tar sands, feeding slurry to autoclaves for thermal processing, or certain emulsionbased slurries.

1-8 SLURRY CIRCUITS IN EXTRACTIVE METALLURGY It would be beyond this book to discuss the principles of extractive metallurgy. Slurry is a very important component in the processing of ores to the final disposal of tailings and shipping of concentrate. Chapter 7 is dedicated to equipment for slurry processing. There are three main processes used for extractive metallurgy: 1. Hydrometallurgy, which implies processing the ore using a liquid medium 2. Electrometallurgy, which involves the application of electric and electro-chemical processes to extract the ore 3. Pyrometallurgy, which involves the use of heat (roasting, smelting, etc.) for processing the ore The most common minerals of high metric tonnage (iron, aluminum, copper, titanium, nickel, chromium, magnesium, zinc, etc.) are found in nature as oxides and sulfides and as a combination of both. Ores are sometimes a mixture of rich metal composition and poorer compositions called gangue. The gangue can be acidic or alkaline, and determines the type of flux used for pyrometallurgy. Since ores come in all levels of complexity, various methods of processing have been developed over the years. The first process ore undergoes is called classification or ore dressing. The purpose here is to separate the richer components of a mixture from the unuseful soils. Mineral processing may be used to produce a single stream, as is the case with taconite circuits, where iron extraction is the main activity. It may also create two streams, such as copper concentrate and gold concentrate, when both minerals are found in the same ore body . Mineral processing is usually undertaken at the mine. Its purpose is to separate some of the gangue before shipment of a concentrate. The concentrate is richer in the desired mineral than the original soil.

1-8-1 Crushing In Figure 1-10, a block diagram for crushing and grinding is presented. These are two of the fundamental steps taken to start a slurry circuit. Large rocks are first crushed to an acceptable size. Depending on the type of equipment used, crushing may be done in a single step to feed a semiautogeneous mill, or in three steps (primary, secondary, and tertiary crushing). Rocks are transported from one crusher to another by conveying in a dry form. Their initial size of 300–600 mm (1–2 ft) is reduced to 100–150 mm (4–6 in). Jaw, gyrator, and cone crushers are commonly used during these stages. The crushed material is transported by conveyors to a storage area called the stockpile. From the stockpile, crushed rocks are transported to the grinding and milling circuit.

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stockpile

Primary Crusher

Secondary Crusher

Tertiary Crusher

Crushing

Ore <2% metal

stockpile solids

slurry

een scr cyclones

tank

Lake

Tailings Dam

FIGURE 1-10

fines

Flotation Cells

Electrostatic Separators

Water Ball Mill

Ball Mill

Tailings pipes 30-40% solids

Reclaim Water

cyclones Ball or Pebble Mill

Rod Mill

fines

Magnetic Separators

slurry

Gravity Separators

Milling and Grinding

Autogeneous Mill

Classification

water

wells

Ri ve r

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Crushing and milling are the starting points of a slurry system.

1-8-2 Milling and Primary Grinding Milling and primary grinding serve the purpose of reducing the size of particles from 100–150 mm (⬇ 4–6 in) down to less than 6 mm (⬇ 1–4 in). This is done by using steel or ceramic balls and rods in rotating cylindrical and conical mills. If the process has undergone primary, secondary, and tertiary crushing, a process of screening is undertaken to separate the very coarse from finer particles. The particles are then further reduced in size by milling in rod mills with the addition of water. The still dilute slurry is transported to a ball mill where steel balls achieve further reduction in size. The mixture of slurry and steel balls undergoes a separation at the exit from the ball mill (by using the difference in momentum between the two materials). Slurry is then diverted into a pump box, which is connected to slurry pumps. The slurry is pumped to cyclones, where the coarse particles are further separated from the fine particles. In the last 20 years, efforts have been made to eliminate secondary and tertiary crushing, by feeding the material directly to an autogenous or a semiautogenous mill (SAG mill). The technology has evolved in size. Today, a single line featuring a SAG mill and two ball mills can handle 65,000 metric tons (short tons) per day. An example of a single line plant is the concentrator of ASARCO in Ray, Arizona (USA). In the 1990s, SIEMENS and ABB introduced the concept of the wrap-around motor,

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which helped eliminate complicated gearboxes for SAG mills. Some of the largest SAG mills are now built with a diameter of 12.2 m (40 ft). The process of crushing is essentially a dry process but the process of milling is a wet process in which slurry comes to play an important role. The use of water eliminates the dangerous generation of dust associated with dry grinding. It is not possible to undertake milling or wet grinding in a single step. The slurry is recirculated as follows. Initially, the coarse and fine particles are separated through a coarse screen or a special cyclone. Then the coarse particles are returned to the SAG mill and the intermediate sized particles are sent to the ball mill. The fine sized particles are taken from the cyclone overflow to a magnetic separation, electrostatic separation, or flotation plant. It is therefore not uncommon to recirculate 250–350% of the feedstock through the circuit of grinding and milling. Attention should be paid to the presence of clays in the ore, and associated dynamic viscosity. Flow from the ball mills can reach a concentration of 40–50% by weight and certain non-Newtonian rheology may manifest itself. Over the years, a plant that started in rocky ground may encounter more clay as it proceeds into deeper depths.

1-8-3 Classification This is essentially a process to separate the particles according to their sinking rates in water. Wet classifiers are used with grinding and may include rubber-lined or ceramic cyclones (called hydrocyclones) and spiral mechanical classifiers. The principle of the cyclone is to feed the slurry tangentially and force it to rotate. By centrifugal forces, the coarser particles sink to the bottom of the cone while the finer particles float to the top. Both streams separate. The underflow, which consists of coarse material, and the overflow, which consists of fines, are then directed to other circuits. The underflow is fed back to the ball mills and the overflow is directed to the flotation circuit or other types of separators. The spiral mechanical classifier is used in pools. The heavier particles are allowed to settle in the pool while the finer particles float and flow out of the pool. The heavier particles are then removed from the bottom of the pool with a spiral or mechanical device. From cyclones or from the grinding circuit, the slurry may pass through different types of separators such as flotation circuits, electrostatic separators, and magnetic separators. Their purpose is to separate by chemical, electrical, or magnetic forces the minerals from the gangue. These steps occur before further thickening prior to feeding the pipeline. The gangue is diverted to tailings circuits (see Figure 1-11).

1-8-4 Concentration and Separation Circuits After a considerable effort to reduce the sizes of particles, it is important to separate the richer soils from the slimes or gangue. This step is achieved by using the properties of the ore itself. Gravity devices work on the principle that the ore (such as gold or diamonds) is heavier than the gangue. These devices include shaking units, the classic miner’s pan, rocking cradle devices, or more sophisticated gold concentrators or mineral sand concentrators. The drawback of these systems is that they may not necessarily be able to treat the fine particles produced by grinding and milling. In diamond extraction plants, X-ray machines are used in conjunction with gravity separation to detect the diamonds. Gravity devices are also used in a number of dry processes as well as slurry processes

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Crushing, grinding, milling (see fig 1- 10)

Electrostatic Separators

Ma gnetic Separators

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Flotation Cells

Gravity Separators

wells

Lake

Ri ve

r

Tailings Dam

Tailings pipes 30-40% solids

tailings thickener Reclaim Water

Page 1.27

minerals

gangue

tank

concentrate thickener

tailings sump

concentrate at about 20% mineral slurry pipeline

reclaim water

filtering drying smelting burning

1.27

FIGURE 1-11 Block for thickening and disposal of tailings, used as the basis for the design of slurry concentrate and fi 1diagram 11 tailings pipelines.

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for mineral sand and placer mining. In the case of mineral sands, the presence of a wide range of heavy oxides allows the miner to separate the various components by using gravity in conjunction with the magnetic or electrostatic properties. Magnetic devices are especially useful, since iron is one of the most common minerals and it is possible to separate iron oxides from gangue by applying a magnet. This is usually done by introducing the slurry over a rotating drum, as shown in Figure 1-12; the drum picks up the magnetic concentrate while the remaining soils are diverted away by the flow of the remaining slurry. Magnetic devices are common in taconite processing plants, iron ore plants, as well as in mineral sand plants. Electrostatic devices were developed in Australia to process beach and mineral sands. The sand ore is fed to a conducting and grounding rotor and is exposed to ionization. The particles, which have certain electrostatic properties, are attracted by the electric charge and are separated from the other particles. The nonconducting particles drop on the rotor and are brushed away into a separate container (Figure 1-13). Flotation devices use the principle of flotation to separate particles that are wettable from other particles. This is a very common process with sulfides but is less efficient with oxides. The cyclone overflow or the fine particles in the slurry after undergoing milling and grinding are fed to a series of flotation tanks (or a flotation machine). An agitator provides vigorous mixing. Air is introduced from a separate compressor line and chemical reagents are added to create froth. The nonwettable minerals float on top of the froth and are pumped away by froth-handling pumps or scraped away by mechanical devices. The wettable particles, such as the gangue, do not float on top of the froth and sink to the bottom of the tank. Special tailing pumps may then pump away the gangue, sometimes for further grinding and processing (particularly gangue from the first flotation tank) and sometimes to the final tailing box (see Figure 1.14). The process of froth generation and flotation is more efficient when carried out in steps. A series of up to six tanks may be constructed to drop gradually, with launders in between. Without reagents, only graphite or molybdenum would be nonwettable. Reagents have been produced by the industry for different grades of sulfides, to depress or activate the extraction of certain minerals, to control pH, etc.

rotating magnet

Feed

Magnetic concentrate

Other soils

FIGURE 1-12 A drum-type magnetic separator. The drum is sometimes replaced by a magnetic belt on a special table.

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Feed Nonionizing electrode fine-wire electrode ionizing Conductors concentrate

Nonconductor soils FIGURE 1-13

An electrostatic separator.

The secondary grinding that is applied to the underflow of the circuits from flotation is useful for extracting secondary ores, and has been applied successfully in copper–gold ores for further extraction of the gold. A mineral process plant includes gravity flows from hydrocyclones to ball mills and pumped flows from SAG mills to hydrocyclones. A good plant layout must allow space for repairs and long bends, and provide the ability to join and split flows. The use of three dimensional computer modeling is a very useful tool for the design engineer to determine the slope of launders and physical constraints to the layout of the plant (Figure 1-15)

froth bubbles with mineral concentrate

aeration agitator pump

secondary grinding FIGURE 1-14 A flotation circuit.

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1-8-5 Piping the Concentrate From these processes of classification, a concentrate is obtained. The slurry can be further thickened or dewatered using thickeners to a concentration of 50–60% by weight. The concentrate can then be pumped for hundreds of kilometers to a port from which it can be shipped to a pyrometallurgy plant. At the port, a filtering plant can provide further dewatering. Long pipelines are used to transport concentrate. At Cuajone, in Peru, an open launder is used to transport tailings from an altitude of 3000 m (10,000 ft) down to sea level. The potential energy drop is used to overcome the friction losses of the launder. In Escondida, Chile, copper concentrate flows by gravity from an altitude in excess of 2500 m (8200 ft) above sea level over a distance in excess of 200 km (125 mi) to a port at sea level. Thus, in a typical copper extraction process the rocks are reduced to very fine particles through vertimills, semiautogeneous mills, and ball mills. Further separation occurs through flotation circuits and grinding.

1-8-6 Disposal of the Tailings Once the concentrated ore has been extracted, the plant is left with the sands and slimes. These are dewatered to an acceptable concentration by weight of 35–45%. Using thicken-

FIGURE 1-15 Three-dimensional computer representation of a grinding circuit with one central SAG mill, a ball mill on each side, hydrocyclones at the top left, and pumps (at the floor level). Courtesy of Hatch & Associates, Vancouver, Canada.

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ers, the flow separates into clarified water that is returned to the plant for use in the milling and grinding circuits, and into an underflow of concentrated tailings. The underflow from the tailings is then pumped away to a large disposal pond, called a tailings dam. The sand sinks to the bottom of the tailings pond and the water from the top is returned back to the plant for further use. During the process of disposing of the tailings, spigots and other devices are used to separate coarse from fine particles at the discharge point; the coarse particles are used to build the beaches or the wall of the dam. Dams have been built up to a height of 200 m (656 ft). The environmental engineer must make sure that no dangerous chemicals seep through the ground. If the tailings contain dangerous substances, a plastic or clay (which tends to create a seal) lining for the pond may be recommended to prevent seepage. Some dangerous collapses of tailing dams have been reported over the years, with detrimental consequences when they contained cyanide products, as in the case of certain gold mines. The technology used in tailing pumps has improved. The casings can be designed to withstand 6.9 MPa (1000 psi) of pressure. However, these are essentially single-stage centrifugal pumps installed in series. Up to seven pumps have been installed in series in mines such as National Steel in Minnesota and Kelian Gold in Indonesia.

1-9 CLOSED AND OPEN CHANNEL FLOWS, PIPELINES VERSUS LAUNDERS The reader will find that over the years the majority of references on slurry flows have focused on pipelines because the interest in the field has concentrated on the ability to haul coal, sand, and phosphate hydraulically. Many mines, particularly in Chile and Peru, are located at very high altitudes. This demand has increased interest in gravity flows. In the early 1970s, Southern Peru Copper installed one of the first long, open launders to dispose of tailings to the sea. The launder was of a concrete and fiberglass design, with a U-shaped cross section. Another example of a long gravity pipeline for copper concentrate is the Escondida concentrate pipeline in Chile, which is longer than 200 km (125 mi). Despite the increasing importance of long gravity pipelines, equipment has not kept up with the expanding need. Cave (1980) described tests on slurry turbines. A 350 mm × 300 mm (14⬙ × 12⬙) was reported by Burgess and Abulnaga (1991). Launders play a very important role in slurry flows of plants. Cyclone underflow is directed to ball mills then to SAG mills by gravity. Flows in these circuits can cause tremendous wear if provision is not made to control speeds. Launders in plants are typically rubber lined. Long-distance pipelines are manufactured of rubber-lined steel or extra-thick, high-density polyethylene (HDPE). Because of the importance of open launders, gravity flows, and drop boxes, Chapter 6 is dedicated to these complex flows. Obviously, not all mines are located on mountaintops, and slurry pipeline flow will continue to be the main emphasis of researchers. In long pipelines, centrifugal pumps can be installed at regular intervals; these require power to be brought in. For long-distance pumping, positive displacement pumps compete well with centrifugal pumps. The positive displacement pumps are of a diaphragm or hose design. They are extremely expensive. A 17.3 MPa (2500 psi) pump may range in price between U.S. $600,000 and $1,200,000 in year 2000 dollars. The higher capital investment required for positive displacement pumps is offset by their higher efficiency. These pumps are built to much smaller flow capacity than are large centrifugal pumps.

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1-10 HISTORICAL DEVELOPMENT OF SLURRY PIPELINES One of the first large engineering projects that involved transportation of solids by liquid was the dredging for the Suez Canal in the 1860s in Egypt. It was reported to have used conduits to dispose of the sand–water mixture. Nora Blatch in 1906 was probably the first person to conduct a systematic investigation of the flow of solid–water mixtures. She used a 25 mm (1 in) horizontal pipe and measured the pressure gradients as a function of flow, density, and solid concentration. As a result, between 1918 and 1924, a 200 mm (8 in) pipeline was installed in the Hammersmith power station, in London, England to transport coal slurry over a distance of 660 yd. In 1948, in France, the Institute of Research SOGREAH began a series of tests on transporting sand and gravel in pipes with a diameter from 38–250 mm (1.5–10 in). These extensive tests were the basis for the formulation by Durand of a number of equations that will be reviewed in Chapter 4. These equations have been subject to further refinements over the last 50 years. In 1952, in the United Kingdom, the British Hydromechanic Research Association (BHRA) started to study the hydraulic transport of lump coal, sand, gravel, and limestone. Limestone pipelines were constructed in Trinidad and England in 1960. The Trinidad pipeline had a diameter of 204 mm (8 in), a length of 9.6 km (6 mi), and was designed to operate in a laminar flow regime. The limestone pipeline in England had a diameter of 250 mm (10 in) and was 112 km (70 mi) long. In 1950, the Consolidated Coal Co. in the United States started to conduct research on the hydrotransport of fine “nonsettling” slurries. Concentrated coal with a weight concentration of 60% and particle size between minus 1168 m (14 mesh) and minus 43 m (325 mesh) was transported. The pipeline transported 1.5 million tons of coal each year between 1957 and 1964. The pipeline stretched 176 km (110 mi) from Cadiz, Ohio to Eastlake in Cleveland, Ohio. In 1957, the Colorado School of Mines collaborated with the American Gilsonite Company and designed a pipeline with a diameter of 200 mm (8 in) to transport crushed gilsonite. The pipeline was constructed between Bonanza, Utah and Grand Junction, Colorado. The particle size was minus 4.7 mm (4 mesh) and solids were pumped at a weight concentration of 48%. Two other pipelines were built in Georgia to transport kaolin in the 1960s. In 1967, an iron ore concentrate slurry pipeline started to operate in Tasmania, Australia. The pipeline had a diameter of 245 mm (95–8 in). Concentrate was transported at a weight concentration of 60% with an average particle size of minus 149 m (100 mesh) over a distance of 85 km (53 mi) through extremely rugged terrain (see Figure 1-16). In 1970, the Black Mesa Pipeline, one of the longest pipelines ever built up to that time, started operation between the Black Mesa Coal fields in Arizona and the Mohave Power Plant in Nevada. Coal was ground to a particle size of minus 1168 m (14 mesh), and transported in a pipe with a diameter of 457 mm (18 in) over a distance of 437 km (273 mi). Coal was dewatered at the end of the line through a mill before combustion with preheated air. Since the 1970s, a number of short and long slurry pipelines have been constructed. Table 1-9 lists a number of such achievements. Now, at the beginning of the 21st century, new complex, multiphase tar–sand pipelines are planned for northern Alberta, Canada.

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FIGURE 1-16 Long slurry pipelines must travel through isolated areas over long distances and may involve pressures up to 2500 psi that require special positive displacement pumps. Courtesy of Wirth Pumps, Germany.

1.11 SEDIMENTATION OF DAMS— THE ROLE OF THE SLURRY ENGINEER In the last 150 years, world population has grown fast and our modern standards of living depend on the production of electricity, and production of food for at least two seasons a year. In an effort to meet these demands, engineers have built small as well as very large dams. In certain areas, very large man-made lakes have been dug in the earth, such as behind the Aswan High Dam in Egypt, the Ataturk Dam in Turkey, and new dams on the Yellow River in China. Some large rivers transport silt that tends to separate from water when the speed of the flow is interrupted by a dam. This phenomenon is called siltation of dams. The problem

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TABLE 1-9 Examples of Slurry Pipelines Built Since 1957

Ore

Site of pipeline, or name of pipeline

Solids Pipe transported, Pipeline length diameter million short inch Mile km tons/yr

Start-up date

Coal

Consolidation, USA Black Mesa ETSI ALTON Belonovo–Novosibirsk, Siberia, Russia

10 18 38 24

108 273 1036 180 158

175 440 1675 112 256

1.3 4.8 25 10 3.4

1957 1970 1979 1981 1985

Iron concentrate

Savage River Waipipi (Iron Sands) Pena, Colorado Las Truchas, Mexico Sierra Grande, Argentina Samarco, Brazil Chongin, North Korea New Zealand Sands, NZ Jian Shan, China La Parla–Hercules, Mexico

9 8,12 8 10 8 20 ? 12 10 8/14

53 6 28 16 20 244 61 5 62 52/182

86 9.7 45 27 32 395 98 8 100 85/295

2.25 1.0 1.8 1.5 2.1 12 4.5

1967 1971 1974 1976 1976 1977 1975

4.5

1982

Copper ore

Los Bronces

24

35

56

Copper concentrate

Bougainville, PNG West Irian, Indonesia Pinto Valley OK Tedi, Papua New Guinea Escondida, Chile (gravity line) Collahausi, Chile Freeport, Indonesia Batu Hijau, Indonesia Alumbrera, Argentina

6 4 4 6 9 7 5 6 6

20 69 11 96 102 125 71 11 194

32 111 17 155 165 203 115 18 314

1.0 0.3 0.4 ? 1.0

1972 1972 1974 1987 1994 1999

0.8

1999 1998

Copper tailings

Bougainville, Papua NG Southern Peru Copper (gravity)

34

31

50 150

Limestone

Rugby Calaveras Michigan Limestone Tailings

10 7 20

57 17 1.2

92 27 2

1.7 1.5

1964 1971

Phosphate Chevron, Vernal, Wyoming ore concentrate Simplot Wenglu

10 8 8

94 89 28

152 145 45

1.3–2.5

1986

Dredging

24

33

21

11,000 gpm 1998

Dallas White Rock Lake, USA

1972

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of siltation of dams has not been well documented or studied. Chanson (1998) and Chanson and James (1999) examined the siltation of Australian dams. Certain dams in Australia became gradually fully silted between 1890 and 1960. They reported that the Koorawatha dam in New South Wales (Figure 1-17) became fully silted with bed-load material. The Cunningham Creek Dam in New South Wales, Australia was well studied by Hellstrom (1941) according to Chanson. Sedimentation problems are more acute with small dams than with medium-size and large reservoirs (Chanson and James,1999). Siltation at Eildon in the State of Victoria occurred in 1940 after torrential rainfalls following bushfires that had destroyed 50% of the catchment forest. The siltation at Eppalock in Victoria followed extensive gold mining, tree clearing, and hydraulic mining between 1851 and 1890. There were some extreme siltation cases. The Quipolly Reservoir No. 1, in Australia underwent very rapid sedimentation between 1941 and 1943 at a rate in excess of 1143m3/km2/year (9600 ft3/mi2/y). The Korrumbyn Creek Dam sedimented in less than 7 years. Since the 1950s, improvements in land management practices and a better understanding of the problems of soil erosion have resulted in better approaches to the protection of dams.

FIGURE 1-17 The Koorawatha Dam in Australia—fully silted. [From Chanson (1999). Reprinted by permission of Butterworth-Heinemann.]

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Chanson and James (1998) discussed the hazards of fully silted dams. The weight of silt pressing against the concrete structure introduces a medium with a specific gravity larger than the water for which the dam was designed. It is important in these cases to monitor the structure. In the 1970s and 1980s, the drought in Ethiopia and the Sudan reduced the flow of the waters to the Nile to dramatically low levels. Egypt avoided famine by using its massive man-made lake behind the Aswan High Dam (Lake Nasser). On the other hand, any specialist who visits Egypt can feel that the fellahin (farmers) are speaking with nostalgia of the “Tamye” or silt that used to come with the annual flood and enrich the land. This raw material was the basis of natural nutrients, as well as mud for the construction of houses and manufacture of bricks. The Egyptian case is not unique. Certain dams in the United States are now the subject of discussions on decommissioning and some were removed in the 1990s. The slurry engineer can offer some much needed solutions. The twenty-first century will see slurry engineers providing adequate solutions in terms of dredging the lakes that are sedimenting and transporting the dredged silt to traditional lands, or to arid lands via special slurry pipelines. A simple concept for such a solution is presented in Figure 1-18. It is proposed that in certain areas, particularly where the accumulation of silt is likely to apply pressure on the dam structure, submersible slurry pumps be installed on a permanent basis. Dredging boats with dredging arms or submersible pumps and cutters would be used on the rest of the lake. Where the capital investment does not justify it, small dredgers with submersible pumps should be used. The slurry from these operations will be dilute, it would be pumped to the shore through a floating plastic pipe. It may be pumped in pipelines and diverted to canals for agricultural purposes. It may also be pumped to brick plants, where it would be dewatered and the silt used as a raw material.

Agriculture dredger

Brick manufacture Dam

Submersible pump silt

FIGURE 1-18 Simplified flow sheet to remove silt behind dams. Silt would be dredged using submersible Fi pumps 1 18at predetermined locations or in association with boat dredgers. The silt would be pumped in slurry form to agricultural farmlands or to special plants for the manufacture of bricks.

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Slurry engineers can provide economic solutions to the siltation of dams. This effort should be made in conjunction with environmental engineers, as new ecosystems often form around large dams. The author hopes that this handbook will be useful to the decision makers who have to deal with siltation of dams, while satisfying the concerns of environmental engineers, as decommissioning is not always the solution.

1-12 CONCLUSION In this first chapter, some of the basic properties of solids, which are important to the composition of slurries, were reviewed. Their importance will be emphasized in the next few chapters. They may lead to Newtonian as well as more complex non-Newtonian flows that require special equations to determine the friction factors, velocity of flow, pipe sizes, head, and efficiency losses in pumps. Wear is an important cost to be paid for transporting solids by liquids. This will be discussed later in the book when exploring slurry pumps and pipelines. The modern slurry engineer can serve the mining and power industries by making possible the transportation of minerals, coal, coal–crude oil mixtures over very long distances, and also play a major role in dredging sediments behind dams to avoid dam failure and to provide arid lands with much needed silt.

1-13 NOMENCLATURE A B Cc Cu Cv Cw Cp d10 d30 d50 d60 d80 K1, K2, K3, K4 k LL PI PL Re VD VT

Constant Constant Coefficient of curvature Coefficient of uniformity Concentration by volume of the solid particles in percent Concentration by weight of the solid particles in percent Heat capacity Grain size at which 10% of the soil is finer Grain size at which 30% of the soil is finer Grain size at which 50% of the soil is finer Grain size at which 60% of the soil is finer Grain size at which 80% of the soil is finer polynomial coefficients in Einstein’s equation for dynamic viscosity Thermal conductivity Liquid limit of clay and silt soils Plastic index of clay and silt soils Plastic limit of clay and silt soils Reynolds number Deposition critical velocity Viscous transition critical velocity Concentration by volume in decimal points Absolute (or dynamic) viscosity Density

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Subscripts l m s

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Liquid Mixture Solids

1-14 REFERENCES The American Society for Testing of Materials. 1993. Practice for Description and Identification of Soils (Visual–Manual Aggregate Mixtures). Standard D2488. Philadelphia: The American Society for Testing of Materials. The American Society for Testing of Materials. 1983. Test Method for Lime Content of Uncured Soil–Lime Mixtures. Standard D3155. Philadelphia: The American Society for Testing of Materials. The British Standard Institute. 1999. Code for Practice of Site Investigation. Standard BS 5930. London: The British Standard Institute. The British Standard Institute. 1999. Aggregate Abrasion Value. Standard BS 812. Pt 113. London: The British Standard Institute. Broch, E., and J. A. Franklin. 1972. The Point-Load Strength Test. International Journal for Rock Mechanics and Mineral Sciences, 9, 669–697. Burgess K. E, and B. E. Abulnaga.1991.The Application of Finite Element Methods to Warman Pumps and Process Equipment. Paper presented to the Fifth International Conference on Finite Element Analysis in Australia, University of Sydney, Australia (July). Cave I. 1980. Slurry Turbines for Energy Recovery. In Seventh International Conference on the Hydraulic Transport of Solids in Pipelines, Sendai, Japan, pp. 9–15, Cranfield, United Kingdom: BHRA Group. Chanson H. 1998. Extreme Reservoir Sedimentation in Australia: A Review. International Journal of Sediment Research, UNESCO-IRTCES, 13, 3, 55–63. Chanson H. 1999. The Hydraulics of Open Channel Flows—An Introduction. Oxford, UK: Butterworth-Heinemann. Chanson H., and D. P. James.1999. Siltation in Australian Reservoirs: Some Observations and Dam Safety Implications.” In Proceedings 28th IAHR Congress, Graz, Austria, Paper B5. Guth, E., and A. R. Simha. 1936. Viscosity of suspensions and solutions. Kolloid-Z, 74, 266. Quoted in Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid–Liquid Flow—Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans. Tech. Publications. Hellstrom B. (1941) Nagra Lakttagelser over Vittring Erosion och Slambidning i Malaya och Australien.” Geografiska Annaler (Stockholm, Sweden), Nos. 1–2, pp. 102–124 (in Swedish). Herbrich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill Inc. The International Journal for Rock Mechanics and Mineral Sciences, 1979, 16, 141–156. The International Society for Rock Mechanics. 1977. Suggested Methods for Determining the Strength of Rock Materials in Triaxial Compression. Lisbon, Portugal: The International Society for Rock Mechanics. The International Society for Rock Mechanics. 1978. Suggested Methods for Determining the Deformability of Rock. Lisbon, Portugal: The International Society for Rock Mechanics. Metzner, A. B., and P. S. Friend. 1959. Heat Transfer to Turbulent Non-Newtonian Fluids. Ind. & Eng. Chem., 51, 7 (July), 879–882. Metzner, A. B., and D. F. Gluck. 1960. Heat Transfer to Non-Newtonian Fluids Under Laminar Flow Conditions. Chem. Eng. Science, 12, 3 (June), 185–190. Nunez, G. A., M. Briceno, C. Mata, and H. Rivas. 1996. Flow Characteristics of Concentrated Emulsions of Very Viscous Oil in Water. Journal of Rheology, 40, 3 (May/June), 405–423. Orr, C., and J. M. Dalla Valle. 1954. Heat-Transfer Properties of Liquid–Solid Suspensions. Chem. Eng. Prog., Symp. Series No. 9, 50, 29–45. The Permanent International Association of Navigation Congresses. 1972. Classification of Soils to be Dredged. In Bulletin No. 11, Vol. I. The Permanent International Association of Navigational Congresses.

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Sargent, J. H. 1984. Classification of Soils to be Dredged. In Supplement to Bulletin No 47. The Permanent International Association of Navigation Congresses. Sulzer Pumps. 1998. Centrifugal Pump Handbook. New York: Elsevier. Thomas, D. G. 1960. Heat and Momentum Transport Characteristics of Non-Newtonian Aqueous Thorium Oxide Suspensions. AIChE Journal, 6 (December), 631–639. Thomas D. G. 1965. Transient Characteristics of Suspensions: Part VIII. A Note on the Viscosity of Newtonian Suspensions of Uniform Spherical Particles. Journal Colloid Science, 20, 267. Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid–Liquid Flow—Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans Tech Publications. Wilson, K. C., G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. New York: Elsevier Applied Sciences. Further Reading: Wilson, G. 1976. Construction of Solids-Handling Centrifugal Pumps. In Pump Handbook. Edited by J. Karassik et al. New York: McGraw-Hill.

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FUNDAMENTALS OF WATER FLOWS IN PIPES

2-0 INTRODUCTION The mechanics of pipe flow is a topic well dealt with in the scientific literature. In this chapter, some of the concepts are reviewed in a simplified manner as they relate to slurry flows. The friction factor for single phase flows is presented in terms of boundary layer theory. Losses in pipes are summarized in terms of fittings and conduits commonly used on large engineering projects. Numerous books have been written for single-phase flows. This chapter limits itself to a brief introduction. The equations in this chapter are based on SI units for consistency. They can be readily used in USCS (United States Customary System) units if the reader assumes the use of slugs and not pounds for mass units. There is considerable confusion about the use of slugs, which are units of mass, and pounds, which are units of force. The conversion factor between these is often called gc and is equal to 32.2 ft/sec. Many other references input gc in their equations to achieve such a conversion, but it was not deemed necessary in this book. The worked examples show how slugs should be used as a unit of mass. Certain models of Newtonian slurry flows are attempts to apply correction factors to the friction losses of the carrier liquid on the basis of the volumetric concentration of solids. The reader is encouraged to examine this chapter before proceeding with more complex themes.

2-1 SHEAR STRESS OF LIQUID FLOWS Modern fluid mechanics is based on the concept of a controlled volume. Mass momentum and energy must be conserved when a particle enters and leaves the volume. Considering flow through a section of pipe of a constant diameter between two locations 1 and 2 as in Figure 2-1, the hydraulics force associated with the drop of pressure is

F12 = ᎏ D i2(P1 – P2) 4

2.1

(2-1)

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w

P

P

2

1

L

FIGURE 2-1 Shear stress and pressure for flow in a pipe.

This force is balanced by the friction force Fr Fr = wDiL

(2-2)

where L is the distance between points 1 and 2 and w is the wall shear stress

ᎏ D 2i (P1 – P2) – wDL = 0 4 or Ri(P1 – P2) Di(P1 – P2) w = ᎏᎏ = ᎏᎏ 4L 2L

(2-3)

The shear stress at any radius and from the center of the pipe is r Ri⌬P = ᎏ = w ᎏ 2L R where L = length Ri = the pipe inner radius (at the inside wall of the pipe) r = local radius The shear stress is calculated. At the center of the pipe there is no shear stress. Example 2-1 Homogeneous slurry is tested in a pipe with an inner diameter of 53 mm (2.086 in). The pressure drop due to friction is measured between two points (A and B), which are separated by a distance of 1.8 m (5.9 ft). The pressure drop is recorded as 3000 Pa (0.435 psi). To appreciate the shear stress distribution from the wall to the center of the pipe, determine the shear stress distribution at the wall and at three points: at a radius of 20 mm (0.787⬙), at a radius of 12 mm (0.472⬙), and at the center of the pipe. Solution This problem will be solved in SI units [Système International (metric)] and in USCS units.

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Solution in SI Units Using Equation 2-3, the shear stress at the wall is 0.053(3000) w = ᎏᎏ = 22 Pa 4 × 1.8 since the inner radius of the pipe is 26.5 mm. At a radius of 20 mm the shear stress is

冢 冣

冢

冣

20 r = w ᎏ = 22 ᎏ = 16.6 Pa Ri 26.5 At a radius of 12 mm the shear stress is

冢 冣

冢

冣

12 r = w ᎏ = 22 ᎏ = 10 Pa Ri 26.5 At the center of the pipe the shear stress is

冢 冣

冢

冣

0 r = w ᎏ = 22 ᎏ = 0 Pa Ri 26.5 Solution in USCS Units Using Equation 2-3, the shear stress at the wall is 2.086 in (0.435 psi) w = ᎏᎏᎏ = 0.0032 psi 4 × 5.9 × 12 since the inner radius of the pipe is 1.043 in. At a radius of 20 mm (0.787 in) the shear stress is

冢 冣

冢

冣

r 0.787 = w ᎏ = 0.0032 ᎏ = 0.0024 psi Ri 1.043 At a radius of 12 mm (0.472 in) the shear stress is

冢

冣

0.472 = 0.0032 ᎏ = 0.00145 psi 1.043 At the center of the pipe = 0

2-2 REYNOLDS NUMBER AND FLOW REGIMES Determining the magnitude of friction was historically a controversial topic until the end of the 19th century. The great disagreement was between the practical engineers and the theory of hydrodynamics. Hager (in 1839) and Poiseville (in 1840) demonstrated that under certain conditions friction was a linear function of the speed of flow. In 1858, Darcy demonstrated that under other conditions friction was in fact proportional to the square of the mean speed of the flow. By 1883, Reynolds had demonstrated that both Poiseville and Darcy were correct, as the mechanics of flows were fundamentally different at very low speeds and at high speeds.

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CHAPTER TWO

Through nondimensional analysis, Reynolds demonstrated that under certain fixed conditions, the transition from a laminar Poiseville flow to a turbulent Darcy flow was based on the ratio of the inertia forces to the viscous forces. In his honor, such a ratio is now called the Reynolds number: UDi Inertia forces UDi Re = ᎏᎏ = ᎏ = ᎏ Viscous forces

(2-4)

where = density of the fluid = absolute or “dynamic” viscosity = kinematic viscosity (defined as the absolute viscosity divided by the density of the liquid) U = average velocity of the flow The kinematic viscosity is the absolute (or dynamic viscosity) divided by the density. Its unit of measurement are, strictly speaking, m2/s or ft2/sec. Another unit used is the centistokes, which is obtained by dividing the absolute viscosity in centipoises by the specific gravity of the fluid. A centipoise is equivalent to one milli-Pascal-second. One unit for kinematic viscosity used in the oil industry (but of limit use in the mining industry) is the seconds Saybolt universal (or SSU). For values of kinematic viscosity larger than 70 centistokes (cst), the following formula is recommended by the Hydraulic Institute (1990): SSU = centistokes × 4.635 In simplified terms, it may be said that two geometrically similar bodies immersed in a fluid will develop inertia and viscous forces in a constant ratio when body forces are negligible. Since Reynolds developed his theory, his approach has been has been extended to other fluids. Modern aerodynamics uses the chord of the wing aerofoil instead of the pipe diameter as the distance parameter for Equation 2-4. In Chapter 3, the concept of the particle Reynolds number based on a characteristic particle diameter shall be introduced. Figure 2.2 presents the equations of the Reynolds number for different shapes and flows. For pipe flows, the critical Reynolds number is considered to be between 2300 and 2800. In many pipes, flow becomes unstable above a Reynolds number of 2300 and slides into a transition regime before converting into turbulent motion.

2-3 FRICTION FACTORS The Fanning friction factor is a nondimensional number defined as the ratio of the wall shear stress to the dynamic pressure of the flow:

w fN = ᎏ U 2/2

(2-5)

Users of USCS units should use slugs/ft3 for density and not the more commonly used units of pounds per cubic feet. The conversion between these two is gc or 32.2 ft/sec2. Substituting Equation 2-3 into Equation 2-5

U 2 ⌬PDI fN = ᎏ / ᎏ 4L 2

冢

冣冢

冣

(2-6)

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FUNDAMENTALS OF WATER FLOWS IN PIPES

c h

Airfoil chord "c" Re = U c /

Close parallel plates Re = [U h/2 ] 32/3

DI dp Annular flow

Re= ( U/

)*2 r 20 + r2i - r 2m 2

2

where rm = (r 0 - r i )/2.3 log10 (r0 /rI )

Use inner diameter for pipe flow Re= U D i /

for sphere use diameter Re= U d p /

FIGURE 2-2 Definition of Reynolds number for various shapes.

Equation 2-6 clearly indicates that the friction factor is dependent on the flow. It will be demonstrated that there is a relationship between the friction factor and the Reynolds number of the flow. In USCS units, density is expressed in slugs/ft3. Example 2-2 The slurry in Example 2-1 has a specific gravity of 1.2 , or a density of 1200 kg/m3. If the speed of the flow is 2 m/s (6.56 ft/s), determine the Fanning friction factor. Solution in SI Units Using the shear stress calculated in Example 2-1 as 22 Pa and inserting it into Equation 25, the Fanning friction factor can be calculated as 22 = 0.0092 fN = ᎏᎏ 1200 × 22 × 0.5 Solutions in USCS Units Assume the density of water to be 62.3 lbm/ft3. Since specific gravity equals 1.2, the density of the slurry is 1.2 × 62.3 = 74.76 lbs/ft3. To convert lbs/ft3, into slugs/ft3 complete the following equation: 74.76 ᎏ = 2.32 slugs/ft3 32.2 In Example 2-1, the wall shear stress was determined to be 0.0032 psi. Using equation 2.5 0.0032 × 144 = 0.0092 fN = ᎏᎏ 2.32 × 6.562 × 0.5

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CHAPTER TWO

2-3-1 Laminar Friction Factors The friction or resistance of a body to motion is confined to a viscous layer at the wall or surface of the body. This layer is called the boundary layer. Understanding this phenomenon remained illusive until the end of the 19th century. At the turn of the 20th century, Prandtl developed the principles of boundary layer theory. To understand the basic principles of the boundary layer, let us start by examining the flow between closely related plates at low speed. With one plate stationary and the other moving at a speed U, a linear velocity profile is established. The wall shear stress w is established as

U w = ᎏ h by Newton’s law, where h is the spacing between plates and

冢 冣

U u=y ᎏ h

(2-7)

With y as the vertical ordinate from the stationary plate, the velocity gradient is defined as du U ᎏ = ᎏ = rate of shearing strain or shear rate d␥ H

(2-8)

Thus, the dynamic viscosity is Shear Stress = ᎏ = ᎏᎏᎏ du/d␥ Rate of Shear Strain

(2-9)

Equation 2-8 is the basis for boundary layer theory. Instead of a moving plate at a velocity U, the velocity of the flow outside the boundary layer is studied (see Figure 2-3). The shear rate is not necessarily linear. For example flow around an aircraft airfoil can be attached, stagnant at a point, or can even reverse flow after separation. Laminar flow in a pipe is described by the Hagen–Poiseville equation: ⌬P 32U ᎏ=ᎏ D i2 L

(2-10)

Upper plate moves at speed U

h y FIGURE 2-3 ary plate.

y u=U h

w

Linear velocity distribution due to a plate moving at a speed U above a station-

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2.7

which can be rearranged as ⌬P = ᎏ L

Di⌬P

= ᎏ ᎏ 冢 冣冢 ᎏ 32U 冣 冢 4L 冣冢 8U 冣 D 2i

Di

w = ᎏ 8U/Di

(2-11)

(2-12)

or the ratio of the wall shear stress and the mean velocity gradient. The term (8U/Di) is often called the pipe (or pipeline) shear rate in laminar flow. From Equation 2-12 8U w = ᎏ Di

(2-13a)

Substituting Equation 2-12 into Equation 2-5, the Fanning friction factor for a laminar flow can be expressed as 8U fN = ᎏ Di

冢

U 2

冣 冢ᎏ 2 冣

/

16 16 fN = ᎏ = ᎏ VDi Re

(2-14)

(2-15)

The Fanning friction factor is more commonly found in reference publications on chemical engineering. Another friction factor used by mechanical engineers is the Darcy friction factor: fDarcy = 4 × fFanning To avoid confusion, in the book the symbols fD will be used for Darcy friction factor and fN will be used for Fanning friction factor. Flow in a laminar regime is considered to be independent of pipe roughness. Example 2-3 A viscous fluid is flowing in a laminar flow at a speed of 1m/s (or 3.28 ft/s) in a pipe with an inner diameter of 336.6 mm (13.25 in). The measured pressure drop over a distance of 200 m (656 ft) is 8400 Pa (1.22 psi). The density of the fluid is 855 kg/m3 (SG = 0.855). Determine an equivalent viscosity for the pipeline fluid, the Reynolds number, and the friction factor. Solution in SI Units From Equation 2-3, the shear stress at the wall is 0.3366 × 8400 w = ᎏᎏ = 3.53 Pa 4 × 200 The Fanning Friction Factor from Equation 2.5 is 2w 2 × 3.53 fN = ᎏ2 = ᎏ2 = 0.0083 V 855 × 1

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CHAPTER TWO

The equivalent pipeline viscosity from Equation 2-12 is

w 3.53 × 0.3366 = ᎏ = ᎏᎏ = 0.148 Pa-s 8U/D 8×1 The Reynolds Number from Equation 2-4

UD 855 × 1 × 0.3366 Re = ᎏ = ᎏᎏ = 1938 0.148 Check on friction factor: 16 fN = ᎏ = 0.00823 1938 Solution in USCS Units 0.8555 × 62.3 density = ᎏᎏ = 1.654 slugs/ft3 32.2 Since it was stated that the pressure drop = 1.215 psi, over a distance of 656.2 ft the shear stress is: 13.25 × 1.2 w = ᎏᎏ = 0.0005 psi 4 × 656.2 × 12 The Fanning Friction Factor from Equation 2.5 is 2w 2 × 0.0005 × 144 fN = ᎏ2 = ᎏᎏ = 0.0081 V 1.654 × 3.282 The equivalent pipeline viscosity converting the shear stress from psi to lbf/ft2, 0.0005 × 144 = 0.072 lbf/ft2, is (13.25/12) × 0.072 Diw = ᎏ = ᎏᎏ = 0.003 lbf-sec/ft2 8V 8 × 3.28 The Reynolds number is 1.654 × 3.28 × 13.25/12 Re = ᎏᎏᎏ = 1997 0.003 Check on friction factor: 16 fN = ᎏ = 0.008 1997

2-3-2 Transition Flow Friction Factor The transition from a laminar to a turbulent flow is difficult to describe. For Reynolds numbers up to 3 × 106, Wasp et al. (1977) recommended the use of Nikuradse equation for the Fanning coefficient: 0.0553 fN = 0.0008 + ᎏ Re0.237

(2-16)

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FUNDAMENTALS OF WATER FLOWS IN PIPES

2-3-3 Friction Factor in Turbulent Flow In turbulent flow, the roughness of the pipe becomes an important factor in determining the friction factor. The roughness is measured in units of length. Up to a certain limiting Reynolds number Re, the friction factor can be expressed by the following Colebrook equation:

冢 冣

冢

Di Di 1 ᎏ = 4 loge ᎏ + 3.48 – 4 loge 1 + 9.35 ᎏᎏ 2 兹f苶N 2 Re兹f苶N

冣

(2-17)

A more simplified equation for the Darcy factor is called the Prandtl–Colebrook equation: 2.51 1 ᎏ = –2 log10 ᎏ + ᎏ 兹f苶 3.7 Di Re兹f 苶 D N

冢

冣

(2-18)

A popular equation used by mechanical engineers (Lindeburg, 1997) as it is explicit and does not require tedious iterations is the Swamee–Jain equation. It is suitable for the range of Reynolds numbers between 5000 and 100,000,000: fD =

冢冦

冥冧 冣

0.25 ᎏᎏᎏ 2 /D log10 ᎏᎏ + (5.74/Re0.9) 3.7

冤

(2-19)

Because the slurry flows occur at Reynolds Numbers smaller than 100,000,000, Equation 2.19 is satisfactory in the context of this handbook. Example 2-4 Using the Swamee–Jain equation, determine the friction factor for a flow of 3500 US gpm in an 18⬙ OD pipe with a wall thickness of 0.375 in. The fluid has a specific gravity of 1.02 and a dynamic viscosity of 2.7 × 10–5 lbf-sec/ft2. Solutions in SI Units (For conversion factors refer to the Appendix at the end of this book.) 3500 × 3.785 Q = ᎏᎏ = 0.221 m3/s 60000 ID of pipe = (18 – 2 × 0.375) = 17.25 in (0.438 m) Area of flow = × 0.25 × 0.4382 = 0.1506 m2 0.221 Average velocity of flow = ᎏ = 1.467m/s 0.1506 Density = 1.02 × 1000 = 1020 kg/m3

= 2.7 × 10–5 × 47.88 = 0.00129 Pa.s 1020 × 1.467 × 0.438 Re = ᎏᎏᎏ = 506975 0.00129

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The absolute roughness of steel pipes is 6 × 10–5 m. Relative roughness is 6 × 10–5 /Di = ᎏ = 0.000137 0.438 0.25 fD = ᎏᎏᎏᎏᎏ = 0.01486 [log10(0.000137/3.7 + 5.74/5069750.9)]2 Solutions in USCS Units Q = 3500 × 0.1337 = 467.95 ft3/min ID of pipe = (18 – 2 × 0.375) = 17.25 in or 1.4375 ft Area of flow = × 0.25 × 1.43752 = 1.623 ft2 467.95 Velocity of flow = ᎏ = 255.3 ft/min or 4.80 ft/s 1.623 1.02 × 62.3 Density = ᎏᎏ = 1.973 slugs/ft3 32.2 1.973 × 4.80 × 1.4375 Re = ᎏᎏᎏ = 504,779 2.7 × 10–5 The absolute roughness of steel is 0.0002 ft. Relative roughness is 0.0002 ᎏ = 0.000139 1.4375 fD = 0.25/[log10(0.000139/3.7 + 5.74/5047790.9)]2 = 0.0149 The Colebrook and Prandtl–Colebrook formulas are limited to a certain range of Reynolds number magnitude. At high Reynolds numbers, the friction factor becomes independent of the Reynolds number. The value of the Reynolds number beyond which the friction factor is independent is calculated using the following equation: Di Re = 70 ᎏ

ᎏ 冪莦ᎏf = 70 ᎏ 冪莦 f 2

N

Di

8

D

The region for which equation 2-19 applies is shown on the Moody diagram to be to the right of the the dashed curve (Figure 2-4). The equations established so far have been developed for clear water and do not apply for plastic fluids or liquids carrying coarse particles. They can apply for any other singlephase Newtonian liquids. (These terms will be explained in Chapter 3). Most fluid dynamics books publish data for linear roughness based on Moody’s work. Such values are applicable to water. However, tests conduced on slurry pipelines can yield different values, due to the erosion of pipe, wear and tear of rubber linings, etc. The Moody diagram is a general graph for the Darcy factor versus the Reynolds number. It is applicable to a very large number of different pipe materials. There are four principal pipe materials associated with slurry flows: plastic pipes [high-density polyethylene (HDPE)], plain steel pipes, rubber-lined steel pipes, and concrete pipes. The absolute

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FIGURE 2-4 Moody diagram for the friction factor versus the Reynolds number for pipe flow (Reproduced from V. L. Streeter, Fluid Mechanics, McGraw-Hill, 1971. Reproduced by permission of McGraw-Hill, Inc.)

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roughness of these materials is presented in Table 2-1. Plain steel pipes and rubber-lined steel pipes are the most common, but HDPE and HDPE-lined steel pipes have gained in importance in the last quarter of the 20th century. One of the concentrate pipelines used in Escondida, Chile featured a long section of gravity flow in HDPE pipe. Certain reference books show a roughness of steel of 0.045–0.05 mm. This is difficult to maintain in steel pipes carrying slurries as they are often subject to erosion and corrosion. For this reason, the author recommends the use of a slightly higher roughness of the order of 0.06 mm in friction calculations. The dimensions of plain steel pipes, their pressure ratings, and relative roughness are presented in Table 2-2. It is obvious that steel pipes are limited in pressure rating to 3000 psi. This criterion is essential when considering location of booster pump stations or chokes. In the case of slurry pipelines, the thickness of the pipes is selected on the basis of 앫 Pressure 앫 Corrosion allowance 앫 Wear allowance Because of the wear allowance, erosion, and corrosion, the rating of slurry pipes is lower than presented in Table 2-2. Steel pipes may be hardened for carrying coarse slurry particles (larger than 6.5 mm or 1–4⬙), or sacrificial thickness is used to resist abrasion. The pressure rating as presented in Table 2-2 should be used as a starting point for the design calculations, and the appropriate allowance should be made for wear. Reclaimed water pipelines use the pressure ratings as in Table 2-2. The roughness and inner diameter of reclaimed water pipes may change due to scaling and deposition of lime. Steel pipes are rubber lined to a typical thickness of 6 mm (or 0.25⬙) for small sizes of pipes [< 150 mm (6⬙), 9.5 mm (3–8⬙)], and 13 mm (1–2⬙) for pipe sizes up to 24⬙. Larger pipes may be custom lined. Lining is done in an autoclave and the rubber is cured under steam. Rubber lining is limited to pumping coarse material up to a size of 6 mm (⬇ 1–4⬙). Rubber does not contribute to the pressure rating of steel pipes. Table 2-3 presents the dimensions and relative roughness of rubber-lined steel pipes. The dimensions of plain HDPE pipes (not HDPE-lined steel) for pressures up to 110 psi (760 kPa) are listed in Table 2-4. The dimensions of HDPE pipes for pressures in the range of 125 to 300 psi (863–2070 kPa) are presented in Table 2-5. These dimensions are slightly different than metric pipes. HDPE is not a magic material but can withstand the abrasion of taconite and some coarse laterites. As with rubber, there must be a cut-off size of particle size beyond which the use of HDPE is not acceptable. Very little has been published on this subject. The use of concrete pipes is often associated with gravity flows.

TABLE 2-1 Absolute Roughness of New Materials Used in Slurry Pipes Description Plastic pipes, PVC, ABS, HDPE Steel pipes Rubber-lined pipes Concrete pipes

Roughness (m)

Roughness (ft)

1.5 × 10–6 6.0 × 10–5 0.00015 0.0012

0.000004921 0.000197 0.000492 0.00394

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TABLE 2-2 Size, Rating, and Relative Roughness of Plain Steel Pipes to U.S. Dimensions* Outside diameter (inch)

Wall thickness (inch)

Pipe internal diameter (inch)

Allowed working pressure (psi) to 650 °F

Relative roughness for new pipe

2⬙ Sch 40 2⬙ Sch 80 2⬙ Sch 160 XX

2.375

0.154 0.218 0.344 0.436

2.067 1.939 1.687 1.503

1159 2038 3890 5356

0.001143 0.001212 0.001400 0.001572

3⬙ Sch 40 3⬙ Sch 80 3⬙ Sch 160 XX

3.500

0.216 0.300 0.438 0.600

3.068 2.900 2.624 2.300

1341 2129 3495 5252

0.000770 0.000815 0.000900 0.001027

4⬙ Sch 40 4⬙ Sch 80 4⬙ Sch 120 4⬙ Sch 160 XX

4.500

0.237 0.337 0.438 0.531 0.674

4.026 3.826 3.624 3.438 3.152

1191 1905 2663 3387 4553

0.000587 0.000617 0.000652 0.000687 0.000749

5⬙ Sch 40 5⬙ Sch 80 5⬙ Sch 120 5⬙ Sch 160 XX

5.633

0.257 0.375 0.500 0.625 0.750

5.117 4.883 4.633 4.383 4.133

1071 1950 2502 3284 4098

0.000461 0.000484 0.000510 0.000539 0.000572

6⬙ Sch 40 6⬙ Sch 80 6⬙ Sch 120 6⬙ Sch 160 XX

6.625

0.280 0.432 0.562 0.719 0.864

6.065 5.761 5.501 5.187 4.897

1000 1739 2394 3215 4004

0.000389 0.000410 0.000429 0.000455 0.000482

8⬙ Sch 20 8⬙ Sch 30 8⬙ Sch 40 8⬙ Sch 60 8⬙ Sch 80 8⬙ Sch 100 8⬙ Sch 120 8⬙ Sch 140 XX 8⬙ Sch 160

8.625

0.250 0.277 0.322 0.406 0.500 0.594 0.719 0.812 0.875 0.906

8.125 8.071 7.981 7.813 7.625 7.437 7.187 7.001 6.875 6.813

655 752 916 1225 1577 1935 2422 2792 3046 3173

0.000291 0.000293 0.000296 0.000302 0.000310 0.000318 0.000329 0.000337 0.000344 0.000347

0.250 0.307 0.365 0.500 0.594 0.719 0.844 1.000 1.125

10.250 10.136 10.020 9.750 9.562 9.312 9.062 8.750 8.500

523 688 856 1255 1537 1918 2308 2804 3211

0.000230 0.000233 0.000236 0.000243 0.000247 0.000254 0.000261 0.000270 0.000278 (continued)

Denomination (inch)

10⬙ Sch 20 10⬙ Sch 30 10⬙ Sch 40 S 10⬙ Sch 60 X 10⬙ Sch 80 10⬙ Sch 100 10⬙ Sch 120 10⬙ Sch 140 XX 10⬙ Sch 160

10.75

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TABLE 2-2 Continued Outside diameter (inch)

Wall thickness (inch)

Pipe internal diameter (inch)

Allowed working pressure (psi) to 650 °F

Relative roughness for new pipe

12⬙ S 12⬙ Sch 40 12⬙ X 12⬙ Sch 60 X 12⬙ Sch 80 12⬙ Sch 100 12⬙ Sch 120 XX 12⬙ Sch 140 12⬙ Sch 160

12.750

0.250 0.330 0.375 0.406 0.500 0.562 0.688 0.844 1.00 1.125 1.312

12.250 12.090 12.000 11.938 11.750 11.626 11.374 11.062 10.750 10.500 10.126

440 634 744 820 1052 1207 1526 1927 2337 2672 3183

0.000193 0.000195 0.000197 0.000198 0.000201 0.000203 0.000208 0.000214 0.000220 0.000225 0.000233

14⬙ Sch 10 14⬙ Sch 20 14⬙ Sch 30 S 14⬙ Sch 40 14⬙ X 14⬙ Sch 60 14⬙ Sch 80 14⬙ Sch 100 14⬙ Sch 120 14⬙ Sch 140 14⬙ Sch 160

14.000

0.250 0.330 0.375 0.400 0.500 0.594 0.750 0.938 1.062 1.250 1.406

13.500 13.376 13.250 13.124 13.000 12.812 12.500 12.124 11.876 11.500 11.188

401 537 676 817 956 1169 1528 1969 2265 2724 3112

0.000175 0.000177 0.000178 0.00180 0.000182 0.000184 0.000189 0.000195 0.000199 0.000205 0.000211

16⬙ Sch 10 16⬙ Sch 20 16⬙ Sch 30 S 16⬙ Sch 40 X 16⬙ Sch 60 16⬙ Sch 80 16⬙ Sch 100 16⬙ Sch 120XX 16⬙ Sch 140 16⬙ Sch 160

16.000

0.250 0.312 0.375 0.500 0.656 0.844 1.031 1.219 1.438 1.594

15.500 15.376 15.250 15.000 14.688 14.312 13.938 13.562 13.124 12.812

350 469 590 834 1142 1520 1903 2296 2764 3104

0.000152 0.000154 0.000155 0.000157 0.000161 0.000165 0.000169 0.000176 0.000180 0.000184

18⬙ Sch 10 18⬙ Sch 20 18⬙ S 18⬙ Sch 30 18⬙ X 18⬙ Sch 40 18⬙ Sch 60 18⬙ Sch 80 18⬙ Sch 100 18⬙ Sch 120 18⬙ Sch 140 18⬙ Sch 160

18.000

0.250 0.312 0.375 0.438 0.500 0.562 0.750 0.938 1.156 1.375 1.562 1.781

17.500 17.376 17.250 17.124 17.000 16.876 16.500 16.124 15.688 15.225 14.876 14.438

312 416 524 632 739 847 1178 1514 1911 2318 2673 3096

0.000135 0.000136 0.000137 0.000138 0.000139 0.000140 0.000143 0.000147 0.000151 0.000155 0.000159 0.000164

Denomination (inch)

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TABLE 2-2 Continued

Denomination (inch)

Outside diameter (inch)

Wall thickness (inch)

Pipe internal diameter (inch)

Allowed working pressure (psi) to 650 °F

Relative roughness for new pipe

20.000

0.250 0.375 0.500 0.594 0.812 1.031 1.281 1.500 1.750 1.969 0.250 0.375 0.500 0.562 0.688 0.969 1.219 1.513 1.812 2.062 2.344 0.312 0.375 0.500 0.625 0.312 0.375 0.500 0.625 0.750 0.375 0.500 0.375 0.500

19.500 19.250 19.000 18.812 18.376 17.938 17.438 17.000 16.500 16.082 23.500 23.250 23.000 22.876 22.624 22.062 21.562 20.938 20.376 19.875 19.312 29.376 29.250 29.00 28.750 35.376 35.250 35.000 34.750 34.500 41.250 41.000 47.250 47.000

280 471 664 811 1155 1507 1917 2284 2710 3091 233 392 552 635 795 1165 1500 1927 2319 2674 3083 254 313 440 568 207 260 366 473 580 223 313 195 274

0.000121 0.000123 0.000124 0.000126 0.000129 0.000132 0.000135 0.000139 0.000143 0.000147 0.0001005 0.0001016 0.0001027 0.000103 0.000104 0.000107 0.000109 0.000113 0.000116 0.000119 0.000122 0.0000804 0.0000807 0.0000814 0.0000822 0.0000668 0.0000670 0.0000675 0.0000679 0.0000685 0.0000573 0.0000576 0.0000499 0.0000503

20⬙ Sch 10 20⬙ Sch 20 S 20⬙ Sch 30 X 20⬙ Sch 40 20⬙ Sch 60 20⬙ Sch 80 20⬙ Sch 100 20⬙ Sch 120 20⬙ Sch 140 20⬙ Sch 160 24⬙ Sch 10 24⬙ Sch 20 S 24⬙ X 24⬙ Sch 30 24⬙ Sch 40 24⬙ Sch 60 24⬙ Sch 80 24⬙ Sch 100 24⬙ Sch 120 24⬙ Sch 140 24⬙ Sch 160 30⬙ Sch 10 30⬙ S 30⬙ Sch 20 X 30⬙ Sch 30 36⬙ Sch 10 36⬙ S 36⬙ Sch 20 X 36⬙ Sch 30 36⬙ Sch 40 42 S

42.000

48 S

48.000

24.000

30.000

36.000

*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of steel is taken as 60 m.

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TABLE 2-3 Size and Relative Roughness of Rubber-Lined Steel Pipes to U.S. Dimensions*

Denomination (inch)

Outside diameter (inch)

Wall thickness (inch)

Rubber thickness (inch)

Pipe internal diameter (inch)

Relative roughness for rubber-lined pipe

2⬙ Sch 40 2⬙ Sch 80 2⬙ Sch 160 XX

2.375

0.154 0.218 0.344 0.436

0.25

1.559 1.431 1.179 0.995

0.003788 0.004127 0.005009 0.005935

3⬙ Sch 40 3⬙ Sch 80 3⬙ Sch 160 XX

3.500

0.216 0.300 0.438 0.600

0.25

2.560 2.392 2.116 1.792

0.002307 0.002469 0.002790 0.003295

4⬙ Sch 40 4⬙ Sch 80 4⬙ Sch 120 4⬙ Sch 160 XX

4.500

0.237 0.337 0.438 0.531 0.674

0.25

3.518 3.138 3.116 2.930 2.664

0.001679 0.00178 0.001895 0.002015 0.002233

5⬙ Sch 40 5⬙ Sch 80 5⬙ Sch 120 5⬙ Sch 160 XX

5.633

0.257 0.375 0.500 0.625 0.750

0.25

4.609 4.375 4.125 3.875 3.625

0.001281 0.00135 0.001431 0.001524 0.001629

6⬙ Sch 40 6⬙ Sch 80 6⬙ Sch 120 6⬙ Sch 160 XX

6.625

0.280 0.432 0.562 0.719 0.864

0.25

5.557 5.253 4.993 4.679 4.389

0.001063 0.001124 0.001183 0.001262 0.001345

8⬙ Sch 20 8⬙ Sch 30 8⬙ Sch 40 8⬙ Sch 60 8⬙ Sch 80 8⬙ Sch 100 8⬙ Sch 120 8⬙ Sch 140 XX 8⬙ Sch 160

8.625

0.250 0.277 0.322 0.406 0.500 0.594 0.719 0.812 0.875 0.906

0.375

7.375 7.732 7.231 7.063 6.875 6.687 6.437 6.251 6.125 6.063

0.000801 0.000807 0.000817 0.000836 0.000859 0.000883 0.000917 0.000945 0.000964 0.000974

0.250 0.307 0.365 0.500 0.594 0.719 0.844 1.000 1.125

0.375

9.500 9.386 9.27 9.00 8.812 8.562 8.312 8.000 7.750

0.000621 0.000629 0.000637 0.000656 0.000670 0.000689 0.000710 0.000738 0.000762

10⬙ Sch 20 10⬙ Sch 30 10⬙ Sch 40 S 10⬙ Sch 60 X 10⬙ Sch 80 10⬙ Sch 100 10⬙ Sch 120 10⬙ Sch 140 XX 10⬙ Sch 16

10.75

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TABLE 2-3 Continued Outside diameter (inch)

Wall thickness (inch)

Rubber thickness (inch)

Pipe internal diameter (inch)

Relative roughness for rubber-lined pipe

12⬙ S 12⬙ Sch 40 12⬙ X 12⬙ Sch 60 X 12⬙ Sch 80 12⬙ Sch 100 12⬙ Sch 120 XX 12⬙ Sch 140 12⬙ Sch 160

12.750

0.250 0.330 0.375 0.406 0.500 0.562 0.688 0.844 1.00 1.125 1.312

0.375

11.500 11.340 11.250 11.188 11.000 10.870 10.624 10.312 10.000 9.750 9.376

0.000513 0.000521 0.000525 0.000528 0.000537 0.000543 0.000556 0.000572 0.000591 0.000606 0.000629

14⬙ Sch 10 14⬙ Sch 20 14⬙ Sch 30 S 14⬙ Sch 40 14⬙ X 14⬙ Sch 60 14⬙ Sch 80 14⬙ Sch 100 14⬙ Sch 120 14⬙ Sch 140 14⬙ Sch 160

14.000

0.250 0.330 0.375 0.400 0.500 0.594 0.750 0.938 1.062 1.250 1.406

0.375

12.750 12.626 12.500 12.374 12.250 12.062 11.750 11.374 11.126 10.750 10.438

0.000463 0.000468 0.000472 0.000477 0.000482 0.000489 0.000503 0.000519 0.000531 0.000549 0.000566

16⬙ Sch 10 16⬙ Sch 20 16⬙ Sch 30 S 16⬙ Sch 40 X 16⬙ Sch 60 16⬙ Sch 80 16⬙ Sch 100 16⬙ Sch 120XX 16⬙ Sch 140 16⬙ Sch 160

16.000

0.250 0.312 0.375 0.500 0.656 0.844 1.031 1.219 1.438 1.594

0.375

14.750 14.626 14.500 14.250 13.938 13.562 13.188 12.668 12.374 12.062

0.000400 0.000404 0.000407 0.000414 0.000424 0.000435 0.000448 0.000466 0.000477 0.000489

18⬙ Sch 10 18⬙ Sch 20 18⬙ S 18⬙ Sch 30 18⬙ X 18⬙ Sch 40 18⬙ Sch 60 18⬙ Sch 80 18⬙ Sch 100 18⬙ Sch 120 18⬙ Sch 140 18⬙ Sch 160

18.000

0.250 0.312 0.375 0.438 0.500 0.562 0.750 0.938 1.156 1.375 1.562 1.781

0.375

16.750 16.626 16.500 16.374 16.250 16.126 15.750 15.374 14.938 14.500 14.126 13.688

0.000353 0.000355 0.000358 0.000361 0.000363 0.000366 0.000375 0.000384 0.000395 0.000407 0.000418 0.000431 (continued)

Denomination (inch)

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TABLE 2-3 Continued Outside diameter (inch)

Wall thickness (inch)

Rubber thickness (inch)

Pipe internal diameter (inch)

Relative roughness for rubber-lined pipe

20⬙ Sch 10 20⬙ Sch 20 S 20⬙ Sch 30 X 20⬙ Sch 40 20⬙ Sch 60 20⬙ Sch 80 20⬙ Sch 100 20⬙ Sch 120 20⬙ Sch 140 20⬙ Sch 160

20.000

0.250 0.375 0.500 0.594 0.812 1.031 1.281 1.500 1.750 1.969

0.375

18.750 18.500 18.250 18.062 17.626 17.188 16.688 16.250 15.750 15.312

0.000315 0.000319 0.000324 0.000327 0.000335 0.000344 0.000354 0.000363 0.000375 0.000386

24⬙ Sch 10 24⬙ Sch 20 S 24⬙ X 24⬙ Sch 30 24⬙ Sch 40 24⬙ Sch 60 24⬙ Sch 80 24⬙ Sch 100 24⬙ Sch 120 24⬙ Sch 140 24⬙ Sch 160

24.000

0.250 0.375 0.500 0.562 0.688 0.969 1.219 1.513 1.812 2.062 2.344

0.375

22.750 22.500 22.250 22.126 21.874 21.312 20.812 20.224 19.626 19.126 18.562

0.000259 0.000260 0.000263 0.000265 0.000267 0.000270 0.000277 0.000284 0.000292 0.000309 0.000318

Denomination (inch)

*Dimensions for steel are based on ANSI B36.1 and B31.1. Absolute roughness of rubber is taken as 150 m.

2-3-4 Hazen–Williams Formula In the United States, the Hazen–Williams formula is often used by civil engineers because it is independent of the Reynolds number. Speed is calculated as: U = 1.319 CRH0.63S 0.54 in ft/s

(2-20)

1.85 Qgpm S = Hv /L = ᎏᎏᎏ 1.67 × C 1.85 × RH1.17

(2-21)

where Qgpm is expressed in US gallons per minute, and S = slope or head loss per unit length U = average velocity of fluid in ft/sec RH = hydraulic radius = area of pipe/perimeter of pipe C = Surface roughness coefficient (refer to Table 2-6) In SI units:

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FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-4 Dimensions of North American HDPE Pipes for Pressure Ratings 50 psi to 110 psi at a Temperature of 23°C (73.4 °F) Pressure rating

DR 32.5 50 psi

DR 26 64 psi

DR 21 80 psi

DR 17 100 psi

DR 15.5 110 psi

Pipe size (inch)

Average outside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

3 4 5 6 7 8 10 12 13 14 16 18 20 22 24 26 28 30 32M 36 40M 42 48M

3.500 4.500 5.563 6.625 7.125 8.625 10.750 12.750 13.375 14.000 16.000 18.000 20.000 22.000 24.000 26.000 28.000 30.000 31.594 36.000 39.469 42.000 47.382

3.214 4.133 5.109 6.084 6.544 7.921 9.874 11.711 12.285 12.859 14.696 16.533 18.370 20.206 22.043 23.880 25.717 27.554 29.054 33.054 36.225 38.576 43.526

3.147 4.046 5.001 5.957 6.406 7.754 9.665 11.463 12.025 12.586 14.385 16.183 17.982 19.778 21.577 23.375 25.174 26.971 28.414 32.366 35.469 37.760 42.616

3.063 3.938 4.870 5.798 6.237 7.550 9.410 11.160 11.707 12.253 14.005 15.755 17.507 19.257 21.007 22.759 24.508 26.258 27.663 31.510 34.561 36.761 41.489

3.021 3.885 4.802 5.720 6.150 7.446 9.279 11.005 11.545 12.086 13.812 15.539 17.265 18.992 20.718 22.445 24.171 25.898 27.288 31.075

6.193 6.661 8.063 10.048 11.919 12.502 13.086 14.967 16.826 18.696 20.565 22.435 24.304 26.173 28.043 29.541 33.651 35.898 39.261 44.302

U = 0.8492 CRH0.63S 0.54 in m/s

(2-22)

All parameters in Equation 2-22 must be in SI units. There is no consistency in using the Hazen–Williams formula from small to large pipes. Despite the fact that commercial publications from pipe suppliers sometimes use Hazen–Williams equations to determine pressure loss of slurries, this method is highly discouraged for long pipelines.

2.4 THE HYDRAULIC FRICTION GRADIENT OF WATER IN RUBBER-LINED STEEL PIPES Equations 2-15 to 2-19 have established a relationship between the friction factor and the Reynolds number. To compute the latter, the properties of water as a carrier fluid are presented in Tables 2-7 and 2-8.

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TABLE 2-5 Dimensions of North American HDPE Pipes for Pressure Ratings 128 psi to 300 psi at a Temperature of 23°C (73.4 °F) Pressure rating

DR 13.5 128 psi

DR11 160 psi

DR 9 200 psi

DR 7.3 254 psi

DR6.3 300 psi

Pipe size (inch)

Average outside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

3 4 5 6 7 8 10 12 13 14 16 18 20 22 24 26 28 30 32M

3.500 4.500 5.563 6.625 7.125 8.625 10.750 12.750 13.375 14.000 16.000 18.000 20.000 22.000 24.000 26.000 28.000 30.000 31.594

2.951 3.795 4.690 5.584 6.006 7.270 9.062 10.749 11.274 11.802 13.488 15.174 16.880 18.544 20.231 21.917 23.603 25.289 26.645

2.826 3.633 4.490 5.349 5.751 6.693 8.679 10.293 10.797 11.301 12.915 14.532 16.145 17.780 19.374 20.988 22.606 24.219 25.527

2.675 3.440 4.253 5.065 5.446 6.594 8.219 9.745 10.225 10.701 12.231 13.760 15.289 16.819 18.346 19.875 21.405 22.934

2.485 3.194 3.948 4.700 5.056 6.119 7.627 9.046 9.491 9.934 11.853 12.772 14.191 —

2.321 2.986 3.690 4.395 4.727 5.723 7.133 8.459 8.674 9.289 10.615 — — —

TABLE 2-6 Hazen–Williams Roughness Coefficients Description Extremely smooth pipe Very smooth pipe Concrete pipe Riveted new pipes and tiled channels Normal cast pipes, 10 year old steel pipes, masonry channels Very rough pipes

Roughness coefficient 140 130 120 110 100 60

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TABLE 2-7 Physical Properties of Water in SI Units

Temperature T (°C) 0 5 10 15 20 25 30 40 50 60 70 80 90 100

Density L (kg/m3)

Dynamic viscosity, kinematic viscosity (mPa·s)

Kinematic viscosity, dynamic viscosity (km2/s)

Surface tension in contact with air (N/m)

Vapor pressure at atmospheric pressure (kN/m2)

999.8 1000 999.7 999.1 998.2 997.0 995.7 992.2 988.0 983.2 977.8 971.8 965.3 958.4

1.781 1.518 1.307 1.139 1.002 0.890 0.798 0.653 0.547 0.466 0.404 0.354 0.315 0.282

1.785 1.519 1.306 1.139 1.003 0.893 0.800 0.658 0.553 0.474 0.413 0.364 0.326 0.294

0.0756 0.0749 0.0742 0.0735 0.0728 0.0720 0.0712 0.0696 0.0679 0.0662 0.0644 0.0626 0.0608 0.0589

0.61 0.87 1.23 1.70 2.34 3.17 4.24 7.38 12.33 19.92 31.16 47.34 70.10 101.33

TABLE 2-8 Physical Properties of Water in USCS Units Temperature T

Density L

Kinematic viscosity

Dynamic viscosity

Surface tension in contact with air

Vapor pressure at atmospheric pressure

(°F)

(slug/ft3)

(lbf-sec/ft2)

(ft2/sec)

(lbf/ft)

(psia)

32 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 212

1.940 1.940 1.940 1.938 1.936 1.934 1.931 1.927 1.923 1.918 1.913 1.908 1.902 1.896 1.890 1.883 1.876 1.868 1.860

0.00003746 0.00003229 0.00002735 0.00002359 0.0000205 0.00001799 0.00001595 0.00001424 0.00001284 0.00001168 0.00001069 0.00000981 0.00000905 0.00000838 0.0000078 0.00000726 0.00000678 0.00000637 0.00000593

0.00001931 0.00001664 0.00001410 0.00001217 0.00001059 0.00009300 0.00008260 0.00007390 0.00006670 0.00006090 0.00005580 0.00005140 0.00004760 0.00004420 0.00004130 0.00003850 0.00003620 0.00003410 0.00003190

0.00518 0.00614 0.00509 0.00504 0.00498 0.00492 0.00486 0.00480 0.00473 0.00467 0.00460 0.00454 0.00447 0.00441 0.00434 0.00427 0.00420 0.00413 0.00404

0.09 0.12 0.18 0.26 0.36 0.51 0.70 0.95 1.27 1.69 2.22 2.89 3.72 4.74 5.99 7.51 9.34 11.52 14.70

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CHAPTER TWO

The friction losses are expressed by the following equation: fDV 2L H= ᎏ 2gDI

(2-23)

where H = head due to losses (in meters for SI units, in ft for USCS units) L = length of the pipe (in meters for SI units, in ft for USCS units) fD = Darcy friction factor DI = pipe inner diameter V = speed of flow in the pipe (m/s or ft/s) g = acceleration due to gravity (9.81 m/s or 32.2 ft/s) The hydraulic friction gradient is defined as the head loss per unit length. It is defined as fDV 2 H iw = ᎏ = ᎏ 2gDI L

(2-24)

This is a very important parameter that will be used in Chapter 4 to evaluate the friction loss of Newtonian flows. Calculations of the friction factor by Equation 2-18 require iterations. The Moody diagram is a logarithmic scale that is rather difficult to use and prone to reading errors. With modern computers, a simple program will give more accurate numbers for the Darcy friction factor than from reading the Moody curve on a difficult logarithmic scale. The following program was written for plain steel and rubber-lined steel pipes. It uses standard US pipe sizes and applies the Swamee–Jain equation (2-19). DIM PIP(300), t(300), a(300), q(300), ep(300) pi = 4 * ATN(1) DEF fnlog10 (X) = LOG(X) * .4342944 ‘ p refers to nominal size ‘ outside diameter is stated p2 = 2.375 p3 = 3.5 p4 = 4.5 p5 = 5.633 p6 = 6.625 p8 = 8.625 p10 = 10.75 p12 = 12.75 p14 = 14 p16 = 16 p18 = 18 p20 = 20 p24 = 24 p30 = 30 p36 = 36 p42 = 42 p48 = 48 INPUT “choose between steel (1) and rubber (2)”, ch IF ch = 1 THEN tr1 = 0 IF ch = 1 THEN tr2 = 0

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FUNDAMENTALS OF WATER FLOWS IN PIPES

IF IF IF IF ef IF IF

ch = 2 THEN ch = 2 THEN ch = 1 THEN ch = 2 THEN = e / .0254 ch = 1 THEN ch = 2 THEN

tr1 tr2 e = e =

2.23

= .254 = .375 .00006 .00015

LPRINT “steel pipe” LPRINT “rubber lined pipe”

‘2 FOR I = 1 TO 4 t(1) = .154 t(2) = .218 t(3) = .344 t(4) = .436 PIP(I) = p2 - 2 * t(I) - 2 * tr1 ep(I) = ef / PIP(I) LPRINT USING “ppe od = ##.### in pip id = ##.#### in thick = #.### in k/d = #.########”; p2; PIP(I); t(I); ep(I) GOSUB swamee NEXT I LPRINT LPRINT The program is repeated for all US sizes of pipes (not shown here) swamee: d1 = PIP(I) * .0254 a = .25 * pi * d1 ^ 2 FOR K = 1 TO 5 q = a * K * 1000 qus = (q * 60 / 3.7854) fd = .4 em = ep(I) Re = 1000 * K * d1 / .001 110 z = (em / 3.7) + (5.74 / Re ^ .9) y = fnlog10(z) fd = .25 / y ^ 2 1111 hl = fd * K ^ 2 / (2 * 9.81 * d1) PRINT “revised swamee factor “; fd LPRINT USING “veloc = ##.### m/s; flow q = ####.#### L/s; flow = ######.## gpm “; K; q; qus LPRINT USING “RE = #########; fd = #.#####; hL = #####.####”; Re; fd; hl LPRINT PRINT “iteration error in swamee friction factor “; dg

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CHAPTER TWO

TABLE 2-9 Flow and Hydraulic Friction Gradient for Steel Pipes at a Speed of 1 m/s to 5m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m3 (62.3 lbs/ft3) and a Kinematic Viscosity of 1 cP (2.09 × 10–5 lbf-sec/ft2) Using the Swamee–Jain Equation* Relative Wall roughness Speed Speed Denomination thickness for new of flow Flow of flow (inch) (inch) pipe (m/s) L/s (ft/s)

Flow Darcy US Reynolds friction gpm number factor

Friction gradient (m/m) or (ft/ft)

2⬙ Sch 40

0.154

0.001143

1 2 3 4 5

2.2 4.3 6.5 8.6 10.8

3.3 6.6 9.9 13.2 16.5

34.3 69 103 137 172

52,502 105,004 157505 210007 262509

0.0244 0.0227 0.0221 0.0217 0.0214

0.0237 0.0883 0.1927 0.3367 0.5204

2⬙ Sch 80

0.218

0.00121

1 2 3 4 5

1.9 3.8 5.7 7.6 9.5

3.3 6.6 9.9 13.2 16.5

30.2 60.4 90.6 121 151

49,251 98,501 147,752 197002 246253

0.0248 0.0231 0.0224 0.0220 0.0218

0.0257 0.0956 0.2087 0.3648 0.5637

2⬙ Sch 160

0.344

0.001400

1 2 3 4 5

1.4 2.9 4.3 5.8 7.2

3.3 6.6 9.9 13.2 16.5

22.9 45.7 68.6 91.4 114

42,850 85,700 128,549 171,399 214249

0.0258 0.0239 0.0232 0.0228 0.023

0.0306 0.114 0.249 0.434 0.671

3⬙ Sch 40

0.216

0.000770

1 2 3 4 5

4.77 9.5 14.3 19.08 23.8

3.3 6.6 9.9 13.2 16.5

75.6 151 227 303 378

77,927 155,854 233,782 311709 389636

0.02213 0.0206 0.0200 0.0197 0.0195

0.0145 0.054 0.118 0.206 0.319

3⬙ Sch 80

0.300

0.000815

1 2 3 4 5

4.3 8.5 12.8 17.1 21.3

3.3 6.6 9.9 13.2 16.5

67.5 135 203 270 337

73,660 147,320 202,980 294,640 368,300

0.0224 0.0209 0.0203 0.0199 0.0197

0.0155 0.0579 0.1264 0.2209 0.3415

3⬙ Sch 160

0.438

0.000900

1 2 3 4 5

3.5 7 10.5 13.9 17.5

3.3 6.6 9.9 13.2 16.5

55.3 111 166 221 277

66,650 133,299 199,949 266,598 332,248

0.0230 0.0214 0.0208 0.0204 0.0202

0.0176 0.0655 0.1431 0.2501 0.3866

4⬙ Sch 40

0.237

0.000587

1 2 3 4 5

8.21 16.4 24.6 32.8 41.1

3.3 6.6 9.9 13.2 16.5

130.2 260.4 391 521 651

102,260 204,521 306,781 409,042 511,302

0.0207 0.0193 0.0188 0.0185 0.0183

0.0103 0.0386 0.0843 0.1473 0.2278

4⬙ Sch 80

0.337

0.000617

1 2 3 4 5

7.4 14.8 22.3 29.7 37.1

3.3 6.6 9.9 13.2 16.5

117 253 353 470 588

97,180 194,361 291,541 388,722 485,902

0.021 0.01957 0.019 0.0187 0.0185

0.011 0.041 0.09 0.157 0.243

4⬙ Sch 160

0.531

0.000687

1 2

6 12

3.3 6.6

95 190

87,325 0.0215 174,650 0.0201

0.0126 0.047

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FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-9 Continued Relative Wall roughness Speed Speed Denomination thickness for new of flow Flow of flow (inch) (inch) pipe (m/s) L/s (ft/s) 4⬙ Sch 160

Flow Darcy US Reynolds friction gpm number factor

3 4 5

18 24 30

9.9 13.2 16.5

285 380 475

261,976 0.0195 349,301 0.0192 436,626 0.019

Friction gradient (m/m) or (ft/ft) 0.102 0.179 0.277

5⬙ Sch 40

0.257

0.000461

1 2 3 4 5

13.3 26.5 39.8 53 66.3

3.3 6.6 9.9 13.2 16.5

210 421 631 841 1052

129,972 259,944 389,915 519,887 649,859

0.0196 0.0183 0.0178 0.0175 0.0173

0.0077 0.0287 0.0628 0.1098 0.1697

5⬙ Sch 80

0.375

0.000484

1 2 3 4 5

12.1 24.1 36.2 48.4 60.4

3.3 6.6 9.9 13.2 16.5

192 383 575 766 958

124,028 248,056 372,085 496,113 620,141

0.1981 0.0185 0.018 0.0177 0.0175

0.0081 0.0304 0.0665 0.1163 0.1797

5⬙ Sch 160

0.625

0.000539

1 2 3 4 5

9.7 19.5 29.2 38.9 48.7

3.3 6.6 9.9 13.2 16.5

154.3 308.6 463 617 772

111,328 222,656 333,985 445,313 556,641

0.0203 0.0189 0.0184 0.0181 0.0179

0.0093 0.0347 0.0759 0.1327 0.2052

6⬙ Sch 40

0.280

0.000389

1 2 3 4 5

18.7 37.3 55.9 74.6 93.2

3.3 6.6 9.9 13.2 16.5

295.4 591 886 1182 1477

154,051 308,102 462,153 616,204 770,255

0.0189 0.0176 0.0171 0.0169 0.0167

0.0062 0.233 0.051 0.0892 0.138

6⬙ Sch 80

0.432

0.000410

1 2 3 4 5

16.8 33.6 50.5 67.3 84

3.3 6.6 9.9 13.2 16.5

267 533 800 1066 1333

146,329 292,659 438,988 585,318 731,647

0.0191 0.0178 0.0173 0.0170 0.0169

0.0066 0.0248 0.0543 0.095 0.147

6⬙ Sch 160

0.719

0.000455

1 2 3 4 5

13.6 27.3 40.9 54.5 68.2

3.3 6.6 9.9 13.2 16.5

216 432 648 864 1,080

131,750 263,500 395,249 526,999 658,749

0.0195 0.0183 0.0177 0.0174 0.0173

0.0076 0.0282 0.0617 0.108 0.167

8⬙ Sch 40

0.322

0.000296

1 2 3 4 5

32.3 65.6 96.8 129.1 161.4

3.3 6.6 9.9 13.2 16.5

512 202,717 1,023 405,435 1,535 608,152 2,046 810,870 2,558 1,013,587

0.0177 0.0166 0.0161 0.0159 0.0157

0.0045 0.0167 0.0365 0.0639 0.0988

8⬙ Sch 80

0.500

0.000310

1 2 3 4 5

29.4 58.9 88.4 117.8 147.3

3.3 6.6 9.9 13.2 16.5

467 934 1,401 1,868 2335

0.0179 0.0047 0.0168 0.0176 0.0163 0.0386 0.0160 0.0675 0.0158 0.1044 (continued)

193,675 387,350 581,025 774,700 968,375

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CHAPTER TWO

TABLE 2-9 Continued Relative Wall roughness Speed Speed Denomination thickness for new of flow Flow of flow (inch) (inch) pipe (m/s) L/s (ft/s)

Flow Darcy US Reynolds friction gpm number factor

Friction gradient (m/m) or (ft/ft)

8⬙ Sch 160

0.906

0.00347

1 2 3 4 5

23.5 47 70.6 94.1 117.6

3.3 6.6 9.9 13.2 16.5

373 746 1,118 1,491 1,864

0.0184 0.0172 0.0167 0.0164 0.0163

0.0054 0.0202 0.0443 0.0774 0.1197

10⬙ Sch 40 S

0.365

0.000236

1 2 3 4 5

51 102 153 204 254

3.3 6.6 9.9 13.2 16.5

806 254,508 0.0169 1,613 509,106 0.0158 2,419 763,524 0.0154 3,226 1,018,032 0.0151 4,032 1,272,540 0.0150

0.0034 0.0127 0.0277 0.0485 0.0750

10⬙ Sch 60 X

0.500

0.000243

1 2 3 4 5

48 96 145 193 241

3.3 6.6 9.9 13.2 16.5

764 247,650 1,527 495,300 2,291 742,950 3,054 990,600 3,818 1,238,250

0.017 0.0159 0.0155 0.0152 0.0151

0.0035 0.0131 0.0286 0.0501 0.0775

10⬙ Sch 120 XX 1.000

0.000269

1 2 3 4 5

39 78 116 155 194

3.3 6.6 9.9 13.2 16.5

615 222,250 1,230 444,500 1,845 666,750 2,460 889,000 3,075 1,111,250

0.0174 0.0163 0.0158 0.0156 0.0154

0.004 0.0149 0.0327 0.0571 0.0883

12⬙ S

0.375

.000197

1 2 3 4 5

73 146 219 292 365

3.3 6.6 9.9 13.2 16.5

1,157 304,800 0.0163 2,313 609,600 0.0152 3,470 914,400 0.0148 4,626 1,219,200 0.0146 5,783 1,524,000 0.0144

0.0027 0.0102 0.0223 0.0390 0.0603

12⬙ X

0.500

0.000201

1 2 3 4 5

59 117 177 235 293

3.3 6.6 9.9 13.2 16.5

928 273,050 0.0166 1,856 546,100 0.0156 2,784 819,150 0.0152 3,713 1,099,000 0.0149 4,641 1,365,250 0.0148

0.0031 0.0116 0.0255 0.0445 0.0689

12⬙ Sch 120 XX 1.000

0.00022

1 2 3 4 5

59 117 177 235 293

3.3 6.6 9.9 13.2 16.5

928 273,050 0.0166 1856 546,100 0.0156 2784 819,150 0.0152 3713 1,099,000 0.0149 4641 1,365,250 0.0148

0.0031 0.0116 0.0255 0.0445 0.0689

14⬙ S

0.375

0.000178

1 2 3 4 5

89 178 267 356 445

3.3 6.6 9.9 13.2 16.5

1,410 336,550 0.0159 2,820 673,100 0.0149 4,231 1,009,650 0.0145 5,640 1,346,200 0.0143 7,050 1,682,750 0.0141

0.0024 0.0090 0.0198 0.0346 0.0536

14⬙ X

0.500

0.000182

1 2 3

86 171 257

3.3 6.6 9.9

1,357 2,175 4,072

0.0025 0.0093 0.0202

173,050 346,100 519,151 692,201 865,251

330,200 0.016 660,400 0.015 990,600 0.0146

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FUNDAMENTALS OF WATER FLOWS IN PIPES

2.27

TABLE 2-9 Continued Relative Wall roughness Speed Speed Denomination thickness for new of flow Flow of flow (inch) (inch) pipe (m/s) L/s (ft/s) 14⬙ X

Flow Darcy US Reynolds friction gpm number factor

Friction gradient (m/m) or (ft/ft)

4 5

343 428

13.2 16.5

5,429 1,320,800 0.0144 6,787 1,651,000 0.0142

0.0354 0.0548

14⬙ Sch 120

1.062

0.000531

1 2 3 4 5

72 143 214 286 357

3.3 6.6 9.9 13.2 16.5

1,133 301,650 0.0163 2,266 603,301 0.0153 3398 904,951 0.01485 4531 1,206,602 0.0146 5664 1,508,251 0.0145

0.0028 0.0103 0.0226 0.0395 0.0611

16⬙ Sch 30 S

0.375

0.000155

1 2 3 4 5

118 236 354 471 589

3.3 6.6 9.9 13.2 16.5

1,868 387,350 0.0155 3,736 774,700 0.0145 5,604 1,162,050 0.0141 7,471 1,549,400 0.0139 9,339 1,936,750 0.0138

0.002 0.0076 0.0167 0.0292 0.0452

16⬙ X

0.500

0.0001575

1 2 3 4 5

114 228 342 456 570

3.3 6.6 9.9 13.2 16.5

1,807 381,000 0.0155 3,614 762,000 0.0146 5,421 1,143,000 0.0142 7,228 1,524,000 0.0139 9,035 1,905,000 0.0138

0.0021 0.0078 0.0170 0.0298 0.0461

16⬙ Sch 120

1.291

0.000176

1 2 3 4 5

91 182 274 365 456

3.3 6.6 9.9 13.2 16.5

1,446 340,817 0.0159 2,892 681,634 0.0149 4,338 1,022,452 0.0145 5784 1,363,269 0.0143 7230 1,704,086 0.0141

0.0024 0.0089 0.0195 0.0341 0.0527

18⬙ S

0.375

0.000137

1 2 3 4 5

151 302 452 603 754

3.3 6.6 9.9 13.2 16.5

2,390 438,150 0.0151 4,780 876,300 0.0142 7,170 1,314,450 0.0138 9,560 1,752,600 0.0136 11,949 2,190,750 0.0134

0.0018 0.0066 0.0144 0.0252 0.0390

18⬙ X

0.500

0.000139

1 2 3 4 5

146 293 439 586 732

3.3 6.6 9.9 13.2 16.5

2,321 431,800 0.0151 4,642 863,600 0.0142 6,963 1,295,400 0.0138 9,284 1,727,200 0.0136 11,605 2,159,000 0.0135

0.0018 0.0067 0.0147 0.0257 0.0397

18⬙ Sch 120

1.375

0.000155

1 2 3 4 5

118 236 354 471 589

3.3 6.6 9.9 13.2 16.5

1,868 387,350 0.0155 3,734 774,700 0.0145 5,604 1,162,050 0.0141 7,471 1,549,000 0.0139 9340 1,966,750 0.0138

0.0020 0.0076 0.0167 0.0292 0.0452

20⬙ Sch 20S

0.375

0.000123

1 2 3 4 5

188 375 563 751 939

3.3 2,976 488,950 0.0148 0.0015 6.6 5,952 977,900 0.0139 0.0058 9.9 8,929 1,466,850 0.0135 0.0126 13.2 11,905 1,955,800 0.0133 0.0221 16.5 14,881 2,444,750 0.0131 0.0342 (continued)

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CHAPTER TWO

TABLE 2-9 Continued Relative Wall roughness Speed Denomination thickness for new of flow (inch) (inch) pipe (m/s)

Speed Flow Flow of flow US L/s (ft/s) gpm

Darcy Reynolds friction number factor

Friction gradient (m/m) or (ft/ft)

20⬙ Sch 30 X

0.500

0.000124

1 2 3 4 5

183 366 549 732 915

3.3 6.6 9.9 13.2 16.5

2,899 5,799 8,698 11,598 14,497

482,600 965,200 1,447,800 1,930,400 2,413,000

0.0148 0.0139 0.0135 0.0133 0.0132

0.0016 0.0059 0.0128 0.0225 0.0348

24⬙ Sch 20 S

0.375

0.000102

1 2 3 4 5

274 548 822 1096 1370

3.3 6.6 9.9 13.2 16.5

4,341 8,683 13,025 17,366 21,707

590,550 1,181,100 1,771,650 2,362,200 2,952,750

0.0142 0.0134 0.013 0.0128 0.01267

0.0012 0.0046 0.0101 0.0177 0.0273

24⬙ Sch S

0.500

0.000102

1 2 3 4 5

268 536 804 1072 1340

3.3 6.6 9.9 13.2 16.5

4,249 584,200 0.01425 0.0012 8,497 1,1684,000 0.01338 0.0047 12,746 1,752,600 0.01302 0.0102 16,995 2,336,800 0.01282 0.0179 21,243 2,921,000 0.01269 0.0277

*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of steel is taken as 60 m.

‘INPUT “hit any key to continue”; m$ CLS NEXT K RETURN The range of speeds used to carry solids in Newtonian flows is typically between 1.5 m/s and 5m/s. Tables 2-9 and 2-10 present friction factor and head losses for water as a carrier fluid for plain and rubber-lined steel pipes. There is no point in tabulating other fluids here as they are rarely used for slurry mixtures. The hydraulic friction gradient of water in rubber-lined pipes in the range of 2⬙ to 18⬙ is presented in Figures 2-5 to 2-13. Rubber thickness of 6.4 mm (0.25⬙) was assumed for 2⬙, 3⬙, 4⬙, and 6⬙ (up to 150 mm) pipes. Rubber thickness of 9.5 mm (0.375⬙) was assumed for 8⬙ to 24⬙ (200 to 610 mm NB) pipes. HDPE friction head was plotted for similar sizes at SDR11 (suitable for 100 psi pressure), to mark the advantages of reduced friction at these sizes using HDPE instead of rubber-lined pipes, wherever it may be appropriate. The design engineer must take in account the pressure limitations of HDPE pipes versus rubber-lined steel pipes. The hydraulic friction gradient for HDPE pipes up to a size of 20⬙ (508 mm), and for speeds in the range of 1 to 5 m/s (3.3 to 16.5 ft/sec) is presented in Table 2-11 These curves and tables allow an easier and accurate determination of the hydraulic friction gradient of water than the Moody diagram.

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FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-10 Flow and Hydraulic Friction Gradients for Rubber-Lined Steel Pipes at a Speed of 1 m/s to 5 m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m3 (62.3 lbs/ft3) and a Kinematic Viscosity of 1 cP (2.09 × 10–5 lbf-sec/ft2) Using the Swamee–Jain Equation*

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

39,599 79,197 118,796 158,394 197,993

0.0309 0.0297 0.289 0.0287 0.0287

0.0399 0.153 0.338 0.595 0.925

2⬙ Sch 40

Steel 0.154 0.003788 Rubber 0.250

1 2 3 4 5

1.2 2.5 3.7 4.9 6.2

3.3 6.6 9.9 13.2 16.5

2⬙ Sch 80

Steel 0.218 0.004123 Rubber 0.250

1 2 3 4 5

1 2.1 3.1 4.2 5.2

3.3 6.6 9.9 13.2 16.5

16.5 33 49.4 65.8 82.2

36,347 72,695 109,042 145,390 181,737

0.0317 0.0299 0.0296 0.0295 0.0337

0.045 0.171 0.377 0.665 1.033

2⬙ Sch 160

Steel 0.344 0.005008 Rubber 0.250

1 2 3 4 5

0.7 1.4 2.1 2.8 3.5

3.3 6.6 9.9 13.2 16.5

11.2 22.3 33.5 44.7 55.8

29,947 59,893 89,840 119,786 149,733

0.0337 0.0322 0.0317 0.0314 0.0312

0.0574 0.2195 0.4854 0.8551 1.3284

3⬙ Sch 40

Steel 0.216 Rubber 0.250

1 2 3 4 5

3.3 6.6 10 13.3 16.6

3.3 6.6 9.9 13.2 16.5

53 105 158 211 263

65,024 130,048 195,072 260,096 325,120

0.0269 0.0257 0.0253 0.0251 0.0250

0.0211 0.0808 0.1788 0.3150 0.4895

3⬙ Sch 80

Steel 0.300 0.002487 Rubber 0.250

1 2 3 4 5

2.9 5.8 8.7 11.6 14.5

3.3 6.6 9.9 13.2 16.5

46 92 138 184 230

60,757 121,514 182,270 243,027 303,784

0.0274 0.0262 0.0258 0.0256 0.0255

0.0230 0.088 0.1949 0.3435 0.5337

0.002791 3⬙ Sch 160 Steel 0.438 Rubber 0.250

1 2 3 4 5

2.3 4.5 6.8 9.1 11.3

3.3 6.6 9.9 13.2 16.5

36 72 108 144 180

53,746 107,493 161,239 214,986 268,732

0.0283 0.0272 0.0267 0.0265 0.0263

0.0269 0.103 0.228 0.402 0.624

4⬙ Sch 40

Steel 0.237 0.001679 Rubber 0.250

1 2 3 4 5

6.3 12.5 18.8 25.1 31.4

3.3 6.6 9.9 13.2 16.5

99 199 298 398 497

89,357 178,714 268,072 357,429 446,786

0.0247 0.0237 0.0233 0.0231 0.0229

0.0141 0.054 0.1195 0.2106 0.3273

4⬙ Sch 80

Steel 0.337 0.00178 Rubber 0.250

1 2 3 4 5

5.6 11.2 16.7 23.3 27.8

3.3 6.6 9.9 13.2 16.5

88.4 177 265 354 442

84,277 168,554 252,832 337,109 421,386

0.0251 0.0151 0.0240 0.0581 0.0236 0.1287 0.0234 0.2268 0.0233 0.352 (continued)

19.5 39 58.6 78 97.6

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Page 2.30

2.30

CHAPTER TWO

TABLE 2-10 Continued

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

4⬙ Sch 160

Steel 0.531 0.002015 Rubber 0.250

1 2 3 4 5

04.4 8.7 13.1 17.4 21.8

03.3 6.6 9.9 13.2 16.5

069 138 207 276 345

074,422 148,844 223,266 297,688 372,110

0.0259 0.0248 0.0244 0.0242 0.0241

0.0178 0.068 0.151 0.265 0.412

5⬙ Sch 40

Steel 0.257 0.001281 Rubber 0.250

1 2 3 4 5

10.8 21.5 32.3 43.1 53.8

3.3 6.6 9.9 13.2 16.5

171 341 512 683 853

117,069 234,137 351,206 468,274 585,343

0.023 0.0221 0.0217 0.0215 0.0214

0.01 0.038 0.085 0.150 0.233

5⬙ Sch 80

Steel 0.375 0.001349 Rubber 0.250

1 2 3 4 5

9.7 19.4 29.1 38.8 48.5

3.3 6.6 9.9 13.2 16.5

154 308 461 615 769

111,125 222,250 333,375 444,500 555,625

0.0233 0.0224 0.022 0.0218 0.0217

0.0107 0.0410 0.0901 0.160 0.249

5⬙ Sch 160

Steel 0.625 0.00152 Rubber 0.250

1 2 3 4 5

7.61 3.3 15.2 6.6 22.8 9.9 30.4 13.2 38.0 16.5

121 241 362 482 603

98,425 196,850 295,275 393,700 492,125

0.024 0.0231 0.0227 0.0225 0.0224

0.0124 0.0478 0.1058 0.1865 0.2898

6⬙ Sch 40

Steel 0.280 0.001063 Rubber 0.250

1 2 3 4 5

15.6 31.3 47 62.5 78.2

3.3 6.6 9.9 13.2 16.5

248 496 744 992 1240

141,148 282,296 423,443 564,591 705,739

0.0219 0.0211 0.0207 0.0206 0.0204

248 496 744 992 1240

6⬙ Sch 80

Steel 0.432 0.001124 Rubber 0.250

1 2 3 4 5

14 28 42 56 70

3.3 6.6 9.9 13.2 16.5

222 443 665 886 1108

133,426 266,852 400,279 533,705 667,131

0.0222 0.0214 0.0210 0.0208 0.0207

0.0085 0.0326 0.0723 0.1274 0.198

6⬙ Sch 160

Steel 0.719 0.001262 Rubber 0.250

1 2 3 4 5

11.1 22.2 33.3 44.4 55.5

3.3 6.6 9.9 13.2 16.5

176 352 528 703 879

118.847 237,693 356,540 475,386 594,233

0.0229 0.0219 0.0215 0.0215 0.0213

0.0098 0.0377 0.0835 0.1472 0.2288

8⬙ Sch 40

Steel 0.322 0.000817 Rubber 0.375

1 2 3 4 5

26.5 53 79.5 106 132

3.3 6.6 9.9 13.2 16.5

420 840 1,260 1,680 2,100

183,667 367,335 551,002 734,670 918,337

0.0205 0.0198 0.0195 0.0193 0.0192

0.0057 0.0219 0.0486 0.0856 0.1331

8⬙ Sch 80

Steel 0.500 0.000859 Rubber 0.375

1 2 3

24 48 72

3.3 6.6 9.9

380 760 1,139

174,625 349,250 523,875

0.0208 0.0199 0.0197

0.006 0.0233 0.0517

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2.31

FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-10 Continued

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

8⬙ Sch 80

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

4 5

96 120

13.2 16.5

1,519 1898

698,500 0.0195 873,125 0.0194

0.0912 0.142

19 37 6 75 93

3.3 6.6 9.9 13.2 16.5

295 590 886 1,181 1,476

154,000 308,000 462,001 616,001 770,001

0.0215 0.0206 0.0203 0.0201 0.020

0.0071 0.0273 0.0604 0.1065 0.1656

8⬙ Sch 160

Steel 0.906 0.000974 Rubber 0.375

1 2 3 4 5

10⬙ Sch 40 S

Steel 0.365 0.000637 Rubber 0.375

1 2 3 4 5

43.5 3.3 87 6.6 131 9.9 174 13.2 218 16.5

690 235,458 1,380 470,916 2,071 706,374 2,761 941,832 3,451 1,177,290

0.0194 0.0186 0.0183 0.0182 0.0181

0.0042 0.0161 0.0357 0.063 0.098

10⬙ Sch 60 X

Steel 0.500 0.000656 Rubber 0.375

1 2 3 4 5

41 82 123 164 205

3.3 6.6 9.9 13.2 16.5

651 228,600 1,301 457,200 1,952 658,800 2,602 914,400 3,253 1,143,000

0.0195 0.0188 0.0185 0.0183 0.0182

0.0044 0.0167 0.0371 0.0653 0.1016

10⬙ Sch 120 XX

Steel 1.000 0.000738 Rubber 0.375

1 2 3 4 5

32 65 97 130 162

3.3 6.6 9.9 13.2 16.5

514 203,200 1,028 406,400 1,542 609,600 2,056 812,800 2,570 1,016,000

0.0201 0.0198 0.0189 0.0188 0.0187

0.0050 0.0193 0.0429 0.0756 0.1174

12⬙ S

Steel 0.375 0.000525 Rubber 0.375

1 2 3 4 5

64.1 3.3 128.3 6.6 192.4 9.9 256.5 13.2 320.7 16.5

1,017 285,750 0.01853 2,033 571,500 0.0178 3,049 857,250 0.01755 4,066 1,143,000 0.0174 5,082 1,428,750 0.0173

0.0033 0.0127 0.0282 0.0497 0.0722

12⬙ X

Steel 0.500 0.000537 Rubber 0.375

1 2 3 4 5

61 123 184 245 307

3.3 6.6 9.9 13.2 16.5

972 279,400 0.0186 1,943 558,800 0.0179 2,915 838,200 0.0176 3,887 1,117,600 0.0175 4,859 1,397,000 0.0174

0.0034 0.0131 0.029 0.051 0.079

12⬙ Sch 120 XX

Steel 1.000 0.000591 Rubber 0.375

1 2 3 4 5

51 101 152 203 253

3.3 6.6 9.9 13.2 16.5

803 254,000 0.0190 1606 508,000 0.0183 2409 762,000 0.0180 3212 1,016,000 0.0179 4015 1,270,000 0.0179

0.038 0.0147 0.0326 0.0574 0.0892

14⬙ S

Steel 0.375 0.000472 Rubber 0.375

1 2 3 4 5

79 158 238 317 396

3.3 6.6 9.9 13.2 16.5

1,255 317,500 0.0181 2,510 635,000 0.0174 3,765 952,500 0.0171 5,020 1,270,000 0.0170 6,275 1,587,500 0.0169

0.0029 0.0112 0.0248 0.0437 0.0679

(continued)

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Page 2.32

2.32

CHAPTER TWO

TABLE 2-10 Continued

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

14⬙ X

Steel 0.500 0.000482 Rubber 0.375

1 2 3 4 5

76 152 228 304 380

3.3 6.6 9.9 13.2 16.5

1,205 311,150 2,410 622,300 3,616 933,450 4,820 1,244,600 6,026 1,555,750

0.0182 0.0175 0.0172 0.0171 0.0170

0.003 0.0115 0.0254 0.0447 0.0695

14⬙ Sch 120

Steel 1.062 0.000531 Rubber 0.375

1 2 3 4 5

63 125 188 251 314

3.3 6.6 9.9 13.2 16.5

997 282,600 1988 565,201 2983 847,801 3977 1,130,402 4971 1,413,002

0.0186 0.0179 0.0176 0.0175 0.0174

0.0034 0.0129 0.0286 0.0503 0.0783

16⬙ Sch 30 S

Steel 0.375 0.000407 Rubber 0.375

1 2 3 4 5

107 213 319 426 532

3.3 6.6 9.9 13.2 16.5

1,689 368,300 3,377 736,600 5,066 1,104,900 6,755 1,473,200 8,443 1,841,500

0.0175 0.0168 0.0166 0.0165 0.0164

0.0024 0.0093 0.0207 0.0364 0.0566

16⬙ X

Steel 0.500 0.000414 Rubber 0.375

1 2 3 4 5

103 206 309 412 515

3.3 6.6 9.9 13.2 16.5

1,631 361,950 3,262 713,900 4,893 1,085,850 6,524 1,447,800 8,155 1,809,750

0.0176 0.0169 0.0167 0.0165 0.0164

0.0025 0.0095 0.0211 0.0372 0.0578

16⬙ Sch 120

Steel 1.291 0.000466 Rubber 0.375

1 2 3 4 5

81 162 243 324 405

3.3 6.6 9.9 13.2 16.5

1,289 321,767 2,578 643,534 3867 965,302 5155 1,287,069 6444 1,608,836

0.0180 0.0174 0.0171 0.0169 0.0169

0.0029 0.011 0.0244 0.0429 0.0668

18⬙ S

Steel 0.375 0.000358 Rubber 0.375

1 2 3 4 5

138 276 414 552 690

3.3 6.6 9.9 13.2 16.5

2,187 419,100 4,373 838,200 6,560 1,257,300 8,746 1,676,400 10,933 2,095,500

0.0170 0.0164 0.0161 0.0160 0.0159

0.002 0.008 0.0176 0.0311 0.0484

18⬙ X

Steel 0.500 0.000363 Rubber 0.375

1 2 3 4 5

134 268 401 535 669

3.3 6.6 9.9 13.2 16.5

2,121 412,750 4,241 825,500 6,362 1,238,250 8,483 1,651,000 10,604 2,063,750

0.0171 0.0164 0.0162 0.0160 0.0159

0.0021 0.0081 0.0180 0.0317 0.0493

18⬙ Sch 120

Steel 1.375 0.000407 Rubber 0.375

1 2 3 4 5

107 213 320 426 533

3.3 6.6 9.9 13.2 16.5

1,689 368,300 3,377 736,600 5,066 1,104,900 6,755 1,473,200 8,443 1,841,500

0.0175 0.0168 0.0166 0.0165 0.0164

0.0024 0.0093 0.0207 0.0364 0.0566

20⬙ Sch 20S

Steel 0.375 0.000319 Rubber 0.375

1 2 3

173 346 520

3.3 6.6 9.9

2,7495 469,900 497 939,800 8,246 1,409,700

0.0166 0.0159 0.0157

0.0018 0.0069 0.0154

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Page 2.33

2.33

FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-10 Continued

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

20⬙ Sch 20S

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

4 5

694 867

13.2 16.5

10,995 1,879,600 0.0156 13,744 2,349,500 0.0155

0.0271 0.0421

20⬙ Sch 30 X

Steel 0.500 0.000324 Rubber 0.375

1 2 3 4 5

169 338 506 675 844

3.3 6.6 9.9 13.2 16.5

2,675 463,550 0.0166 5,350 927,100 0.0160 8,025 1,390,650 0.0158 10,670 1,854,200 0.0156 13,375 2,317,750 0.0155

0.0018 0.0070 0.0156 0.0275 0.0428

24⬙ Sch 20 S

Steel 0.375 0.000262 Rubber 0.375

1 2 3 4 5

257 513 780 1,026 1,283

3.3 6.6 9.9 13.2 16.5

4,066 8,132 12,198 16,284 20,330

571,500 1,143,000 1,714,500 2,286,000 2,857,500

0.0159 0.0153 0.0151 0.0150 0.0149

0.0014 0.0055 0.0121 0.0214 0.0332

0.000265 24⬙ Sch S Steel 0.500 Rubber 0.375

1 2 3 4 5

251 502 753 1,003 1,254

3.3 6.6 9.9 13.2 16.5

3,976 7,952 11,930 15,904 19,880

565,150 1,130,300 1,695,450 2,260,600 2,825,700

0.0159 0.0154 0.0151 0.0150 0.0149

0.0014 0.0055 0.0123 0.0216 0.0336

*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of rubber is input as 150 m (0.000492 ft).

2-5 DYNAMICS OF THE BOUNDARY LAYER Boundary layer theory has been extensively covered by a number of authors. A book by Schlichting (1968) is considered one of the classical references on this subject. When a uniform flow approaches a plate, the particles at the wall of the plate are slowed down by the dynamic viscosity of the fluid. A layer called the boundary layer develops. When the flow enters a pipe, effects develop at the entrance until the flow is uniform.

2-5-1 Entrance Length Flow in a pipe at relatively low speed when the Reynolds number is smaller than 2500 is characterized by a certain distance called “entrance length,” over which the velocity profile takes the final parabolic shape shown in Figure 2-14. The length Le is expressed as Le = 0.028 DiRe For turbulent flows, the entrance length is equivalent to 50 times the inner diameter.

9:14 AM

Page 2.34

2.34

0

40

5 m/s

sch

0.8 0.7

4 m/s

0.6 0.5 0.4 3 m/s

0.3 0.2

2.0 4.0 6.0 Flow Rate (L/s)

rometers (0.000492 ft)

0.8

16.5 ft/sec 40

1.0 0.9

8.0

0

0.0

1 m/s

sch 8

0.0

2 m/s

sch

0.1

Hydraulic friction gradient (ft/ft)

sch 8

0.9

60

1.0

sch 1

Hydraulic friction gradient (m/m)

CHAPTER TWO

60

2/28/02

sch 1

abul-2.qxd

0.7 13.2 ft/sec

0.6 0.5 0.4 0.3

9.9ft/sec

0.2 0.1 0.0

6.6 ft/sec 3.3 ft/sec

0.0 20 40 60 80 100 120 Flow Rate (US gallons/min) FIGURE 2-5 Hydraulic friction gradient for water in a rubber-lined 2⬙ pipe, Sch 40, Sch 80 and Sch 160. Caculations for 2⬙ steel pipe. Rubber thickness = 0.250⬙ (6.4 mm). Rubber roughness = 150 m (0.000492 ft).

rometers (0.000492 ft)

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Page 2.35

2.35

Hydraulic friction gradient (m/m) (m/m)

FUNDAMENTALS OF WATER FLOWS IN PIPES

0.7 5 m/s

0.6 0.5

h sc 4 m/s

0.4

h sc

Rubber Lined Steel Pipes

80

40 sch

0.3

4 m/s

3 m/s

2 m/s

0.1

HDPE SDR 11 5 m/s

3 m/s

0.2

2 m/s

1 m/s

0.0 0.0

Hydraulic friction (ft/ft)gradient (ft/ft)

0 16

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0 18.0 20.0 Flow Rate (L/s)

0.7 0.6 0.5 13.2 ft/sec

0.4 0.3

16.5 ft/sec 0 16 h sc

Rubber Lined Steel Pipes 0 h8 sc

40 sch

9.9ft/sec

0.2 0.0 0

16.5 ft/sec

9.9 ft/sec

6.6 ft/sec

0.1

HDPE SDR 11 13.2 ft/sec

6.6 ft/sec

3.3 ft/sec

20

40

60

80

100 120 140 160 180 200 220 240 260 220 240 260 Flow Rate (US gallons/min)

FIGURE 2-6 Hydraulic friction gradient for water in a rubber-lined 3⬙ pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.25⬙ (6.4 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE line 3⬙ SDR11 roughness 1.5 m (0.00000492 ft).

2-5-2 Friction Velocity Prandtl proposed a concept of friction velocity: Uf =

w

ᎏ =U ᎏ 冪莦 冪莦2 fN

(2-25)

Blasius conducted tests on turbulent flows in pipes and developed an equation for the shear wall stress in terms of the maximum velocity outside the boundary layer.

冢 冣

r w = 0.0225 (Umax)7/4 ᎏ R

1/4

(2-26)

The local magnitude of the velocity in a boundary layer at a height y above the wall is u ᎏ = 8.73(y+)n Uf

(2-27)

where n = 1/7 to 1/9 and where the nondimensional height parameter y+ is defined as the relative distance from the wall:

abul-2.qxd 2/28/02

5 m/s 4 m/s

0.1

2 m/s 3 m/s 1 m/s

0.0 0.0

2 m/s

10

20

30

Flow Rate (L/s)

40

HDPE SDR 11

sc h 80 40

13.2 ft/sec

16.5 ft/sec

sch

0.3

sch

Energy gradient (ft/ft)

5 m/s

0.2 3 m/s

160

0.4

sc h

sch 80 40

160 4 m/s

sch

0.3

Rubber Lined Steel Pipes

Page 2.36

2.36

Energy gradient (m/m)

0.4

9:14 AM

Rubber Lined Steel Pipes

0.2 16.5 ft/sec

9.9 ft/sec

0.1

13.2 ft/sec

6.6 ft/sec

9.9 ft/sec

0.0

6.6 ft/sec 3.3 ft/sec

0.0 100 200 300 400 500

HDPE SDR 11

Flow Rate (US gallons/min)

FIGURE 2-7 Hydraulic friction gradient for water in a rubber-lined 4⬙ pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.25⬙ (6.4 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE line 3⬙ SDR11 roughness 1.5 m (0.00000492 ft).

2/28/02

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Page 2.37

2.37

FUNDAMENTALS OF WATER FLOWS IN PIPES

Energygradient (m/m)

0.30 Rubber Lined Steel

0.25 0.20

ch 4 m/s s

0.15 0.10

5 m/s 0 16

40 sch

80 sch

3 m/s

0.05

1 m/s

0.0

2 m/s

HDPE SDR11 0.0

10

20

30

40

50

60

70

80

Flow Rate (L/s) 0.30 0.25 Energy gr adient (ft/ft)

abul-2.qxd

16.5 ft/sec 60 1 ch s 13.2 ft/sec 80 sch

0.20 0.15

40 sch

0.10 0.05 0.0

9.9 ft/sec 6.6 ft/sec 3.3 ft/sec

HDPE SDR11 0.0

200

400

600

800

1000

1200

Flow Rate ( US gallons/sec) FIGURE 2-8 Hydraulic friction gradient for water in a rubber-lined 6⬙ pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.25⬙ (6.4 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE line 3⬙ SDR11 roughness 1.5 m (0.00000492 ft).

Uf␥ y+ = ᎏ

(2-28)

and where u = velocity of the flow at distance y. The boundary layer can be divided into a number of sections. At small values of y+ up to 5, the velocity profile is linear in a sublayer (see Figure 2-15). The flow is considered to be laminar in the sublayer. Above y+ = 5, a buffer zone develops up to y+ = 50 and turbulence develops. The thickness of the boundary viscous sublayer ␦ is usually expressed as 11.6 11.6 ␦= ᎏ = ᎏ 兹苶 (苶 w苶) U

ᎏ 冪莦 8 fD

(2-29)

In the turbulent region, the velocity profile is established as Uf y u ᎏ = 5.75 log10 ᎏ + 5.5 Uf

冢 冣

(2-30)

abul-2.qxd 2/28/02

0.08 3 m/s

0.06 0.04 0.02

HDPE SDR11 2 m/s 1 m/s

0.0

20

40

60

80

100 120 140 Flow Rate (L/s)

h

40

sc

0.08 9.9 ft/sec

0.06 0.04 0.02

HDPE SDR11 6.6 ft/sec 3.3 ft/sec

0.0

0.0

h

13.2 ft/sec

0.10

sc

0.12

80

160

0.14

sch

Hydraulic friction gradient (ft/ft)

80 h

sc

4 m/s

0.10

40

sc

0.12

h

sch

160

0.14

16.5 ft/sec

0.16

Page 2.38

2.38

Hydraulic friction gradient (m/m)

5 m/s

0.16

Rubber Lined Steel 0.18

0

400

800 1200 1600 Flow Rate (US gallons/min)

9:14 AM

Rubber Lined Steel 0.18

2000

FIGURE 2-9 Hydraulic friction gradient for water in a rubber-lined 8⬙ pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.375⬙ (9.5 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE roughness 1.5 m (0.00000492 ft).

2/28/02

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Page 2.39

2.39

FUNDAMENTALS OF WATER FLOWS IN PIPES

Hydrau lic friction gradient (m/m)

0.12 0.10

5 m/s

0.08

5 m/s

0.06

h sc " 10 3 m/s

0.04 0.02 0.0

2 m/s 1 m/s

0.0 40

80

40

4 m/s 4 m/s

40 ch s " 12 3 m/s

2 m/s 1 m/s

120 160 200 240 280 320 Flow Rate (L/s)

0.12 16.5 ft/sec

Hydrau lic friction gradient (ft/ft)

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0.10 0.08 0.06 0.04 0.02

9.9 ft/sec 6.6 ft/sec

16.5 ft/sec

13.2 ft/sec 13.2 ft/sec

" 12

40 sch

9.9 ft/sec

0.0 0.0

3.3 ft/sec

ch "s 0 1

40

800

1600 2400 3200 4000 4800 Flow Rate ( US gallons/sec) 6.6 ft/sec

FIGURE 2-10 Hydraulic friction gradient for water in a rubber-lined 10⬙ pipe, Sch 40 and 12⬙, Sch 40 pipes. Rubber thickness = 0.375⬙ (9.5 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE roughness 1.5 m (0.00000492 ft).

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CHAPTER TWO

Hydraulic friction gradient (m/m)

0.08 5 m/s

0.06 DR "S

0.04

11

5 m/s

10

0.02

DR "S

11

12

0.0 0.0 40

80

120 160 200 240 280 320 Flow Rate (L/s)

Hydraulic friction gradient (ft/ft)

0.08 0.06 0.04

" 10

0.02

R SD

16.5 ft/sec 16.5 ft/sec

11

DR "S

11

12

0.0 0.0

800

1600 2400 3200 4000 4800 Flow Rate ( US gallons/sec)

FIGURE 2-11 Hydraulic friction gradient of water in 10⬙ and 12⬙ SDR11 HDPE pipes. Roughness 1.5 m (0.00000492 ft).

The reader is encouraged to review the work of Schlichling (1968) for details of boundary layer theory. Example 2-5 Determine the boundary viscous sublayer thickness and friction velocity of Example 2-4. Solution in SI Units In Example 2-4, the Darcy friction factor was determined to be 0.0178. In Section 2.31 it was stated that fD = 4fN, therefore 0.0178 fN = ᎏ = 0.00445 4 Since the velocity of the flow is 1.467 m/s,

冪莦

冪莦

ff 0.00445 Uf = U ᎏ = 1.467 ᎏ = 0.0692 m/s 2 2 From Equation 2.28 the thickness of the viscous sublayer is calculated as 11.6 ␦= ᎏ U

ᎏ 冪莦 8 fD

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FUNDAMENTALS OF WATER FLOWS IN PIPES

30 ) (S ch

0.08

16 "S

S

0.07

14 "

0.06

" 18 5 m/s

4 m/s

0.05 0.04

S

3 m/s

0.03 0.02 2 m/s

0.01

1 m/s

0.0 0.0

100 200 300 Flow Rate (L/s)

0 09

400

500

600

Flow Rate (L/s) 30 )

0.09

(S ch

0.08 14 "

S

0.07 0.06 0.05

" 18

13.2 ft/sec

0.04

S

16.5 ft/sec

0.03

9.9 ft/sec

0.02 0.01

700

16 "S

Hydraulic friction gradient (m/m)

0.09

Hydraulic friction gradient (ft/ft)

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6.6 ft/sec 3.3 ft/sec

0.0 0.0

2000

4000 6000 8000 Flow Rate (US gallons/min)

10000

FIGURE 2-12 Hydraulic friction gradient for water in rubber-lined 14⬙ pipe, Sch 40 and 16⬙ S and 18⬙ S pipes. Wall thickness = 0.375⬙ (9.5 mm). Rubber thickness = 0.375⬙ (9.5 mm). Rubber roughness = 150 m (0.000492 ft).

11.6 (0.00129) ␦ = ᎏᎏ 1020 (1.467)

ᎏ = (4.72) 10 冪莦 8 0.0178

Solution in USCS Units 0.0178 fN = ᎏ = 0.00445 4 Since the average velocity flow in 4.8 ft/s,

冪莦

0.00445 Uf = 4.8 ᎏ 0.2264 ft/s 2

–7

m

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CHAPTER TWO

0.08 11

0.07

0.05

5 m/s

18

R

11

0.04

16

"S

D

R

0.06

SD

5 m/s

11

5 m/s

"

0.03

"

R SD

14

Hydraulic friction gradient (m/m)

0.09

0.02 0.01 0.0 0.0

100

200

300

400

500

600 700 Flow Rate (L/s)

0.08 0.07 D

R

11

0.06 "S

0.05

16

18

"

R

11

SD

R

11

0.04

SD

16.5 ft/sec

"

0.03 14

Hydraulic friction gradient (ft/ft)

0.09

0.02 0.01 0.0 0.0

2000

4000 6000 8000 Flow Rate (US gallons/min)

10000

FIGURE 2-13 Hydraulic friction gradient of water in 14⬙, 16⬙ and 18⬙ SDR 11 HDPE pipes. Absolute roughness 1.5 m (0.00000492 ft).

g 2-

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FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-11 Flow and Hydraulic Friction Gradient for HDPE Pipes SDR11 at a Speed of 1 m/s to 5 m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m3 (62.3 lbs/ft3) and a Kinematic Viscosity of 1 cP (2.09 × 10–5 lbf-sec/ft2) Using the Swamee–Jain Equation Pipe inner Speed Pipe size diameter Relative of flow Flow (in) (in) roughness (m/s) L/s

Speed of flow (ft/s)

Flow US gpm

Reynolds number

Darcy friction factor

Friction gradient (m/m) or (ft/ft)

3

2.826

0.0000053

1 2 3 4 5

4.0 8.0 12.1 16.2 20.2

3.3 6.6 9.9 13.2 16.5

64 128 192 257 321

71,780 143,561 215,341 287,122 358,902

0.01917 0.01659 0.01532 0.0145 0.01391

0.0136 0.0471 0.0979 0.167 0.247

4

3.633

00000041

1 2 3 4 5

6.7 13.4 20.1 26.8 33.4

3.3 6.6 9.9 13.2 16.5

106 212 318 424 530

92,278 184,556 276,835 369,113 461,391

0.0182 0.0158 0.0146 0.0138 0.01329

0.01 0.035 0.0726 0.1223 0.1835

5

4.49

0.0000033

1 2 3 4 5

10.2 204 306 408 510

3.3 6.6 9.9 13.2 16.5

162 324 486 648 810

114,046 228,092 342,138 456,184 570,230

0.01738 0.01514 0.01403 0.01331 0.01279

0.0078 0.0271 0.0564 0.0592 0.1429

6

5.349

0.0000028

1 2 3 4 5

15 29 44 58 73

3.3 6.6 9.9 13.2 16.5

230 460 689 919 1,150

135,865 271,729 407,594 543,458 679,323

0.01677 0.01465 0.01359 0.01290 0.01241

0.0063 0.0220 0.0459 0.0774 0.1164

7⬙

5.7510

0.0000026

1 2 3 4 5

17 34 51 67 84

3.3 6.6 9.9 13.2 16.5

266 531 797 1,063 1,328

146,075 292,151 438,226 584,302 730,377

0.01653 0.01445 0.01341 0.01274 0.01225

0.0058 0.0202 0.0421 0.0711 0.1069

8⬙

7.270⬙

0.0000022

1 2 3 4 5

25 49 74 98 123

3.3 6.6 9.9 13.2 16.5

389 779 1,168 1,558 1,947

176,860 353,720 530,581 707,441 884,301

0.0159 0.0139 0.01296 0.01232 0.01186

0.0046 0.0161 0.0336 0.0568 0.0854

10⬙

8.675⬙

0.0000017

1 2 3 4 5

38 76 114 153 191

3.3 6.6 9.9 13.2 16.5

605 1,210 1,815 2,420 3,025

220,447 440,893 661,340 881,786 1,102,233

0.0152 0.0134 0.0125 0.0119 0.0114

0.0035 0.0124 0.0259 0.0439 0.0660

12⬙

10.293⬙ 0.0000015

1 2 3 4 5

54 107 161 215 268

3.3 6.6 9.9 13.2 16.5

851 1,702 2,552 3,404 4,254

261,442 522,884 784,327 1,045,769 1,307,211

0.01475 0.0130 0.0121 0.0115 0.0111

0.0029 0.0101 0.0212 0.0359 0.0541 (continued)

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TABLE 2-11 Continued Pipe inner Speed Pipe size diameter Relative of flow Flow (in) (in) roughness (m/s) L/s

Speed of flow (ft/s)

Flow US gpm

Reynolds number

Darcy friction factor

Friction gradient (m/m) or (ft/ft)

13⬙

10.797⬙ 0.0000014

1 2 3 4 5

59 118 177 236 295

3.3 6.6 9.9 13.2 16.5

936 1,872 2,808 3,745 4,681

274,244 548,488 822,731 1,096,975 1,371,219

0.0146 0.0129 0.0120 0.0114 0.0110

0.0027 0.0096 0.0201 0.0340 0.0512

14⬙

11.301⬙ 0.0000013

1 2 3 4 5

65 129 194 259 324

3.3 6.6 9.9 13.2 16.5

1,026 2,051 3,077 4,103 5,129

287,045 574,091 861,136 1,148,182 1,435,227

0.0145 0.0128 0.0119 0.0113 0.0109

0.0026 0.0091 0.0190 0.0322 0.0485

16⬙

12.915⬙ 0.0000012

1 2 3 4 5

85 169 253 338 423

3.3 6.6 9.9 13.2 16.5

1,340 2,679 4,020 5,359 6,698

328,041 656,082 984,123 1,312,164 1,640,205

0.0141 0.0125 0.0116 0.0110 0.0107

0.0022 0.0078 0.0163 0.0276 0.0416

18⬙

14.532⬙ 0.000001

1 2 3 4 5

107 214 321 428 535

3.3 6.6 9.9 13.2 16.5

1,696 3,392 5,088 6,784 8,480

369,113 738,226 1,107,338 1,476,451 1,845,564

0.0138 0.0122 0.0114 0.0109 0.0105

0.0019 0.0068 0.0142 0.0240 0.0362

20⬙

16.146⬙ 0.0000009

1 2 3 4 5

132 264 396 528 660

3.3 6.6 9.9 13.2 16.5

2,094 4,187 6,281 8,375 10,469

410,108 820,217 1,230,325 1,640,434 2,050,542

0.0136 0.0120 0.0112 0.0107 0.0103

0.0017 0.0060 0.0125 0.0213 0.0321

From the Equation 2.28, the thickness of the viscous sub-layer is 11.6 × 2.7 × 10–5 ␦ = ᎏᎏ 1.973 × 4.8

ᎏ = 1.56 × 10 冪莦 8 0.0178

–6

ft

2-6 PRESSURE LOSSES DUE TO CONDUITS AND FITTINGS The sizing of pumps is based on determining pressure losses between the starting point (A) and the final delivery point (B) (Figure 2-16). It is important to know that the static pressure at (A) includes atmospheric pressure or the pressure of any pressurizing gas, and

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2.45

Le

V

Di

FIGURE 2-14

Entrance length for flows in pipes.

the pressure due to the height of liquid above the centerline of the first impeller or impeller at suction. It is also important to know that static pressure at (B) includes any pressurizing gas at (B) and the height of liquid at (B) above the centerline of the pump’s impeller. The additional pressure losses between (A) and (B) include the friction losses and pressure losses in all the pipe fittings such as valves, elbows, expansions, contraction branches, and bypasses. Pressure is also lost at entry and exit as well. Such pressure losses are expressed in terms of the Darcy–Weisbach equation and in terms of pressure loss factors for each fitting. Total pressure loss due to friction:

U 2 Lj U 2 Hf = fD ᎏ ᎏ + ⌺Kf ᎏ 2 Dij 2 where Lj = length of the conduit j Kf = pressure loss of the fitting f

Turbulent layer

V + +

Buffer layer Viscous sublayer FIGURE 2-15 Boundary layer of flow over a plate.

(2-31)

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g

CHAPTER TWO

6

H 2

5 Pipe Diameters H 1

FIGURE 2-16

Simplified pumping system between two tanks.

TABLE 2-12 Equivalent Length of Valves for Friction Loss of Calculations for Single-Phase Turbulent Flow

Fitting Gate valves Globe valves Angle valves Ball valves Butterfly valves Plug valves—straightway Plug valves—3 way through flow Plug valve—branch flow Stop check valve—straight through Stop check valve—angular 90 deg Swing check valve Lift check valve Tilting disc check valve Foot valve with strainer—poppet disc Foot valve with strainer—hinged disc

Equivalent length/diameter ratio 8 340 55 3 16 18 30 90 400 200 300 55 5–15 420 75

Minimum recommended speed for full disc lift m/s

Ft/s

2.12 2.89 2.32 5.4 1.16–3.08 0.58 1.35

6.96 9.49 7.59 17.7 3.80 to 10.13 1.90 4.43

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2.47

0.5 0.4

Loss Factor K

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0.3 0.2 0.1 0.0 0 2 4 6 8 10 Ratio R/D (bend radius/pipe diam)

FIGURE 2-17 Loss Factors for rough wall bends for pipes 1–7⬙ (after Crane Technical Bulletin No 410).

The differential head that a pump must deliver to pump a liquid between (A) and (B) is therefore TDH = (PB – PA)/g + (ZB – ZA) + HfOB + HfOA where HfOA = the pressure losses due to conduits and fittings between the tank (A) and the pump, including entry loss HfOB = the losses due to conduits and fittings between the pump and tank (B), including exit losses Table 2-12 presents examples of loss coefficients for fittings. Some practical considerations limit the use of fittings in slurry circuits. For example, elbows should have a minimum radius of three pipe diameters to avoid short turns (see Figure 2-17). Such an approach minimizes wear. Example 2-6 The fluid of Example 2-4 is pumped at a flow rate of 3500 gpm through a 16 × 14 pump. The steel fittings include 14⬙ × 18⬙ reducer, an 18⬙ knife gate valve, three long radius 90° elbows with a diameter to radius ratio of 3. The length of the pipe is 355 ft. Determine the total dynamic head if the liquid level in the suction tank is 10 ft and the level in the discharge tank is 50 ft above the centerline of the impeller. Ignore the length of the suction pipe as negligible. Solution in SI Units Net static head: (50 ft – 10 ft) 0.3048 = 12.2 m Dynamic head at the entry of the pump: Di = 15.25 in or 0.387 m Suction Area = 0.1178 m2

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0.221 Speed = ᎏ = 1.875 m/s 0.1178 Dynamic head at suction: 1.8752 U2 ᎏ = ᎏ = 0.18 m 2g 2 × 9.81 For a sharp entrance pipe, the recommended K factor is 0.5. Friction losses at the entry to the pump are calculated as follows: 0.5 × 0.18 = 0.09 m On the discharge of the pump for a 14 × 18 reducer, the loss factor is calculated from the area ratio as

冢

d 21 K = 1 – ᎏ2 d2

= 0.1681 冣 = 冢1 – ᎏ 17.25 冣 2

13.252

2

2

앫 For an18 ft full-bore gate valve, the loss factor K = 0.10 앫 For the elbow r/D = 3, the loss factor K = 0.14 앫 For the reentry pipe L/D = 65 Total friction losses:

冤

冥

355 × 0.3048 + 65 × 0.438 0.09 m + ᎏᎏᎏ × 0.0178 + (0.1681 + 0.10 + 3 × 0.14) 0.438 1.4672 × ᎏ = 0.774 m 2 × 9.81 TDH = 0.774 m + 12.2 m = 12.97 m Solution in USCS Units Net static head = 50 ft – 10 ft = 40 ft Speed in suction pipe is calculated as follows: ID = 15.25⬙ = 1.27 ft Flow Rate = 7.79 ft3/s Area = 1.267 ft2 7.79 Velocity = ᎏ = 6.15 ft/s 1.267 Dynamic head at suction: 6.152 ᎏ = 0.587 ft 2 × 32.2

The K factor for sharp entrance in 0.5. Loss during suction is 0.5 × 0.587 = 0.293 ft. On the discharge of the pump the K factor is determined as in the SI unit solution. Total friction losses are:

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FUNDAMENTALS OF WATER FLOWS IN PIPES

冤

冢

冣

冥

355 + 65 × 1.27 4.82 0.293 ft + 0.0178 ᎏᎏ + 0.1681 + 0.10 + 0.42 ᎏ = 2.44 ft 1.27 2 × 32.2 TDH = 40 + 2.44 = 42.44 ft

2-7 ORIFICE PLATES, NOZZLES, AND VALVE HEAD LOSSES The flow through an orifice plate, a nozzle, or a valve is reduced from the ideal theoretical value by a discharge coefficient: Q = Cd Qideal

(2-32)

The ideal theoretical flow is considered the product of speed and area at the opening of the orifice, valve, or nozzle. The theoretical velocity through the orifice is calculated as Vth = 兹2苶g苶h苶 =

ᎏ 冪莦 R 2⌬P

(2-33)

However, the velocity is typically smaller than the theoretical velocity: Vo = CveVth where Cve = velocity coefficient. The flow through an opening contracts from the full area. This is known as the vena contracta effect.

Reentrant tube

Sharp Edged

Square Edged

Reentrant tube

Length = 1/2 to

Stream clears

Length = 2-1/2

1 diameter

sides

diameters

V

V

C =0.52 d

C =0.61 d

V

C =0.61 d

C =0.73 d

FIGURE 2-18 Discharge coefficients of orifice plates and nozzles.

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For a thin plate or sharp-edged orifice, the vena contracta is assumed to be one half of an orifice diameter d1 downstream from the orifice, but in reality the distance may be from 30% to 80% of d1. For flow of water at a high Reynolds number through a small orifice diameter, Lindeburg (1998) reported that the contracted area is approximately 61% to 63%. The coefficient of contraction is defined as area of vena contracta Cc = ᎏᎏᎏ orifice area

(2-34).

The total discharge is the product of the reduced velocity and the contracted area: Q = CveVth(Cc A1) The product of the velocity coefficient by the area contraction coefficient is called the discharge coefficient: Cd = CveCc

(2-35)

Q = Cd A1兹(2 苶g 苶h 苶)苶

(2-36a)

Q = Cd A1兹(2 苶⌬ 苶P 苶苶 /)苶

(2-36b)

actual discharge Cd = ᎏᎏᎏ theoretical discharge Typical values for discharge coefficients from nozzles and orifices are shown in Figure 2-18. The Cameron Hydraulic Handbook (1977) recommends a further correction for large openings when d2/d1 > 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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2.51

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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2.53

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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2.55

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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FUNDAMENTALS OF WATER FLOWS IN PIPES

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 2kthL

(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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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

Drag coefficient, CD

Particle Reynolds number, Rep

Drag coefficient, CD

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

3.5

0.6

4

0.4

2

0.2

8000

2000

80

100

60

40

20

6000

0

0

4000

30

6

CD

CD

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS

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

Terminal velocity Vt in mm/s

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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MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS

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

Terminal velocity Vt in mm/s

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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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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MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS

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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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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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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= 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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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

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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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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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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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MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS

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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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 (<1.0) Pressure drop Flow rate Radius Radius Radius of the container in the coaxial cylinder rotary viscometer Reynolds number

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Ren Rep Ri R0 sp

CHAPTER THREE

0 s w

A c T

Reynolds Number of a particle based on dn Reynolds Number of a sphere particle based on its diameter Inner radius of a pipe or tube Radius of the bob in the coaxial cylinder rotary viscometer The surface area per unit volume of a sphere of equivalent dimensions or 6/dg, also called specific surface of a particle Front area of a particle orthogonal to the direction of flow Surface area of a wing along the direction of flight Applied torque for the cylinder rotary viscometer Absolute temperature Average velocity of the flow Terminal velocity at very low volume concentration of solids Terminal velocity at given volume concentration of solids The terminal (or free settling) speed Weight the ratio of immobilized dispersing fluid to the volume of solids related approximately to the particle and floc apparent diameter A coefficient used to express to the shear stability of a pseudoplastic mixture Concentration by volume in decimal points Shear strain Wall shear rate or rate of shear strain with respect to time Coefficient of rigidity of a non-Newtonian fluid, also called Bingham viscosity A structural parameter for thixotropic fluids, which do not possess a yield stress value Carrier liquid absolute viscosity Apparent viscosity of a pseudoplastic fluid Effective viscosity Apparent viscosity of a pseudoplastic fluid at zero shear rate Bingham plastic limiting viscosity, or apparent viscosity of a pseudoplastic fluid at very high shear rate Pythagoras number (ratio of circumference of a circle to its diameter) Duration of the shear for a time-dependent fluid Density Shear stress at a height y or at a radius r Yield stress for a Bingham plastic or yield pseudoplastic Structural stress of a thixotropic fluid Wall shear stress Kinematic viscosity Angular velocity of particle Angular velocity of complete system The logarithmic standard deviation Albertson shape factor Clift shape factor Thomas shape factor

Subscripts g L m p s

Equivalent sphere Liquid Mixture Particle Solids

Sf Sw T Ta V V0 Vc Vt W

d/dt

a e 0

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3.41

3–10 REFERENCES Albertson, M. L. 1953. Effects of shape on the fall velocity of gravel particles. Paper read at the 5th Iowa Hydraulic Conference, Iowa University, Iowa City, Iowa. Allen, H. S. 1900. The motion of a sphere in a viscous fluid. Phil. Mag., 50, 323–338, 519–534. Boger, D. V., and Q. D. Nguyen. 1987. The Flow Properties of Weipa #3 and #4 Plant Tailings. Internal study conducted by Comalco Aluminium Ltd, Weipa, Australia, quoted in Darby, R., R. Mun, and D. V. Boger. 1992. Predict Friction Loss in Slurry Pipes. Chem. Engineering, 99, 9 (September), 117–211. Brown, G. G. 1950. Unit Operations. New York: Wiley. Brown, N. P. 1991. The settling behavior of particles in fluids. In Slurry Handling, Edited by N. P. Brown and N. I. Heywood. New York: Elsevier Applied Sciences. Caldwell, D. H., and H. E. Babitt. 1941. Flow of muds, sludge and suspensions in circular pipe. Am. Inst. Chem. Engrs. Trans., 37, 2 (April 25), 237–266. Caldwell, D. H., and H. E. Babitt. 1942. Pipeline flow of solids in suspension. Symp. Am. Soc. of Civ. Eng. Proc., 68, 3 (March), 480–482. Cheng, D. C. H., and W. Whitaker. 1972. Applications of the Warren Spring Laboratory pipeline design method to settling suspension. Paper read at the 2nd Annual Hydrotransport Conference, Bedford, England. Chilton, R. A., and R. Stainsby. 1998. Pressure loss equations for laminar and turbulent non-Newtonian pipe flow. Journal of Hydraulic Engineering, 124, 5 (May), 522–529. Clift, R., J. R. Grace, and M. E. Weber. 1978. Bubbles, Drops and Particles. New York: Academic Press. Concord, S., and J. F. Tassin. 1998. Rheoptical study of thixotropy in synthetic clay suspensions. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Cross, M. M. 1965. Rheology of non-Newtonian fluids—New flow equation for pseudoplastic systems. Journal of Colloid Science, 20, 417. Dahlgreen, S. E. 1958. Eyring model of flow applied to thixotropic equilibrium. Journal of Colloid Science, 13 (April), 151–158. Dedegil, M. Y. 1987. Drag coefficient and settling velocity of particles in non-Newtonian suspensions. Journal of Fluids Engineering, 109 (September), 319–323. Dick, R. I., and B. B. Ewing. 1967. Rheology of activated sludge. Journal of Water Pollution Control Federation, 39, 543. Dahlgreen, S. E. 1958. Eyring model of flow applied to thixotropic equilibrium. Journal of Colloid Science, 13 (April), 151–158. Fredrickson, A. G. 1970. A model for the thixotropy of suspensions. American Inst. of Chem. Eng. Journal, 16, 436. Gay E. D., P. A. Nelson, and W. P. Armstrong. 1969. Flow properties of suspensions with high solids concentration. American Inst. of Chem. Eng. Journal, 15, 6, 815–822. Goodrich and Porter. 1967. Govier, G. W., C. A. Shook, and E. O. Lilge. 1957. Rheological Properties of water suspensions of finely subdivided magnetite, galena, and ferrosilicon. Trans. Can. Inst. Mining Met., 60, 157. Govier, G. W., and K. Aziz. 1972. The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold. Hedstrom, B. O. A. 1952. Flow of plastic materials in pipes. Ind. Eng. Chem., 33, 651–656. Herbrich, J. 1968. Deep ocean mineral recovery. Paper read at the World Dredging Conference II, Rotterdam, the Netherlands. Herbrich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill. Heywood, N. I. 1991. Rheological characterisation of non-settling slurries. In Slurry Handling, Edited by N. P. Brown and N. I. Heywood. New York: Elsevier Applied Sciences. Heywood, N. I. 1996. The performance of commercially available Coriolis mass flowmeters applied to industrial slurries. Paper read at the 13th International Hydrotransport Symposium on Slurry Handling and Pipeline Transport. Johannesburg, South Africa. Cranfield, UK: BHRA Group. Inter-Agency Committee on Water Resources. 1958. Report 12. Internal report by the Subcommittee on Sedimentation, Minneapolis, Minnesota.

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Kearsey, H. A., and L. E. Gill. 1963. Study of sedimentation of flocculated thorium slurries using gamma ray technique. Trans. Inst. Chem. Engrs., 41, 296. Kherfellah, N., and K. Bekkour. 1998. Rheological characteristics of clay suspensions. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Krusteva, E. 1998. Viscosmetric and pipe flow of inorganic waste slurries. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Lapassin, R., S. Pricl, and M. Stoffa. 1998. Viscosity of aqueous suspensions of binary and ternary alumina mixtures. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Metzner, A. B., and M. Whitlock. 1958. Flow behavior of concentrated (dilatant) suspensions. Trans. Soc. Rheology, 2, 239–254. Moore, F. 1959. Rheology of Ceramic Slips and Bodies. British Ceramic Society Transactions, 58, 470. Mun, R. 1988. The Pipeline Transportation of Suspensions with a Yield Stress. Master’s Thesis, University of Melbourne, Australia. Parzonka, W. 1964. Determination of the maximum concentration of homogeneous mixtures (in French). Journal of the French Academy of Science, 259, 2073. Pilpel, N. 1965. Flow properties of non-cohesive powders. Chemical Process Eng. 46, 4, 167–179. Prokunin, A. N. 1998. Particle-wall interaction in liquids with different rheology. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Richards, R. H. 1908. Velocity of Galena and Quartz Falling in Water. Trans AIME, 38, 230–234. Ritter, R. A., and G. W. Govier. 1970. The development and evaluation of a theory of thixotropic behavior. Can. Journal Chem. Eng., 48, 505. Rubey, W. W. 1933. Settling velocities of gravel, sand and silt particles. Amer. Journal of Science, 25, 148, 325–338. Skelland, A. H. P. 1967. Non-Newtonian Flow and Heat Transfer. New York: Wiley. Teixeira, M. A. O. M., R. J. M. Craik, and P. F. G. Banfill. 1998. The effect of wave forms on the vibrational processing of fresh concrete. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Thomas, D. G. 1961. Transport characteristics of suspensions: Part II. Minimum transport velocity for flocculated suspensions in horizontal pipes. AIChE Journal, 7 (September), 423–430. Thomas, D. G. 1963. Transport characteristics of suspensions. Ch. E. Journal, 9, 310. Thomas, A. D. 1981. Slurry pipeline rheology. Paper presented at the National Conference on Rheology. Second Annual Conference of the British Society of Rheology, Australian Branch, University of Sydney, Australia. Turton, R., and O. Levenspiel. 1986. A short note on drag correlation for spheres. Powder Technology Journal, 47, 83. Valentik, L., and R. L. Whitemore. 1965. Terminal velocity of spheres in Bingham plastics. British Journal of Applied Phys., 16, 1197. Vlasak, P., Z. Chara, and P. Stern. 1998. The effect of additives on flow behaviour of kaolin–water mixtures. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid-Liquid Flow—Slurry Pipeline Transportation. Trans-Tech Publications. Wells, P. J. 1991. Pumping non-Newtonian slurries. Technical Bulletin 14. Sydney, Australia: Warman International. Whorlow, R. W. 1992. Rheological Techniques, 2d. ed. New York: Ellis Horwood. Wilson, K. C., G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. New York: Elsevier Applied Sciences. Worster, R. C., and D. E. Denny. 1955. Hydraulic transport of solid materials in pipes. Proceedings of the Institute of Mechanical Engineers (UK), 38, 230–234. Further Reading: Caldwell, D. H., and H. E. Babitt. 1942. Pipeline flow of solids in suspension. Symp. Am. Soc. of Civ. Eng. Proc., 68, 3 (March), 480–482. Goodrich, J. E., and R. S. Porter. 1967. Rheological interpretation of torque—Rheometer data. Polymer Eng & Science, 7 (January), 45–51.

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Lazerus, J. H., and P. T. Slatter. 1988. A method for the rheological characterization of tube viscometer data. Journal of Pipelines, 7, 165–176. Thomas, D. G. 1960. Heat and momentum transport characteristics of non-Newtonian aqueous thorium oxide. AIChE Journal, 7, 431. Wilson, K. C. 1991. Pipeline design for settling slurries. In Slurry Handling. Edited by N. P. Brown, and N. I. Heywood. New York: Elsevier Applied Sciences. 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.

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CHAPTER 4

HETEROGENEOUS FLOWS OF SETTLING SLURRIES

4-0 INTRODUCTION A practical engineer sifting through the literature on slurry flows would be astonished by the number of different equations. Since the work of the French scientists Durand and Condolios in 1952, and British scientists Newitt et al. in 1955, engineers and scientists have continued to develop new equations for deposition velocity and friction losses. This chapter reviews the evolution of models of Newtonian slurry flows from wellgraded and uniform particle sizes to complex mixtures of coarse and fine particles. For this purpose, some equations are listed in a historical context, to demonstrate their evolution to the reader. The sheer number of equations demonstrates how complicated heterogeneous flows are. A number of factors interact in a horizontal pipe. The flow takes the form of different regimes, and includes everything from a simple stationary bed at low speed to a pseudohomogeneous flow at high speeds. For each regime, equations have been developed over the years to account for the mean particle diameter, the diameter of the conduit, the density of the particles, their drag coefficient, the speed of flow, etc. As a result, there are many angles from which a heterogeneous flow of settling solids can be examined. The followers of Durand and Condolios put great emphasis on the drag coefficient of the solid particles, whereas the followers of Newitt prefer to focus on the terminal velocity. As we have clearly demonstrated in Chapter 3, the drag coefficient and the terminal velocity are interrelated. Examining a flow of slurry is often an exercise of playing with a murky liquid in which very little can be seen. Sand does not behave like coal and there is not a single universal law that may apply to the transportation of solids by liquids. It is therefore important to rely on database, historical information, and empirical data. In the past, engineers have tried to simplify the complexity of slurry flows by defining certain transition velocities. With the use of modern research tools, there is an emerging approach of rejecting the concept of an abrupt change from one state of flow to another, and a tendency to consider such a change over a band of the speed. Different approaches have been developed to examine the mixture of coarse and fine particles from superimposed layers to two-layer models. This book is intended for engineers, and various examples are included in the text. The purpose of such examples is to simplify the use of complex equations. With modern personal computers, which use simple languages such as quick basic, an engineer can efficiently calculate friction losses for a heterogeneous slurry flow. 4.1

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4-1 REGIMES OF FLOW OF A HETEROGENEOUS MIXTURE IN HORIZONTAL PIPE The history of slurry pipelines was briefly presented in Section 1-10 of Chapter 1. Two schools are credited for laying the foundations of modern hydro-transport engineering; SOGREAH in France, and the British Hydro-mechanic Research Association of the United Kingdom. Starting in 1952, Durand and Condolios of SOGREAH published a number of studies on the flow of sand and gravel in pipes up to 900 mm (35.5 in) in diameter. Based on the specific gravity of particles with a magnitude of 2.65, they proposed to divide the flows of nonsettling slurries in horizontal pipes into four categories based on average particle size as follows: 앫 Homogeneous suspensions for particles smaller than 40 m (mesh 325) 앫 Suspensions maintained by turbulence for particle sizes from 40 m (mesh 325) to 0.15 mm (mesh 100) 앫 Suspension with saltation for particle sizes between 0.15 mm (mesh 100) and 1.5 mm (mesh 11) 앫 Saltation for particles greater than 1.5 mm (mesh 11) This initial classification was refined over the next 18 years by Newitt et al. (1955), Ellis and Round (1963), Thomas (1964), Shen (1970), and Wicks (1971). Due to the interrelation between particle sizes and terminal and deposition velocities, the original classification proposed by Durand has been modified to four flow regimes based on the actual flow of particles and their size. Referring to Figures 4-1 and 4-2, there are four main regimes of flow in a horizontal pipe 1. 2. 3. 4.

Flow with a stationary bed Flow with a moving bed and saltation (with or without suspension) Heterogeneous mixture with all solids in suspension Pseudohomogeneous or homogeneous mixtures with all solids in suspension

velocity

d nde spe u s ly Ful

suspended with moving bed

suspended with saltation

lenticular deposits

stationary deposits with ripples

blocked pipe

FIGURE 4-1 Flow regimes of heterogeneous flows in terms of speed versus volumetric concentration (after Newitt et al., 1955).

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Flow

with

a sta tion ary be

F d wi low w th or ith a wi tho mov ut sus ing b pen ed sio ns

HETEROGENEOUS FLOWS OF SETTLING SLURRIES

particle size

abul-4.qxd

th

wi

4.3

all

low sf n u o ne sio ege en ter susp e H s in id sol Flow as a homogeneous or

pseudohomogeneous suspension

Mean velocity

FIGURE 4-2 Flow regimes of heterogeneous flows in terms of particle size versus mean velocity (after Shen, 1970).

Two special cases shown in Figure 4-1 are not considered to be at the limits of these regimes of flows. They are lenticular deposits at very low speeds but low solid concentration, and blocked pipes at high solid concentration. Slurry flows that have some form of segregation or separation of solids in layers are called “heterogeneous” flows, whereas the slurries themselves are called “settling” slurries. 4-1-1 Flow With a Stationary Bed When the slurry flow speed is low, the bed thickens. As the fluid above the bed tries to move the solids by entrainment, they tend to roll and tumble. The particles with the lowest settling speed move as an asymmetric suspension, whereas the coarser particles build up the bed. As the speed drops even further, the pressure to maintain the flow becomes quite high and eventually the pipe blocks up. Flow with saltation and asymmetric suspension occurs above the speed of blockage. This means that the coarser particles “sand up,” whereas the finer particles continue to move. Certain tailing lines have exhibited this phenomenon. In fact, when a process plant is built with a tailing line too large to handle the initial flow, the operator may choose to let the bottom of the pipe sand up to reduce the effective cross-sectional area of the pipe. This principle has been successfully applied to pipelines in a variety of countries. Saltation can eventually lead to blockage of a pipe. It may result in a number of problems, such as water hammer, wear, and freezing in cold climates. Most engineering specifications require that the pipeline be designed to operate at speeds higher than those associated with saltation. 4-1-2 Flow With a Moving Bed When the speed of the flow is low and there are a large number of coarse particles, the bed moves like desert sand dunes. The top particles are entrained in the moving fluid

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above the bed. Consequently, the upper layers of the bed move faster than the lower layers in a horizontal pipe. If the mixture were composed of a wide range of particles with different sizes and settling velocities, the bed would be composed of the particles with the highest settling speed. Particles with a moderate settling speed are maintained in an asymmetric suspension, with most particles concentrated in the lower half of the pipe, whereas the particles with the lowest settling speed move as a symmetric suspension

4-1-3 Suspension Maintained by Turbulence As the flow speed increases, turbulence is sufficient to lift more solids. All particles move in an asymmetric pattern with the coarsest at the bottom of a horizontal pipe covered with superimposed layers of medium- and fine-sized particles. Many particles may strike the bottom of the pipe and rebound. Wear on the bottom of the pipe must be taken into account in maintenance schedules and the pipes must be rotated at intervals suggested by the slurry engineer in order to maintain an even wear pattern of the internal wall of the pipe. Although the flow is not symmetric, from the point of view of power consumption, this regime may be the most economical for transporting a certain mass of solids. Wilson (1991) calls all flows below V3 fully stratified flows and all flows above V3 fully suspended flows. The transition from fully stratified to fully suspended flows is considered by this author to be fairly complex and should be represented by sigmoid or ogee curves. It is a transition over a range of the speed and not an abrupt transition at a single value of the speed. The work of Wilson and colleagues will be examined in Section 4-4-5.

4-1-4 Symmetric Flow at High Speed At speeds in excess of 3.3 m/s (10 ft/s), all solids may move in a symmetric pattern (but not necessarily uniformly). Sometimes this flow is called pseudohomogeneous because of its symmetry around the pipe axis. Power consumption is a linear relationship of the static head multiplied by the velocity, but is proportional to the cube of velocity needed to overcome friction losses. Power consumption in pseudohomogeneous mixtures of coarse and fine particles may be excessive for long pipelines. Pseudohomogeneous mixtures of fine or ultrafine particles may occur at speeds as low as 1.52 m/s (5 ft/s). One definition of fine and coarse particles was explained Govier and Aziz (1972), who proposed the following: 앫 Ultrafine particles: dp < 10 m (mesh 1250), where gravitational forces are negligible. 앫 Fine particles: 10 m < dp < 100 m (mesh 1250 < dp < mesh 140), usually carried fully suspended but subject to concentration gradients and gravitational forces. 앫 Medium sized particles: 100 m < dp < 1000 m (mesh 240 < dp < mesh 15), will move with a deposit at the bottom of the pipe and with a concentration gradient. 앫 Coarse particles: 1000 m < dp < 10,000 m (0.039 in < dp < 0.394 in). These are seldom fully suspended and form deposits on the bottom of the pipe. 앫 Ultracoarse particles are larger than 10 mm (0.4 in). These particles are transported as a moving bed on the bottom of the pipe. Considering particle sizes while ignoring their density is meaningless. Practical engineers do shift the boundaries between different sizes based on the density of the particles. There is no question that beads of high-density polyethylene will behave differently than

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sand particles with the same average diameter because the former is lighter than water while the latter is 2.65 times heavier than water.

4-2 HOLD-UP The previous section describes how different layers of solids move with different speeds, from the bottom, coarser particles, to the finer particles at the top of the horizontal pipe. The theory of hold-ups complicates this process, however. Hold-ups are due to velocity slip of layers of particles of larger sizes, particularly in the moving bed flow pattern. Newitt et al. (1962) conducted speed measurements of a slurry mixture in a horizontal pipe. In the case of light Plexiglas pipe, zircon or fine sand did not result in local slip; particles and water moved at the same speed. However, for coarse sand and gravel, they observed asymmetric suspension and a sliding bed. They also observed that in the upper layers of the horizontal pipe, the concentrations of larger particles were the same as for finer solids, but were marked by differences in the magnitude of the discharge rate of the lower layers.

4.3 TRANSITIONAL VELOCITIES The four regimes of flow described in Section 4-1 can be represented by a plot of the pressure gradient versus the average speed of the mixture (Figure 4-5). The transitional velocities are defined as 앫 V1: velocity at or above which the bed in the lower half of the pipe is stationary. In the upper half of the pipe, some solids may move by saltation or suspension.

1.0

Ratio distanc e from bottom of pipe to the inner diameter (y/D) I

0.8 0.6 0.4

C

0.0 3

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v 7 0.0

0.10

4 0.1

0 0

0.05

0.1

0.15

0.20

Discharge solids concentration C y FIGURE 4-3 Distribution of concentration of solids in a pipe versus average volumetric concentration.

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앫 V2: velocity at or above which the mixture flows as an asymmetric mixture with the coarser particles forming a moving bed. 앫 V3 or VD: velocity at or above which all particles move as an asymmetric suspension and below which the solids start to settle and form a moving bed. 앫 V4: velocity at or above which all solids move as a symmetric suspension.

Velocity (ft/se c)

Volumetric Concentration (%)

0

5

10

15

20

direction of flow 30 20 10 0 0

1

2

3

4

5

6

Velocity (m/s ) FIGURE 4-4 Simplified concept of particle distribution in a pipe as a function of volumetric concentration and speed.

1

at w asymmetric flow

V2 V 3

stationary bed

V1

3

V4

symmetric flow

2

er

4

moving bed

Pressure drop per unit of length

slurry

Speed of flow

FIGURE 4-5 Velocity regimes for heterogeneous slurry flows.

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

V3 is effectively the deposition velocity, often called in the past the Durand velocity for uniformly sized coarse particles. It is no longer recommended that it be called the Durand velocity, as tests in the last 20 years have led to new equations that include the effects of particle size and composition of the slurry. The magnitude of the velocity depends on the volumetric concentration (Figure 4-7).

4-3-1 Transitional Velocities V1 and V2 The transitional velocity V1 is obviously not used for the operation of slurry lines. It may be of interest in lab research, instrumentation, and monitor of start-up. The transitional velocity V2 is determined individually from pressure measurements of the pressure gradient. The main focus of the tests is to determine the height of the bed and to derive a stratification ratio. Wilson (1970) developed a model for the incipient motion of granular solids at V2. He assumed a hydrostatic pressure exerted by the solids on the wall and proposed the following equation: 1 ⌬P ᎏ ᎏ L L

– sin cos

s

+ ᎏ 冢sin – ᎏ 冣冥 冢 冣冤 ᎏᎏ D tan 4 Rw i

r

s(S – 1) Cvb(sin – cos )g = ᎏᎏᎏ 2

(4-1)

where (⌬P/L)2 = pressure gradient at 2 = half the angle subtended at the pipe center due to the upper surface of the bed, in radians s = coefficient of static friction of the solid particles against the wall of the pipe Rw = cross-sectional area of the bed divided by the bed width r = angle of repose of the solid particles

100

10

0

0

10

100

d 50

Cumulative passing (%)

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1000

Particle mesh diameter ( m) FIGURE 4-6 Concept of d50 by cumulative passing percentage versus particle size.

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S = ratio of density of solids to density of liquid Cvb = volume fraction solids in the bed (When USCS units are used, express density in slugs/ft3 rather than lbm/ft3). For 0.7 mm (mesh 24) sand with water in a 90 mm (3.5 in) pipe, Wilson measured = 0.35 and concluded that the assumptions of hydrostatic distribution of the granular pressure were correct.

4-3-2 The Transitional Velocity V3 or Speed for Minimum Pressure Gradient The transitional velocity V3 is extremely important because it is the speed at which the pressure gradient is at a minimum. Although there is evidence that solids start to settle at slower speeds in complex mixtures, operators and engineers often referred to transitional velocity as the speed of deposition. Durand and Condolios (1952) derived the following equation for uniformly sized sand and gravel: VD = V3 = FL{2 · g · Di[(s – L)/L]}1/2

(4-2)

where FL = is the Durand factor based on grain size and volume concentration V3 = the critical transition velocity between flow with a stationary bed and a heterogeneous flow Di = pipe inner diameter (in m) g = acceleration due to gravity (9.81 m/s) s = density of solids in a mixture (kg/m3) L = density of liquid carrier The Durand factor FL is typically represented in a graph for single or narrow graded particles, as in Figure 4-7 after the work of Durand (1953). However, since most slurries

Durand Velocity Factor FD

2.0

1.0

For single or narrow graded slurries

CV = 15% CV = 10%

CV = 5%

C = 2% V

Based on Schiller equation using d50

CV = 15% CV = 5%

0 0

1.0

2.0

3.0

Particle diameter (mm)

FIGURE 4-7 Durand velocity factor versus particle size—comparison between the conventional values for single graded slurries and Schiller’s equation using d50 for wide graded slurry.

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

are mixtures of particles of different sizes, this plot is considered too conservative. The Durand velocity factor has been refined by a number of authors. In an effort to represent more diluted concentrations, Wasp et al. (1970) proposed including a ratio between the particle diameter and the pipeline diameter. Wasp proposed the use of a modified factor F⬘ L so that

S – L VD = V3 = FL⬘ 2gDi ᎏ L

冤

冥 冤ᎏ D 冥 1/2

dp

1/6

(4-3)

i

Schiller and Herbich (1991) proposed the following equation for the Durand velocity factor based on the d50 of the particles: )[1 – exp (–6.9 d50)]} FL = {(1.3 × C 0.125 v

(4-4)

where d50 is expressed in mm. Some reference books define a Froude number as Fr = FL · 兹2苶. The particle size d50 is the statistically determined particle size below which half (or 50%) would be equal or smaller to that set size. The following example illustrates the concept of d50. Example 4-1 A sample of slurry is sieved for particle size. The data is collected in the laboratory (see Table 4-1). Plot the data on a logarithmic graph and determine the d50. Solution The data is plotted in Figure 4-6; the d50 is determined to be 145 m. Example 4-2 A slurry mixture has a d50 of 300 m. The slurry is pumped in a 30 in pipe with an ID of 28.28⬙. The volumetric concentration is 0.27. Using Equations 4-4 and 4-2, determine the speed of deposition for a sand–water mixture if the specific gravity of sand is 2.65. Solution in SI Units From Equation 4-4: FL = (1.3 × 0.270.125)(1-exp (–6.9 × 0.3)) FL = 1.1 × 0.8738 FL = 0.964 From Equation 4-2 the deposition velocity is V3 = 0.964 (2 × 9.81 × 28.25 × 0.0254 × 1.65)1/2 V3 = 4.64 m/s

TABLE 4-1 Data for Example 4-1 Particle size (m)

425

300

212

150

106

75

53

45

38

–38

Cumulative passing (%)

97.2

87.1

68.3

51.3

35.9

20.5

14.5

11.8

10.8

—

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Solution in USCS Units From Equation 4-4: FL = (1.3 × 0.270.125)(1 – exp (–6.9 × 0.3)) FL = 1.1 × 0.8738 FL = 0.964 From Equation 4-2 the deposition velocity is V3 = 0.964 (2 × 32.2 × 28.25/12 × 1.65)1/2 V3 = 15.25 ft/sec Various curves have been published for the magnitude of FL. They are often plotted for a single graded size and use difficult to read logarithmic scales. For the sake of accuracy, Table 4-2 tabulates the magnitude of FL between 0.08 mm < d50 < 5mm on the basis

TABLE 4-2 The Coefficient FL Based on Schiller’s Equation Using the d50 of the Particles for Particles Between 0.080 and 5 mm for Volumetric Concentration up to 30%. FL = {(1.3 × C v0.125)[1 – exp(–6.9 d50)]} d50 (mm)

CV = 0.05

CV = 0.10

CV = 0.15

CV = 0.20

CV = 0.25

CV = 0.30

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 1.5 2 2.5 3.0 3.5 4.0 5.0

0.379 0.446 0.503 0.554 0.598 0.636 0.669 0.735 0.781 0.814 0.837 0.854 0.866 0.874 0.880 0.884 0.887 0.889 0.890 0.891 0.892 0.893 0.8939 0.8940 0.8940 0.8940 0.8940 0.8940 0.8940

0.414 0.486 0.549 0.604 0.652 0.693 0.730 0.801 0.852 0.888 0.913 0.931 0.944 0.953 0.959 0.964 0.967 0.969 0.971 0.972 0.973 0.974 0.9748 0.9749 0.9749 0.9749 0.9749 0.9749 0.9749

0.435 0.511 0.577 0.635 0.686 0.729 0.768 0.843 0.896 0.934 0.961 0.980 0.993 1.002 1.009 1.014 1.017 1.020 1.021 1.023 1.023 1.0245 1.0255 1.0255 1.0255 1.0255 1.0255 1.0255 1.0255

0.451 0.530 0.599 0.658 0.711 0.756 0.796 0.874 0.929 0.968 0.996 1.015 1.029 1.039 1.046 1.051 1.055 1.057 1.059 1.060 1.061 1.062 1.063 1.063 1.063 1.063 1.063 1.063 1.063

0.464 0.545 0.616 0.677 0.731 0.777 0.818 0.898 0.955 0.995 1.024 1.044 1.058 1.069 1.076 1.081 1.084 1.087 1.089 1.090 1.091 1.092 1.0931 1.0932 1.0932 1.0932 1.0932 1.0932 1.0932

0.474 0.557 0.630 0.693 0.748 0.795 0.837 0.919 0.977 1.018 1.048 1.068 1.083 1.093 1.101 1.106 1.109 1.112 1.114 1.115 1.116 1.1172 1.1183 1.1184 1.1184 1.1184 1.1184 1.1184 1.1184

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of Schiller’s equation. The magnitude of FL based on d50 is smaller than values published in the literature for single graded slurry mixtures (lab mixtures using a uniform size of particles). A number of authors have confirmed that this is the case (Warman International Inc., 1990). In order to compare the conventional magnitude of FL based on single and narrow graded particles to the Schiller equation, both ranges of FL are plotted in Figure 4-7. With a more complex approach that takes into account the actual viscosity of the slurry mixture and the density of the particles, Gillies et al. (1999) developed an equation for the Froude number F in terms of the Archimedean number (which we will discuss in Section 4-4-5 for stratified coarse flows): 4 Ar = ᎏ d 3 ( – )g 3L2 p L s L

(4-5)

To estimate the deposition velocity V3, Gilles et al. (1999) developed an equation for the Froude number based on the Archimedean number: Fr = aArb

(4-6)

where Fr = FL · 兹2苶 For Ar > 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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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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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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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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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 = Km 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 V2L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 2L兹苶 C苶 D

fDm – fDL ᎏ CV fDL

gDi (s – L) ᎏᎏ V 2L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 CdVL

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)

Average particle size (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 3LV 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

Average particle size (mm)

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

Iterated concentration C/CA

–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)

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

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

Average particle size (mm)

Drag coefficient for a particle with shape factor of 1

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

Iterated concentration C/CA

–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 “ <<<<<<<<<<<<<<<<<<<” IF rt$ = “Y” OR rt$ = “y” THEN PRINT “ <<<<<<<<<<<<<<<<<<<” IF rt$ = “Y” OR rt$ = “y” THEN PRINT “ calculations for another flow rate” IF ch = 1 THEN INPUT “state the flow rate in m3/HR”; qM IF ch = 1 THEN q1m = qM/3600 IF ch = 2 THEN INPUT “state the flow rate in US gpm”; qus IF ch = 2 THEN q1m = qus * 3.785/60000 IF rt$ = “Y” OR rt$ = “y” THEN GOTO 14 INPUT “specific gravity of carrier liquid”; sgl INPUT “specific gravity of solids”; sgs PRINT PRINT “ please choose between input of weight or volume concentration” PRINT “ 1- weight concentration” PRINT “ 2- volume concentration” PRINT 12 INPUT “ 1 or 2”; cwe

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IF cwe = 1 THEN INPUT “weight concentration in percent”; cwin IF cwe = 2 THEN INPUT “volume concentration in percent”; cvin IF cwe = 0 OR cwe > 2 THEN GOTO 12 PRINT IF cwe = 1 THEN cw = cwin/100 IF cwe = 1 THEN sgm = sgl/(1 - cw * (1 - sgl/sgs)) IF cwe = 1 THEN cv = (sgm - sgl)/(sgs - sgl) IF cwe = 1 THEN PRINT USING “specific gravity of mixture = ##.##, cv #.###”; sgm; cv IF cwe = 2 THEN cv = cvin/100 IF cwe = 2 THEN sgm = cv * (sgs - sgl) + sgl IF cwe = 2 THEN cw = cv * sgs/sgm IF cwe = 2 THEN PRINT USING “specific gravity of mixture = ##.##, cw = #.###”; sgm; cw INPUT “hit any key to continue”; jk$ CLS PRINT 22 INPUT “pipe outside diameter in inches”; d0 IF d0 = 0 THEN GOTO 22 INPUT “wall thickness in inches”; tw INPUT “liner thickness in inches”; tl D1 = d0 - 2 * (tw + tl) PRINT “inside pipe diameter in inches”; D1 d1m = D1 * .0254 a1 = pi * d1m ^ 2/4 14 v1 = q1m/a1 v1us = v1/.3048 IF rt$ = “Y” OR rt$ = “y” THEN GOTO 18 INPUT “viscosity of slurry in cPoise or mPa-s”; viscp visc = viscp/1000 18 ReL = sgl * 1000 * d1m * v1/visc PRINT “Reynolds Number of carrier liquid”; ReL IF rt$ = “Y” OR rt$ = “y” THEN GOTO 20 IF ch = 1 THEN INPUT “pipe roughness in meters”; em IF ch = 2 THEN INPUT “pipe roughness in feet”; ef IF ch = 2 THEN em = ef * .3048 edi = em/d1m PRINT “relative roughness”; edi 20 a = (edi/3.7 + 5.7/ReL ^ .9) b = fnlog10(a) fd = .25/b ^ 2 PRINT “darcy factor for carrier liquid”; fd fan = fd/4 dpl = fd * v1 ^ 2 * sgm * 1000/(2 * d1m) slopliq = 2 * fan * v1 ^ 2/(9.81 * d1m) PRINT USING “head loss per length = ####.##### = “; slopliq PRINT USING “press drop due to carrier liquid = #####.## Pa/m”; dpl PRINT Ub = 9.81 * d1m * (sgm/sgl - 1)/v1 ^ 2 PRINT “Ub”; Ub INPUT “hit any key to continue”; jk$ INPUT “hit any key to continue”; jk$ PRINT “ starting from the top size you are asked to input particle size for “ PRINT “ each fraction and its volumetric concentration as part of the solids” FOR i = 1 TO 50 IF i = 1 THEN PRINT “top fraction” PRINT “size”; i

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IF rt$ = “Y” OR rt$ = “y” THEN GOTO 140 95 INPUT “particle size (in microns) and cumulative volume conc.(%)”; dp(i), cvdp(i) 140 IF cvdp(i) > 100 THEN GOTO 95 IF i = 1 THEN GOTO 85 cvind(i) = –cvdo + cvdp(i) dpav(i) = (dpo + dp(i))/2 GOTO 90 85 cvind(i) = cvdp(i) dpav(i) = dp(i) 90 PRINT USING “average particle size = ###### micron,av.volume conc = ###.#### %”; dpav(i); cvind(i) cvdo = cvdp(i) dpo = dp(i) rep(i) = dpav(i) * 10 ^ -3 * v1 * sgm/visc PRINT “particle reynolds number”; rep(i) IF (rep(i) > 2) AND (rep(i) < 500) THEN cd(i) = 18.5 * rep(i) ^ -.6 IF rep(i) < 2 THEN cd(i) = 27 * rep(i) ^ (–.84) IF (rep(i) > = 500) AND (rep(i) < 200000) THEN cd(i) = .44 vt(i) = (4 * 9.81 * dpav(i) * .000001 * (sgs – sgl)/(3 * cd(i))) ^ .5 IF vt(i) >(v1/2) THEN PRINT “warning deposit velocity is higher than half the average flow velocity” PRINT USING “drag coef for av.diam = #.###;terminal velocity = #.#### m/s”; cd(i); vt(i) dfbed(i) = 82 * cvind(i) * cv * (Ub ^ 1.5) * (cd(i) ^ -.75)/100 PRINT “ correction to friction factor”; dfbed(i) ‘PRINT “pressure drop due this fraction”; dpbed(i) hm = hm + dfbed(i) PRINT USING “ TOTAL correction to friction factor = ###.###”; hm IF dp(i) = 0 THEN GOTO 130 IF cvdp(i) = 100 THEN GOTO 130 ‘INPUT “DO YOU WANT TO CONTINUE (Y/N)”; LKJ$ ‘IF LKJ$ = “n” OR LKJ$ = “N” THEN np = i ‘IF LKJ$ = “n” OR LKJ$ = “N” THEN GOTO 130 120 np = i NEXT i 130 fannew = fan * (1 + hm) PRINT “total friction factor for slurry”; fannew pressure = fannew * 2 * sgm * 1000 * v1 ^ 2/d1m slope = pressure/(9810 * sgm) PRINT USING “pressure = #####.## Pa/m”; pressure PRINT “slope of slurry or head per unit length”; slope 150 INPUT “DO YOU WANT TO DO ITERATION BASED ON CONCENTRATION C/CA (y/n)”; HT$ IF HT$ = “N” OR HT$ = “n” THEN GOTO 325 DFANNEW = fannew uf = v1 * SQR(DFANNEW/2) PRINT USING “ FRICTION VELOCITY = ##.### m/s”; uf FOR i = 1 TO np IF i = 1 THEN dhm = 0 z(i) = –1.8 * vt(i)/(.38 * uf) cca(i) = 10 ^ z(i) PRINT “size,av diam and c/ca “; i, dpav(i), cca(i) dpnew(i) = dfbed(i) * cca(i) dhm = dpnew(i) + dhm NEXT i DFANNEW = fan * (dhm + 1) PRINT “REVISED FANNING FACTOR = “; DFANNEW pressureit = DFANNEW * 2 * sgm * 1000 * v1 ^ 2/d1m

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PRINT USING “iterated pressure = #######.## Pa/m = “; pressureit slopeit = pressureit/(9810 * sgm) PRINT “revised slope “; slopeit 325 INPUT “do you want to repeat the calculation for another flow rate (Y/N)”; rt$ IF rt$ = “Y” OR rt$ = “y” THEN GOTO 10 LPRINT “REVISED FANNING FACTOR = “; DFANNEW LPRINT USING “iterated pressure = #######.## Pa/m = “; pressureit LPRINT “revised slope “; slopeit 525 RETURN

When it was developed in the early 1970s, the Wasp method was considered state of the art. It does, however, ignore or minimize one important parameter, namely the shear stress between the different superimposed layers. This is a topic that the two-layer method attempts to tackle, as we shall see in Section 4-10. It does, therefore, tend to predict pressure losses higher than those from stratified flows in certain circumstances of bimodal (fine and coarse) distribution. Nevertheless, the Wasp method remains a very useful method to this day for the design of pipelines, particularly when it is supported by lab tests, as we showed in Example 4-11.

4-7 SALTATION AND BLOCKAGE Most modern engineering specifications for the design of slurry pipelines categorically forbid flow at speeds below V3. However, the instrumentation engineer needs to know the pressure rise in saltation or at blockage. Herbich (1991) argued that motors should be sized to handle the flow in saltation, and there are incidences where it may be economical to reduce the cross-sectional area of the pipe by allowing flow over a stationary bed. 4.7.1 Pressure Drop Due to Saltation Flows In saltation, there is a bed at the bottom part of the horizontal pipe (Figure 4-11) and different approaches are use to evaluate the pressure losses.

Fines in suspension

Saltation bed FIGURE 4-11

Concept of Saltation.

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Newitt et al. (1955) derived the following equation for saltation flow with d50 > 0.025 mm (0.001 in) and for pipes smaller than 25.4 mm (1 in): im – i Z = ᎏ = 66{[s/L – 1]gDi/Vm2} Cvi

(4-39)

Babcock (1968) conducted tests on water suspensions of coarse sand and steel shot. He expressed a nondimensional ratio of the square of the speed to the product of the pipe inner diameter and the acceleration of gravity. His tests indicated that in the range of 5 < (V 2/gDi) < 40 the friction losses could be expressed as gDi s Z = 60.6 ᎏ – 1 ᎏ L Vm2

冢

冣

(4-40)

But for a water suspension of taconite, in the range of 1 < (V2/gDi) < 12: gDi s Z = 66 ᎏ – 1 ᎏ L Vm2

冢

冣

(4-41)

Newitt’s equation (Equation 4-39) is the most commonly used for saltation flows. Example 4-12 The slurry described in Example 4-7 is in saltation at 1.5 m/s. Determine the resultant friction factor. Solution in SI Units 1.5 × 1,000 × 0.5778 Re = ᎏᎏᎏ = 671,860 0.00129 0.25 fd1 = ᎏᎏᎏᎏᎏ = 0.01572 [log10{(0.0002596/3.7) + (5.74/671,8600.9)}]2 Using the Newitt equation (4-39), which was derived for small pipes, would have given

冤

冥

im – iL 1.65 × 9.81 × 0.5778 = 274 Z = ᎏ = 66 ᎏᎏᎏ CviL 1.52 im ᎏ – 1 = 60.35 iL im = 61.35i Calculating the density of the mixture as:

m = Cv(s – L) + L = 0.22 (1,650) + 1,000 m = 1363 kg/m3 or S.G. = 1.363 Using the Newitt approach with fL = 0.01572

冤

冥

0.01572 × 1.52 im = ᎏᎏ 61.35 = 0.1914 m/m 2 × 0.5778 × 9.81

␦P ᎏ = m gim = 1363 · 9.81 · 0.914 = 2560 Pa/m ␦z

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

A more advanced but less known method to estimate the friction gradient for saltation flows is the Graf–Acaroglu method that is presented in Chapter 6, which we will also discuss in the next section with a worked example.

4.7.2 Restarting Pipelines after Shut-Down or Blockage The loss of power, a water hammer situation, or the freezing of a pipeline are occurrences that require adequate understanding of the power and pressure needed to restart a pipeline. The concept of the hydraulic radius is defined in Chapter 6 and is essential to understand blockage. Vallentine (1955) studied the blockage of a mixture of 0.53 mm (No. 1) and 1.05 mm (No. 2) sand in 50 mm (2 inch) and 150 mm (6 inch) pipe. He proposed to write the Darcy equation in terms of a function for blockage: fD · V 2L fD · Q2L fD · Q2L H = ᎏᎏ = ᎏᎏ2 = ᎏ (B) (4 · RH) 2 · g (8 · g · RH) A 8gD 5I

(4-42)

where B is the blocked area of pipe and (B) is a function of blockage. Vallentine proposed that the blocked area B is a function of the total flow, pipe diameter, flow rate of solids, and flow rate of mixture: 1/3 Q 1/2 L Qm ᎏᎏ B = fn ᎏ 1/2 1/3 2.5 D I [s – L) Q s

冤

冥

(4-43)

Herbich (1991) plotted these functions. They are presented in Figures 4-12 and 4-13. In Chapter 6, the work of Graf and Acaroglu is examined in Section 6.5.4. Schiller (1991) proposed to apply their equation (Equation 6.66b) to the problem of flow over a

1.0 0.9

BLOCKAGE B

abul-4.qxd

0.8

No. 1 Nonuniform (Vallentine)

0.7

No. 2

0.6

Uniform Sands (Craven)

''

(

“

)

0.5 0.4 0.3 0.2 0.1 0.0

0

1

2

3 Q ᎏ d2.5

4

5

6

7

8

冣 冪莦 ␥ ␥ 冢 Qs ᎏ ᎏ – Q s w

–1/3

FIGURE 4-12 Blockage factor B. (From Herbich, 1991, reprinted with permission from McGraw-Hill, Inc.)

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4.46

10 % 10 0. 5% 0.0

BLOCKAGE Q (cfs)

9 8 7 6 5

2.75 50% 2.50 % 0 4 2.25 30% 2.00 20% % 1.75 10 5% 1.50 2% 1.25 1.00

4 3 2 1

PIPE DIAMETER d (ft)

% 3.00 60 0. 25 %

Qs ᎏ Q

1.0 0.7 % 5% 0. 5%

CHAPTER FOUR

2.0 %

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Qs ᎏ = Relative Rate of Sediment Q Transport B = Average Pipe Area Blocked

0.75 0.50

0

FIGURE 4-13 Blockage chart. (From Herbich, 1991, reprinted with permission from McGraw-Hill, Inc.)

stationary bed. Schiller established the hydraulic radius of the bed in terms of the ratio of the mean velocity of flow to the deposition velocity V3 as 苶苶·苶g苶·苶 R苶) 苶苶 g苶 V/兹(4 H苶 = V3/兹(D I ·苶) or RH = 0.25 [(V/V3)2]DI Combining these two equations with the Graf–Acaroglu equation (6.64), Schiller proposed to replace the slope by head losses per unit length. Schiller proposed to replace dp by d50 for a mixture of particles of different sizes: {0.25 [(V/V3)2]DI} (s – L)d50] = 10.39 ᎏᎏᎏ CV · V · ᎏᎏ 3 L(h/L) [0.25 [(V/V3)2]DI] 兹[( 苶苶苶 苶L苶–苶苶)g 1苶苶 d 50 苶]苶 s/

冤

冥

–2.52

(4-44)

Example 4-13 Considering that the slurry of Example 4.1 is partially blocked at a speed of 12 ft/sec. Determine the resultant head per unit length to maintain flow: V3 = 15.25 ft/sec Solution in SI Units V = 3.66 m/s DI = 0.718 m The hydraulic radius is therefore [0.25 [(.787)2]0.718] = 0.111 m.

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冦

1.65(3 × 10–4) 0.27 · 3.66 · 0.111 ᎏᎏᎏ = 10.39 ᎏᎏ –4 3 (h/L) 0.111 兹苶 [(苶1苶 .6苶5苶 )9苶 .8苶1苶(苶3苶 ×苶1苶0苶苶 ) 苶]

冧

–2.52

5247 = 10.39[(h/L)2.52 [839090.5] h/L = 0.0527 Solution in USCS Units V = 12 ft/sec DI = 2.357 ft The hydraulic radius is therefore [0.25 [(12/15.25)2]2.357] = 0.365 ft. d50 = 0.000984 ft

冦

1.65(0.000984) 0.27 · 12 · 0.365 ᎏᎏᎏ = 10.39 ᎏᎏ 3 (h/L) 0.365 兹苶 [(苶1苶 .6苶5苶 )3苶2苶 .2苶 (0苶 .0苶0苶0苶9苶8苶4苶 ) 苶]

冧

–2.52

5256 = 10.39[(h/L)2.52 [844444] h/L = 0.0526 Wood 1979 attempted to simplify this complex problem by proposing that the pressure gradient between the incipient motion velocity V2 (when particles start to leave the bed) and the actual deposition velocity V3 (when the particles start to move as a heterogeneous flow) is essentially composed of three components: 1. A component required to overcome the boundary shear stress and turbulence 2. A component required to accelerate the slurry due to changes in mean velocity of the slurry 3. A component required to accelerate the particles that leave the bed The third component is the principle source of increase in the pressure gradient. The particles are assumed to be eroded from the bed at a rate based on flow conditions. If this rate exceeds the capability of the flow to suspend the solids, they tend to fall back into the bed, until the process is restarted. The analysis of Wood is based on open channel flow, which is the topic of Chapter 6. Wood treats the stationary bed by considering its hydraulic diameter, and proceeds to apply a modified Zandi and Govatos approach.

4-8 PSEUDOHOMOGENEOUS OR SYMMETRIC FLOWS Above V4, the flow is pseudohomogeneous or symmetric (but not necessarily uniform). With negligible hold-up, Govier and Aziz (1972) proposed to express the pressure loss in terms of the ratio of the friction fanning factor for the slurry mixture and the liquid mixture or as im – i fNmm – fNLL ᎏ = ᎏᎏ Cv iL fNLL

(4-45)

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CHAPTER FOUR

where fNm = fanning factor of mixtures fNL = fanning factor of liquid at equivalent volume In Chapter 1, the increase in dynamic viscosity due to the volumetric concentration of solids was discussed; it can be expressed as the Einstein–Thomas equation:

m = L[1 + ACv + BC v2 + C exp (DCv)] where A, B, C, and D are constants. Since the fanning friction factor is a function of the Reynolds number, the ratio of the Reynolds numbers of the mixture and liquid would be

mVmDi Rem = ᎏ m LVmDi ReL = ᎏ m The ratio of the Reynolds number helps to establish a ratio of friction factors: Rem L Cv(s – L) ᎏ = ᎏ = {1 + ACv + BC v2 + C exp(DCv)} ᎏᎏ + 1 ReL m L

冢

冣

(4-46)

The next step consists of using the Blasius equation for transition flows: fm ᎏ = {ReL/Rem)0.25 fL

(4-47)

Although Govier and Aziz (1972) did not discuss turbulent flow, their analysis can be extended. For turbulent flows up to Reynolds numbers from 5000 to 100,000,000, which is well outside the range used for mining, the Swamee–Jain equation (Equation 2.19) can be used to obtain the ratio of friction factors: {log10(0.27 /Di + 5.74/ReL0.9}2 fm/fL = ᎏᎏᎏᎏ {log10(0.27 /Di + 5.74/Re m0.9}2

(4-49)

4-9 STRATIFIED FLOWS In Section 4.6, it was clearly indicated that Wasp et al. (1977) extended the Durand and Zandi approach to a concept of multiple and superimposed layers of particles of different sizes and volumetric concentration, with a logarithmic concentration distribution. This method uses the drag coefficient of the particles as an important parameter. Another school of researchers, particularly lead by Shook, Wilson, and Gilles in Canada, focused on refining the original models of Newitt, which were based on the terminal velocity. These authors proposed that the compound mixture of coarse and fine particles can be simplified to what they called “stratified flows,” in which the fines slide over a moving bed of coarse solids. Newitt, as expressed by Equation 4-5, had proposed that the deposition velocity was a mere factor of 17 times the terminal velocity, Wilson (1991) indicated that this approach was not confirmed by tests for large pipes. The concept he proposed was that the flow was going through a gradual transition from a fully stratified to a fully suspended flow, with a gradual change in the pressure gradient.

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

4.49

Defining a parameter in terms of the pressure gradient: im – iL = ᎏᎏ {m/L – 1}

(4-50)

Moreover, plotting versus the speed, as in Figure 4-14, allows one to find a speed V50 halfway between a fully stratified flow and fully suspended flow. The slope on this plot is defined as M. In the region of partially stratified flow, the logarithmic plot confirms that the pressure gradient is a function of the velocity raised to the power (–M). M is calculated from the slope at V50 considered to be the point at which half-stratified flow occurs, as in Figure 4-14. Using V50 as a reference velocity, and considering that the value of at V50 is effectively half the mechanical sliding coefficient p, Wilson (1992) proposed that = fn(V, V50, p, M) Determining V50 is no easy task. In this partially stratified model, it is argued that the lifting of particles by turbulence is strongly influenced by their diameter. Thus, the fines tend to be transported better by the fluid than the coarse solids at the bottom of the pipe. These particles are transported by eddies, and the largest eddy would be equal to the pipe diameter. The pipe diameter was therefore proposed as a reference. The resistance to motion of the solid particles is proposed to be the result of a contact load associated with mechanical sliding friction in the coarser bed and fluid friction associated with the friction velocity of the carrier fluid. In this complex environment, Wilson et al. (1992) defined a new settling velocity as [{s – L}gL]1/3 Vs = 0.9 Vt + 2.7 ᎏᎏ L2/3

冤

冥

(4-51)

fully stratified flow

gradient M

p

m

L

-1)

partially stratified

m

L

log (i - i )/(

abul-4.qxd

Fully suspended flow V 50

Speed

FIGURE 4-14 Concept of V50 for stratified flows.

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4.50

CHAPTER FOUR

The two constants 0.9 and 2.7 were obtained from test data on sand. The velocity V50 is then expressed in terms of this settling velocity: 苶/f 苶dL 苶)苶 cosh (60 dp/DI) V50 = Vs兹(8

(4-52)

where cosh = the hyperbolic trigonometric function fdL = Darcy friction factor for an equal volume of water In tests conducted for sand in water at 20° C, is 0.1 m/s at a particle size of 0.40 mm, indicated that the magnitude of p, the mechanical friction coefficient, was 0.44. From Figure 4-14:

= 0.5 p(V/V50)–M = 0.22(V/V50)–M

(4-53)

For a mixture of particles of different diameters, a settling velocity Vs85 at d85 and Vs50 at d50 are used to calculate a standard deviation s:

冤

Vs85 cosh (60 d85/DI) s = log ᎏᎏᎏ Vs50 cosh (60 d50/DI)

冥

(4-54)

The coefficient M is expressed as M = (0.25 + 13s2)–1/2. The magnitude of Vm/V50 or ratio of mean velocity of the mixture to V50 is defined as the stratification ratio. The friction losses may be expressed in terms of the stratification ratio. When d85/d50 < 2, the slurry is considered to be narrowly graded, and M is set at 1.7. For 2 < d85/d50 < 5, 1.7 > M > 0.4. There is no question that the approach of Wilson et al. (1992) is extremely interesting, but it is more complex than the methods proposed by Equations 4-3 to 4-7. It requires the support of testing, a database, a computer software, and a personal computer.

4.10 TWO-LAYER MODELS Khan and Richardson (1996) explained that Shook and Roco (1991) developed a two-layer model for stratified slurry flows, which may be summarized as follows: 앫 The slurry flow of heterogeneous mixtures is considered to consist of two layers, each with its own velocity of motion and volumetric concentration; but it is assumed that there is no slip between liquid and solid phases. 앫 The solids in the upper layer are fully suspended, are at a volumetric concentration CVu, and move at a velocityVU. 앫 The coarser solids in the lower layer are considered to be packed. However, because of their irregular shape, there is a certain void fraction between the particles. For sand, the lower layer is considered to be at a maximum volumetric concentration CVU of 60% and move at a velocityVB. The total area of flow for the upper layer of fines and the lower layer of coarser particles is A = AB + AU For the mixture, the mass balance VA = VB AB + VU AU For the liquid phase, (1 – CV)VA = (1 – CVU) VU AU + (1 – CVB)VB AB For the solid phase, CVVA = CVBVB AB + CVUVU AU = CVU AV + (CVB – CVU)VB AB

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4.51

Referring to Figure 4-15: AB = 0.25 D 2I ( – sin cos ) AU = 0.25 D 2I ( – + sin cos ) The upper wet perimeter is WPU = DI ( – ) WPB = DI At the interface WPi = DI sin The momentum and force balance is expressed as dP –AU ᎏ = UWPU + iWPi dx

for the upper layer

dP –AB ᎏ = LWPB – iWPi + fc⌺F dx

for the lower layer

dP –A ᎏ = UWPU + LWPB + fc⌺F dx

for the entire pipe

fc is the friction factor due to the Coulombic friction between the particles and the pipe and ⌺F is the total force per unit length exerted normal to the pipe (e.g., weight of the bed per unit length, etc.) At low to moderate concentrations B = 0.5LL fNBV B2. Certain solids are at a concentration CVB – CVU. They are considered to be of a buoyant weight that is supported at the wall as a result of interparticle contacts. Averaged over the entire cross-sectional area of the pipe, these particles define the “contact load” CC, which is calculated as follows: AB CC = [CVB – CVU] ᎏ A

(4-55)

where AB = cross-sectional area of the lower layer A = cross-sectional area of the entire pipe The upper layer volumetric concentration CVU is obtained from the mean in-situ concentration CX and the total contact load CC as CVU = Cr – CC. A parameter CX is defined as the in-situ concentration in the x-direction, as CX = CVB + CVU

(4-56)

The relationship between CC and CX is established as an experimental correlation: CC ᎏ = e–⌫ CX

(4-57a)

⌫ = 0.124 Ar–0.061[(V 2/(gdp))0.28][(dp/Di)–0.431](S/L– 1)–0.272

(4-57b)

where CVB = 0.60. And for sand slurries, according to SAC (2000): ⌫ = –0.122 Ar–0.12[(V/VD)0.30][(dp/Di)–0.51](S/L– 1)–0.255 The Archimedean number Ar is defined in Equation 4-5.

(4-57c)

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CHAPTER FOUR

The density of the suspended concentration CVU and the density of the carrier liquid L are then used to compute the shear stresses in the two layers (Figure 4-15). A fanning friction factor is then obtained for each layer based on some average velocity of the flow in the pipe, density of the carrier liquid, and dynamic viscosity of the carrier liquid, and using the relative roughness of the pipe /Di. The Churchill (1977) equation is then used to calculate the friction factor as fn = 2[8Re–12 + (A + B)–1.5)1/12

(4-58)

where A = [–2.457 ln[(7/Re)0.9 + 0.27 /Di)]16 B = (37530/Re)16 The Reynolds number is calculated on the basis of the average pipe flow speed, since the speed of flow in each layer is not known at this point. An equivalent “sand roughness” is then defined as the ratio of particle size to pipe diameter (dp/DI). At the interface between the bottom and top layer, a friction factor is derived by modifying the Colebrook equation (Equation 2-17):

WPU WPUB AU DI

2

WPB AB

VB

FIGURE 4-15

VU Speed

above bottom quadrant

Vertical distance (y)

above bottom quadrant

Vertical distance (y)

typical real distribution

C VB C VU Solids volumetric concentration

Two-layer modeling of coarse and fine particle mixtures.

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

2(1 + Y) fNi = ᎏᎏᎏ2 [4 log10(dp/Di) + 3.36]

(4-59)

where Y=0 for dp/Di < 0.0015 Y = 4 + 1.42 log10 (dp/Di) for 0.0015 < dp/Di < 0.15 and Ar (the Archimedean number) < 3 × 105 For bimodal mixtures (mixtures of fine and coarse particles), the Saskatchewan Research Council (SRC) (2000) suggested that the effects of the fines on the viscosity of the carrier liquid should be used to calculate the apparent viscosity f. However, the actual density of the liquid should be used without corrections for volumetric concentration of fines. Khan and Richardson (1996) pointed out that the concept of an equivalent sand roughness is very speculative as the transition between one layer and the other. By subsequent iterations, the friction factor and velocity for each of the two layers are obtained from the shear stress as the product of the friction factor and the dynamic pressure:

U = 0.5fNUUVU2 B = 0.5fNBBV B2 fNBBV B2WPB D 2i gfc(s – L)(CVL – CVU)(1 – CVB)(sin – cos ) BWPB = ᎏᎏ + ᎏᎏᎏᎏᎏᎏ (4-60) 2(1 + CVU – CVB) 2 where fc is the coefficient of kinematic friction between the particles and the pipe. The total normal force per unit length was defined by Shook as D 2i g(s – L)(sin – cos ) (CVB – CVU) (1 – CVB) ⌺F = ᎏᎏᎏ ᎏᎏᎏ 1 – (CVB – CVU) 2

冦

冧

(4-61)

where is the half angle formed by the bottom layer with respect to the center of the pipe. To appreciate the complexity of this approach, SRC (2000) indicated that this approach yielded six unknowns (VB, VU, , Cr, Cc, and –dP/dx). The numbers of iterations that are required are better dealt with on a computer. The two-layer models have gained wide acceptance in the oil–sand industry. The following example is an illustration at the first level of iteration. Example 4-14 Sand slurry in a pipe is flowing at 6.5 m/s (21.3 ft/sec). The pipe diameter is 717 mm (28.35⬙) pipe and the sand particle diameter dp = 360 m (0.0142⬙).The volumetric concentration was presented to be 0.27. Upon review of the composition of the sand, it was noticed that 15% of the solids were fines smaller than 74 m. If the lower bed is packed at 60%, the contact load Cr = 0.30, and the Columbian friction factor fc is 0.50, determine the pressure gradient (assume water dynamic viscosity 1 cP, and sand S.G. 2.65). Volumetric concentration in the upper layer consists essentially of fines: CVU = 0.15 · 0.27 = 0.0405 CVB = 0.85 · 0.27 = 0.23 By the Einstein–Thomas equation, the dynamic viscosity of the carrier liquid needs to be corrected for a concentration of 0.0405 in the upper layer:

= L(1 + (2.5 · 0.0405) + (10.05 · 0.04052) + 0.00273 exp (16.6 · 0.0405)] = 1.123 L = 1.123cP

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CHAPTER FOUR

4 × 9.81 (3.6 × 10–4)3 × 1000 (1650) Ar = ᎏᎏᎏᎏ = 666 3 (1.123 × 10–3)2 ⌫ = 0.124 Ar–0.061 [(V2/(gdp))0.28][(dp/Di)–0.431](S/L – 1)–0.272 = 0.124 (666–0.061) (6.52/(9.81 · 3.6 × 10–4)0.28][0.0005–0.431]1.65–0.272 = 395.52 CC ᎏ = e–395.52 = 0 CX Density of the liquid and fines in the upper layer is U = .0401 · 2650 + (1 – .0401) · 1000 = 1066 kg/m3 If it is assumed that, due to the void fractions, the lower layer is full at 60%, and since the volumetric concentration is 0.23, then the area used is 0.23/0.6 = 0.383. area = 2[0.25D I2( – 0.5 sin cos )] = 0.383 × 0.25DI2

⬇ 80 degrees ⬇ 0.444 AB = 0.25

D I2(

– sin cos ) = 0.25 × 0.7172x(1.396 – 0.171) = 0.1574 m2

AU = 0.25 D I2( – + sin cos ) = 0.25 × 0.7172x(0.556 + 0.171) = 0.246 m2 WPU = DI( – ) = 1.252 m WPB = DI = 1 m The total normal force per unit length was defined by Shook as

冦

0.7172 9.81 (2650 – 1066)(0.985 – 1.39 × 0.1736) (0.23 – 0.0405) (1 – 0.23) ⌺F = ᎏᎏᎏᎏᎏᎏ ᎏᎏᎏ 2 1 – (0.23 – 0.0405) = 632 N/m

冧

fNBBV B2WPB BWPB = ᎏᎏ + fc ⌺F 2 Since WPB = 1 m, 2 (1) fNL 1066V LL B(1) = ᎏᎏ + 0.5 · 632 = 533 fNBV B2 + 316 2

dp/DI = 0.00035/0.717 = 0.000488, so Y = 0 fNI = 2/[4 log10(0.717/0.00035) +3.36]2 = 0.00725 To determine the friction factors for the upper and lower layers it is required to determine the speed in each layer. To obtain the difference between the velocity in the upper and lower layers, it is essential to obtain the shear stress i between the two layers, or to make some assumptions and to proceed with further iterations. In a first iteration, it shall be assumed that VU = 1.1 VLL. AV = ABVB + AUVU 0.25 · · 0.717 · 6.5 = 0.1574 m2(VL)+ 0.246 m2(1.1VL) 2

VB = 6.14 m/s VU = 6.75 m/s

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

The following calculations require the reader to review some of the concepts of Chapter 6. The hydraulic diameter (Equation 6-2) for the upper layer is 4AU/WPU = 4 × 0.246/1.252 = 0.786 m. The Reynolds Number for the upper layer is ReU = 1066 · 6.75 · 0.786/(1.123 · 10–3) = 5,036,209. The pipe is rubber coated to a roughness of 0.00015 m, /Di = 0.0002. Applying Churchill’s equation to the upper layer: A = {–2.457 ln[(7/Re)0.9 + 0.27 /Di)}16 = 1.192 · 1022 B = (37530/Re)16 = 9.045 · 10–35 fnu = 2[8Re–12 + (A + B)–1.5)1/12 = 2[3 · 10–80 + 7.684 · 10–34]1/12 = 0.0035

U = 0.5fNU UV U2 = 0.5 · 0.0035 · 1066 · 6.752 = 85 Pa For the lower layer the hydraulic diameter is 4ABWPB = 0.6296 m. The Reynolds number for the lower layer is ReU = 1066 · 6.14 · 0.6296/(1.123 · 10–3) = 3,669,531. The pipe is rubber coated to a roughness of 0.00015 m, /Di = 0.000238. Applying Churchill’s equation to the upper layer: A = {–2.457 ln[(7/Re)0.9 + 0.27 /Di)]}16 = 1.006 · 1021 B = (37530/Re)16 = 1.433 · 10–32 fnu = 2[8Re–12 + (A + B)–1.5)1/12 = 2[1.34 · 10–78 + 3.134 · 10–32]1/12 = 0.0047

B = 0.5fNBBV B2 = 0.5 · 0.0047 · 1066 · 6.142 = 94 Pa At the interface i, 0.5fNiU (V U2 – V B2) = 0.5 · 0.0035 · 1066 · (6.742 – 6.142) = 14.68 Pa. dP –AU ᎏ = UWPU + iWPi = 85 · 1.252 + 14.68 · 14.68 · 0.717 · 0.985 = 117 Pa/m dx Since AU = 0.246 m2, then dP/dx = 475.6 Pa/m. Equation 4-60 is not applicable at high concentration of fines, as the slurry starts to behave as a non-Newtonian mixture. For particles with d50 finer than 74 m, the method does not give very reliable results. For flows above a deposited bed (flows with saltation), SRC (2000) proposed to treat the upper layer as a noncircular flow. This means that the hydraulic diameter must be determined from the wetted area. The difference in roughness between these two surfaces (upper surface of the bed) and pipe roughness is not well discussed. The hydraulic diameter is calculated as DHB = 4AB/(WPB). It is considered that for pipe flows, the deposit velocity is a function of the pipe diameter raised to the power of 0.4 for noncircular channels and the friction loss gradient is a function of the ratio V2/DH. By defining VU as the velocity of the upper layer and V3 the critical velocity at which the bed deposits, the following equation is established: 0.4 VU DHB ᎏ=ᎏ 0.4 V3 D

(4-62)

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CHAPTER FOUR

d

WP

U

WP

UL

AU

Deposited solids

Friction loss per unit length

deposited bed

upper layer

i1

Flow rate FIGURE 4-16

Flow above a deposited bed—two-layer model.

The ratio of friction gradients is D im V U2 D ᎏ = ᎏ2 ᎏ = ᎏ i3 V 3 DHB DHB

冢

冣

0.2

(4-63)

where i1 is pressure gradient at V1. At deposition, the flow rate in the upper layer QU is lower than the flow rate through the deposited bed QB, and from the ratio of velocities as per Equation 4-57: QU = AUVU Q3 = AV3 QU DHB ᎏ= ᎏ QB D

冢

冣

0.2

AU ᎏ A

(4-64)

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4.57

4.11 VERTICAL FLOW OF COARSE PARTICLES Newitt et al. (1961) conducted tests for flows of solids in vertical pipes. For fine solids they derived the following empirical equation: Di

S

p

L

冢 冣 冢 ᎏd 冣冢 ᎏ 冣

im – iL gDi ᎏ = 0.0037 ᎏ iLCv V2

1/2

2

(4-65)

In a vertical flow, it would not be possible to develop dunes, a bed, or saltation. There is no concentration gradient and the flow may be treated as pseudohomogeneous for friction loss calculations, as discussed in Section 4-4-3. Since the flow in a vertical pipe is pseudohomogeneous, a simple instrument to measure flow rate of slurry is the inverted U column, which consists of 4 elbows and 2 vertical pipe spools (Figure 4-17). The first vertical branch must be sufficiently high to eliminate entrance effects (previously discussed in Chapter 2). Toward the top of the pipe, a pressure tap measures the static pressure. Pressure loss occurs through the two elbows. Another pressure tap is added on the downward section of the pipe. The inverted U column is calibrated on water and the pressure loss is a function of the flow rate as well as the density of the slurry:

V 2 ⌬P = K ᎏ 2 VmD 2i Q = ᎏ Cd 4 where Cd is the discharge coefficient, or 苶苶 ⌬P 苶苶 /m 苶)苶 Q = CdDi 兹(2

The density of the mixture may be calculated from the input data or measured using a nuclear radiation density gage.

2 3 zA zB 1 4

flow

FIGURE 4-17

Inverted U tube piping for measuring flow of a slurry mixture.

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In a vertical flow, in addition to the friction losses as discussed for a pseudohomogeneous flow, there is a hydrostatic pressure gradient, so that the total pressure drop is: ⌬P = mg[⌬z + fmV m2 ⌬z/(2gDi)] ⌬P = pressure loss between two points ⌬z = height difference between two points For further reading, see the work of Einstein and Graf (1966) who described such a flow and a concentration meter for water–sand mixtures.

4-12 INCLINED HETEROGENEOUS FLOWS Between the two types of horizontal and vertical flows that we discussed in this chapter, there is a range of inclined flows that are important but are not the subject of extensive studies. In fact, it may be argued that in many plants, most flows are either horizontal or vertical, with elbows and fittings between them. An understanding of inclined heterogeneous flows is essential for certain long overland pipelines, thickener feed systems, pumpbox feed systems, and, in some respects, to shed light on open channel flows. Until very recently, attention was focused on uniformly graded slurries. Worster and Denny (1955) indicated that if the pipe is inclined from the horizontal, the fraction loss is i␥ = i + (im – i) cos ␥

(4.65)

where i␥ = pressure gradient at ␥ im = pressure gradient of the mixture in the horizontal pipe This equation suggests that the pressure loss is lower in an inclined pipe than in a horizontal pipe. It also suggests that the pressure gradient is the same upward or downward. The experimental work of Kao and Hawang (1979) indicates that this is not correct. In fact, they noticed that the friction losses for upflows increased up to a certain magnitude of the angle of inclination and then decreased. In the case of downflows, they measured a reduction of friction losses from the values of the horizontal pipe. Wilson et al. (1992) have discussed the effect of pipe inclination on their V50, and suggest that Worster and Denny’s equation be modified by using (cos ␥)1.85 instead of cos ␥. They also published data on certain particles with a diameter between 1 mm and 6 mm. The test data indicated that the Durand factor for deposition velocity FL (see Equation 4-2) increased up to an angle of inclination of 30 degrees and by as much as 38%. They also noticed that the deposition velocity V3 increased by 50% at 30 degrees pipe inclination, but then they noticed a drop at an angle of 40 degrees. They did not conduct further tests. Interestingly, they noticed a reduction of the Durand factor FL by 0.3 at a negative inclination of 20 degrees. There is a dearth of information on flow in inclined pipes, and as overconservative as it may be, Worster and Denny’s equation continues to be used. This approach should change, particularly when the angle of inclination is less than 30 degrees or up to –20 degrees.

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4.12.1 Critical Slope of Inclined Pipes Pipeline have to go through dunes and hills. Certain slurry pipelines may follow the topography (Figure 4-18) but others require pipe bridges to avoid sedimentation of the slurry after a shutdown or power failure (Figure 4-19). A question often raised in designing a pipeline is determining the critical slope of the pipe to avoid areas of blockage once the flow is shut down. To avoid blockage of the line, it is necessary to eliminate areas of steep slopes. A commonly used design restriction of 10–16% (5.7–9°) is often adopted when there is no knowledge of the critical slope. The idea is that the slurry would settle without segregation to a “soft” consistency and not migrate down to the steepest slope in the pipeline during shutdown. Kao and Hwang (1979) criticized this approach as being extremely stringent because it adds to construction costs and capital expenses. They implied that it would be better to properly understand the critical slope rather than to use a rule of thumb. Shook et al. (1974) measured the maximum inclination of a 50 mm (2 in) ID Perspex pipe on a sand–water mixture. They reported that: 앫 The maximum rising angle is 14°, or slope of 24%, before the solids start to slide back. 앫 The sliding bed was at the interface of the solid bed and the pipe wall rather than within the settling bed.

FIGURE 4-18 Tailings pipelines follow the slope of the hills and use soil friction produced by partial burial as an anchor.

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FIGURE 4-19 Pipeline carrying taconite tailings with an important portion of coarse particles required pipe bridges to avoid blockage after shutdown or power failure.

앫 The critical angles for the sand bed increases from 22–26°(slope of 40–48.7%) with the decrease of the size of particles. The critical angle is neither affected by the ratio dp/DI nor by the concentration of the solids. Durand and Gilbert (1960) derived the following equation for inclined pipe: im␥ = iL + iB(cos ␥) where im = energy gradient for mixture iL = energy gradient for liquid iB = energy gradient for solid bed Kao and Hwang (1979) observed that the critical slope for glass beads and for sand occurred at 23°(42% slope) from the horizontal. For other substances such as coal and walnut shells, the initial motion appeared to occur at the interface between the particle bed and the pipe wall. This suggested that the internal friction between irregularly shaped coarse particles was higher than the friction at the wall of the pipe. The critical slope Kao and Hwang (1979) defined was the value for initial particle motion. For sand or glass beads it was 27° ± 2°, and for coal and walnut shells it was 37° ± 2°. Craven and Ambrose (1953) investigated the effect of tube inclination on the head loss for a pipe partially blocked with sediments and for a pipe flowing full. They found that at

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a given average speed, an adverse slope in excess of 10% increased the pressure losses by 25% or more by comparison with a horizontal pipe.

4-12-2 Two-Layer Model for Inclined Flows Matsouk (1996) developed a two-layer model for inclined pipes. His tests were conducted in a 150 mm (6⬙) pipe at angles of inclination between –35 and +35 degrees. The fundamental equations for this model for the upper layer are

UWPU + UBWPUB d(P – Ugh) ᎏᎏ = ᎏᎏ AU dx

(4-67)

For the lower layer they are

BWPB – iWPi + fc⌺FN cos ␥ + FW sin ␥ d(P – Bgh) ᎏᎏ = ᎏᎏᎏᎏᎏ AL dx

(4-68)

where FW is the submerged weight of the sediments in the lower layer. The force balance for the whole pipe is then

BWPB + UWPU + fc⌺FN cos ␥ + FW sin ␥ d(P – Ugh) ᎏᎏ = ᎏᎏᎏᎏᎏ dx A

(4-69)

Equations 4-67 to 4-69 are then solved in a similar manner as presented in Section 4-10. Matsouk pointed out that his approach was different than the models of Shook and Roco (1991) (the SRC model), Lazarus (1989), and Lazarus and Cook (1993), which did not include the pipe axis component of the submerged weight (due to buoyancy) FW sin ␥. The tests of Matsouk did not confirm the Worster and Denny equation. At pipe inclinations close to the angle of internal friction of the transported solids, the behavior of the solids was different for inclining and descending pipes under the same speed and volumetric concentration. The difference was larger with coarse than with fine solids. Matsouk con-

x

P2 submerged lower layer of coarse solids

z

P

1

L

P+ g z FIGURE 4-20

Concept of the two-layer bipolar flow of slurry at an angle of inclination.

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cluded that the deformation of the lower layer was essentially due to the submerged weight of the solids at the bottom of the pipe.

4-13 CONCLUSION In this chapter, we have shown that two principal schools of heterogeneous slurry flows have developed over the last 50 years—one around the Durand–Condolios approach and the other around the Newitt approach. The former evolved gradually until Wasp modified it for multilayer compound systems. The latter gradually evolved to yield the two-layer model. There is no consensus as to which model to use. Some have argued that the Wasp method was more suitable for coal, whereas the two-layer model was better for sand. This is based on the number of papers published on two-layer models for sand slurry mixtures, emanating from the great interest in Canadian oil sands. The science of slurry flows is continuously evolving and needs to be understood in the context of its evolution. It would be a mistake to favor one school of thought over another. Many successful pipelines have been designed using either model. The slurry engineer should appreciate that these models are effective tools to be used with correct empirical coefficients obtained by experimental testing of samples in pumping loops. After reading this chapter, some readers may get the feeling that designing a slurry flow is a combination of science and art. Slurry dynamics may appear to be an exercise of examining each mixture for its properties, much as the physician must examine each patient before administering the cure. The flow of coarse particles in water or mixtures of coarse and fine solids in a liquid is complex. When data is not well accumulated, it is recommended to conduct slurry tests in a pump test loop. The mixture of coarse and fine particles can lead to a concentration gradient of solids in horizontal pipe. The coarse particles tend to flow in the lower layers, whereas the fines flow in the upper layers. Certain authors recommend determining the pressure losses of fine solids separately from coarse solids at the corresponding volumetric concentration and particle diameter of each size. Methods based on concentration ratio for each layer have been developed. A process of iteration is needed to achieve a final estimation of the pressure loss due to the bed. New models for inclined flows are appearing, such as the work of Matsouk (1996) on inclined two-layer models The correct design of inclined flow must be based on empirical data on the critical slope. The principles reviewed in this chapter apply to dredging and transporting sand, gravel, coal, steel shot, and rocks from SAG, rod, or ball mills, cyclone underflows, and tailings, etc., which often have particle sizes larger than 70 m (mesh 200). The practical engineer needs to appreciate the limitations of each method and that models have often been developed for a certain range of particle sizes based on experimental data. It is wise to check the original data. Because the new methods of stratified flows or two-layer models use the actual hydraulic diameter of the bed, whereas the Wasp and Durand methods use the actual pipe diameter, it is easy to get confused. In fact, some of the proponents of the two-layer models leave the impression that Wasp and Durand are using the “wrong pipe diameter.” This is not the approach to take. It is wiser to recognize that the Wasp and Durand methods are useful tools for the range of slurries for which they were developed. This includes concentrations of coarse particles up to a volumetric fraction of 20%. This covers, in fact, most dredged gravels and sands, coal in a certain range of sizes, as well as crushed rocks. It is also important to appreciate that the work of Zandi and Govatos (1967) was based on

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sands up to a volumetric concentration up to 22%, and it would be erroneous to push the envelope of application of their equations beyond such a range. In the last thirty years, considerable progress has been made with stratified flows. The new two-layer models push the envelope of understanding beyond the limits of the models of Durand, Zandi, and Wasp into the range of volumetric concentrations of 30%. But these models have their limitations too. Certainly it would be unwise to use the two-layer model when the d50 is smaller than 74 micrometers. There is still a considerable amount of experimental data associated with these stratified models needed to obtain the Conlombic friction factor and to determine the difference in velocity between the upper and lower layers. When the particles are not too coarse, such a difference is not too large, and approximations such as those proposed by Richardson and Khan are justified, but when the d50 is around 400 micrometers or higher, the difference in velocity and shear between the upper and lower layers become important. In Chapter 6, the analysis presented in this chapter will be extended to include open channel flows. The Acaroglu–Graf equation presented in Chapter 6 was applied here to flows with saltation.

4-14 NOMENCLATURE a A AB Ap Ar AU b B C CA CC CD CE Ct Cv Cvb Cv bed Cvi CVL CVU Cw CX C1 C3 dg dp d85 d50 DH Di dsp

Height of layer A above bottom of conduit Cross-sectional area of the entire pipe Cross-sectional area of the lower layer surface area of particle The Archimedean number Area of upper layer of flow in the two-layer model Factor used to calculate the Archimedean number blocked area of pipe volumetric concentration of the particle diameter under consideration Concentration of solid particles at a reference plane A (usually at 0.08 DI) Contact load in the Shook–Roco two-layer model Drag coefficient Coefficient of discharge In-situ concentration Concentration of solids by volume Volume fraction of solids in the bed Concentration of solids in the moving bed Concentration of solids in the moving bed of fraction i Concentration of solids by volume in the lower layer in the Shook–Rocco model Concentration of solids by volume in the upper layer in the Shook–Rocco model Weight concentration In-situ concentration in the two-layer model Constant Constant Diameter of spherical particle Average diameter of the particle Sieve passage diameter for 85% of the particles Sieve passage diameter for 50% of the particles Hydraulic diameter of a noncircular flow Inner diameter of pipe diameter of equivalent sphere using the sphericity factor

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Es fc fD fDL FL fN fNL fNm fNB fNU Fr g i im i i1 i3 K Ke Kf Kx K2 K3 L Ne m mi mt M P PW Q QB QU Ri r Re Rem ReL RH R1 s Sf Sm U Uf Ufc Uf 0 V VB VD

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The mass transfer coefficient coefficient of kinematic friction between particles and pipe Darcy–Weisbach friction factor Darcy friction factor for equivalent volume of water Durand–Condolios coefficient to determine the deposition velocity Fanning friction factor Fanning friction factor for equivalent volume of water Fanning friction factor for mixture Fanning friction factor for the bottom layer in the two-layer model Fanning friction for the top layer in the two-layer model Froude number Acceleration of falling objects due to gravity (9.78–9.82 m/s2) Equivalent pressure gradient of water at the same volume as the slurry Energy gradient of slurry mixture in equivalent m of water per m of pipe length Pressure gradient for pipe at inclination Pressure gradient at V1 Pressure gradient at V3 Coefficient or constant Experimental factor for the pressure gradient Coefficient in the Durand equation for pressure drop Von Karman constant An experimentally determined constant Coefficient proportional to the mechanical friction factor ␥ Length of conduit Index number Power coefficient in Zandi’s models The mass fraction of solids with particle diameter of dp Total mass of particles Slope of the log scale of pressure gradient versus velocity of a stratified flow Pressure loss Wetted perimeter Flow rate Flow rate in the lower layer of the two-layer model Flow rate in the upper layer of the two-layer model Inner diameter of a pipe Local radius for a point in the flow Reynolds number Reynolds number of the mixture Reynolds number of the liquid carrier Hydraulic radius Cross-sectional area of the bed divided by the bed width Specific gravity of the solids Specific gravity of liquid Specific gravity of slurry mixture Average velocity Friction velocity Critical friction velocity at which the solids start depositing Friction velocity at deposition for limiting case of infinite dilution Velocity Velocity in the bottom layer Deposition velocity (also called V3)

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Vf VL Vm Vmin VR Vs Vt VU V1 V2 V3 V4 V50 WP y Z

4.65

Velocity of carrier fluid Velocity of the lower layer in the two-layer model of Shook and Roco Mean velocity of a mixture Velocity at minimum pressure drop Ratio of solid volume concentration of solids to liquid concentration Settling velocity in the stratified flow model Terminal velocity of the solid particle Velocity of the upper layer in the two-layer model of Shook and Roco Velocity for which flow with a stationary bed is started Velocity for which flow with a moving bed is started Deposition velocity or velocity above which solids start to move Velocity above which all solids move as a pseudohomogeneous mixture Velocity at 50% stratification Wetted perimeter Distance from the lower boundary (pg 33) Nondimensional parameter to express difference between friction losses due to the slurry and an equivalent volume of water

Subscripts B Bottom layer bed Due to the moving bed i Fraction i m Mixture U Upper layer Greek letters  Constant of proportionality Roughness Angle from the vertical starting at the lowest quadrant point ␣ The angle from the horizontal Pressure loss factor r The angle of repose of solid particles L Density of liquid carrier m Density of the slurry mixture s Density of solid sediments U Density of suspended fines and carrier liquid in the upper layer of the two-layer model W Density of water B Shear stress for the lower layer in the two-layer model U Shear stress for the upper layer in the two-layer model ⌫ factor used to compute ratio of concentrations in the two-layer model Factor to determine speed at 50% stratification in the Wilson model L Dynamic viscosity of the carrier liquid m Dynamic viscosity of the slurry mixture p Mechanical friction coefficient s Granular stress of the solid particles Kinematic viscosity of water L Viscosity of carrier liquid at equal volume M Viscosity of slurry mixture Product of the index number and the volumetric concentration s Coefficient of static friction of the solid particles against the wall of the pipe

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4-15 REFERENCES Acaroglu, E. R. 1968. Sediment Transport in Conveyance Systems. Ph.D. dissertation, Cornell University. Acaroglu, E. R., and W. Graf. 1968. Designing conveyance systems for solid–liquid flows. Paper presented at the International Symposium on Solid–Liquid Flow in Pipes, March 3–7, University of Pennsylvania, Philadelphia. Babcock, H. A. 1967. Head losses in pipeline transportation of solids. Paper presented at First World Dredging Conference, WODCON I, the Netherlands. Babcock, H. A. 1968. Heterogeneous flow of heterogeneous solids. Paper presented at International Symposium on Solid Liquid Flow in Pipes, March 3–7, University of Pennsylvania, Philadelphia. Babcock, H. A. 1971. Heterogeneous flow of heterogeneous solids. In I. Zandi (Ed.), Advances in Solid–Liquid Flows in Pipes and its Applications. pp. 125–148. Oxford: Pergamon Press. Bagnold, R. A. 1954. Gravity-free dispersion of large spheres in a Newtonian fluid under shear. Proc. Royal Soc. A, 225, 49–63. Bagnold, R. A. 1955. Some flume experiments on large grains but little denser than the transporting fluid, and their implications. Proc. Inst. Civ. Eng., 4, 3, 174–205. Bagnold, R. A. 1957. The flow of cohesionless grains in fluids. Phil. Trans. Roy. Soc., 249, 235–297. Blatch, N. S. 1906. Water filtration at Washington, DC, discussion trans. Amer. Soc. Civ. Eng. 57, 400–408. Charles, M. E., and G. S. Stevens. 1972. The pipeline flow of slurries—transitional velocities. Paper presented at the Second International Conference on Hydraulic Transport of Solids and Pipes. Second conference of the British Hydromechanic Research Association. Cranfield, England. Churchill, S. W. 1977. Friction factor equation spans all fluid flow regimes. Chemical Engineering 84, no. 7: 91–92. Craven, J. P., and H. H. Ambrose. 1953. The transportation of sand in pipes. Engineering Bulletin (University of Iowa), 34, 67–88. Durand R. 1953. Basic Relationship of the transportation of solids in experimental research. Proc. of the International Association for Hydraulic Research—University of Minnesota, September 1953. Durand, R., and E. Condolios. 1952. Experimental investigation of the transport of solids in pipes. Paper presented at Deuxieme Journée de l’hydraulique, Societé Hydrotechnique de France. Durand, R., and R. Gilbert. 1960. Transport hydraulique et refoulement des mixtures en conduites. Transactions École des Ponts et Chaussees, 130, 3–4. Einstein, H. A., and W. H. Graf. 1966. Loop systems for measuring sand–water mixtures. Journal of Hydraulic Division, Am. Soc. Civ. Eng. 92, HY1, paper 4608, 1–12. Ellis, H. S., and G. F. Round. 1963. Laboratory studies on the flow of nickel–water suspensions. Canadian Mining and Metallurgical Bulletin, 56, 773–781. Faddick R. R. 1982. Ship Loading Coarse Coal Slurries. In The 8th International Conference on Hydraulic Transport of Solids in Pipes, Johannesburg, South Africa. Cranfield, UK: BHRA Group. Gaessler, H. 1967. Experimentelle und Theoretische Untersuchungen uber die Stromungsvorgange Beim Transport von Festoffen in Flassigkeiten durch Horizontale Rohrleitungen. Doctoral thesis. Technische Hochschule, Karlsruhe, Germany. Quoted in G. W. Govier and K. Aziz, The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold Co., 1972, pp. 668 –670. Geller, L. B, and W. M. Gray. 1986. Selected Theoretical Studies Made in Conjunction with the Joint Canada/FRG Research Project on Coarse Slurry, Short Distance Pipeline. CANMET SP 8616 E–Government of Canada Publications Gillies, R. G. J. Schaan, R. J. Sumner, M. J. McKibben, and C. A. Shook. 1999. Deposition velocities for Newtonian slurries in turbulent flows. Paper presented at the Engineering Foundation Conference, Oahu, HI. Submitted for publication in the Canadian J. Chem. Eng. Reference cited by Saskatchewan Research Council (2000). Slurry pipeline course handout. Govier, G. W., and K. Aziz. 1972. The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold.

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Hayden, J. W., and T. E. Stelson. 1968. Hydraulic conveyance of solids in pipes. Paper presented at the International Symposium on Solid–Liquid Flow in Pipes, March 3–7, University of Pennsylvania, Philadelphia. Herbich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill. Hill, R. A., P. E. Snock, and R. L. Gandhi. 1986. Hydraulic transport of solids. In The Pump Handbook, 2nd ed. Edited by I. J. Karassik et al. New York: McGraw-Hill. Hsu, S. T., A. V. Beken, L. Landweber, and J. F. Kennedy. 1971. The distribution of suspended sediment in turbulent flows in circular pipes. Paper presented at the American Institute of Chemical Engeering Conference on Solids Transport in Slurries, Atlantic City, NJ. Hunt, I. N. 1969. Turbulent transport of heterogeneous sediment. Quarterly Journal Mechanics and App. Maths., 22, 234–246. Ippen, A. T. 1971. A new look at sedimentation in turbulent streams. Journal of Boston Soc. Civil Eng., 58, 3, 131–163. Ismail, H. M. 1952. Turbulent transfer mechanism and suspended sediment in closed channels. Trans. Amer. Soc. Chem. Eng., 117, 2500, 409–447. Kao, D. T. Y., and L. Y. Hwang. 1979. Critical slope for slurry pipelines. Paper presented at the Hydrotransport 6 Conference of the British Hydromechanical Research Association, Cranfield, England. Khan, A. R., and J. F. Richardson. 1996. Comparison of coarse slurry pipeline models. In Proceedings of Hydrotransport 13, pp. 259–281. Cranfield, UK: BHR Group. Lazarus, J. H. 1989. Mixed-regime slurries in pipelines. I. Mechanistic model. Journal of Hydraulic Engineering, ASCE, 115, 11, 1496–1509. Lazarus, J. H. & Cooke, R. 1993. Generalised mechanistic model for heterogeneous flow in a nonNewtonian vehicle. In Proceedings of Hydrotransport 12, pp. 671–690. Cranfield, UK: BHR Group. Matsouk V. 1996. Internal structure of slurry flow in inclined pipe—Experiments and mechanistic modeling. In Proceedings of Hydrotransport 13, pp. 187–210. Cranfield, UK: BHR Group. Newitt, D. M., J. F. Richardson, M. Abbott, and R. B. Turtle. 1955. Hydraulic conveying of solids in horizontal pipes. Trans Inst. of Chem. Eng., 33, 93–113. Newitt, D. M., J. R. Richardson, and J. B. Glibbon. 1961. Hydraulic conveying of solids in vertical pipes. Trans. Inst. of Chem. Eng., 39, 93–100. Newitt, D. M., J. R. Richardson, and C. A. Shook (Eds.). 1962. Symposium on Interaction between Fluids and Particles, London. London: Institution of Chemical Engineers. Raj, R. S. 1972. Pressure loss in hydraulic transport of solids in inclined pipes. Paper presented at Hydrotransport 2, Coventry, England. Saskatchewan Research Council. 2000. Slurry Pipeline Course—SRC Pipe Flow Technology Center, Saskatoon, Canada, May 15–16. Schiller, R. E., and P. E. Herbich. 1991. Sediment transport in pipes. In Handbook of Dredging, Edited by P. E. Herbich. New York: McGraw-Hill. Shen, H. W. 1970. Sediment transportation mechanism—Transportation of sediments in pipes. Journal Hydraulics Division Am. Soc. Civ. Eng., 96, 1503–1538. Shook, C. A. 1981. Lead Agency Report for MTCM Cooperative Research Project. Report E-725-6C-81. Report prepared for the Saskatchewan Research Council, Saskatchewan, Canada. Shook, C. A., J. R. Rollins, and G. S. Vassie. 1974. Sliding in Inclined Slurry Pipelines and Shutdown. Report IX. Report prepared for the Saskatchewan Research Council, Saskatchewan, Canada. Shook, C. A., and M. C. Roco. 1991. The two layer model. In Slurry Flow: Principles and Practice. Newton, MA: Butterworth-Heinemann. Spells, K. E. 1955. Correlation for use in transport of aqueous suspensions of fine solids through pipes. Trans Inst. Chem. Eng., 33, 79–84. Thomas, D. G. 1963. Transport characteristics of suspensions: Relation of hindered settling floc characteristics to rheological parameters. Am. Inst. Chem. Eng. Journal, 9, 310–319. Thomas, D. G. 1962. Transport Characteristics of suspensions, Part IV. Am. Inst. Chem Eng. Journal, 8, 373–378. Thomas, D. G. 1964. Transport characteristics of suspensions, Part IX. Am Inst. Chem. Eng. Journal, 10, 303–308. Traynis, V. V. 1970.Parameters and Flow Regimes for Hydraulic Transport of Coal by Pipelines,

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Translated and Edited by W. C. Cooley and R. R. Faddick. Terraspace Inc. report, April 1970, pp 17–19. Turian, R. M., and T. Yuan. 1977. Flow of slurries and pipelines. Am. Inst. Chem. Eng. Journal, 23, 3, 232. Vallentine, H. R.1955. Transportation of Sands in Pipelines. Commonwealth Engineer (Australia), April, 349–355. Warman International Inc. 1990. Slurry Handbook. Madison, WI: Warman International Inc. Wasp, E. J. et al. 1970. Deposition velocities, transition velocities, and spatial distribution of solids in slurry pipelines. Paper read at 1st International Conference on Hydraulic Transportation of Solids in Pipes, Cranfield, England. Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid–Liquid Flow—Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans-Tech Publications. Wicks, M. 1971. Transportation of solids of low concentrations in horizontal pipes. In Advances in Solid–Liquid Flow in Pipes and Its application, edited by I. Zandi. New York: Pergamon Press. Wilson, K. C. 1970. Slip point of beds in solid–liquid pipe flow. Am. Soc. Chem. Eng. Hydraulic Division, no. HY1, paper 6992: 1–12. Wilson, K. C. 1991. Pipeline Design for Settling Slurries. In Slurry Handling, edited by N. P. Brown and N. I. Heywood. New York: Elsevier Applied Sciences. Wilson, K. C., and D. G. Judge. 1976. Paper presented at the International Symposium on Freight Pipelines, Washington, DC Wilson, K. C., G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. New York: Elsevier Applied Sciences. Wilson, W. E. 1942. Mechanics of flow of non-colloidal solids. Trans. Am. Soc. of Chem. Eng., 107, 1576. Wood D. J. 1979. Pressure gradient requirements for re-establishment of slurry flow. In Sixth International Conference on Hydraulic Transport of Solids in Pipes, p. 217. Cranfield, UK: BHRA Group. Worster, R. C., and D. E. Denny. 1955. Hydraulic transport of solid materials in pipes. Proceedings of the Institute of Mechanical Engineers (UK), 38, 230–234. Zandi, I., and G. Govatos. 1967. Heterogeneous flow of solids in pipeline. Proceedings of the Hydraulic Division of Am. Soc. Civ. Eng., 93, no. HY3, paper 5244, 145–159. Zandi, I. 1971. Hydraulic transport of bulky materials. In Advances in Solid–Liquid Flow in Pipes and Its applications, pp. 1–38, I. Zandi (Ed.). Oxford: Pergamon Press.

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CHAPTER 5

HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

5-0 INTRODUCTION The rheology of non-Newtonian flows and homogeneous flows was examined in detail in Chapter 3. With modern methods of grinding, the size of particles can be reduced to values smaller than 70 m (0.0028 in). As shown in Table 3-8, a wide range of metal concentrates and tailings are pumped at a sufficiently high concentration with a small enough particle size for the mixture to behave as a Bingham plastic. There are other clays and slurries that may behave as pseudoplastics. In certain circuits of oil sands processing plants with tar at the level of flotation, the slurry may behave as a thixotropic mixture. With non-Newtonian flows, it is important to take into account the rheology, yield stress, power law exponent, coefficient, and even the time response. Different models have evolved over the years for Bingham and pseudoplastic slurries. Some of these models put more emphasis on the laminar flow regime, in which roughness effects are negligible. Some other models extend to the transition and turbulent regimes. The effect of pipe roughness on friction loss factors in non-Newtonian flows remains a topic worth investigating and researching. For thixotropic slurries, methods are used to predict start-up pressure after a shutdown of the pipeline, and the time required to clean the conduit of gelled material before resuming pumping. The equations for friction factors of non-Newtonian fluids are fairly complex and require iteration and data on the rheology. Throughout the years, different authors have developed equations for “modified” Reynolds numbers, Hedstrom numbers, etc. In this chapter, equations developed by different authors will be reviewed. Through worked examples, the reader will be shown methods of calculating the friction factor. It is the purpose of this chapter to focus on the engineering side of the problem. The reader is strongly advised to read through Chapter 3, to gain the fundamentals for this chapter. For the practical engineer, who is more concerned with the actual design of a pipeline or a pumping system, the numerous and different definitions of the so-called “modified Reynolds number” can be very confusing. Every few years, an author develops a new definition of the “modified Reynolds number” and claims to have found a relationship with the friction factor. The cautious approach for an engineer is to assume that such a universal relationship is illusive, and for every type of slurry there may be a model to use. 5.1

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Sometimes the use of two different methods can yield differences of 25–35% in the estimation of the friction factor. One important difference with the slurries described in Chapter 4 is that most nonNewtonian slurries do not exhibit the stratification of solids, which is usual with coarse particles.

5-1 FRICTION LOSSES FOR BINGHAM PLASTICS Bingham plastics were defined in Chapter 3. They are characterized by a yield stress that must be overcome to start the flow. Examples of Bingham plastics are listed in Table 3-9

5-1-1 Start-up Pressure The start-up pressure for pumping a Bingham plastic is expressed in terms of the true yield stress: 40 L Pst = ᎏ Di

(5-1)

The start-up pressure per unit length is obtained by dividing Equation 5-1 by the length of the pipeline: 40 Pst ᎏ=ᎏ Di L Examples for the starting pressure per unit length for slurries in 3⬙, 6⬙, 12⬙, and 18⬙ pipes are presented in Table 5-1. The Reynolds Number for a Bingham slurry is expressed as: DiVm ReB = ᎏ

(5-2)

The coefficient of rigidity was defined in Equation 3-29 as

0 = ᎏ + ⬁ (d␥/dt)

(3-29)

For Bingham slurries a nondimensional coefficient is defined as the plasticity number:

0Di PL = ᎏ V

(5-3)

The Hedstrom number is the product of the plasticity number and the Reynolds number and is calculated as D 2i m0 He = ᎏ 2

(5-4)

Table 5-2 shows examples of Bingham slurry mixtures and the magnitude of the Hedstrom number for flows in rubber-lined 6⬙ (150 mm NB), 12⬙ (300 mm NB) and 18⬙ (450

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TABLE 5-1 Starting Pressure per Unit Length for Certain Slurries in Pa/m

5.3

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 Aqueous clay suspension III Aqueous clay suspension V 21.4% Bauxite 18% Iron oxide 7.5 % Kaolin 58% Limestone 52.4% Fine liminite 14% Sewage sludge

3⬙ Pipe Sch 40, rubber lined, ID = 2.560⬙ (65 mm)

6⬙ Pipe Sch 40, rubber lined, ID = 5.557⬙ (141.2 mm)

12⬙ Pipe S, rubber lined, ID = 11.500⬙ (292 mm)

18⬙ Pipe S, rubber lined, ID = 16.500⬙ (419 mm)

6.86

234

108

52

37

Particle size, d50

Density, kg/m3

Yield stress, Pa

92% under 74 m

1520

3.8

80% under 1 m

1280

59

13.1

3631

1671

808

567

80% under 1 m

1207

25

6.7

1538

708

342

240

80% under 1 m

1149

7.8

4.0

480

221

107

75

1520 1440 1360 1163 1170 1103 1530 2435 1060

34.5 20 6.65 8.5 0.78 7.5 2.5 30 3.1

44.7 32.8 19.4 4.1 4.5 5 15 16 24.5

2123 1231 409 523 49 462 154 1846 191

977 567 188 241 23 213 71 849 88

473 274 91 116 11 103 34 411 43

332 192 64 82 7.7 72 24 209 30

<200 m <50 m Colloidal <160 m <50 m

*References on rheology data were presented in Table 3-9.

Page 5.3

Slurry

Coefficient of rigidity, mPa · s (cP)

9:17 AM

Starting pressure per unit length (Pa/m)*

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TABLE 5-2 Examples of Hedstrom Numbers for Bingham Slurries in 3⬙, 6⬙, 12⬙, and 18⬙ Rubber-Lined Pipes Hedstrom number*

Density, kg/m3

6⬙ Pipe Sch 40, rubber lined, ID = 5.557⬙ (141.2 mm)

12⬙ Pipe S, rubber lined, ID = 11.500⬙ (292 mm)

18⬙ Pipe S, rubber lined, ID = 16.500⬙ (419 mm)

518,568

2,445,272

1.046 × 107

2.124 × 107

92% under 74 m

1520

80% under 1 m

1280

59

13.1

1,859,285

8,773,821

3.752 × 107

7.616 × 107

80% under 1 m

1207

25

6.7

2,840,039

1.472 × 107

6.078 × 107

1.234 × 108

80% under 1 m

1149

7.8

4.0

2,366,581

1.117 × 107

4.776 × 107

9.694 × 107

1520 1440 1360 1163 1170 1103 1530 2435 1060

34.5 20 6.65 8.5 0.78 7.5 2.5 30 3.1

44.7 32.8 19.4 4.1 4.5 5 15 16 24.5

110,885 113,102 101,528 2,484,607 190,407 1,398,052 71,825 1,205,610 23,129

523,259 533,721 479,100 1.172 × 107 898,514 6,597,299 338,937 5,689,180 109,145

2,237,759 2,282,499 2,048,910 5.014 × 107 3,842,564 2.821 × 107 1,449,488 2.433 × 107 466,768

4,541,865 4,632,671 4,158,568 1.017 × 108 7,799,056 5.726 × 107 2,941,952 4.938 × 107 947,375

<200 m <50 m Colloidal <160 m <50 m

*References on rheology data were presented in Table 3-9.

3.8

6.86

Page 5.4

5.4

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 Aqueous clay suspension III Aqueous clay suspension V 21.4% Bauxite 18% Iron oxide 7.5 % Kaolin clay 58% Limestone 52.4% Fine liminite 14% Sewage sludge

Particle size, d50

3⬙ Pipe Sch 40, rubber lined, ID = 2.560⬙ (65 mm)

9:17 AM

Slurry

Yield stress, Pa

Coefficient of rigidity, mPa · s (cP)

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5.5

mm) pipes. Laminar flows in large pipes are considered for certain slurries at relatively high weight concentration (⬇ 60%), or certain high-energy mixtures (e.g., crude oil with fine and ultrafine coal). Example 5-1 A slurry consists of a clay and water mixture. It is tested and classified as a Bingham mixture with a yield stress of 17 Pa. The pipe inner diameter is 63 mm. The pipe length is 6500 m. Determine the start-up pressure, ignoring any static head. Solution in SI Units Using Equation 5-1: Pst = 40L/Di Pst = 4 × 17 × 6500/0.063 = 7,015,873 (1018 psi) Solution in US units Pst = 40L/Di

0 = 17 Pa/6895 = 2.465 × 10–3 psi L = 6500 m/0.0254 = 255,905 in Di = 63/25.4 = 2.48 in 4 × 2.465 × 10–3 × 255,905 Start-up pressure Pst = ᎏᎏᎏ = 1017.6 psi 2.48 5-1-2 Friction Factor in Laminar Regime Buckingham (1921) was the first to develop an equation for a fully developed laminar flow. This equation has since been modified by Hedstrom (1952) and others to express the friction factor as a function of the Hedstrom and Reynolds numbers: He He4 fNL 1 ᎏ = ᎏ – ᎏ2 + ᎏ 3 ReB 6Re B 3 f NL Re B8 16

(5-5)

or

冤

He4 16 He fNL = ᎏ 1 + ᎏ – ᎏ 3 ReB 6ReB 3 f NL ReB7

冥

(5-6)

This would occur below the critical Reynolds number or in transition between laminar and turbulent flow. The last term between brackets in the equation is often considered second order. Example 5-2 A Bingham slurry with a concentration of 50% by weight is tested in a plastic-lined pipe with an inner diameter of 2.5 in. The tests indicate a yield stress of 1.5 Pa, a slurry mixture specific gravity of 1.54, and a coefficient of rigidity of 0.4 Pa · s. Assuming a flow speed of 4 ft/s in a laminar regime, determine the friction factor by Buckingham’s equation.

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Solution in SI Units Pipe ID = 2.5⬙ or 63.5 mm Speed = 4 ft/s or 1.219 m/s Reynolds number = 0.0635 × 1.219 × 1540/0.4 = 298 Hedstrom number = (0.0635)2 × 1540 × 1.5/(0.42) = 916.8 Using Equation 5-6 and ignoring the higher-order terms: fNL ⬇ [16/ReB][1 + He/(6ReB)] = 0.0812 This a fanning factor, so the Darcy friction factor is 0.3248. Figure 5-1 presents values of the friction factor versus the Reynolds number for a wide range of Hedstrom numbers from 0 to 109. The transition between laminar and turbulent flows is shown by the dotted curve of the critical Reynolds number. From the point of view of engineering, the most practical flows in pipes are in a range of Hedstrom numbers between 105 and 108, as shown in Table 5-2. The transition to turbulent flow will be examined in more detail throughout this chapter. To appreciate the practical magnitude of the laminar friction factor, Table 5-3 presents cases at a speed of 1 m/s (3.3 ft/sec) for rubber-lined pipes in sizes of 3⬙ (80 mm N.B.), 6⬙ (150 mm N.B), 12⬙ (300 mm N.B.), and 18⬙ (450 mm N.B.). The Fanning friction factor based on the Buckingham equation is in the range of 0.001 to 0.15.

FIGURE 5-1 Friction factor versus Reynolds number and Hedstrom number. (From Hill R. A, P. E. Snoek, and R. L. Ghandi, Hydraulic transport of solids, in The Pump Handbook, 2nd ed., I. J. Karassik et al. (Eds.), New York: McGraw-Hill. Reproduced by permission.)

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TABLE 5-3 Friction Factor at a Speed of 1 m/s (3.3 ft/sec) for Bingham Mixtures in Rubber-Lined Pipes*

5.7

54.3% Aqueous suspension of cement rock Flocculated aqueous China clay suspension No. 1 Flocculated aqueous China clay suspension No. 4 Flocculated aqueous China suspension No. 6 Aqueous clay suspension I Aqueous clay suspension III Aqueous clay suspension V 21.4% Bauxite 18% Iron oxide 7.5 % Kaolin clay 58% Limestone 52.4% Fine liminite 14% Sewage sludge

3⬙ Pipe Sch 40, rubber lined, ID = 2.560⬙ (65 mm)

6⬙ Pipe Sch 40, rubber lined, ID = 5.557⬙ (141.2 mm)

12⬙ Pipe S, rubber lined, ID = 11.500⬙ (292 mm)

18⬙ Pipe S, rubber lined, ID = 16.500⬙ (419 mm)

0.00778

0.00718

0.00691

0.00684

Particle size, d50

Density, kg/m3

Yield stress, Pa

92% under 74 m

1520

3.8

80% under 1 m

1280

59

13.1

0.12541

0.12407

0.12348

0.12331

80% under 1 m

1207

25

6.7

0.00566

0.05586

0.05554

0.05545

80% under 1 m

1149

7.8

4.0

0.0186

0.0185

0.0183

0.0182

1520 1440 1360 1163 1170 1103 1530 2435 1060

34.5 20 6.65 8.5 0.78 7.5 2.5 30 3.1

44.7 32.8 19.4 4.1 4.5 5 15 16 24.5

0.0677 0.0426 0.01655 0.0204 0.0272 0.01925 0.0046 0.0345 0.0135

0.0639 0.0396 0.0146 0.0199 0.0221 0.01864 0.00447 0.0336 0.0104

0.0621 0.0382 0.01382 0.0197 0.0199 0.01838 0.00441 0.0332 0.0009

0.0616 0.0379 0.01359 0.0196 0.0193 0.0183 0.00439 0.0331 0.00087

<200 m <50 m Colloidal <160 m <50 m

*References on rheology data were presented in Table 3-9.

6.86

Page 5.7

Slurry

Coefficient of rigidity, mPa · s (cP)

9:17 AM

Fanning friction factor fN at a speed of 1 m/s (3.3 ft/sec)

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CHAPTER FIVE

5-1-3 Transition to Turbulent Flow Regime Hanks and Pratt (1967) analyzed extensive experimental data on critical Reynolds numbers and proposed a relationship between the Reynolds and Hedstrom Numbers at transition as

冢

冣

He 4 1 ReBc = ᎏ 1 – ᎏ xc + ᎏ x 4c xc 3 3

(5-7)

where xc = 0/wc = the ratio of the yield stress to the wall shear stress at the transition from laminar to turbulent flow. At the transition, xc He = 16,800 ᎏ3 (1 – xc)

(5-8)

Figure 5-2 plots the magnitude of critical Reynolds number versus the Hedstrom number for a number of Bingham slurries. Example 5-3 Using Figure 5-2, determine the critical Reynolds number for a clay slurry at a Hedstrom number of 10,000. Solution From Figure 5-2 Rec ⬇ 3700. Wasp et al. (1977) defined the effective pipeline viscosity for laminar flow as

w e = ᎏ 8V/Di

Critical Reynolds Number

10

10

10

(5-9)

5

4

3

10

3

10

4

10

5

10

6

10

8

Hedstrom Number

FIGURE 5-2 The critical Reynolds number versus the Hedstrom number for flow in pipes. (After Hanks, R. W., and D. R. Pratt. 1967.)

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HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

In Chapter 3, in the description of the capillary tube test, the Buckingham equation was derived. When ignoring the fourth-power term, the equation reduces to

w ⬇ 8V/Di + 4/3 0

(3-62)

or 8V/Di + 4/3 0 DI0 e ⬇ ᎏᎏ ⬇ 1 + ᎏ 8V/Di 6V

冤

冥

(5-10)

In many laminar flow pipelines the term Di0/(6V) is much smaller than unity. In such cases Equation 5-10 is then simplified to

e ⬇ Di0/(6V)

(5-11)

At transition, Wasp et al. (1977) point out that this equation can be approximated to

e ⬇ [Di0/(6VTR)] using the effective pipeline viscosity, the critical Reynolds number is expressed as 2 /0 ReBC = 6V TR

At high shear rate for Bingham plastics, e ⬇ ⬁ ⬇ (see Figure 3-10): VTR =

冪莦 ReBC0 ᎏ 6

(5-12)

The Wasp method is based on numerous assumptions, and usually terminates by assuming that the transition Reynolds number is in the range of 2000 to 3000 for numerous Bingham slurries. In some respects, it is a useful tool for hand calculations. A more widely accepted method since the mid 1980s is to compute the transition velocity that was proposed by Wilson and Thomas, to be discussed in Section 5-4-3. It is then assumed that the transition from laminar to turbulent flow occurs when the Wilson–Thomas and the Buckingham equations intersect.

5-1-4 Friction Factor in the Turbulent Flow Regime Hanks and Dadia (1971) developed a semiempirical equation for the turbulent flow of Bingham slurries in closed conduits. These equations were modified by Darby (1981) and Darby et al. (1992) to give a friction factor for the turbulent regime as fNT = 10aReBb

(5-13)

where a = –1.47[1 + 0.146 exp(–2.9 × 10–5He)] b = –0.193 The values of the parameters a and b are based on empirical data for closed conduits. Bingham slurries do not exhibit a sudden change from laminar to turbulent flow. Darby et al. (1992) reviewed the work of previous authors and proposed to combine the laminar and turbulent fanning friction factors into the following equation: m m (1/m) fN = ( f NL + f NT )

(5-14)

m = 1.7 + 40,000/ReB.

(5-15)

where

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CHAPTER FIVE

Equations 5-12 and 5-13 do not account for pipe roughness and are essentially for very smooth pipes such as glass and high-density polyethylene pipes. Studies have been published in the past on flow of Bingham plastics in the laminar regime, where roughness effects are neglected. Thomas and Wilson (1987) have even argued that the non-Newtonian fluids form a viscous sublayer in the boundary layer, that is usually thicker than with Newtonian flows. This viscous sublayer is considered to suppress the contribution of roughness and, in effect, Wilson and Thomas do support the assumptions made by Darby (1981). Many concentrates are pumped at high volumetric concentrations but also at a relatively moderate speed of 2–2.5 m/s (6.6–8.2 ft/s). We will, however, discuss the effects of roughness in Section 5-7. Example 5-4 In Figure 3-9, the relationship between the Bingham plastic apparent viscosity and the shear rate was presented as

= (d␥/d) + ⬁ at high shear rate ⬇ ⬁. Considering an aqueous clay suspension III (Table 3-9) with a mixture density of 1440 kg/m3 (SG = 1.44), a yield stress of 20 Pa, and a coefficient of rigidity of 32.8 mPa · s as reported by Caldwell and Babbitt (see references of Chapter 3), determine the friction factor for flow at 2.5 m/s in a 63 mm ID pipe, using the Darby method. Determine also the pressure drop per unit length. Solution SI Units Reynolds number:

VDI 1440 × 2.5 × 0.063 Re = ᎏ = ᎏᎏ = 6914.6 0.0328 Hedstrom number: (0.063)2 × 1440 × 20 D20 ᎏᎏᎏ He = ᎏ = = 106,249 2 (0.0328)2 m = 1.7 + 40,000/Re m = 1.7 + 40,000/6914.6 = 7.49 a = –1.47[1 + 0.146 exp(–2.9 × 10–5He)] = –1.479 fT = 10aRe–0.193 fT = 10–1.479 × 6914.6–0.193 = 0.006

冢

冣

冢

Re 16 Re4 16 Re ⬇ ᎏ 1+ ᎏ fL = ᎏ 1 + ᎏ – ᎏ 7 3 3f L He Re 6He Re 6He

冢

冣

16 6914.6 fL = ᎏ 1 + ᎏᎏ = 0.00234 6914.6 6 × 106,249 fn = ( f Lm + f Tm)1/m Therefore, fn = (0.002347.49 + 0.0067.49)1/7.49 = 0.006 is the fanning friction factor

冣

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HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

5.11

Darcy factor: fD = 4fN = 0.006 × 4 = 0.024 Pressure drop per unit length: dP/dz = fDV 2/(2Di) dP/dz = 1440 × 0.024 × 2.52/(2 × 0.063) = 1,714 Pa/m (0.816 psi/ft)

5-2 FRICTION LOSSES FOR PSEUDOPLASTICS Pseudoplastic rheology was extensively discussed in Chapter 3, Section 3.4-2. Examples of pseudoplastics are listed in Table 3-10.

5-2-1 Laminar Flow A number of models have been developed for pseudoplastic flows. These treat the fluid as a continuum. 5-2-1-1 The Rabinowitsch–Mooney Relations Herzog and Weissenburg (1928) developed an equation for laminar time-independent, viscous non-Newtonian flows. It was subject to further refinements by Rabinowitsch (1929) and Mooney (1931). For a circular pipe, a relationship is established between the shear stress and the absolute value of the rate of shear ␥ = –du/dr f() = –du/dr Rabinowitsch and Mooney derived a general relationship for the shear rate at the wall:

冢 冣

du – ᎏ dr

w

8V 1 + 3 =ᎏ ᎏ DI 4

冢

冣

(5-16)

where d[ln(Di ⌬P/4L)] = ᎏᎏ d[ln(8V/D)]

(5-17)

5-2-1-2 The Metzner and Reed Approach Metzner and Reed (1955) developed an equation for the Reynolds number in laminar flow as D IV 2– ReMR = ᎏ ␥ where is defined by Equation 5-16 and

␥ = K⬘8(–1) ␥ = gcK⬘8

(–1)

in SI units in USCS units

(5-18)

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CHAPTER FIVE

冢

冣

In SI units

(5-19a)

冢

冣

In USCS units

(5-19b)

1 + 3n K⬘ = K ᎏ 4n 1 + 3n K⬘ = K\gc ᎏ 4n

n

n

The fanning friction factor is then expressed in the laminar flow regime in the conventional manner but using the modified Reynolds number: 16 fNL = ᎏ ReMR

(5-20)

Example 5-5 The pressure drop in a 80 mm ID pipe is to be determined for a slurry with S.G. = 1.37. The power law exponent had been previously determined to be 0.4, and the power law factor K as 16 dynes-spcn/cm2. The speed of the flow is 1.35 m/s. Use the Metzner and Reed approach to calculate the friction factor, assuming a shear rate of 600 s–1. Solution From Equation 5-15: 8V 1 + 3 –600 = ᎏ ᎏ Di 4

冢

冣

8 × 1.35 1 + 3 –600 = ᎏ ᎏ 0.08 4

冢

冣

–4.45 = (1 + 3)/4 –1.11 = 1/ + 3

= 0.529 DV 2– ReMR = ᎏ ␥

冢

1 + 3n K⬘ = K ᎏ 4n

= 18.174 冣 = 16冢 ᎏ 1.6 冣 n

1 + 1.2

n

␥ = K⬘8–1 = 18.174 × 8–0.47 = 6.84 0.080.529 × 1.351.471 × 1370 ReMR = ᎏᎏᎏ 6.84 ReMR = 81.87 16 fn = ᎏ = 0.195 ReMR The Metzner and Reed approach has become a classical method of dealing with time-independent non-Newtonian fluids. It has been extended to Bingham slurries but the opinion of the author is that this approach is fairly difficult to use for Bingham slur-

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HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

5.13

ries, and a more practical method would be the Darby approach, described in Section 51-4. The Metzner and Reed approach requires the engineer to assume a value of the shear stress at the wall w to calculate x. Such assumptions are very difficult to make for engineers outside a research lab. It may be more practical to send samples of the slurry to a rheology lab and to go from plots of yield stress versus weight concentration, as well as from plots of viscosity versus weight concentration and shear rates to a more straightforward computation of friction factors (Figure 5-3). 5.2.1.3 The Tomita Method Tomita (1959) defined a fanning friction factor for power law fluids as 2Di⌬P 1 + 2n fPL = ᎏ2 ᎏ 3LmV 1 + 3n

冢

冢

6 [1/n + 3]1–n RePL = ᎏn ᎏᎏ 2 1/n + 2

冣

D inV 2–nm

冣冢 ᎏᎏ 冣 K

(5-21)

(5-22)

In the laminar flow regimes:

FIGURE 5-3 Friction factor versus Reynolds number for power law factors. [From Hill R. A, P. E. Snoek, and R. L. Ghandi, Hydraulic transport of solids, in The Pump Handbook, 2nd ed., I. J. Karassik et al. (Eds.), New York: McGraw-Hill. Reproduced by permission.]

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CHAPTER FIVE

16 fPL = ᎏ Remod

(5-23)

5-2-1-3 Heywood Method Heywood (1991) proposed to define a modified Reynolds number for pseudoplastics as

mVDi 4n Remod = ᎏ ᎏ K 1 + 3n

冢

冣 冢ᎏ 8V 冣 Di

n

n–1

(5-24)

where K and n are the consistency coefficient and flow behavior indexes for pseudoplastic flows previously defined in Chapter 3. In the laminar flow regime, Heywood (1991) used the conventional method of defining the fanning friction factor in terms of the Reynolds number in the laminar flow regime as previously discussed in Chapter 2, or fNPL = 16/Remod

(5-25)

The effective pipeline viscosity is expressed as

冢

4n e = K ᎏ 1 + 3n

冣 冢ᎏ D 冣 n

8V

n–1

(5-26)

i

5-2-2 Transition Flow Regime Ryan and Johnson (1959) defined a critical Reynolds number for purely viscous pseudoplastics as 6464n (n + 2)(n+2)/(n+1) Rec = ᎏᎏᎏ (1 + 3n)2

(5-27)

The friction factor at the transition from laminar to turbulent, flow called the critical friction factor is 1 (1 + 3n)2 fNc = ᎏ ᎏᎏ 404n (n + 2)(n+2)/(n+1)

(5-28)

Table 5-4 tabulates the critical Reynolds number and fanning friction factor versus the power factor “n.” The minimum friction factor is 0.0067 at n = 0.5. However, Heywood (1991) deducted from various test data that the minimum value for fNc = 0.004, which is even lower than the values indicated by Equation 5-28 (Figure 5.4).

5-2-3 Turbulent Flow Various equations have been developed over the years for turbulent flow of pseudoplastics in smooth pipes. These equations are based on empirical data and semitheoretical models. Using the modified Reynolds number as per Equation 5-17, Dodge and Metzner (1959) developed the following semitheoretical equation for turbulent flow:

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HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

TABLE 5-4 Critical Reynolds Number and Fanning Friction Factor versus the Flow Behavior Index “n” According to the Ryan and Johnson Method Flow behavior index “n”

Critical Reynolds number

Critical fanning friction factor, fNC

Flow behavior index “n”

Critical Reynolds number

Critical fanning friction factor, fNC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1577 2143 2345 2396 2381 2337 2280 2219

0.01015 0.00747 0.00682 0.00668 0.00672 0.00685 0.00702 0.0072

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

2158 2099 2043 1990 1941 1895 1852 1812

0.00741 0.00762 0.0783 0.00804 0.00824 0.00844 0.00864 0.00883

1 0.4 4 (1– /2) ᎏ=ᎏ log10[Remod f NT ]– ᎏ 0.75 1.2 兹f苶N苶T

(5-29)

2500 f NCR

( from equation 5-28)

0.008

2300

0.006

2100 Re CR

0.004 0.002 0.0 0.0

1900 1700

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1500

(from Equation 5-27)

0.010

Modified nritical Reynolds number

Although Equation 5-29 has been extensively used, it has its own limitations. Measuring the power exponent “n” in laminar flow tests and then trying to apply it to turbulent flows is asking for trouble, particularly for cases when n < 0.5. Heywood and Richardson (1978) showed that pumping flocculated clays yielded higher experimental values of friction coefficient than those predicted by Dodge and Metzner (1959), particularly when the value of “n” had been obtained at low shear stress. Note: Equation 5-20 does not incorporate the effects of roughness. Govier and Aziz (1972) indicated that Equation 5-20 gives excellent agreement between calculated and experimental data in the range of modified Reynolds numbers ReMR of 2900–36,000 and

Critical fanning friction factor

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Flow index "n" FIGURE 5-4 Values of critical Reynolds number and critical fanning factor versus the flow index “n” for pseudoplastic slurries based on Equation 5-27.

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modified power exponent s of 0.36–1.0. Equation 5-20 requires repeated iteration. The use of a personal computer is recommended. For power law fluids, Tomita (1959) extended his laminar flow model (discussed in Section 5.2.1.3) to turbulent flows in smooth pipes by applying Prandtl’s mixing length concept, and developed a different implicit equation: 1 ᎏ = 4 log10(RePL 兹苶 fP苶 LT) – 0.40 兹苶 fP苶 LT

(5-30)

where RePL and fPLT were already expressed for the laminar flow in Equations 5-21 and 522. Tomita’s equation was supported by 40 experimental data points on starch pastes and lime slurries. Irvine (1988) published the following equation: F⬘(n) fn = ᎏᎏ Re[1/(3n+1)] mod

(5-31)

where

冢

冣

8nn 2 ᎏᎏ F⬘(n) = ᎏ n–1 7n 8 7 (1 + 3n)n

1/(3n+1)

(5-32)

Example 5-6 At a volumetric concentration of 30%, a magnetite suspension has a power law coefficient K of 12 dynes-secn/cm2 and a power law exponent of 0.2. If the slurry is homogeneous and nonsettling at a speed of 1.5 m/s, determine the friction factor in a 101 mm ID pipe, at a slurry density of 1600 kg/m3. Solution From Equation 5-24, the modified Reynolds number is calculated as

冢

1600 × 1.5 × 0.101 4 × 0.2 Remod = ᎏᎏ ᎏᎏ 12 × 0.1 1 + 3 × 0.2

冣 冢ᎏ 8 × 1.5 冣 0.2

0.101

–0.8

= 202 × 0.5 × 45,697

Remod = 4615 It is necessary to check if the flow is turbulent. From Equation 5-27: 6464 n(n + 2)(n+2)/(n+1) Rec⬘ = ᎏᎏᎏ (1 + 3n)2 6464 × 0.2(2.2)(2.2/1.2) = 2143 Rec⬘ = ᎏᎏᎏ (1 + 0.6)2 Since Remod > Rec⬘, flow is therefore turbulent. Two different approaches will be used. Irvine Method From Equation 5-32:

冢

2 8 × 0.20.2 ᎏᎏ F⬘(n) = ᎏ –0.8 1.4 8 7 (1 + 0.6)0.2 From Equation 5-31, the fanning friction factor is

冣

1/1.6

= 0.862

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0.862 fn = ᎏ = 0.00442 46151/1.6 Tomita Method From Equation 5-22: RePL = 4615 × 80.8 × 20.2 × 6(1.6/0.2)0.8/2.8 = 316,457 From Equation 5-30: 1 ᎏ = 4 log10(316,457 × 兹苶 fP苶 LT) – 0.40 兹苶 fP苶 LT Iteration 1 starts by assuming at fPLT ⬇ 0.004 correction fPLT = 0.0035 Iteration 2 starts at fPLT ⬇ 0.0042 fPLT = 0.00352 Iteration 3 starts at fPLT ⬇ 0.0045 fPLT = 0.003498 Iteration 4 starts at fPLT ⬇ 0.0035 fPLT = 0.00359 So by theTomita method we obtain a friction factor of 0.0035. The Irvine method yields a friction factor 23% higher than the Tomita method. Both methods do not account for roughness of the pipe wall.

5.3 FRICTION LOSSES FOR YIELD PSEUDOPLASTICS Yield pseudoplastics were described extensively in Chapter 3. Examples were listed in Table 3.11.

5-3-1 The Hanks and Ricks Method In the laminar flow regime, Hanks and Ricks (1978), defined the fanning friction factor in terms of the modified Reynolds number: 16 fNPL = ᎏ Remod

(5-33)

where

冢

(1 – x)2 2x(1 – x) x2 = (1 + 3n)n(1 – x)1+n ᎏ + ᎏ + ᎏ 1 + 3n 1 + 2n 1+n 2 yp yp x = ᎏ = ᎏᎏ2 w fN · · V where yp is the yield stress for pseudoplastic.

冣

n

(5-34)

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For yield-pseudoplastics, Hanks and Ricks (1978), Heywood (1991) proposed to define a modified Hedstrom Number as D i2m yp Hemod = ᎏ ᎏ K K

冢 冣

2–n/n

(5-35)

The critical Reynolds number is established in terms of the modified Reynolds number and Hedstrom number, as in Figure 5-5.

5-3-2 The Heywood Method For the laminar flow regime fanning friction factor, Heywood (1991) modified the Buckingham equation to

冢

16 2Hemod fNLY = ᎏ 1 + ᎏᎏ2 Remod fNLY (Remod)

ᎏᎏᎏ 冣冦1 – (2n + 1) f (Re ) 2Hemod NLY

mod

2

1 + ᎏᎏ 冣冥冧 冤1 + ᎏᎏᎏ (n + 1) f (Re ) 冢 f (Re ) 2nHemod

4nHemod NLY

mod

2

NLY

mod

(5-36)

2

5-3-3 The Torrance Method Defining x = y/w, Torrance (1963) derived the following equation for turbulent friction factor of a yield pseudoplastic:

Critical (Hanks & Ricks) Reynolds Number

3200 2800 2400

4

He = 10 He = 0

2000 1600

6

He = 10 1200 800 400 0

0

0.1 0..2 0.3

0.4 0.5 0.6

0.7 0.8

0.9 1.0

Flow Behavior Index "n"

FIGURE 5-5 Values of the critical value of the “Hanks and Ricks Reynolds number” versus the flow index “n” for yield pseudoplastic slurries.

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1 ᎏᎏ =

兹f苶

– 2.95冥 + ᎏ log 冤冢 ᎏ n 冣 n 2.69

log 冢ᎏ n 4.53

4.53

10RePLC( f

10(1

– x)

冣

0.68 ) + ᎏ (5n – 8) n

[1–n/2]

(5-37)

where RePLC = DnV 2–n/K8n–1. The work of Torrance was essentially an exercise in algebra, and has not been substantiated by experimental data. Nevertheless it has been substantially quoted in the absence of suitable confirming data.

5-4 GENERALIZED METHODS Various models have been developed for complex non-Newtonian flows. The most important ones are listed, but most were derived from empirical data.

5-4-1 The Herschel–Bulkley Model Herschel and Bulkley (1928) developed a model that has been extensively applied to sewage sludge, kaolin slurries, and mine tailings:

= y + j(␥)n

(5-38)

y a = ᎏ = ᎏ + j(␥)n–1 ␥ ␥

(5-39)

The apparent viscosity is

where j is called the Herschel–Bulkley parameter.

5-4-2 The Chilton and Stainsby Method Chilton and Stainsby (1998) indicated that the accuracy of the Herschel–Bulkley model deteriorated at high shear rates. However, this may or may not be significant, depending on the application. At high strain rates the model predicts that the viscosity tends to zero, which is obviously incorrect. In Chapter 3, Section 3-4-2-2 , the Sisko, Cross, Meter, and Bird rheological models were presented. Chilton and Stainsby (1998) stressed the limitations of these models and the shear rates at which they are valid. In an effort to solve the Rabinowitsch and Mooney equations presented in Equations 5-15 and 5-16, Chilton and Stainsby (1998) proposed to express the pressure drop for a Herschel–Bulkley fluid as: ⌬P 4j 8V ᎏ=ᎏ ᎏ L D D

ᎏ ᎏᎏ 冢 冣 冢ᎏ 4n 冣 冢 1 – x 冣冢 1 – ax – bx – cx 冣 n

3n + 1

n

1

1

2

where 4L0 0 x= ᎏ = ᎏ w Di⌬P

3

n

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1 a= ᎏ (2n + 1) 2n b = ᎏᎏ (n + 1)(2n + 1) 2n2 c = ᎏᎏ (n + 1)(2n + 1) For a Bingham slurry, j = or the coefficient of rigidity. For a power law pseudoplastic fluid, 0 = 0, j = p or the power law viscosity. For a Newtonian, Hagen–Poiseuille fluid, 0 = 0, = 1, j = or viscosity. 32V ⌬P 4 8V ᎏ=ᎏ ᎏ =ᎏ L D D D2

冢 冣

Defining an effective viscosity as ᎏ ᎏᎏ 冢 冣 冢ᎏ 4n 冣 冢 1 – x 冣冢 1 – ax – bx – cx 冣

8V * = j ᎏ Di

n–1

3n + 1

1

n

1

2

n

3

(5-40)

Chilton and Stainsby proposed their equation for a generalized Reynolds number as

VDi ReMR = ᎏ *

(5-41)

Using the value (defined by Equation 5-16), the authors indicated that 3 + 1 * = L ᎏ 4

冢

冣

and a modified Reynolds number is defined as 4VDi ReMR = ᎏᎏ L(3 + 1)

(5-42)

if the wall viscosity could be measured. The friction factor in the laminar regime is then expressed as 16 fn = ᎏ ReMR

(5-43)

In the turbulent regime, for Herschel–Bulkley fluids Chilton and Stainsby derived

冢

ReMR fn = 0.079 ᎏᎏ n2(1 – x)4

冣

–0.25

(5-44)

Chilton and Stainsby then proposed another modified Reynolds number: ReMR R⬘ = ᎏᎏ n2(1 – x)4

(5-45)

This indicates that Equation 5-44 is a modified Blasius equation (see Chapter 2): fn = 0.079(R⬘)–0.25

(5-46)

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5.21

Without further proof, they proposed to follow the Prandtl equation of Newtonian fluids as if it could be applied to non-Newtonian fluids: f n–0.5 = 4.0 log10(R⬘f n0.5) – 0.4

(5-47)

Deviations from experimental data were noticed at high Reynolds numbers due to the limitations of the Herschel–Bulkley model on which basis this model was developed. (See Figures 5-6 and 5-7.) Example 5-7 Lab tests are conducted on sewage sludge in a 6⬙ pipe. The density is 1018 kg/m3 (S.G. = 1.018). The yield stress is measured as 1.28 Pa. The slurry is a Herschel–Bulkley fluid mixture with a parameter j = 0.2. The power law coefficient is determined to be 0.74. Calculate the friction factor for a flow of 350 L/s in a 18⬙ pipe of 0.375⬙ thickness, assuming a wall shear stress of 1.6 Pa. Pipe inner diameter = (18 – 2 × 0.375) × 0.0254 = 0.438 m Pipe inner area = 0.25 × × 0.4382 = 0.151m2 Pipe inner speed = 0.35/0.151 = 2.31 m/s

0 1.28 x = ᎏ = ᎏ = 0.8 w 1.6

FIGURE 5-6 The friction factor versus the Chilton–Stainsby Reynolds number for Bingham mixtures. (From A. Chilton and R. Stainsby, 1998. Reproduced with permission of Journal of Hydraulic Engineering, ASCE.)

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FIGURE 5-7 The friction factor versus the Chilton–Stainsby Reynolds number for pseudoplastic mixtures. (From A. Chilton and R. Stainsby, 1998. Reproduced with permission of Journal of Hydraulic Engineering, ASCE.)

冢

8 × 2.31 * = 0.2 ᎏ 0.438

冣

(0.74–1)

ᎏ ᎏᎏ 冢 ᎏᎏ 4 × 0.74 冣 冢 1 – 0.8 冣冢 1 – ax – bx – cx 冣 3 × 0.74 + 1

冢

冢

1

2

1 = 0.366 ᎏᎏ 1 – ax – bx2 – cx3 8 × 2.31 * = 0.2 ᎏ 0.438

1

0.7

3

冣

冣 冢 ᎏᎏ 2.96 冣 –0.26

(2.12 + 1)

0.75

1 5 ᎏᎏᎏᎏᎏᎏ 1 – 0.403 × 0.064 – 0.343 × 0.0642 – 0.254 × 0.0643

冢

冣

* = 0.3725 ReMR = 1018 × 2.31 × 0.438/0.3725 = 2765 This is transition Reynolds Number between laminar and turbulent flow. In laminar regime fn = 16/ReMR = 0.0057.

fnV22 ⌬P Pressure drop ᎏ = ᎏ = 72 Pa/m Di L 5-4-3 The Wilson–Thomas Method The Wilson–Thomas Method was developed in the 1980s for yield pseudoplastic and power law slurries. Wilson (1985) and Thomas and Wilson (1987) assumed that the fluid

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5.23

is a continuum, but that in non-Newtonian cases the viscous sublayer was thicker than with the water. (See Chapter 2 for the definition of viscous sublayer.) Defining a velocity VN for a Newtonian fluid at the same wall shear stress w as for the flow of a non-Newtonian fluid, a bulk velocity V is defined as

冤

冢

冣

冢

1–n n+1 V = VN + Uf 2.5 ln ᎏ + 11.6 ᎏ 2 n+1

冣冥

(5-48)

with VN Di Uf ᎏ = 2.5 loge ᎏ Uf

冢

冣

(5-49)

Since Uf = (w/m)1/2, the Wilson–Thomas model requires the designer to assume a value of wall shear stress. The effective viscosity is defined as

冢

4n eff = ᎏ 3n + 1

冣 冢ᎏ D 冣 n–1

8V

n–1

(5-50)

i

which is slightly different than expressed by Equation 5-26. In the laminar flow regime, the shear rate is expressed as

w 8V 4n ᎏ=ᎏ ᎏ Di 3n + 1 K

冢 冣

1/n

(5-51)

For Bingham fluids, the Wilson–Thomas equation is written as

冤

冢

冣

冥

1–x V = VN + Uf 2.5 ln ᎏ + x (14.1 + 1.25x) 1+x

(5-52)

where x = 0/w ReMR = mVDi(1 – x)/ In the laminar flow regime 8V w 1 4 ᎏ = ᎏ 1 – ᎏ x + ᎏ x4 Di 3 3

冤

冥

(5-53)

(where p = plastic viscosity), which is essentially the Buckingham equation. For Bingham fluids in the laminar regime, if x Ⰶ 0.5 then 4 8V w = ᎏ + ᎏ 0 Di 3 To obtain the transition velocity from laminar to turbulent flow, the following approach based on the Wilson–Thomas method is recommended. In the turbulent regime, a Reynolds number based on the friction velocity Uf is defined as

DIUf Ref = ᎏ p

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The Wilson–Thomas velocity is defined by modifying Equation 5.52 to

冤

冢

冣冥

1–x V = Uf 2.5 ln Re + 2.5 ln ᎏ + x (14.1 + 1.25x) 1+x

(5-54)

The transition occurs at the intersection of Equations 5-53 and 5-54. The Wilson–Thomas method was derived from first principles and does not use empirical correction coefficients. It has proved to be correct for many slurries, but it sometimes overpredicts losses for the Carson slurries (described in Equation 3-52). Example 5-10 shows the method of calculation using the Wilson–Thomas equation. Figure 5-8 compares the Wilson–Thomas method with the Hanks–Dadia friction factor. Figure 5-9 compares it with experimental data.

5-4-4 The Darby Method: Taking into Account Particle Distribution Professor R. Darby (2000) from Texas A&M University recently published a new method to predict the friction factor of power law non-Newtonian slurries. It is, however, much closer to the domain of slurries and takes into account such concepts as the drag coefficient, to which the reader was exposed in Chapter 3. In a method reminiscent of the work of Wasp for compound heterogeneous flows (see Chapter 3), Darby stated that the overall pressure drop for a non-Newtonian fluid is essentially the pressure drop of the liquid phase plus the pressure drop due to the solids: ⌬Pm = ⌬Pf + ⌬Ps

(5-55)

for each fraction i of solids with an average diameter dpi, and with a volumetric concentration Cv, the individual pressure drop must be computed. In a first iteration, the pressure drop for the liquid phase is computed by treating it as homogeneous non-Newtonian liquid by the various methods described in this chapter. A Froude number for the solid particles is defined as

FIGURE 5-8 Comparison between the Wilson–Thomas and the Hanks–Dadia models for friction factor of Bingham slurries. (From A. D. Thomas and K. C. Wilson, 1985. Reproduced by permission of Canadian Journal of Engineering.)

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FIGURE 5-9 Comparison between the Wilson–Thomas and experimental data on pressure losses for a limestone slurry. (From A. D. Thomas and K. C. Wilson, 1985. Reproduced by permission of Canadian Journal of Engineering.)

V 2t V2 Fr2 = ᎏᎏ = ᎏᎏ (s/L – 1)d g (s/L – 1)gDi

(5-56)

where Vt is the terminal velocity of falling particles. At slip velocity Vr between the solids and the liquid phases, with V as the carrier speed of the suspension, a nondimensional pressure drop is defined as Vt ⌬s X = ᎏᎏ ᎏ Cv L(s/L – 1)gL V

冢 冣

2

(5-57)

If Vr = Vf – Vs, W r2 X= ᎏ 1 – Wr

(5-58)

where Wr = Vr /V. For small volume fraction 0 < Cv < 0.25, X = X0. For other concentrations, X = X0 + 0.1F R2(Cv – 0.25)

(5.59)

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Procedures for Newtonian Mixtures: 1. Calculate the Froude number. 2. Determine Vt from the Archimedean number: d 3p f g(s/L – 1) Ar = ᎏᎏ L2 with f being the liquid viscosity. If Ar < 15, the terminal velocity can be obtained from the Stokes equation If Ar > 15, the particle Reynolds number is obtained after rearranging the Dallavalle equation: Rep = [(14.42 + 1.827兹F 苶r苶)0.5 – 3.798]2 = dpVt L/L

(5-60)

3. Calculate the Froude number from Equation 5-56. 4. The value of ratio Wr is then calculated from the Molerus diagram (Figure 5-10). The values of X and ⌬Ps/L are then calculated from Equations 5-59 and 5-60. This step may be repeated for each range of particle size and summed up with other particle sizes to get an overall pressure drop for solids. However for Non-Newtonian mixtures additional procedures are needed.

FIGURE 5-10 Molerus diagram. (From R. Darby, 2000. Reproduced by permission of Chemical Engineering.)

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5.27

Terminal Velocity for Power Law Fluids For a solid particle settling in a non-Newtonian fluid, defining 1.33 + 0.37n Z = ᎏᎏ 1 + 0.7n3.7

(5.61)

as a nondimensional power law parameter and C1 = [(1.88/n)8 + 34]–1/8 as another nondimensional power law parameter, Darby expressed the modified Reynolds number as

冢

4.8 RePL = Z ᎏ 1/2 C d – C1

d nV 2–n t f

冣 =ᎏ K 2

(5.62)

with Cd the drag coefficient: 4g(m/L – 1)d Cd = ᎏᎏ 3V 2t

(5.63)

Friction Factors The transition from laminar to turbulent flow occurs at a critical Reynolds number: RePLC = 2100 + 875(1 – n)

(5.64)

Defining fL as the fanning factor in the laminar regime, and fT as the fanning factor in the turbulent regime, Darby derived the following equations:

␣ fn = (1 – ␣) fL + ᎏᎏ ( f –8 + f L–8)1/8 T

(5.65)

where the laminar flow friction factor is 16 fL = ᎏ RePL

(5.66)

0.0682n–1/2 fT = ᎏᎏ 1/(1.87+2.39n) RePL

(5.67)

At the transition from laminar to turbulent flow, the friction factor is expressed as: (0.414+0.757n) fTR = 1.79 × 10–4 exp(–5.24n)RePL

(5.68)

1 ␣= ᎏ 1 + 4⌬

(5.69)

⌬ = RePL – RePLC

(5.70)

and

Example 5-8 Slurry is required to flow in a 250 mm ID pipe and has the following characteristics:

0 = 15 Pa = 43 mPa · s = 1450 kg/m3

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K = 0.4 Pa · sn n = 0.8 d85 = 65 m V = 1.76 m/s Determine the friction factor. Check on critical Reynolds number: RePLC = 2100 + 875(1 – 0.8) = 2275 Z = 1.13 C1 = 0.4236 Calculations will be done by a series of iterations. Iteration 1. Assume the drag coefficient is Cd = 0.4, then

冢

4.8 RePL = 1.13 ᎏᎏ 0.41/2 – 0.4236

冣 = 597 2

16 16 fL = ᎏ = ᎏ = 0.026 RePL 597 0.0682 × 0.8–1/2 fT = ᎏᎏ = 0.0141 5970.264 ⌬ = 597 – 2275 = –1678 1 ␣= ᎏ 1 + 4–⌬

␣⬵0 ␣ fn = (1 – ␣)fL + ᎏᎏ ( f T–8 + f L–8)1/8 fn = fL = 0.026 By further iteration, the magnitude of the friction factor is refined.

5-5 TIME-DEPENDENT NON-NEWTONIAN SLURRIES Time-dependent non-Newtonian flows are difficult to model. There is a certain lapse of time to overcome before a stable friction factor can be obtained. Govier and Aziz (1972) have published cases in which the lapsing time was as high as 1000 minutes with Prembina crude oil. During this initial lapse of time, the pressure gradient for friction losses dropped from an initial value of 72.5 Pa/m (0.0032 psi/ft) down to 18 Pa/m (0.0008 psi/ft) by the time the flow had stabilized. Such a ratio of 4:1 is certainly not negligible. Govier and Aziz (1972) indicated that once the initial period of stabilization is reached, the general form of pressure loss equations are the same as for time-independent non-Newtonian fluids. At the entry to a pipe, the flow may be laminar, but at a certain distance, once the entrance effects are overcome, the flow can transit to turbulence. The start-up pressures for thixotropic slurries may be quite high, particularly when these slurries coagulate into gels inside the pipeline. During an initial of period of time, gels must be expelled from the pipeline. Crude oils, fuel oils, bentonite clay, and drilling

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5.29

muds are a concern to engineers. Water may be used as an expelling medium to clear up the pipeline. Positive displacement pumps are preferred for thixotropic slurries to overcome the high starting pressures. Various models have been developed for thixotropic fluids. These slurries are sometimes treated as Bingham plastics and sometimes as pseudoplastics, based on the experimental data from test work. The problem of pumping thixotropic froth in oil sand plants was a challenge to manufacturers of pumps, and for a while the belief was that only positive displacement pumps could be used. In 2000, the operators of oil-sand processing plants invited the manufacturers of slurry pumps to develop new appropriate pumps. Research is being conducted at the Saskatchewan Research Institute in Canada and is starting to yield new concepts of pump design.

5-6 EMULSIONS The concept of emulsions was introduced in Chapter 1. In an emulsion, one phase is suspended as droplets rather than particles. Tar or bitumen, for example, can be suspended in water up to a volumetric concentration of 70%. In some of the oil sand extraction processes in Canada, the situation can be further complicated by the addition of flocculants. The Venezuelan corporation, PDVSA, and its research branch, INTEVEP, conducted considerable research on Orimulsion™, a proprietary synthetic fluid composed of surfactants, bitumen droplets, and water. The surfactants keep the bitumen droplets in suspension. Nunez et al. (1996) published a comprehensive paper on the flow characteristics of concentrated emulsions in water with a volumetric concentration in the range of 70–85%.

5-7 ROUGHNESS EFFECTS ON FRICTION COEFFICIENTS Szilas et al. (1981) published the following equation for pseudoplastic oil flow in rough pipes at high Reynolds numbers:

冢

冣

8.03 1 4 1.414 ᎏ = ᎏ log10 (Remod(4fn)1–n/2) + 1.511/n 4.24 + ᎏ – ᎏ – 2.114 兹苶 fn n n n

(5-71)

However, this equation does not include any terms for wall roughness. Torrance (1963) developed a theoretical equation for yield pseudoplastics in fully turbulent flow (at high Reynolds numbers) in rough pipes as

冢

冣

2.65 1 RePLC ᎏ = 4.07 log10 ᎏ + 6.0 – ᎏ 兹苶 fn n

(5-72)

where RePLC = DnV 2–n /K8n–1. This applies to the region where the friction factor is independent of the Reynolds number, or at high Reynolds numbers. Govier and Aziz (1972) suggest the following procedure: 앫 Compute the friction factor for the slurry in a smooth tube using one of the methods described in Sections 5-2, 5-3, and 5-4. 앫 Using the Moody diagram determine the ratio of the friction factor for rough to smooth pipe at the value of the Reynolds number (using ReB, RePLC, or Remod).

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Aral and Kaylon (1994) focused on highly concentrated suspensions and investigated the effects of temperature as well as surface roughness. To take into account the pipe roughness in laminar and turbulent flows, SRC (2000) recommended the use of the equation of Churchill (1977):

冤冢 ᎏ Re 冣

fn = 2

8

12

+ (A + B)–1.5

冥

1/12

(5-73)

where

冦

冤冢 ᎏ Re 冣 7

A = –2.457 ln

冢

0.9

冥冧

+ 0.27 /Di

37,530 B= ᎏ Re

16

冣

16

(5.74) (5.75)

Example 5-9 Assuming turbulent flow and using the Wilson–Thomas model, calculate the bulk velocity for a flow with the following characteristics:

0 = 20 Pa w = 24 Pa = 0.016 Pa · s = 3 × 10–6 m Di = 141 mm = 1350 kg/m3 x = /w = 20/24 = 0.833 Iteration 1. Assume VN = 2.5 m/s, then Re = VDi/

= /(1 – x) = 0.016/(1 – 0.833) = 0.096 Re = VDi/ Re = 1350 × 2.5 × 0.141/0.096 = 4967 Relative roughness /Di = 3 × 10–6/0.141 = 0.000021 From Equations 5-74 and 5-75: A = {–2.457 ln[(7/4967) )0.9 + 0.27 × 0.000021)]16 = 3.86 × 1018 B = 1.1286 × 1014 fn = 0.00949 Iteration 2. Checking the value of VN. From Chapter 2:

w = fnV 2/2 (2 × 24)/(0.00949 × 1350) = 3.75 V = 1.935 m/s The friction velocity from Equation 2-5:

w苶 / = 0.133 m/s Uf = 兹苶

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Using Equation 5-52:

冤

1 – 0.833 V = 1.935 + 0.133 2.5 ln ᎏ + 0.833(14.1 + 1.25 × 0.833) 1.833

冥

2.815 m/s = 1.935 + 0.133[–5.989 + 12.61] Therefore, the equivalent Newtonian fluid velocity is 1.935 m/s, and the mean velocity of the Bingham slurry will effectively be 2.815 m/s. Up until 1990, one author after another tried to treat non-Newtonian slurries as homogeneous mixtures. The arbitrary assumption that the particle diameter was absent in many equations when the diameter was smaller than 44 m or 74 m (depending on the author) is now being challenged. The advent of new experimental methods, such as laser velocimeters, is helping engineers understand the complexity of turbulent non-Newtonian fluids. The work of Park et al. (1989) using laser velocimeters, and Pokryvalio and Grozberg (1995) using electro-diffusion techniques on non-Newtonian slurries was analyzed by Slatter et al. (1996), who confirmed the significance of high-intensity turbulence at the wall. This encouraged them to postulate a theory of particle roughness. Slatter et al. proposed that the d85 particle size be used for particle roughness. (This may be an attempt to correlate with the Nikrudase particle roughness of Newtonian slurries.) For large pipe, when the actual roughness exceeds the value d85, is used. For

< d85

dx = d85

> d85

dx =

For a yield pseudoplastic, Slatter et al. (1996) defined their Reynolds number, Rer, in terms of the friction velocity, consistency factor K, and power coefficient n, as well as roughness dx: 8U 2f Rer = ᎏᎏn 0 + K(8Uf /dx)

(5.76)

If Rer > 3.32, then smooth wall turbulence occurs, and the mean bulk velocity V is expressed as

冢 冣

V Di ᎏ = 2.5 ln ᎏ – 2.5 ln Rer + 1.75 Uf 2dx

(5.77)

If Rer ( 3.32, then fully developed rough turbulent flow occurs, and the mean bulk velocity V is expressed as

冢 冣

Di V ᎏ = 2.5 ln ᎏ + 4.75 Uf 2dx

(5.78)

This correlation produces an abrupt transition from smooth turbulent to fully turbulent flow at the wall of the pipe. We have already explained that the Wilson–Thomas model was based on the assumption that the viscous sublayer in non-Newtonian flows was thicker than with water, thus suppressing the effect of roughness. The work of Slatter, Thorscalden, and Petersen (1996) on mixtures of kaolin clay and sand indicates that this is not very achievable (Figure 5-11). The resultant pressure losses (Figure 5-12) are therefore higher. Figure 5-12 does indicate that the Torrance and the Wilson–Thomas models correlate well. Both models were based on mathematical assumptions at 20 year intervals.

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FIGURE 5-11 A comparison between the Wilson–Thomas and Slatter, Thorscalden, and Petersen models for the viscous sublayer. (From P. T. Slatter et al., 1996. Reproduced by permission of BHR Group.)

FIGURE 5-12 A comparison of pressure drop per unit length between the Slatter, Thorscalden, and Petersen and Wilson–Thomas models. (From P. T. Slatter et al., 1996. Reproduced by permission of BHR Group.)

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Example 5-10 A sand–kaolin mixture with a volumetric concentration of 19.5% is flowing in a 431 mm ID pipe at an average speed of 1.8 m/s. The yield stress is 5.5 Pa, the coefficient of consistency K = 0.124 Pa · sn, and the power coefficient n = 0.64. The specific gravity of the solids are 2.65 and the d85 = 131 m. Using the Slatter method, determine the friction factor. Solution Density of mixture:

m = Cv(s – L) + L m = 0.19(1265) + 1000 = 1240 kg/m3 Iteration 1. Assume fully turbulent flow:

冢

冣

V 0.5Di ᎏ = 2.5 ln ᎏᎏ = 4.75 = 23.26 Uf 131 × 10–6 fD/8 苶. If V = 1.8 m/s, then Uf = 0.0774 m/s, since Uf = V 兹苶 fD = 0.0147 8 × 1240 × 0.07742 = 1.78 Rer = ᎏᎏᎏᎏᎏ 5.5 + 0.124(8 × 0.0774/0.131 × 10–3)0.64 Iteration 2. Since Rer < 3.32, the equation to use is

冢

冣

V 0.5Di ᎏ = 2.5 ln ᎏ + 2.5 ln Rer + 1.75 Uf d85 V ᎏ = 18.51 + 1.44 + 1.75 Uf V ᎏ = 21.70 Uf 0.046 = (fD/8) fD = 0.01698 Uf = 0.0829 m/s

5-8 WALL SLIPPAGE A phenomenon encountered with non-Newtonian mixtures is a tendency for the low-viscosity constituent to migrate to regions of high shear and to lubricate the flow. One example is the core annular flow of crude oil in water, where the more viscous material is lubricated by the less viscous material. In the case of emulsions and certain non-Newtonian slurries, lubrication occurs by a slip layer of water on the wall. Mathematically, the concept of slip can be treated as a discontinuity. Heywood (1991) proposed to represent slip by the slip velocity Vs. In the laminar flow regime, the total flow in a pipe would be

冢 冣冕

D 2i Di Q = Vs ᎏ + ᎏ ᎏ 4 8 w

d␥ 2 ᎏ d dt

3 w 0

(5-79)

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Heywood (1991) noticed that there are no methods to evaluate slip in turbulent flows. Evaluation of slip in laminar flow is conducted using the coaxial cylinder described in Chapter 3. Nunez et al. (1996) indicate that the migration of viscous droplets from the wall in an emulsion exhibit the Segré–Silberberg effect. However, they pointed out that not all emulsions experience slip and that the phenomena of slip appeared to be characteristic of the crude oil or viscous component. In some respects, slip does reduce the friction factor by lubrication from the least viscous phase. However, Aral and Kaylon (1994) found that increasing the surface roughness tended to reduce or eliminate slip. One particular problem with emulsions is the fracture of the droplets under high shear rates. A form of comminution occurs as large droplets come in contact with other ones. Degradation of non-Newtonian slurries under high shear rates is not well documented. There is evidence that high shear rates occur in centrifugal pumps. Adequate clearance may reduce the degradation of the emulsion or slurry but tends to reduce the efficiency of the pump. Degradation can include a form of coalescence or formation of colloids and larger droplets or particles. Clay ball formation is encountered in dredging operations after passage through the pump.

5-9 PRESSURE LOSS THROUGH PIPE FITTINGS The method of the two K-factor was presented in Chapter 2 in Section 2.8. It was explained that Hooper had established a general relationship

冢

1 K1 K = ᎏ + K⬁ 1 + ᎏ DI-in Re

冣

where K1 is the value of K at a Reynolds number of 1 K⬁ is the value of K at high Reynolds number Di-in is the internal pipe diameter in inches Johnson (1982) reviewed some of the problems of pumping non-Newtonian mixtures. In his assessment of fittings for sewage, he indicated important discrepancies in the laminar regime with losses 2–4 times as much as those for water flows. In the turbulent regime, the losses were either of the order of those for water or higher. He recommended that further studies be conducted for laminar flow, but for turbulent flows, the concept of equivalent length be used. In Chapter 2, concepts of pressure losses for Newtonian liquids were examined. Edwards et al. (1985) reviewed the flow of non-Newtonian slurries in laminar regimes. They recommended that the modified Reynolds number (using ReB or Remod) be used to correlate with the loss factor Kf of Newtonian flows in laminar regimes. In certain fittings such as globe valves, turbulence is enhanced by the geometry of the fitting. Although the flow may be laminar in a straight pipe up to a transition to a Reynolds number of 2000, turbulence in the globe valve may actually start at much lower Reynolds number such as 900. Much more work is needed on loss factors for non-Newtonian flows in transition and turbulent regimes. Govier and Aziz (1972) had noticed the lack of any methodology to compute loss from pipe fittings with non-Newtonian flows. They proposed that the

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5.35

method of equivalent length be used, i.e., that the equivalent length of pipe fittings for Newtonian liquids be added to computations of total length for pressure loss of the nonNewtonian slurry.

5-10 SCALING UP FROM SMALL TO LARGE PIPES Non-Newtonian flows are complex. A lab test in a pumping loop can yield very useful rheology data about the pressure drop. Scaling up to larger pipes is one method of predicting pipeline flows. Heywood et al. (1992) proposed the following methods: 앫 For laminar flow, the plot of the shear rate (8V/Di) against the shear stress (Di⌬P/4L) is independent of the pipe diameter and the pipe roughness, so that experimental data could be converted directly into practical data for pipeline design. 앫 For turbulent flows, the Bowen method, which is essentially a modification of the Blasius method, should be used. Bowen (1961) suggested the following modification to the Blasius equation: Dix = kV w

(5.80)

The shear stress is plotted against the flow rate Q to obtain the magnitude of the exponent w. The result is then used to plot /V w against the diameter Di to obtain the values of k and x. The intersection of the turbulent and laminar flow curves gives the transition point. Kenchington (1972) showed that this method showed great discrepancy when the ratio of diameter between the large field pipe and the lab pipe exceeded 6 to 12 folds, so that large pipe tests may be needed. The Bowen method assumed the same range of roughness between a lab and a field pipe. It is therefore important to apply correction factors for roughness when using this method.

5-11 PRACTICAL CASES OF NON-NEWTONIAN SLURRIES The equations presented in the previous sections of this chapter are fairly complex and are based on so many assumptions that the practical engineer may feel lost. Some real examples are needed to guide the designer of non-Newtonian systems.

5-11-1 Bauxite Residue Want et al. (1982) reviewed the design of a bauxite residue pipeline for Alcoa Australia. The plant disposed 4.75 Mtpy (million tonnes per year) of alumina. Tests conducted on samples confirmed that the rheology of the slurry at concentrations in excess of 45% by weight could be expressed by the Carson equation:

冤 冢 冣冥

du w1/2 = 1/2 0 + ⬁ ᎏ dr

1/2

where du/dr is expressed by Equation 5-15 and Equation 5-16.

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In the case of Kwina red mud, tests indicated that the modified consistency coefficient K⬘ (Equation 5-18) was a complex function of the weight concentration: K⬘ = 3.88 × 10–18 (100 Cw)10.57 Pa/s and that the value of was

= 2.55 × 106(Cw × 100)–4.19 for Cw ⱕ 51% = 6.685 (Cw × 100)–0.926 for Cw > 51% Applying the Dodge–Metzner model Want et al. (1982) applied Equation 5-20 to express the consumed power under turbulent conditions as: (Q)3 150 – 100 Cw ᎏᎏ eT = 1.44 × 10–4 fn ᎏ D 5i (100 Cw)1.5

冤

冥

2

in kW/km, and in laminar conditions as: 64mQ(150 – 100Cw) eL = 0.393 K⬘ ᎏᎏᎏ 3 × 104 Cw

冤

冥 冤ᎏ D 冥 1+

1+3

1

in kW/km

i

Want et al (1982) discussed the thixotropic nature of red mud at high concentration, and the importance of flocculants and dispersants on the rheology of this slurry. Referring to Figure 5-13, it is clear that there is a change in the pressure drop per unit length as the weight concentration is increased and the flow changes from turbulent to laminar. This drop in pressure to a minimum at such a transition is often misunderstood, particularly because it goes against the concepts examined in Chapter 4. The important parameters include the diameter of the pipe, so that there is an optimum diameter, and an optimum weight concentration for a given tonnage of fine solids to be transported. Slurries may therefore be pumped at very high concentrations (Figure 5-14) using positive displacement pumps (Figure 5-15) over long distances, provided that the correct weight concentration is used near the turbulent to laminar transition region. Example 5-11 Using the data obtained by Want, examine pumping red mud bauxite residues at a speed of 1.74 m/s in a 141 mm I.D. pipe at weight concentrations of 45% and 60%. Determine the required power for a horizontal pipeline, 3 km long. Assume a density of 1350 at 45% and 1800 at 60%. Solution At Cw = 45%: K⬘ = 3.88 × 10–18 (100Cw)10.57 K⬘ = 1.157

= 2.55 × 106 (100 Cw)–4.19 = 0.302 Q = AV = × 0.25 × 0.1412 × 1.74 = 0.0272 m3/s

冤

64 × 1350 × 0.0272(150 – 45) eT = 0.393 × 1.157 ᎏᎏᎏᎏ 3 × 104 × 0.45 eT = 188 kW/km For 3 km, this is equivalent to 565 kw or 758 hp.

冥 冤ᎏ 0.141 冥 1.3

1

1.906

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FIGURE 5-13 Pressure drop per unit length for red mud tailings. (From F. M. Want et al., 1982. Reproduced by permission of BHR Group.)

At Cw = 60%: K⬘ = 3.88 × 10–18 (60)10.57 = 24.2

= 6.685(60)–0.926 = 0.151

冤

64 × 1800 × 0.0272 (150 – 60) eL = 0.393 × 24.2 ᎏᎏᎏᎏ 3 × 104 × 0.6 eT = 1041.3 kW/km For 3 km, eL = 3123.9 kW or 4186 hp.

冥 冤ᎏ 0.141 冥 1.3

1

1.906

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FIGURE 5-14 Slurry can be pumped in a non-Newtonian regime at high volumetric concentrations. (Courtesy of Geho Pumps.)

5-11-2 Kaolin Slurries Slatter et al. (1996) reported that Kemblowski and Kolodziejski (1973) found that the Dodge and Metzner model did not well represent the flow of kaolin slurries. They derived the following empirical equation: 0.3164 4fn = ᎏ 0.25 ReMR and more generally: E 1/ReMR 4fn = ᎏ m ReMR

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FIGURE 5-15 The pumping of high concentration slurries and pastes may require positive displacement pumps. (Courtesy of Geho Pumps.)

where E, m, and are empirical parameters and functions of the apparent flow behavior index (defined in Equation 5-16) and the modified Reynolds Number ReMR is defined by Equation 5-17.

5-12 DRAG REDUCTION Ippolito and Sabatino (1984) showed that the addition of 3% salt to water tended to reduce the friction factor of bentonite suspensions. Sauermann (1982) indicated that the addition of 0.2 kg/ton of tripolyphsophate (Na5P3O10) to gold slimes at a weight concentration of 67.7% , and with particles smaller than 50 m, reduced the pressure gradient in laminar flow by as much as 30%. Some viscosity reducing agents were discussed in

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Chapter 3.There are, however, very little data published on other methods for reducing friction losses of non-Newtonian slurries. Schowalter (1977) discussed some aspects of drag reduction in non-Newtonian slurries and reported certain cases of mixtures with a pressure drop actually lower than that of water.

5-13 PULP AND PAPER Pulp and paper pump slurries behave as non-Newtonian slurries. The following equations have been reported by the Cameron Hydraulic Data book of IDP (1995), based on work at the University of Maine. A modified Reynolds number is defined as D 0.205 Vsg I ReMR = ᎏᎏ C1.157

(5-81)

where C = % consistency of the pulp, oven dry g = 32.2 ft/s = density in slugs/ft3 Vs = velocity, defined in ft/sec A modified friction factor is defined as 3.97 f= ᎏ 1.636 ReMR

(5-82)

A special equation for friction losses is therefore defined as fV 2LK0 Hf = ᎏ DI

(5-83)

The correction factor K0 depends on the type of pulp. It is considered to be 1.00 for unbleached sulfite softwood, 0.90 for bleached sulfite softwood, 0.90 for unbleached kraft softwood, 0.90 for soda hardwood, 0.90 for reclaimed fiber, 1.0 for presteamed groundwood–softwood, and 1.42 for stoned groundwood–softwood. (IDP 1995). The following program calculates friction factors. 10 PRINT “program for stock , pulp and paper” PRINT “based on the curves of the University of Maine” PRINT “ which are correlations to the data of Brecht and Heller” pi = 4 * ATN(1) INPUT “name of project”; pr$ INPUT “name of client “; nc$ INPUT “date of calculations”; dat$ INPUT “ type of pulp”; pul$ INPUT “consistency of pulp in percent”; CS C = CS/100 15 INPUT “ choose between (1) SI units (m) and (2) US units (feet) “; NB IF ABS(NB) < 1 THEN GOTO 15 IF ABS(NB) > 2 THEN GOTO 15 IF (NB > 1) AND (NB < 2) THEN GOTO 15 GOSUB conversion 18 IF NB = 1 THEN INPUT “inner diameter (m)”, D1M IF NB = 2 THEN INPUT “inner diameter (ft)”; D1US

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IF NB = 1 THEN d1 = D1M * dl IF NB = 1 THEN D1US = D1M/.3048 IF NB = 2 THEN d1 = D1US * dl IF NB = 1 THEN INPUT “pulp flow rate (m3/hr)”, qm IF NB = 2 THEN INPUT “pulp flow rate in USgpm”; qus IF NB = 1 THEN q = qm/60000 IF NB = 2 THEN q = qus * 3.785/60000 a = .25 * pi * d1 ^ 2 v = q/a PRINT USING “pulp speed = ###.### m/s”; v vus = v/.3048 PRINT USING “PULP SPEED = ##.### ft/s”; vus IF (C > .03) AND (vus > 8) THEN PRINT “speed exceeds 8 ft/s please use larger pipe size” IF C > .03 THEN GOTO 25 IF (C < .02) AND (vus > 10) THEN PRINT “speed exceeds 10 ft/s please use larger” IF C < .02 THEN GOTO 25 IF vus > 9 THEN PRINT “SPEED EXCEEDS 9 FT/S, PLEASE USE LARGER PIPE” 25 INPUT “DO YOU WANT TO USE A LARGER PIPE (Y/N)”; p$ IF p$ = “Y” OR p$ = “y” THEN GOTO 18 IF NB = 1 THEN INPUT “DENSITY OF STOCK IN KG/M3 (OFTEN ASSUMED TO BE 1000 KG/M3 “; DENSM IF NB = 1 THEN DENSUS = DENSM * (62.4/1000) IF NB = 2 THEN INPUT “DENSITY OF STOCK IN LBS/CU.FT (OFTEN ASSUMED TO BE 62.4 LBS/CU.FT”; DENSUS REM0 = D1US ^ .205 * vus * DENSUS/CS ^ 1.157 FM0 = 3.97/REM0 ^ 1.636 PRINT USING “MODIFIED REYNOLDS NUMBER = #######”; REM0 PRINT USING “MODIFIED FRICTION FACTOR = #.####”; FM0 PRINT “ please choose between the following pulps” PRINT “ 1- unbleached sulfite “ END conversion: IF NB = 1 THEN dl = 1 IF NB = 2 THEN dl = .3048 IF NB = 1 THEN ql = 3875/60 IF NB = 2 THEN ql = 60/3875 RETURN

5-14 CONCLUSION The world of non-Newtonian flows is very complex and encompasses very different flows, including pulp and paper, food, plastics, and clays. The use of various equations developed by researchers do not yield the same values of the friction coefficient. The problem is compounded by the fact that many rheological tests are conducted in tubes and in laminar flows, yielding values of consistency factor K and exponent n outside the range of turbulent flows. The practical engineer is often left to use his engineering judgment to use the appropriate equation. Proper tests of the slurry flow at the correct range of shear stresses are essential to avoid errors. Various reference books on the subject have equations similar to Equations 5-20 to 5-25 without emphasizing the limitations to their use. Because many of the equations for friction loss factors are implicit and require iteration methods, the use of personal computers is encouraged. The flow of non-Newtonian slurries is complex and requires a significant energy in-

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put. Methods have been developed over the years to reduce friction losses. These include dilution, reduction of volumetric concentration of solids, removal of all flocculants, provisions for air purging of the pipeline, addition of high-aspect-ratio fibers, addition of deflocculants (soluble ionic compounds), reduction of the angularity of particles, and addition of viscosity reducing agents. Not all these methods are always possible, and some require capital investment. It is hoped that the various worked examples in this book will help the practical engineer to design slurry pipelines. It may be necessary to use more than one method and compare results. There are many advantages to pumping slurries at high concentrations, such as concentrates from process plants, food pastes, and ceramic slurry for the manufacture of new materials.

5-15 NOMENCLATURE a A Ar b B C1 CD Cv Cw d85 dp dP/dz dx Di eT E fD fL fN fNC fNL fNLY fNPL fPLT fT fTR Fr g gc He Hemod j K K⬘

Nondimensional parameter and function of Hedstrom number Factor for friction in the Churchill equation Archimedean number Nondimensional parameter Factor for friction in the Churchill equation Nondimensional power law parameter Drag coefficient Volume fraction of solid particles in the slurry mixture Weight concentration Particle diameter passing 85% (m) Diameter of particle Pressure gradient per unit length Equivalent roughness Conduit inner diameter (m) Consumed energy Empirical coefficient Darcy friction factor Laminar component of fanning friction factor for a Bingham plastic Fanning friction factor Fanning friction at transition between laminar and turbulent flow Laminar component of fanning friction factor Laminar component of fanning friction factor for a yield pseudoplastic Fanning friction factor for a pseudoplastic in a laminar regime Tomita laminar friction factor Turbulent component of fanning friction factor Darby friction factor for transition from laminar to turbulent flows with power law fluids Froude number Acceleration due to gravity (9.8 m/s2) Conversion from slugs to pounds mass in USCS units Hedstrom number for Bingham plastic Modified Hedstrom for yield pseudoplastic Herschel–Bulkley parameter Coefficient of consistency for power law fluids Modified coefficient of consistency for power law fluids

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L m n P PL Pst Q R⬘ Re ReB ReBc Rec Remod ReMR RePL RePLC Rer Rep Uf V VN Vr Vs Vt VTR Wr x xc Z

5.43

Length of conduit or pipe Exponent in Darby’s equation for fanning friction factor calculations Flow behavior index for pseudoplastic flows Pressure Plasticity number Start-up pressure to pump a non-Newtonian slurry Flow rate Chilton and Stainsby proposed modified Reynolds number Reynolds number Reynolds number for a Bingham plastic using the coefficient of rigidity for viscosity Critical transition Reynolds number for a Bingham plastic using the coefficient of rigidity for viscosity Reynolds number at transition Modified Hedstrom number for yield pseudoplastic Modified Reynolds number for a power law fluid Tomita Reynolds number for a power law fluid Tomita Reynolds number for a power law fluid at transition Slatter Reynolds number Particle Reynolds number Friction velocity Speed Newtonian velocity Slip velocity Velocity of solids Terminal velocity of falling particles Transition velocity from laminar to turbulent flows Ratio of slip velocity to slurry speed Ratio of the yield stress to the wall shear stress Ratio of the yield stress to the wall shear stress at the transition from laminar to turbulent flow Settling factor for a non-Newtonian fluid

Subscripts L Liquid m Mixture p Particle Greek symbols ␣ Function for use of laminar and friction factors ⌬ Increment Concentration by volume in decimal points ⌬P Pressure drop ⌬m Density change for the mixture Density L Density of liquid carrier in kg/m3 m Density of slurry mixture in Kg/m3 s Density of solids Modified flow behavior index for pseudoplastic flows ␥ Shear strain d␥/dt Wall shear rate or rate of shear strain with respect to time

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⬁ a e e L p * ⬁ 0 w yp

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CHAPTER FIVE

Linear roughness (m) Coefficient of rigidity of a non-Newtonian fluid, also called Bingham plastic viscosity Coefficient of rigidity at high shear rate Carrier liquid dynamic viscosity Apparent viscosity of a pseudoplastic fluid Effective pipeline viscosity Wilson–Thomas effective viscosity Viscosity of liquid carrier Effective pipeline viscosity for pseudoplastic Effective viscosity for Hagen-Poiseuille fluid Bingham plastic limiting viscosity of slurry mixtures (Poise) Empirical function of the power exponent n Pythagoras number (ratio of circumference of a circle to its diameter) Shear stress at a height y or at a radius r Yield stress for a Bingham plastic Wall shear stress Yield stress for pseudoplastic

5-16 REFERENCES Abulnaga, B. E. 1997. Slurcal—Computer Program for Non-Newtonian Flows. Fluor Daniel Wright Engineers, Vancouver, BC, Canada (unpublished). Aral, B. K., and D. M. Kaylon. 1994. Effects of temperature and surface roughness on time dependent development of wall slip in steady torsional flow of concentrated Suspensions. Journal of Rheology, 38, 957–972. Bowen, R. L. 1961. Chemical Engineering, 143–150. Buckingham, E. 1921. On plastic flow through capillary tubes. ASTM Proceedings, 21, 1154. Chilton, R. A., and R. Stainsby. 1998. Pressure loss equations for laminar and turbulent nonNewtonian pipe flow. Journal of Hydraulic Engineering, 124, 5, 522–529. Clifton, R. A, and R. Stainsby. 1998. Pressure loss for laminar and turbulent non-Newtonian pipe flow. Journal of Hydraulic Engineering, 124, 5, 522–529. Churchill, S. W. 1977. Friction factor equation spans all fluid flow regimes. Chemical Engineering 84, 7, 91–92. Darby, R. 2000. Pressure drop of non-Newtonian slurries, a wider path. Chemical Engineering, 107, 5, 64–67. Darby, R. 1981. How to predict the friction factor for flow of Bingham plastics. Chemical Engineering, 88, 26, 59–61. Darby, R., R. Mun, and V. Boger. 1992. Prediction friction loss in slurry pipes. Chemical Engineering, September. Dodge, D. W., and A. B. Metzner. 1959. Turbulent flow of non-Newtonian systems. Am. Inst. Chem. Engr., 5, 2, 189–204. Edwards, M. F., M. S. M. Jadallah, and R. Smith. 1985. Head losses in pipe fittings at Low Reynolds Number. Chem. Engr, Res. Des., 63, 1, 43–50. Govier, G. W., and K. Aziz. 1972. The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold. Hanks, R. W. 1962. A Generalized Criterion for Laminar–Turbulent Transition in the Flow of Fluids. Union Carbide report. Hanks, R. W., and D. R. Pratt. 1967. On the flow of Bingham plastic slurries in pipes and between parallel plates. Soc. Petr. Eng. Journal, 7, 342–346. Hanks, R. W., and B. H. Dadia. 1971. Theoretical analysis of the turbulent flow of non-Newtonian slurries in pipes. American Journal of Chemical Engineering, 17, 554–557.

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Hanks, R. W., and B. L. Ricks. 1975. Transitional and turbulent pipeflow of pseudoplastic fluids. Journal of Hydronautics, 9, 39–44. Hedstrom, B. O. A. 1952. Flow of plastic materials in pipes. Ind. Eng. Chem., 44, 651–656. Herschel, W. H., and R. Bulkley. 1928. Measurements of consistency as applied to rubber benzene solution. Proc. ASTM, 26, Part 2, 621–633. Herzog, R. L., and K. Weissenberg. 1928. Kolloid Z, 46, 277. Heywood, N. I. 1991. Pipeline design for non-settling slurries. In Slurry Handling, Brown, N. P., and N. I. Heywood (Eds.). New York: Elsevier Applied Sciences. Heywood, N. I., D. C. H. Cheng, and A. J. Carlton. 1992. Slurry systems. In Piping Design Handbook, McKetta, J. J. (Ed.). New York: Marcel Dekker, pp. 585–622 Heywood, N. I., and J. F. Richardson. 1978. Rheological behavior of flocculated and dispersed kaolin suspensions in pipe flow. Journal of Rheology, 22, 6, 559–613. IDP (now called Flowserve). 1995. Cameron Hydraulic Data. NJ: IDP. Ippolito, M., and C. Sabatino. 1984. Rheological behavior and friction resistance of colloidal aqueous suspensions. In Proceedings of the IXth International Congress on Rheology, Mexico. Mexico: Universidad Nacional Autonoma de Mexico. Irvine. 1988. Experimental measurements of isobaric thermal expansion coefficients of Non-Newtonian fluids. Heat Transfer, 1, 2, 155–163. Johnson, M. 1982. Non-Newtonian fluid system design. Some problems and their solutions. In 8th International Conference on the Hydraulic Transport of Solids in Pipe. Johanesburg, South Africa. Cranfield, UK: BHR Group. Kemblowski, Z., and J. Kolodziejski. 1973. Flow resistances of non-Newtonian fluids in transitional and turbulent flow. Int. Chem. Eng., 13, 265–279. Kenchington, J. M. 1972. In Proceedings of the 2nd International Conference on Hydraulic Transportation of Solids.Cranfield, UK: BHR Group. Metzner, A. B., and J. C. Reed. 1955. Flow of non-Newtonian laminar, transition and turbulent regions. Am. Inst. Chem. Eng. Journal, 1, 4, 434. Molerus, O. 1993. Principles of Flow in Disperse Systems. London: Chapman and Hall. Mooney, M. J. 1931. Explicit formulas for slip and fluidity. Journal of Rheology, 2, 2, 210–222. Nunez, G. A., M. Briceno, C. Mata, and H. Rivas. 1996. Flow characteristics of concentrated emulsions of very viscous oil in water. The Journal of Rheology, 40, 3, 405–423. Park, J. T., R. J. Munnheimer, T. A. Grimley, and T. B. Morrow. 1989. Pipe flow measurements of a transparent Non-Newtonian slurry. Journal of Fluids Engineering, 111, 331–336. Porkryvailo, N. A., and Y. G. Grozberg. 1995. Investigation of structure of turbulent wall flow of clay suspensions in channel with electro diffusion method. In Proceedings of the 8th International Conference on Transport and Sedimentation of Solid Particles, Prague, Czech Republic. Rabinowitsch, B. 1929. Veber die viskositat und elastizitat von solen. Z. Phisik Chem. Ser. A., 145, 1–26. Ryan, N. M., and M. M. Johnson. 1959. Transition from laminar to turbulent flows in pipes. Amer. Inst. of Chem. Engr., 5, 433–435 Sauermann, H. B. 1982. The Influence of particle diameter on the pressure gradients of gold slimes pumping In Proceedings of the 8th International Conference on the Hydraulic Transport of Solids in Pipes. Jahannesburg, South Africa, August 1982, Paper E1, pp. 241–246. Cranfield, UK: BRHA Group. Schowalter, W. R. 1977. Mechanics of Non-Newtonian Fluids. New York: Pergamon Press. Slatter, P. T., G. S. Thorvaldsen, and F. W. Petersen. 1996. Particle roughness turbulence. In Proceedings of the 13th International Hydrotransport Symposium on Slurry Handling and Pipeline Transport, Johannesburg, South Africa. Cranfield, UK: BRHA Group. SRC. 2000. Slurry Pipeline Course, May 15–16, 2000, Saskatchewan Research Centre, Saskatoon, Canada. Szilas, A. P., E. Bobok, and L. Navratil. 1981. Determination of turbulent pressure loss Non-Newtonian oil flow in rough tubes. Rheol Acta, 20, 487–496. Thomas, A. D., and K. C. Wilson. 1987. New analysis of non-Newtonian turbulent flow, yield power law fluids. Canadian Journal of Chemical Engineering, 65, 335–338. Tomita, Y. 1959. On the fundamental formula of non-Newtonian flow. Bulletin of the Japan. Soc. Mech.. Engr., 2, 7, 469–474.

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Torrance, B. McK. 1963. Friction factors for turbulent non-Newtonian fluid flow in circular pipes. South African Mechanical Engineer, 13, 4, 89–91. Want, F. M., P. M. Colombera, Q. D. Nguyen, and D. V. Boger. 1982. Pipeline design for the transport of high-density bauxite residue slurries. In Proceedings of the 8th International Conference on the Hydraulic Transport of Solids in Pipes, Johannesburg, South Africa. Cranfield, UK: BRHA Group. Wasp, E., J. Penny, and R. Ghandi. 1977. Solid-liquid flow slurry pipeline transportation. Clausthal, Germany: Trans Tech Publications. Wilson, K. C. 1985. A new analysis of turbulent flow of non-Newtonian fluids. Canadian Journal of Chemical Engineering, 63, 539–546. Further readings Abulnaga, B. E. 1997. Channel 1.0 Computer Program For an Open Channel Slurry Flow. Fluor Daniel Wright Engineers. Vancouver, BC, Canada. Internal report. Al Fariss, T. and K. L. Pinder. 1987. Flow through porous media of a shear-thinning liquid with yield stress. Canadian Journal of Chemical Engineering, 65, 391–405. Bouzaiene, R., and D. Hassani-Ferri. 1992. A selection of pressure loss predictions based on slurry/backfill characterization and flow conditions. C.I.M. Bulletin, 85, 959, 63–68. Chlabra, R. P., J. F. Richardson, and R. Darby. 2000. Non-Newtonian flow in process industry, fundamentals and engineering application. Chemical Engineering, 107, 4, Draad, A. A., G. D. C. Kuiken, and F. T. M. Nieuwstadt. 1998. Laminar turbulent transition in pipe flow for Newtonian and non-Newtonian fluids. Journal of Fluid Mechanics, 377, D25, 267–312. Hedstrom, B. O. A. 1952. Flow of plastic materials in pipes. Journal of Industrial Engineering Chemistry, 44, 651–656. Sandall, O. C., O. T. Hanna, and K. Amurath. 1986. Experiments on turbulent non-Newtonian mass transfer in a circular tube. Am. Inst. Chem. Eng. Journal, 32, 2095–2098. Steffe, J. F., and R. G. Morgan. 1986. Pipeline design and pump selection for non-Newtonian fluid foods. Food Technology, 40, 78–85. Wheeler, J. A., and E. H. Wissler. 1965. Friction factor: Reynolds number relation for the steady flow of pseudoplastic fluids through rectangular ducts. Am. Inst. Chem. Eng. Journal, 11, 207–216.

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CHAPTER 6

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

6-0 INTRODUCTION The design of mineral processing plants and tailings disposal systems often includes gravity flows in open channels. Such flows are often called slack flows. They involve a free boundary to the atmosphere. In the past, many launders were designed using rules of thumb; however, the development of large mines requires a rigorous scientific approach. Most of the published papers on sediment transportation in open channels dwell extensively on the geophysics of canals and rivers. The field of open channel hydraulics is so vast and complex that the reader may have to consult various reference books such as Graf (1971). One subject of great interest to civil engineers is the carrying capacity of the channel for sediments. This is often termed the sediment load and is measured as a function of the flow rate, width of the channel, and sediment concentration. Using channels to transport solids has limitations due to the fact that no pumps are used to force the flow. Many papers have been published over the years on the geophysics of rivers and the maximum permissible speeds used to avoid scouring and removal of bed materials. Transferring the knowledge about scouring speeds of canals and rivers into useful information for a designer of a hydrotransport system is not a straightforward process. In fact there, is not a single unified mathematical model to represent slurry flows in open channels. In this chapter, a methodology is presented to estimate the friction losses for slurry flows in open channels, cascades, drop boxes, and distribution boxes. In the last twenty years, new developments in thickeners encouraged various operators to develop the concept of adding flocculants to launders. Tailings and concentrate slurries are thereby allowed to flow at higher and higher concentrations in gravity modes. The engineer must take into account the rheology, particularly certain aspects of high yield stress and nonNewtonian characteristics. Mineral processing plants often divide or combine flows in drop boxes, distribution boxes, and plunge pools. The design principles of such entities are presented at the end of the chapter. 6.1

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6-1 FRICTION FOR SINGLE-PHASE FLOWS IN OPEN CHANNELS The words flume, launder, open channel, and slack flow are often used to express the same thing. In the following discussion, these words will be used interchangeably. Even though launders are crucial to mining, very little research on the subject has been published. Despite the lack of reference material on launders for slurries, it is important to start from basic principles. The analysis will focus initially on water flows. The reader will then be introduced to the complexity of slurry flows. The reader should appreciate that an upper practical limit on these flows is a 65% concentration of solids by weight. Since the flow does not fill the launder or pipe, the hydraulic diameter is the defined as the equivalent diameter of flow for an open channel. The hydraulic radius is defined as the ratio of the area of the flow to the wetted perimeter. It is also called the hydraulic mean depth in certain European books. A RH = ᎏ P

(6-1)

FIGURE 6-1 Large concrete structures offer a method of conveying large quantities of slurry. This structure was built to convey 150,000 tons per day of soft high clay tailings at a Peruvian copper mine.

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And the hydraulic diameter is 4A DH = ᎏ P

(6-2)

Figure 6-2 shows various possible shapes for open launders with methods to estimate wetted area and hydraulic radius. The most common launders in mining and dredging are, however, circular and rectangular in shape. Sometimes a circular pipe is opened and vertical walls are added to produce a U-shape. The friction loss for a closed channel and steady-state single-phase flow was examined in Chapter 2. Using the Darcy factor, the head loss for an open launder can be expressed in terms of the hydraulic radius: fDLU 2 h= ᎏ 2g(4RH)

(6-3)

Since the Darcy friction coefficient fD is usually accepted as four times the fanning friction coefficient fN, Equation 6-3 for a slurry may be rewritten in terms of the fanning factor fN, discussed in Chapter 2. fNLU 2 h= ᎏ 2gRH

(6-4)

For a fully developed and uniform flow, the slope or energy gradient of an open launder is established in terms of the head loss per unit of length (Henderson, 1990): fNU 2 H S= ᎏ = ᎏ 2gRH L

2

2 < A = R2( – sin cos ) P = 2R R( – sin cos ) RH = ᎏᎏᎏ 2

(6-5)

A = BH H P = 2H + B BH RH = ᎏ 2H + B

B

2 = RH = DI/4

2

2 > =– A = R2( – sin  cos ) P = 2R R( – sin  cos ) RH = ᎏᎏᎏ 2

H

if H > R A = R[2(H – R) + R/2] R = 2(H – R) + R R[2(H – R) + R/2] RH = ᎏᎏᎏ 2(H – R) + R

2R

FIGURE 6-2 Hydraulic radius for shapes of open channels.

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or fDU 2 H S= ᎏ = ᎏ 8gRH L

(6-6)

Many models for open channel flows of water are based on the Chezy number and the Manning number. The Chezy number is inversely proportional to the square root of the friction factor: Ch =

ᎏ 冪莦 f

(6-7)

ᎏ 冪莦 f

(6-8)

8g D

or Ch =

2g N

The Manning number is a function of both the hydraulic radius and the friction factor: RH1/6 n= ᎏ Ch

(6-9)

or RH1/6 n= ᎏ 2g ᎏᎏ fN

(6-10)

冪 莦莦

Some experimental values for the Manning number “n” are shown in Table 6-1 as derived for water flows. These values are not correct for transportation of solids, particularly solids that introduce a new roughness factor that we will discuss. This table is presented as a reference for dirty water, very dilute mixtures, or decant water that are present in mining and tailings circuits but do not constitute real slurries. Green et al. (1978) summarized the research activities of the U.S. Army Corps of Engineers who derived the following relationship between the hydraulic radius and effective roughness of the channel (in USCS units): RH0.1667 n = ᎏᎏᎏ 23.85 + 21.95 log(RH/ks)

(6-11)

where RH = hydraulic radius in feet k = effective linear roughness in feet n = Manning number in ft–1/3/sec The linear roughness ks (Table 6-2) is also used to compute the flow of water in open channels. The Ministry of Transport (1969) recommends the following equation for flow of viscous liquids in open channels: 1.225(/) k 苶2苶R 苶H 苶S苶 log ᎏ + ᎏᎏ V = – 兹3 14.8 RH RH 兹3苶2苶R 苶苶S H苶

冤

where = absolute viscosity of fluid = density of fluid S = slope or energy gradient g = acceleration due to gravity

冥

(6-12)

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TABLE 6-1 Typical Values for the Manning Number “n” for Water Flows (Do Not Use for Slurries) Channel surface Glass, plastic, machined metal surface Smooth steel surface Sawn timber, joints uneven Corrugated metal Smooth concrete Cement, plaster Concrete culvert (with connection) Glazed brick Concrete, timber forms, unfinished Untreated gunite Brickwork or dressed masonry Rubble set in cement Earth excavation, clean, no weeds Earth, some stones and weeds Natural stream bed, clean and straight Smooth rock cuts Channels not maintained Winding natural channels with pools and shoals Very weedy, winding, and overgrown natural rivers Clean alluvial channels with sediments

Manning factor “n,” ft–1/3 s–1

Manning factor “n,” m–1/3 s–1

0.011 0.008 0.014 0.016 0.0074 0.011 0.009 0.009 0.014 0.015–0.017 0.014 0.017 0.020 0.025 0.020 0.024 0.034–0.067 0.033–0.040 0.075–0.150 0.031 (d75)1/6 using d75 size in feet

0.016 0.012 0.021 0.024 0.011 0.016 0.013 0.013 0.0208 0.022–0.0252 0.0208 0.0252 0.022 0.037 0.030 0.035 0.050–0.1 0.049–0.059 0.111–0.223 0.0561 (d75)1/6 using d75 size in m

After Manning (1895) and Henderson (1990).

Having read Chapters 1–3, the reader must have become aware that sizing pipe involves a straightforward relationship between the flow rate, the cross-sectional area of the pipe, and the required velocity. In the case of open launders, particularly those of rectangular and U-shape, the main concern is to avoid spills. At certain bends, around certain obstacles, or at a sudden reduction of physical slope, the flow may slow down considerably and even spill out of the launder. For straight runs away from such bends and junctions of launders, open conduits are designed to be one-third full. When pipes are used as open launders, they are typically sized to be 50% full, but in the case of tenacious froth they may be sized to be 25% or 30% full (Figure 6-3). Designers often prefer to have a steep launder rather than to suffer a loss of time unblocking settled slurry. As the above equations indicate, the friction loss factor does depend on the hydraulic radius, and larger launders tend to require less slope than small launders. Excessively steep launders tend to lose their liners through fast wear. Obtaining the correct slope without an excessive margin of safety is the correct approach to engineering. Example 6-1 A slurry of unknown properties is flowing in a half full 457 mm (18 in) pipe with wall thickness of 9.5 mm (0.375 in). The measured flow rate is 0.189 m3/s (3000 US gpm). The launder is inclined at a slope of 2%. Determine the friction factor and the Chezy and Manning numbers.

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TABLE 6-2 Recommended Values of Absolute Roughness in mm

Classification (assumed clean and new unless otherwise stated) Smooth Drawn nonferrous pipes of aluminum, brass, copper, alkathene, glass, Perspex, HDPE Plastic pipe, welded joints Plastic pipe, flanged or coupled joints Fiberglass pipe (FRP), flanged or coupled Metal Asbestos cement Spun bitumen-lined pipe Spun concrete-lined pipe Wrought iron pipe Rusty wrought iron pipe Uncoated steel pipe Coated steel pipe Rubber-lined steel pipe Plastic-lined steel pipe Galvanized iron Coated cast iron Tate relined pipes Riveted steel pipes (untuberculated)—(good = girth, riveted only; normal = full riveted, taper, or cylinder joints; poor = full riveted, butt-strap joints) Riveted steel pipes (untuberculated)—plates < 6 mm Riveted steel pipes (untuberculated)—plates > 6 mm Concrete Class 4—Monolithic construction against oiled steel surface with no surface irregularities, smooth-surfaced precast pipelines with no shoulders or depressions at the joints Class 4a—Monolithic construction in units of 2 m or over with spigot and socket joints, or ogee joints pointed internally Class 3—Monolithic construction against steel, wet-mix, or spun pre-cast pipes, or with cement or asphalt coating Class 2—Monolithic construction against rough texture precast pipe or cement gun surface (for very coarse textures, take = size of aggregate in evidence) Class 1—Precast pipes with mortar squeeze at joints Smooth trowel led surface Lined concrete pipe Unlined concrete pipe

Values of roughness ks, mm (*) Good

Normal

Poor

Average effective roughness of launder mm (**)

0.003 0.146 2.292 2.292

0.03 0.15 0.015 0.03

0.015 0.03 0.03 0.06 0.6 0.03 0.06

0.15 3 0.06 0.15

0.06 0.06 0.15

0.15 0.15 0.3

0.30 0.30 0.6

0.6 1.5

1.5 3

3 6

0.06

0.15

0.15

0.3

0.3

0.6

1.5

0.6

1.5

0.3

3 0.6

6 1.5

0.725 1.35 0.350 0.726

3.63 1.35 0.73 1.35

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TABLE 6-2 Continued

Classification (assumed clean and new unless otherwise stated) Steel construction Welded sections, unlined Rolled sections, unlined Rubber lined Plastic lined Plastic construction, free formed Clayware Pitch fiber Pitch fiber Glazed vitrified clay, very accurately lined joints Glazed vitrified clay in 1 m under 600 mm diameter Clayware, glazed vitrified clay in 1 m under 600 mm diameter Clayware, glazed vitrified clay in 0.6 m under 300 mm diameter Clayware, glazed vitrified clay in 0.6 m over 300 mm diameter Clayware, butt jointed drain tile Clayware, glazed brickwork Clayware, brickwork, well pointed Clayware, old brickwork in need of pointing Mature foul sewers constructed of materials with roughness when new, not exceeding those given for mature sewers Slimed not more than 6 mm Lime incrustations, grease, or slime not more than 25 mm thick, or even layer of fine sludge Gritty solids, lying unevenly in inverts (higher figures relate to shoals of debris at Froude number of order of 0.3 to 0.5) Unlined rock tunnels Granite and other homogeneous rocks Diagonally bedded slates (use values with design diameter) Earth channels Straight uniform artificial channels Straight natural channels, free from shoals, boulders, and weeds

Values of roughness ks, mm (*) Good

Normal

Poor

Average effective roughness of launder mm (**) 1.35 0.73 1.35 0.73 0.73

0.003

0.03

0.06 0.15 0.3

0.3 0.6

0.15

0.3

0.3

0.6

0.6 0.6 1.5

1.5 1.5 3 15

3 3 6 30

0.6 6

1.5 15

3 30

60

150

300

60

150 300

300 600

15 150

60 300

150 600

1.35

*Data from the Ministry of Technology, United Kingdom (1969), Hydraulics Research Paper 4, Tables for the Hydraulic Design of Storm-drains, Sewers and Pipe-Lines. **Data from Green (1978).

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H = Z/3

H = Z/3

H=R Z

Z

H 2R

B

FIGURE 6-3 Recommended degree of fill for open channel slurry flows (does not apply to junction boxes).

Solution in Metric Units Q = 3,000 US gpm × 3.785 = 11,355 L/min = 0.1893 m3/s ID of pipe = 438.15 mm Area for a half full pipe = /8 × 0.438152 = 0.0754 m2 Velocity = 0.1893/0.0754 = 2.51 m/s Wetted perimeter = × 0.43815/2 = 0.688 m Hydraulic radius = A/P = 0.0754/0.688 = 0.1095 Slope = 2% 2

fN(2.51)2 fNV 0.02 = ᎏ ᎏᎏ = 2.93 fN 2gRH 2 × 9.81 × 0.1095 Fanning friction factor, fN = 0.0068 The Darcy factor, fD = fN × 4 = 0.027 The Chezy number, Ch =

ᎏ = ᎏ = 53.71 m 冪莦 冪莦 f 0.0068 2g

2 × 9.81

1/2

/s

N

The Manning number, n = RH1/6/Ch = 0.10951/6/53.714 = 0.0129 m–1/3/s Solution in USCS Units Q = 3000 US gpm = 3000/7.4805 = 401.04 ft3/min ID of a pipe = 17.25 in = 17.25/12 = 1.4375 ft Area of a half full pipe A = (/8)1.43752 = 0.8115 ft2 Velocity = P/A = 401.04/0.8115 = 494.21 ft/min (8.237 ft/s) P = D/2 = × 1.4375/2 = 2.258 ft Hydraulic radius = A/P = 0.8115/2.258 = 0.359 ft

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SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

Slope = 2% fN × 8.2372 fNV 2 0.02 = ᎏ = ᎏᎏ = 2.935 fN 2gRH 2 × 32.2 × 0.359 The fanning friction factor, fN = 0.0068 The Darcy factor, fD = fN × 4 = 0.027 The Chezy number, Ch =

ᎏ = ᎏ = 97.2 ft 冪莦 冪莦 f 0.0068 2g

2 × 32.2

1/2

/s

N

The Manning number, n = RH1/6/Ch = 0.3591/6/97.2 = 0.0087 ft–1/3/s The reader should be careful when using the Manning roughness “n.” Contrary to common belief, it is not a nondimensional number and its value changes from SI units to USCS units by the ratio of conversion from feet to meters to the power of 1/3 or 0.673.

6-2 TRANSPORTATION OF SEDIMENTS IN AN OPEN CHANNEL Determining the Chezy number for slurries is a method of approaching the design of launders. Julian et al. (1921) measured an average Chezy number of 80 ft1/2/s for rectangular launders (with width = twice the depth) of minimum wetted perimeter for carrying slime overflow and average stamp-battery pulp in cyaniding gold and silver circuits. Classical theories of suspended solids in open channels are based on two-dimensional turbulent flow. Consider a two-dimensional turbulent flow with a velocity U in the horizontal x-direction and V in the vertical y-direction. Reynolds (1895) defined the shear stress parallel to x on a plane normal to y as

= – (U⬘V⬘)average

(6.13)

where

= density of fluid and (U⬘V⬘)average = average of the fluctuations of the turbulent velocities Boussinesq (1877) developed an equation in the form of dU = m ᎏ dy

(6-14)

where m = the eddy viscosity, analogous to the dynamic viscosity discussed in Chapter 2. m = the coefficient of exchange of momentum between neighboring streams of the fluid, expressed in m2/s or ft2/sec. Von Karman (1935) developed the following equation: dU = V⬘Lmix ᎏ dy where  = correlation coefficient ⬵ 1.0 (see Section 6.2.3) V⬘ = average of absolute values of fluctuations normal to the main flow Lmix = mixing length

(6-15)

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By substituting Equation 6-15 into Equation 6-13 it is concluded that

m = V⬘Lmix

(6-16)

A rate of transfer of mass of suspended particles per unit area is defined as dC dm ᎏ = –V⬘Lmix ᎏ dt dy

(6-17)

But the values of , V⬘, and Lmix are not necessarily equal in magnitude in Equations 6-14 and Equation 6-17. C = concentration of suspended solids O’Brien (1933) studied the suspension of sediments in an open channel flow. He developed a theory that the rate of transfer of solids upward must be in equilibrium with the downward exchange of momentum due to gravitational forces: dC VtCy = –V⬘Lmix ᎏ dy

(6-18a)

dC VtCy = s ᎏ dy

(6-18b)

where Cy = volume concentration of solids at level y y = distance from the lower boundary s = mass transfer coefficient for sediments, similar to m but not necessarily equal to it. Solving Equation 6-18 yields C loge ᎏ = Ca

冕

dC ᎏ dy

y

a

(6-19)

where Ca is the concentration of solids at an arbitrary reference plane of height “a.” If s is constant over the depth, then s(y) = constant. Equation 6-18 is then solved to give C ᎏ = e–J Ca

(6-20)

where J = (y – a)(Vt/s). The correct procedure consists of establishing a relationship between s and the vertical coordinate y before solving Equation 6-18. In the absence of detailed information about the relationship between s and m, they are assumed to be equal so that

m = –V⬘Lmix = ᎏ dU/dy

(6-21)

Substituting Equation 6-21 into Equation 6-18 yields the concentration of distribution C loge ᎏ = –Vt Ca

冕

y

a

dU/dy ᎏ dy

(6-22)

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6.11

In a uniform open channel with a large ratio of width to depth, the shear stress is expressed as

冢

ym – y = w ᎏ ym

冣

(6-23)

where w = shear stress at the wall ym = distance from the boundary to the liquid surface Substituting Equation 6-23 into Equation 6-21 yields C loge ᎏ = –Vt Ca

冕

y

a

dU/dy ᎏᎏ dy (1 – y/ym)w

(6-24)

The velocity gradient is expressed in the form of the universal defect law as

冢 冣

U – Umax 1 2y ᎏ = ᎏ loge ᎏ 兹苶 (w苶/苶苶) Kx DI

(6-25)

For a pipe, Kx = 0.4 and y is the distance from the internal wall at the bottom of a horizontal pipe. Keulegan (1938) demonstrated that Equation 6-24 applied to open channels. For a wide-open channel, the value of ym, or depth of flow, is used:

冢 冣

U – Umax 1 y ᎏ = ᎏ loge ᎏ 兹苶 苶 苶 (苶 / ) K y w x m

(6-26)

In Chapter 2, the friction velocity Uf was defined as Uf =

w

ᎏ =U ᎏ 冪莦 冪莦2 fN

Substituting Equation 6-6 yields 苶R 苶H 苶S 苶)苶 Uf = 兹(g

(6-27)

w = gRHS

(6-28)

where fN = the fanning friction factor U = the mean velocity of the flow S = slope Equation (6-28) is called the DuBoys equation (Wood, 1980). It clearly establishes that for a slurry to move at a density , a minimum level of shear stress must be available: ᎏ 冤冢 ᎏ y 冣y –a冥

C Vt loge ᎏ = ᎏ loge Ca KxUf

(6-29)

m

冢 冣

C h ᎏ= ᎏ Ca ha

a

ym – 1

Z

(6-30)

where Vt Z = ᎏᎏ Kx兹g苶y苶S m苶

(6-31)

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ym h= ᎏ –1 y

(6-32)

ym – a ha = ᎏ a

(6-33)

Equation 6-30 establishes that the relative concentration of solids depends on their vertical position and on the factor Z, which is a function of the ratio of the terminal velocity of the particles Vt to the group KxUf. It is therefore a measure of the intensity of turbulence.

6-2-1 Measurements of the Concentration of Sediments Determining the magnitude of the plane “a” is the starting point to solve Equation 6-28 and 6-29. Rouse (1937) suggested the height “a” be equal to the height of the roughness elements ks. He suggested using Equation 6-19, assuming an interval from y = 0 to y = ks, and by assuming that (y) = (ks) = constant. Rouse indicated that at y = 0 the solids concentration corresponding to the bed of sediments should be used. Richardson (1937) reported test results for a 305 mm × 305 mm × 1830 mm (1 ft × 1 ft × 6 ft) flume and indicated that in the boundary region the concentration of sediments was inversely proportional to the vertical coordinate y, but that in open streams it conformed better with Equation 6-30. Vanoni (1946) conducted a series of tests in an 838 mm (33 in) wide by 18.29 m (60 ft) long flume to validate Equation 6-29 (Figure 6-4). Average velocity was noticed to occur at 0.368 ym or the depth of the liquid. The velocity profile followed a logarithmic function of depth. The following results were obtained by Vanoni (1946). 앫 The sediment concentration profile followed the pattern set by Equation 6-30. However, the exponent was smaller than the value of Z expressed by Equation 6-31 when the sediments became coarser. 앫 A random turbulence was observed and slip between fluid and sediment was suspected as the sediment accelerated. Thus, the assumption that mass and momentum transfer were equal was not satisfied, as the theory did not account for slip and random turbulence. 앫 For fine materials, the coefficient of sediment mass transfer was smaller than the coefficient of momentum transfer. 앫 For coarse materials, the coefficient of sediment mass transfer was larger than the coefficient of momentum transfer. 앫 Suspended load decreased the coefficient of mass transfer. The reduction was more important with fine solids than with coarse solids. 앫 Suspended load reduced the value of the Von Karman constant. Values of Kx between 0.314 and 0.342 were measured (by comparison with 0.4 for full pipes). The reduction of the Von Karman coefficient indicated a reduced level of mixing and a tendency by the sediments to suppress turbulence. 앫 The suspended load tended to reduce the resistance to flow. Sediment-laden water moved faster than clear water and the Manning roughness number decreased with the sediment load, as shown in Figure 6-5.

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FIGURE 6-4 Velocity profile for the flow of a sand–water mixture in a rectangular open channel. (From Vanoni, 1946, by permission of ASCE.) 6.13

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FIGURE 6-5 Variation of the equivalent roughness, Von Karman coefficient, and Z1 with the weight concentration of the sand–water mixture. (From Vanoni, 1946, by permission of ASCE.)

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앫 Suspended load tended to cause a flow to become unevenly distributed. Unsymmetrical sediment distribution within the flow caused secondary circulation. 앫 The velocity distribution near the center of the flume followed Von Karman universal defect law. Example 6-2 Iron sand is flowing in a rectangular open channel that is 300 mm wide × 120 mm high at a speed of 3 m/s. Assume the Von Karman coefficient K = 0.33. The average particle size is 0.3 mm. Using Einstein’s approaches, it is assumed that the reference layer “a” is to be twice the particle size diameter. The flume is one-third full (i.e., ym = 40 mm). The slurry weight concentration is 45%, the dynamic viscosity is 1 cP, and the specific gravity of sand is 4.1. Calculate C/Ca if the slope is 3% at depth with 2% increments of depth. Ignore any dunes. Assume  = 1.0. Solution in Metric Units The first approach is to determine the terminal velocity of the sand. The Particle Reynolds number is: dpVm/ = (0.3 × 10–3 × 3 × 1000/1 × 10–3) = 900 This is turbulent flow. By Newton’s law (Equation 3-13): Vt = 1.74[g(s – L/L)]0.5dp0.5 Vt = 1.74[9.81 × 3.1 × 0.3 × 10–3]0.5 Vt = 0.167 m/s Using Equation 6-31: Z = 0.167/[0.33 × (9.81 × 0.04 × 0.03)0.5] Z = 4.664 Applying Equation 6-29: C/Ca = (h/ha)4.664 As it will be explained later in this chapter, Einstein proposed that the value of “a” be equal to twice the grain diameter, or in this case 0.6 mm = twice the particle size of sand. Let us calculate the concentration of solids at 2% of the depth of the flume. 2% of depth = 0.02 × 40 = 0.8 mm: ym h = ᎏ – 1 = h = (1/0.02) – 1 = 49 y ym – a ha = ᎏ a ha = (40/0.6) – 1 = 65.67 h ᎏ = 49/65.67 = 0.735 ha C ᎏ = 0.7354.664 = 0.2378 Ca

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4% of depth: h = (1/0.04) – 1 = 24 h ᎏ = 24/65.67 = 0.3655 ha C ᎏ = 0.36554.664 = 9.15 × 10–3 Ca So most of the solids will be in the bottom 4% of the launder. Example 6-3 Iron sand (SG = 4.1) is flowing in a rectangular channel 600 mm wide × 120 mm high. The liquid level is 60 mm high. The slope is 1% and the liquid is flowing at 1.5 m/s. The particle average diameter is 0.5 mm. The specific gravity of the solids is 4.1 and the dynamic viscosity of the mixture is 1.5 cP. Calculate C/Ca at 4% intervals. Assume Von Karman Kx = 0.33. Ignore any dunes. Assume  = 1.0. Solution The particle Reynolds number is : 0.5 × 10–3 × 1.5 m/s × 1000/1.5 × 10–3 = 500 This is transition flow. From Chapter 3, Allen’s law would apply:

s/L Vt = 0.20 g ᎏ L

冢

冣

0.72

d 1.8 ᎏ (/)0.45

(0.5 × 10–3)1.8 ᎏ Vt = 0.20(9.81 × 3.1)0.72 ᎏᎏ (1.5 × 10–6)0.45 Vt = 1.116 mm/s Using Equation 6-30: Z = 1.116 × 10–3/[0.33 (9.81 × 60 × 10–3 × 0.01)0.5] Z = 3.3822 × 10–3/0.0767 = 0.044 The magnitude of “a” is assumed to be twice the particle’s diameter: a = 2dp = 2 × 0.5 mm = 1 mm C ᎏ = (h/ha)0.044 Ca 60 mm ha = ᎏᎏ = 59 1 mm – 1 Let us calculate the concentration at 2% depth: y = 0.02 × 60mm = 1.2 mm 1 h = ᎏ = 49 0.02 – 1

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h ᎏ = 49/59 = 0.83 ha C ᎏ = 0.830.044 = 0.99 Ca 4% depth: y = 2.4 mm 1 h = ᎏ = 24 0.04 – 1 h ᎏ = 24/59 = 0.4067 ha C ᎏ = 0.40670.044 = 0.96 Ca 8% depth: y = 4.8 mm 1 h = ᎏ = 11.5 0.08 – 1 C ᎏ = (11.5/59)0.044 = 0.931 Ca 12% depth: y = 7.2 mm C ᎏ = (7.33/59)0.044 = 0.912 Ca 16% depth: y = 9.6 mm C ᎏ = (5.25/59)0.044 = 0.899 Ca 20% depth: y = 12 mm C ᎏ = (4/59)0.044 = 0.888 Ca 24% depth: y = 14.4 mm C ᎏ = (3.17/59)0.044 = 0.879 Ca

6.17

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28% depth: y = 16.8 mm C ᎏ = 0.871 Ca 32% depth: y = 19.2 mm C ᎏ = 0.864 Ca Examples 6-2 and 6-3 show the effect of slope on the distribution of solids. In the case of Example 6-2, the slope is higher at 3% and most of the solids move at the bottom of the channel. In Example 6-3, the slope is low at 1% and all the sand is mixed with the water. Graf (1971) reviewed the experiments conducted by various researchers and a tendency developed to measure an empirical Z1 as a substitute for Z. He summarized the work of Einstein and Chien (1955) who developed an approximate relationship between Z and Z1: Z Z1 = ᎏᎏᎏᎏᎏ LZ兹(2 苶 苶)苶 2 Z L exp(–L2Z 2/) + ᎏᎏ exp(–x2/2)dx (2苶 兹苶 苶) 0

冕

(6-34)

where x = loge y L = loge(1 + RKx) The best fit occurs when RKx = 0.3. Einstein (1950) called the flow layer right on top of the bed the “bed layer,” and indicated that it would be impossible to have suspension of the solids there. He measured a thickness of layer t = 2dp. The material within this layer was the source of the suspended load and established the lower limit for Ca. Einstein then proceeded to derive very complex equations that require numerical integration. It would be beyond the scope of this book to dwell on such equations.

6-2-2 Mean Concentrations for Dilute Mixtures (Cv < 0.1) Celik and Rodi (1991) published data suggesting that Einstein’s equations were overestimating the concentration at a = 2d. For fairly dilute suspensions with a volumetric concentration of less than 10%, which is common in a lot of applications, they proposed a more simplified approach, defining the suspended sediment load qbs as qbs = qbCT =

冕

RH

␦a

where qbs = flow rate of sediment per unit width qb = total flow rate of mixture per unit width C = time averaged (mean) concentration CT = mean transport capacity concentration U = time averaged velocity in x-direction RH = hydraulic radius or mean depth of liquid

LUCdy

(6-35)

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␦a = depth of bed load layer y = vertical coordinate above bottom of channel Defining parameter Cm as the depth-averaged concentration, Celik and Rodi (1991) used their results from their previous paper (1984) to conclude that Cm/CT ⬇ 1.13, and derived the following equation:

w Um CT = ᎏᎏ ᎏ (s – L)gym Vt

(6-36)

where is a constant of proportionality (⯝ 0.034 from tests on sand). The exercise consists of calculating CT. For this purpose, these two authors discussed the problem of the effective shear stress. Reviewing the work of other authors and their own, Celik and Rodi (1991) pointed out that there are certain important factors in the turbulent regime, such as: 앫 When there are elements causing increased roughness, the flow separates, and only a part of the shear stress determined from the energy slope is effective in moving the particles in suspension. The work of gravity is then used partly to overcome friction at the bed. 앫 The turbulent energy, which is produced in a turbulent regime, is related to the total shear stress, but the drag force is then not effective in maintaining the particles in suspension. 앫 The turbulence energy of the upper layers needs to be transferred by convection and diffusion first to the region above the bed. Smaller quantities of energy are then available to suspend the bed in the presence of large amounts of roughness. 앫 The mean velocities and the wall shear stresses in separated regions are much smaller than in the areas where the flow is still attached. 앫 The presence of separated regions in the flow and stagnant areas cause the particles to settle in dead water zones and it becomes very difficult to resuspend them. 앫 The permeability of a bed increases the resistance of the bed. (Zippe and Graf, 1983). 앫 For rivers, a typical value of the ratio of friction velocity to average velocity (Uf /U) is 0.05. Tests conducted by Van Rijn (1981) and by Apmann and Rumer (1967) indicate that the flow over an approximately flat bed of loose sand shows great similarity to the characteristics of flow over a rough surface. To take in to account all these effects in the turbulent regime, Celik and Rodi (1991) proposed an equation for the effective shear rate:

e = [1 – (ks/ym)]w

(6-37)

where ks = the equivalent resistance parameter (in most cases the roughness height or absolute roughness) = empirical constant ( = 0.06 in tests obtained by these authors) ym = average depth of liquid in flume In conclusion, Celik and Rodi (1984) established a simplified relationship between roughness and friction velocity for dilute mixtures as

冤

ks Um ᎏ = Er exp –1 – Kx ᎏ ym Uf

冥

(6-37)

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where Er ⬇ 30 Kx = the Von Karman constant Substituting Equation 6-37 into Equation 6-36 and using the effective shear stress yields

w Um CT = [1 – (ks/ym)] ᎏᎏ ᎏ (s – L)gym Vt

(6-39)

Equation 6-39 is plotted in Figure 6-6. From test data, Celik and Rodi (1991) obtained a value where = 0.034 and = 0.06 for flow over a flat bed of loose sand without large dunes or antidunes. The particle diameter was between 0.005 mm and 0.6 mm and volumetric concentration was limited to 10%. On a logarithmic scale, the slope he obtained was 0.034 in the range of CT from

ks F= 1– ᎏ ym

Um2

Um

ᎏ 冤 冢 冣 冥 ᎏᎏ ( Ⲑ – 1)gy V s

L

m

t

FIGURE 6-6 The volumetric capacity CT from Celik and Rodi (1991). (Reprinted by permission of ASCE.)

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0.00001 to 0.10. Nevertheless the authors pointed that the value of is very empirical and strongly dependent on the friction velocity Uf. Unfortunately, the did not provide the correlation, and this leaves the engineer facing a situation of measuring such a value or iterating from case to case.

Example 6-4 Fine sand with an average particle diameter dp of 0.3 mm (0.000984 ft) and SG = 2.625 is transported at a volumetric concentration CV = 7.91%. The volume flow rate is 1200 m3/hr (42,378 ft3/hr). The launder is 600 mm (1.97 ft) wide and the height of the liquid is 200 mm. Determine the slope of the launder using the Celik–Rodi method. Assume Kx = 0.33, = 0.06, and = 1.5 cP (or 3.13 × 10–5 lbf-sec/ft2). Solution in SI Units Cv Ct = ᎏ = 0.07 1.3 The average speed is Q Um = ᎏ = (1200/3600)/(0.2 × 0.6) = 2.79 m/s A Assuming the roughness of the bed to be equal to twice the average particle diameter, ks = 0.6 mm From equation 6-38:

冤

ks Um ᎏ = Er exp –1 – Kx ᎏ ym Uf

冥

冤

2.79 = 30 exp –1 – 0.33 ᎏ Uf

冥

0.9207 ln(0.001) = 1 – ᎏᎏ Uf Uf = 0.112 m/s Using Equation 6-27: Uf = 兹g 苶R 苶苶S H苶 The hydraulic radius for this rectangular channel is (0.6 · 0.2) RH = ᎏᎏ = 0.12 m (0.6+0.2+0.2) Uf = 0.1395 m/s = (9.81 × 0.12 × S)1/2 S = 0.0107 or 1.07%

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Solution in USCS Units Cv Ct = ᎏ = 0.07 1.3 Area of flow = 1.97 × 0.656 = 1.293 ft2 The average speed is Q Um = ᎏ = [(42378 ft3/hr)/3600]/1.293 = 9.1 ft/sec A Assuming the roughness of the bed to be equal to twice the average particle diameter, ks = 0.0236 in If the depth is 8 in: ks ᎏ = 0.0236/8 = 0.003 ym From Equation 6-38:

冤

ks Um ᎏ = Er exp –1 – Kx ᎏ ym Uf

冥

冤

9.1 = 30 exp –1 – 0.33 ᎏ Uf

冥

3.003 ln(0.001) = 1 – ᎏ Uf Uf = 0.38 ft/sec Using Equation 6-27: Uf = 兹g 苶R 苶H 苶S苶 The hydraulic radius for this rectangular channel is RH = (1.987 · 0.65)/(1.987 + 0.65 + 0.65) = 0.391 ft Uf = 0.38 ft/sec = (32.2 × 0.391 × S)1/2 S = 0.011 or 1.1%

6-2-3 Magnitude of  Carstens (1952) demonstrated that  never exceeds unity (1.0). For fine particles,  ⬇ 1.0 or s ⬇ . For coarse particles,  < 1.0 or s < .

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Brush et al. (1962), Matyukhin an Prokofyev (1966) and Majumdar and Carstens (1967) have confirmed that  = 1.0 or s ⯝ for fine particles. For coarse particles,  < 1.0 or .s < .

6-3 CRITICAL VELOCITY AND CRITICAL SHEAR STRESS Graf (1971) established a relationship between the forces for the incipient movement of a set of loose, cohesionless solid particles and the angle of repose as FT tan = ᎏ FN where FN = force normal to angle of repose FT = force tangential to angle of repose These two forces are the resultants of the lift and drag forces (discussed in Chapter 3) referred to in Figure 6-7: FN = W cos ␣ – L FT = W cos ␣ + D W cos ␣ + D tan = ᎏᎏ W cos ␣ – L

(6-40)

The surface area resisting motion is expressed in terms of a shape factor 1 and the particle diameter d. The surface area associated with lift is expressed in terms of a shape factor 2 and the particle diameter d: L = 0.5CLU b2 2d 2

(6-41)

D = 0.5CDU b2 1d 2

(6-42)

where Ub is the bed velocity. The submerged weight of the particle is expressed as a shape factor and the diameter of the particle: W = 3gd 2(s – )

(6-43)

Substituting Equations 6-40, 6-41, and 6-42 into 6-43 establishes the relationship between the critical velocity and the actual shape and density of the particle: 2 23(tan cos ␣ – sin ␣) Ubc ᎏᎏ = ᎏᎏᎏ (s/L – 1) CD1 + CL 2 tan

(6-44)

where Ubc is the the critical bed velocity to start the motion of the particles. Graf (1971) defined the right-hand part of Equation 6-44 as the sediment coefficient . Fortier and Scobey (1926) conducted extensive experiments on permissible canal velocities to understand the erosion and transportation of sediments. Their results are presented in Table 6-3. Their main conclusions which are still valid today, were

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ity

loc

Ve

Lift

Drag Weight

FIGURE 6-7 Lift and drag forces on sediments in an open channel.

앫 The laws governing the transport of silt and detritus in open channels are very distantly related to the laws governing scouring of the canal bed and are not directly applicable. 앫 The material of seasoned canal beds consists of solids of different shapes and sizes. When the fines fill the interstices between the coarser solids, they form a dense and stable mass that is more resistant to the erosion of water. 앫 The velocity required to scour the bedded canal is much higher than the velocity required to suspend particles outside the bed. 앫 Colloids tend to cement clay, sand, and gravel in such a way that the compound mixture resists erosion. 앫 The grading of materials ranging from fine to coarse, coupled with the adhesion between colloids and these solids, makes it possible to operate at high velocities without appreciable scouring effects. Neill (1967) derived the following equation to estimate the critical velocity for coarse particles in launders: 2 dP U bc ᎏᎏ = 2.50 ᎏ (s/ – 1)gdP ym

冤 冥

–0.20

(6-45)

The term dP/ym is sometimes called “relative sand roughness.” In Chapters 3 and 4 we examined definitions of velocity during all periods of movement from settling to deposition, etc. Similarly, in open channels the critical scour velocity is well above the sedimentation velocity (or equal to the terminal velocity in full pipe flow). Sometimes engineers confuse these two terms, although they are quite different in magnitude. The critical shear stress is at the point of the incipient motion, or at which the motion starts, and is expressed as

cr ᎏᎏ = (s – )d

(6-46)

where is the the sediment coefficient. Thomas (1979) argued that sand particles finer than 0.15 mm would be completely enveloped in the viscous sublayer so that the critical shear stress could be simplified to

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cr = 1.21 m[g (s/L – 1)]2/3

(6-47)

With = kinematic viscosity. Wilson (1980) indicated that this correlated well with the work of Ambrose (1953), who had indicated a change of resistance as the grains of sand became of the order of magnitude of the roughness of the flume. Substituting Equation 6-47 into the DuBoys equation (6-28), the critical slope to initiate motion of a bed of small sand particles in an open channel is

w = mgRH S = 1.21 m[g (s/L – 1)]2/3

TABLE 6-3 Permissible Canal Velocities (after Fortier et al., 1925) Velocity, after aging, of canals at a depth of 910 mm (3 ft) or less

Original material excavated for the canal Fine sand (noncolloidal) Sandy loam (noncolloidal) Silt loam (noncolloidal) Alluvial silts when noncolloidal Ordinary firm loam Volcanic ash Fine gravel Stiff clay (very colloidal) Graded loam to cobbles, when noncolloidal Alluvial silts when colloidal Graded, silt to cobbles, when colloidal Coarse gravel (noncolloidal) Cobbles and shingles Shales and hard pans

Clear water, no detritus ____________ m/s ft/s 0.45 0.54 0.61 0.61 0.84 0.84 0.84 1.15 1.15 1.15 1.22 1.22 1.5 1.83

1.5 1.75 2.0 2.0 2.5 2.5 2.5 3.75 3.75 3.75 4.0 4.0 5.0 6.0

Water transporting colloidal silts ____________ m/s ft/s 0.84 0.84 0.91 1.07 1.07 1.07 1.52 1.52 1.52 1.52 1.67 1.83 1.67 1.83

2.5 2.5 3.0 3.5 3.5 3.5 5.0 5.0 5.0 5.0 5.5 6.0 5.5 6.0

Water transporting noncolloidal silts, sands, gravel, or rock fragments _____________ m/s ft/s 0.45 0.61 0.61 0.61 0.69 0.61 1.15 0.91 1.52 0.91 1.52 1.98 1.98 1.5

TABLE 6-4 Effects of Sediment Load on Equivalent Roughness (after Vanoni, 1946) Average sediment load in grams/liter

Ratio of equivalent roughness to size of bottom sand

0 0.17 3.21 7.36 16.2

0.328 0.282 0.190 0.110 0.072

1.5 2.0 2.0 2.0 2.25 2.00 3.75 3.0 5.0 3.0 5.0 6.5 6.5 5.0

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The critical slope to start motion is 1.21[(s/L – 1)]2/3 Scrit = ᎏᎏᎏ RHg1/3

(6-48)

Equation (6-47) confirms the correlation between the average depth of the slurry, the weight of solids, the viscosity and density of the mixture, and the minimum slope. Example 6-5 A slurry mixture of fine particles and water has a specific gravity of solids 4.1 and is flowing in a partially filled rectangular launder 600 mm wide. Determine the critical slope if the launder is to flow one-third full. The density of the mixture is 1250 kg/m3 (2.42 slugs/ft3) and the dynamic viscosity is 20 mPa · s (4.17 × 10–4 lbf-sec/ft2). Solution in Metric Units The kinematic viscosity of the mixture is 20 × 10–3/1250 = 0.000016 m2/s. If the launder is one-third full, the hydraulic radius is 0.6 × 0.2 RH = ᎏ = 0.12 m 0.6 + 0.4 From Equation 6-47: 1.21[ (s/L – 1)]2/3 1.21[16 × 10–6(4.1 – 1)]2/3 ᎏᎏᎏ = 0.0063 or 0.63% Scrit = ᎏᎏ 0.12 × 9.811/3 RHg1/3 is the minimum slope to start motion of the slurry in these conditions. Solution in USCS Units The width of the launder is 1.97 ft, the height of the liquid would be 0.66 ft, and the hydraulic radius is 1.97 × 0.66 RH = ᎏᎏ = 0.395 ft 1.97 + 0.66 The kinematic viscosity of the mixture is 4.17 × 10–4 lbf-sec/ft2 ᎏᎏᎏ = 1.723 × 10–4 lbf-sec-ft/slugs 2.42 slugs/ft3 From Equation 6-47 1.21[(s/L – 1)]2/3 1.21[1.723 × 10–4 (4.1 – 1)]2/3 ᎏᎏᎏᎏ Scrit = ᎏᎏᎏ = = 0.0063 or 0.63% RHg1/3 0.395 × 32.21/3 is the minimum slope to start motion of the slurry in these conditions. This analysis was extended by Wilson (1980) for sand flowing in partially filled pipes. Wilson defined three regimes for sand flowing in open launders with particle diameters in the range of 0.02 mm to 4 mm (mesh 625–5):

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1. Homogeneous flow occurs at a slope S < 0.0006/DI 2. Optimum slope for heterogeneous flow S = 0.10(dp/1000)1.5D I–0.7 3. Fully blocked pipe occurs at S > 0.42 For rocks with solid specific gravity between 1.5 and 6.0, Wilson (1980) extended the analysis and stated, for homogeneous flow: 0.6 [ (s/L – 1)]0.6 S1 ⱕ ᎏᎏᎏ × 10–3 DI 1650

(6-49)

31.6 × 10–3 d p1.5[(s/L – 1)]1.2 S1 < S2 ⱕ ᎏᎏᎏᎏ D I0.7 1.651.2

(6-50)

0.42[(s/L – 1)]0.35 S3 ⱖ ᎏᎏ 1.650.35

(6-51)

or heterogeneous flow:

for blocked flow:

Example 6-6 A slurry mixture of fine particles and water of d85 = 0.08 mm with a density of solids 4100 kg/m3 is flowing in a partially filled circular pipe with an inner diameter of 438 mm (17.25 in). The pipe inner diameter is 438 mm (17.25 in). Determine the minimum slope for flow as a heterogeneous mixture and the slope for a blocked pipe. Solution from Equation 6-50 For heterogeneous flow: 31623d p1.5[(s/L – 1)]1.2 S2 ⱕ ᎏᎏᎏ D I0.7 1.651.2 316230.000081.5 [(4.1- 1)]1.2 S2 = ᎏᎏᎏ 0.4380.7 1.651.2 S2 = 0.086 or 8.6% The slope for a blocked pipe is determined from Equation 6-51 as 52.4%.

6-4 DEPOSITION VELOCITY The terrains over which tailing flumes are built do not always have an appropriate slope. If the flow operates in a subcritical flow regime, the engineer must calculate a realistic estimation of the deposition velocity. Dominguez et al. (1996) published an equation based on experimental data measured at Codelco and the Chilean Research Center of Mining and Metallurgy. For cases where the viscosity effects are negligible

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8gRH (S – m) VD = 1.833 ᎏᎏ m

冤

冥 冢ᎏ R 冣 1/2

d85

0.158

(6-52)

H

However, in cases where the dynamic viscosity of the carrier liquid is instrumental, such as with alkaline water, Dominguez et al. (1996) derived the following equation: 8gRH (S – m) VD = 1.833 ᎏᎏ m

冤

冥 冢ᎏ R 冣 1/2

d85

0.158

1.2(3,100/J)

(6-53)

H

where J = RH(gRH)1/2/m m = the absolute viscosity of the mixture A comparison between the deposition velocity as calculated by Equation 6-52 and experimental data is presented in Figure 6-8. Example 6-7 A slurry mixture of coarse particles of d85 = 12 mm with Cw = 40%, density of solids 4100 kg/m3, and a specific gravity 4.1 is flowing in a circular pipe with an inner diameter of 438 mm (17.25 in). Assuming that the pipe is to be half full, determine the deposition velocity. Solution RH = D/4 = 0.438/4 = 0.1105 m (4.35 in)

m = 1000/[1 – 0.4(4100 – 1000)/4100] = 1433kg/m3 Using Equation 6-51:

FIGURE 6-8 Correlation between calculated and experimental measurements of the deposition velocity of coarse slurries. (From Dominguez et al., 1996, reprinted by permission of BHR Group.)

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VD = 1.833[8 × 9.81 × 0.1105 (4100 – 1433)/1433]0.5 (0.012/0.1105)0.158 VD = 1.29 × 4.0174 = 5.183 m/s Green et al. (1978) and the ASCE and WPCF (1977) proposed that Camp’s equation for the self-cleaning speed of sewers be used in the design of slurry launders: 8Ke s – m Vsc = ᎏ dpg ᎏ fD m

冤

冢

冣冥

1/2

(6-54)

where Ke = an experimental constant (the ASCE recommends a constant of 0.06 for grit chambers, whereas Green recommends a constant of 0.8 for sewers) Vsc is expressed in m/s (ft/s) d is expressed in m (ft) g = 9.81 m2/s (32 ft/s) To use this equation, one must first determine the Darcy friction factor and then solve by iteration. Because the average speed is a logarithmic distribution of depth, it would be wise to design launders with a mean speed equal to twice the self-cleaning speed. Example 6-8 Calculate the self-cleaning speed of Example 6-1, assuming a sewer flow where Ke = 0.8, solids at a SG of 3.1, Cw = 20%, and dp = 2 mm. Solution in Metric Units In Example 6-1, the Darcy factor was calculated to equal fD = fN × 4 = 0.027.

m = 1000/[1 – 0.2(3100 – 1000)/3100] = 1157 kg/m3 Vsc = [8 × 0.8 × 2 × 10–3 × 9.81(3100 – 1157)/(1157 × 0.027)]1/2 Vsc = 2.79 m/s Solution in USCS Units In Example 6-1, the Darcy factor was calculated to equal fD = fN × 4 = 0.027. dp = 6.56 × 10–3 ft Sm = 1/[1 – 0.2(3.1 – 1)/3.1] = 1.157 Vsc = [8 × 0.8 × 6.56 × 10–3 × 32.2(3.1 – 1.157)/(1.157 × 0.027)]0.5 Vsc = 9.2 ft/s

6-5 FLOW RESISTANCE AND FRICTION FACTOR FOR HETEROGENEOUS SLURRY FLOWS Equations 6-6 to 6-10 established the parameter for friction loss of single-phase Newtonian flows in open channels. Equation 2-25 established the correlation between friction factor and friction velocity. Two approaches are often used by designers of launders; one is based on the friction factor and the other on the velocity.

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6-5-1 Flow Resistances in Terms of Friction Velocity Liu (1957) and Acaroglu (1968) indicated that the ratio of the friction velocity to the settling speed of a particle were implicitly related and proposed the following function:

冤

Uf dUf ᎏ = fct ᎏ Vt

冥

(6.55)

Graf (1971) proposed including the Froude number to reflect the degree of turbulence:

冤

Uf Uav dUf ᎏ = fct ᎏ ’ ᎏ Vt 兹(g 苶y苶 苶 m)

冥

where ym is the average depth of the fluid. This is confirmed in studies by Garde and Dattari (1963) and Bogardi (1965). The presence of sand dunes at the bottom of a channel with a typical wavelength is a function of the Froude number (Kennedy, 1963). As shown in Figure 6-9, antidunes are formed in the regime of critical flow (0.8 < Fr < 1.5). The presence of dunes increases the effective wall shear stress in the form of profile drag. Graf (1971) proposed to establish a hydraulic radius RH⬘ due to the grain roughness and a separate value RH⬘⬘ based on the bed forms so that:

w = gS(RH⬘ + RH⬘⬘) where S is the physical slope. He defined a friction velocity Uf as a combination of the component due to grain roughness Uf⬘ and due to the bed form as Uf⬘⬘: Uf2 = Uf⬘2 + Uf⬘⬘2

2.8 2.4

Froude Froude numbe numberFr F

2.0 antidunes 1.6 1.2 dunes 0.8 0.4 0.0

0

2

4

6

8

10

= wavelength

12

14

16

18

2 * * D/

FIGURE 6-9 Wavelength of dunes and dunes versus the Froude number in open channel flows. (After Kennedy, 1963.)

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6-5-2 Friction Factors The previous paragraph demonstrates that the presence of the bed forms (dunes and antidunes) tends to increase the friction velocity and therefore the friction factors. 6-5-2-1 Effect of Roughness There is a dearth of information on the effect of roughness factors on slurry flow in closed or open conduits. The Ministry of Technology of the United Kingdom (1969) recommended modifying the Colebrook equation by using the hydraulic radius for the single-phase fluids. Green et al. (1978) concurred with this approach and proposed that the Darcy friction factor be obtained from the following equation by replacing the diameter with four times the hydraulic radius and using the Reynolds number based on the hydraulic radius: ks 1 2.51 ᎏ = –2 log10 ᎏ + ᎏ 14.8R 兹f苶 Re 兹f苶 H D D

冤

冥

(6-56)

where ks = the linear roughness (measured in the same units as the hydraulic radius, e.g., meters) Re = 4RHV/, the Reynolds number expressed in terms of the hydraulic radius This definition of the linear roughness is difficult to calculate. In a fast flow, the roughness of the pipe or channel wall may be used. Attempts have been made to define a (Nikuradse) sand roughness for closed conduits, such as the ratio of the particle diameter to the inner diameter of the conduit (dp/DI) but very little has been published for open channels. The problem is far from simple, and the roughness is often taken as twice the grain diameter. The presence of dunes at low Froude number tends to complicate the picture by introducing another parameter for roughness. Example 6-9 Considering Example 6-1, assume that the roughness is 0.0045 mm. Reiterate the friction factor using Equation 6-56. Assume m = 1350 kg/m3 and = 2.8 cP. fD = 0.027 RH = 0.1095 m V = 2.51 m/s Re = V × 4RH/ = 1350 × 2.5 × 4 × 0.1095/2.8 × 10–3 Re = 131,987 Iteration 1 0.027–0.5 = –2log10[0.0045/(14.8 × 0.1095) + (2.51/(131,987 · 0.0270.5)] 6.085 ⫽ 5.077 Iteration 2 Assume fD = 0.038, then 0.038–0.5 = –2log10(2.777 × 10–3 + 3.707 × 10–6) 5.13 ⬇ 5.11 Therefore the Darcy factor is iterated to 0.038. 6-5-2-2 Effect of Particle Concentration on Slurry Viscosity Green et al. (1978) proposed to incorporate the effect of particle concentration in the form of increased effective viscosity by using the Einstein–Thomas equation (Equation 1-9) to

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define an effective viscosity. This effective viscosity is then used to compute the Darcy factor by using the hydraulic radius in the Colebrook equation. This approach is, however, essentially limited to Newtonian slurries in pseudohomogeneous flow well above the deposition velocity. This approach has been covered by Equation 6-53. It does not take in account any dunes or partial deposition at the bottom of the bed . It is, however, a useful and straightforward approach for flows at supercritical Froude number. 6-5-2-3 Effects of Particle Sizes on the Chezy Coefficient Richardson et al. (1967) derived the following equations. For a plane with little or no sediment transportation: Ch ym ᎏ = 5.9 log ᎏ + 5.44 兹苶 g d85

冢 冣

(6-57)

For a plane with appreciable sediment transport: Ch ym ᎏ = 7.4 log ᎏ 兹苶 g d85

(6-58)

冣 冢 冣

(6-59)

冢 冣

For ripples (in English units):

冢

Ch 0.3 ym 0.13 ᎏ = 7.66 – ᎏ log ᎏ + ᎏ + 11 兹苶 g Uf d85 Uf For dunes and antidunes: ⌬RHS

– ᎏ冣 冢 冣冪冢莦1莦莦莦 R S 莦

Ch ym ᎏ = 7.4 log ᎏ 兹苶 g d85

(6-60)

H

where ⌬RHS is the increase of RHS due to the form roughness. Example 6-10 A launder is designed to transport appreciable coarse sediments over a plane with d85 = 4 mm. The height of the slurry must be limited to 150 mm. Using Equation 6-60, determine the required slope if the flow rate is 850 m3/hr and the width of the channel is 450 mm. Solution Ch/兹g苶 = 7.4 log(0.15/0.004) = 11.65 Ch = 36.5 m1/2/s Area of flow = 0.15 × 0.45 = 0.0675 m2 Q = 850 m3/hr = 0.236 m3/s V = 3.498 m/s fD = 8g/Ch2 = 8 × 9.81/36.52 = 0.0589 Slope = fDV 2/8gRH RH = A/P = (0.15 × 0.450)/(0.450 + 0.15 + 0.15) RH = 0.793 m S = 0.0589 × 3.4982/[8 × 9.81 × 0.793] = 0.0116 or 1.16%

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6-5-2-4 Effect of Bed Form on the Friction Graf (1971) discussed the importance of Equation 6-52 and proposed to write the total friction factor for flow in a channel in the presence of dunes or bed forms. He suggested the following equation for the overall friction factor: fD = f D⬘ + f D⬘⬘

(6-61)

where f D⬘ = Darcy friction factor for the channel without bed forms f D⬘⬘ = Darcy friction factor due to bed forms In concordance with Silberman (1963) and Vanoni and Hwang (1967), Graf (1971) indicated that f D⬘ can be estimated from conventional pipe equations by substituting the diameter of the pipe with four times the hydraulic radius. This has already been presented in Equation 6-54. A similar approach was developed by Lovera and Kennedy (1969). From lab tests, Vanoni and Hwang (1967) derived the following equation for f D⬘⬘: 1 RH ᎏ = 3.5 log ᎏ – 2.3 兹苶 f D苶 ⬘⬘ e⌬Hav

(6-62)

where ⌬Hav = mean height of the bed form e = A/Ab = (where A = total area and Ab = the horizontal projection of the lee face of the bed forms) When the magnitude of “e” cannot be determined, Equation 6-62 can be written in terms of the wavelength of the bed form as:

RH 1 ᎏ = 3.3 log ᎏ2 – 2.3 兹苶 f D苶 ⬘⬘ (⌬Hav)

(6-63)

where is the length of the dune. In this section, the importance of dunes was well emphasized. The designer of a slurry flume should avoid these troublesome regimes by designing for supercritical flows wherever the topography allows it.

6-5-3 The Graf–Acaroglu Relation Starting from basic principles of lift and drag forces and buoyancy and weight on a solid particle, Acaroglu (1968), Graf and Acaroglu (1968) proceeded to develop a methodology that applies equally well to both closed conduits and open channels. They considered that the drag force was a function of the Reynolds number based on the bed velocity Ub and the shape factor 1: D = 0.5CDLUb21d 2 However, they chose a different shape factor D for the drag coefficient:

冢

冣

Ubdp CD = f1 ᎏ , D

(6-64)

The bed velocity for the solids-water mixture is expressed as

冢

Uf ks y Ub = Uf f2 ᎏ , CV, ᎏ ks

冣

(6-65)

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where ks = the Nikuradse’s equivalent sand roughness (d/DI) CV = the volumetric concentration of solids The shear friction velocity is expressed as Uf =

w

ᎏ= 冪莦

兹R 苶H 苶S 苶g 苶

(6-66)

By assuming that the absolute roughness of the bed is equal to the particle diameter, Acaroglu and Graff proceeded to define the shear intensity parameter as (s – L)d ⌿A = ᎏᎏ LSRH

(6-67)

At a critical value of the shear intensity parameter ⌿Acr, the shear stress is equal to the critical shear stress previously discussed. When ⌿A > ⌿Acr or w < cr, no movement of sediments occurs. When ⌿A < ⌿Acr or w > cr, movement of sediments occurs. The power consumed with friction or head losses in the open channel is expressed in terms of the energy slope (head loss per unit length) and a nondimensional transport parameter is derived as CVUavRH A = ᎏᎏ3 兹苶 (苶 L苶苶–苶1苶 )g苶 d 苶p s/苶

(6-68)

By examining data from various authors and by regression analysis, Graf and Acaroglu extrapolated the following relationship:

A = 10.39 (⌿A)–2.52

(6-69a)

or CVUavRH (s – L)d ᎏᎏ3 = 10.39 ᎏᎏ 兹苶 (苶 L苶苶–苶1苶 )g苶 d 苶 LSRH s/苶

冤

冥

–2.52

(6-69b)

Equation 6-66 was obtained for finely graded sand with a particle diameter between 0.091 mm and 2.70 mm (0.0036 – 0.1063 in) and was studied in rivers and open channel flumes. This equation applied to both closed conduits and open channels (Figure 6-10). Graf (1971) pointed out that this equation was based on extensive data that was often difficult to analyze. For some unknown reasons, there has not been much research since 1970 to refine the Graf–Acaroglu equation. This is probably due to the fact that practically all research on slurry flows tends to limit itself to full pipes. Example 6-11 A mine is located at high altitude. The tailings need to be transported by gravity over a long distance. The estimated particle size is 6 mm. The volumetric concentration is set at 25%. The specific gravity of the solids is 3.1. The carrier liquid is water. The channel is rectangular in shape with a width of 750 mm. Assuming an average speed of 2 m/s, determine the minimum slope for a flow rate of 1100 m3/hr using the Graf–Acaroglu method. Solution in Metric Units Q = 1100 m3/hr or (1100/3600) = 0.3055 m3/s For a speed of 2 m/s, the height of the liquid would be: 0.3055/(2 × 0.75) = 0.204 m

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SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

S

L

L

SRH

A

dp

100.0

10.0

1.00

0.10 0.01

0.1

1.0

10

100

1000

CV U RH A S

L

gd

3 p

FIGURE 6-10 The Graf–Acaroglu relationship for flow of sand in open and closed channels, assuming a particle roughness equal to the grain size. Adapted from Graf and Acaroglu, 1968.

The hydraulic radius is RH = (0.75 × 0.204)/(0.75 + 2 × 0.204) = 0.132 m From Equation 6-68: 0.0662 0.25 × 2 × 0.132 A = ᎏᎏᎏ = ᎏ = 0.188 兹苶 [(苶3苶 .1苶–苶1苶 ) 苶9苶 .8苶1苶 ×苶6苶 ×苶1苶0–3 苶苶] 0.3515 From Equation 6-69a:

A = 10.39 ⌿ A–2.52 or ⌿A = 4.911 From Equation 6-67: 4.911 × 0.132 = 3.1 × 6 × 10–3s–1 S = 0.0287 or 2.87%. The Graf-Acaroglu method is very useful to determine the bed depth of a full pipe with saltation. Equation 4-43 of Chapter 4 uses this method.

6-5-4 Slip of Coarse Materials Kuhn (1980) conducted velocity measurements on transportation of coarse coal in trapezoidal flumes under controlled laboratory conditions. He reported slip between the coarse solids and liquid. At the inclination of 2.5°, the speed of the coarse material was of 10% slower than the liquid speed of 4.5 m/s, but it decreased gradually to a slip of 8.5% of the speed of 6 m/s at an inclination of 6°. Such a degree of slip is reminiscent

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of the two-layer models of heterogeneous flow extensively discussed in Chapter 4 for full pipe flows.

6-5-5 Comparison between Different Models Blench et al. (1980) conducted a comparison between the different equations to calculate the sediment discharge versus the water discharge in open channels and plotted them as in Figure 6-11. It is obvious that different equations yield different results. The sediments are transmitted in different patterns. When the flow is tranquil (Froude number Fr < 1), two kinds of sand waves may develop: dunes and ripples. They are similar in shape, with an upstream surface and a gentle and gradually varying slope, finishing with an abrupt downstream slope (Figure 6-12). Although similar in shape, ripples are independent of the magnitude of flow, whereas dunes are strongly dependent on flow. At Froude number larger than unity (sometimes referred to as supercritical flows), the flow suppresses the formation of bed forms. Anti-dunes, which are more symmetrical than ripples, form. They move in the same direction as the flow or even opposite to the flow.

FIGURE 6-11 Comparison between the different models for transport of sediments in open channels. (From ASCE, 1975. Reprinted by permission of ASCE.)

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liquid level Flow direction

ym

wavelength FIGURE 6-12 Sand dunes at low Froude number (Fr < 1).

Anti-dunes accentuate the deformation of the free surface (Figure 6-13). They do not occur in closed conduits and their motion is at a lower speed than the fluid. Tournier and Judd (1945) reported that the specific gravity of the ore is an important factor to consider. Heavier ores require more slope to be transported in an open channel, as shown in Figure 6-14. Tournier and Judd (1945) reported that the size of the particles play an important role, and that larger particles require more slope, as shown in Figure 6-15. Figures 6-12 and 6-13 clearly demonstrate that it may be erroneous to use conventional Manning formulae for water flow depending on the roughness of the pipe, as these ignore the resultant roughness due to sand dunes and anti-dunes. Figures 6-14 and 6-15 clearly in-

liquid level ym

Flow direction

wavelength FIGURE 6-13 Sand anti-dunes at high Froude number (Fr > 1).

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2. 7 =

cg ra vi ty

sp ec ifi

sp ec ifi

cg ra vi ty

=

3. 8

slope of launder (%)

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0

20 40 60 80 Weight concentration (%)

slope of launder (%)

FIGURE 6-14 The slope is a function of the specific gravity of the ore as well as the weight concentration (after Tournier and Judd, 1945). The magnitude of the slope is not shown here as it depends also on the hydraulic radius or shape of the launder.

0

4

8 12 16 particle size (mm)

18

20

FIGURE 6-15 Larger particles require more slope (after Tournier and Judd, 1945). The magnitude of the slope is not shown here as it depends also on the hydraulic radius or shape of the launder.

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dicate that the density of the ore as well as the size of the solids do increase the slope. These important factors are unfortunately too often ignored by some engineers who rely on the Manning equation.

6-6 FRICTION LOSSES AND SLOPE FOR HOMOGENEOUS SLURRY FLOWS Green et al. (1978) attempted to estimate the effect of particle sizes and concentration in the form of increased effective viscosity. In this approach, however, they essentially limited their studies to Newtonian slurries and did not take into account the effects of yield stress and plasticity, effects often encountered at high concentrations of fine particles. Geophysicists prefer to talk about cohesive bed forms when exploring the movement of clays in rivers. For certain soils, cohesive forces develop between the solid particles. Graf (1971) indicated that Equation 6-43 should be modified to include X0, a coefficient of cohesion of the material. For a Bingham slurry, the yield stress is added to Equation 6-45 to express the critical shear stress at the point of the incipient motion:

cr ᎏᎏ = + X0 (s – )gd

(6-70)

Cohesive (soil) materials include clay-sized (colloidal) particles, silt-sized particles, and sometimes sand-sized particles. Graf (1971) classified clays into the following three main categories: 1. Kaolinites 2. Montmorillonites 3. Illites Other more minor clays include halloysites, chlorites, and vermiculites. Clay materials have residual electrostatic forces that attract cations and anions. This is measured as a cation exchange capacity in milliequivalents per 100 grams. Grim (1962) stated that Kaolinites have an exchange capacity of 3–15 milliequivalent per 100 grams, whereas illites rated higher at 10–40, and montmorillonites at 80–150. In very simple terms, Grim explained that there are two main structures for clays. One structure consists of two close sheets of packed hydroxyl molecules, in which aluminum, iron, and magnesium atoms are embedded in an octahedral coordination equidistant from the six oxygen atoms in the hydroxyls. The second structure consists of silica tetrahedrons. In the tetrahedron, the silicon atom is equidistant from the hydroxyls. The silica tetrahedral groups are arranged to form a hexagonal network, which is repeated from sheet to sheet. It would be beyond the scope of this book to discuss the physical aspects of clays. The slurry engineer should appreciate that these electrostatic forces and arrangement of chemical groups influence the yield stress at certain concentrations that may be at either the lower end or the higher end. Cohesive soils have the ability to absorb water and develop plasticity; they also have limit liquid. All three characteristics were briefly defined in Chapter 1. Cohesive colloids can form lumps. Forces on such lumps are shown in Figure 6-16. Graf (1971) summarized the test work of Karasev (1964) who proposed that the erosion of cohesive material beds in rivers is due to aggregate-to-aggregate rather than by

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CHAPTER SIX

Lift Force

Cohesion Force

Drag + Cohesion forces

Weight Force FIGURE 6-16

Forces on a colloidal lump moving in an open channel.

particle-to-particle contact. Karasev (1964) derived the following equation for the average scouring velocity: ym 2d50(s – L) + 3⌳ Ucr = 0.142 Ch ᎏᎏᎏ 1.2 + 8 ᎏ L Patm

冤

冢

冣冥

0.5

(6-71)

where ⌳ = information about cohesion Patm = atmospheric pressure A comparison between the computed and measured values of a scouring velocity is presented in Table 6-5. This value of critical speed should be considered the speed for minimum transportation of clay by hydrotransport. The flow of water in canals containing sand, cohesive soils, and cohesive banks was examined by Graf (1971) on the basis of the work of Simons et al. (1963). The graph in Figure 6-17 suggests that the hydraulic radius can be correlated to the flow rate by the following equation: RH = 0.43 · Q0.361 6-6-1 Bingham Plastics Whipple (1997) developed numerical models for open channel flow of Bingham fluids but did not provide a methodology to calculate friction losses. This paper is important for a geophysicist but provides no useful tools for the slurry engineer. For a homogeneous slurry, there are two important numbers to calculate: the Reynolds number and the plasticity number (defined in Chapter 5). In the absence of well-defined models for friction losses of Bingham slurries, Abulnaga (1997) proposed a methodology to modify some of the equations of full-pipe flows by expressing the Reynolds and Hedstrom numbers in terms of the hydraulic radius. At high shear rate, the coefficient of rigidity is taken as the viscosity for a Bingham plastic: Re = 4RHV/

(6-72)

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TABLE 6-5 Scouring Velocity of Clays (after Karasev, 1964) Scouring velocity in streams ______________________________ Diameter of Karasev’s aggregate Adhesion ⍀ Experimental formula ______________ _____________ _______________ _____________ mm in kPa psi m/s ft/sec m/s ft/sec

#

Material

1 2 3 4 5 6 7 8 9 10 11

Aggregate materials Medium loam Rammed clay Rammed clay Heavy loam Heavy loam Clay Clay Clay Clay Compacted clay Heavy loam

2.2 3.0 4.0 2.7 4.0 4.0 4.0 6.0 4.0 0.8 1.0

0.087 0.118 0.157 0.106 0.157 0.157 0.157 0.24 0.157 0.03 0.04

225.6 550 202 373 225.6 245 285 196 285 510 304

12 13 14 15 16 17

Dispersed materials Medium loam Clay Clay Clay Clay Clay

4.0 1.5 3.2 4.0 0.8 4.0

0.157 0.059 0.126 0.157 0.03 0.157

746 706 785 432 785 549

32.7 80 29.3 54 32.7 35.5 41.3 28.4 41.3 73.9 44.1

1.74 2.16 2.06 2.04 1.20 2.20 1.30 1.43 1.54 2.23 2.14

3.28 7.09 6.76 6.69 3.94 7.22 4.26 4.69 5.05 7.32 7.02

1.83 2.82 2.36 2.22 1.70 1.75 1.86 1.45 1.86 3.20 2.35

6 9.3 7.74 7.28 5.58 5.74 6.1 4.76 6.10 1.05 7.71

108 102 114 62.6 114 79.6

1.91 3.06 2.35 2.87 2.40 1.54

6.27 10.04 7.71 9.42 7.87 5.05

2.88 2.68 2.4 1.93 2.42 2.50

9.45 8.79 7.87 6.33 7.94 8.2

FIGURE 6-17 Correlation between the hydraulic radius and the discharge flow rate. (From Graf, 1971, reprinted by permission of McGraw-Hill.)

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And the Plasticity number is written in terms of the hydraulic radius as

04RH PL = ᎏ V

(6-73)

The Hedstrom number is the product of the Plasticity number and the Reynolds number and is calculated as 16RH20 He = ᎏ 2

(6-74)

In Chapter 3, the different categories of non-Newtonian flows were reviewed. Methods to calculate friction losses for Bingham slurries and power law slurries were presented in Chapter 5. Modified Reynolds and Hedstrom numbers for pseudoplastics were also introduced in Chapter 5. In Chapter 5, the Buckingham equation was introduced as He He4 fNL 1 ᎏ = ᎏ – ᎏ2 + ᎏ 3 ReB 6 ReB 3 f NL ReB8 16

(5-5)

Modifying it for an open channel in laminar flow yields

16RH20/2 fNL ᎏ ⬇ ᎏ – ᎏᎏ2 4RHVm 6(4RHVm/) 16 0 fNL ᎏ⬇ᎏ–ᎏ 4RHV 6V 2m 16

(6-75)

The Darby equation for the friction factor in turbulent regime is fNT = 10aReBb where a = –1.47[1 + 0.146 exp(–2.9 × 10–5He)] b = –0.193 The values of the parameters “a” and “b” were based on empirical data for closed conduits. They may be tentatively modified to open channels to yield 4RHVm fNT = 10a ᎏ

冢

冣

b

(5-10)

where a = –1.47 [1 + 0.146 exp{–2.9 × 10–5(16RH2m0/2)}] b = –0.193 Equations for the empirical parameters “a” and “b” should be confirmed by extensive testing in open channels. It is unfortunate that very little research is conducted in this extremely important field of fluid dynamics. Darby et al. (1992) proposed to combine the laminar and turbulent fanning friction factors into the following equation: m (1/m) fN = (f mNL + f NT )

(5-11)

40,000 m = 1.7 + ᎏ ReB

(5-12)

where

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For an open channel this becomes 10,000 m = 1.7 + ᎏ RHVm

(6-76)

Example 6-12 The soft high clay tailing from a copper concentrator develops a Bingham viscosity of 400 mPa · s at a weight concentration of 45% as well as a yield stress of 5 Pa. The slurry density is 1350 kg/m3. The tailings flow rate is 1600 m3/hr. If the maximum allowed slope is 1.5%, determine the suitable size for a half-full smooth HDPE pipe (i.e., ignore roughness factor). Solution in SI Units Iteration 1 Assume a speed of 1.8 m/s. Required area for flow is A = (1600/3600)/1.8 = 0.247 m2 Pipe ID = [(8/)A]0.5 = 0.789 m (31.09 in)

= 0.4 Pa · s RH = DI/4 = 0.789/4 = 0.19725 ⬇ 0.198 m Re = 4RHmV/ = 4 × 0.198 × 1350 × 1.8/0.4 = 2673 He = 16 × 0.1982 × 1350 × 5/0.42 = 26,463

冤

冥

冤

26,463 16 He 16 fL ⬇ ᎏ 1 + ᎏ ⬇ ᎏ 1 + ᎏ Re 6Re 2673 6 × 2673

冥

fL ⬇ 0.0158 a = –1.47[1 + 0.146 exp(– 2.9 × 10–5Re)] a = –1.7 a

fT = 10 Re0.193 = 0.00435 m = 1.7 + 40,000/Re = 16.67 fN = ( f Lm + f Tm)1/m = 0.016 Using Equation 6-5: fNV 2 S= ᎏ 2gRH S = (0.016 × 1.82)/(2 × 9.81 × 0.198) = 0.0132 S = 1.32% This approach was used by Abulnaga (1997) at Fluor Daniel Wright Engineers to design a tailings launder (see Figure 6-1) to transport tailings rich in soft high clay for a Peruvian copper mine. The tailings system functioned well. The Wilson–Thomas method for full flow in closed channels does not rely on empirical coefficients such as the Darby method, but is based on the assumption that a thick sublayer lubricates the wall surface of the pipe. It has not been modified yet for open channel flows. This is a topic well worth further research.

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6-7 FLOCCULATION LAUNDERS Hydraulic flocculation is sometimes conducted in launders feeding a thickener. The G gradient is a measure of the average shear rate for a flocculation tank. For a tank flocculated using a mixer, Camp (1955) defined the G gradient as: G=

ᎏ 冪莦 V P

(6-77)

ol

where P = power applied by the mixer Vol = volume of the liquid in the tank = average viscosity of the solution Camp (1955) as well as the American Society of Civil Engineers (1977) modified this equation for launders, conduits, and flocculation basins by proposing that the power term P be replaced by the power due to friction in the launder: P = Qg⌬H

(6-78)

where = density of the mixture g = acceleration due to gravity Q = flow rate ⌬H = head loss due to friction As a measure to control ⌬H, ASCE proposed to install a baffle system in the flume. They defined this process as hydraulic flocculation: G=

Qg⌬H

ᎏ= ᎏ 冪莦 V 冪莦 V P

(6-79)

ol

Assuming that no baffles are installed in the launder, it would be possible to express the volume V as the product of the wetted area by the length of the launder: Vol = AL Assuming that the slope S of the launder is equal to the energy gradient (as per Equation 6-5), or friction loss per unit length (or S = ⌬H/L), then G=

Qg⌬H

QgS

ᎏ= ᎏ 冪莦 冪莦 VA A

(6-80)

Equation 6-80 is valid only if no baffles or other artificial means are added to the launder to increase friction losses. Example 6-13 A tailing is flowing to a thickener. In-line flocculation is applied to raise the viscosity to 20 mPa · s. The flow rate is 1800 m3/hr. The density of the mixture is 1420 kg/m3. The launder is rectangular with a slope of 1.2%. The chemical process requires that the G gradient be smaller than 100. Determine a suitable size for the launder. Solution From Equation 6-79: 100 = [1420(1800/3600) 9.81 × 0.012/(A × 0.02)]0.5

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100 = 64.66 (1/A)0.5 A0.5 = 0.646 m A = 0.4179 m2 If the depth of the liquid is one-third the width of the launder, A = w2/3 W = 1.12 m Depth = 0.373 m Velocity = Q/A = 0.5 m3/s/0.4179 m2 = 1.196 m/s

6-8 FROUDE NUMBER AND STABILITY OF SLURRY FLOWS A measure of the stability of a flow in an open channel can be expressed in the form of the Froude number (Fr), a ratio of inertia to gravity forces: V Fr = ᎏ 兹苶 gy苶 m

(6-81)

In the case of slurry flows, it is important to avoid subcritical flows (Fr < 0.8), as they often cause settling problems. Critical flows (0.8 < Fr < 1.5) may be associated with a degree of instability and wavy motion, leading in some cases to working problems and overflows. Kennedy (1963) reported test work on sand and suggested that antidunes occur at a Froude number of 0.8 to 1.4 (critical regime). Green et al. (1978) recommended that slurry launders be designed for Froude numbers in excess of 1.5, to avoid regimes of instability of flow. On the other hand, excessively high Froude numbers (Fr > 5) are associated with steep slopes. Steep slope instability was discussed by Niepelt and Locher (1989). Slug flow is reported at high Froude numbers, causing working instability in the form of roll waves. Although economics may often dictate maintaining gentle slopes of < 2% on many long-distance tailings projects, the design of plants must too often accommodate tight spaces. The use of steep slopes (> 8%) may often cause high speeds, in excess of 6 m/s (20 ft/s). To avoid premature wear of liners or pipes, excessive slopes in plants should be avoided. If it is not possible to avoid steep slopes, drop boxes, pressurized boxes, chokes, or full-flow closed conduits should be used wherever wear is a major concern. There are certain conditions in an open channel that may lead to a sudden change from a supercritical regime to a subcritical regime. This is often associated with the so-called “hydraulic jump,” an increase in the depth of the liquid due to the lower speed of motion.

6-9 METHODOLOGY OF DESIGN The design of open launders is complex, with many implicit functions. Abulnaga (1997) suggested starting the calculations assuming a speed of 2 m/s (6.56 ft/s). For many types of slurries, this is a good starting point. The flow rate is divided by the assumed speed to

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obtain an area of the flow A. If a pipe is selected, the flow area A is taken as ⬇ 45–55% of the cross-sectional area of commercial pipes to obtain a pipe diameter. 1. If a rectangular launder is to be manufactured, the width is used to compute the height of the liquid. For these launders, the height of the liquid is assumed to be 1/3 the width. 2. For the area of the flow, the perimeter and hydraulic radius are then obtained. 3. The Froude number is then calculated. If it appears that the flow will be in a subcritical or critical regime, the speed should be increased. 4. Repeat steps 1 to 4 until the Froude number is larger than 1.5. 5. Input the rheology of the slurry to obtain the plasticity, Reynolds, or Hedstrom number based on the hydraulic radius of the flow. 6. Compute the friction factor in accordance with one of the equations listed. 7. Obtain the friction loss per unit length and equate it to the slope, as per Equation 6-5. 8. If the energy gradient or slope from Step 9 exceeds the physical contour of the terrain where the launder is to be installed, reiterate assuming a slower flow. 9. Check on the deposition velocity or self-cleaning abilities of the solids. If the deposition velocity is more than 50% of the average velocity, speed up the flow by changing the cross section of the launder or the physical slope. The following computer program, “Non-Newt-Channel,” uses the Darby method to design open channel flow for non-Newtonian fluids on the basis of modifications to the Darby method. Computer Program “Non-Newt Channels” CLS PRINT “CHANNEL FLOW PROGRAM FOR NON-NEWTONIAN FLOWS PRINT “****************************************” PRINT c = 0 pi = 4 * ATN(1) DEF fnlog10 (x) = LOG(x) * .43242944# DEF FNASN (x) = x + x ^ 3/6 + 3 * x ^ 5/(2 * 4 * 5) + 15 * x ^ 7/(2 * 4 * 6 * 7) + (15 * 7)/(48 * 7 * 8) * x ^ 9 DEF fnacos (x) = pi/2 – (x + x ^ 3/6 + (3/(2 * 4 * 5)) * x ^ 5 + 15/(8 * 6 * 7) * x ^ 7 + 15 * 7 * x ^ 9/(48 * 7 * 8)) g = 9.81 INPUT “PROJECT “; proj$ PRINT INPUT “DATE “; d$ CLS INPUT “NAME OF ENGINEER “; e$ ‘e$ CLS 5 INPUT “AREA”; a$ INPUT “LINE NUMBER “; li$ CLS INPUT “DO YOU INTEND TO USE US UNITS (y/n)”; U$ GOSUB CONVERSION INPUT “INITIAL FLOW RATE (m3/s)”; q ‘INPUT “INITIAL FLOW RATE (m3/HR)”; QH ‘q = QH/3600

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q = q * qul PRINT USING “flow rate = ##.### m3/s”; q PRINT INPUT “DESIGN FACTOR “; SF c = 0 6 IF c = 1 THEN GOTO 19 PRINT INPUT “specific gravity of carrier liquid”; sgl INPUT “specific gravity of solids”; sgs PRINT PRINT “ please choose between input of weight or volume concentration” PRINT “ 1- weight concentration” PRINT “ 2- volume concentration” PRINT 12 INPUT “ 1 or 2”; cwe IF cwe = 1 THEN INPUT “weight concentration in percent”; cwin IF cwe = 2 THEN INPUT “volume concentration in percent”; cvin IF cwe = 0 OR cwe > 2 THEN GOTO 12 PRINT IF cwe = 1 THEN cw = cwin/100 IF cwe = 1 THEN sgm = sgl/(1 - cw * (1 - sgl/sgs)) IF cwe = 1 THEN cv = (sgm - sgl)/(sgs - sgl) IF cwe = 1 THEN cvin = 100 * cv IF cwe = 1 THEN PRINT USING “specific gravity of mixture = ##.##, cv = #.###”; sgm; cv IF cwe = 2 THEN cv = cvin/100 IF cwe = 2 THEN sgm = cv * (sgs - sgl) + sgl IF cwe = 2 THEN cw = cv * sgs/sgm IF cwe = 2 THEN cwin = cw * 100 IF cwe = 2 THEN PRINT USING “specific gravity of mixture = ##.##, cwin = ##.##%”; sgm; cw dens = sgm * 1000 PRINT INPUT “DO YOU KNOW THE VISCOSITY (y/n)”; ZS$ IF ZS$ = “Y” OR ZS$ = “y” THEN INPUT “VISCOSITY (mPa.s)”; vu1 CLS IF ZS$ = “N” OR ZS$ = “n” THEN KRAT = 1 + 2.5 * cv + 10.05 * cv ^ 2 + .00273 * EXP(16.6 * cv) ‘ASSUMED VISCOSITY OF WATER 1 mPa · s IF ZS$ = “N” OR ZS$ = “n” THEN vu = KRAT * .001 IF ZS$ = “Y” OR ZS$ = “y” THEN vu = vu1/1000 CLS PRINT USING “VISCOSITY = ##.##### Pa.s”; vu INPUT “hit any key”; t$ CLS INPUT “yield stress in dynes/cm2 “; y1 y = y1/10 CLS PRINT “you can let the program iterate for itself or you can input an initial speed” PRINT v1 = 2 PRINT “iteration starts at 2 m/s (6.6 ft/s)” INPUT “do you prefer to input an initial speed (Y/N)”; b$

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IF b$ = “N” OR b$ = “n” THEN GOTO 18 17 PRINT USING “ current speed = ##.## m/s “; v1 IF us = 1 THEN INPUT “ initial speed in ft/s “; v1 IF us = 2 THEN INPUT “ initial speed in m/s “; v1 v1 = v1 * leg 18 INPUT “maximum allowed slope in percent “; s1 19 IF c = 1 THEN q = q * SF IF c = 1 THEN GOTO 50 21 PRINT “choose shape of launder from following list “ PRINT PRINT “1- rectangular “ PRINT “2- circular” PRINT “3- U shape” INPUT “which choice (1,2,3 etc.....)”; ck 50 IF ck = 1 THEN GOSUB rect IF ck = 2 THEN GOSUB circ IF ck = 3 THEN GOSUB ushape PRINT “v1” re = v1 * 4 * rh * dens/vu he = (4 * rh/vu) ^ 2 * dens * y PRINT USING “Reynolds No = #########, Hedstrom No = ######## “; re; he GOSUB friction IF c = 0 THEN GOSUB settling IF v2m > (v1/2) THEN PRINT “to avoid settling, flow should be speeded up” IF v2m > (v1/2) THEN GOSUB increase IF sm > s1 THEN PRINT “slope exceeds maximum allowed slope” IF sm > s1 THEN GOTO 17 ‘INPUT “do you want to print out these results as a minimum slope”; min$ ‘IF min$ = “Y” OR min$ = “y” THEN min = 1 ‘IF min = 1 THEN GOSUB print1 ‘IF min = 1 THEN GOTO 999999 GOSUB gradient INPUT “do you want a hard copy (Y/N)”; mv$ IF mv$ = “y” OR mv$ = “Y” THEN GOSUB print1 999999 IF SF > 1 THEN c = c + 1 IF c = 1 THEN GOTO 19 END increase: v1 = v1 * 1.03 PRINT USING “velocity = ##.## m/s”; v1 area = q/v1 RETURN decrease: v1 = v1 * .97 area = q/v1 RETURN are: area = q/v1 PRINT “flow area “; area RETURN

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CONVERSION: IF U$ = “Y” OR U$ = “y” THEN us = 1 IF U$ = “n” OR U$ = “N” THEN us = 2 IF us = 1 THEN PRINT “for us units use foot for length, US gallon for flow” IF us = 1 THEN PRINT “speed in ft/s” ft = .3048 gal = 3.785 inch = .0254 IF us = 2 THEN GOTO 888 leg = ft qul = gal/60000 GOTO 890 888 leg = 1 qul = 1 890 RETURN rect: CLS IF c = 1 THEN GOTO 1600 PRINT “you have chosen a rectangular launder - do you want to continue (Y/N)”; mre$ IF mre$ = “n” OR mre$ = “N” THEN GOTO 21 14 IF us = 1 THEN INPUT “width of channel (ft) “; w IF us = 2 THEN INPUT “width of channel (m) “; w IF w = 0 THEN GOTO 14 w = w * leg PRINT INPUT “do you want the program to calculate height of walls (Y/N)”; hr$ IF hr$ = “N” OR hr$ = “n” THEN INPUT “height of walls “; hl IF hr$ = “y” OR hr$ = “Y” THEN hl = .5 * w hl = hl * leg 1600 area = q/v1 dep = area/w IF c = 1 THEN GOTO 1606 IF p$ = “y” OR p$ = “Y” THEN GOTO 1606 INPUT “what ratio of fill is acceptable (0.333, 0.5, 0.75)”; fill 1605 IF hr$ = “N” OR hr$ = “n” THEN hl = dep/fill hfill = hl * fill 1606 IF (dep > hfill) THEN PRINT “depth of liquid exceeds preferred fill ratio” IF dep > hfill THEN v1 = v1 * 1.01 IF dep > hfill THEN area = q/v1 IF dep > hfill THEN dep = area/w IF dep > hfill THEN GOTO 1605 ‘calculation of hydraulic radius rh = area/(w + 2 * dep) mhd = dep GOSUB froude IF nf < = 1.5 THEN PRINT “froude number too low at “; nf IF nf < = 1.5 THEN INPUT “flow is unstable do you want to stabilize the flow “; p$ IF p$ = “N” OR p$ = “n” THEN GOTO 1612 IF nf > 1.5 THEN GOTO 1612 v1 = v1 * 1.01

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area = q/v1 GOTO 1600 1612 RETURN circ: rf = 0 PRINT “calculation for a circular launder” ‘RINT “ITERATION STARTS FOR 1/2 FULL PIPE” INPUT “degree of fullness as a ratio of area of flow to pipe area (.5,.6 etc..)”; full1 1999 PRINT “speed (m/s)”; v1 INPUT “do you want to change speed (Y/N)”; lk$ IF lk$ = “y” OR lk$ = “Y” THEN INPUT “new speed in m/s “; v1 ‘ IF lk$ = “n” OR lk$ = “N” THEN GOTO 2001 2095 area = q/v1 IF c = 1 THEN GOTO 456 dia = SQR(area * 4/(full1 * pi)) IF us = 1 THEN diapus = dia/.0254 IF us = 1 THEN PRINT USING “recommended inner pipe diameter = ##.## in “; diapus IF us = 2 THEN PRINT USING “recommended inner pipe diameter = ###.####m”; dia 2001 IF us = 2 THEN GOTO 2100 IF nff = 0 THEN GOTO 2002 PRINT USING “present pipe i.d = ##.### m”; id IF us = 1 THEN idus = id/.0254 IF us = 1 THEN PRINT “present pipe id = ###.##inches”; idus 2002 INPUT “ pipe outer diameter in inches”; dout IF rf > 0 THEN GOTO 2004 INPUT “pipe thickness in inches “; thickus INPUT “pipe liner thickness in inches “; linus 2004 idin = dout - 2 * thickus - 2 * linus PRINT USING “pipe i.d = ###.### in “; idin id = idin/12 PRINT USING “pipe id = ##.## ft”; id GOTO 2105 2100 INPUT “pipe outer diameter in mm”; d2m IF nff = 1 THEN GOTO 2101 IF rf > 1 THEN GOTO 2101 INPUT “pipe thickness in mm”; thick INPUT “pipe liner thickness in mm”; lin 2101 idm = d2m - 2 * thick - 2 * lin id = idm/1000 2105 id = id * leg r1 = id/2 a2 = pi * r1 ^ 2 456 IF area < a2 THEN GOSUB depth1 IF area > a2 THEN GOSUB depth2 IF a2 = area THEN PRINT “DEPTH = RADIUS” RATIO1 = dep/(2 * r1) INPUT “HIT ANY KEY TO CONTINUE”; l$ RATIO1 = dep/id 457 PRINT “RATIO OF LIQUID DEPTH TO DIAMETER”; RATIO1 IF RATIO1 < 1.05 * full1 AND RATIO1 > .95 * full1 THEN GOTO 470 IF RATIO1 < .2 THEN GOTO 490

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IF RATIO1 > .85 THEN GOTO 492 IF RATIO1 < .48 THEN INPUT “ DO YOU WANT TO INCREASE THE DEPTH OF LIQUID TO REDUCE SLOPE (y/n)”; n$ IF RATIO1 > .52 THEN INPUT “do you want to decrease liquid depth “; dp$ IF RATIO1 < .48 AND n$ = “N” OR n$ = “n” THEN GOTO 470 IF RATIO1 < .48 AND n$ = “y” OR n$ = “y” THEN GOTO 465 IF RATIO1 > .52 AND dp$ = “N” OR dp$ = “n” THEN GOTO 470 IF RATIO1 > .52 AND dp$ = “Y” OR dp$ = “y” THEN GOTO 467 GOTO 470 490 PRINT “ please reduce pipe diameter as depth is less than 20% of diameter” INPUT “do you want to decrease the pipe diameter “; kjl$ IF kjl$ = “n” OR kjl$ = “N” THEN GOTO 456 IF kjl$ = “Y” OR kjl$ = “y” THEN GOTO 2001 492 PRINT “please increase pipe diameter as depth is more than 90% of diameter” INPUT “do you want to change pipe diameter (Y/N)”; qg$ IF qg$ = “Y” OR qg$ = “y” THEN GOTO 465 IF qg$ = “n” OR qg$ = “N” THEN GOTO 470 465 GOSUB decrease GOSUB are GOSUB depth1 GOTO 456 467 GOSUB increase GOSUB are GOSUB depth2 GOTO 456 470 PRINT “speed (m/s)”; v1 RATIO1 = dep/id INPUT “hit any key to continue”; l$ GOSUB angle 2120 mhd = dep mhd = id * (fnacos(1 - 2 * RATIO1) - (2 - 4 * RATIO1) * SQR(ABS (RATIO1 - RATIO1 ^ 2)))/(8 * SQR(ABS(RATIO1 - RATIO1 ^ 2))) mhds = mhd/.0254 PRINT USING “mean hydraulic depth = ##.##m ##.### in”; mhd; mhds GOSUB froude PRINT “FROUDE NUMBER”, nf INPUT “HIT ANY KEY TO CONTINUE”; lk$ IF nf < = .8 THEN PRINT “a new diameter is recommended” IF nf < = 1.4 THEN GOTO 1999 IF nf < = .8 THEN GOSUB increase IF nf < = .8 THEN GOSUB are IF nf < = .8 THEN PRINT “flow is subcritical” 3000 IF (nf > .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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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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SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

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

d

2

L

d

r L >L r b

L

FIGURE 6-26

3

Geometry of the nappe from a waterfall.

subcritical flow

Lj

ventilation Dd df

d1 d d3

Lp Ld

d < d3

Lr

Ld < Lj

Lr < Lb Case (A)

FIGURE 6-27 1955.)

Patterns of flow with free fall with entry in a subcritical regime. (After Rand, 6.62

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In Case C (Figure 6-28), the nappe is submerged, and the depth of the tail water is higher than d2. Compared with the previous two cases, this turbulent under-nappe region is the deepest of the three cases. The turbulent roller extends much further and is less intense than the hydraulic jump. Referring to Figure 6-29, if Dd is the depth of the fall from the bottom to the brink of the bottom of the drop box, a drop number Dr can be defined as q b2 Dr = ᎏ3 gD d

(6-89)

In real life there is always a sort of churning area of liquid under the nappe, but from a theoretical point of view, which ignores this pool of liquid, the location of the centerline of the nappe intersecting with the bottom of the apron or the drop box would be expressed as Lp ᎏ = 1.98[Dr1/3 + 0.357 Dr2/3]1/2 Dd

(6-90)

This point is also called the toe of the nappe. The hydraulic jump occurs at a distance Ld, which may be smaller, equal to, or even larger than Lp (Figure 6-26) The ratio of Ld/Lp is maximum at 1.87 when Dr = 1. Tests reported by Rand (1955) indicate that the value of Ld ᎏ = 4.30Dr0.27 Dd

(6-91)

In cases where the hydraulic jump starts at the toe of the nappe, the experimental work of Rand (1955) indicates that the reference depth d2 for the tail water can be expressed as a constant: Ld ᎏ = 2.60 d2

(6-92)

ventilation Dd d2

df d1 d = d2 Ld

Lr

Ld = Lj

Lr = Lb Case (B)

FIGURE 6-28 Rand, 1955.)

Geometry of nappe from a free fall with entry in a critical regime. (After

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ventilation Dd df

d

d

Lj Ld

Lr

d > 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 22

(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 + 22 = 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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-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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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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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 <1% by volume 2 Wet grinding with the addition of >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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hydrocyclone

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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7.21

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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TABLE 7-4 Selection Guide for Grinding Mills

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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 (<350 mm, 10⬙) Large feed (<25 mm, 1⬙) Intermediate feed (<12 mm, 0.5⬙) Fine feed (<1 mm, 14 mesh) Coarse product (<3.4 mm, 6 mesh) Fine product (0.4 mm, 35 mesh) Maximum production of fines Minimum production of fines Production of cubical particles Primary mill of two-stage circuit Secondary mill of two-stage circuit Operation in open or closed circuit Operation in closed circuit only

2/28/02

Mineral

Rod __________________ Autogenous Ball Vertical _________________ Peripheral ______________________ _____________ Primary Secondary Overflow Discharge Overflow Grate Pebble Spindle Tower Vibrating Hammer

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7.25

A special chamber on the tip is installed to feed the material and the grinding medium. The liners on the inside are designed to be ribbed either along the length of the cylinder or in spiral shaped ribs. Ni-hard is the most commonly used alloy for liners (Figure 7-19), but manganese steel and chrome steel are also used. In some designs, rubber is used as the liner material (Figure 7-18) It is important to cut down maintenance costs and periods of outage to replace liners. Many mines open their mills once a month for maintenance and to replace the worn liners. The grinding medium is steel rods in the case of rod mills and steel balls in the case of ball mills. As the cylinder or tumbler rotates, the heavy rods and balls are lifted by the ribs of the liners. The rods and balls fall by gravity after a certain angle of rotation is reached. The impact in turn fractures and grinds the rocks into smaller stones. The spacing between the ribs of the liner is critical. Too narrowly spaced ribs may jam the coarser rocks and delay their fracture. Speed of rotation is extremely important. At a certain speed, the material, which is lifted by friction against the liner, starts to fall down. The cascading effects of stone against stone causes grinding by attrition. The material output is fine but wear is high. As the speed of the mill is increased, grinding takes place by impact of the rods or balls against the rocks. As the speed increases even further, centrifugal forces become sufficient for the material to centrifuge. This speed is called “the critical speed of the mill.” Mills are designed to operate at 75% of their critical speed. The diameter of the rods is often 50 mm (2 in) but can be set by the designer of the mill. It is, however, important to separate the rods or balls from the slurry at the discharge of the mill before they enter the slurry pump. The successful separation of steel balls from the slurry involves proper design of trommels, a mechanism to catch the balls, and screens on top of the pump box. Ideally, the balls should be recycled back to the feed of the milling unit.

FIGURE 7-19

Worn-out metal liners removed during monthly maintenance of a SAG mill.

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7-4-1 Rod Mills Rod mills are a type of fine crusher and can reduce the size of rocks down to 1 mm (0.04 in). They perform better than a fine crusher in less than optimum conditions when the feed is damp or contains clay. Typically, the length to diameter ratio of the rod mills is 1.5 to 2.5. Milling occurs by impact of rod against rod. The stones are trapped between the rods and disintegrate. The coarser stones are the first to break. The finer escape milling. Rod mills are not used on closed circuits. In the last 20 years, the mining industry has tended to replace rod mills with large autogenous and semiautogenous mills 7-4-2 Ball Mills In ball mills, metal balls are used as the grinding media. The balls are made of a variety of materials. Steel balls are forged. High chrome balls are cast with 28% chrome and are available from special foundries. About 1 kg of balls is used per ton of stone. Small balls with a diameter of about 25 mm (1 in) are preferred to larger ones in order to maximize the area of contact between balls and stones. The slurry weight concentration in a ball mill is 65–80%. Excessive concentration will cause the particles to stick to the balls and will decrease the effectiveness of grinding. The ball mill may then “freeze” and spill out its contents, causing costly downtime to empty the mill. For this reason, the weight concentration should not be allowed to exceed 80%. A trunnion at the discharge of the ball mill separates balls from slurry. The balls are then conveyed back to the feed. Balls gradually wear out through repeated feeding to the mill and must be replaced. Ball mills are built in different diameters up to a maximum of 6.5 m (21 ft), and in power drives up to 9650 kW (13,000 hp). Their shape is determined by the type of output (Figure 7-20). 7-4-3 Autogenous and Semiautogenous Mills Autogenous and semiautogenous (AG and SAG) mills are extremely large mills with a maximum diameter of 12.2 m (40 ft). In the last few years, Siemens and ABB have devel-

feed

feed

discharge

balls

slurry

Cascade mills (wet and dry grinding) - Used for autogneous and semiautogeneous milling in closed circuit - Primary Grinding with minimum retention time for very fine output - Diameter to length ratio 2:1

balls

discharge

slurry

(b) Conical shape mill - Suitable for fine discharge

FIGURE 7-20 The shape of ball grinding mills is determined by the type of discharge and ore.

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7.27

FIGURE 7-21 SAG mill with wrap-around or ring motor. [Courtesy of Metso Minerals (formerly known as the companies Nordberg and Svedala).]

oped and installed “wraparound” motors. In this design, the outside diameter of the tumble on one side is part of the rotor of the motor (Figure 7-21). These motors are manufactured up to a power size of 7000 kW (9400 hp). The large diameter of these mills maximizes the impact forces. Although the feed is typically 150–180 mm (6–7 in) in diameter, the output can be as fine as 0.3 mm (0.012 in). Particles tend to cleave along their natural grain boundaries. Six to ten percent of steel balls are added on a continuous basis to the feed to assist grinding through a separate entry. Wet milling and grinding is less dusty and less noisy than dry grinding. The feed and output trunnions are on opposite sides. The trommel on one side catches the steel or high chrome balls to prevent them from falling into the pump box.

7-5 AGITATED GRINDING Agitation is another method of grinding. The whole body of the mill may sit on springs and be agitated by crankshafts or an eccentric mechanism driven by a motor. Another ap-

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proach is to have a rotor, an agitator, and a rotating hammer inside the mill to impart energy.

7-5-1 Vertical Tower Mills A vertical mill was developed in Japan for grinding fine minerals. The tower mill is a combination of vertical tanks, a screw mixer, a mill, and a classifier. From a chute on the side of the mill, the rocks, steel balls, and liquid are introduced. The vertical screw rotates around a vertical shaft and creates an upward vertical counterflow. The finer materials float to the top and are led to a side chamber for classification, while the heavier and coarser solids sink with the steel balls (Figure 7-22). The diameter of the balls is 6–32 mm (0.25–1.25 in). Size reduction of the ore is limited to 5 mm (0.197 in) due to limited grinding. Vertical tower mills are manufactured for a maximum output of 100 tph. They require limited floor space and have a low consumption of power.

7-5-2 Vertical Spindle Mills The vertical spindle mill (also known as SAM) uses a central vertical multistage mixer (Figure 7-23). Each stage consists of a number of wolfram carbide pins fixed on a hollow shaft. They provide horizontal stirring. This machine operates with feed smaller than 1 mm (16 mesh) for fine and ultrafine wet or dry grinding. The units are small and compact and can be relocated within the plant. Maximum power is 75 kW per unit.

7-5-3 Roller Mills Roller mills are used for soft grinding of industrial minerals in a dry state. The mill consists of a rotating table on a vertical axis. Two rollers rotate around their own shafts at an angle with respect to each other. The rollers are spring loaded. The output is diverted to dry cyclones and the oversized material is fed back to the roller mill. A new generation of high-pressure roller mills has appeared on the market since the 1980s. A very high level of torque is transmitted to the rollers to maximizing the crushing loads. High-pressure rollers are mainly used in cement plants, diamond processing (when the extraction is from rocks, as it is in Canada), and to a certain extent in the field of metalliferrous minerals.

7-5-4 Vibrating Ball Mills The body of the mill consists of a central feed chamber and two side chambers. The feed is from the top and the discharge from the central chamber is at the opposite end. The whole body of the mill sits on four strong springs. Two electric motors synchronized by V-belts rotate an eccentric mechanism linked to each of the side chambers (Figure 7-24). This machines uses fine feed smaller than 5 mm (mesh 4) and is particularly suited for difficult material with an energy index Wi > 30 kWh/sh.ton. These are essentially small machines with maximum motor size rated at 55 kW or 75 hp. However, they are often chosen over tumbling mills for lower installation cost, lower operating cost, less floor space, increased grinding flexibility, and improved product control within the limitation of their size. Rods or balls may be used as grinding media within open or closed circuits with these machines.

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classifier box

vertical bars for wall protection and help to grinding

7.29

recirculation of coarse material helix for upwards pumping while mixing and grinding slurry pump

FIGURE 7-22

Slurry circuit of vertical grinding tower mill for solids with a maximum diameter of 6.4 mm (1/4⬙).

Page 7.29

launder for fines (output)

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Page 7.30

vertical bars for wall protection and help to grinding 7.30

recirculation of coarse material helix for upwards pumping while mixing and grinding slurry pump 5

K

M

A Z D A

A

FIGURE 7-23

Vertical spindle mill slurry circuit.

P

U -

2

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two motors in parallel at each end

7.31

one side chamber at each end feed

discharge

FIGURE 7-24

Vibrating mill.

7-5-5 Hammer Mills In the hammer mill, a central rotor arm is fitted with rings of arms that crush and mill the feed against the wall of the mill. It is essentially used for dry milling of low-abrasive and friable minerals such as cement, coal, gypsum, and limestone. It is considered by some engineers a crusher rather than a mill.

7-6 SCREENING DEVICES In Chapters 1 and 3, the concept of d50 was introduced as the particle size diameter below and at which 50% of the particles can pass through the opening of a sieve. The same concept applies to the definition of screen size. The screen size aperture is equal to d50. An ideal screen would let all particles equal to or smaller than d50 pass through. This is not always the case, as the performance of the screen depends on a variety of factors: 앫 Screen deck size. In order for all particles to use the screen effectively, the layer of solids above the screen needs to be very thin. This means a large deck size for a given mass of solids. For economical reasons, this is not possible and a thick layer of solids forms on the smaller screens. 앫 Vibration. To move away the coarse particles that block the passage of the finer ones, it is essential to oscillate the screen. The amplitude of the oscillation must match the specifics of the solids. Too much vibration could cause the solids to float as a cloud without passing through the screens. 앫 Presentation angle. Ideally, the solids should be fed as normal as possible to the screen. This means that the solids should come in at a 90° angle. Unfortunately, this is not always possible. 앫 Screen material. Screens are manufactured of metal, rubber, and even fiberglass. Metal screens have a wider aperture than rubber, which is more flexible and less prone to particle binding. 앫 Moisture content. Sprays are sometimes added to screens to improve their efficiency and flush the solids. Sprays suppress clouds of fine particles.

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7-6-1 Trommel Screens Trommel screens are essentially rotating cylindrical mesh. These trommels rotate at a slight angle of inclination to facilitate the removal of material. Trommel screens can be supplied as a set of concentric screens of different aperture. The finest screens are the quickest to show wear. 7-6-2 Shaking Screens Shaking screens move in a horizontal reciprocal motion along the length of the screen. Solids are fed in a horizontal circular movement. The discharge moves in the direction of the horizontal movement of the screen. The motion of the solids changes from circular at the feed to eccentric, and finally to horizontal shaking. 7-6-3 Vibrating Screens These screens are set at an angle with respect to the horizontal. The vibration occurs at a right angle to the screen by the rotation of unbalanced counterweights on a shaft above the screen. Vibration can also be induced by electromagnets and oscillating currents. Vibration levels are high and the screens must be mounted on vibration isolation rubber pads. These screens are extremely noisy and exceed 100 dBA levels of noise, nevertheless, they are the most widely used. 7-6-4 Banana Screens Banana screens are essentially stationary screens. The sieve is bent around a curved screen. The top of the screen is vertical and solids are fed from the top. Particles pass successive wedge bars and solids are removed between them based on the trigonometric opening normal to the fall. To avoid clogging, the bars are pneumatically tilted at regular intervals. These screens can be designed to sieve particles as fine as 50 m.

7-7 SLURRY CLASSIFIERS Classification is the process that separates coarse from fine. Various methods use the effect of size, density, and magnetic and electrostatic properties of the solids. When the weight concentration is smaller than 15%, particles settle in a “free settling” mode. When the weight concentration increases, turbulence promotes settling of the heavier particles faster than the lighter particles. Two families of density classifiers are available: 1. Classifiers that use the principle of free settling to achieve size separation 2. Classifiers that use the particles’ hindered settling speed for density separation and for concentration of a particular mineral 7-7-1 Hydraulic Classifiers In a hydraulic classifier, solids are fed at the top through a chamber that leads into a column. Water is pumped from the bottom of the column. The counterflow moves into a

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number of successive columns or stages. Products settle in accordance with the principles described in Chapter 3. Solids are removed at the bottom through a restriction such as a spigot. The spigot valve opens and closes in accordance with the applied pressure from the accumulation of solids. This is the principle of operation of the hydrosizer, which is often connected to other gravity and magnetic separators.

7-7-2 Mechanical Classifiers A mechanical classifier is a combination of an open channel flow, a weir, and a mechanical device to remove solids. The trough to which the slurry is directed is inclined with respect to the horizontal. On one side, a weir is constructed opposite to the direction of the flow. The heavier solids deposit upstream of the weir, whereas the finer solids pass over it. This system operates on the principle of the sliding bed described in Chapters 4 and 6. The weir is designed to minimize turbulence. A mechanical rake or rotating Archimedean screw removes the coarse solids. Water drains away as the solids are removed. The speed of the rake or rotating shaft is critical to the efficiency of separation. The height of the weir must be adjustable to change the depth of the pool, the rising velocity, and the cut point between coarse and fine. The addition of water controls the density of the slurry. Dilution may be required for effective separation, but excessive water dilution may have to be followed by thickening after classification. Mechanical classifiers are expensive to install but in some applications they are selected for high-density valuable minerals as they assist in their immediate recovery without requiring further complicated flotation circuits. In some respects, flotation circuits use some of the principles of mechanical classifiers by using a circular internal weir, a mixer, an underflow pump, and a separate froth pump.

7-7-3 Hydrocyclones Hydrocyclones (Figure 7-25) are the most common classifiers in the mining industry. They require little space and operate on the pressure from the mill discharge pumps (typically 104–152 kPa, 15–22 psi). They are typically used to classify solids from a size of 40 to 400 m (mesh 325 to 35). The principle of the cyclone relies on creation of a vortex; sometimes primary and secondary vortexes are created by feeding the material tangentially. In a vortex, a certain pressure field is created to counterbalance centrifugal forces. In the cyclone, the applied pressure is converted into a swirling motion. The intensity of the swirl is measured as the swirl number: angular momentum S = ᎏᎏᎏ axial momentum If the swirl flow number exceeds 0.5, the swirl is classified as a strong swirl. Strong swirl is associated with a low-pressure zone at the core. A strong swirl is associated with an important pressure drop. The cyclone feed gauge pressure at the inlet flange is often of the order of 70–100 kPa (10 to 15 psi). The swirling chamber of the cyclone is where the separation starts. The inlet area to the cyclone is often of the order of 5–7% of the chamber area, or the inlet diameter is be-

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FIGURE 7-25 A number of hydrocyclones may be used in parallel to classify slurry flow in a grinding circuit.

tween 20% and 25% of the swirling chamber diameter. The inlet nozzle is rectangular in shape in some sheet metal fabricated cyclones, or circular in fiberglass cyclones. Inside the swirling chamber, a pipe section protrudes from the top of the cyclone. It is called the vortex finder. It must extend below the feed entrance to avoid shortcuts of unclassified slurry to the top discharge or overflow. The diameter of the vortex finder is typically 32% to 36% of the swirling chamber diameter. The finer and lighter particles flow out of the hydrocyclone through the vortex finder (Figure 7-26). In some of the earlier metal fabricated designs, the swirling chamber consisted of a single cylinder. In fiberglass designs, it is split into two halves, which are individually lined with removable rubber liners. The cyclone chamber is followed by a cylindrical chamber with a depth approximately equal to its diameter. This chamber provides some retention time. The cylindrical transition chamber is followed by a conical chamber, often designed with an included angle between 10 and 20 degrees. It provides further retention time. At the bottom of the cyclone, the apex is installed. It acts as a sort of nozzle or orifice. For different applications, different orifice diameters may be used, and for different apex diameters, different pressures are required. The apex is therefore a sort of controlling element to the cyclone. The minimum apex orifice diameter is on the order of 10% of the swirling chamber diameter, and the largest orifice diameter is on the order of 35%. In either case, the apex must allow the flow of the coarse materials. At the bottom of the apex, the discharge is called the cyclone underflow. At the top of the vortex finder, the discharge, which consists of fines, is called the cyclone overflow. For primary grinding circuits, the underflow typically contains 50 to 53% by volume

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COMPONENTS OF SLURRY PLANTS

discharge of finer particles (cyclone overflow)

Involute design vortex finder

inlet pipe

cylindrical feed chamber (top half)

cylindrical chamber (bottom half)

fiberglass body Conical section angle from 10 to 20 degrees discharge pipe for coarse solids (cyclone underflow)

Rubber liner Apex

FIGURE 7-26 A cross-sectional representation of a rubber-lined cyclone. (Courtesy of Mazdak International Inc.)

of solids, whereas for regrinding circuit, the underflow typically delivers 40 to 45% solids by volume. Hydrocyclones can be manufactured from dough-molded compound fiberglass, cast iron, or sheet metal lined with polyurethane. Metal and fiberglass cyclones are lined with rubber or with hard metal (Ni-hard or 28% chrome white iron). Burgess and Abulnaga (1991) presented a finite element analysis of fiberglass cyclones. The performance of the hydrocyclone is calculated by using a partition curve similar to a screen curve. This gives the d50 size, or 50% probability at the cut point. This cut point is defined as the condition for which 50% of the feed will be discharged as coarse particles in the cyclone underflow and 50% as fines or cyclone overflow. For every cyclone design, there is a base d50C or cut-off for the recovery (Figure 7-27). Cyclones are usually operated in a steady mode with constant pressure. Surges can lead to unfavorable air entrainment. To maintain constant pressure from the pumps, the pump box must have a constant level of slurry. To adjust the slurry level, the sump must be provided with a water addition mechanism. Normal feed to cyclones consists of a slurry at 30% solids concentration by weight. Some mines operate with slurry weight concentrations as high as 35%. Higher concentration by weight imposes higher pressures of operation, which can cause a reduction in efficiency of operation of the hydrocyclone while coarsening the cut point. Vortex finders are changed in accordance with the required cut. A larger diameter vortex finder tends to coarsen the overflow while increasing its discharge flow rate at a constant pressure.

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CHAPTER SEVEN

100 ACTUAL RECOVERY

CORRECTED RECOVERY

50

Recovery to underflow, %

d 50c

0 Diameter of particles in micrometers FIGURE 7-27

Typical particle recovery curve for the underflow from a hydrocyclone.

The size of the discharge spigot is selected in order to maximize the flow of solids at high density and to reduce the flow of water in the underflow. Too small a spigot tends to dewater the underflow and to break the air core while reducing the overall efficiency. Because of the wide range of slurries with different particle sizes, the cutoff d50C is used to normalize the particle size (Arterburn, 1982). The actual particle diameter from a recovery is divided by the d50C size and a parameter X is defined as: X = particle diameter/d50C particle diameter The recovery to the underflow (Arterburn, 19xx) on a corrected basis is defined as: e4X – 1 Rr = ᎏᎏ 4X e + e4 – 2

(7-15)

Using the base d50C, Arterburn (1982) proposed to use three correction factors for an application, C1, C2, C3, or d50C(application) = d50C(base) × C1 × C2 × C3

(7-16)

The base d50C is defined as a polynomial function of the cyclone swirling chamber diameter. Arterburn proposed d50C = 2.84(D/100)0.66

(7-17)

where D is the cyclone chamber diameter in meters. The correction factor C1 is based on the volumetric concentration of solids fed to the cyclone:

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COMPONENTS OF SLURRY PLANTS

冢

53 – 100CV C1 = ᎏᎏ 53

冣

–1.43

(7-18)

Equation 7-18 applies in a range CV < 0.4. The pressure drop between the feed nozzle and the cyclone overflow ⌬P is used to compute the second correction factor C2: C2 = 3.27(⌬P)–0.28

(7-19)

where ⌬P is expressed in kPa. To minimize energy losses, ⌬P should be in the range of 40 to 70 kPa, Arterburn (1982), particularly for coarse separation in grinding circuits. The final correction factor C3 is based on the density of the solids with respect to the liquid density: C3 =

ᎏ 冪莦 – 1650 S

(7-20)

L

Tables 7-5 and 7-6 show the typical size ranges for cyclones. Example 7-2 A hydrocyclone with a diameter of 250 mm is selected for a flow rate of 15 L/s at a pressure of 140 kPa. The specific gravity of the solids is 4.8 and the volumetric concentration is 0.3. If the pressure drop between the feed and the overflow is maintained at 50 kPa, determine the corrected d50C. Assuming a discharge coefficient of 0.5 and a remaining pressure of 20 kPa at the apex, determine the underflow capacity for an apex diameter of 80 mm if the underflow density is 2000 kg/m3. From Equation 7-17 the base d50C = 2.84 (D/100)0.66 = 23.77 m. From Equation 7-18 the correction factor C1 is

冢

53 – 30 C1 = ᎏ 53

冣

–1.43

= 3.3

From Equation 7-19 the correction factor C2 is C2 = 3.27(⌬P)–0.28 = 3.27 (50)–0.28 = 1.0935

TABLE 7-5 Typical Range of Sizes for Cyclones Operating at Pressures from 20 to 500 kPa (3–72 psi) Diameter (of swirling chamber) in mm 100 150 250 380 510 660 760

Capacity in L/s

Diameter (of swirling chamber) in inches

Capacity in USgpm

1 L/s @ 20 kPa–6 L/s @500 kPa 3 L/s @ 20 kPa–15 L/s @500 kPa 7 L/s @ 20 kPa–35 L/s @500 kPa 12 L/s @ 20 kPa–60 L/s @500 kPa 26 L/s @ 20 kPa–140 L/s @500 kPa 50 L/s @ 20 kPa–250 L/s @500 kPa 85 L/s @ 20 kPa–450 L/s @500 kPa

4 6 10 15 20 26 30

16 [email protected] 3psi–96 gpm @ 72 psi 48 gpm @ 3psi–240 gpm @ 72 psi 110 [email protected] 3psi–555 gpm @ 72 psi 190 [email protected] 3psi–950 gpm @ 72 psi 410 [email protected] 3psi–2200 gpm @ 72 psi 793 [email protected] 3psi–3963 gpm @ 72 psi 1350 [email protected] 3psi–7100 gpm @ 72 psi

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TABLE 7-6 Typical Cyclone Size versus Particle Cut Diameter of hydrocyclone

Discharge cut size

25 mm (1⬙) 100 mm (4⬙) 250 mm (10⬙) 500 mm (20⬙)

5 m (mesh 2500) 40 m (mesh 380) 75 m (mesh 200) 150 m (mesh 100)

The final correction factor C3 is based on the density of the solids with respect to the liquid density: C3 =

ᎏᎏ = 0.66 冪莦 4800 – 1000 1650

d50C application = 23.77 × 3.3 × 1.0935 × 0.66 = 57 m The flow rate in the underflow is determined from nozzle equations: 苶P 苶苶 / = 0.5 · 0.00503 兹2 苶0苶 = 0.01124 m3/s = 11.24 L/s = 178 USgpm Q = CD · A兹2苶⌬

7-7-4 Magnetic Separators In beach and mineral sand plants as well as in taconite processing plants, minerals have magnetic properties. The presence of a magnet would attract the ferrous ores and separate them from other solids. This is the principle of magnetic separation. Magnetic separators work on two principles. 1. An electromagnetic drum set in a stream 2. A belt driven by an electromagnetic drum on which solids in a dry state or slurry form are allowed to pass to separate the ferrous ores

7-8 FLOTATION CIRCUITS Flotation is a method of separating solids from streams by creating a froth to which they are attracted. Thus in a slurry circuit, flocculants are added to create a froth rich with the metal concentrate. The trick is to make mineral particles hydrophobic, or water repellant. Flotation involves the selected “adsorption” of hydrocarbons (e.g., ethyl xanthate) on liberated minerals (e.g., chalcopyrite), which can then be attached to and transported by air bubbles in the slurry to a so-called froth layer and then separated from the hydrophilic (wetted) particles. For flotation to be efficient, it must be repeated a few times in a circuit that includes a rougher, a scavenger, and a cleaner as shown in Figure 7-28. The collector in a flotation circuit consists of a hydrophobic hydrocarbon chain of melecules (grease or wax) that repels the mineral-laden water and causes it to attach itself to the passing air bubble. The surface chemistry is divided into three categories:

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COMPONENTS OF SLURRY PLANTS

Air bubble

Wa ter

particles

circulating concentrate

feed

rougher

tails

scavenger

tails

concentrate circulating tails

cleaner concentrate

FIGURE 7-28

Fig 7-28

Flow chart for flotation circuit with rougher, scavenger, and cleaner.

1. Physical adsorption with a free energy of the collector smaller than 5 kcal/mol 2. Chemisorption with a free energy of the collector larger than 30 kcal/mol 3. Intermediary stages between adsorption and chemisorption Sulfide minerals are relatively easier to separate by chemisorption because they can use the major collectors such as xanthates and dithiosphophates. Certain special additives with high surface energy capabilities can also be added to separate different grades of sulfides (e.g., to sink pyrite while floating chalcopyrite). Oxide minerals (e.g., hematite, apatite. etc.) and silicate minerals are more difficult to separate by flotation than sulfide minerals. For oxides and silicate minerals, flotation is difficult because it is done by adsorption with minimal free surface energy using anionic fatty acids and cationic amines, which operate essentially by electrostatic forces. When various ores are present, flotation may be done in stages using tanks in series. In each tank, a different pH level may be set or different collectors may be added, with the output from each tank going to a different circuit for further treatment. Depressants are chemicals that make the particle surface hydrophilic and nonfloatable. Typical depressants include bichromate, cyanide, zinc sulphate, and lime. Activators are chemicals that make the surface of nonfloating particles active for collector attachment. Typical activators are copper sulphate and sodium sulphide. The pH value is a determining factor in many flotation circuits. It is adjusted by using various chemicals such as lime, caustic soda, sulfuric acid, etc. Frothers are chemicals that are used to decrease the surface tension of water in order to 앫 Develop improved stability in the pulp 앫 Achieve smaller and better bubble size

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앫 Create a suitable froth layer 앫 Help destroy froth, after which it is removed Typical frothers include alcohols and pine oil. The size of the flotation tank is based on the required flow rate as well as the required retention time, an aeration factor, and a scale factor: Q · tr · sc volume = ᎏ a

(7-21)

where sc = scale factor = 0.85 for plants = 1.0 for pilot plants = 1.7 for lab batch Q = flow rate tr = retention time a = aeration factor = 0.85 Example 7-3 A flow rate of 500 m3/hr requires a retention time of 20 minutes. Size the tank assuming four tanks. volume = (500/60) · 20 · 1/0.85 = 196 m3 196/4 = 49 m3 per tank The number of cells in a flotation circuit is determined by the degree of metallurgical control and the concern for short-circuiting. It used to be believed that the correct approach would consist of small cells and longer banks. However, with the advent of special mixers and good aeration techniques, it is now possible to use larger tanks (Figure 7-29). Flotation circuit can be very simple or very complex. A simple circuit such as used with coal, achieves floatation in a single step and does not involve cleaning of the froth. In a more complex circuit, an initial stage, called the rougher, is added; it acts as a preconcentrator. The flocculated output goes then to a second stage, called cleaning, that is done at higher dilution and is sometimes associated with regrinding at various stages. When the ore grade is fairly low but the mineral is of high value, a scavenger is used for additional preconcentration. Froth is a real challenge in the design of pumps. This will be reviewed in Chapter 8.

7-9 MIXERS AND AGITATORS Mixers or agitators (Figure 7-30) are very important components of mineral and chemical process plants. They are used in various stages such as flotation circuits, leaching circuits, gold adsorption on carbon, preparation of special chemicals such as milk of lime, preparation of feed for pipelines, and final storage where sedimentation is likely to occur. Mixers are used in the gold leaching processes. Special tanks for carbon in pulp (CIP) or carbon in leach (CIL) are built with mixers. The largest diameter of these tanks is approximately 17 m (56 ft). For large plants, the process of flotation or leaching in a single tank is not very efficient. To increase productivity, tanks are installed in series (up to five stages or five tanks in a series), thus eliminating possible short-circuiting.

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COMPONENTS OF SLURRY PLANTS

concentrate air

feed

agitator impeller gangue FIGURE 7-29

FIGURE 7-30

Simplified flotation circuit.

Top-entry agitator. (Courtesy of Hayward Gordon, Canada.)

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There are processes that require a single mixing tank with one or two agitator mixers; an example is the preparation milk of lime, where solids are simply mixed with water and the mixers are used to prevent sedimentation. Special processes in solvent extraction plants require gentle mixing while pumping the organic solution over a weir. For example, in a copper SX-EW plant the organic solution is kerosene-based; it absorbs the copper sulfates and separates them from other solids. For these applications, manufacturers have developed special pump mixers, which are essentially large, open shrouded pump impellers with a large number of vanes. Most tanks used in mining are built with a height to diameter ratio around unity and use a single-stage mixer. The mixer is of a vertical shaft design (Figure 7-30). The impeller diameter is in the range of 30% to 45% of the tank diameter. The impeller is usually situated at about 30% of the depth of the tank. Baffles at the wall of the tank break the vortex that is formed by the agitator. These baffles have a width of 8% of the tank diameter. Certain processes use tall, concentric tanks (sometimes called Pechuka tanks) with two agitators in a series on a single shaft. These are more common in South Africa than in North America. Horizontal agitators are installed on the periphery of very large tanks, particularly in the pulp and paper industry. They have not been popular in slurry mixing tanks in mineral processes as they are difficult to maintain. A vertical mixing tank (Figure 7-31) is fit with baffles at the walls to break the vortex generated by the agitator. A certain gap is left between the baffles and the wall of the tank. In some respects, the propeller-type mixer causes continuous mass flow against a stationary flat bottom. If the speed of rotation is not sufficient, a stagnation area develops. The levels of turbulence in the tank, as well as the shape of the bottom of the tank,

b = DT/12 to D T /10 feed

H = DT

gap = (1/72) tank diameter

C = D A = DT /3

D A = 0.3 D T to 0.45 D T DT

FIGURE 7-31

Typical dimensions for the design of mixing tanks.

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are important parameters. Sometimes a conical deflector is installed at the bottom of the tank. Mixing can be done by using an outside pump. The tank must then have a conical bottom. The discharge of the tank at the bottom flows to a pump. The discharge of the pump is returned to the tank and passes through a jet mixer. For tanks with a flat bottom, the discharge may be from the bottom or through a pipe on the side (Figure 7-32). For very viscous mixtures, anchor agitators are recommended. The blades are vertical and rotate fairly close to the wall surface of the tank. For some difficult and frothy pulps in biological slurry treatment, helicoidal mixers are installed (Figure 7-33). For some complex mixtures, the agitator may incorporate a hollow shaft to sparge oxygen, an impeller to break up the froth at a high level, and one at the bottom of the shaft to mix the slurry (Figure 7-34). The propeller-type agitator is the most common in the mining industry. Its design can be examined from various angles: mechanical strength, speed of operation, hydrofoil shape of the blades, etc. The shaft is designed for the “jamming” condition or “start-up” in a settled tank. The main force is taken at 75% of the maximum radius blade span meas-

feed

bottom and side discharge

feed

Side discharge

feed

bottom and central discharge

feed

Top discharge

FIGURE 7-32 Various patterns of discharge from the mixing tank in accordance with the required degree of agitation.

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Anchor mixer for very viscous mixtures FIGURE 7-33

Helicoidal mixer for very Helicoidal viscousmixer mixtures for v

Anchor and helicoidal mixers are used for particularly viscous mixtures.

hollow shaft hollow shaft vertical verticalmotor motor Hollow shaftfor for Ho llow shaft central oxygen oxygen flow flow

to su support SSole o lepplate late to pport mixer Topoof structure Top f vvessel essel stru cture

threaded threaded coupling coupling Open housingfor for Op en housing breaker foam break er

oTTop p column column (flanged) (flanged)

foambreaker break er foam Grease lubricatedsleeve sleeve Gre ase lubricated bearing & packing bearing & pack ing

Bo ttom column Bottom column(flanged) (flanged) Mixer Mixer

oxygendischarge discharge oxygen

FIGURE 7-34 Complex mixer for biological slurries with central injection of oxygen and with a baffle to skim the froth. 7.44

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ured from the center of the shaft. For severe duty application, the shaft is designed for 2.5 times the rated motor torque without exceeding the yield stress (or 0.2% proof stress) of the shaft material. For light-duty mixers, the shaft is designed for 1.5 times the rated motor torque without exceeding the yield stress of the shaft material. The shaft must not operate at speeds higher then 70% of the first critical speed if the impeller is not dynamically balanced, or higher than 85% of the first critical speed if the impeller is dynamically balanced. The deflection of the shaft should be limited, particularly in the case of anchor agitators, so that the blades do not hit the walls or baffles. The shaft is supported between two bearings in a cantilever arrangement. The bearings may be part of a vertical gearbox or an independent bearing assembly. For large agitators, the gearbox and bearing assembly are integral. In Figure 7-35, the flow between the two levels z1 and z2 contracts across the propeller, which induces a velocity Vi. Since the flow at the free surface as well as at the bottom of the tank is negligible, this flow resembles the ground effect of a hovering helicopter. From airscrew theory, the thrust across the propeller is: T = 2AV 2i

(7-22)

where Vi = the induced velocity A = area of flow across the propeller 苶/2 苶A 苶 苶 Vi = 兹T

(7-23)

power = TVi = 2AVi3

(7-24)

where T = thrust Another important theory used to calculate thrust is the blade theory. In Chapter 3, the concept of lift and drag around an aerofoil or an aircraft wing was introduced. The blade of a propeller mixer is essentially a rotating wing exposed to a flow velocity V and a rotating speed in rpm. Due to the contraction of the stream across the propeller, the flow velocity is half the induced velocity. Since the blades are set at a certain pitch angle (quite often 40–45°), they are at an angle of attack with respect to the relative speed. The relative speed is the vectorial addition of these two perpendicular speeds (Figure 7-35). 2 2 苶V 苶2苶 苶苶r苶 苶 W= 兹¼ i +

(7-25)

where the angular speed = 2N/60 and N is rotations per minute. When the flow approaches a blade at a relative speed W and an angle of incidence ␣ with respect to the chord of the blade, a certain pressure distribution develops around the blade. The result is a lift force L perpendicular to W and a resistance drag force D tangential to it. A good designer keeps the angle of attack at a value that corresponds to the maximum lift-to-drag ratio. For every airfoil, a plot of lift-to-drag curve (Figure 7-36) is obtained. If the blade is pitched at an angle , then the vertical force is Y = lift cos – drag sin = L cos – D sin

(7-26)

and the horizontal force X = L sin + D cos

(7-27)

The vertical force Y is opposite to thrust, whereas the horizontal force X multiplied by the radius gives the resistant torque. Near the tips of the propeller, the flow degrades due to the presence of tip vortices.

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b = DT /12 to D T /10 feed

Y = L cos – D sin

7.46

X = L sin + D cos angle of incidence

z1

V /2

i

V

Lift

W

pitch angle of blade Drag

i

z2

W

V /2

i

U=r FIGURE 7-35

Induced velocity and hydrodynamic forces for a propeller-type mixer.

Page 7.46

W

gap

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angle of incidence Lift coefficient C L 1.3

FIGURE 7-36 flow.

Lift W

Drag

Drag coefficient C D 0.1

stall

stall

10 o Angle of attack (or incidence )

10o Angle of attack (or incidence )

Lift and drag forces as a function of the angle of incidence with respect to the

The load distribution for many propellers is a maximum at 75% of the radius (Figure 737) so that the effective torque is measured at this region. In order to predict the performance of a full-scale mixer, a test may be conducted on a reduced scale model under laboratory conditions. Performance is scaled up to large units using nondimensional factors such as the power factor from the theory of rotating equipment. In basic terms, it means that two mixers of the same geometrical design (but differ-

radial distribution of load

FIGURE 7-37

mixer propeller blade

Distribution of the total force as a function of the span of the blade.

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ent sizes) will behave in similar ways with the same fluid and tank conditions. They would have the same power number. The power number is defined as P Cp = ᎏ N3D5

(7-28)

where D = tip diameter of the mixer N = rotational speed (rpm) = density of the slurry P = power The mixer Reynolds number is defined as ND2 Rem = ᎏ 2

(7-29)

The relationship between the power number and the Reynolds number is shown in Figure 7-38. Examples of the power number are presented in table Table 7-7. The ability of the mixer impeller to pump or induce flow is measured and defined by a nondimensional flow factor: Q CQ = ᎏ3 ND

(7-30)

It is also a function of the Reynolds number, as shown in Figure 7-39. Gates et al. (1976) examined the use of mixerss to maintain solids in suspension. An equivalent volume Voleq is defined is defined as

FIGURE 7-38 Power coefficient versus Reynolds number. The top curve is typical of flatblade mixers with wide blades. The middle curve is typical of flat-blade mixers with narrow blades. The bottom curve is typical of pitched-blade mixers. (Reproduced by permission of Hayward Gordon.)

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TABLE 7-7 Typical Power Number for Mixers in Turbulent Flows Type

3 Blades

4 Blades

1.6 0.7 0.2

1.7 0.8 0.5

45° flat-pitched blades Propeller (marine) Hydrofoil

Voleq = Sm Vol

(7-31)

where Sm is the specific gravity of the slurry mixture. The terminal velocity for spheres was discussed in Section 3-1-3-1. A design settling velocity Vd is correlated to Vt by a correction factor fw: Vd = fwVt

(7-32)

where Vd = design settling velocity Vt = the terminal (or free settling) speed The correction factor fw is presented in Table 7-8 as a function of weight concentration. This empirical coefficient was developed by Chemineer Inc., based on experimental work. It is often difficult to predict the nature of the flow or the drag coefficient near an impeller blade. At weight concentrations in excess of 15%, the solids start to interact, hindering settling so that the settling velocity must be adjusted. The level of agitation is very important to the mechanics of suspension. Chemineer Inc. developed a scale of agitation from 1 to 10, summarized by Gates and al. (1976) as in Table 7-9. Figure 7-40 shows the level of suspension of solids in correlation with the Chemineer scale. Often, manufactures define mixing as simple, mild, medium, vigorous, or violent (Figure 7-40). The science of mixing and keeping solids in suspension is highly empirical. The engineer should take into account existing similar installations as well as lab work results. Because of wear associated with slurries, a simple flat blade system is used to design

Flow Coefficient

D/H = 0.40 D/H = 0.45 D/H = 0.50 Ratio of Impeller diameter to tank height

C = Q 3 Q ND

Turbulent Laminar

Transition

Reynolds Number

2

Re = ND /(2

)

FIGURE 7-39 Flow coefficient versus Reynolds number.

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TABLE 7-8 The Correction Factor fw Presented as a Function of Weight Concentration Solids weight concentration (%)

Factor fw

2 5 10 15 20 25 30 35 40 45 50

0.8 0.84 0.91 1.0 1.1 1.2 1.3 1.42 1.55 1.70 1.85

From Gates et al., 1976, reprinted by permission from Chemical Engineering.

TABLE 7-9 Chemineer Scale for Agitation of Solids in Suspension Scale of agitation 1–2

3–5

6–8

9–10

Description At levels 1–2, agitation is required for minimal suspension of solids. Agitators capable of working at an agitation level of 1–2 will: 앫 Produce motion of all of the solids of the design-settling velocity in the vessel 앫 Permit moving fillets of solids on the bottom, which are periodically suspended Agitation levels 3–5 characterize most chemical process industries solids suspension applications. This scale range is typically used for dissolving solids. Agitators capable of working at an agitation level of 3–5 will: 앫 Suspend all of the solids of design velocity completely off the vessel bottom 앫 Provide slurry uniformity to at least one-third of the fluid batch height 앫 Be suitable for slurry draw-off at low exit-nozzle elevations Agitation levels 6–8 characterize applications where the solids suspension level approaches uniformity. Agitators capable of scale level 6 will: 앫 Provide concentration uniformity of solids to 95% of the fluid batch height 앫 Be suitable for slurry draw-off up to 80% of the fluid batch height Agitation levels 9–10 characterize applications where the solids suspension uniformity is the maximum practical. Agitators capable of scale 9 will: 앫 Provide concentration uniformity of solids to 98% of the fluid batch height 앫 Be suitable for slurry draw-off by means of overflow

From Gates et al., 1976. Reprinted by permission from Chemical Engineering.

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a. Unstable particles are on vessel bottom (Scale of agitation = 1)

b. Particles swept off vessel bottom (Scale of agitation = 3)

c. Solids are homogeneously distributed (Scale of agitation = 9)

FIGURE 7-40 Intensity of agitation yields different patterns of solid suspension. (From Gates et al., 1976. Reproduced by permission from Chemical Engineering.)

the mixer. The blades are often pitched at an angle of 45° with respect to the horizontal plane. To size such a flat-bladed propeller in a mixing tank, with baffles at 90° to each other, baffle width of 1/12th of the tank diameter, and offset from the wall with a gap of 1/72nd of the tank diameter, Gates et al. (1976) proposed the following empirical equation in USCS units:

冢

HP Din = 394 ᎏ SmN 3n

冣

0.2

(7-33a)

where Din = diameter in inches HP = power in hp n = number of impellers N = speed in rev/min Sm = specific gravity of the slurry mixture To express Equation 7-33a in SI units:

冢

P Dimp = 37.57 ᎏ SmN3n

冣

0.2

(7-33b)

where Dimp = diameter in meters P = power in Watts To correlate between the levels of agitation in a tank, the speed of rotation, and diameter of the impeller, Gates et al. (1976) plotted the scale of agitation versus , a factor defined in U.S. units as:

= N3.75Dimp2.81/Vd with Vd expressed in ft/min, D in inches, and N in rev/min (Figure 7-41).

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FIGURE 7-41 Chart to determine speed and diameter of a mixer versus the solids suspension scale. [From Gates et al. (1976). Reprinted by permission of Chemical Engineering.]

Example 7-4 A tank contains 600 ft3 of a slurry mixture. The specific gravity of the solids is 4.1 and the average particle size d50 is 0.01 ft. It is required to be designed for overflow output at an agitation scale of 9. The weight concentration of the mixture is 20%. Size the mixer, assuming a single impeller with an impeller-to-tank diameter of 0.4, baffles 1/12th the tank diameter and baffle gap of 1/72nd the tank diameter; use Figure 7-34. Assume viscosity of 1 cP. Compare the power consumption with a mixer operating at a Chemineer scale of 5. Solution in USCS Units Since most tank have a diameter equal to the height, and assuming a volume of 90% the tank volume, the effective volume occupied by the slurry is Vol = 0.9 × 0.25 × DT3 = 600 ft3 Hence, DT = 9.47 ft T = 9.47/0.9 = 10.52 Dimp = 0.4 × 10.52⬘ = 4.21⬘ or approximately 50.5 inch For a scale of 9 and D/T = 0.4, = 30 × 1010, so

= N3.75D2.81/Vd We must determine Vd. The particles are coarse enough to assume a drag coefficient CD = 0.44. Substituting into Equation 3-7 4(3.1) × 32.2 × 0.01 4(S – L)gdg = ᎏᎏᎏ CD = ᎏᎏ 3 × V 2t 3LV 2t Vt = 1.5 ft/s = 90 ft/min From Table 7-8, the correction factor fw = 1.0 and Vd = VT, or

=30 × 1010 = N3.75D2.81/Vd = N3.7550.52.81/90 = 679.5 N3.75 N = 202 rev/min To determine the required horsepower, Equation 7-33a is used:

冢

HP Din = 394 ᎏ SmN 3n

冣

0.2

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The specific gravity of the mixture is obtained by using Equation 1-4: 100 m = ᎏᎏᎏ = 1128 kg/m3 15/4100 + (100 – 15)/1000 or Sm = 1.128. [50.5/394] × (1.128 × 2023)0.2 = HP0.2 or HP = 322 hp. For a scale of 5, and D/T = 0.4, = 5 × 1010, so

= 5 × 1010 = N 3.75D2.81/Vd = N 3.7550.52.81/90 = 679.5 N 3.75 N = 125 rev/min To determine the required horsepower, Equation 7-33a is used: [50.5/394] × (1.128 × 1253)0.2 = HP0.2 or HP = 76 hp. Going from a mild level of agitation at a 5 on the Chemineer scale to level 9 increases the power consumption 4.24 fold. The shear rate was previously defined in Chapter 2 as the rate of change of the velocity with respect to height above the wall. More generally for a mixer, the shear rate is the change of velocity with respect to depth. The induced velocity is essentially created by the impeller, and the maximum shear rate is experienced at the tip of the blade. There is, however, an average shear rate estimated for the impeller zone, and an average in bulk of the tank. For a radial impeller, all the flow is discharged at the tip of the impeller, and all the solids are subject to the same speed and head. In the case of the propeller or axial turbine, the velocity distribution is proportional to the radius. All solids pass through a higher shear zone in the case of the radial machine. The closer the impeller is to the bottom of the tank, the more the induced velocity is suppressed. A proximity factor hc/D is defined as the ratio of the gap of the impeller to the diameter. This principle is similar to those in the world of aeronautics. The ground effect is essentially the pressure exerted by a helicopter or airplane. The closer it is to the ground, the more pronounced is the effect and, eventually, the induced drag is reduced when the machine flies very close to the ground. In the case of the mixers, the flow is restricted as the impeller gets closer to the bottom of the tank. This affects power consumption. A radial mixer tends to induce flow tangentially, whereas an axial machine tends to induce it vertically. Propeller and radial mixers tend to create different patterns of recirculation around their blades. Mounting more than one impeller on the same shaft gives different patterns of performance based on the mutual interference that the different inpellers exert on each other. The power from two axial turbines is not twice the power from a single propeller, as the induced velocity of the second propeller is not twice the induced velocity of the first propeller. In fact, in some applications, the first propeller is smaller in diameter than the bottom propeller. However, two radial impellers closely spaced consume more than twice the power of a single unit. In sewage treatment plant processes as well as in chemical and mining processes, gas is sparged in the liquid. The impeller of the mixer is then used to provide dispersion of the gas and circulation of the tank contents. For radial machines, a small radial impeller is installed at the bottom to mix the gas. In the case of hydrofoils, different approaches are used. If the shaft and blades are hollow, gas may be pumped through the shaft and blades. In other applications, the impeller is contained within a cylinder that is within the tank. This prevents flooding the impeller with gas.

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A hydrofoil type of mixer would have the highest pumping capacity but would develop the lowest head or shear at constant horsepower or torque. Hydrofoil mixers are often chosen for high-flow applications. For heat transfer, solid suspension, blending, or solid dissolving, bulk pumping is critical. A hydrofoil mixer would be the preferred machine. However, for gas–liquid contacting, molecular mixing, solid dispersion (reduction of agglomerates), an axial flow, flatblade machine or radial mixer are preferred options. The mixing of non-Newtonian slurries is fairly complex and must rely on experimental data. Wasp et al. (1977) proposed that the power consumption for mixing non-Newtonian slurries is a function of the Reynolds number, the Hedstrom number, and the Froude number: P 2 0 N2Dimp ᎏ = fn ᎏ ,ᎏ ,ᎏ 3 5 2 2 2 N D imp ND imp N D imp g

冢

冣

(7-34)

In other words, the power consumption is based on the Reynolds number, Hedstrom number, and Froude number based on the impeller diameter. The mixing of non-Newtonian fluids is common in the manufacture of polymers and considerable data is available. It is, however, not very wise to extrapolate to non-Newtonian slurry mixtures. In the last 25 years of the 20th century, larger and larger mixers have been built. Certain shaft failures and blade failures have occurred on some large agitators. Some were due to corrosion of the bolts holding the blades and others due to jamming of the shaft. It is important to understand that starting an agitator from fully settled conditions can be very stressful to the machine. The reader should consult appropriate reference books on machine design and gearbox design. A service factor of 1.5 is a minimum for sizing the gearbox. It would be beyond the scope of this book to explore the selection of gearboxes for agitators. However, the designer of slurry mixers should be aware of certain important mechanical criteria, such as critical speed. To examine the distribution of loads on the blade, the program “Agitblade” may be used. Program “Agitblade” for Propeller Blade Loads CLS REM propeller design for AGITATOR PI = 3.1415 DIM R(20), PITCH(20), V(20), W(20), ALPHA(20), BETA(20), BLPHA(20) DIM C(20), CL(20), CD(20), CT(20), CU(20), T(20), INCIA(20) DIM P(20), TK(20), VV(20), VH(20), VS(20), PK(20) INPUT “FLUID DENSITY”; DENS INPUT “radius at hub “; RHUB INPUT “ RADIUS AT TIP”; RTIP AP = PI * RTIP ^ 2 100 PRINT “OPTIONS FOR COMPUTATION” PRINT “1-CALCULATIONS ASSUMING KNOWLEDGE OF INDUCED VELOCITY” PRINT “ USING MOMENTUM THEORY” PRINT “2- BLADE THEORY CALCULATIONS FOR BLADEWISE DISTRIBUTION OF FORCES” PRINT “ AND FLOW CHARACTERISTICS” PRINT “3- VORTEX FLOW CALCULATIONS”

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INPUT “YOUR OPTION “; OPT IF OPT = 1 THEN 300 IF OPT = 2 THEN 2000 IF OPT = 3 THEN 6000 300 INPUT “NUMBER OF PROPELLERS ON THE SAME SHAFT”; NU IF NU > 1 THEN PRINT “CALCULATIONS FOR FIRST PROPELLER” INPUT “VELOCITY UPSTREAM THE PROPELLER “; VS IF VS = 0 THEN 305 INPUT “IS THIS VELOCITY PARALLEL TO SHAFT (Y/N)”; V$ IF V$ = “Y” OR V$ = “y” THEN 305 INPUT “INCLINATION OF U/STREAM VELOCITY WRT SHAFT AXIS IN DEGREES”; INC INCI = INC * PI/180 305 FOR M = 1 TO NU VS(M) = VS PRINT PRINT PRINT “CALCULATION FOR PROPELLER “; M INCIA(M) = INCI * 180/PI VV(M) = VS * COS(INCI) VH(M) = VS * SIN(INCI) GOTO 320 310 VV(M) = VS 320 IF VV(M) = 0 THEN PRINT “COMPONENT OF U/S VEL PARALLEL TO PROP IS NIL” 321 INPUT “IS THE HYDROSTATIC PRESSURE EQUAL ON BOTH SIDES OF THE PROP (Y/N)”; PH$ IF PH$ = “Y” OR PH$ = “y” THEN 340 INPUT “HYDROSTATIC PRES UPSTREAM PROP “; PHUS INPUT “HYDROSTATIC PRES DOWNSTREAM PROP “; PHDS 340 INPUT “DO YOU KNOW THE INDUCED VELOCITY (Y/N)”; IV$ IF IV$ = “N” OR IV$ = “n” THEN 600 INPUT “INDUCED VELOCITY “; VU(M) VU = VU(M) T(M) = DENS * AP * SQR((VV(M) + VU) ^ 2 + VH(M) ^ 2) * VU ^ 2 + (PHUS – PHDS) * AP P(M) = T(M) * VV(M) + .5 * T(M) * VU(M) TK(M) = T(M)/1000 PRINT USING “ THRUST = ###########.#### kN”; TK(M) PK(M) = P(M)/1000 PRINT USING “POWER = #########.### kW”; PK(M) VH = VH(M) VS = SQR((VV(M) + VU) ^ 2 + VH(M) ^ 2) INCI = ATN(VH/(VV(M) + VU(M))) NEXT M LPRINT “CALCULATIONS BASED ON MOMENTUM THEORY” FOR M = 1 TO NU

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LPRINT LPRINT LPRINT “CALCULATION FOR PROPELLER “; M LPRINT USING “VELOCITY UPSTREAM SHAFT = ###.## m/s”; VS(M) LPRINT USING “ITS COMPONENT PARRALLEL TO SHAFT= ###.## m/s “; VV(M) LPRINT USING “ITS COMPONENT PERPANDICULAR TO SHAFT = ###.## m/s “; VH(M) LPRINT USING “ITS INCLINATION WRT TO SHAFT = ###.# deg “; INCIA(M) LPRINT USING “INDUCED VELOCITY = ###.## m/s “; VU(M) LPRINT USING “ RESULTANT THRUST = #####.## KN”; TK(M) LPRINT USING “ INDUCED POWER CONSUMPTION = #####.## KW”; PK(M) NEXT M GOTO 8000 600 PRINT “YOU MAY HAVE TO CALCULATE THE INDUCED VELOCITY BY ASSUMING A CERTAIN THRUST MAGNITUDE” 2000 INPUT “VERTICAL SPEED”; V INPUT “PITCH AT HUB”; PITCHR INPUT “PITCH AT TIP”; PITCT INPUT “REQUIRED TIP SPEED “; VTIP RPS = VTIP/RTIP PRINT USING “THE ROTATIONAL SPEED IN ######.## RAD/S “; RPS RDIV = (RTIP - RHUB)/10 PITDIV = (PITCHR - PITCT)/10 INPUT “chord at the root”; CR INPUT “TIP CHORD “; CT REM CALCULATE THE ADVANCE RATIO OF THE PROPELLER J = V/(2 * PI * RTIP) PRINT “ADVANCE RATIO OF PROP “; J REM CALCULATE BLADE AREA AND HENCE ASPECT RATIO SB = (RTIP - RHUB) * .5 * (CT + CR) AR = (RTIP - RHUB) ^ 2/SB PRINT “BLADE ASPECT RATIO”; AR PRINT “BLADE AREA “; SB CD0 = .015 K = .1/(1 + 2/AR) KD = K ^ 2/3 * AR REM K IS THE LIFT COEFFICIENT SLOPE DCL/DALPHA IN THE LINEAR RANGE PRINT “LIFT SLOPE PER DEGREE IN THE LINEAR RANGE “; K ACR = (CR - CT)/(RTIP - RHUB) CDIV = (CR - CT)/10 DTIP = 2 * RTIP INPUT “POWER NUMBER “; CP P = CP * DENS * RPS ^ 3 * DTIP ^ 5 * CP/2 * PI PK = P/1000 TOR = PK/RPS PRINT USING “REQUIRED TORQUE = #####.## KNm”; TOR PRINT USING “REQUIRED POWER #####.## KW”; PK FOR N = 1 TO 11 R(N) = RHUB + (N - 1) * RDIV PITCH(N) = PITCHR - (N - 1) * PITDIV

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V(N) = R(N) * RPS W(N) = SQR(V(N) ^ 2 + V ^ 2) BLPHA(N) = ATN(V/V(N)) ALPHA(N) = BLPHA(N) * 180/(2 * PI) BETA(N) = PITCH(N) - ALPHA(N) C(N) = CR - ACR * (R(N) - RHUB) CL(N) = K * BETA(N) CD(N) = CD0 + KD * CL(N) CT(N) = CL(N) * COS(BLPHA(N)) - CD(N) * SIN(BLPHA(N)) CU(N) = CL(N) * SIN(BLPHA(N)) + CD(N) * COS(BLPHA(N)) T(N) = CT(N) * .5 * W(N) ^ 2 * DENS * C(N) * RDIV NEXT N RPM = RPS * 60/(2 * PI) LPRINT “AGITATOR PROPELLER” LPRINT USING “VERTICAL SPEED = #####.### m/s”; V LPRINT USING “HUB RADIUS = ###.#### m TIP Radius = ####.#### m “; RHUB, RTIP LPRINT USING “TIP SPEED = ####.### M/S”; VTIP LPRINT USING “ANGULAR SPEED = ######.## RPM”; RPM LPRINT USING “VERTICAL SPEED = ####.## m/s “; V LPRINT USING “PITCH AT TIP= ####.## ; PITCH AT HUB = ####.##”; PITCT, PITCHR LPRINT “BLADE AREA”; SB LPRINT “BLADE ASPECT RATIO”; AR LPRINT “APPROX LIFT SLOPE COEF IN LINEAR RANGE”; K LPRINT “APPROX LIFT/DRAG POLAR RATIO “; KD LPRINT USING “PROPELLER AREA ######.## m^2”; AP LPRINT “LOCAL RADIUS PITCH ANGLE ANGLE OF INCIDENCE TOTAL VELOCITY ALPHA” FOR N = 1 TO 11 LPRINT R(N), PITCH(N), BETA(N), W(N), ALPHA(N) NEXT N LPRINT “ LOCAL RADIUS,CHORD, LIFT COEF, DRAG COEF, THRUST COEF, TANG FORCE COEF” FOR N = 1 TO 11 LPRINT R(N), C(N), CL(N), CD(N), CT(N), CU(N) LPRINT NEXT N 6000 8000 INPUT “DO YOU WANT TO PROCEED WITH OTHER THEORIES OF DESIGN (Y/N) “; D$ IF D$ = “Y” OR D$ = “y” THEN 100 END REM CT=THRUST COEFFICIENT PARALLEL TO PROPELLER SHAFT REM CU= FORCE COEFFICIENT PERPANDICULAR TO SHAFT AND CAUSING RESISTANCE

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The designer of mixers with hydrofoils should be aware that the center of pressure does not always correspond with the center of gravity of the blade. The center of pressure could be as far as 25% of the blade chord on high-aspect-ratio blades or those with a high ratio of blade length to blade chord. A pitching–bending moment results; it must be considered when sizing the bolts of the blade. It is important to check the stress on the shaft of an agitator. The following program, written in QuickBasic, allows one to check for the case of a shaft with two impellers in series. Program “Dblagit.Bas” for Two Agitators Mounted on a Shaft PRINT “double agit” pi = 4 * ATN(1) ‘INPUT “are you using SI units (Y/N)”; l$ l$ = “n” IF l$ = “n” OR l$ = “N” THEN conv = .0254 IF l$ = “y” OR l$ = “Y” THEN conv = 1! ‘INPUT “distance from bottom bearing to first coupling”; ab ab = 9.41 ‘INPUT “shaft O.D. for first section of shaft ab “; dab0 ‘INPUT “shaft I.D. for first section of shaft ab “; dabi dab0 = 5 jab = (pi/32) * (dab0 ^ 4 - dabi ^ 4) * conv ^ 4 ‘INPUT “distance from first coupling to second coupling”; bc bc = 135.64 ‘INPUT “shaft O.D. for second section of shaft ab “; dbc0 ‘INPUT “shaft I.D. for second section of shaft ab “; dbci dbc0 = 8.625 dbci = 7.625 ‘sch 80 jbc = (pi/32) * (dbc0 ^ 4 - dbci ^ 4) * conv ^ 4 ‘INPUT “distance from second coupling to first impeller”; cd cd = 48 ‘INPUT “shaft O.D. for third section of shaft cd “; dcd0 ‘INPUT “shaft I.D. for first section of shaft cd “; dcdi ‘dcd0 = 6.625 ‘dcdi = 6.065 dcd0 = dbc0 dcdi = dbci jcd = (pi/32) * (dcd0 ^ 4 – dcdi ^ 4) * conv ^ 4 ‘INPUT “distance from first to second impeller”; de de = 150 dde0 = dcd0 ddei = dcdi jed = jcd ‘INPUT “ power on top impeller”; p1 p1 = 21 * 746 ‘INPUT “ power on bottom impeller”; p2 p2 = 35 * 746 INPUT “rpm”; rpm

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7.59

w = rpm * 2 * pi/60 ‘assume start up torque factor 250% ts = 2.5 t1 = p1/w t2 = p2/w PRINT “torque from top impeller”; t1 ‘INPUT “diameter of first impeller”; dimp1 dimp1 = 84 ‘INPUT “diameter of second impeller”; dimp2 dimp2 = 66 r1 = dimp1 * conv/2 r2 = dimp2 * conv/2 f1 = t1/(.75 * r1) PRINT “ tangential force on top impeller”; f1 f2 = t2/(.75 * r2) PRINT “tangential force on bottom impeller”; f2 ma = (f1 * conv * (ab + bc + cd) + f2 * conv * (ab + bc + cd + de))/10 ‘assume unsymmetry of forces at 10% radius may = SQR(ma ^ 2 + .75 * (t1 + t2) ^ 2) mayn = ma/1000000! PRINT USING “total bending moment at a = ####.## MNm”; mayn syab = mayn/jab PRINT USING “shaft stress syab = ###.## MPa”; syab IF syab > 100 THEN PRINT “warning the required stress limit is 100 MPa” mb = (f1 * conv * (bc + cd) + f2 * conv * (bc + cd + de))/10 mby = SQR(mb ^ 2 + .75 * (t1 + t2) ^ 2) mbyn = mby/1000000! PRINT USING “total bending moment at b = ####.## MPa”; mbyn sybc = mbyn/jbc PRINT USING “shaft stress syab = ###.## MPa”; sybc IF sybc > 100 THEN PRINT “warning the required stress limit is 100 MPa” mcd = (f1 * conv * cd + f2 * conv * (cd + de))/10 mcd = SQR(mcd ^ 2 + .75 * t2 ^ 2) mcdn = mcd/1000000! PRINT USING “total bending moment at c = ####.## MPa”; mcdn sycd = mcdn/jab PRINT USING “shaft stress sycd = ###.## MPa”; sycd IF sycd > 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 = mR2

(7-38)

where = 2N/60 R = radius of rotation The ratio of the centrifugal force to the weight is called the centrifugal number Nc: Nc = mR2/mg = R2/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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FIGURE 7-44

7.63

Centrifugal separator. (Courtesy of Knelson Concentrators.)

冪莦

R Ks = 2N ᎏ g

(7-40)

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

2RNdp 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

冣

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 = 2R0Hus = CvQ Separation will occur when us > CvQ/2R0H.

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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

7.65

Liquid Mixture

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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

앫 앫 앫 앫 앫

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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water connection packings rings

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bearings cartridge

shaft sleeve gland plate

8.3

adjustment bolt

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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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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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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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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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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8.9

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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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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8.17

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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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

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

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

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CHAPTER EIGHT

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

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

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

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

190 280 420 760 874

231 462 693 924 1063

297 594 891 1188 1366

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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

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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CHAPTER EIGHT

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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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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CHAPTER EIGHT

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 = 2RN/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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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

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

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

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

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

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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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 222 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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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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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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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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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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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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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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8.41

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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CHAPTER EIGHT

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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8.43

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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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

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 = 0.25d 2SLPA On the suction side, there is suction pressure Ps, so the thrust force is: TS = 0.25d 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 – 0.1252imp{[R 22 – R 2b] – [(1 + tb/xb)2 · (R 22 – R 2b)]}

(8-40)

where P2 = 0.75g(TDH) + PS.

Ps

FIGURE 8-31

R1

P1 dA

d sl

PA

Axial loads on an impeller with plain shrouds.

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CHAPTER EIGHT

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 – 0.1252imp{[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 = PAR 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 – 0.1252imp{[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 = PsRS2

(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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8.47

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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CHAPTER EIGHT

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) <= 35 THEN nr(i) = n/1.354 Rem radial load equation is in SI unit Fr(i)=k(i) * sg * 1000 * 9.81 * d2 * b2 * h(i) u1 = nr(i) * pi * (dr/304.5)/60 u2 = nr(i) * pi * (d2/304.5)/60 u3 = nr(i) * pi * (dsl/304.5)/60 u4 = nr(i) * pi * (ds/304.5)/60 u5 = nr(i) * pi * (db/304.5)/60 PRINT USING “u5 = ####.## ft/s u3= ####.## ft/s”; u5; u3 h1 = h2 - (nr(i)/1000) ^ 2/13.55 * ((1 + tfv/sf) ^ 2 * (d2 ^ 2 - dr ^ 2))/c ^ 2 PRINT “h1= “; h1 INPUT “hit any key t continue “; l$ h5 = h2 - (nr(i)/1000) ^ 2/13.55 * ((1 + tbv/sb) ^ 2 * (d2 ^ 2 - db ^ 2))/c ^ 2 PRINT “h5= “; h5 INPUT “hit any key t continue “; l$ h3 = h5 - (u5 ^ 2 - u3 ^ 2)/(8 * 32.2) PRINT “h3= “; h3 INPUT “hit any key t continue “; l$ t21 = .5 * (a2 - a1) * (h2 + h1) * (sg/2.31) PRINT “t21 = “; t21 INPUT “hit any key t continue “; l$ t14 = .5 * (a1 - a4) * (h4 + h1) * (sg/2.31) PRINT “t14= “; t14 INPUT “hit any key t continue “; l$ t25 = .5 * (a2 - a5) * (h2 + h5) * (sg/2.31) PRINT “t25= “; t25 INPUT “hit any key t continue “; l$ t53 = .5 * (a5 - a3) * (h5 + h3) * (sg/2.31) PRINT “t53= “; t53 INPUT “hit any key t continue “; l$ fa(i) = t21 + t14 - t25 - t53 fan(i) = fa(i)/(9.81 * 2.2) f(i)=fr(i)/(9.81 * 2.2) PRINT USING “flow= ##### L/s head = #### m speed = ##### rpm”; Q(i); h(i); nr(i) PRINT USING “axial load = ####### lbs ####### N “; fa(i); fan(i) 998 END

8-5-3 Thread Pull Force Most modern slurry pumps feature a threaded shaft assembly. The torque from the operation of the pump is ultimately transmitted to the pump assembly through the shaft. The

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8.49

TABLE 8-13 Limiting Dimensions of American National Standard General Purpose Single Start ACME Threads. External Threads (for Shafts), Class 2G Nominal diameter (inch)

Threads per inch

Major diameter, min/max, in inch

Minor diameter, min/max, in inch

Pitch diameter, min/max, in inch

1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 4.50 5.00

4 4 4 3 3 3 2 2 2 2 2

1.4875–1.5000 1.7375–1.7500 1.9875–2.0000 2.2333–2.2500 2.4833–2.5000 2.7333–2.7500 2.9750–3.0000 3.4750–3.5000 3.9750–4.0000 4.4750–4.5000 4.9750–5.0000

1.1965–1.2300 1.4456–1.4800 1.6948–1.7300 1.8572–1.8967 2.1065–2.1467 2.3558–2.3967 2.4326–2.4800 2.9314–2.9800 3.4302–3.4800 3.9291–3.9800 4.4281–4.4800

1.3429–1.3652 1.5916–1.6145 1.8402–1.8637 2.0450–2.0713 2.2939–2.3207 2.5427–2.5700 2.7044–2.7360 3.2026–3.2350 3.7008–3.7340 4.1991–4.2330 4.6973–4.7319

For more information consult ANSI standard B1.5-1977.

ACME external thread (Table 8-13) is used for the shaft and the ACME internal thread (Table 8-14) is used for the impeller. Because the impeller thread is cast, particularly with hard metals, Class 2G is suggested because it has a wider range of tolerances than the 3G, 4G, and 5G Classes. BSW shaft threads are used on the smallest sizes. Figure 8-32 represents a typical ACME shaft thread. In order to determine the shaft stresses and the axial pull due to torque, the first step is to assess the torque due power: Tq = 60PW/(2N)

(8-51)

Example 8-5 A pump is sized for 200 m3/hr, at a TDH of 36 m and specific gravity of 1.4. The pump speed is 600 rpm and the hydraulic efficiency is 67%. Determine the power and the torque.

TABLE 8-14 Limiting Dimensions of American National Standard General Purpose Single Start ACME Threads, Internal Threads (for Impellers), Class 2G Nominal diameter (inch)

Threads per inch

Major diameter, min/max, in inch

Minor diameter, min/max, in inch

Pitch diameter, min/max, in inch

1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 4.50 5.00

4 4 4 3 3 3 2 2 2 2 2

1.5200–1.5400 1.7700–1.7900 2.0200–2.0400 2.2700–2.2900 2.5200–2.5400 2.7700–2.7900 3.0200–3.0400 3.5200–3.5400 4.0200–4.0400 4.5200–4.5400 5.0200–5.0400

1.2500–1.2625 1.5000–1.5125 1.7500–1.7625 1.9167–1.9334 2.1667–2.1834 2.4167–2.4334 2.5000–2.5250 3.0000–3.0250 3.5000–3.5250 4.0000–4.0250 4.5000–4.5250

1.3750–1.3973 1.6250–1.6479 1.8750–1.8985 2.0833–2.1096 2.3333–2.3601 2.5833–2.6106 2.7500–2.7816 3.2500–3.2824 3.7500–3.7832 4.2500–4.2839 4.7500–4.7846

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CHAPTER EIGHT

pitch p

pitch dia

2 = 29˚

minor dia

major dia

p/2

h = p/2

bt

FIGURE 8-32 ACME thread for pump shafts.

Solution in SI Units power = (200/3600) · 1.4 · 9810 · 36/0.67 = 40,997 W torque = power/rotational speed = 40,997/(2 · · N/60) = 652.5 N-m The helix angle of the thread is defined as

冤

L tan = ᎏ dm

冥

(8-52)

where L = length of a full turn = pitch for single-start threads L = 2 × pitch for double start threads pitch = distance between two consecutive threads measured at the thread diameter dm = pitch diameter The axial load transmitted through the thread from the torque is expressed as

dm cos ␣n – fL Fth = 2 · Tq ᎏᎏᎏ dm( fdm + L cos ␣n)

冤

冥

(8-53)

tan ␣n = tan ␣ cos For ACME threads it equals 14.5°. For square threads it is nil. For modified square it is 5°. For buttress threads it is 7°. So for an ACME thread: tan ␣n = 0.968 cos The coefficient of friction f is measured between the shaft and the impeller. In some pumps, the shaft is of steel but the impeller may be of bronze. Slurry pumps are essentially steel against iron and the coefficient of friction is considered to be in the range of 0.14 to 0.15:

dm cos ␣n – fL Fth = 2 · Tq ᎏᎏᎏ dm( fdm + L cos ␣n)

冤

冥

(8-54)

If n is the number of engaged threads, the axial load from this thread pull force creates a bending stress Sb and a shear stress Ss at the root of the shaft thread:

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8.51

冤

冥

(8-55)

冤

冥

(8-56)

3Fthh Sb = ᎏ2 dmnb t Fth Ss = ᎏ dmnb t

where h = height of the thread tooth = (major diameter – minor diameter)/2; in the case of ACME threads h = p/2 bt = thread width at the root 8-5-4 Radial Force on the Drive End When pulleys are installed to drive the pump, the torque transmitted through a pulley diameter Dp results in a force. Different equations are available, but the simplest expresses the resultant pulley force as:

冤 冥

4Tq Fp = ᎏ Dp

(8-57)

8-5-5 Total Forces from the Wet End The total radial force transmitted by the impeller to the shaft is due to the combination of the hydraulic radial thrust and the weight of the impeller: F1 = FR + Wimp

(8-58)

It is assumed that F1 is acting on the center of gravity of the impeller. The total axial load is: F2 = ±FA as the axial force may change direction as the number of stages exceeds two pumps in series. The torque, a source of torsion stress, was defined in Equation 8-51. On the drive side, the pulley force is upward for overhead-mounted motors or sideways for sidemounted motors. Calculations are often made on the assumption of overhead-mounted motors: F3 = Fp – Wp

(8-59)

On this basis, the shaft of the pump is designed. Due to fatigue considerations, the maximum stress should be smaller than the lesser of 18% yield, or 30% ultimate tensile strength. Referring to Figure 8-33, the equilibrium of forces shows that the reaction force at the wet end is RW and at the drive end it is RD: RW – F1 – RD + F3 = 0 Taking moments at the point of contact load of the wet end bearing: –A · F1 + RD · B – F3(B + C) = 0

(8-60)

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FR + Wimp Fp - Wpulley RD (with belt drive)

dA

Torque

d

d WE

FA

d L th

k

RW B

A

d D.E

C

FIGURE 8-33 Loads on the shaft of a horizontal slurry pump.

冤

冥

F3(B + C) A · F1 RD = ᎏᎏ + ᎏ B B

冤

冥

F1 · (A + B) F3C RW = ᎏ + ᎏᎏ B B

(8-61)

(8-62)

The reader should refer to specialized books on machine design that detail all aspects of the design of shafts, stress concentration, and bearing life calculations from the reaction forces at both the drive end (outboard) and wet end (inboard) bearings. The manufacturers of bearings have their own detailed factors for type of lubricant and ratio of axial to radial force. Some manufacturers of slurry pumps offer grease lubricated bearing assemblies and reserve the oil version for high-speed and high-thrust loads (as in pumps in series), whereas some use oil all across their range of pumps. 8-5-6 Flange Loads A common misconception is that the flanges of slurry pumps can take the same loads as water pumps. The fact that the discharge flange is split radially to allow access to rubber or metal liners by itself is an indication that this is not the case at all. The casing of a slurry pump can be distorted by excessive pipe loads on the flange. The consultant engineer is therefore well advised to contact the manufacturer for allowed flange loads. It is also necessary to provide proper pipe supports at the discharge of the slurry pump, and not to use the pump by itself as an anchor block to piping. The common error is to apply a large expansion at the discharge of the pump, such as from a 4⬙ pump discharge to an 8⬙ pipe. Doubling the diameter is effectively multiplying by four the area exposed to the full pressure, of which a quarter is absorbed by the pump, leaving three quarters to be balanced by a pipe fitting such as a properly supported dead end bulkhead, an anchor block, or, whenever possible, by soil friction, as is the case with pipeline pumps.

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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

8-6 ADJUSTMENT OF THE WET END The wear of the impeller front vanes, the throatbush, is believed to cause a drop in efficiency. The impeller must be readjusted by moving it forward. To move the impeller relative to the casing, the shaft assembly must be moved relative to the pump frame, as the latter is bolted to the pump casing. Two different methods are available: 1. A special bolt under the bearing cartridge (Figure 8-34) 2. Push and pull bolts at the drive end (Figure 8-35) Once the bearing cartridge is moved, it is fixed in place by clamping bolts that are tightened against the frame.

8-7 VERTICAL SLURRY PUMPS The vertical sump (Figure 8-36) complements the horizontal pump. The vertical pump is particularly suitable for floor sumps in mill discharge areas and in dealing with flotation circuits. The vertical sump pump may be supplied as: 앫 A stand-alone pump with double suction impeller (Figure 8-37) to be installed in a concrete or metal sump, particularly with flotation columns

bearing cartridge

pump frame

clamping bolts for bearing cartridge

adjustment bolt for bearing cartridge

FIGURE 8-34 Adjustment of the pump impeller by a special bolt between the bearing cartridge and the frame.

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clamping bolts bearing cartridge

push bolts pump frame FIGURE 8-35 Bearing assembly of the ZJ slurry pump (made in China) with adjusting push and pull bolts. (Courtesy of AJP Services Inc. The distributor for Canada.)

FIGURE 8-36 Inc.)

Sump pump with double suction impeller. (Courtesy of Mazdak International

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(a)

(b) Rubber-Lined, Acid-Proof Pumps with Double Suction Impellers Pump Size Frame Units

A

SP2 BV inch 26 mm 660 SP3 CV inch 32 mm 813 SP4 DV inch 37 mm 940 SP6 EV inch 48 mm 1220 SP8 FV inch 52 mm 1321

B

C

D

E

F

11 280 14 356 14 356 14 356 14 356

32 813 36 915 48 1220 60 1524 60 1524

36 915 48 1220 60 1524 72 1829 84 2987

12 305 16 406 20 508 24 610 14 356

20 508 20 508 26 660 35 889 35 889

G

H

J

K

L

6 16 20 20 20 152 406 508 508 508 8 22 22 26 26 203 559 559 660 660 10 26 26 30 30 254 660 660 762 762 14 34 34 38 38 356 864 864 965 965 16 47 47 52 52 406 1194 1194 1321 1321

N

P

40 1016 60 1524 72 1829 84 2134 96 2438

8 200 12.6 320 14.8 376 19.7 500 23 584

*D is the standard depth—other shaft length are available in 12⬙ increments—consult the plant for critical speed *E is the minimum priming level *C and N are typical sump dimensions for the sump

FIGURE 8-37 Dimensions for sump pump and corresponding sump. (Courtesy of Mazdak International Inc.) 8.55

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앫 A single suction impeller with an auger or agitator below the impeller to agitate settled solids in floor sumps (Figure 8-38) 앫 A top suction pump supplied integrally with a metal conical tank, called a “tank pump” (Figure 8-39) The vertical slurry pump is designed to have all its bearings above the baseplate so as to be well protected from slurry ingestion (Figure 8-40). Due to the depth of the sump, the design engineer must pay particular attention to the critical speed of the pump. For this reason, the shaft of these pumps can be as large as 200 mm (8⬙) to offer the necessary rigidity. Vertical slurry pumps are particularly popular in froth handling circuits. To handle the combination of solids, air, and liquids, a double suction impeller is often recom-

motor & pulleys

bearing assembly baseplate 2" discharge eyebolt

column wearplate casing

shaft

impeller screen

agitator

fig 8-38

FIGURE 8-38 Sump pump with single suction impeller and auger to agitate settled solids particularly suited for mill discharge floor. (Courtesy of Mazdak International Inc.)

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motor & pulleys

bearing assembly

baseplate eyebolt

inlet tank shaft impeller casing discharge

93 FIGURE 8-39 Tank sump pump with top suction impeller and integral tank for particularly difficult frothy slurries.

mended with vertical sump pumps. As an alternative, tank pumps with top suction (Figure 8-39) are used. In either configuration, the impeller must be designed to resist air biting. A special type of process used to extract gold is based on cyanide leaching. Leached gold is then separated by adsorption, the property of certain materials such as carbon to fix gold on their surface. Carbon spheres are used as an adsorption material. This process is done in special “carbon in leach” or “carbon in pulp” circuits with mixing tanks. The transfer of these solutions requires recessed or vortex impellers that can pump without breaking the carbon lumps. The impeller is recessed out of the flow as shown in Figure 8-41. A design that is gaining popularity in plants for recycling newspaper is the vertical pump with a recessed impeller and a chopper blade. It is not uncommon that the recycling bins for paper now found in every suburb of North America end up containing milk cartons, plastic bottles, toys, and even pieces of wood. These materials are not very good for conventional stainless steel pumps with mechanical seals. The cantilever sump pump (Figure 8-41) with the chopper offers the ability to pump long fibers while chopping them and eliminating the maintenance problems of mechanical seals.

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FIGURE 8-40 Components of a vertical slurry pump showing that the bearings are above the baseplate. 1. Shaft sea. 2. Top bearing cover. 3. Top bearing. 4. Bearing assembly. 5. Crease nipple. 6. Bearing locknut. 7. Bearing washer. 8. Bottom bearing—spherical roller for heavy duty. 9. Discharge pipe. 10. Baseplate. 11. Shaft seal. 12. Bottom hub. 13. Pedestal— Open structure. 14. Top suction strainer. 15. Wear plate. 16. Shaft. 17. Shaft sleeve. 18. Double suction impeller for minimum thrust loads. 19. Pump casing. 20. Lower suction strainer. (Courtesy of Mazdak International Inc.) 8.58

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stator

Rotor

Wet end with cutter

FIGURE 8-41 Vertical slurry pump with a recessed impeller. This pump is suitable for carbon transfer in gold cyanide circuits and for wastewater treatment applications. The addition of a cutter (rotor and stator) renders this pump particularly suitable for certain applications such as sewage treatment and newspaper recycling plants.

8-8 GRAVEL AND DREDGE PUMPS Hard metal pumps play an important role in dredging lakes and ports. Some sizes are presented in Table 8-12. A typical construction of a dredge pump is presented in Figure 8-42. Dredge pumps are designed to handle particularly large boulders and lumps of clay. Some of the largest dredge pumps are designed to handle 6.3 m3/s or 100,000 US gpm. A special low-pressure, high-flow pump called the ladder pump is designed to be mounted at the tip of the suction arm. Its purpose is essentially to move the material up to the boat hopper or up to a booster pump on a hopper. The booster pump is designed for higher discharge head. A particular type of pump is the phosphate-matrix-handling pump. It does resemble in many aspects a sort of dredge pump, but is built of materials to handle both corrosion and

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One piece shell and engine side door/liner for minimum parts replacement

Integral stuffing box shell mount

Solid impeller hub eliminates problems with threaded or bolted inserts

OPTIONAL ONE PIECE DOOR/SIDE LINER DESIGN AVAILABLE

Precision machined heavy duty shell Adjusting bolt allows for easy adjustment of impeller for proper operating clearances

Separate thrust bearing

Heavy duty bearing assembly for high power & loading conditions

Advanced design impeller. Good hydraulic performance without sacrificing spherical clearance

FIGURE 8-42 Components of the Marathon dredge pump. (Courtesy of Mobile Pulley and Machine Works.)

wear. It is often driven by a diesel engine through a gearbox. The complete baseplate with driver and pump are relocated from one area to another as mining is done.

8-9 AFFINITY LAWS Affinity laws are used to predict the effects of changing the speed of a pump, trimming an impeller, and extrapolating the performance of a pump from case (A) to case (B). They state that: HA/HB = N A2/N B2

(8-63)

HA/HB = D A2/DB2

(8-64)

HA/HB = N A2/N B2

(8-65)

QA/QB = DA/DB

(8-66)

QA/QB = NA/NB

(8-67)

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8.61

8-10 PERFORMANCE CORRECTIONS FOR SLURRY PUMPS Slurry pumps are designed and tested for hydraulics using water as a reference fluid. However, they are designed to handle large spherical rocks. Often, four-vane impellers are less efficient but outlast other types, particularly on abrasive slurries. Attempts have been made to quantify and qualify the spacing of vanes on the performance of dredge slurry pumps. As early as 1932, Fischer and Thoma conducted tests on a pump built of a transparent material. Although the flow was observed to be close to the designed value at best efficiency point, it quickly deviated at other values. They observed a large area of flow separation on the trailing edge of the vane, with reverse flow in certain instances. Understanding the effects of solids on centrifugal pumps has been a slow process. Fairbanks (1941) developed a theory to correlate the head developed by a pump for a slurry mixture with the volumetric concentration and specific gravity of the solids. He explained that the fundamental Euler equation could be modified to account for the density and flow rate of the mixture as: T = mQm(r2Vt2m – r1Vt1m) The power needed to pump the mixture by an ideal pump (at 100% hydraulic efficiency) is then expressed as: T = mQmgHm where is the angular speed . The mixture head is then expressed as two components for solids and carrier fluid: Hm = (/gm) · [s · Cv · (r2Vt2s – r1Vt1s) + (1 – Cv) · (r2Vt2L – r1Vt1L)

(8-68)

where Cv = volumetric concentration of solids m = density of mixture s = density of solids Fairbanks concluded from his tests on a single pump that : 앫 The drop in the head-capacity curve varies not only as the concentration increases, but also as the particle size of the material in suspension increases. 앫 The fall velocity of the suspended material is the most important parameter for predicting the effect of solids on pump performance. 앫 The power input is a linear relationship of the apparent specific gravity of the solids in suspension

8-10-1 Corrections for Viscosity and Slip Viscosity must be taken in account when pumping viscous slurries. Viscosity reduces the efficiency of pumping and the head developed by a pump (Figure 8-43). The Hydraulic Institute Standards provides correction curves for viscous fluids pumping, but warns against extrapolating to other pumps or fluids. The Institute does not publish curves for viscous slurries. Duchham and Aboutaleb (1976) derived equations to predict the effects of viscosity

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1.4 1.2

Head (Wate

1.0

Ef fic Ef ie nc fic y i (v en (w isc cy at ou er sf ) lu id )

1.0 0.8 H/HN

r)

us fluid)

0.6 0.4

0.8

N

Head (Visco

0.6 0.4

0.2

0.2

0.0 0.0

0.5 Q/Q

Fig 8 43 FIGURE 8-43

0.0 1.5

1.0 N

Effect of viscosity on the performance of centrifugal pumps.

and density on the flow rate, head, and power consumption by comparing particle Reynolds number and power factor. Their analysis did not present a definite appreciation of the effects of viscosity. Sheth et al. (1987) investigated slip factors for slurry pumps by conducting tests on a Wilfley pump. The pump had a 267 mm (10.5 in) diameter, 27 mm (1.06 in) blade width, and a discharge angle of 31°. The following equation was derived by Sheth et al. (1987) to account for the effects of the slurry mixture carrier densities:

s ᎏᎏ2 L · N · D

冢

冣

0.12

m ND2 = 0.0989 – 0.00157 ᎏ ᎏ L Q

where s = slip factor = dynamic (absolute viscosity) of liquid carrier Dimp = impeller diameter N = rotating speed of impeller m = density of slurry mixture L = density of liquid carrier Q = flow rate

冢

冣

0.5

(8-69)

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The above equation is empirical and the exponents and coefficients may change for different pump designs. More research work on different designs would have to be published before a universal formula is adopted. Example 8-6 A slurry pump is to be designed to pump slurry under the following conditions: maximum speed at intake 4 m/s (13 ft/s) flow rate 120 L/s (1858 USGPM) head 40 m (131 ft) slurry density 1470 kg/m30 (SGm = 1.47) water carrier slurry viscosity 100 mPa · s max solid particle size 25 mm (1 in) Using the Sheth formula, determine the geometry of the impeller. Solution 0.120 m3/s suction area = ᎏᎏ = 0.04 m2 3 suction diameter = 0.225 m (8.85 in) suction area at 4 m/s = 0.03 m2 suction diameter = 0.195m (7.7 in)

s ᎏᎏ2 L · N · D

冢

冣

0.12

m ND2 = 0.0989 – 0.00157 ᎏ ᎏ L Q

冢

Or, calling a = N · D2:

s · 0.331 = 0.0989a0.12 – 0.0067a0.62 s = 0.298a0.12 – 0.020a0.62 If A = 30, then:

s = 0.298 × 1.5 – 0.0202 × 8.23 s = 0.28 If a = 40, then:

s = 0.464 – 0.198 = 0.265 If a = 20, then:

s = 0.427 – 0.129 = 0.297 If a = 15, then:

s = 0.4124 – 0.108 = 0.304 If a = 10, then:

s = 0.393 – 0.084 = 0.31

冣

0.5

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Let us assume a = 10, then: 10 = N · D2 Since the particle size passage is 25 mm (⬇ 1⬙), assume discharge width = 30 mm or 0.03 m (1.18 in). The head ratio = 2gH/u2 (in the United States) is:

= 2H[1 – (cm/u2) cot 2] If we assume D = 0.4m (⬵ 16u), then: 10 = N × 0.16 ⇒ N = 62.5 rev/s U = 78.5 m/s Cm = Q/ADis ⇒ discharge area = × 0.4??(1 – zt/sin 2) If b = 30 mm (1.181⬙) then: A = × 0.4 × 0.03 (1 – zt/sin 2) = 0.037 (1 – zt/sin 2) If Z = 4 vanes and t = 30 mm then: A = 0.037 (1 – 0.12/sin 2) If 2 = 15° then: A = 0.0198 m2 Cm = 0.12/0.0198 = 6.05 m/s Cm/U2 = 0.077

= 2Hs(1 – (Cm/U2) cot 2) = 0.44H 2gH 2 × 9.81 × 40 = ᎏ = ᎏᎏ = 0.127 2 U2 78.52 0.127 = 0.44 H ⇒ H = 0.289 This is not a very efficient pump due to the combination of viscosity and solid density: consumed power = gQH/H = 9.81 × 470 × 0.120 × 40/0.289 ⬵ 240 kw or 327 hp It is recommended to install a 400 hp motor. 8-10-2 Concepts of Head Ratio and Efficiency Ratio When Pumping Solids Stepanoff (1969) explained that when pumping solids in suspension, a pump impeller imparts energy to the carrier liquid. For a homogeneous mixture, he explained, the impeller will be able to impart as many feet of mixture as it would have been able to impart head of water. The performance of the impeller is not impaired but the power consumption increases linearly with the specific gravity of the mixture. In reality, at best efficiency point, the presence of solids tends to reduce the hydraulic head by the energy wasted to move them through the impeller passageways. Similarly, the efficiency of the pump when handling the mixture will be reduced by the presence of solids. Two factors can be defined head ratio and efficiency ratio.

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8.65

The head ratio is HR = Hm/Hw

(8-70)

or the ratio of head developed when pumping slurry to the head developed when pumping water. The efficiency ratio is ER = Em/Ew

(8-71)

or the ratio of efficiency developed when pumping slurry to the efficiency developed when pumping water. Stepanoff (1969) indicated that at best efficiency: HR = ER Stepanoff (1969) reported work by Japanese investigators who indicated that tests on carbide slurries tended to show that the head–capacity ratio may increase or decrease depending on whether the solids concentration tended to cause the slurry to behave as a Newtonian or non-Newtonian mixture. Reviewing published data between 1941 and 1971, Hunt and Faddick (1971) reported that various tests in different labs and field applications confirmed that: 앫 The drop in head (in feet of mixture flowing) developed for a given volumetric discharge rate decreased as the concentration of the solids in suspension increased. 앫 The required brake horsepower for a given pump operating at a given capacity increased as the concentration of solid material in suspension increased. 앫 The efficiency at a given capacity decreased as the concentration of the solid material in suspension increased. Hunt and Faddick (1971) simulated the performance of centrifugal pumps pumping wood chips by tests using rectangular plastic parts with an average specific gravity of 1.02. They used four different impeller designs in two different volute designs. There was no consistency in the extent of head drop or efficiency with solid concentration, and the results indicated that the actual design of the pump was a very important factor. A difference of head and efficiency of 5 to 7% was noticed for the different designs. The authors therefore discouraged applying head and efficiency ratio factors for one pump to another pump of a different geometry, but encouraged further research into the mechanisms of flow through the rotating passages of these pumps. It is important to appreciate the work of Hunt and Faddick. Often, a pump vendor will produce a chart or curve to obtain the head and efficiency ratio. The limitations of such curves are that they apply only to pumps of similar geometrical design. The discrepancy of 5–10% between one design and another may have to be absorbed by the motor. McElvain (1974) published data on the effects of solids on pump performance. He worked on the concept of the head and efficiency reduction factors defined as: RH = 1 – HR

(8-72)

R = 1 – ER

(8-73)

He tested impellers up to a diameter of 35 cm (13.78 in) and on various concentrations of silica and one grade of heavy mineral. He developed a set of curves and established a relationship between volumetric concentration and the head and efficiency reduction factors as: RH = R = 5 · K · CV

(8-74)

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1.2 Head (Slurry)

Head (Water) 1.0

Ef fic Ef ie nc fic y( ie w nc at y er (s ) lu rry )

1.0

H/H N

0.8 0.6 0.4

0.8

N

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0.6 0.4

0.2

0.2

0.0 0.0

0.5

1.0 Q/Q

Fig 8-44 FIGURE 8-44

0.0 1.5

N

Effect of solids on the performance of centrifugal pumps.

The K factor was then plotted against the d50 and for solids of various specific gravity (see Figure 8-45). The assumption that RH = R was accepted to hold true for slurry volumetric concentrations smaller than 20%. This covers a substantial number of pump applications. Example 8-7 Heavy metal oxide slurry is to be pumped at a volumetric concentration of 18%. The specific gravity of the solids is 5.0 and the d50 is 400 m. The calculated head on slurry is 35 m. Determine the head ratio and the equivalent water head on the pump performance curve. Solution Using the McElvain equation, the value K is determined from the lower curve at 0.38. Substituting in Equation 8-74, at a volumetric concentration of 18% (less than 20%): RH = R = 5 · K · CV = 5 · 0.38 · 0.18 = 0.342 Substituting into Equation 8-72: HR = 1 – RH = 0.658 Since the calculated head for friction, the equivalent value on water is Hw = Hm/HR = 35 m/0.658 = 53.2 m The engineer must therefore select the appropriate pump speed from the pump curve that would develop 53.2 m.

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0.1 K-Factor

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1.5

0.2 2.65

0.3

4.0

0.4

S = 5.0 s

0.5 10

100

1000

10000

Particle Size d 50 ( m) FIGURE 8-45 Correction to the head K factor of centrifugal pumps on the basis of the specific gravity and particle size Ss = specific gravity of solids. (After McElvain, 1974.)

Sellgren and Vappling (1986) reported that at high volumetric concentration the efficiency ratio was smaller than the head ratio, thus indicating a more pronounced loss of efficiency. Sellgen and Addie (1993) reported losses as low as half of those predicted by McElvain. The curves of McElvain do not take into account another important factor, namely the ratio of particle size to impeller diameter. Burgess and Reizes (1976) proposed that the head ratio and efficiency ratio were a function of three parameters: 1. Weight concentration 2. Ratio of d50 to impeller diameter 3. Specific gravity of the solid particles Sellgen and Addie (1993) indicated that there is a size effect and that head and efficiency losses were less drastic in large pumps than in small pumps (Figure 8-46). This clearly demonstrates the importance of pump design on performance. The importance of pump design on the head and efficiency ratio was confirmed by Czarnota et al. (1996) through tests on ITT-Flygt submersible pumps. Their work confirmed that high-efficiency pumps suffered from less degradation of performance than less efficient pumps. Head reduction was confirmed to be a linear function of the volumetric concentration of solids. Larger particles were found to slip more than smaller particles. An important factor they reported is that settling or separation can occur due to centrifugal forces. These forces are proportional to the square of the radius, and in the presence of large particles can lead to partial blockage, higher water velocity, and more slip between solids and liquid. A well-mixed particle distribution tended to decrease derating of pumps. Russian engineers developed a very advanced mathematical model based on full screen analysis instead of the average d50, which has been the focus of most equations in

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FIGURE 8-46 Effect of the size of the pump impeller on the correction factor for head RH for slurry at a weight concentration of 42%. (From Sellgren and Addie, 1992.).

Australia, Europe, and North America. The work of Kuznetsov and Samoilovich (1985, 1986) was summarized by Angle et al. (1997). These advanced mathematical models permit corrections based on the number of vanes, discharge angle of the vanes, and volumetric concentration of each range of diameter of solids in the slurry. It would be very appropriate to explore these models; however, when examining a worn-out impeller, as in Figure 8-47, the reader may wonder how practical such models may be. 8-10-3 Concepts of Head Ratio and Efficiency Ratio Due to Pumping Froth Flotation froth is complex slurry and may contain an important amount of air and gases (Figure 8-48). The industry uses the froth factor as a measure. Basically, it is determined by filling a column of flotation slurry and measuring the height H0. It is then left for 24 hr to rest. The height of the slurry H⬘ is then measured. The froth factor F is defined as: F = H0/H⬘

(8-75)

Using this concept of froth factor to size pumps must be done very carefully. Different grades of froth leads to different levels of entrained gases, as shown in Table 8-15. Conventional centrifugal pumps can not handle excessive amounts of entrained gases. A very common misunderstanding in the industry is that the flow rate of slurry must be multiplied by the froth factor to size the pump. This violates a very fundamental principle that gases or air are compressible fluids. In other words, as the bubbles pass through the impeller they are compressed and reduced in size. In fact, the proper sizing of slurry pumps to handle froth must be based on a full examination of the system. For example, if

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8.69

FIGURE 8-47 Worn-out impeller, showing gradual degradation of the impeller tip diameter and vanes. Deterioration of hydraulics and head efficiency may occur throughout the wear life of the impeller and pump.

FIGURE 8-48 Flotation slurry froth contains sufficient air bubbles to degrade the performance of centrifugal pumps. (Courtesy of EOMCO Process Equipment Co.)

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TABLE 8-15 Correlation between Froth Factor and Percentage of Entrained Air Froth factor

% entrained air

1.5 2.0 2.5 3.0

2–3% 3–5% 5–7% > 7%

Example Normal flotation tailings Flotation tailings with minimum retention time Tenacious flotation tailings with minimum retention time Froth with very fine particles

the flotation cells are away from the pump and the froth is transported by gravity in launders, it may be argued that the surface area of the launders acts as a deaerator for air removal. In that case, does a 24 hr tube test apply well? The correct approach is in fact to remove as much of the air as possible before the froth enters the pump feed sump. This sump may also be designed in a conical shape to maximum surface area at the top. Certain forms of froth are very difficult to pump, such as the type associated with tar sands, in which viscosity plays a major role. Efficiency as low as 10% was reported with conventional pumps. Cappelino et al. (1992) presented a very thorough study on the performance of centrifugal pumps with open impellers with emphasis on pulp and paper flotation circuits and deinking cells. High-consistency stock (12%) can have as much as 20–28% entrained air. At the inlet to the impeller, the pressure drop tends to cause an expansion of the air and gases, and this indicates well that the concept of the froth factor can be misleading Example 8-8 The height of the liquid in a froth cell is 30 m above the pump. The depression at the inlet to the pump is about 6 m. Determine the expansion of gases, assuming a barometric pressure of atmospheric air at 9.5 m. The pump is designed to deliver a total head of 54 m. Determine the final volume of the gases. Solution The effective absolute head in the sump is: 30 + 9.5 = 39.5 m Due to the depression of 6 m, the absolute pressure is then: 39.5 – 6 = 33.5 m The expansion ratio at constant temperature is: 39.5/33.5 = 1.179 The absolute discharge head is: suction head + TDH + atmospheric barometric height = 30 + 54 + 9.5 = 93.5 m Ratio of discharge to suction absolute head is: 93.5/39.5 = 2.37 The size of the air or gas bubbles will then shrink by the inverse of this ratio, or 42.2%. Since the laws of thermodynamics apply, the concept of a constant froth factor is illusive. It would be a grave error to size piping and equipment based on the suction froth factor.

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FIGURE 8-49 Head and power correction factors for entrained gas due to flotation circuits. (From Cappellino, Roll, and Wilson, 1992. Reproduced by permission of Texas A&M University.)

The performance of slurry pumps deteriorates in the presence of entrained gases. Cappelino et al. (1992) have therefore proposed to define appropriate head and power correction factors as: head measured with entrained gas HF = ᎏᎏᎏᎏ head measured without entrained gas

(8-76)

power measured with entrained gas PF = ᎏᎏᎏᎏ power measured without entrained gas

(8-77)

or

Special pumps are available for handling froth and entrained gases. One interesting design is the Sulzer–Ahlstrom ART pump. It features holes through the impeller leading

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straight to an expeller at the back of the impeller. This expeller discharges the air through a separate discharge flange at the back of the casing. In some other designs, the stuffing box is connected to an external liquid ring vacuum pump that can remove any entrained air in the slurry. The manufacturers of pulp and paper pumps have also designed special impeller with protruding vanes that extend into the suction pipe to break up any large air particles. This concept is gaining popularity in some oil sand applications to handle particularly thixotropic and viscous froth. Dredge pumps sometimes face a similar problem. Gases are disturbed or released (particularly methane) during certain phases of dredging and end up accumulating in the ladder pump. Herbich and Miller (1970) conducted extensive test work on the effect of air on the development of head. Herbich (1992) proposed that special air removal systems be installed on the suction side of the pump with an ejector, as this is better than a vacuum pump.

8-11 CONCLUSION In this chapter, some of the important parameters that give slurry pumps their final shape were examined. It is obvious that slurry pumps are different from water pumps and that considerable research should be undertaken in fields such as pump-out vanes, expeller design, and effects of wide impellers on performance. The successful performance of these pumps depends on their resistance to wear. The slurry—in terms of its composition and concentration, and in terms of any froth-induced gases—is the determining factor for power consumption and the final hydraulics across the impeller and casing. These parameters are extremely important for the successful installation of these pumps.

8-12 NOMENCLATURE A Ap bt b2 B2 BHP C Cm Cv Cw Cp D d2 dSL dm ER Fth FA Fp

Factor to calculate slip, depending on the use of volute or diffuser Equivalent casing area for stress calculations Thread width at the root Width of the impeller at the impeller tip diameter Width of the casing at the impeller tip diameter Power in bhp Constant Meridional velocity across the impeller volumetric concentration of solids Concentration by weight of the solid particles in percent Heat capacity Equivalent diameter of casing The tip diameter of the impeller vanes Diameter of shaft sleeve Pitch diameter Efficiency ratio Force due to thread pull Axial thrust on shaft due to hydraulic forces Force on casing due to design pressure

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Fp FR G H H1 H2 H30 HE HIMP SV HR HVOL SV HSV HV K Kx Ky L Lth m M N NPSH Nq NUS NSS PA Pb Pe Ps PD PW PV Q R1 R2 R3 R4 Rb RC RH R RMD Rr Rs Rv0 R1 R2 Rv0 RD

8.73

Force due to belts at the drive end of the pump Radial thrust Acceleration due to gravity (9.78 to 9.81 m/s2 or 32.2 ft/sec) Height of the thread tooth = (major diameter – minor diameter)/2; in the case of ACME threads h = p/2 Head at the medium diameter of the eye of the impeller Head at the tip diameter of vane of the impeller Head at 30% of best efficiency capacity Euler ideal head for an impeller Shut-off head due to the impeller Head ratio Shut-off head due to the volute Total shut-off head Vapor head Correction factor for the head ratio Coefficient to determine Xv Coefficient to determine Yv length of a full turn in a shaft thread = pitch for single start threads The length of the shaft thread Exponent in vortex equation Number of pumps in series Rotational speed of the pump in rev/min Net Positive Suction Head Specific speed in SI units Specific speed in US units Suction specific speed Atmospheric pressure Pressure at the root of the pump-out vanes Rb Pressure at the surface of the liquid in absolute terms on the suction side Pressure at the suction diameter d1 Pressure losses between the surface of the liquid and the pump, due to friction, valves, etc. Pump power in Watts Vapor pressure Flow rate Root radius of the vanes of the impeller Tip radius of the vanes of the impeller Radius of the smaller circle of a twin circle volute Radius of the larger circle of a twin circle volute Radius at the root of the pump out vanes Cutwater radius Head correction factor Efficiency correction factor Meridional radius of the volute at the throat Tip radius of the pump-out vanes Tip radius of the shaft sleeve Local radius of vanes Radius of the root of the impeller vane Tip radius of the vanes of an impeller Radius of vortex Reaction force at the drive end bearing

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RW s S Sb Ss Tq t tC tL TS TSL TDH U V W Wimp Wp X XV YV Z Z1 Ze

CHAPTER EIGHT

Reaction force at the wet end bearing Gap between the edge of the pump-out vanes and the back wear plate of the casing Static moment, obtained by graphical integration along the meridional plane of the vanes Bending stress at the root of the shaft thread Shear stress at the root of the shaft thread Torque Depth of the pump-out vanes Thickness of the pump casing Thickness of the liner Thrust on suction side Thrust on shaft sleeve Total Dynamic Head Tip speed Absolute velocity across the impeller Relative velocity across an impeller Weight of impeller Weight of pulleys Total gap between the pump-out vanes’ impeller surface and the pump back plate Width of the volute in the x-direction Width of the volute in the y-direction Number of vanes Geodetic elevation of liquid surface above the centerline of the pump impeller Geodetic elevation of the centerline of the pump impeller

Greek Letters  Angular inclination of the vane with respect to the tangent Slip factor Density of liquid Cavitations parameter Angular velocity liq Angular velocity of the liquid in the gap between the pump-out vanes and the pump back plate imp Rotational velocity of the impeller or expeller SI Head coefficient to SI convention US Head coefficient to US convention Subscripts 1 At the root of the vane 2 At the tip of the vane E Eye of the impeller F Front shroud R Rear shroud Imp Impeller Liq Liquid Sl Sleeve

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8-13 REFERENCES Abulnaga, B. E. 2001. Recommendations for the design of mill discharge slurry pumps. Mazdak International Inc. Internal Report 02/2001 (unpublished). Addie, G. R. and F. W. Helmly. 1989. Recent improvements in dredge pump efficiencies and suction performances. Europort Dredging Seminar, Central Dredging Association, Delft, Netherlands. Anderson, H. H. 1938. Mine pumps. J. Mining Soc. Durham, United Kingdom. Anderson, H. H. 1977. Statistical records of pump and water turbine effectiveness. International Mechanical Engineers Conference on Scaling for Performance Prediction in Rotodynamic Pumps. September, pp. 1–6. Anderson, H. H. 1980. Centrifugal Pumps. Trade and Technical Press: UK. Anderson, H. H. 1984. The area ratio system. World Pumps, 201. Angle, T. and J. Crisswell (Editors). 1977. Slurry Pump Manual. Salt Lake City, Utah: Envirotech. Angle, T. and A. Rudonov. 1999. Slurry Pump Manual. Salt Lake City, Utah: Envirotech. ANSI/ASME B106.1. 1985. Design of Transmission Shafting. American Society of Mechanical Engineers, New York. Burgess, K. E. and B. E. Abulnaga. 1991. The application of finite element methods to Warman pumps and process Equipment. Paper presented to the Fifth International Conference on Finite Element Analysis in Australia, University of Sydney, Australia, July 1991. Burgess, K. E. and J. A. Reizes. 1976. The effect of sizing, specific gravity and concentration on the performance of centrifugal slurry pumps. Proc. Inst. Mech. Eng., 190, 36. Cappellino, C. A., D. Roll, and G. Wilson. 1992. Design considerations and application guidelines for pumping liquids with entrained gas using open impeller centrifugal pumps. Proceedings of the Ninth International Pump Users Symposium, Texas A&M University. Czarnota, Z., M. Fahlgren, M. Grainger, and S. Saunders. 1996. The effects of slurries on the performance of submersible pumps. BHR Group Hydrotranport, 13, 643–655. Duchham C. D. and Y. K. A. Aboutaleb. 1976. Some tests in a single stage semi-open impeller centrifugal pump handling coal dust slurries. In Proceedings Pumps and Turbine Conferences, Vol 1. Fairbanks, L. C. Jr. 1941. Effects on the characteristics of centrifugal pumps. Solids in Suspension Symposium, Proc. Am. Soc. Civ. Eng., 129, 129. Fischer, K. and D. Thoma. 1932. Investigation of the flow conditions in a centrifugal pump. Transactions ASME, 54. Frost, T. H. and E. Nielsen. 1991. Shut-off head of centrifugal pumps and fans. Proc. Inst. Mech. Eng., 205, 217–223. Herbrich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill. Herbich, J. B. and R. J. Christopher. 1963. Use of high speed photography to analyze particle motion in a model dredge pump. In Proceedings of the International Association for Hydraulic Research, London England. Herbich, J. B. and R. E. Miller. 1970. Effect of air content on performance of a dredge pump. In Proceedings of the World Dredging Conference, Wodcon 70, Singapore. Hunt, A. W and R. F. Faddick. 1971. The effects of solids on centrifugal pump characteristics. In Advances in Solid–Liquid Flow in Pipes and Its Application, I. Zandi (Ed.), New York: Pergamon Press. Jekat, W. K. 1992. Centrifugal pump theory. Section 2.1 in Pump Handbook, J. Karassik et al. (Eds.), New York: McGraw Hill. Kuznetsov, O. V. and D. C. Samoilovich. 1986. Increase of Reliability of Slurry Pumps in Service (in Russian). Moscow: CINTIchimneftemash, ser.XM-4. McElvain, R. E. 1974. High pressure pumping. Skillings Mining Review, 63, 4, 1–14. Pfeiderer, C. 1961. Die Kreiselpumpen. Berlin: Springler-Verlag. Samoilovich, D. C. 1986. Experimental Study of Slurry Pumps Performances (in Russian). Moscow: CINTIchimneftemash, ser.XM-4. Sellgren, A. and L. Vappling. 1986. Effects of highly concentrated slurries on the performance of centrifugal pumps. Proceedings of the International Symposium on Slurry Flows, FED Vol 38, ASME, USA, pp. 143–148. Sellgren, A. and G. R. Addie. 1992. Effects of solids on the performance of centrifugal slurry pumps.

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Paper presented at the 10th Colloquium: Massenguttransport durch Rohrietungen in Meschede, Germany, May 20–22. Sellgren, A. and G. R. Addie. 1993. Solids effect on the characteristics of centrifugal slurry pumps. Paper presented at the 12th International Conference on Slurry Handling and Pipeline Transport, Brugge, Belgium. Sheth, K. K., G. L. Morrison, and W. W. Peng. 1987. Slip factors of centrifugal slurry pumps. A.S.M.E. Journal of Fluids Engineering, 109, 313–318. Stepanoff, A. J. 1969. Gravity flow of bulk solids and transportation of solids in suspension. New York: Wiley. Stepanoff, A. J. 1993. Centrifugal and Axial Flow Pumps. Melbourne, FL: Krieger. Sulzer Pumps. 1998. Centrifugal Pump Handbook. New York: Elsevier. Turton, R. K. 1994. Rotodynamic Pump Design. Cambridge: Cambridge University Press. K. C. Wilson, G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. London: Elsevier Applied Sciences. Wilson, G. 1976. Construction of solids-handling centrifugal pumps. In Pump Handbook, J. Karassik et al. (Eds.) New York: McGraw Hill. Worster, R. C. 1963. The flow in volutes and effect on centrifugal pump performance. Proc. Inst. Mech. Eng., 177, 843. Further Reading Kazim, K. A. and B. Maiti. 1997. A correlation to predict the performance characteristics of centrifugal pumps handling slurries. In Proceedings of the Institution of Mechanical Engineers. Part A. Journal of Power and Energy, 211, A2, 147–157. Cader, T., O. Masbernat, and M. C. Rocco. 1994. Two phase velocity distributions and overall performance of a centrifugal slurry pump. Journal of Fluid Engineering, 116, 316–323. Gandhi, B. K., S. N. Singh, and V. Seshadri. 2000. Improvements in the prediction of performance of centrifugal slurry pumps handling slurries. Proceedings of the Institution of Mechanical Engineers. Part A. Journal of Power and Energy, 214, 5, 473–486.

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CHAPTER 9

POSITIVE DISPLACEMENT PUMPS

9-0 INTRODUCTION Positive displacement slurry pumps and mud transfer pumps play a major role in a number of industries such as mining and metallurgical processes, chemicals, power generation, porcelain and ceramics, and sugar refining. These pumps are versatile, efficient, and suitable for pressures up to 17.3 MPa (2500 psi). Positive displacement pumps have gained acceptance on long-distance mineral concentrate pipelines as their high capital cost is recuperated through lower installation cost of electric systems, elimination of booster stations, and high hydraulic efficiency, which is superior to centrifugal pumps. Plunger or diaphragm pumps do not handle large flow rates in excess of 100 m3/hr (4400 US gpm), but they are suitable for a wide range of applications at higher volumetric concentrations than centrifugal pumps. Positive displacement pumps can pump slurries with a weight concentration of 70%.

9-1 SOLID PISTON PUMPS Positive displacement slurry pumps are used in a number of industries (Table 9-1). Solid piston pumps are reserved for the pumping of slurries of a low to medium abrasiveness (Miller Number <50) such as chalk slurry, fine coal, flotation material, and drilling mud sludge. In these pumps, the slurry comes into contact with the piston and with the packings. In a duplex pump, the flow splits into two cylinders inside the pump, whereas in a triplex it splits into three cylinders. Table 9-2 compares both designs. A special feature of the duplex pumps is that they can be built in a single- or doubleacting configuration. In the case of the double-acting pump, the slurry flows on both sides of the piston. On one side, there is a piston rod that goes through the packing before connecting to the connecting rod and crankshaft (Figure 9-1). Because the flow is divided in a duplex double-acting piston pump to the two sides of

9.1

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TABLE 9-1 Applications of Positive Displacement Pumps Industry

Application

Mining

Coal transportation (e.g., Novo Siberski pipeline, Black Mesa Pipeline) Flotation material Washery refuse Deep mine dewatering of water with solid particles Limestone, milk of lime Potash rock salt, phosphate, iron ore, nickel ore concentrate Bauxite, red mud, gold mud Sand, pyrite, REA gypsum Backfilling Underground drainage Filter press feed Autoclave feed

Chemicals

Salt slurry Porcelain slurry Pastes Detergent slurry Combustion furnace feed Filter press feed

Power generation

Coal and coal slurry Flue Pressurized fluidized bed combustion Wet ash removal Ship loading Long-distance pipelines

Construction

Bentonite, clay mash, cement

Porcelain

Clay slurry Filter press feed

Sugar

Carbonation slurry Sugar beet washing

Information provided by courtesy of Wirth-Maschinen and Bohrgerate, Germany.

TABLE 9-2 Comparison between Duplex and Triplex Pumps Duplex Single Acting 2 cylinder liners 2 piston gaskets 2 cylinders 4 valves Slurry does not come in contact with packing

Duplex Double Acting 2 cylinder liners 4 piston gaskets 2 cylinders 2 piston rods with packing 8 valves Slurry does come in contact with packing

Triplex Single Acting 3 cylinder liners 3 piston gaskets 3 cylinders 6 valves Slurry does not come in contact with packing

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9.3

FIGURE 9-1 Concept of the double-acting duplex piston pump. (Courtesy of Wirth Pumps.)

the piston, these pumps operate at lower speeds than single-acting duplex and triplex pumps (Figure 9-2). The single-acting duplex pump seems to have disappeared from the world of manufacturing. For proper balancing, the pistons of duplex pumps are 180° out of phase (Figure 9-3), but for triplex pumps they are 120° out of phase with each other (Figure 9-4). Triplex pumps have a lower degree of oscillation than duplex units. The degree of

FIGURE 9-2 Concept of the triplex piston pump. (Courtesy of Wirth Pumps.)

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FIGURE 9-3 Concept of gear mechanism for duplex piston pumps. (Courtesy of Wirth Pumps.)

FIGURE 9-4 Pumps.)

Concept of gear mechanism for triplex piston pumps. (Courtesy of Wirth

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FIGURE 9-5 Pulsation diagram for triplex pump. (Courtesy of Wirth Pumps.)

variation of flow in the former is 23% (Figure 9-5) compared to 46% in the latter (Wallrafen, 1983; Figure 9-6). Duplex and triplex slurry pumps are manufactured to a power frame of approximately 1500 kW (2000 bhp). The Black Mesa Pipeline featured 13 duplex pumps, each with a driving power of 1250 kW (1675 bhp) to transport 4.8 million tons of coal over a distance of 440 km (275 miles) (Wallrafen, 1983). Some of these pumps were manufactured by Wilson-Snyder in the United States. Piston slurry pumps are used extensively as mud transfer pumps. Gardner-Denver in the United States offers a range of duplex pumps in the power range of 12–76 kW (16–102 hp).

FIGURE 9-6 Pulsation diagram for duplex pump. (Courtesy of Wirth Pumps.)

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FIGURE 9-7 Triplex pump TPK 7⬙ × 12⬙/1600. Driving power 1200 kW (1600 hp). (Courtesy of Wirth Pumps.)

9-2 PLUNGER PUMPS For a long time, plunger pumps were not considered to be suitable for slurry transportation, but in the late 1970s, manufacturers developed a suitable flushing system to minimize wear of the plunger. Plunger pumps are single acting. They use plungers instead of pistons (Figure 9-8) with valves. They operate at 80–120 cycles per minute. Plunger pumps are prone to wear. They are less expensive to purchase than diaphragm pumps but have a higher maintenance cost. These pumps use three types of valves: 1. Free-floating valves 2. Spring-loaded spherical (Rollo) valves 3. Spring-loaded elastomer-seal (mud) valves These valves are shown in Figure 9-9. It is important to minimize packing wear with piston pumps. Smith (1985) proposed four methods: 1. Use of conventional packing of plungers at low speed with slurries of low abrasiveness 2. Provision of a clean, slurry-free environment for the packing rubbing surface (by synchronized or continuous injection of water or cleaning fluid) 3. Separation of the slurry from the pumping element 4. Total isolation of the slurry from the packing (by providing a separate diaphragm chamber) The SAMARCO pipeline in Brazil used 14 plunger pumps with a driver power of 920 kW each to deliver 12 million metric tons of iron oxide ore concentrate over a distance of

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FIGURE 9-8 Schematic representation of a plunger pump.

400 km (250 mi) (Wallrafen, 1983). These triplex plunger pumps were manufactured by Wilson-Snyder. The Wilson-Snyder line of plunger slurry pumps features 21 different sizes from 45–1250 kW (60–1700 hp). They have also been used in Georgia, U.S.A. on a kaolin pipeline. Kaolin is not very abrasive. These pumps have also been used for mine dewatering from a depth of 1036 m (3400 ft). The water contained solids.

FIGURE 9-9 Categories of valves for slurry pumps. (a) Free floating; (b) spring-loaded spherical (Rollo); (c) spring-loaded elastomer seal. (From Smith, 1985. Reprinted by permission of McGraw-Hill.)

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Some of the triplex plunger pumps are rated at 41.4 MPa (6000 psi) (Wilson-Snyder, 1977), such as the model 85-25. The volume capacity is rated from 363–2941 L/min (96–777 US gpm).

9-3 PISTON DIAPHRAGM PUMPS To handle abrasive slurries that piston pumps would find difficult, manufacturers such as Geho Pumps (Netherlands), Wirth (Germany), and Gorman-Rupp (United States) have developed pumps to use a diaphragm or a sort of flexible piston that comes in contact with the slurry or sludge. Feluwa of Germany added a hose so that there is an isolating bath of oil between the hose and the diaphragm. The pumps from Geho, Wirth, and Feluwa feature a crankshaft mechanism to move the diaphragm but the Gorman-Rupp pump uses an air cylinder to actuate the diaphragm. Diaphragm piston pumps use a sort of oil chamber between a reciprocating piston (operated by a crankshaft) and the diaphragm (Figure 9-10). Provided that no puncture occurs in the diaphragm, slurry does not come in contact with the piston. A special control system is installed to detect diaphragm rupture. Wallrafen (1983) reported that Wirth manufactured its first piston diaphragm pump in 1969 to pump sand slurries. The first unit lasted 1000 hrs without having to replace worn parts. By the early 1980s, wear life of 6000 hrs was achieved with rubber materials and

FIGURE 9-10 Pumps.)

Concept of double-acting piston duplex diaphragm pump. (Courtesy of Wirth

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9.9

proper design of the diaphragm. Diaphragm piston pumps are designed as duplex doubleacting or as triplex single-acting pumps in a similar concept rather to solid piston pumps (Figures 9-10 and 9-11). Piston diaphragm pumps (Figure 9-12) are more expensive than plunger pumps. For autoclave feed pumps, Geho developed a special design to handle slurries as hot as 200°C (392°F) at a high flow rate. Solids concentrations can be as high as75% and pumps can operate at slurry temperatures up to 200°C. Typical uses in the mining industry include: 앫 앫 앫 앫 앫 앫 앫

Long-distance slurry (mineral concentrate) pipelines (up to 300 km long) Clean and efficient tailings disposal High-pressure bauxite digester feed Autoclave and reactor feed Mine backfilling Mine dewatering (single stage) Hydraulic ore hoisting

With more than 400 piston diaphragm pumps installed on some of the world’s most demanding long-distance pipeline applications, Geho Pumps has taken the opportunity, through a significant research and development effort, to constantly improve piston diaphragm pump design. This has resulted in numerous proprietary design improvements and technical innovations relevant to severe slurry pumping.

FIGURE 9-11 Pumps.)

Concept of single-acting piston triplex diaphragm pump. (Courtesy of Wirth

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FIGURE 9-12

Geho pump at Freeport. (Courtesy of Geho.)

Geho Piston diaphragm are used for tailings disposal and pumping at very high concentration so that: 앫 Amount of free water is virtually eliminated, allowing slurry to be stacked or distributed in layers 앫 Dry stacking requires less storage space 앫 Prevents contamination of the environment by leakage 앫 Rain and wind do not affect the solidified tailings 앫 Mechanical stability of tailings allows a high stack with rehabilitation possibilities after use Typical tailings applications of Geho pumps include: 앫 앫 앫 앫 앫 앫

Bayswater Power Station—fly ash disposal (Australia) Nabalco, Gove Refinery—red mud disposal (Australia) Ledvice Power Station—fly ash disposal (Czech Republic) Pingguo Aluminium Company—red mud disposal (Peoples Republic of China) Khaperkheda Ash Handling Plant—fly ash disposal (India) National Aluminum Company—red mud disposal (India)

TABLE 9-3 Examples of Installation of Piston Diaphragm Pumps on Slurry Pipeline and Tailings Applications Location

Manufacturer

Installation application flow rate stated for each pump

Alsen Zementwerke

Geho

Antamina, Peru

Wirth

Ashanti Goldfields Ghana

Wirth

3 piston diaphragm pumps to pump limestone slurry over 10 km (6.3 mi) 4 piston pumps, 100 m3/hr (440 gpm), 25.2 MPa (3650 psi), copper concentrate 1 pump, 215 m3/hr (947 gpm), 6 MPa (870 psi) backfill slime (gold tailings)

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9.11

TABLE 9-3 (continued) Location

Manufacturer

Installation application flow rate stated for each pump

Bajo Alumbrera, Argentina

Geho

Cameco, Canada

Wirth

Cia Minera Disputada de Las Condes, Chile Codelco, Chile

Wirth

Course Nickel , Australia

Wirth

Doña Ines Collahuasi, Chile

Geho

ECPSA, Cuba

Wirth

Empresa Minera Yauliyacu, Peru Eskay Creek, Canada

Wirth Wirth

Freeport, Indonesia

Geho

Goldmine, South Africa

Wirth

ISCOR, Hillendake Mine, South Africa Jian Shan

Wirth

Nabalco, Australia

Geho

Norilsk Nickel Combinat

Geho

Pasminco, Australia

Wirth

Los Pelambres, Chile

Geho

Sicartsa, Mexico

Geho

J. R. Simplot, USA

Geho

Batu Hijau, Indonesia

Geho

Cockburn Cement, Australia

Geho

New Zealand Steel

Geho

Rio Capim, Brazil

Geho

Minera Escondid, Chile

Geho

OEMK, Ukraine

Geho

6 piston diaphragm pumps in 3 booster stations to transport copper concentrate over a distance of 320 km (225 mi) in a 150 mm (6 in) line, 91 m3/h at 217 bar 2 pumps, 80 m3/hr (352 gpm), 12.5 MPa (1810 psi), uranium ore 3 pumps 115 m3/hr (510 gpm), 2.5 MPa (360 psi) copper tailings 3 pumps, 140 m3/hr (616 gpm), 3.5 MPa (507 psi), c opper tailings 2 pumps, 193 m3/hr (850 gpm), 5 MPa (725 psi), lateritic nickel ore 2 piston diaphragm pumps for 203 km of copper concentrate transport, 117 m3/h at 217 bar 10 pumps, 200 m3/hr (800gpm), 6.4 MPa (920 psi), iron–nickel slurry 2 pumps, 90 m3/hr (400 gpm), 14.4 MPa (2080 psi), tailings 1 pump, 25 m3/hr (110 gpm), 11.7 MPa (1700 psi) for a 6.5 km (4 mi) pipeline 2 piston diaphragm pumps to transport copper concentrate over 120 km (75mi), 159 m3/h at 40 bar 1 pump, 12 m3/hr (66 gpm), 12 MPa (1714 psi), backfilling 2 pumps 450 m3/hr (1980 gpm), 7.4 MPa (1075 psi), heavy mineral tailings 100 kms iron ore concentrate transport (PR of China), 2 piston diaphragm pumps, 216 m3/h at 153 bar 3 piston diaphragm pumps for a highly concentrated red mud slurry to a disposal area, 200 m3/h at 160 bar 9 piston diaphragm pumps, 55 km of multimetallic ore transport (North Siberia, Russia), 400 m3/h at 80 bar 3 pumps, 161 m3/hr (710 gpm), 12.5 MPa (1810 psi) zinc and lead concentrate 2 piston diaphragm pumps for 120 km pipeline transportation of copper concentrate slurry, 165 m3/h at 150 bar 1 piston diaphram pump for iron ore concentrate slurry, 380 m3/h at 110 bar 4 piston diaphragm pumps for 100 km transportation of phosphate slurry, 97 m3/h at 228 bar 2 piston diaphragm pumps for 120 km copper concentrate slurry transportation, 123 m3/h at 228 bar 3 piston diaphragm pumps for shell and slur

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Page iii

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

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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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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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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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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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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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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.

1.3

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CHAPTER ONE

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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GENERAL CONCEPTS OF SLURRY FLOWS

1.5

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.

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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

<12500

<1

<0.0004 1.14

Mud

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TABLE 1-7 Testing Parameters on Soils to Determine Dredgeability, Suitability for Placer Mining, or Slurry Preparation Type of soil

Testing parameters

Sand

Density Water content Specific gravity of grains Grain size Water permeability Frictional properties Lime content Organic content

Silt

Density Water content Water permeability Shear strength or sliding resistance Plasticity Lime content Organic content

Clay

Density Water content Sliding resistance Consistency ranges (plasticity) Organic content

Peat

Same parameters as clay

Gravel

Same parameters as sand

The difference between the liquid and plastic limits is defined as the plasticity index: PI = LL – PL

(1-3)

1-2 SLURRY FLOWS A slurry mixture is essentially a mixture of a carrying fluid and solid particles held in suspension. The most commonly used fluid is water, but over the years, attempts have been made to use crude oils with milled coal, and even air in pneumatic conveying. The flow of slurry in a pipeline is much different from the flow of a single-phase liquid. Theoretically, a single-phase liquid of low absolute (or dynamic) viscosity can be allowed to flow at slow speeds from a laminar flow to a turbulent flow. However, a twophase mixture, such as slurry, must overcome a deposition critical velocity or a viscous transition critical velocity. The analogy can be made here in terms of an airplane: if the speed drops excessively, the airplane stalls and stops flying. If the slurry’s speed of flow is not sufficiently high, the particles will not be maintained in suspension. On the other hand, in the case of highly viscous mixtures, if the shear rate in the pipeline is excessively low, the mixture will be too viscous and will resist flow. Sections 1-2-1 and 1-2-2 define the two basic slurry flows.

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1-2-1 Homogeneous Flows Solids are uniformly distributed throughout the liquid carrier. An example of homogeneous flows is copper concentrate slurry after undergoing a process of grinding and thickening. Particles are then very fine and the mixture is at a high concentration (50–60% by weight). As the concentration of particles is increased (beyond 40% by weight for many slurries), the mixture becomes more viscous and develops non-Newtonian properties. Apart from rich concentrate slurries, drilling mud, sewage sludge, and fine limestone (cement kiln feed slurry) behave as homogeneous flows. Typical particle sizes for homogeneous mixtures are smaller than 40 m to 70 m (325–200 mesh), depending on the density of the solids. The presence of clays in certain circuits must not be ignored. If clay is not separated, the slurry can be quite viscous. Pumps and pipes must be sized properly to handle the resultant absolute (or dynamic) viscosity. Certain mines in Peru contain material called soft high clay, which can increase the absolute (or dynamic) viscosity of the slurry up to 400 mPa at weight concentrations in excess of 45%. Dilution to lower concentration and changes to recycling load are solutions to such a problem.

1-2-2 Heterogeneous Flows In a heterogeneous flow, solids are not uniformly mixed in the horizontal plane. A gradient of concentration exists in the vertical plane. Dunes or a sliding bed may form in the pipe, with the heavier particles at the bottom and the lighter ones in suspension, particularly at the critical deposition velocity. The different phases retain their properties and the largest particles do not necessarily cause the biggest problems; it really depends on the ratio that they are mixed with finer particles. Heterogeneous slurries are encountered in many placer mining, phosphate rock mining, and dredging applications. Concentration of particles remains low, typically less than 25% by weight in many dredging applications and below 35% by weight in many tailing disposal applications. Heterogeneous flows require a minimum carrier velocity. In some tailing applications of the Taconite mines of Minnesota, the typical deposition velocity is in excess of 3.4–4 m/s (11–13 ft/s). Nature being complex, flows are encountered that have the characteristics of heterogeneous or homogeneous flows. The concept of pseudohomogeneous flows is also used when a large fraction of particles are fine but there remain a sufficient fraction of coarse particles that may deposit as the flow speed is reduced below a minimum value.

1-2-3 Intermediate Flow Regimes Intermediate regimes occur when some of the particles are homogeneously distributed and others are heterogeneously distributed. Intermediate regime flows include tailings from mineral processing plants and a wide range of industrial slurries.

1-2-4 Flows of Emulsions Strictly speaking, an emulsion is not slurry. An emulsion is a mixture of two phases at certain temperatures resulting in an essentially homogeneous flow. An example of an emulsion is a mixture of bitumen at 70% by volume with water at 30% by volume. If sur-

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factants are used, the bitumen remains well mixed with water in a certain temperature range. Emulsions can become unstable under certain high shear rates or through very tight clearances of pumps according to Nunez et al. (1996). In the 1990s, PDVSA-BITOR constructed a 300 km (188 mi) pipeline in Venezuela with a diameter of 660–915 mm (26–36 in) for transporting their ORIMULSION fuel, a mixture of highly concentrated bitumen and water. The fuel is a substitute for coal in thermal plants. Emulsions do not encounter the deposition velocity of slurries, but as flows become unstable, fine droplets of heavy oils or bitumen may coalesce into larger ones, causing changes of flow.

1-2-5 Flows of Emulsions—Slurry Mixtures A mixture of an emulsion and solids such as fine coal could be used to produce a fluid with a high calorific value. Coal must be very fine to burn readily in combustion furnaces. The flow is similar to a homogeneous flow with a high absolute (or dynamic) viscosity component.

1-3 SINKING VELOCITY OF PARTICLES AND CRITICAL VELOCITY OF FLOW Various parameters of speed determine whether a mixture may separate or continue to flow. In fact, the designer of a thickener or a mixer is often more interested in the sinking velocity of particles. On the other hand, the designer of a pipeline has to pay attention to the critical velocity of flow, settling speed, and whether the flow is vertical or horizontal, particularly in the case of heterogeneous flows.

1-3-1 Sinking or Terminal Velocity of Particles This is the minimum speed needed to maintain particles in suspension, particularly in a process of mixing or thickening. This velocity is not identical with the critical velocity of flow, and should not be confused with it. Table 1-8 presents examples of sinking velocity of various soils. The designer of a mixing system or a thickener is encouraged to conduct lab tests, since clays may be mixed with sands in some areas, or the soil may be stratified, with layers of different materials.

1-3-2 Critical Velocity of Flows In Chapters 3, 4, and 5 the mechanics of solid suspensions are described in detail. An important parameter to introduce in this chapter is the critical velocity of a slurry flow. Figure 1-8 plots the pressure loss per unit length on the y-axis, versus the velocity V of a slurry flow on the x-axis. Five points are shown for flow at a constant volume concentration. For this slurry of moderate viscosity, the flow is stationary and the solids clog the pipeline below point 1. There is insufficient speed to move the particles. As the flow is accelerated, the speed reaches point 1, which is called the deposition critical velocity VD, or minimum speed to start the flow. Between points 1 and 2, the bed builds up, dunes form, and the different phases are well separated. Between points 2 and 3 the flow is streaking but

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TABLE 1-8 Sinking Velocity of Soil Particles (after Sulzer Pumps, 1998, with permission of Elsevier) Particle diameter, Mesh size, micrometers US fine

Pressure drop per unit of length

0.2 0.6 1 2 5 6 20 50 60 100 200 250 300 500 600 1000 2000

Soil grain size identification Sinking velocity, m/s

Sinking velocity, ft/s

3 × 10–8 2.8 × 10–7 7 × 10–6 9.2 × 10–6 17 × 10–6 25 × 10–6 28 × 10–5 17 × 10–4 25 × 10–4 0.07 0.021 0.026 0.032 0.053 0.063 0.10 0.17

270 230 150 70 60 35 30 18 10

Grain size by ASTM

Fine clay Coarse clay

Silt

Fine silt

Fine sand

Coarse silt Intermediate silt to sand Fine sand Medium coarse sand

Coarse sand Coarse sand Very coarse sand

slurry

4-5 Pseudohomogeneous

5

3-4 Jumping and rolling

4

2-3 Streaking

3 1

1-2 Bedding

2

wa

Grain size, international

Clay

ter

y/D concentration

Below 1 Stationary and clogging

Speed of flow FIGURE 1-8 Pressure drop versus velocity for water and for a slurry mixture. (After Sulzer Pumps, 1998, with permission of Elsevier.)

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momentum is building up. Between points 3 and 4, there is sufficient speed to cause jumping and rolling of the coarse particles. Above point 4, the speed is sufficiently high to allow a pseudohomogeneous flow in which the fine particles act as a carrier for the coarse particles. These stages are extensively reviewed in Chapter 4. When absolute (or dynamic) viscosity is an important factor, such as in clayish slurries or homogeneous flows, another parameter, the viscous transition critical velocity VT must be determined. There are two regimes for flow of homogeneous mixtures. Flow at speeds less than VT is associated with laminar flows, whereas flows above VT are characteristic of turbulent flows. Flow in the laminar regime is often characterized by a friction loss factor, which is 64/Re for the Darcy factor or 16/Re for the Fanning factor (this topic will be discussed in more details in Chapter 2). As a result, the losses in the laminar regime appear to be a linear function of speed, whereas in the turbulent regime they are proportional to the square of the speed. As we will see in Chapter 5, researchers have struggled with special definitions of a modified Reynolds number for non-Newtonian flows. Emulsions have been pumped over long distances in laminar flow. Nunez et al. (1996) demonstrates the existence of certain effects similar to comminution, i.e., breakup of large droplets into finer ones, as well as coalescence of small particles into larger ones under different flow regimes, shear rates, and constraints. A question often asked is what the relationship between the sinking speed (as per Table 1-8) and the deposition of critical velocity VD? This question comes up when the rheology laboratory produces the results of thickening tests, and when there is not enough money or time to conduct proper slurry loop tests. This point will be examined in Chapter 4 and various approaches have been adopted over the years, from the simplest assumption that the critical deposition velocity is 17 times as large as the terminal or sinking velocity, to more complex mathematical formulae. Often in a lab test, the coarse particles deposit rapidly while the fine particles are still in suspension. Proper pump tests are often recommended, particularly when a multimillion dollar pipeline is being designed. Tests can be conducted at a number of universities, provincial and state research centers, or with the help of manufacturers of slurry pumps. Examples include the Saskatchewan Science Research Center in Canada; the GIW research lab of KSB Pumps (USA), described by Wilson et al. (1992); the Slurry Research Lab of Mazdak International Inc. (U.S.A.); Texas A&M University (U.S.A.); and Melbourne University (Australia), among others.

1-4 DENSITY OF A SLURRY MIXTURE The density of a slurry mixture is a function of 앫 The density of the carrier fluid 앫 The density of the solid particles 앫 The concentration by volume of the solid phase The density of the solid particles is determined carefully by various experimental methods. Fine particles tend to entrap air, which the lab technician must remove by proper agitation or by adding a small quantity of wetting agent. Some materials exhibit a change of packing abilities and therefore density as a function of particle size. If the solids are to be passed through a comminution process, a SAG (semiautogeneous), or ball mill, they can occupy more volume per unit mass as they be-

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come finer. The slurry engineer is therefore encouraged to measure the solid’s density at the proposed size of particles to be transported in slurry form. Certain errors can occur in evaluating the density of solids with heterogeneous mixtures. If the heavier slurry particles settle out and a sample is taken, it may reflect a greater density of finer particles. Due to these possible sources of error, the engineer is encouraged to measure the density of the slurry mixture after proper mixing, and to use the data on concentration by weight or by volume to work back to the density of the solids. The density of a slurry mixture is expressed as 100 m = ᎏᎏᎏ Cw/s + (100 – Cw)/L

(1-4)

where Cw = concentration by weight m = density of the mixture phase l = density of the liquid phase s = density of the solid phase Engineers use the term concentration by weight, as it is easier to convert back into the total tonnage of solids to be transported through a pipeline or across an extractive metallurgy plant. However, the characteristics of the mixture, the mechanics of flow, and the resultant physical properties are more related to the concentration by volume. The concentration by volume of solids in a mixture is expressed as Cwm 100 Cw/s Cv = ᎏ = ᎏᎏᎏ s Cw/s + (100 – Cw)/L

(1-5)

The concentration by weight of solids in a mixture is expressed as Cvs Cvs Cv = ᎏ = ᎏᎏ m Cvs + (100 – Cv)

(1-6)

Example 1-A A pipeline is designed to transport 140 metric ton (308,000 lb) of sand per hour. The specific gravity of the sand particles is 2.65 (or the density is 2650 kg/m3). The concentration by weight is 30%. Determine the density of the mixture if the carrier fluid is water and determine the resultant flow rate. 앫 Weight of sand in the mixture over a period of 1 hr is 140 metric ton (308,000 lb) 앫 Weight of equivalent volume of water equivalent to the sand content is 140,000 kg/2,650 kg/m3 = 52,800 앫 Weight of water in the slurry mixture at a concentration of 30% by weight = 140,000 kg (100 – 30)/30 or 326,600 kg 앫 Total weight of slurry mixture transported in 1 hr is the sum of the weight of water and sand or 140,000 + 326,600 = 466,600 kg/hr 앫 Total weight of equivalent volume of water is 326,000 + 52,800 = 378,800 kg or 378 m3 of liquid, since density of water is 1000 kg/m3 The density of the slurry mixture is therefore 466,600 kg/378 m3 = 1,230 kg/m3. Alternatively, the specific gravity of the slurry compared to water is 1.23. The flow rate, being the volume per unit of time, is equivalent to 378 m3/hr.

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1.21

1-5 DYNAMIC VISCOSITY OF A NEWTONIAN SLURRY MIXTURE Although density is essentially a static property, absolute (or dynamic) viscosity is a dynamic property and tends to reduce in magnitude as the shear rate in a pipeline increases. Thus, engineers have had to define different forms of viscosity over the years, everything from dynamic viscosity, to kinematic viscosity, to effective pipeline viscosity. The effective pipeline viscosity will be discussed in detail in Chapters 3, 4, and 5. In this chapter, the reader is introduced to basic concepts of the mixture of slurry in a stationary state. This is effectively what the pump, or a mixer, might see at the start-up of a plant. As is often the case, when the driver cannot deliver enough torque to overcome the absolute (or dynamic) viscosity, the operator is forced to dilute the slurry mixture. Plasticity as defined in Section 1-1-4 is an important parameter in determining overall absolute (or dynamic) viscosity of a mixture of clay and water. There are, however, numerous soils in nature, such as sand and water or gravel and water, in which the solids contribute little to the overall absolute (or dynamic) viscosity, except in terms of their concentration by volume.

1-5-1 Absolute (or Dynamic) Viscosity of Mixtures with Volume Concentration Smaller Than 1% For such solid–liquid mixtures in diluted form, Einstein developed the following formula for a linear relationship between absolute (or dynamic) viscosity and volume concentration:

m ᎏ = 1 + 2.5 L

(1-7)

where m = absolute (or dynamic) viscosity of the slurry mixture L = absolute (or dynamic) viscosity of the carrying liquid This is a very simple equation that is based on the following assumptions: 앫 Particles are fairly rigid 앫 The mixture is fairly dilute and there is no interaction between the particles Such a flow is not encountered, except in laminar regimes of very dilute concentrations (below a volume concentration of 1%).

1-5-2 Absolute (or Dynamic) Viscosity of Mixtures with Solids with Volume Concentration Smaller than 20% Thomas (1965) took the equation of Einstein further by calculating for higher volumetric concentrations of Newtonian mixtures:

m ᎏ = 1 + K1 + K22 + K33 + K44 + . . . L where K1, K2, K3, and K4 are constants

(1-8)

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K1 is the Einstein constant of 2.5 (from Equation 1-7), and K2 has been found to be in the range of 10.05–14.1 according to Guth and Simha (1936). It is difficult to extrapolate the higher terms K3 and K4 in Equation 1-8. They are ignored with volumetric concentrations smaller than 20%.

1-5-3 Absolute (or Dynamic) Viscosity of Mixtures with High Volume Concentration of Solids For higher concentrations, Thomas (1965) proposed the following equation with an exponential function:

m ᎏ = 1 + K1 + K22 + A exp(B) L

(1-9)

where K2 = 10.05 A = 0.00273 B = 16.6 Figure 1-9 is based on Equation 1-9 and is widely accepted in the slurry industry for heterogeneous mixtures of a Newtonian rheology.

1-6 SPECIFIC HEAT Thomas (1960) derived an equation for the specific heat of a mixture as a function of the specific heat of the liquid and solid phases: CpmCws + CpmCwL Cpm= ᎏᎏ 100

(1-10)

1-7 THERMAL CONDUCTIVITY AND HEAT TRANSFER Thermal conductivity is difficult to measure, as solids may settle during the test. Sometimes it is recommended to apply a small quantity of gel to maintain the solids in suspension. Orr and Dalla Valle (1954) derived the following equation for the thermal conductivity of slurry mixtures: 2kl + ks – 2(kl – ks) km = kl ᎏᎏᎏ 2kl + ks – 2(kl + ks)

冢

where k = thermal conductivity and subscripts l = liquid m = mixture s = solids.

冣

(1-11)

L m 70 60 50 40 30 20 10 0

0

10

20

30

40

50

60

70

80

90

CV Volumetric concentration of solids

10 12 14 16 18 20 22 25 27 29 31 33 35 37 39 42 43 44

CV [%] 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

m

L

8.950 9.932 11.07 12.40 13.94 15.75 17.86 20.33 23.22 26.62 30.61 35.29 40.80 47.28 54.91 63.89 74.47 86.94 101.63 118.95 139.4

FIGURE 1-9 Ratio of viscosity of mixture versus viscosity of carrier in accordance with the Thomas equation for coarse slurries.

Page 1.23

viscosity of carrier liquid

80

L

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3 5 7

90

m 1.029 1.089 1.156 1.233 1.365 1.465 1.575 1.696 1.83 1.978 2.142 2.426 2.649 2.907 3.210 3.573 4.017 4.570 5.273 6.734 7.37 8.103

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Metzner et al. (1959, 1960) published articles on heat transfer for slurries. The applications for heat transfer problems have been confined to the nuclear industry, the processing of tar sands, feeding slurry to autoclaves for thermal processing, or certain emulsionbased slurries.

1-8 SLURRY CIRCUITS IN EXTRACTIVE METALLURGY It would be beyond this book to discuss the principles of extractive metallurgy. Slurry is a very important component in the processing of ores to the final disposal of tailings and shipping of concentrate. Chapter 7 is dedicated to equipment for slurry processing. There are three main processes used for extractive metallurgy: 1. Hydrometallurgy, which implies processing the ore using a liquid medium 2. Electrometallurgy, which involves the application of electric and electro-chemical processes to extract the ore 3. Pyrometallurgy, which involves the use of heat (roasting, smelting, etc.) for processing the ore The most common minerals of high metric tonnage (iron, aluminum, copper, titanium, nickel, chromium, magnesium, zinc, etc.) are found in nature as oxides and sulfides and as a combination of both. Ores are sometimes a mixture of rich metal composition and poorer compositions called gangue. The gangue can be acidic or alkaline, and determines the type of flux used for pyrometallurgy. Since ores come in all levels of complexity, various methods of processing have been developed over the years. The first process ore undergoes is called classification or ore dressing. The purpose here is to separate the richer components of a mixture from the unuseful soils. Mineral processing may be used to produce a single stream, as is the case with taconite circuits, where iron extraction is the main activity. It may also create two streams, such as copper concentrate and gold concentrate, when both minerals are found in the same ore body . Mineral processing is usually undertaken at the mine. Its purpose is to separate some of the gangue before shipment of a concentrate. The concentrate is richer in the desired mineral than the original soil.

1-8-1 Crushing In Figure 1-10, a block diagram for crushing and grinding is presented. These are two of the fundamental steps taken to start a slurry circuit. Large rocks are first crushed to an acceptable size. Depending on the type of equipment used, crushing may be done in a single step to feed a semiautogeneous mill, or in three steps (primary, secondary, and tertiary crushing). Rocks are transported from one crusher to another by conveying in a dry form. Their initial size of 300–600 mm (1–2 ft) is reduced to 100–150 mm (4–6 in). Jaw, gyrator, and cone crushers are commonly used during these stages. The crushed material is transported by conveyors to a storage area called the stockpile. From the stockpile, crushed rocks are transported to the grinding and milling circuit.

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GENERAL CONCEPTS OF SLURRY FLOWS

stockpile

Primary Crusher

Secondary Crusher

Tertiary Crusher

Crushing

Ore <2% metal

stockpile solids

slurry

een scr cyclones

tank

Lake

Tailings Dam

FIGURE 1-10

fines

Flotation Cells

Electrostatic Separators

Water Ball Mill

Ball Mill

Tailings pipes 30-40% solids

Reclaim Water

cyclones Ball or Pebble Mill

Rod Mill

fines

Magnetic Separators

slurry

Gravity Separators

Milling and Grinding

Autogeneous Mill

Classification

water

wells

Ri ve r

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Crushing and milling are the starting points of a slurry system.

1-8-2 Milling and Primary Grinding Milling and primary grinding serve the purpose of reducing the size of particles from 100–150 mm (⬇ 4–6 in) down to less than 6 mm (⬇ 1–4 in). This is done by using steel or ceramic balls and rods in rotating cylindrical and conical mills. If the process has undergone primary, secondary, and tertiary crushing, a process of screening is undertaken to separate the very coarse from finer particles. The particles are then further reduced in size by milling in rod mills with the addition of water. The still dilute slurry is transported to a ball mill where steel balls achieve further reduction in size. The mixture of slurry and steel balls undergoes a separation at the exit from the ball mill (by using the difference in momentum between the two materials). Slurry is then diverted into a pump box, which is connected to slurry pumps. The slurry is pumped to cyclones, where the coarse particles are further separated from the fine particles. In the last 20 years, efforts have been made to eliminate secondary and tertiary crushing, by feeding the material directly to an autogenous or a semiautogenous mill (SAG mill). The technology has evolved in size. Today, a single line featuring a SAG mill and two ball mills can handle 65,000 metric tons (short tons) per day. An example of a single line plant is the concentrator of ASARCO in Ray, Arizona (USA). In the 1990s, SIEMENS and ABB introduced the concept of the wrap-around motor,

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which helped eliminate complicated gearboxes for SAG mills. Some of the largest SAG mills are now built with a diameter of 12.2 m (40 ft). The process of crushing is essentially a dry process but the process of milling is a wet process in which slurry comes to play an important role. The use of water eliminates the dangerous generation of dust associated with dry grinding. It is not possible to undertake milling or wet grinding in a single step. The slurry is recirculated as follows. Initially, the coarse and fine particles are separated through a coarse screen or a special cyclone. Then the coarse particles are returned to the SAG mill and the intermediate sized particles are sent to the ball mill. The fine sized particles are taken from the cyclone overflow to a magnetic separation, electrostatic separation, or flotation plant. It is therefore not uncommon to recirculate 250–350% of the feedstock through the circuit of grinding and milling. Attention should be paid to the presence of clays in the ore, and associated dynamic viscosity. Flow from the ball mills can reach a concentration of 40–50% by weight and certain non-Newtonian rheology may manifest itself. Over the years, a plant that started in rocky ground may encounter more clay as it proceeds into deeper depths.

1-8-3 Classification This is essentially a process to separate the particles according to their sinking rates in water. Wet classifiers are used with grinding and may include rubber-lined or ceramic cyclones (called hydrocyclones) and spiral mechanical classifiers. The principle of the cyclone is to feed the slurry tangentially and force it to rotate. By centrifugal forces, the coarser particles sink to the bottom of the cone while the finer particles float to the top. Both streams separate. The underflow, which consists of coarse material, and the overflow, which consists of fines, are then directed to other circuits. The underflow is fed back to the ball mills and the overflow is directed to the flotation circuit or other types of separators. The spiral mechanical classifier is used in pools. The heavier particles are allowed to settle in the pool while the finer particles float and flow out of the pool. The heavier particles are then removed from the bottom of the pool with a spiral or mechanical device. From cyclones or from the grinding circuit, the slurry may pass through different types of separators such as flotation circuits, electrostatic separators, and magnetic separators. Their purpose is to separate by chemical, electrical, or magnetic forces the minerals from the gangue. These steps occur before further thickening prior to feeding the pipeline. The gangue is diverted to tailings circuits (see Figure 1-11).

1-8-4 Concentration and Separation Circuits After a considerable effort to reduce the sizes of particles, it is important to separate the richer soils from the slimes or gangue. This step is achieved by using the properties of the ore itself. Gravity devices work on the principle that the ore (such as gold or diamonds) is heavier than the gangue. These devices include shaking units, the classic miner’s pan, rocking cradle devices, or more sophisticated gold concentrators or mineral sand concentrators. The drawback of these systems is that they may not necessarily be able to treat the fine particles produced by grinding and milling. In diamond extraction plants, X-ray machines are used in conjunction with gravity separation to detect the diamonds. Gravity devices are also used in a number of dry processes as well as slurry processes

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Crushing, grinding, milling (see fig 1- 10)

Electrostatic Separators

Ma gnetic Separators

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Flotation Cells

Gravity Separators

wells

Lake

Ri ve

r

Tailings Dam

Tailings pipes 30-40% solids

tailings thickener Reclaim Water

Page 1.27

minerals

gangue

tank

concentrate thickener

tailings sump

concentrate at about 20% mineral slurry pipeline

reclaim water

filtering drying smelting burning

1.27

FIGURE 1-11 Block for thickening and disposal of tailings, used as the basis for the design of slurry concentrate and fi 1diagram 11 tailings pipelines.

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for mineral sand and placer mining. In the case of mineral sands, the presence of a wide range of heavy oxides allows the miner to separate the various components by using gravity in conjunction with the magnetic or electrostatic properties. Magnetic devices are especially useful, since iron is one of the most common minerals and it is possible to separate iron oxides from gangue by applying a magnet. This is usually done by introducing the slurry over a rotating drum, as shown in Figure 1-12; the drum picks up the magnetic concentrate while the remaining soils are diverted away by the flow of the remaining slurry. Magnetic devices are common in taconite processing plants, iron ore plants, as well as in mineral sand plants. Electrostatic devices were developed in Australia to process beach and mineral sands. The sand ore is fed to a conducting and grounding rotor and is exposed to ionization. The particles, which have certain electrostatic properties, are attracted by the electric charge and are separated from the other particles. The nonconducting particles drop on the rotor and are brushed away into a separate container (Figure 1-13). Flotation devices use the principle of flotation to separate particles that are wettable from other particles. This is a very common process with sulfides but is less efficient with oxides. The cyclone overflow or the fine particles in the slurry after undergoing milling and grinding are fed to a series of flotation tanks (or a flotation machine). An agitator provides vigorous mixing. Air is introduced from a separate compressor line and chemical reagents are added to create froth. The nonwettable minerals float on top of the froth and are pumped away by froth-handling pumps or scraped away by mechanical devices. The wettable particles, such as the gangue, do not float on top of the froth and sink to the bottom of the tank. Special tailing pumps may then pump away the gangue, sometimes for further grinding and processing (particularly gangue from the first flotation tank) and sometimes to the final tailing box (see Figure 1.14). The process of froth generation and flotation is more efficient when carried out in steps. A series of up to six tanks may be constructed to drop gradually, with launders in between. Without reagents, only graphite or molybdenum would be nonwettable. Reagents have been produced by the industry for different grades of sulfides, to depress or activate the extraction of certain minerals, to control pH, etc.

rotating magnet

Feed

Magnetic concentrate

Other soils

FIGURE 1-12 A drum-type magnetic separator. The drum is sometimes replaced by a magnetic belt on a special table.

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GENERAL CONCEPTS OF SLURRY FLOWS

Feed Nonionizing electrode fine-wire electrode ionizing Conductors concentrate

Nonconductor soils FIGURE 1-13

An electrostatic separator.

The secondary grinding that is applied to the underflow of the circuits from flotation is useful for extracting secondary ores, and has been applied successfully in copper–gold ores for further extraction of the gold. A mineral process plant includes gravity flows from hydrocyclones to ball mills and pumped flows from SAG mills to hydrocyclones. A good plant layout must allow space for repairs and long bends, and provide the ability to join and split flows. The use of three dimensional computer modeling is a very useful tool for the design engineer to determine the slope of launders and physical constraints to the layout of the plant (Figure 1-15)

froth bubbles with mineral concentrate

aeration agitator pump

secondary grinding FIGURE 1-14 A flotation circuit.

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1-8-5 Piping the Concentrate From these processes of classification, a concentrate is obtained. The slurry can be further thickened or dewatered using thickeners to a concentration of 50–60% by weight. The concentrate can then be pumped for hundreds of kilometers to a port from which it can be shipped to a pyrometallurgy plant. At the port, a filtering plant can provide further dewatering. Long pipelines are used to transport concentrate. At Cuajone, in Peru, an open launder is used to transport tailings from an altitude of 3000 m (10,000 ft) down to sea level. The potential energy drop is used to overcome the friction losses of the launder. In Escondida, Chile, copper concentrate flows by gravity from an altitude in excess of 2500 m (8200 ft) above sea level over a distance in excess of 200 km (125 mi) to a port at sea level. Thus, in a typical copper extraction process the rocks are reduced to very fine particles through vertimills, semiautogeneous mills, and ball mills. Further separation occurs through flotation circuits and grinding.

1-8-6 Disposal of the Tailings Once the concentrated ore has been extracted, the plant is left with the sands and slimes. These are dewatered to an acceptable concentration by weight of 35–45%. Using thicken-

FIGURE 1-15 Three-dimensional computer representation of a grinding circuit with one central SAG mill, a ball mill on each side, hydrocyclones at the top left, and pumps (at the floor level). Courtesy of Hatch & Associates, Vancouver, Canada.

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1.31

ers, the flow separates into clarified water that is returned to the plant for use in the milling and grinding circuits, and into an underflow of concentrated tailings. The underflow from the tailings is then pumped away to a large disposal pond, called a tailings dam. The sand sinks to the bottom of the tailings pond and the water from the top is returned back to the plant for further use. During the process of disposing of the tailings, spigots and other devices are used to separate coarse from fine particles at the discharge point; the coarse particles are used to build the beaches or the wall of the dam. Dams have been built up to a height of 200 m (656 ft). The environmental engineer must make sure that no dangerous chemicals seep through the ground. If the tailings contain dangerous substances, a plastic or clay (which tends to create a seal) lining for the pond may be recommended to prevent seepage. Some dangerous collapses of tailing dams have been reported over the years, with detrimental consequences when they contained cyanide products, as in the case of certain gold mines. The technology used in tailing pumps has improved. The casings can be designed to withstand 6.9 MPa (1000 psi) of pressure. However, these are essentially single-stage centrifugal pumps installed in series. Up to seven pumps have been installed in series in mines such as National Steel in Minnesota and Kelian Gold in Indonesia.

1-9 CLOSED AND OPEN CHANNEL FLOWS, PIPELINES VERSUS LAUNDERS The reader will find that over the years the majority of references on slurry flows have focused on pipelines because the interest in the field has concentrated on the ability to haul coal, sand, and phosphate hydraulically. Many mines, particularly in Chile and Peru, are located at very high altitudes. This demand has increased interest in gravity flows. In the early 1970s, Southern Peru Copper installed one of the first long, open launders to dispose of tailings to the sea. The launder was of a concrete and fiberglass design, with a U-shaped cross section. Another example of a long gravity pipeline for copper concentrate is the Escondida concentrate pipeline in Chile, which is longer than 200 km (125 mi). Despite the increasing importance of long gravity pipelines, equipment has not kept up with the expanding need. Cave (1980) described tests on slurry turbines. A 350 mm × 300 mm (14⬙ × 12⬙) was reported by Burgess and Abulnaga (1991). Launders play a very important role in slurry flows of plants. Cyclone underflow is directed to ball mills then to SAG mills by gravity. Flows in these circuits can cause tremendous wear if provision is not made to control speeds. Launders in plants are typically rubber lined. Long-distance pipelines are manufactured of rubber-lined steel or extra-thick, high-density polyethylene (HDPE). Because of the importance of open launders, gravity flows, and drop boxes, Chapter 6 is dedicated to these complex flows. Obviously, not all mines are located on mountaintops, and slurry pipeline flow will continue to be the main emphasis of researchers. In long pipelines, centrifugal pumps can be installed at regular intervals; these require power to be brought in. For long-distance pumping, positive displacement pumps compete well with centrifugal pumps. The positive displacement pumps are of a diaphragm or hose design. They are extremely expensive. A 17.3 MPa (2500 psi) pump may range in price between U.S. $600,000 and $1,200,000 in year 2000 dollars. The higher capital investment required for positive displacement pumps is offset by their higher efficiency. These pumps are built to much smaller flow capacity than are large centrifugal pumps.

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1-10 HISTORICAL DEVELOPMENT OF SLURRY PIPELINES One of the first large engineering projects that involved transportation of solids by liquid was the dredging for the Suez Canal in the 1860s in Egypt. It was reported to have used conduits to dispose of the sand–water mixture. Nora Blatch in 1906 was probably the first person to conduct a systematic investigation of the flow of solid–water mixtures. She used a 25 mm (1 in) horizontal pipe and measured the pressure gradients as a function of flow, density, and solid concentration. As a result, between 1918 and 1924, a 200 mm (8 in) pipeline was installed in the Hammersmith power station, in London, England to transport coal slurry over a distance of 660 yd. In 1948, in France, the Institute of Research SOGREAH began a series of tests on transporting sand and gravel in pipes with a diameter from 38–250 mm (1.5–10 in). These extensive tests were the basis for the formulation by Durand of a number of equations that will be reviewed in Chapter 4. These equations have been subject to further refinements over the last 50 years. In 1952, in the United Kingdom, the British Hydromechanic Research Association (BHRA) started to study the hydraulic transport of lump coal, sand, gravel, and limestone. Limestone pipelines were constructed in Trinidad and England in 1960. The Trinidad pipeline had a diameter of 204 mm (8 in), a length of 9.6 km (6 mi), and was designed to operate in a laminar flow regime. The limestone pipeline in England had a diameter of 250 mm (10 in) and was 112 km (70 mi) long. In 1950, the Consolidated Coal Co. in the United States started to conduct research on the hydrotransport of fine “nonsettling” slurries. Concentrated coal with a weight concentration of 60% and particle size between minus 1168 m (14 mesh) and minus 43 m (325 mesh) was transported. The pipeline transported 1.5 million tons of coal each year between 1957 and 1964. The pipeline stretched 176 km (110 mi) from Cadiz, Ohio to Eastlake in Cleveland, Ohio. In 1957, the Colorado School of Mines collaborated with the American Gilsonite Company and designed a pipeline with a diameter of 200 mm (8 in) to transport crushed gilsonite. The pipeline was constructed between Bonanza, Utah and Grand Junction, Colorado. The particle size was minus 4.7 mm (4 mesh) and solids were pumped at a weight concentration of 48%. Two other pipelines were built in Georgia to transport kaolin in the 1960s. In 1967, an iron ore concentrate slurry pipeline started to operate in Tasmania, Australia. The pipeline had a diameter of 245 mm (95–8 in). Concentrate was transported at a weight concentration of 60% with an average particle size of minus 149 m (100 mesh) over a distance of 85 km (53 mi) through extremely rugged terrain (see Figure 1-16). In 1970, the Black Mesa Pipeline, one of the longest pipelines ever built up to that time, started operation between the Black Mesa Coal fields in Arizona and the Mohave Power Plant in Nevada. Coal was ground to a particle size of minus 1168 m (14 mesh), and transported in a pipe with a diameter of 457 mm (18 in) over a distance of 437 km (273 mi). Coal was dewatered at the end of the line through a mill before combustion with preheated air. Since the 1970s, a number of short and long slurry pipelines have been constructed. Table 1-9 lists a number of such achievements. Now, at the beginning of the 21st century, new complex, multiphase tar–sand pipelines are planned for northern Alberta, Canada.

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FIGURE 1-16 Long slurry pipelines must travel through isolated areas over long distances and may involve pressures up to 2500 psi that require special positive displacement pumps. Courtesy of Wirth Pumps, Germany.

1.11 SEDIMENTATION OF DAMS— THE ROLE OF THE SLURRY ENGINEER In the last 150 years, world population has grown fast and our modern standards of living depend on the production of electricity, and production of food for at least two seasons a year. In an effort to meet these demands, engineers have built small as well as very large dams. In certain areas, very large man-made lakes have been dug in the earth, such as behind the Aswan High Dam in Egypt, the Ataturk Dam in Turkey, and new dams on the Yellow River in China. Some large rivers transport silt that tends to separate from water when the speed of the flow is interrupted by a dam. This phenomenon is called siltation of dams. The problem

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TABLE 1-9 Examples of Slurry Pipelines Built Since 1957

Ore

Site of pipeline, or name of pipeline

Solids Pipe transported, Pipeline length diameter million short inch Mile km tons/yr

Start-up date

Coal

Consolidation, USA Black Mesa ETSI ALTON Belonovo–Novosibirsk, Siberia, Russia

10 18 38 24

108 273 1036 180 158

175 440 1675 112 256

1.3 4.8 25 10 3.4

1957 1970 1979 1981 1985

Iron concentrate

Savage River Waipipi (Iron Sands) Pena, Colorado Las Truchas, Mexico Sierra Grande, Argentina Samarco, Brazil Chongin, North Korea New Zealand Sands, NZ Jian Shan, China La Parla–Hercules, Mexico

9 8,12 8 10 8 20 ? 12 10 8/14

53 6 28 16 20 244 61 5 62 52/182

86 9.7 45 27 32 395 98 8 100 85/295

2.25 1.0 1.8 1.5 2.1 12 4.5

1967 1971 1974 1976 1976 1977 1975

4.5

1982

Copper ore

Los Bronces

24

35

56

Copper concentrate

Bougainville, PNG West Irian, Indonesia Pinto Valley OK Tedi, Papua New Guinea Escondida, Chile (gravity line) Collahausi, Chile Freeport, Indonesia Batu Hijau, Indonesia Alumbrera, Argentina

6 4 4 6 9 7 5 6 6

20 69 11 96 102 125 71 11 194

32 111 17 155 165 203 115 18 314

1.0 0.3 0.4 ? 1.0

1972 1972 1974 1987 1994 1999

0.8

1999 1998

Copper tailings

Bougainville, Papua NG Southern Peru Copper (gravity)

34

31

50 150

Limestone

Rugby Calaveras Michigan Limestone Tailings

10 7 20

57 17 1.2

92 27 2

1.7 1.5

1964 1971

Phosphate Chevron, Vernal, Wyoming ore concentrate Simplot Wenglu

10 8 8

94 89 28

152 145 45

1.3–2.5

1986

Dredging

24

33

21

11,000 gpm 1998

Dallas White Rock Lake, USA

1972

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1.35

of siltation of dams has not been well documented or studied. Chanson (1998) and Chanson and James (1999) examined the siltation of Australian dams. Certain dams in Australia became gradually fully silted between 1890 and 1960. They reported that the Koorawatha dam in New South Wales (Figure 1-17) became fully silted with bed-load material. The Cunningham Creek Dam in New South Wales, Australia was well studied by Hellstrom (1941) according to Chanson. Sedimentation problems are more acute with small dams than with medium-size and large reservoirs (Chanson and James,1999). Siltation at Eildon in the State of Victoria occurred in 1940 after torrential rainfalls following bushfires that had destroyed 50% of the catchment forest. The siltation at Eppalock in Victoria followed extensive gold mining, tree clearing, and hydraulic mining between 1851 and 1890. There were some extreme siltation cases. The Quipolly Reservoir No. 1, in Australia underwent very rapid sedimentation between 1941 and 1943 at a rate in excess of 1143m3/km2/year (9600 ft3/mi2/y). The Korrumbyn Creek Dam sedimented in less than 7 years. Since the 1950s, improvements in land management practices and a better understanding of the problems of soil erosion have resulted in better approaches to the protection of dams.

FIGURE 1-17 The Koorawatha Dam in Australia—fully silted. [From Chanson (1999). Reprinted by permission of Butterworth-Heinemann.]

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Chanson and James (1998) discussed the hazards of fully silted dams. The weight of silt pressing against the concrete structure introduces a medium with a specific gravity larger than the water for which the dam was designed. It is important in these cases to monitor the structure. In the 1970s and 1980s, the drought in Ethiopia and the Sudan reduced the flow of the waters to the Nile to dramatically low levels. Egypt avoided famine by using its massive man-made lake behind the Aswan High Dam (Lake Nasser). On the other hand, any specialist who visits Egypt can feel that the fellahin (farmers) are speaking with nostalgia of the “Tamye” or silt that used to come with the annual flood and enrich the land. This raw material was the basis of natural nutrients, as well as mud for the construction of houses and manufacture of bricks. The Egyptian case is not unique. Certain dams in the United States are now the subject of discussions on decommissioning and some were removed in the 1990s. The slurry engineer can offer some much needed solutions. The twenty-first century will see slurry engineers providing adequate solutions in terms of dredging the lakes that are sedimenting and transporting the dredged silt to traditional lands, or to arid lands via special slurry pipelines. A simple concept for such a solution is presented in Figure 1-18. It is proposed that in certain areas, particularly where the accumulation of silt is likely to apply pressure on the dam structure, submersible slurry pumps be installed on a permanent basis. Dredging boats with dredging arms or submersible pumps and cutters would be used on the rest of the lake. Where the capital investment does not justify it, small dredgers with submersible pumps should be used. The slurry from these operations will be dilute, it would be pumped to the shore through a floating plastic pipe. It may be pumped in pipelines and diverted to canals for agricultural purposes. It may also be pumped to brick plants, where it would be dewatered and the silt used as a raw material.

Agriculture dredger

Brick manufacture Dam

Submersible pump silt

FIGURE 1-18 Simplified flow sheet to remove silt behind dams. Silt would be dredged using submersible Fi pumps 1 18at predetermined locations or in association with boat dredgers. The silt would be pumped in slurry form to agricultural farmlands or to special plants for the manufacture of bricks.

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Slurry engineers can provide economic solutions to the siltation of dams. This effort should be made in conjunction with environmental engineers, as new ecosystems often form around large dams. The author hopes that this handbook will be useful to the decision makers who have to deal with siltation of dams, while satisfying the concerns of environmental engineers, as decommissioning is not always the solution.

1-12 CONCLUSION In this first chapter, some of the basic properties of solids, which are important to the composition of slurries, were reviewed. Their importance will be emphasized in the next few chapters. They may lead to Newtonian as well as more complex non-Newtonian flows that require special equations to determine the friction factors, velocity of flow, pipe sizes, head, and efficiency losses in pumps. Wear is an important cost to be paid for transporting solids by liquids. This will be discussed later in the book when exploring slurry pumps and pipelines. The modern slurry engineer can serve the mining and power industries by making possible the transportation of minerals, coal, coal–crude oil mixtures over very long distances, and also play a major role in dredging sediments behind dams to avoid dam failure and to provide arid lands with much needed silt.

1-13 NOMENCLATURE A B Cc Cu Cv Cw Cp d10 d30 d50 d60 d80 K1, K2, K3, K4 k LL PI PL Re VD VT

Constant Constant Coefficient of curvature Coefficient of uniformity Concentration by volume of the solid particles in percent Concentration by weight of the solid particles in percent Heat capacity Grain size at which 10% of the soil is finer Grain size at which 30% of the soil is finer Grain size at which 50% of the soil is finer Grain size at which 60% of the soil is finer Grain size at which 80% of the soil is finer polynomial coefficients in Einstein’s equation for dynamic viscosity Thermal conductivity Liquid limit of clay and silt soils Plastic index of clay and silt soils Plastic limit of clay and silt soils Reynolds number Deposition critical velocity Viscous transition critical velocity Concentration by volume in decimal points Absolute (or dynamic) viscosity Density

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Subscripts l m s

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Liquid Mixture Solids

1-14 REFERENCES The American Society for Testing of Materials. 1993. Practice for Description and Identification of Soils (Visual–Manual Aggregate Mixtures). Standard D2488. Philadelphia: The American Society for Testing of Materials. The American Society for Testing of Materials. 1983. Test Method for Lime Content of Uncured Soil–Lime Mixtures. Standard D3155. Philadelphia: The American Society for Testing of Materials. The British Standard Institute. 1999. Code for Practice of Site Investigation. Standard BS 5930. London: The British Standard Institute. The British Standard Institute. 1999. Aggregate Abrasion Value. Standard BS 812. Pt 113. London: The British Standard Institute. Broch, E., and J. A. Franklin. 1972. The Point-Load Strength Test. International Journal for Rock Mechanics and Mineral Sciences, 9, 669–697. Burgess K. E, and B. E. Abulnaga.1991.The Application of Finite Element Methods to Warman Pumps and Process Equipment. Paper presented to the Fifth International Conference on Finite Element Analysis in Australia, University of Sydney, Australia (July). Cave I. 1980. Slurry Turbines for Energy Recovery. In Seventh International Conference on the Hydraulic Transport of Solids in Pipelines, Sendai, Japan, pp. 9–15, Cranfield, United Kingdom: BHRA Group. Chanson H. 1998. Extreme Reservoir Sedimentation in Australia: A Review. International Journal of Sediment Research, UNESCO-IRTCES, 13, 3, 55–63. Chanson H. 1999. The Hydraulics of Open Channel Flows—An Introduction. Oxford, UK: Butterworth-Heinemann. Chanson H., and D. P. James.1999. Siltation in Australian Reservoirs: Some Observations and Dam Safety Implications.” In Proceedings 28th IAHR Congress, Graz, Austria, Paper B5. Guth, E., and A. R. Simha. 1936. Viscosity of suspensions and solutions. Kolloid-Z, 74, 266. Quoted in Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid–Liquid Flow—Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans. Tech. Publications. Hellstrom B. (1941) Nagra Lakttagelser over Vittring Erosion och Slambidning i Malaya och Australien.” Geografiska Annaler (Stockholm, Sweden), Nos. 1–2, pp. 102–124 (in Swedish). Herbrich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill Inc. The International Journal for Rock Mechanics and Mineral Sciences, 1979, 16, 141–156. The International Society for Rock Mechanics. 1977. Suggested Methods for Determining the Strength of Rock Materials in Triaxial Compression. Lisbon, Portugal: The International Society for Rock Mechanics. The International Society for Rock Mechanics. 1978. Suggested Methods for Determining the Deformability of Rock. Lisbon, Portugal: The International Society for Rock Mechanics. Metzner, A. B., and P. S. Friend. 1959. Heat Transfer to Turbulent Non-Newtonian Fluids. Ind. & Eng. Chem., 51, 7 (July), 879–882. Metzner, A. B., and D. F. Gluck. 1960. Heat Transfer to Non-Newtonian Fluids Under Laminar Flow Conditions. Chem. Eng. Science, 12, 3 (June), 185–190. Nunez, G. A., M. Briceno, C. Mata, and H. Rivas. 1996. Flow Characteristics of Concentrated Emulsions of Very Viscous Oil in Water. Journal of Rheology, 40, 3 (May/June), 405–423. Orr, C., and J. M. Dalla Valle. 1954. Heat-Transfer Properties of Liquid–Solid Suspensions. Chem. Eng. Prog., Symp. Series No. 9, 50, 29–45. The Permanent International Association of Navigation Congresses. 1972. Classification of Soils to be Dredged. In Bulletin No. 11, Vol. I. The Permanent International Association of Navigational Congresses.

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Sargent, J. H. 1984. Classification of Soils to be Dredged. In Supplement to Bulletin No 47. The Permanent International Association of Navigation Congresses. Sulzer Pumps. 1998. Centrifugal Pump Handbook. New York: Elsevier. Thomas, D. G. 1960. Heat and Momentum Transport Characteristics of Non-Newtonian Aqueous Thorium Oxide Suspensions. AIChE Journal, 6 (December), 631–639. Thomas D. G. 1965. Transient Characteristics of Suspensions: Part VIII. A Note on the Viscosity of Newtonian Suspensions of Uniform Spherical Particles. Journal Colloid Science, 20, 267. Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid–Liquid Flow—Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans Tech Publications. Wilson, K. C., G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. New York: Elsevier Applied Sciences. Further Reading: Wilson, G. 1976. Construction of Solids-Handling Centrifugal Pumps. In Pump Handbook. Edited by J. Karassik et al. New York: McGraw-Hill.

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FUNDAMENTALS OF WATER FLOWS IN PIPES

2-0 INTRODUCTION The mechanics of pipe flow is a topic well dealt with in the scientific literature. In this chapter, some of the concepts are reviewed in a simplified manner as they relate to slurry flows. The friction factor for single phase flows is presented in terms of boundary layer theory. Losses in pipes are summarized in terms of fittings and conduits commonly used on large engineering projects. Numerous books have been written for single-phase flows. This chapter limits itself to a brief introduction. The equations in this chapter are based on SI units for consistency. They can be readily used in USCS (United States Customary System) units if the reader assumes the use of slugs and not pounds for mass units. There is considerable confusion about the use of slugs, which are units of mass, and pounds, which are units of force. The conversion factor between these is often called gc and is equal to 32.2 ft/sec. Many other references input gc in their equations to achieve such a conversion, but it was not deemed necessary in this book. The worked examples show how slugs should be used as a unit of mass. Certain models of Newtonian slurry flows are attempts to apply correction factors to the friction losses of the carrier liquid on the basis of the volumetric concentration of solids. The reader is encouraged to examine this chapter before proceeding with more complex themes.

2-1 SHEAR STRESS OF LIQUID FLOWS Modern fluid mechanics is based on the concept of a controlled volume. Mass momentum and energy must be conserved when a particle enters and leaves the volume. Considering flow through a section of pipe of a constant diameter between two locations 1 and 2 as in Figure 2-1, the hydraulics force associated with the drop of pressure is

F12 = ᎏ D i2(P1 – P2) 4

2.1

(2-1)

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w

P

P

2

1

L

FIGURE 2-1 Shear stress and pressure for flow in a pipe.

This force is balanced by the friction force Fr Fr = wDiL

(2-2)

where L is the distance between points 1 and 2 and w is the wall shear stress

ᎏ D 2i (P1 – P2) – wDL = 0 4 or Ri(P1 – P2) Di(P1 – P2) w = ᎏᎏ = ᎏᎏ 4L 2L

(2-3)

The shear stress at any radius and from the center of the pipe is r Ri⌬P = ᎏ = w ᎏ 2L R where L = length Ri = the pipe inner radius (at the inside wall of the pipe) r = local radius The shear stress is calculated. At the center of the pipe there is no shear stress. Example 2-1 Homogeneous slurry is tested in a pipe with an inner diameter of 53 mm (2.086 in). The pressure drop due to friction is measured between two points (A and B), which are separated by a distance of 1.8 m (5.9 ft). The pressure drop is recorded as 3000 Pa (0.435 psi). To appreciate the shear stress distribution from the wall to the center of the pipe, determine the shear stress distribution at the wall and at three points: at a radius of 20 mm (0.787⬙), at a radius of 12 mm (0.472⬙), and at the center of the pipe. Solution This problem will be solved in SI units [Système International (metric)] and in USCS units.

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Solution in SI Units Using Equation 2-3, the shear stress at the wall is 0.053(3000) w = ᎏᎏ = 22 Pa 4 × 1.8 since the inner radius of the pipe is 26.5 mm. At a radius of 20 mm the shear stress is

冢 冣

冢

冣

20 r = w ᎏ = 22 ᎏ = 16.6 Pa Ri 26.5 At a radius of 12 mm the shear stress is

冢 冣

冢

冣

12 r = w ᎏ = 22 ᎏ = 10 Pa Ri 26.5 At the center of the pipe the shear stress is

冢 冣

冢

冣

0 r = w ᎏ = 22 ᎏ = 0 Pa Ri 26.5 Solution in USCS Units Using Equation 2-3, the shear stress at the wall is 2.086 in (0.435 psi) w = ᎏᎏᎏ = 0.0032 psi 4 × 5.9 × 12 since the inner radius of the pipe is 1.043 in. At a radius of 20 mm (0.787 in) the shear stress is

冢 冣

冢

冣

r 0.787 = w ᎏ = 0.0032 ᎏ = 0.0024 psi Ri 1.043 At a radius of 12 mm (0.472 in) the shear stress is

冢

冣

0.472 = 0.0032 ᎏ = 0.00145 psi 1.043 At the center of the pipe = 0

2-2 REYNOLDS NUMBER AND FLOW REGIMES Determining the magnitude of friction was historically a controversial topic until the end of the 19th century. The great disagreement was between the practical engineers and the theory of hydrodynamics. Hager (in 1839) and Poiseville (in 1840) demonstrated that under certain conditions friction was a linear function of the speed of flow. In 1858, Darcy demonstrated that under other conditions friction was in fact proportional to the square of the mean speed of the flow. By 1883, Reynolds had demonstrated that both Poiseville and Darcy were correct, as the mechanics of flows were fundamentally different at very low speeds and at high speeds.

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Through nondimensional analysis, Reynolds demonstrated that under certain fixed conditions, the transition from a laminar Poiseville flow to a turbulent Darcy flow was based on the ratio of the inertia forces to the viscous forces. In his honor, such a ratio is now called the Reynolds number: UDi Inertia forces UDi Re = ᎏᎏ = ᎏ = ᎏ Viscous forces

(2-4)

where = density of the fluid = absolute or “dynamic” viscosity = kinematic viscosity (defined as the absolute viscosity divided by the density of the liquid) U = average velocity of the flow The kinematic viscosity is the absolute (or dynamic viscosity) divided by the density. Its unit of measurement are, strictly speaking, m2/s or ft2/sec. Another unit used is the centistokes, which is obtained by dividing the absolute viscosity in centipoises by the specific gravity of the fluid. A centipoise is equivalent to one milli-Pascal-second. One unit for kinematic viscosity used in the oil industry (but of limit use in the mining industry) is the seconds Saybolt universal (or SSU). For values of kinematic viscosity larger than 70 centistokes (cst), the following formula is recommended by the Hydraulic Institute (1990): SSU = centistokes × 4.635 In simplified terms, it may be said that two geometrically similar bodies immersed in a fluid will develop inertia and viscous forces in a constant ratio when body forces are negligible. Since Reynolds developed his theory, his approach has been has been extended to other fluids. Modern aerodynamics uses the chord of the wing aerofoil instead of the pipe diameter as the distance parameter for Equation 2-4. In Chapter 3, the concept of the particle Reynolds number based on a characteristic particle diameter shall be introduced. Figure 2.2 presents the equations of the Reynolds number for different shapes and flows. For pipe flows, the critical Reynolds number is considered to be between 2300 and 2800. In many pipes, flow becomes unstable above a Reynolds number of 2300 and slides into a transition regime before converting into turbulent motion.

2-3 FRICTION FACTORS The Fanning friction factor is a nondimensional number defined as the ratio of the wall shear stress to the dynamic pressure of the flow:

w fN = ᎏ U 2/2

(2-5)

Users of USCS units should use slugs/ft3 for density and not the more commonly used units of pounds per cubic feet. The conversion between these two is gc or 32.2 ft/sec2. Substituting Equation 2-3 into Equation 2-5

U 2 ⌬PDI fN = ᎏ / ᎏ 4L 2

冢

冣冢

冣

(2-6)

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FUNDAMENTALS OF WATER FLOWS IN PIPES

c h

Airfoil chord "c" Re = U c /

Close parallel plates Re = [U h/2 ] 32/3

DI dp Annular flow

Re= ( U/

)*2 r 20 + r2i - r 2m 2

2

where rm = (r 0 - r i )/2.3 log10 (r0 /rI )

Use inner diameter for pipe flow Re= U D i /

for sphere use diameter Re= U d p /

FIGURE 2-2 Definition of Reynolds number for various shapes.

Equation 2-6 clearly indicates that the friction factor is dependent on the flow. It will be demonstrated that there is a relationship between the friction factor and the Reynolds number of the flow. In USCS units, density is expressed in slugs/ft3. Example 2-2 The slurry in Example 2-1 has a specific gravity of 1.2 , or a density of 1200 kg/m3. If the speed of the flow is 2 m/s (6.56 ft/s), determine the Fanning friction factor. Solution in SI Units Using the shear stress calculated in Example 2-1 as 22 Pa and inserting it into Equation 25, the Fanning friction factor can be calculated as 22 = 0.0092 fN = ᎏᎏ 1200 × 22 × 0.5 Solutions in USCS Units Assume the density of water to be 62.3 lbm/ft3. Since specific gravity equals 1.2, the density of the slurry is 1.2 × 62.3 = 74.76 lbs/ft3. To convert lbs/ft3, into slugs/ft3 complete the following equation: 74.76 ᎏ = 2.32 slugs/ft3 32.2 In Example 2-1, the wall shear stress was determined to be 0.0032 psi. Using equation 2.5 0.0032 × 144 = 0.0092 fN = ᎏᎏ 2.32 × 6.562 × 0.5

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2-3-1 Laminar Friction Factors The friction or resistance of a body to motion is confined to a viscous layer at the wall or surface of the body. This layer is called the boundary layer. Understanding this phenomenon remained illusive until the end of the 19th century. At the turn of the 20th century, Prandtl developed the principles of boundary layer theory. To understand the basic principles of the boundary layer, let us start by examining the flow between closely related plates at low speed. With one plate stationary and the other moving at a speed U, a linear velocity profile is established. The wall shear stress w is established as

U w = ᎏ h by Newton’s law, where h is the spacing between plates and

冢 冣

U u=y ᎏ h

(2-7)

With y as the vertical ordinate from the stationary plate, the velocity gradient is defined as du U ᎏ = ᎏ = rate of shearing strain or shear rate d␥ H

(2-8)

Thus, the dynamic viscosity is Shear Stress = ᎏ = ᎏᎏᎏ du/d␥ Rate of Shear Strain

(2-9)

Equation 2-8 is the basis for boundary layer theory. Instead of a moving plate at a velocity U, the velocity of the flow outside the boundary layer is studied (see Figure 2-3). The shear rate is not necessarily linear. For example flow around an aircraft airfoil can be attached, stagnant at a point, or can even reverse flow after separation. Laminar flow in a pipe is described by the Hagen–Poiseville equation: ⌬P 32U ᎏ=ᎏ D i2 L

(2-10)

Upper plate moves at speed U

h y FIGURE 2-3 ary plate.

y u=U h

w

Linear velocity distribution due to a plate moving at a speed U above a station-

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2.7

which can be rearranged as ⌬P = ᎏ L

Di⌬P

= ᎏ ᎏ 冢 冣冢 ᎏ 32U 冣 冢 4L 冣冢 8U 冣 D 2i

Di

w = ᎏ 8U/Di

(2-11)

(2-12)

or the ratio of the wall shear stress and the mean velocity gradient. The term (8U/Di) is often called the pipe (or pipeline) shear rate in laminar flow. From Equation 2-12 8U w = ᎏ Di

(2-13a)

Substituting Equation 2-12 into Equation 2-5, the Fanning friction factor for a laminar flow can be expressed as 8U fN = ᎏ Di

冢

U 2

冣 冢ᎏ 2 冣

/

16 16 fN = ᎏ = ᎏ VDi Re

(2-14)

(2-15)

The Fanning friction factor is more commonly found in reference publications on chemical engineering. Another friction factor used by mechanical engineers is the Darcy friction factor: fDarcy = 4 × fFanning To avoid confusion, in the book the symbols fD will be used for Darcy friction factor and fN will be used for Fanning friction factor. Flow in a laminar regime is considered to be independent of pipe roughness. Example 2-3 A viscous fluid is flowing in a laminar flow at a speed of 1m/s (or 3.28 ft/s) in a pipe with an inner diameter of 336.6 mm (13.25 in). The measured pressure drop over a distance of 200 m (656 ft) is 8400 Pa (1.22 psi). The density of the fluid is 855 kg/m3 (SG = 0.855). Determine an equivalent viscosity for the pipeline fluid, the Reynolds number, and the friction factor. Solution in SI Units From Equation 2-3, the shear stress at the wall is 0.3366 × 8400 w = ᎏᎏ = 3.53 Pa 4 × 200 The Fanning Friction Factor from Equation 2.5 is 2w 2 × 3.53 fN = ᎏ2 = ᎏ2 = 0.0083 V 855 × 1

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The equivalent pipeline viscosity from Equation 2-12 is

w 3.53 × 0.3366 = ᎏ = ᎏᎏ = 0.148 Pa-s 8U/D 8×1 The Reynolds Number from Equation 2-4

UD 855 × 1 × 0.3366 Re = ᎏ = ᎏᎏ = 1938 0.148 Check on friction factor: 16 fN = ᎏ = 0.00823 1938 Solution in USCS Units 0.8555 × 62.3 density = ᎏᎏ = 1.654 slugs/ft3 32.2 Since it was stated that the pressure drop = 1.215 psi, over a distance of 656.2 ft the shear stress is: 13.25 × 1.2 w = ᎏᎏ = 0.0005 psi 4 × 656.2 × 12 The Fanning Friction Factor from Equation 2.5 is 2w 2 × 0.0005 × 144 fN = ᎏ2 = ᎏᎏ = 0.0081 V 1.654 × 3.282 The equivalent pipeline viscosity converting the shear stress from psi to lbf/ft2, 0.0005 × 144 = 0.072 lbf/ft2, is (13.25/12) × 0.072 Diw = ᎏ = ᎏᎏ = 0.003 lbf-sec/ft2 8V 8 × 3.28 The Reynolds number is 1.654 × 3.28 × 13.25/12 Re = ᎏᎏᎏ = 1997 0.003 Check on friction factor: 16 fN = ᎏ = 0.008 1997

2-3-2 Transition Flow Friction Factor The transition from a laminar to a turbulent flow is difficult to describe. For Reynolds numbers up to 3 × 106, Wasp et al. (1977) recommended the use of Nikuradse equation for the Fanning coefficient: 0.0553 fN = 0.0008 + ᎏ Re0.237

(2-16)

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FUNDAMENTALS OF WATER FLOWS IN PIPES

2-3-3 Friction Factor in Turbulent Flow In turbulent flow, the roughness of the pipe becomes an important factor in determining the friction factor. The roughness is measured in units of length. Up to a certain limiting Reynolds number Re, the friction factor can be expressed by the following Colebrook equation:

冢 冣

冢

Di Di 1 ᎏ = 4 loge ᎏ + 3.48 – 4 loge 1 + 9.35 ᎏᎏ 2 兹f苶N 2 Re兹f苶N

冣

(2-17)

A more simplified equation for the Darcy factor is called the Prandtl–Colebrook equation: 2.51 1 ᎏ = –2 log10 ᎏ + ᎏ 兹f苶 3.7 Di Re兹f 苶 D N

冢

冣

(2-18)

A popular equation used by mechanical engineers (Lindeburg, 1997) as it is explicit and does not require tedious iterations is the Swamee–Jain equation. It is suitable for the range of Reynolds numbers between 5000 and 100,000,000: fD =

冢冦

冥冧 冣

0.25 ᎏᎏᎏ 2 /D log10 ᎏᎏ + (5.74/Re0.9) 3.7

冤

(2-19)

Because the slurry flows occur at Reynolds Numbers smaller than 100,000,000, Equation 2.19 is satisfactory in the context of this handbook. Example 2-4 Using the Swamee–Jain equation, determine the friction factor for a flow of 3500 US gpm in an 18⬙ OD pipe with a wall thickness of 0.375 in. The fluid has a specific gravity of 1.02 and a dynamic viscosity of 2.7 × 10–5 lbf-sec/ft2. Solutions in SI Units (For conversion factors refer to the Appendix at the end of this book.) 3500 × 3.785 Q = ᎏᎏ = 0.221 m3/s 60000 ID of pipe = (18 – 2 × 0.375) = 17.25 in (0.438 m) Area of flow = × 0.25 × 0.4382 = 0.1506 m2 0.221 Average velocity of flow = ᎏ = 1.467m/s 0.1506 Density = 1.02 × 1000 = 1020 kg/m3

= 2.7 × 10–5 × 47.88 = 0.00129 Pa.s 1020 × 1.467 × 0.438 Re = ᎏᎏᎏ = 506975 0.00129

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The absolute roughness of steel pipes is 6 × 10–5 m. Relative roughness is 6 × 10–5 /Di = ᎏ = 0.000137 0.438 0.25 fD = ᎏᎏᎏᎏᎏ = 0.01486 [log10(0.000137/3.7 + 5.74/5069750.9)]2 Solutions in USCS Units Q = 3500 × 0.1337 = 467.95 ft3/min ID of pipe = (18 – 2 × 0.375) = 17.25 in or 1.4375 ft Area of flow = × 0.25 × 1.43752 = 1.623 ft2 467.95 Velocity of flow = ᎏ = 255.3 ft/min or 4.80 ft/s 1.623 1.02 × 62.3 Density = ᎏᎏ = 1.973 slugs/ft3 32.2 1.973 × 4.80 × 1.4375 Re = ᎏᎏᎏ = 504,779 2.7 × 10–5 The absolute roughness of steel is 0.0002 ft. Relative roughness is 0.0002 ᎏ = 0.000139 1.4375 fD = 0.25/[log10(0.000139/3.7 + 5.74/5047790.9)]2 = 0.0149 The Colebrook and Prandtl–Colebrook formulas are limited to a certain range of Reynolds number magnitude. At high Reynolds numbers, the friction factor becomes independent of the Reynolds number. The value of the Reynolds number beyond which the friction factor is independent is calculated using the following equation: Di Re = 70 ᎏ

ᎏ 冪莦ᎏf = 70 ᎏ 冪莦 f 2

N

Di

8

D

The region for which equation 2-19 applies is shown on the Moody diagram to be to the right of the the dashed curve (Figure 2-4). The equations established so far have been developed for clear water and do not apply for plastic fluids or liquids carrying coarse particles. They can apply for any other singlephase Newtonian liquids. (These terms will be explained in Chapter 3). Most fluid dynamics books publish data for linear roughness based on Moody’s work. Such values are applicable to water. However, tests conduced on slurry pipelines can yield different values, due to the erosion of pipe, wear and tear of rubber linings, etc. The Moody diagram is a general graph for the Darcy factor versus the Reynolds number. It is applicable to a very large number of different pipe materials. There are four principal pipe materials associated with slurry flows: plastic pipes [high-density polyethylene (HDPE)], plain steel pipes, rubber-lined steel pipes, and concrete pipes. The absolute

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2.11

FIGURE 2-4 Moody diagram for the friction factor versus the Reynolds number for pipe flow (Reproduced from V. L. Streeter, Fluid Mechanics, McGraw-Hill, 1971. Reproduced by permission of McGraw-Hill, Inc.)

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CHAPTER TWO

roughness of these materials is presented in Table 2-1. Plain steel pipes and rubber-lined steel pipes are the most common, but HDPE and HDPE-lined steel pipes have gained in importance in the last quarter of the 20th century. One of the concentrate pipelines used in Escondida, Chile featured a long section of gravity flow in HDPE pipe. Certain reference books show a roughness of steel of 0.045–0.05 mm. This is difficult to maintain in steel pipes carrying slurries as they are often subject to erosion and corrosion. For this reason, the author recommends the use of a slightly higher roughness of the order of 0.06 mm in friction calculations. The dimensions of plain steel pipes, their pressure ratings, and relative roughness are presented in Table 2-2. It is obvious that steel pipes are limited in pressure rating to 3000 psi. This criterion is essential when considering location of booster pump stations or chokes. In the case of slurry pipelines, the thickness of the pipes is selected on the basis of 앫 Pressure 앫 Corrosion allowance 앫 Wear allowance Because of the wear allowance, erosion, and corrosion, the rating of slurry pipes is lower than presented in Table 2-2. Steel pipes may be hardened for carrying coarse slurry particles (larger than 6.5 mm or 1–4⬙), or sacrificial thickness is used to resist abrasion. The pressure rating as presented in Table 2-2 should be used as a starting point for the design calculations, and the appropriate allowance should be made for wear. Reclaimed water pipelines use the pressure ratings as in Table 2-2. The roughness and inner diameter of reclaimed water pipes may change due to scaling and deposition of lime. Steel pipes are rubber lined to a typical thickness of 6 mm (or 0.25⬙) for small sizes of pipes [< 150 mm (6⬙), 9.5 mm (3–8⬙)], and 13 mm (1–2⬙) for pipe sizes up to 24⬙. Larger pipes may be custom lined. Lining is done in an autoclave and the rubber is cured under steam. Rubber lining is limited to pumping coarse material up to a size of 6 mm (⬇ 1–4⬙). Rubber does not contribute to the pressure rating of steel pipes. Table 2-3 presents the dimensions and relative roughness of rubber-lined steel pipes. The dimensions of plain HDPE pipes (not HDPE-lined steel) for pressures up to 110 psi (760 kPa) are listed in Table 2-4. The dimensions of HDPE pipes for pressures in the range of 125 to 300 psi (863–2070 kPa) are presented in Table 2-5. These dimensions are slightly different than metric pipes. HDPE is not a magic material but can withstand the abrasion of taconite and some coarse laterites. As with rubber, there must be a cut-off size of particle size beyond which the use of HDPE is not acceptable. Very little has been published on this subject. The use of concrete pipes is often associated with gravity flows.

TABLE 2-1 Absolute Roughness of New Materials Used in Slurry Pipes Description Plastic pipes, PVC, ABS, HDPE Steel pipes Rubber-lined pipes Concrete pipes

Roughness (m)

Roughness (ft)

1.5 × 10–6 6.0 × 10–5 0.00015 0.0012

0.000004921 0.000197 0.000492 0.00394

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TABLE 2-2 Size, Rating, and Relative Roughness of Plain Steel Pipes to U.S. Dimensions* Outside diameter (inch)

Wall thickness (inch)

Pipe internal diameter (inch)

Allowed working pressure (psi) to 650 °F

Relative roughness for new pipe

2⬙ Sch 40 2⬙ Sch 80 2⬙ Sch 160 XX

2.375

0.154 0.218 0.344 0.436

2.067 1.939 1.687 1.503

1159 2038 3890 5356

0.001143 0.001212 0.001400 0.001572

3⬙ Sch 40 3⬙ Sch 80 3⬙ Sch 160 XX

3.500

0.216 0.300 0.438 0.600

3.068 2.900 2.624 2.300

1341 2129 3495 5252

0.000770 0.000815 0.000900 0.001027

4⬙ Sch 40 4⬙ Sch 80 4⬙ Sch 120 4⬙ Sch 160 XX

4.500

0.237 0.337 0.438 0.531 0.674

4.026 3.826 3.624 3.438 3.152

1191 1905 2663 3387 4553

0.000587 0.000617 0.000652 0.000687 0.000749

5⬙ Sch 40 5⬙ Sch 80 5⬙ Sch 120 5⬙ Sch 160 XX

5.633

0.257 0.375 0.500 0.625 0.750

5.117 4.883 4.633 4.383 4.133

1071 1950 2502 3284 4098

0.000461 0.000484 0.000510 0.000539 0.000572

6⬙ Sch 40 6⬙ Sch 80 6⬙ Sch 120 6⬙ Sch 160 XX

6.625

0.280 0.432 0.562 0.719 0.864

6.065 5.761 5.501 5.187 4.897

1000 1739 2394 3215 4004

0.000389 0.000410 0.000429 0.000455 0.000482

8⬙ Sch 20 8⬙ Sch 30 8⬙ Sch 40 8⬙ Sch 60 8⬙ Sch 80 8⬙ Sch 100 8⬙ Sch 120 8⬙ Sch 140 XX 8⬙ Sch 160

8.625

0.250 0.277 0.322 0.406 0.500 0.594 0.719 0.812 0.875 0.906

8.125 8.071 7.981 7.813 7.625 7.437 7.187 7.001 6.875 6.813

655 752 916 1225 1577 1935 2422 2792 3046 3173

0.000291 0.000293 0.000296 0.000302 0.000310 0.000318 0.000329 0.000337 0.000344 0.000347

0.250 0.307 0.365 0.500 0.594 0.719 0.844 1.000 1.125

10.250 10.136 10.020 9.750 9.562 9.312 9.062 8.750 8.500

523 688 856 1255 1537 1918 2308 2804 3211

0.000230 0.000233 0.000236 0.000243 0.000247 0.000254 0.000261 0.000270 0.000278 (continued)

Denomination (inch)

10⬙ Sch 20 10⬙ Sch 30 10⬙ Sch 40 S 10⬙ Sch 60 X 10⬙ Sch 80 10⬙ Sch 100 10⬙ Sch 120 10⬙ Sch 140 XX 10⬙ Sch 160

10.75

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TABLE 2-2 Continued Outside diameter (inch)

Wall thickness (inch)

Pipe internal diameter (inch)

Allowed working pressure (psi) to 650 °F

Relative roughness for new pipe

12⬙ S 12⬙ Sch 40 12⬙ X 12⬙ Sch 60 X 12⬙ Sch 80 12⬙ Sch 100 12⬙ Sch 120 XX 12⬙ Sch 140 12⬙ Sch 160

12.750

0.250 0.330 0.375 0.406 0.500 0.562 0.688 0.844 1.00 1.125 1.312

12.250 12.090 12.000 11.938 11.750 11.626 11.374 11.062 10.750 10.500 10.126

440 634 744 820 1052 1207 1526 1927 2337 2672 3183

0.000193 0.000195 0.000197 0.000198 0.000201 0.000203 0.000208 0.000214 0.000220 0.000225 0.000233

14⬙ Sch 10 14⬙ Sch 20 14⬙ Sch 30 S 14⬙ Sch 40 14⬙ X 14⬙ Sch 60 14⬙ Sch 80 14⬙ Sch 100 14⬙ Sch 120 14⬙ Sch 140 14⬙ Sch 160

14.000

0.250 0.330 0.375 0.400 0.500 0.594 0.750 0.938 1.062 1.250 1.406

13.500 13.376 13.250 13.124 13.000 12.812 12.500 12.124 11.876 11.500 11.188

401 537 676 817 956 1169 1528 1969 2265 2724 3112

0.000175 0.000177 0.000178 0.00180 0.000182 0.000184 0.000189 0.000195 0.000199 0.000205 0.000211

16⬙ Sch 10 16⬙ Sch 20 16⬙ Sch 30 S 16⬙ Sch 40 X 16⬙ Sch 60 16⬙ Sch 80 16⬙ Sch 100 16⬙ Sch 120XX 16⬙ Sch 140 16⬙ Sch 160

16.000

0.250 0.312 0.375 0.500 0.656 0.844 1.031 1.219 1.438 1.594

15.500 15.376 15.250 15.000 14.688 14.312 13.938 13.562 13.124 12.812

350 469 590 834 1142 1520 1903 2296 2764 3104

0.000152 0.000154 0.000155 0.000157 0.000161 0.000165 0.000169 0.000176 0.000180 0.000184

18⬙ Sch 10 18⬙ Sch 20 18⬙ S 18⬙ Sch 30 18⬙ X 18⬙ Sch 40 18⬙ Sch 60 18⬙ Sch 80 18⬙ Sch 100 18⬙ Sch 120 18⬙ Sch 140 18⬙ Sch 160

18.000

0.250 0.312 0.375 0.438 0.500 0.562 0.750 0.938 1.156 1.375 1.562 1.781

17.500 17.376 17.250 17.124 17.000 16.876 16.500 16.124 15.688 15.225 14.876 14.438

312 416 524 632 739 847 1178 1514 1911 2318 2673 3096

0.000135 0.000136 0.000137 0.000138 0.000139 0.000140 0.000143 0.000147 0.000151 0.000155 0.000159 0.000164

Denomination (inch)

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TABLE 2-2 Continued

Denomination (inch)

Outside diameter (inch)

Wall thickness (inch)

Pipe internal diameter (inch)

Allowed working pressure (psi) to 650 °F

Relative roughness for new pipe

20.000

0.250 0.375 0.500 0.594 0.812 1.031 1.281 1.500 1.750 1.969 0.250 0.375 0.500 0.562 0.688 0.969 1.219 1.513 1.812 2.062 2.344 0.312 0.375 0.500 0.625 0.312 0.375 0.500 0.625 0.750 0.375 0.500 0.375 0.500

19.500 19.250 19.000 18.812 18.376 17.938 17.438 17.000 16.500 16.082 23.500 23.250 23.000 22.876 22.624 22.062 21.562 20.938 20.376 19.875 19.312 29.376 29.250 29.00 28.750 35.376 35.250 35.000 34.750 34.500 41.250 41.000 47.250 47.000

280 471 664 811 1155 1507 1917 2284 2710 3091 233 392 552 635 795 1165 1500 1927 2319 2674 3083 254 313 440 568 207 260 366 473 580 223 313 195 274

0.000121 0.000123 0.000124 0.000126 0.000129 0.000132 0.000135 0.000139 0.000143 0.000147 0.0001005 0.0001016 0.0001027 0.000103 0.000104 0.000107 0.000109 0.000113 0.000116 0.000119 0.000122 0.0000804 0.0000807 0.0000814 0.0000822 0.0000668 0.0000670 0.0000675 0.0000679 0.0000685 0.0000573 0.0000576 0.0000499 0.0000503

20⬙ Sch 10 20⬙ Sch 20 S 20⬙ Sch 30 X 20⬙ Sch 40 20⬙ Sch 60 20⬙ Sch 80 20⬙ Sch 100 20⬙ Sch 120 20⬙ Sch 140 20⬙ Sch 160 24⬙ Sch 10 24⬙ Sch 20 S 24⬙ X 24⬙ Sch 30 24⬙ Sch 40 24⬙ Sch 60 24⬙ Sch 80 24⬙ Sch 100 24⬙ Sch 120 24⬙ Sch 140 24⬙ Sch 160 30⬙ Sch 10 30⬙ S 30⬙ Sch 20 X 30⬙ Sch 30 36⬙ Sch 10 36⬙ S 36⬙ Sch 20 X 36⬙ Sch 30 36⬙ Sch 40 42 S

42.000

48 S

48.000

24.000

30.000

36.000

*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of steel is taken as 60 m.

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TABLE 2-3 Size and Relative Roughness of Rubber-Lined Steel Pipes to U.S. Dimensions*

Denomination (inch)

Outside diameter (inch)

Wall thickness (inch)

Rubber thickness (inch)

Pipe internal diameter (inch)

Relative roughness for rubber-lined pipe

2⬙ Sch 40 2⬙ Sch 80 2⬙ Sch 160 XX

2.375

0.154 0.218 0.344 0.436

0.25

1.559 1.431 1.179 0.995

0.003788 0.004127 0.005009 0.005935

3⬙ Sch 40 3⬙ Sch 80 3⬙ Sch 160 XX

3.500

0.216 0.300 0.438 0.600

0.25

2.560 2.392 2.116 1.792

0.002307 0.002469 0.002790 0.003295

4⬙ Sch 40 4⬙ Sch 80 4⬙ Sch 120 4⬙ Sch 160 XX

4.500

0.237 0.337 0.438 0.531 0.674

0.25

3.518 3.138 3.116 2.930 2.664

0.001679 0.00178 0.001895 0.002015 0.002233

5⬙ Sch 40 5⬙ Sch 80 5⬙ Sch 120 5⬙ Sch 160 XX

5.633

0.257 0.375 0.500 0.625 0.750

0.25

4.609 4.375 4.125 3.875 3.625

0.001281 0.00135 0.001431 0.001524 0.001629

6⬙ Sch 40 6⬙ Sch 80 6⬙ Sch 120 6⬙ Sch 160 XX

6.625

0.280 0.432 0.562 0.719 0.864

0.25

5.557 5.253 4.993 4.679 4.389

0.001063 0.001124 0.001183 0.001262 0.001345

8⬙ Sch 20 8⬙ Sch 30 8⬙ Sch 40 8⬙ Sch 60 8⬙ Sch 80 8⬙ Sch 100 8⬙ Sch 120 8⬙ Sch 140 XX 8⬙ Sch 160

8.625

0.250 0.277 0.322 0.406 0.500 0.594 0.719 0.812 0.875 0.906

0.375

7.375 7.732 7.231 7.063 6.875 6.687 6.437 6.251 6.125 6.063

0.000801 0.000807 0.000817 0.000836 0.000859 0.000883 0.000917 0.000945 0.000964 0.000974

0.250 0.307 0.365 0.500 0.594 0.719 0.844 1.000 1.125

0.375

9.500 9.386 9.27 9.00 8.812 8.562 8.312 8.000 7.750

0.000621 0.000629 0.000637 0.000656 0.000670 0.000689 0.000710 0.000738 0.000762

10⬙ Sch 20 10⬙ Sch 30 10⬙ Sch 40 S 10⬙ Sch 60 X 10⬙ Sch 80 10⬙ Sch 100 10⬙ Sch 120 10⬙ Sch 140 XX 10⬙ Sch 16

10.75

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TABLE 2-3 Continued Outside diameter (inch)

Wall thickness (inch)

Rubber thickness (inch)

Pipe internal diameter (inch)

Relative roughness for rubber-lined pipe

12⬙ S 12⬙ Sch 40 12⬙ X 12⬙ Sch 60 X 12⬙ Sch 80 12⬙ Sch 100 12⬙ Sch 120 XX 12⬙ Sch 140 12⬙ Sch 160

12.750

0.250 0.330 0.375 0.406 0.500 0.562 0.688 0.844 1.00 1.125 1.312

0.375

11.500 11.340 11.250 11.188 11.000 10.870 10.624 10.312 10.000 9.750 9.376

0.000513 0.000521 0.000525 0.000528 0.000537 0.000543 0.000556 0.000572 0.000591 0.000606 0.000629

14⬙ Sch 10 14⬙ Sch 20 14⬙ Sch 30 S 14⬙ Sch 40 14⬙ X 14⬙ Sch 60 14⬙ Sch 80 14⬙ Sch 100 14⬙ Sch 120 14⬙ Sch 140 14⬙ Sch 160

14.000

0.250 0.330 0.375 0.400 0.500 0.594 0.750 0.938 1.062 1.250 1.406

0.375

12.750 12.626 12.500 12.374 12.250 12.062 11.750 11.374 11.126 10.750 10.438

0.000463 0.000468 0.000472 0.000477 0.000482 0.000489 0.000503 0.000519 0.000531 0.000549 0.000566

16⬙ Sch 10 16⬙ Sch 20 16⬙ Sch 30 S 16⬙ Sch 40 X 16⬙ Sch 60 16⬙ Sch 80 16⬙ Sch 100 16⬙ Sch 120XX 16⬙ Sch 140 16⬙ Sch 160

16.000

0.250 0.312 0.375 0.500 0.656 0.844 1.031 1.219 1.438 1.594

0.375

14.750 14.626 14.500 14.250 13.938 13.562 13.188 12.668 12.374 12.062

0.000400 0.000404 0.000407 0.000414 0.000424 0.000435 0.000448 0.000466 0.000477 0.000489

18⬙ Sch 10 18⬙ Sch 20 18⬙ S 18⬙ Sch 30 18⬙ X 18⬙ Sch 40 18⬙ Sch 60 18⬙ Sch 80 18⬙ Sch 100 18⬙ Sch 120 18⬙ Sch 140 18⬙ Sch 160

18.000

0.250 0.312 0.375 0.438 0.500 0.562 0.750 0.938 1.156 1.375 1.562 1.781

0.375

16.750 16.626 16.500 16.374 16.250 16.126 15.750 15.374 14.938 14.500 14.126 13.688

0.000353 0.000355 0.000358 0.000361 0.000363 0.000366 0.000375 0.000384 0.000395 0.000407 0.000418 0.000431 (continued)

Denomination (inch)

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TABLE 2-3 Continued Outside diameter (inch)

Wall thickness (inch)

Rubber thickness (inch)

Pipe internal diameter (inch)

Relative roughness for rubber-lined pipe

20⬙ Sch 10 20⬙ Sch 20 S 20⬙ Sch 30 X 20⬙ Sch 40 20⬙ Sch 60 20⬙ Sch 80 20⬙ Sch 100 20⬙ Sch 120 20⬙ Sch 140 20⬙ Sch 160

20.000

0.250 0.375 0.500 0.594 0.812 1.031 1.281 1.500 1.750 1.969

0.375

18.750 18.500 18.250 18.062 17.626 17.188 16.688 16.250 15.750 15.312

0.000315 0.000319 0.000324 0.000327 0.000335 0.000344 0.000354 0.000363 0.000375 0.000386

24⬙ Sch 10 24⬙ Sch 20 S 24⬙ X 24⬙ Sch 30 24⬙ Sch 40 24⬙ Sch 60 24⬙ Sch 80 24⬙ Sch 100 24⬙ Sch 120 24⬙ Sch 140 24⬙ Sch 160

24.000

0.250 0.375 0.500 0.562 0.688 0.969 1.219 1.513 1.812 2.062 2.344

0.375

22.750 22.500 22.250 22.126 21.874 21.312 20.812 20.224 19.626 19.126 18.562

0.000259 0.000260 0.000263 0.000265 0.000267 0.000270 0.000277 0.000284 0.000292 0.000309 0.000318

Denomination (inch)

*Dimensions for steel are based on ANSI B36.1 and B31.1. Absolute roughness of rubber is taken as 150 m.

2-3-4 Hazen–Williams Formula In the United States, the Hazen–Williams formula is often used by civil engineers because it is independent of the Reynolds number. Speed is calculated as: U = 1.319 CRH0.63S 0.54 in ft/s

(2-20)

1.85 Qgpm S = Hv /L = ᎏᎏᎏ 1.67 × C 1.85 × RH1.17

(2-21)

where Qgpm is expressed in US gallons per minute, and S = slope or head loss per unit length U = average velocity of fluid in ft/sec RH = hydraulic radius = area of pipe/perimeter of pipe C = Surface roughness coefficient (refer to Table 2-6) In SI units:

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FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-4 Dimensions of North American HDPE Pipes for Pressure Ratings 50 psi to 110 psi at a Temperature of 23°C (73.4 °F) Pressure rating

DR 32.5 50 psi

DR 26 64 psi

DR 21 80 psi

DR 17 100 psi

DR 15.5 110 psi

Pipe size (inch)

Average outside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

3 4 5 6 7 8 10 12 13 14 16 18 20 22 24 26 28 30 32M 36 40M 42 48M

3.500 4.500 5.563 6.625 7.125 8.625 10.750 12.750 13.375 14.000 16.000 18.000 20.000 22.000 24.000 26.000 28.000 30.000 31.594 36.000 39.469 42.000 47.382

3.214 4.133 5.109 6.084 6.544 7.921 9.874 11.711 12.285 12.859 14.696 16.533 18.370 20.206 22.043 23.880 25.717 27.554 29.054 33.054 36.225 38.576 43.526

3.147 4.046 5.001 5.957 6.406 7.754 9.665 11.463 12.025 12.586 14.385 16.183 17.982 19.778 21.577 23.375 25.174 26.971 28.414 32.366 35.469 37.760 42.616

3.063 3.938 4.870 5.798 6.237 7.550 9.410 11.160 11.707 12.253 14.005 15.755 17.507 19.257 21.007 22.759 24.508 26.258 27.663 31.510 34.561 36.761 41.489

3.021 3.885 4.802 5.720 6.150 7.446 9.279 11.005 11.545 12.086 13.812 15.539 17.265 18.992 20.718 22.445 24.171 25.898 27.288 31.075

6.193 6.661 8.063 10.048 11.919 12.502 13.086 14.967 16.826 18.696 20.565 22.435 24.304 26.173 28.043 29.541 33.651 35.898 39.261 44.302

U = 0.8492 CRH0.63S 0.54 in m/s

(2-22)

All parameters in Equation 2-22 must be in SI units. There is no consistency in using the Hazen–Williams formula from small to large pipes. Despite the fact that commercial publications from pipe suppliers sometimes use Hazen–Williams equations to determine pressure loss of slurries, this method is highly discouraged for long pipelines.

2.4 THE HYDRAULIC FRICTION GRADIENT OF WATER IN RUBBER-LINED STEEL PIPES Equations 2-15 to 2-19 have established a relationship between the friction factor and the Reynolds number. To compute the latter, the properties of water as a carrier fluid are presented in Tables 2-7 and 2-8.

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TABLE 2-5 Dimensions of North American HDPE Pipes for Pressure Ratings 128 psi to 300 psi at a Temperature of 23°C (73.4 °F) Pressure rating

DR 13.5 128 psi

DR11 160 psi

DR 9 200 psi

DR 7.3 254 psi

DR6.3 300 psi

Pipe size (inch)

Average outside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

Average inside diameter (inch)

3 4 5 6 7 8 10 12 13 14 16 18 20 22 24 26 28 30 32M

3.500 4.500 5.563 6.625 7.125 8.625 10.750 12.750 13.375 14.000 16.000 18.000 20.000 22.000 24.000 26.000 28.000 30.000 31.594

2.951 3.795 4.690 5.584 6.006 7.270 9.062 10.749 11.274 11.802 13.488 15.174 16.880 18.544 20.231 21.917 23.603 25.289 26.645

2.826 3.633 4.490 5.349 5.751 6.693 8.679 10.293 10.797 11.301 12.915 14.532 16.145 17.780 19.374 20.988 22.606 24.219 25.527

2.675 3.440 4.253 5.065 5.446 6.594 8.219 9.745 10.225 10.701 12.231 13.760 15.289 16.819 18.346 19.875 21.405 22.934

2.485 3.194 3.948 4.700 5.056 6.119 7.627 9.046 9.491 9.934 11.853 12.772 14.191 —

2.321 2.986 3.690 4.395 4.727 5.723 7.133 8.459 8.674 9.289 10.615 — — —

TABLE 2-6 Hazen–Williams Roughness Coefficients Description Extremely smooth pipe Very smooth pipe Concrete pipe Riveted new pipes and tiled channels Normal cast pipes, 10 year old steel pipes, masonry channels Very rough pipes

Roughness coefficient 140 130 120 110 100 60

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TABLE 2-7 Physical Properties of Water in SI Units

Temperature T (°C) 0 5 10 15 20 25 30 40 50 60 70 80 90 100

Density L (kg/m3)

Dynamic viscosity, kinematic viscosity (mPa·s)

Kinematic viscosity, dynamic viscosity (km2/s)

Surface tension in contact with air (N/m)

Vapor pressure at atmospheric pressure (kN/m2)

999.8 1000 999.7 999.1 998.2 997.0 995.7 992.2 988.0 983.2 977.8 971.8 965.3 958.4

1.781 1.518 1.307 1.139 1.002 0.890 0.798 0.653 0.547 0.466 0.404 0.354 0.315 0.282

1.785 1.519 1.306 1.139 1.003 0.893 0.800 0.658 0.553 0.474 0.413 0.364 0.326 0.294

0.0756 0.0749 0.0742 0.0735 0.0728 0.0720 0.0712 0.0696 0.0679 0.0662 0.0644 0.0626 0.0608 0.0589

0.61 0.87 1.23 1.70 2.34 3.17 4.24 7.38 12.33 19.92 31.16 47.34 70.10 101.33

TABLE 2-8 Physical Properties of Water in USCS Units Temperature T

Density L

Kinematic viscosity

Dynamic viscosity

Surface tension in contact with air

Vapor pressure at atmospheric pressure

(°F)

(slug/ft3)

(lbf-sec/ft2)

(ft2/sec)

(lbf/ft)

(psia)

32 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190 200 212

1.940 1.940 1.940 1.938 1.936 1.934 1.931 1.927 1.923 1.918 1.913 1.908 1.902 1.896 1.890 1.883 1.876 1.868 1.860

0.00003746 0.00003229 0.00002735 0.00002359 0.0000205 0.00001799 0.00001595 0.00001424 0.00001284 0.00001168 0.00001069 0.00000981 0.00000905 0.00000838 0.0000078 0.00000726 0.00000678 0.00000637 0.00000593

0.00001931 0.00001664 0.00001410 0.00001217 0.00001059 0.00009300 0.00008260 0.00007390 0.00006670 0.00006090 0.00005580 0.00005140 0.00004760 0.00004420 0.00004130 0.00003850 0.00003620 0.00003410 0.00003190

0.00518 0.00614 0.00509 0.00504 0.00498 0.00492 0.00486 0.00480 0.00473 0.00467 0.00460 0.00454 0.00447 0.00441 0.00434 0.00427 0.00420 0.00413 0.00404

0.09 0.12 0.18 0.26 0.36 0.51 0.70 0.95 1.27 1.69 2.22 2.89 3.72 4.74 5.99 7.51 9.34 11.52 14.70

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The friction losses are expressed by the following equation: fDV 2L H= ᎏ 2gDI

(2-23)

where H = head due to losses (in meters for SI units, in ft for USCS units) L = length of the pipe (in meters for SI units, in ft for USCS units) fD = Darcy friction factor DI = pipe inner diameter V = speed of flow in the pipe (m/s or ft/s) g = acceleration due to gravity (9.81 m/s or 32.2 ft/s) The hydraulic friction gradient is defined as the head loss per unit length. It is defined as fDV 2 H iw = ᎏ = ᎏ 2gDI L

(2-24)

This is a very important parameter that will be used in Chapter 4 to evaluate the friction loss of Newtonian flows. Calculations of the friction factor by Equation 2-18 require iterations. The Moody diagram is a logarithmic scale that is rather difficult to use and prone to reading errors. With modern computers, a simple program will give more accurate numbers for the Darcy friction factor than from reading the Moody curve on a difficult logarithmic scale. The following program was written for plain steel and rubber-lined steel pipes. It uses standard US pipe sizes and applies the Swamee–Jain equation (2-19). DIM PIP(300), t(300), a(300), q(300), ep(300) pi = 4 * ATN(1) DEF fnlog10 (X) = LOG(X) * .4342944 ‘ p refers to nominal size ‘ outside diameter is stated p2 = 2.375 p3 = 3.5 p4 = 4.5 p5 = 5.633 p6 = 6.625 p8 = 8.625 p10 = 10.75 p12 = 12.75 p14 = 14 p16 = 16 p18 = 18 p20 = 20 p24 = 24 p30 = 30 p36 = 36 p42 = 42 p48 = 48 INPUT “choose between steel (1) and rubber (2)”, ch IF ch = 1 THEN tr1 = 0 IF ch = 1 THEN tr2 = 0

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FUNDAMENTALS OF WATER FLOWS IN PIPES

IF IF IF IF ef IF IF

ch = 2 THEN ch = 2 THEN ch = 1 THEN ch = 2 THEN = e / .0254 ch = 1 THEN ch = 2 THEN

tr1 tr2 e = e =

2.23

= .254 = .375 .00006 .00015

LPRINT “steel pipe” LPRINT “rubber lined pipe”

‘2 FOR I = 1 TO 4 t(1) = .154 t(2) = .218 t(3) = .344 t(4) = .436 PIP(I) = p2 - 2 * t(I) - 2 * tr1 ep(I) = ef / PIP(I) LPRINT USING “ppe od = ##.### in pip id = ##.#### in thick = #.### in k/d = #.########”; p2; PIP(I); t(I); ep(I) GOSUB swamee NEXT I LPRINT LPRINT The program is repeated for all US sizes of pipes (not shown here) swamee: d1 = PIP(I) * .0254 a = .25 * pi * d1 ^ 2 FOR K = 1 TO 5 q = a * K * 1000 qus = (q * 60 / 3.7854) fd = .4 em = ep(I) Re = 1000 * K * d1 / .001 110 z = (em / 3.7) + (5.74 / Re ^ .9) y = fnlog10(z) fd = .25 / y ^ 2 1111 hl = fd * K ^ 2 / (2 * 9.81 * d1) PRINT “revised swamee factor “; fd LPRINT USING “veloc = ##.### m/s; flow q = ####.#### L/s; flow = ######.## gpm “; K; q; qus LPRINT USING “RE = #########; fd = #.#####; hL = #####.####”; Re; fd; hl LPRINT PRINT “iteration error in swamee friction factor “; dg

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CHAPTER TWO

TABLE 2-9 Flow and Hydraulic Friction Gradient for Steel Pipes at a Speed of 1 m/s to 5m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m3 (62.3 lbs/ft3) and a Kinematic Viscosity of 1 cP (2.09 × 10–5 lbf-sec/ft2) Using the Swamee–Jain Equation* Relative Wall roughness Speed Speed Denomination thickness for new of flow Flow of flow (inch) (inch) pipe (m/s) L/s (ft/s)

Flow Darcy US Reynolds friction gpm number factor

Friction gradient (m/m) or (ft/ft)

2⬙ Sch 40

0.154

0.001143

1 2 3 4 5

2.2 4.3 6.5 8.6 10.8

3.3 6.6 9.9 13.2 16.5

34.3 69 103 137 172

52,502 105,004 157505 210007 262509

0.0244 0.0227 0.0221 0.0217 0.0214

0.0237 0.0883 0.1927 0.3367 0.5204

2⬙ Sch 80

0.218

0.00121

1 2 3 4 5

1.9 3.8 5.7 7.6 9.5

3.3 6.6 9.9 13.2 16.5

30.2 60.4 90.6 121 151

49,251 98,501 147,752 197002 246253

0.0248 0.0231 0.0224 0.0220 0.0218

0.0257 0.0956 0.2087 0.3648 0.5637

2⬙ Sch 160

0.344

0.001400

1 2 3 4 5

1.4 2.9 4.3 5.8 7.2

3.3 6.6 9.9 13.2 16.5

22.9 45.7 68.6 91.4 114

42,850 85,700 128,549 171,399 214249

0.0258 0.0239 0.0232 0.0228 0.023

0.0306 0.114 0.249 0.434 0.671

3⬙ Sch 40

0.216

0.000770

1 2 3 4 5

4.77 9.5 14.3 19.08 23.8

3.3 6.6 9.9 13.2 16.5

75.6 151 227 303 378

77,927 155,854 233,782 311709 389636

0.02213 0.0206 0.0200 0.0197 0.0195

0.0145 0.054 0.118 0.206 0.319

3⬙ Sch 80

0.300

0.000815

1 2 3 4 5

4.3 8.5 12.8 17.1 21.3

3.3 6.6 9.9 13.2 16.5

67.5 135 203 270 337

73,660 147,320 202,980 294,640 368,300

0.0224 0.0209 0.0203 0.0199 0.0197

0.0155 0.0579 0.1264 0.2209 0.3415

3⬙ Sch 160

0.438

0.000900

1 2 3 4 5

3.5 7 10.5 13.9 17.5

3.3 6.6 9.9 13.2 16.5

55.3 111 166 221 277

66,650 133,299 199,949 266,598 332,248

0.0230 0.0214 0.0208 0.0204 0.0202

0.0176 0.0655 0.1431 0.2501 0.3866

4⬙ Sch 40

0.237

0.000587

1 2 3 4 5

8.21 16.4 24.6 32.8 41.1

3.3 6.6 9.9 13.2 16.5

130.2 260.4 391 521 651

102,260 204,521 306,781 409,042 511,302

0.0207 0.0193 0.0188 0.0185 0.0183

0.0103 0.0386 0.0843 0.1473 0.2278

4⬙ Sch 80

0.337

0.000617

1 2 3 4 5

7.4 14.8 22.3 29.7 37.1

3.3 6.6 9.9 13.2 16.5

117 253 353 470 588

97,180 194,361 291,541 388,722 485,902

0.021 0.01957 0.019 0.0187 0.0185

0.011 0.041 0.09 0.157 0.243

4⬙ Sch 160

0.531

0.000687

1 2

6 12

3.3 6.6

95 190

87,325 0.0215 174,650 0.0201

0.0126 0.047

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FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-9 Continued Relative Wall roughness Speed Speed Denomination thickness for new of flow Flow of flow (inch) (inch) pipe (m/s) L/s (ft/s) 4⬙ Sch 160

Flow Darcy US Reynolds friction gpm number factor

3 4 5

18 24 30

9.9 13.2 16.5

285 380 475

261,976 0.0195 349,301 0.0192 436,626 0.019

Friction gradient (m/m) or (ft/ft) 0.102 0.179 0.277

5⬙ Sch 40

0.257

0.000461

1 2 3 4 5

13.3 26.5 39.8 53 66.3

3.3 6.6 9.9 13.2 16.5

210 421 631 841 1052

129,972 259,944 389,915 519,887 649,859

0.0196 0.0183 0.0178 0.0175 0.0173

0.0077 0.0287 0.0628 0.1098 0.1697

5⬙ Sch 80

0.375

0.000484

1 2 3 4 5

12.1 24.1 36.2 48.4 60.4

3.3 6.6 9.9 13.2 16.5

192 383 575 766 958

124,028 248,056 372,085 496,113 620,141

0.1981 0.0185 0.018 0.0177 0.0175

0.0081 0.0304 0.0665 0.1163 0.1797

5⬙ Sch 160

0.625

0.000539

1 2 3 4 5

9.7 19.5 29.2 38.9 48.7

3.3 6.6 9.9 13.2 16.5

154.3 308.6 463 617 772

111,328 222,656 333,985 445,313 556,641

0.0203 0.0189 0.0184 0.0181 0.0179

0.0093 0.0347 0.0759 0.1327 0.2052

6⬙ Sch 40

0.280

0.000389

1 2 3 4 5

18.7 37.3 55.9 74.6 93.2

3.3 6.6 9.9 13.2 16.5

295.4 591 886 1182 1477

154,051 308,102 462,153 616,204 770,255

0.0189 0.0176 0.0171 0.0169 0.0167

0.0062 0.233 0.051 0.0892 0.138

6⬙ Sch 80

0.432

0.000410

1 2 3 4 5

16.8 33.6 50.5 67.3 84

3.3 6.6 9.9 13.2 16.5

267 533 800 1066 1333

146,329 292,659 438,988 585,318 731,647

0.0191 0.0178 0.0173 0.0170 0.0169

0.0066 0.0248 0.0543 0.095 0.147

6⬙ Sch 160

0.719

0.000455

1 2 3 4 5

13.6 27.3 40.9 54.5 68.2

3.3 6.6 9.9 13.2 16.5

216 432 648 864 1,080

131,750 263,500 395,249 526,999 658,749

0.0195 0.0183 0.0177 0.0174 0.0173

0.0076 0.0282 0.0617 0.108 0.167

8⬙ Sch 40

0.322

0.000296

1 2 3 4 5

32.3 65.6 96.8 129.1 161.4

3.3 6.6 9.9 13.2 16.5

512 202,717 1,023 405,435 1,535 608,152 2,046 810,870 2,558 1,013,587

0.0177 0.0166 0.0161 0.0159 0.0157

0.0045 0.0167 0.0365 0.0639 0.0988

8⬙ Sch 80

0.500

0.000310

1 2 3 4 5

29.4 58.9 88.4 117.8 147.3

3.3 6.6 9.9 13.2 16.5

467 934 1,401 1,868 2335

0.0179 0.0047 0.0168 0.0176 0.0163 0.0386 0.0160 0.0675 0.0158 0.1044 (continued)

193,675 387,350 581,025 774,700 968,375

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CHAPTER TWO

TABLE 2-9 Continued Relative Wall roughness Speed Speed Denomination thickness for new of flow Flow of flow (inch) (inch) pipe (m/s) L/s (ft/s)

Flow Darcy US Reynolds friction gpm number factor

Friction gradient (m/m) or (ft/ft)

8⬙ Sch 160

0.906

0.00347

1 2 3 4 5

23.5 47 70.6 94.1 117.6

3.3 6.6 9.9 13.2 16.5

373 746 1,118 1,491 1,864

0.0184 0.0172 0.0167 0.0164 0.0163

0.0054 0.0202 0.0443 0.0774 0.1197

10⬙ Sch 40 S

0.365

0.000236

1 2 3 4 5

51 102 153 204 254

3.3 6.6 9.9 13.2 16.5

806 254,508 0.0169 1,613 509,106 0.0158 2,419 763,524 0.0154 3,226 1,018,032 0.0151 4,032 1,272,540 0.0150

0.0034 0.0127 0.0277 0.0485 0.0750

10⬙ Sch 60 X

0.500

0.000243

1 2 3 4 5

48 96 145 193 241

3.3 6.6 9.9 13.2 16.5

764 247,650 1,527 495,300 2,291 742,950 3,054 990,600 3,818 1,238,250

0.017 0.0159 0.0155 0.0152 0.0151

0.0035 0.0131 0.0286 0.0501 0.0775

10⬙ Sch 120 XX 1.000

0.000269

1 2 3 4 5

39 78 116 155 194

3.3 6.6 9.9 13.2 16.5

615 222,250 1,230 444,500 1,845 666,750 2,460 889,000 3,075 1,111,250

0.0174 0.0163 0.0158 0.0156 0.0154

0.004 0.0149 0.0327 0.0571 0.0883

12⬙ S

0.375

.000197

1 2 3 4 5

73 146 219 292 365

3.3 6.6 9.9 13.2 16.5

1,157 304,800 0.0163 2,313 609,600 0.0152 3,470 914,400 0.0148 4,626 1,219,200 0.0146 5,783 1,524,000 0.0144

0.0027 0.0102 0.0223 0.0390 0.0603

12⬙ X

0.500

0.000201

1 2 3 4 5

59 117 177 235 293

3.3 6.6 9.9 13.2 16.5

928 273,050 0.0166 1,856 546,100 0.0156 2,784 819,150 0.0152 3,713 1,099,000 0.0149 4,641 1,365,250 0.0148

0.0031 0.0116 0.0255 0.0445 0.0689

12⬙ Sch 120 XX 1.000

0.00022

1 2 3 4 5

59 117 177 235 293

3.3 6.6 9.9 13.2 16.5

928 273,050 0.0166 1856 546,100 0.0156 2784 819,150 0.0152 3713 1,099,000 0.0149 4641 1,365,250 0.0148

0.0031 0.0116 0.0255 0.0445 0.0689

14⬙ S

0.375

0.000178

1 2 3 4 5

89 178 267 356 445

3.3 6.6 9.9 13.2 16.5

1,410 336,550 0.0159 2,820 673,100 0.0149 4,231 1,009,650 0.0145 5,640 1,346,200 0.0143 7,050 1,682,750 0.0141

0.0024 0.0090 0.0198 0.0346 0.0536

14⬙ X

0.500

0.000182

1 2 3

86 171 257

3.3 6.6 9.9

1,357 2,175 4,072

0.0025 0.0093 0.0202

173,050 346,100 519,151 692,201 865,251

330,200 0.016 660,400 0.015 990,600 0.0146

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FUNDAMENTALS OF WATER FLOWS IN PIPES

2.27

TABLE 2-9 Continued Relative Wall roughness Speed Speed Denomination thickness for new of flow Flow of flow (inch) (inch) pipe (m/s) L/s (ft/s) 14⬙ X

Flow Darcy US Reynolds friction gpm number factor

Friction gradient (m/m) or (ft/ft)

4 5

343 428

13.2 16.5

5,429 1,320,800 0.0144 6,787 1,651,000 0.0142

0.0354 0.0548

14⬙ Sch 120

1.062

0.000531

1 2 3 4 5

72 143 214 286 357

3.3 6.6 9.9 13.2 16.5

1,133 301,650 0.0163 2,266 603,301 0.0153 3398 904,951 0.01485 4531 1,206,602 0.0146 5664 1,508,251 0.0145

0.0028 0.0103 0.0226 0.0395 0.0611

16⬙ Sch 30 S

0.375

0.000155

1 2 3 4 5

118 236 354 471 589

3.3 6.6 9.9 13.2 16.5

1,868 387,350 0.0155 3,736 774,700 0.0145 5,604 1,162,050 0.0141 7,471 1,549,400 0.0139 9,339 1,936,750 0.0138

0.002 0.0076 0.0167 0.0292 0.0452

16⬙ X

0.500

0.0001575

1 2 3 4 5

114 228 342 456 570

3.3 6.6 9.9 13.2 16.5

1,807 381,000 0.0155 3,614 762,000 0.0146 5,421 1,143,000 0.0142 7,228 1,524,000 0.0139 9,035 1,905,000 0.0138

0.0021 0.0078 0.0170 0.0298 0.0461

16⬙ Sch 120

1.291

0.000176

1 2 3 4 5

91 182 274 365 456

3.3 6.6 9.9 13.2 16.5

1,446 340,817 0.0159 2,892 681,634 0.0149 4,338 1,022,452 0.0145 5784 1,363,269 0.0143 7230 1,704,086 0.0141

0.0024 0.0089 0.0195 0.0341 0.0527

18⬙ S

0.375

0.000137

1 2 3 4 5

151 302 452 603 754

3.3 6.6 9.9 13.2 16.5

2,390 438,150 0.0151 4,780 876,300 0.0142 7,170 1,314,450 0.0138 9,560 1,752,600 0.0136 11,949 2,190,750 0.0134

0.0018 0.0066 0.0144 0.0252 0.0390

18⬙ X

0.500

0.000139

1 2 3 4 5

146 293 439 586 732

3.3 6.6 9.9 13.2 16.5

2,321 431,800 0.0151 4,642 863,600 0.0142 6,963 1,295,400 0.0138 9,284 1,727,200 0.0136 11,605 2,159,000 0.0135

0.0018 0.0067 0.0147 0.0257 0.0397

18⬙ Sch 120

1.375

0.000155

1 2 3 4 5

118 236 354 471 589

3.3 6.6 9.9 13.2 16.5

1,868 387,350 0.0155 3,734 774,700 0.0145 5,604 1,162,050 0.0141 7,471 1,549,000 0.0139 9340 1,966,750 0.0138

0.0020 0.0076 0.0167 0.0292 0.0452

20⬙ Sch 20S

0.375

0.000123

1 2 3 4 5

188 375 563 751 939

3.3 2,976 488,950 0.0148 0.0015 6.6 5,952 977,900 0.0139 0.0058 9.9 8,929 1,466,850 0.0135 0.0126 13.2 11,905 1,955,800 0.0133 0.0221 16.5 14,881 2,444,750 0.0131 0.0342 (continued)

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2.28

CHAPTER TWO

TABLE 2-9 Continued Relative Wall roughness Speed Denomination thickness for new of flow (inch) (inch) pipe (m/s)

Speed Flow Flow of flow US L/s (ft/s) gpm

Darcy Reynolds friction number factor

Friction gradient (m/m) or (ft/ft)

20⬙ Sch 30 X

0.500

0.000124

1 2 3 4 5

183 366 549 732 915

3.3 6.6 9.9 13.2 16.5

2,899 5,799 8,698 11,598 14,497

482,600 965,200 1,447,800 1,930,400 2,413,000

0.0148 0.0139 0.0135 0.0133 0.0132

0.0016 0.0059 0.0128 0.0225 0.0348

24⬙ Sch 20 S

0.375

0.000102

1 2 3 4 5

274 548 822 1096 1370

3.3 6.6 9.9 13.2 16.5

4,341 8,683 13,025 17,366 21,707

590,550 1,181,100 1,771,650 2,362,200 2,952,750

0.0142 0.0134 0.013 0.0128 0.01267

0.0012 0.0046 0.0101 0.0177 0.0273

24⬙ Sch S

0.500

0.000102

1 2 3 4 5

268 536 804 1072 1340

3.3 6.6 9.9 13.2 16.5

4,249 584,200 0.01425 0.0012 8,497 1,1684,000 0.01338 0.0047 12,746 1,752,600 0.01302 0.0102 16,995 2,336,800 0.01282 0.0179 21,243 2,921,000 0.01269 0.0277

*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of steel is taken as 60 m.

‘INPUT “hit any key to continue”; m$ CLS NEXT K RETURN The range of speeds used to carry solids in Newtonian flows is typically between 1.5 m/s and 5m/s. Tables 2-9 and 2-10 present friction factor and head losses for water as a carrier fluid for plain and rubber-lined steel pipes. There is no point in tabulating other fluids here as they are rarely used for slurry mixtures. The hydraulic friction gradient of water in rubber-lined pipes in the range of 2⬙ to 18⬙ is presented in Figures 2-5 to 2-13. Rubber thickness of 6.4 mm (0.25⬙) was assumed for 2⬙, 3⬙, 4⬙, and 6⬙ (up to 150 mm) pipes. Rubber thickness of 9.5 mm (0.375⬙) was assumed for 8⬙ to 24⬙ (200 to 610 mm NB) pipes. HDPE friction head was plotted for similar sizes at SDR11 (suitable for 100 psi pressure), to mark the advantages of reduced friction at these sizes using HDPE instead of rubber-lined pipes, wherever it may be appropriate. The design engineer must take in account the pressure limitations of HDPE pipes versus rubber-lined steel pipes. The hydraulic friction gradient for HDPE pipes up to a size of 20⬙ (508 mm), and for speeds in the range of 1 to 5 m/s (3.3 to 16.5 ft/sec) is presented in Table 2-11 These curves and tables allow an easier and accurate determination of the hydraulic friction gradient of water than the Moody diagram.

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FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-10 Flow and Hydraulic Friction Gradients for Rubber-Lined Steel Pipes at a Speed of 1 m/s to 5 m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m3 (62.3 lbs/ft3) and a Kinematic Viscosity of 1 cP (2.09 × 10–5 lbf-sec/ft2) Using the Swamee–Jain Equation*

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

39,599 79,197 118,796 158,394 197,993

0.0309 0.0297 0.289 0.0287 0.0287

0.0399 0.153 0.338 0.595 0.925

2⬙ Sch 40

Steel 0.154 0.003788 Rubber 0.250

1 2 3 4 5

1.2 2.5 3.7 4.9 6.2

3.3 6.6 9.9 13.2 16.5

2⬙ Sch 80

Steel 0.218 0.004123 Rubber 0.250

1 2 3 4 5

1 2.1 3.1 4.2 5.2

3.3 6.6 9.9 13.2 16.5

16.5 33 49.4 65.8 82.2

36,347 72,695 109,042 145,390 181,737

0.0317 0.0299 0.0296 0.0295 0.0337

0.045 0.171 0.377 0.665 1.033

2⬙ Sch 160

Steel 0.344 0.005008 Rubber 0.250

1 2 3 4 5

0.7 1.4 2.1 2.8 3.5

3.3 6.6 9.9 13.2 16.5

11.2 22.3 33.5 44.7 55.8

29,947 59,893 89,840 119,786 149,733

0.0337 0.0322 0.0317 0.0314 0.0312

0.0574 0.2195 0.4854 0.8551 1.3284

3⬙ Sch 40

Steel 0.216 Rubber 0.250

1 2 3 4 5

3.3 6.6 10 13.3 16.6

3.3 6.6 9.9 13.2 16.5

53 105 158 211 263

65,024 130,048 195,072 260,096 325,120

0.0269 0.0257 0.0253 0.0251 0.0250

0.0211 0.0808 0.1788 0.3150 0.4895

3⬙ Sch 80

Steel 0.300 0.002487 Rubber 0.250

1 2 3 4 5

2.9 5.8 8.7 11.6 14.5

3.3 6.6 9.9 13.2 16.5

46 92 138 184 230

60,757 121,514 182,270 243,027 303,784

0.0274 0.0262 0.0258 0.0256 0.0255

0.0230 0.088 0.1949 0.3435 0.5337

0.002791 3⬙ Sch 160 Steel 0.438 Rubber 0.250

1 2 3 4 5

2.3 4.5 6.8 9.1 11.3

3.3 6.6 9.9 13.2 16.5

36 72 108 144 180

53,746 107,493 161,239 214,986 268,732

0.0283 0.0272 0.0267 0.0265 0.0263

0.0269 0.103 0.228 0.402 0.624

4⬙ Sch 40

Steel 0.237 0.001679 Rubber 0.250

1 2 3 4 5

6.3 12.5 18.8 25.1 31.4

3.3 6.6 9.9 13.2 16.5

99 199 298 398 497

89,357 178,714 268,072 357,429 446,786

0.0247 0.0237 0.0233 0.0231 0.0229

0.0141 0.054 0.1195 0.2106 0.3273

4⬙ Sch 80

Steel 0.337 0.00178 Rubber 0.250

1 2 3 4 5

5.6 11.2 16.7 23.3 27.8

3.3 6.6 9.9 13.2 16.5

88.4 177 265 354 442

84,277 168,554 252,832 337,109 421,386

0.0251 0.0151 0.0240 0.0581 0.0236 0.1287 0.0234 0.2268 0.0233 0.352 (continued)

19.5 39 58.6 78 97.6

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Page 2.30

2.30

CHAPTER TWO

TABLE 2-10 Continued

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

4⬙ Sch 160

Steel 0.531 0.002015 Rubber 0.250

1 2 3 4 5

04.4 8.7 13.1 17.4 21.8

03.3 6.6 9.9 13.2 16.5

069 138 207 276 345

074,422 148,844 223,266 297,688 372,110

0.0259 0.0248 0.0244 0.0242 0.0241

0.0178 0.068 0.151 0.265 0.412

5⬙ Sch 40

Steel 0.257 0.001281 Rubber 0.250

1 2 3 4 5

10.8 21.5 32.3 43.1 53.8

3.3 6.6 9.9 13.2 16.5

171 341 512 683 853

117,069 234,137 351,206 468,274 585,343

0.023 0.0221 0.0217 0.0215 0.0214

0.01 0.038 0.085 0.150 0.233

5⬙ Sch 80

Steel 0.375 0.001349 Rubber 0.250

1 2 3 4 5

9.7 19.4 29.1 38.8 48.5

3.3 6.6 9.9 13.2 16.5

154 308 461 615 769

111,125 222,250 333,375 444,500 555,625

0.0233 0.0224 0.022 0.0218 0.0217

0.0107 0.0410 0.0901 0.160 0.249

5⬙ Sch 160

Steel 0.625 0.00152 Rubber 0.250

1 2 3 4 5

7.61 3.3 15.2 6.6 22.8 9.9 30.4 13.2 38.0 16.5

121 241 362 482 603

98,425 196,850 295,275 393,700 492,125

0.024 0.0231 0.0227 0.0225 0.0224

0.0124 0.0478 0.1058 0.1865 0.2898

6⬙ Sch 40

Steel 0.280 0.001063 Rubber 0.250

1 2 3 4 5

15.6 31.3 47 62.5 78.2

3.3 6.6 9.9 13.2 16.5

248 496 744 992 1240

141,148 282,296 423,443 564,591 705,739

0.0219 0.0211 0.0207 0.0206 0.0204

248 496 744 992 1240

6⬙ Sch 80

Steel 0.432 0.001124 Rubber 0.250

1 2 3 4 5

14 28 42 56 70

3.3 6.6 9.9 13.2 16.5

222 443 665 886 1108

133,426 266,852 400,279 533,705 667,131

0.0222 0.0214 0.0210 0.0208 0.0207

0.0085 0.0326 0.0723 0.1274 0.198

6⬙ Sch 160

Steel 0.719 0.001262 Rubber 0.250

1 2 3 4 5

11.1 22.2 33.3 44.4 55.5

3.3 6.6 9.9 13.2 16.5

176 352 528 703 879

118.847 237,693 356,540 475,386 594,233

0.0229 0.0219 0.0215 0.0215 0.0213

0.0098 0.0377 0.0835 0.1472 0.2288

8⬙ Sch 40

Steel 0.322 0.000817 Rubber 0.375

1 2 3 4 5

26.5 53 79.5 106 132

3.3 6.6 9.9 13.2 16.5

420 840 1,260 1,680 2,100

183,667 367,335 551,002 734,670 918,337

0.0205 0.0198 0.0195 0.0193 0.0192

0.0057 0.0219 0.0486 0.0856 0.1331

8⬙ Sch 80

Steel 0.500 0.000859 Rubber 0.375

1 2 3

24 48 72

3.3 6.6 9.9

380 760 1,139

174,625 349,250 523,875

0.0208 0.0199 0.0197

0.006 0.0233 0.0517

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Page 2.31

2.31

FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-10 Continued

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

8⬙ Sch 80

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

4 5

96 120

13.2 16.5

1,519 1898

698,500 0.0195 873,125 0.0194

0.0912 0.142

19 37 6 75 93

3.3 6.6 9.9 13.2 16.5

295 590 886 1,181 1,476

154,000 308,000 462,001 616,001 770,001

0.0215 0.0206 0.0203 0.0201 0.020

0.0071 0.0273 0.0604 0.1065 0.1656

8⬙ Sch 160

Steel 0.906 0.000974 Rubber 0.375

1 2 3 4 5

10⬙ Sch 40 S

Steel 0.365 0.000637 Rubber 0.375

1 2 3 4 5

43.5 3.3 87 6.6 131 9.9 174 13.2 218 16.5

690 235,458 1,380 470,916 2,071 706,374 2,761 941,832 3,451 1,177,290

0.0194 0.0186 0.0183 0.0182 0.0181

0.0042 0.0161 0.0357 0.063 0.098

10⬙ Sch 60 X

Steel 0.500 0.000656 Rubber 0.375

1 2 3 4 5

41 82 123 164 205

3.3 6.6 9.9 13.2 16.5

651 228,600 1,301 457,200 1,952 658,800 2,602 914,400 3,253 1,143,000

0.0195 0.0188 0.0185 0.0183 0.0182

0.0044 0.0167 0.0371 0.0653 0.1016

10⬙ Sch 120 XX

Steel 1.000 0.000738 Rubber 0.375

1 2 3 4 5

32 65 97 130 162

3.3 6.6 9.9 13.2 16.5

514 203,200 1,028 406,400 1,542 609,600 2,056 812,800 2,570 1,016,000

0.0201 0.0198 0.0189 0.0188 0.0187

0.0050 0.0193 0.0429 0.0756 0.1174

12⬙ S

Steel 0.375 0.000525 Rubber 0.375

1 2 3 4 5

64.1 3.3 128.3 6.6 192.4 9.9 256.5 13.2 320.7 16.5

1,017 285,750 0.01853 2,033 571,500 0.0178 3,049 857,250 0.01755 4,066 1,143,000 0.0174 5,082 1,428,750 0.0173

0.0033 0.0127 0.0282 0.0497 0.0722

12⬙ X

Steel 0.500 0.000537 Rubber 0.375

1 2 3 4 5

61 123 184 245 307

3.3 6.6 9.9 13.2 16.5

972 279,400 0.0186 1,943 558,800 0.0179 2,915 838,200 0.0176 3,887 1,117,600 0.0175 4,859 1,397,000 0.0174

0.0034 0.0131 0.029 0.051 0.079

12⬙ Sch 120 XX

Steel 1.000 0.000591 Rubber 0.375

1 2 3 4 5

51 101 152 203 253

3.3 6.6 9.9 13.2 16.5

803 254,000 0.0190 1606 508,000 0.0183 2409 762,000 0.0180 3212 1,016,000 0.0179 4015 1,270,000 0.0179

0.038 0.0147 0.0326 0.0574 0.0892

14⬙ S

Steel 0.375 0.000472 Rubber 0.375

1 2 3 4 5

79 158 238 317 396

3.3 6.6 9.9 13.2 16.5

1,255 317,500 0.0181 2,510 635,000 0.0174 3,765 952,500 0.0171 5,020 1,270,000 0.0170 6,275 1,587,500 0.0169

0.0029 0.0112 0.0248 0.0437 0.0679

(continued)

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Page 2.32

2.32

CHAPTER TWO

TABLE 2-10 Continued

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

14⬙ X

Steel 0.500 0.000482 Rubber 0.375

1 2 3 4 5

76 152 228 304 380

3.3 6.6 9.9 13.2 16.5

1,205 311,150 2,410 622,300 3,616 933,450 4,820 1,244,600 6,026 1,555,750

0.0182 0.0175 0.0172 0.0171 0.0170

0.003 0.0115 0.0254 0.0447 0.0695

14⬙ Sch 120

Steel 1.062 0.000531 Rubber 0.375

1 2 3 4 5

63 125 188 251 314

3.3 6.6 9.9 13.2 16.5

997 282,600 1988 565,201 2983 847,801 3977 1,130,402 4971 1,413,002

0.0186 0.0179 0.0176 0.0175 0.0174

0.0034 0.0129 0.0286 0.0503 0.0783

16⬙ Sch 30 S

Steel 0.375 0.000407 Rubber 0.375

1 2 3 4 5

107 213 319 426 532

3.3 6.6 9.9 13.2 16.5

1,689 368,300 3,377 736,600 5,066 1,104,900 6,755 1,473,200 8,443 1,841,500

0.0175 0.0168 0.0166 0.0165 0.0164

0.0024 0.0093 0.0207 0.0364 0.0566

16⬙ X

Steel 0.500 0.000414 Rubber 0.375

1 2 3 4 5

103 206 309 412 515

3.3 6.6 9.9 13.2 16.5

1,631 361,950 3,262 713,900 4,893 1,085,850 6,524 1,447,800 8,155 1,809,750

0.0176 0.0169 0.0167 0.0165 0.0164

0.0025 0.0095 0.0211 0.0372 0.0578

16⬙ Sch 120

Steel 1.291 0.000466 Rubber 0.375

1 2 3 4 5

81 162 243 324 405

3.3 6.6 9.9 13.2 16.5

1,289 321,767 2,578 643,534 3867 965,302 5155 1,287,069 6444 1,608,836

0.0180 0.0174 0.0171 0.0169 0.0169

0.0029 0.011 0.0244 0.0429 0.0668

18⬙ S

Steel 0.375 0.000358 Rubber 0.375

1 2 3 4 5

138 276 414 552 690

3.3 6.6 9.9 13.2 16.5

2,187 419,100 4,373 838,200 6,560 1,257,300 8,746 1,676,400 10,933 2,095,500

0.0170 0.0164 0.0161 0.0160 0.0159

0.002 0.008 0.0176 0.0311 0.0484

18⬙ X

Steel 0.500 0.000363 Rubber 0.375

1 2 3 4 5

134 268 401 535 669

3.3 6.6 9.9 13.2 16.5

2,121 412,750 4,241 825,500 6,362 1,238,250 8,483 1,651,000 10,604 2,063,750

0.0171 0.0164 0.0162 0.0160 0.0159

0.0021 0.0081 0.0180 0.0317 0.0493

18⬙ Sch 120

Steel 1.375 0.000407 Rubber 0.375

1 2 3 4 5

107 213 320 426 533

3.3 6.6 9.9 13.2 16.5

1,689 368,300 3,377 736,600 5,066 1,104,900 6,755 1,473,200 8,443 1,841,500

0.0175 0.0168 0.0166 0.0165 0.0164

0.0024 0.0093 0.0207 0.0364 0.0566

20⬙ Sch 20S

Steel 0.375 0.000319 Rubber 0.375

1 2 3

173 346 520

3.3 6.6 9.9

2,7495 469,900 497 939,800 8,246 1,409,700

0.0166 0.0159 0.0157

0.0018 0.0069 0.0154

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Page 2.33

2.33

FUNDAMENTALS OF WATER FLOWS IN PIPES

TABLE 2-10 Continued

Pipe size (inch)

Wall thickness (inch)

Relative roughness Speed Speed Flow for new of flow Flow of flow US pipe (m/s) L/s (ft/s) gpm

20⬙ Sch 20S

Reynolds number

Darcy friction factor

Energy gradient (m/m) or (ft/ft)

4 5

694 867

13.2 16.5

10,995 1,879,600 0.0156 13,744 2,349,500 0.0155

0.0271 0.0421

20⬙ Sch 30 X

Steel 0.500 0.000324 Rubber 0.375

1 2 3 4 5

169 338 506 675 844

3.3 6.6 9.9 13.2 16.5

2,675 463,550 0.0166 5,350 927,100 0.0160 8,025 1,390,650 0.0158 10,670 1,854,200 0.0156 13,375 2,317,750 0.0155

0.0018 0.0070 0.0156 0.0275 0.0428

24⬙ Sch 20 S

Steel 0.375 0.000262 Rubber 0.375

1 2 3 4 5

257 513 780 1,026 1,283

3.3 6.6 9.9 13.2 16.5

4,066 8,132 12,198 16,284 20,330

571,500 1,143,000 1,714,500 2,286,000 2,857,500

0.0159 0.0153 0.0151 0.0150 0.0149

0.0014 0.0055 0.0121 0.0214 0.0332

0.000265 24⬙ Sch S Steel 0.500 Rubber 0.375

1 2 3 4 5

251 502 753 1,003 1,254

3.3 6.6 9.9 13.2 16.5

3,976 7,952 11,930 15,904 19,880

565,150 1,130,300 1,695,450 2,260,600 2,825,700

0.0159 0.0154 0.0151 0.0150 0.0149

0.0014 0.0055 0.0123 0.0216 0.0336

*Dimensions are based on ANSI B36.1 and B31.1. Absolute roughness of rubber is input as 150 m (0.000492 ft).

2-5 DYNAMICS OF THE BOUNDARY LAYER Boundary layer theory has been extensively covered by a number of authors. A book by Schlichting (1968) is considered one of the classical references on this subject. When a uniform flow approaches a plate, the particles at the wall of the plate are slowed down by the dynamic viscosity of the fluid. A layer called the boundary layer develops. When the flow enters a pipe, effects develop at the entrance until the flow is uniform.

2-5-1 Entrance Length Flow in a pipe at relatively low speed when the Reynolds number is smaller than 2500 is characterized by a certain distance called “entrance length,” over which the velocity profile takes the final parabolic shape shown in Figure 2-14. The length Le is expressed as Le = 0.028 DiRe For turbulent flows, the entrance length is equivalent to 50 times the inner diameter.

9:14 AM

Page 2.34

2.34

0

40

5 m/s

sch

0.8 0.7

4 m/s

0.6 0.5 0.4 3 m/s

0.3 0.2

2.0 4.0 6.0 Flow Rate (L/s)

rometers (0.000492 ft)

0.8

16.5 ft/sec 40

1.0 0.9

8.0

0

0.0

1 m/s

sch 8

0.0

2 m/s

sch

0.1

Hydraulic friction gradient (ft/ft)

sch 8

0.9

60

1.0

sch 1

Hydraulic friction gradient (m/m)

CHAPTER TWO

60

2/28/02

sch 1

abul-2.qxd

0.7 13.2 ft/sec

0.6 0.5 0.4 0.3

9.9ft/sec

0.2 0.1 0.0

6.6 ft/sec 3.3 ft/sec

0.0 20 40 60 80 100 120 Flow Rate (US gallons/min) FIGURE 2-5 Hydraulic friction gradient for water in a rubber-lined 2⬙ pipe, Sch 40, Sch 80 and Sch 160. Caculations for 2⬙ steel pipe. Rubber thickness = 0.250⬙ (6.4 mm). Rubber roughness = 150 m (0.000492 ft).

rometers (0.000492 ft)

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Page 2.35

2.35

Hydraulic friction gradient (m/m) (m/m)

FUNDAMENTALS OF WATER FLOWS IN PIPES

0.7 5 m/s

0.6 0.5

h sc 4 m/s

0.4

h sc

Rubber Lined Steel Pipes

80

40 sch

0.3

4 m/s

3 m/s

2 m/s

0.1

HDPE SDR 11 5 m/s

3 m/s

0.2

2 m/s

1 m/s

0.0 0.0

Hydraulic friction (ft/ft)gradient (ft/ft)

0 16

2.0

4.0

6.0

8.0

10.0

12.0

14.0

16.0 18.0 20.0 Flow Rate (L/s)

0.7 0.6 0.5 13.2 ft/sec

0.4 0.3

16.5 ft/sec 0 16 h sc

Rubber Lined Steel Pipes 0 h8 sc

40 sch

9.9ft/sec

0.2 0.0 0

16.5 ft/sec

9.9 ft/sec

6.6 ft/sec

0.1

HDPE SDR 11 13.2 ft/sec

6.6 ft/sec

3.3 ft/sec

20

40

60

80

100 120 140 160 180 200 220 240 260 220 240 260 Flow Rate (US gallons/min)

FIGURE 2-6 Hydraulic friction gradient for water in a rubber-lined 3⬙ pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.25⬙ (6.4 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE line 3⬙ SDR11 roughness 1.5 m (0.00000492 ft).

2-5-2 Friction Velocity Prandtl proposed a concept of friction velocity: Uf =

w

ᎏ =U ᎏ 冪莦 冪莦2 fN

(2-25)

Blasius conducted tests on turbulent flows in pipes and developed an equation for the shear wall stress in terms of the maximum velocity outside the boundary layer.

冢 冣

r w = 0.0225 (Umax)7/4 ᎏ R

1/4

(2-26)

The local magnitude of the velocity in a boundary layer at a height y above the wall is u ᎏ = 8.73(y+)n Uf

(2-27)

where n = 1/7 to 1/9 and where the nondimensional height parameter y+ is defined as the relative distance from the wall:

abul-2.qxd 2/28/02

5 m/s 4 m/s

0.1

2 m/s 3 m/s 1 m/s

0.0 0.0

2 m/s

10

20

30

Flow Rate (L/s)

40

HDPE SDR 11

sc h 80 40

13.2 ft/sec

16.5 ft/sec

sch

0.3

sch

Energy gradient (ft/ft)

5 m/s

0.2 3 m/s

160

0.4

sc h

sch 80 40

160 4 m/s

sch

0.3

Rubber Lined Steel Pipes

Page 2.36

2.36

Energy gradient (m/m)

0.4

9:14 AM

Rubber Lined Steel Pipes

0.2 16.5 ft/sec

9.9 ft/sec

0.1

13.2 ft/sec

6.6 ft/sec

9.9 ft/sec

0.0

6.6 ft/sec 3.3 ft/sec

0.0 100 200 300 400 500

HDPE SDR 11

Flow Rate (US gallons/min)

FIGURE 2-7 Hydraulic friction gradient for water in a rubber-lined 4⬙ pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.25⬙ (6.4 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE line 3⬙ SDR11 roughness 1.5 m (0.00000492 ft).

2/28/02

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Page 2.37

2.37

FUNDAMENTALS OF WATER FLOWS IN PIPES

Energygradient (m/m)

0.30 Rubber Lined Steel

0.25 0.20

ch 4 m/s s

0.15 0.10

5 m/s 0 16

40 sch

80 sch

3 m/s

0.05

1 m/s

0.0

2 m/s

HDPE SDR11 0.0

10

20

30

40

50

60

70

80

Flow Rate (L/s) 0.30 0.25 Energy gr adient (ft/ft)

abul-2.qxd

16.5 ft/sec 60 1 ch s 13.2 ft/sec 80 sch

0.20 0.15

40 sch

0.10 0.05 0.0

9.9 ft/sec 6.6 ft/sec 3.3 ft/sec

HDPE SDR11 0.0

200

400

600

800

1000

1200

Flow Rate ( US gallons/sec) FIGURE 2-8 Hydraulic friction gradient for water in a rubber-lined 6⬙ pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.25⬙ (6.4 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE line 3⬙ SDR11 roughness 1.5 m (0.00000492 ft).

Uf␥ y+ = ᎏ

(2-28)

and where u = velocity of the flow at distance y. The boundary layer can be divided into a number of sections. At small values of y+ up to 5, the velocity profile is linear in a sublayer (see Figure 2-15). The flow is considered to be laminar in the sublayer. Above y+ = 5, a buffer zone develops up to y+ = 50 and turbulence develops. The thickness of the boundary viscous sublayer ␦ is usually expressed as 11.6 11.6 ␦= ᎏ = ᎏ 兹苶 (苶 w苶) U

ᎏ 冪莦 8 fD

(2-29)

In the turbulent region, the velocity profile is established as Uf y u ᎏ = 5.75 log10 ᎏ + 5.5 Uf

冢 冣

(2-30)

abul-2.qxd 2/28/02

0.08 3 m/s

0.06 0.04 0.02

HDPE SDR11 2 m/s 1 m/s

0.0

20

40

60

80

100 120 140 Flow Rate (L/s)

h

40

sc

0.08 9.9 ft/sec

0.06 0.04 0.02

HDPE SDR11 6.6 ft/sec 3.3 ft/sec

0.0

0.0

h

13.2 ft/sec

0.10

sc

0.12

80

160

0.14

sch

Hydraulic friction gradient (ft/ft)

80 h

sc

4 m/s

0.10

40

sc

0.12

h

sch

160

0.14

16.5 ft/sec

0.16

Page 2.38

2.38

Hydraulic friction gradient (m/m)

5 m/s

0.16

Rubber Lined Steel 0.18

0

400

800 1200 1600 Flow Rate (US gallons/min)

9:14 AM

Rubber Lined Steel 0.18

2000

FIGURE 2-9 Hydraulic friction gradient for water in a rubber-lined 8⬙ pipe, Sch 40, Sch 80, and Sch 160. Rubber thickness = 0.375⬙ (9.5 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE roughness 1.5 m (0.00000492 ft).

2/28/02

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Page 2.39

2.39

FUNDAMENTALS OF WATER FLOWS IN PIPES

Hydrau lic friction gradient (m/m)

0.12 0.10

5 m/s

0.08

5 m/s

0.06

h sc " 10 3 m/s

0.04 0.02 0.0

2 m/s 1 m/s

0.0 40

80

40

4 m/s 4 m/s

40 ch s " 12 3 m/s

2 m/s 1 m/s

120 160 200 240 280 320 Flow Rate (L/s)

0.12 16.5 ft/sec

Hydrau lic friction gradient (ft/ft)

abul-2.qxd

0.10 0.08 0.06 0.04 0.02

9.9 ft/sec 6.6 ft/sec

16.5 ft/sec

13.2 ft/sec 13.2 ft/sec

" 12

40 sch

9.9 ft/sec

0.0 0.0

3.3 ft/sec

ch "s 0 1

40

800

1600 2400 3200 4000 4800 Flow Rate ( US gallons/sec) 6.6 ft/sec

FIGURE 2-10 Hydraulic friction gradient for water in a rubber-lined 10⬙ pipe, Sch 40 and 12⬙, Sch 40 pipes. Rubber thickness = 0.375⬙ (9.5 mm). Rubber roughness = 150 m (0.000492 ft). 2-HDPE roughness 1.5 m (0.00000492 ft).

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Page 2.40

2.40

CHAPTER TWO

Hydraulic friction gradient (m/m)

0.08 5 m/s

0.06 DR "S

0.04

11

5 m/s

10

0.02

DR "S

11

12

0.0 0.0 40

80

120 160 200 240 280 320 Flow Rate (L/s)

Hydraulic friction gradient (ft/ft)

0.08 0.06 0.04

" 10

0.02

R SD

16.5 ft/sec 16.5 ft/sec

11

DR "S

11

12

0.0 0.0

800

1600 2400 3200 4000 4800 Flow Rate ( US gallons/sec)

FIGURE 2-11 Hydraulic friction gradient of water in 10⬙ and 12⬙ SDR11 HDPE pipes. Roughness 1.5 m (0.00000492 ft).

The reader is encouraged to review the work of Schlichling (1968) for details of boundary layer theory. Example 2-5 Determine the boundary viscous sublayer thickness and friction velocity of Example 2-4. Solution in SI Units In Example 2-4, the Darcy friction factor was determined to be 0.0178. In Section 2.31 it was stated that fD = 4fN, therefore 0.0178 fN = ᎏ = 0.00445 4 Since the velocity of the flow is 1.467 m/s,

冪莦

冪莦

ff 0.00445 Uf = U ᎏ = 1.467 ᎏ = 0.0692 m/s 2 2 From Equation 2.28 the thickness of the viscous sublayer is calculated as 11.6 ␦= ᎏ U

ᎏ 冪莦 8 fD

2/28/02

9:14 AM

Page 2.41

2.41

FUNDAMENTALS OF WATER FLOWS IN PIPES

30 ) (S ch

0.08

16 "S

S

0.07

14 "

0.06

" 18 5 m/s

4 m/s

0.05 0.04

S

3 m/s

0.03 0.02 2 m/s

0.01

1 m/s

0.0 0.0

100 200 300 Flow Rate (L/s)

0 09

400

500

600

Flow Rate (L/s) 30 )

0.09

(S ch

0.08 14 "

S

0.07 0.06 0.05

" 18

13.2 ft/sec

0.04

S

16.5 ft/sec

0.03

9.9 ft/sec

0.02 0.01

700

16 "S

Hydraulic friction gradient (m/m)

0.09

Hydraulic friction gradient (ft/ft)

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6.6 ft/sec 3.3 ft/sec

0.0 0.0

2000

4000 6000 8000 Flow Rate (US gallons/min)

10000

FIGURE 2-12 Hydraulic friction gradient for water in rubber-lined 14⬙ pipe, Sch 40 and 16⬙ S and 18⬙ S pipes. Wall thickness = 0.375⬙ (9.5 mm). Rubber thickness = 0.375⬙ (9.5 mm). Rubber roughness = 150 m (0.000492 ft).

11.6 (0.00129) ␦ = ᎏᎏ 1020 (1.467)

ᎏ = (4.72) 10 冪莦 8 0.0178

Solution in USCS Units 0.0178 fN = ᎏ = 0.00445 4 Since the average velocity flow in 4.8 ft/s,

冪莦

0.00445 Uf = 4.8 ᎏ 0.2264 ft/s 2

–7

m

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0.08 11

0.07

0.05

5 m/s

18

R

11

0.04

16

"S

D

R

0.06

SD

5 m/s

11

5 m/s

"

0.03

"

R SD

14

Hydraulic friction gradient (m/m)

0.09

0.02 0.01 0.0 0.0

100

200

300

400

500

600 700 Flow Rate (L/s)

0.08 0.07 D

R

11

0.06 "S

0.05

16

18

"

R

11

SD

R

11

0.04

SD

16.5 ft/sec

"

0.03 14

Hydraulic friction gradient (ft/ft)

0.09

0.02 0.01 0.0 0.0

2000

4000 6000 8000 Flow Rate (US gallons/min)

10000

FIGURE 2-13 Hydraulic friction gradient of water in 14⬙, 16⬙ and 18⬙ SDR 11 HDPE pipes. Absolute roughness 1.5 m (0.00000492 ft).

g 2-

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TABLE 2-11 Flow and Hydraulic Friction Gradient for HDPE Pipes SDR11 at a Speed of 1 m/s to 5 m/s (3.28 to 16.4 ft/sec) for Water at a Density of 1000 kg/m3 (62.3 lbs/ft3) and a Kinematic Viscosity of 1 cP (2.09 × 10–5 lbf-sec/ft2) Using the Swamee–Jain Equation Pipe inner Speed Pipe size diameter Relative of flow Flow (in) (in) roughness (m/s) L/s

Speed of flow (ft/s)

Flow US gpm

Reynolds number

Darcy friction factor

Friction gradient (m/m) or (ft/ft)

3

2.826

0.0000053

1 2 3 4 5

4.0 8.0 12.1 16.2 20.2

3.3 6.6 9.9 13.2 16.5

64 128 192 257 321

71,780 143,561 215,341 287,122 358,902

0.01917 0.01659 0.01532 0.0145 0.01391

0.0136 0.0471 0.0979 0.167 0.247

4

3.633

00000041

1 2 3 4 5

6.7 13.4 20.1 26.8 33.4

3.3 6.6 9.9 13.2 16.5

106 212 318 424 530

92,278 184,556 276,835 369,113 461,391

0.0182 0.0158 0.0146 0.0138 0.01329

0.01 0.035 0.0726 0.1223 0.1835

5

4.49

0.0000033

1 2 3 4 5

10.2 204 306 408 510

3.3 6.6 9.9 13.2 16.5

162 324 486 648 810

114,046 228,092 342,138 456,184 570,230

0.01738 0.01514 0.01403 0.01331 0.01279

0.0078 0.0271 0.0564 0.0592 0.1429

6

5.349

0.0000028

1 2 3 4 5

15 29 44 58 73

3.3 6.6 9.9 13.2 16.5

230 460 689 919 1,150

135,865 271,729 407,594 543,458 679,323

0.01677 0.01465 0.01359 0.01290 0.01241

0.0063 0.0220 0.0459 0.0774 0.1164

7⬙

5.7510

0.0000026

1 2 3 4 5

17 34 51 67 84

3.3 6.6 9.9 13.2 16.5

266 531 797 1,063 1,328

146,075 292,151 438,226 584,302 730,377

0.01653 0.01445 0.01341 0.01274 0.01225

0.0058 0.0202 0.0421 0.0711 0.1069

8⬙

7.270⬙

0.0000022

1 2 3 4 5

25 49 74 98 123

3.3 6.6 9.9 13.2 16.5

389 779 1,168 1,558 1,947

176,860 353,720 530,581 707,441 884,301

0.0159 0.0139 0.01296 0.01232 0.01186

0.0046 0.0161 0.0336 0.0568 0.0854

10⬙

8.675⬙

0.0000017

1 2 3 4 5

38 76 114 153 191

3.3 6.6 9.9 13.2 16.5

605 1,210 1,815 2,420 3,025

220,447 440,893 661,340 881,786 1,102,233

0.0152 0.0134 0.0125 0.0119 0.0114

0.0035 0.0124 0.0259 0.0439 0.0660

12⬙

10.293⬙ 0.0000015

1 2 3 4 5

54 107 161 215 268

3.3 6.6 9.9 13.2 16.5

851 1,702 2,552 3,404 4,254

261,442 522,884 784,327 1,045,769 1,307,211

0.01475 0.0130 0.0121 0.0115 0.0111

0.0029 0.0101 0.0212 0.0359 0.0541 (continued)

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TABLE 2-11 Continued Pipe inner Speed Pipe size diameter Relative of flow Flow (in) (in) roughness (m/s) L/s

Speed of flow (ft/s)

Flow US gpm

Reynolds number

Darcy friction factor

Friction gradient (m/m) or (ft/ft)

13⬙

10.797⬙ 0.0000014

1 2 3 4 5

59 118 177 236 295

3.3 6.6 9.9 13.2 16.5

936 1,872 2,808 3,745 4,681

274,244 548,488 822,731 1,096,975 1,371,219

0.0146 0.0129 0.0120 0.0114 0.0110

0.0027 0.0096 0.0201 0.0340 0.0512

14⬙

11.301⬙ 0.0000013

1 2 3 4 5

65 129 194 259 324

3.3 6.6 9.9 13.2 16.5

1,026 2,051 3,077 4,103 5,129

287,045 574,091 861,136 1,148,182 1,435,227

0.0145 0.0128 0.0119 0.0113 0.0109

0.0026 0.0091 0.0190 0.0322 0.0485

16⬙

12.915⬙ 0.0000012

1 2 3 4 5

85 169 253 338 423

3.3 6.6 9.9 13.2 16.5

1,340 2,679 4,020 5,359 6,698

328,041 656,082 984,123 1,312,164 1,640,205

0.0141 0.0125 0.0116 0.0110 0.0107

0.0022 0.0078 0.0163 0.0276 0.0416

18⬙

14.532⬙ 0.000001

1 2 3 4 5

107 214 321 428 535

3.3 6.6 9.9 13.2 16.5

1,696 3,392 5,088 6,784 8,480

369,113 738,226 1,107,338 1,476,451 1,845,564

0.0138 0.0122 0.0114 0.0109 0.0105

0.0019 0.0068 0.0142 0.0240 0.0362

20⬙

16.146⬙ 0.0000009

1 2 3 4 5

132 264 396 528 660

3.3 6.6 9.9 13.2 16.5

2,094 4,187 6,281 8,375 10,469

410,108 820,217 1,230,325 1,640,434 2,050,542

0.0136 0.0120 0.0112 0.0107 0.0103

0.0017 0.0060 0.0125 0.0213 0.0321

From the Equation 2.28, the thickness of the viscous sub-layer is 11.6 × 2.7 × 10–5 ␦ = ᎏᎏ 1.973 × 4.8

ᎏ = 1.56 × 10 冪莦 8 0.0178

–6

ft

2-6 PRESSURE LOSSES DUE TO CONDUITS AND FITTINGS The sizing of pumps is based on determining pressure losses between the starting point (A) and the final delivery point (B) (Figure 2-16). It is important to know that the static pressure at (A) includes atmospheric pressure or the pressure of any pressurizing gas, and

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Le

V

Di

FIGURE 2-14

Entrance length for flows in pipes.

the pressure due to the height of liquid above the centerline of the first impeller or impeller at suction. It is also important to know that static pressure at (B) includes any pressurizing gas at (B) and the height of liquid at (B) above the centerline of the pump’s impeller. The additional pressure losses between (A) and (B) include the friction losses and pressure losses in all the pipe fittings such as valves, elbows, expansions, contraction branches, and bypasses. Pressure is also lost at entry and exit as well. Such pressure losses are expressed in terms of the Darcy–Weisbach equation and in terms of pressure loss factors for each fitting. Total pressure loss due to friction:

U 2 Lj U 2 Hf = fD ᎏ ᎏ + ⌺Kf ᎏ 2 Dij 2 where Lj = length of the conduit j Kf = pressure loss of the fitting f

Turbulent layer

V + +

Buffer layer Viscous sublayer FIGURE 2-15 Boundary layer of flow over a plate.

(2-31)

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CHAPTER TWO

6

H 2

5 Pipe Diameters H 1

FIGURE 2-16

Simplified pumping system between two tanks.

TABLE 2-12 Equivalent Length of Valves for Friction Loss of Calculations for Single-Phase Turbulent Flow

Fitting Gate valves Globe valves Angle valves Ball valves Butterfly valves Plug valves—straightway Plug valves—3 way through flow Plug valve—branch flow Stop check valve—straight through Stop check valve—angular 90 deg Swing check valve Lift check valve Tilting disc check valve Foot valve with strainer—poppet disc Foot valve with strainer—hinged disc

Equivalent length/diameter ratio 8 340 55 3 16 18 30 90 400 200 300 55 5–15 420 75

Minimum recommended speed for full disc lift m/s

Ft/s

2.12 2.89 2.32 5.4 1.16–3.08 0.58 1.35

6.96 9.49 7.59 17.7 3.80 to 10.13 1.90 4.43

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0.5 0.4

Loss Factor K

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0.3 0.2 0.1 0.0 0 2 4 6 8 10 Ratio R/D (bend radius/pipe diam)

FIGURE 2-17 Loss Factors for rough wall bends for pipes 1–7⬙ (after Crane Technical Bulletin No 410).

The differential head that a pump must deliver to pump a liquid between (A) and (B) is therefore TDH = (PB – PA)/g + (ZB – ZA) + HfOB + HfOA where HfOA = the pressure losses due to conduits and fittings between the tank (A) and the pump, including entry loss HfOB = the losses due to conduits and fittings between the pump and tank (B), including exit losses Table 2-12 presents examples of loss coefficients for fittings. Some practical considerations limit the use of fittings in slurry circuits. For example, elbows should have a minimum radius of three pipe diameters to avoid short turns (see Figure 2-17). Such an approach minimizes wear. Example 2-6 The fluid of Example 2-4 is pumped at a flow rate of 3500 gpm through a 16 × 14 pump. The steel fittings include 14⬙ × 18⬙ reducer, an 18⬙ knife gate valve, three long radius 90° elbows with a diameter to radius ratio of 3. The length of the pipe is 355 ft. Determine the total dynamic head if the liquid level in the suction tank is 10 ft and the level in the discharge tank is 50 ft above the centerline of the impeller. Ignore the length of the suction pipe as negligible. Solution in SI Units Net static head: (50 ft – 10 ft) 0.3048 = 12.2 m Dynamic head at the entry of the pump: Di = 15.25 in or 0.387 m Suction Area = 0.1178 m2

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0.221 Speed = ᎏ = 1.875 m/s 0.1178 Dynamic head at suction: 1.8752 U2 ᎏ = ᎏ = 0.18 m 2g 2 × 9.81 For a sharp entrance pipe, the recommended K factor is 0.5. Friction losses at the entry to the pump are calculated as follows: 0.5 × 0.18 = 0.09 m On the discharge of the pump for a 14 × 18 reducer, the loss factor is calculated from the area ratio as

冢

d 21 K = 1 – ᎏ2 d2

= 0.1681 冣 = 冢1 – ᎏ 17.25 冣 2

13.252

2

2

앫 For an18 ft full-bore gate valve, the loss factor K = 0.10 앫 For the elbow r/D = 3, the loss factor K = 0.14 앫 For the reentry pipe L/D = 65 Total friction losses:

冤

冥

355 × 0.3048 + 65 × 0.438 0.09 m + ᎏᎏᎏ × 0.0178 + (0.1681 + 0.10 + 3 × 0.14) 0.438 1.4672 × ᎏ = 0.774 m 2 × 9.81 TDH = 0.774 m + 12.2 m = 12.97 m Solution in USCS Units Net static head = 50 ft – 10 ft = 40 ft Speed in suction pipe is calculated as follows: ID = 15.25⬙ = 1.27 ft Flow Rate = 7.79 ft3/s Area = 1.267 ft2 7.79 Velocity = ᎏ = 6.15 ft/s 1.267 Dynamic head at suction: 6.152 ᎏ = 0.587 ft 2 × 32.2

The K factor for sharp entrance in 0.5. Loss during suction is 0.5 × 0.587 = 0.293 ft. On the discharge of the pump the K factor is determined as in the SI unit solution. Total friction losses are:

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FUNDAMENTALS OF WATER FLOWS IN PIPES

冤

冢

冣

冥

355 + 65 × 1.27 4.82 0.293 ft + 0.0178 ᎏᎏ + 0.1681 + 0.10 + 0.42 ᎏ = 2.44 ft 1.27 2 × 32.2 TDH = 40 + 2.44 = 42.44 ft

2-7 ORIFICE PLATES, NOZZLES, AND VALVE HEAD LOSSES The flow through an orifice plate, a nozzle, or a valve is reduced from the ideal theoretical value by a discharge coefficient: Q = Cd Qideal

(2-32)

The ideal theoretical flow is considered the product of speed and area at the opening of the orifice, valve, or nozzle. The theoretical velocity through the orifice is calculated as Vth = 兹2苶g苶h苶 =

ᎏ 冪莦 R 2⌬P

(2-33)

However, the velocity is typically smaller than the theoretical velocity: Vo = CveVth where Cve = velocity coefficient. The flow through an opening contracts from the full area. This is known as the vena contracta effect.

Reentrant tube

Sharp Edged

Square Edged

Reentrant tube

Length = 1/2 to

Stream clears

Length = 2-1/2

1 diameter

sides

diameters

V

V

C =0.52 d

C =0.61 d

V

C =0.61 d

C =0.73 d

FIGURE 2-18 Discharge coefficients of orifice plates and nozzles.

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For a thin plate or sharp-edged orifice, the vena contracta is assumed to be one half of an orifice diameter d1 downstream from the orifice, but in reality the distance may be from 30% to 80% of d1. For flow of water at a high Reynolds number through a small orifice diameter, Lindeburg (1998) reported that the contracted area is approximately 61% to 63%. The coefficient of contraction is defined as area of vena contracta Cc = ᎏᎏᎏ orifice area

(2-34).

The total discharge is the product of the reduced velocity and the contracted area: Q = CveVth(Cc A1) The product of the velocity coefficient by the area contraction coefficient is called the discharge coefficient: Cd = CveCc

(2-35)

Q = Cd A1兹(2 苶g 苶h 苶)苶

(2-36a)

Q = Cd A1兹(2 苶⌬ 苶P 苶苶 /)苶

(2-36b)

actual discharge Cd = ᎏᎏᎏ theoretical discharge Typical values for discharge coefficients from nozzles and orifices are shown in Figure 2-18. The Cameron Hydraulic Handbook (1977) recommends a further correction for large openings when d2/d1 > 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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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 2kthL

(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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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

Drag coefficient, CD

Particle Reynolds number, Rep

Drag coefficient, CD

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

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0.2

8000

2000

80

100

60

40

20

6000

0

0

4000

30

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CD

CD

MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS

Rep

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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

Terminal velocity Vt in mm/s

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

Terminal velocity Vt in mm/s

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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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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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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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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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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= 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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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

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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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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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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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MECHANICS OF SUSPENSION OF SOLIDS IN LIQUIDS

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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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 (<1.0) Pressure drop Flow rate Radius Radius Radius of the container in the coaxial cylinder rotary viscometer Reynolds number

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Ren Rep Ri R0 sp

CHAPTER THREE

0 s w

A c T

Reynolds Number of a particle based on dn Reynolds Number of a sphere particle based on its diameter Inner radius of a pipe or tube Radius of the bob in the coaxial cylinder rotary viscometer The surface area per unit volume of a sphere of equivalent dimensions or 6/dg, also called specific surface of a particle Front area of a particle orthogonal to the direction of flow Surface area of a wing along the direction of flight Applied torque for the cylinder rotary viscometer Absolute temperature Average velocity of the flow Terminal velocity at very low volume concentration of solids Terminal velocity at given volume concentration of solids The terminal (or free settling) speed Weight the ratio of immobilized dispersing fluid to the volume of solids related approximately to the particle and floc apparent diameter A coefficient used to express to the shear stability of a pseudoplastic mixture Concentration by volume in decimal points Shear strain Wall shear rate or rate of shear strain with respect to time Coefficient of rigidity of a non-Newtonian fluid, also called Bingham viscosity A structural parameter for thixotropic fluids, which do not possess a yield stress value Carrier liquid absolute viscosity Apparent viscosity of a pseudoplastic fluid Effective viscosity Apparent viscosity of a pseudoplastic fluid at zero shear rate Bingham plastic limiting viscosity, or apparent viscosity of a pseudoplastic fluid at very high shear rate Pythagoras number (ratio of circumference of a circle to its diameter) Duration of the shear for a time-dependent fluid Density Shear stress at a height y or at a radius r Yield stress for a Bingham plastic or yield pseudoplastic Structural stress of a thixotropic fluid Wall shear stress Kinematic viscosity Angular velocity of particle Angular velocity of complete system The logarithmic standard deviation Albertson shape factor Clift shape factor Thomas shape factor

Subscripts g L m p s

Equivalent sphere Liquid Mixture Particle Solids

Sf Sw T Ta V V0 Vc Vt W

d/dt

a e 0

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3.41

3–10 REFERENCES Albertson, M. L. 1953. Effects of shape on the fall velocity of gravel particles. Paper read at the 5th Iowa Hydraulic Conference, Iowa University, Iowa City, Iowa. Allen, H. S. 1900. The motion of a sphere in a viscous fluid. Phil. Mag., 50, 323–338, 519–534. Boger, D. V., and Q. D. Nguyen. 1987. The Flow Properties of Weipa #3 and #4 Plant Tailings. Internal study conducted by Comalco Aluminium Ltd, Weipa, Australia, quoted in Darby, R., R. Mun, and D. V. Boger. 1992. Predict Friction Loss in Slurry Pipes. Chem. Engineering, 99, 9 (September), 117–211. Brown, G. G. 1950. Unit Operations. New York: Wiley. Brown, N. P. 1991. The settling behavior of particles in fluids. In Slurry Handling, Edited by N. P. Brown and N. I. Heywood. New York: Elsevier Applied Sciences. Caldwell, D. H., and H. E. Babitt. 1941. Flow of muds, sludge and suspensions in circular pipe. Am. Inst. Chem. Engrs. Trans., 37, 2 (April 25), 237–266. Caldwell, D. H., and H. E. Babitt. 1942. Pipeline flow of solids in suspension. Symp. Am. Soc. of Civ. Eng. Proc., 68, 3 (March), 480–482. Cheng, D. C. H., and W. Whitaker. 1972. Applications of the Warren Spring Laboratory pipeline design method to settling suspension. Paper read at the 2nd Annual Hydrotransport Conference, Bedford, England. Chilton, R. A., and R. Stainsby. 1998. Pressure loss equations for laminar and turbulent non-Newtonian pipe flow. Journal of Hydraulic Engineering, 124, 5 (May), 522–529. Clift, R., J. R. Grace, and M. E. Weber. 1978. Bubbles, Drops and Particles. New York: Academic Press. Concord, S., and J. F. Tassin. 1998. Rheoptical study of thixotropy in synthetic clay suspensions. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Cross, M. M. 1965. Rheology of non-Newtonian fluids—New flow equation for pseudoplastic systems. Journal of Colloid Science, 20, 417. Dahlgreen, S. E. 1958. Eyring model of flow applied to thixotropic equilibrium. Journal of Colloid Science, 13 (April), 151–158. Dedegil, M. Y. 1987. Drag coefficient and settling velocity of particles in non-Newtonian suspensions. Journal of Fluids Engineering, 109 (September), 319–323. Dick, R. I., and B. B. Ewing. 1967. Rheology of activated sludge. Journal of Water Pollution Control Federation, 39, 543. Dahlgreen, S. E. 1958. Eyring model of flow applied to thixotropic equilibrium. Journal of Colloid Science, 13 (April), 151–158. Fredrickson, A. G. 1970. A model for the thixotropy of suspensions. American Inst. of Chem. Eng. Journal, 16, 436. Gay E. D., P. A. Nelson, and W. P. Armstrong. 1969. Flow properties of suspensions with high solids concentration. American Inst. of Chem. Eng. Journal, 15, 6, 815–822. Goodrich and Porter. 1967. Govier, G. W., C. A. Shook, and E. O. Lilge. 1957. Rheological Properties of water suspensions of finely subdivided magnetite, galena, and ferrosilicon. Trans. Can. Inst. Mining Met., 60, 157. Govier, G. W., and K. Aziz. 1972. The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold. Hedstrom, B. O. A. 1952. Flow of plastic materials in pipes. Ind. Eng. Chem., 33, 651–656. Herbrich, J. 1968. Deep ocean mineral recovery. Paper read at the World Dredging Conference II, Rotterdam, the Netherlands. Herbrich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill. Heywood, N. I. 1991. Rheological characterisation of non-settling slurries. In Slurry Handling, Edited by N. P. Brown and N. I. Heywood. New York: Elsevier Applied Sciences. Heywood, N. I. 1996. The performance of commercially available Coriolis mass flowmeters applied to industrial slurries. Paper read at the 13th International Hydrotransport Symposium on Slurry Handling and Pipeline Transport. Johannesburg, South Africa. Cranfield, UK: BHRA Group. Inter-Agency Committee on Water Resources. 1958. Report 12. Internal report by the Subcommittee on Sedimentation, Minneapolis, Minnesota.

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Kearsey, H. A., and L. E. Gill. 1963. Study of sedimentation of flocculated thorium slurries using gamma ray technique. Trans. Inst. Chem. Engrs., 41, 296. Kherfellah, N., and K. Bekkour. 1998. Rheological characteristics of clay suspensions. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Krusteva, E. 1998. Viscosmetric and pipe flow of inorganic waste slurries. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Lapassin, R., S. Pricl, and M. Stoffa. 1998. Viscosity of aqueous suspensions of binary and ternary alumina mixtures. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Metzner, A. B., and M. Whitlock. 1958. Flow behavior of concentrated (dilatant) suspensions. Trans. Soc. Rheology, 2, 239–254. Moore, F. 1959. Rheology of Ceramic Slips and Bodies. British Ceramic Society Transactions, 58, 470. Mun, R. 1988. The Pipeline Transportation of Suspensions with a Yield Stress. Master’s Thesis, University of Melbourne, Australia. Parzonka, W. 1964. Determination of the maximum concentration of homogeneous mixtures (in French). Journal of the French Academy of Science, 259, 2073. Pilpel, N. 1965. Flow properties of non-cohesive powders. Chemical Process Eng. 46, 4, 167–179. Prokunin, A. N. 1998. Particle-wall interaction in liquids with different rheology. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Richards, R. H. 1908. Velocity of Galena and Quartz Falling in Water. Trans AIME, 38, 230–234. Ritter, R. A., and G. W. Govier. 1970. The development and evaluation of a theory of thixotropic behavior. Can. Journal Chem. Eng., 48, 505. Rubey, W. W. 1933. Settling velocities of gravel, sand and silt particles. Amer. Journal of Science, 25, 148, 325–338. Skelland, A. H. P. 1967. Non-Newtonian Flow and Heat Transfer. New York: Wiley. Teixeira, M. A. O. M., R. J. M. Craik, and P. F. G. Banfill. 1998. The effect of wave forms on the vibrational processing of fresh concrete. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Thomas, D. G. 1961. Transport characteristics of suspensions: Part II. Minimum transport velocity for flocculated suspensions in horizontal pipes. AIChE Journal, 7 (September), 423–430. Thomas, D. G. 1963. Transport characteristics of suspensions. Ch. E. Journal, 9, 310. Thomas, A. D. 1981. Slurry pipeline rheology. Paper presented at the National Conference on Rheology. Second Annual Conference of the British Society of Rheology, Australian Branch, University of Sydney, Australia. Turton, R., and O. Levenspiel. 1986. A short note on drag correlation for spheres. Powder Technology Journal, 47, 83. Valentik, L., and R. L. Whitemore. 1965. Terminal velocity of spheres in Bingham plastics. British Journal of Applied Phys., 16, 1197. Vlasak, P., Z. Chara, and P. Stern. 1998. The effect of additives on flow behaviour of kaolin–water mixtures. In Proceedings of the Fifth European Rheology Conference. Ljubljana, Slovenia: University of Ljubljana. Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid-Liquid Flow—Slurry Pipeline Transportation. Trans-Tech Publications. Wells, P. J. 1991. Pumping non-Newtonian slurries. Technical Bulletin 14. Sydney, Australia: Warman International. Whorlow, R. W. 1992. Rheological Techniques, 2d. ed. New York: Ellis Horwood. Wilson, K. C., G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. New York: Elsevier Applied Sciences. Worster, R. C., and D. E. Denny. 1955. Hydraulic transport of solid materials in pipes. Proceedings of the Institute of Mechanical Engineers (UK), 38, 230–234. Further Reading: Caldwell, D. H., and H. E. Babitt. 1942. Pipeline flow of solids in suspension. Symp. Am. Soc. of Civ. Eng. Proc., 68, 3 (March), 480–482. Goodrich, J. E., and R. S. Porter. 1967. Rheological interpretation of torque—Rheometer data. Polymer Eng & Science, 7 (January), 45–51.

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Lazerus, J. H., and P. T. Slatter. 1988. A method for the rheological characterization of tube viscometer data. Journal of Pipelines, 7, 165–176. Thomas, D. G. 1960. Heat and momentum transport characteristics of non-Newtonian aqueous thorium oxide. AIChE Journal, 7, 431. Wilson, K. C. 1991. Pipeline design for settling slurries. In Slurry Handling. Edited by N. P. Brown, and N. I. Heywood. New York: Elsevier Applied Sciences. 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.

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CHAPTER 4

HETEROGENEOUS FLOWS OF SETTLING SLURRIES

4-0 INTRODUCTION A practical engineer sifting through the literature on slurry flows would be astonished by the number of different equations. Since the work of the French scientists Durand and Condolios in 1952, and British scientists Newitt et al. in 1955, engineers and scientists have continued to develop new equations for deposition velocity and friction losses. This chapter reviews the evolution of models of Newtonian slurry flows from wellgraded and uniform particle sizes to complex mixtures of coarse and fine particles. For this purpose, some equations are listed in a historical context, to demonstrate their evolution to the reader. The sheer number of equations demonstrates how complicated heterogeneous flows are. A number of factors interact in a horizontal pipe. The flow takes the form of different regimes, and includes everything from a simple stationary bed at low speed to a pseudohomogeneous flow at high speeds. For each regime, equations have been developed over the years to account for the mean particle diameter, the diameter of the conduit, the density of the particles, their drag coefficient, the speed of flow, etc. As a result, there are many angles from which a heterogeneous flow of settling solids can be examined. The followers of Durand and Condolios put great emphasis on the drag coefficient of the solid particles, whereas the followers of Newitt prefer to focus on the terminal velocity. As we have clearly demonstrated in Chapter 3, the drag coefficient and the terminal velocity are interrelated. Examining a flow of slurry is often an exercise of playing with a murky liquid in which very little can be seen. Sand does not behave like coal and there is not a single universal law that may apply to the transportation of solids by liquids. It is therefore important to rely on database, historical information, and empirical data. In the past, engineers have tried to simplify the complexity of slurry flows by defining certain transition velocities. With the use of modern research tools, there is an emerging approach of rejecting the concept of an abrupt change from one state of flow to another, and a tendency to consider such a change over a band of the speed. Different approaches have been developed to examine the mixture of coarse and fine particles from superimposed layers to two-layer models. This book is intended for engineers, and various examples are included in the text. The purpose of such examples is to simplify the use of complex equations. With modern personal computers, which use simple languages such as quick basic, an engineer can efficiently calculate friction losses for a heterogeneous slurry flow. 4.1

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4-1 REGIMES OF FLOW OF A HETEROGENEOUS MIXTURE IN HORIZONTAL PIPE The history of slurry pipelines was briefly presented in Section 1-10 of Chapter 1. Two schools are credited for laying the foundations of modern hydro-transport engineering; SOGREAH in France, and the British Hydro-mechanic Research Association of the United Kingdom. Starting in 1952, Durand and Condolios of SOGREAH published a number of studies on the flow of sand and gravel in pipes up to 900 mm (35.5 in) in diameter. Based on the specific gravity of particles with a magnitude of 2.65, they proposed to divide the flows of nonsettling slurries in horizontal pipes into four categories based on average particle size as follows: 앫 Homogeneous suspensions for particles smaller than 40 m (mesh 325) 앫 Suspensions maintained by turbulence for particle sizes from 40 m (mesh 325) to 0.15 mm (mesh 100) 앫 Suspension with saltation for particle sizes between 0.15 mm (mesh 100) and 1.5 mm (mesh 11) 앫 Saltation for particles greater than 1.5 mm (mesh 11) This initial classification was refined over the next 18 years by Newitt et al. (1955), Ellis and Round (1963), Thomas (1964), Shen (1970), and Wicks (1971). Due to the interrelation between particle sizes and terminal and deposition velocities, the original classification proposed by Durand has been modified to four flow regimes based on the actual flow of particles and their size. Referring to Figures 4-1 and 4-2, there are four main regimes of flow in a horizontal pipe 1. 2. 3. 4.

Flow with a stationary bed Flow with a moving bed and saltation (with or without suspension) Heterogeneous mixture with all solids in suspension Pseudohomogeneous or homogeneous mixtures with all solids in suspension

velocity

d nde spe u s ly Ful

suspended with moving bed

suspended with saltation

lenticular deposits

stationary deposits with ripples

blocked pipe

FIGURE 4-1 Flow regimes of heterogeneous flows in terms of speed versus volumetric concentration (after Newitt et al., 1955).

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Flow

with

a sta tion ary be

F d wi low w th or ith a wi tho mov ut sus ing b pen ed sio ns

HETEROGENEOUS FLOWS OF SETTLING SLURRIES

particle size

abul-4.qxd

th

wi

4.3

all

low sf n u o ne sio ege en ter susp e H s in id sol Flow as a homogeneous or

pseudohomogeneous suspension

Mean velocity

FIGURE 4-2 Flow regimes of heterogeneous flows in terms of particle size versus mean velocity (after Shen, 1970).

Two special cases shown in Figure 4-1 are not considered to be at the limits of these regimes of flows. They are lenticular deposits at very low speeds but low solid concentration, and blocked pipes at high solid concentration. Slurry flows that have some form of segregation or separation of solids in layers are called “heterogeneous” flows, whereas the slurries themselves are called “settling” slurries. 4-1-1 Flow With a Stationary Bed When the slurry flow speed is low, the bed thickens. As the fluid above the bed tries to move the solids by entrainment, they tend to roll and tumble. The particles with the lowest settling speed move as an asymmetric suspension, whereas the coarser particles build up the bed. As the speed drops even further, the pressure to maintain the flow becomes quite high and eventually the pipe blocks up. Flow with saltation and asymmetric suspension occurs above the speed of blockage. This means that the coarser particles “sand up,” whereas the finer particles continue to move. Certain tailing lines have exhibited this phenomenon. In fact, when a process plant is built with a tailing line too large to handle the initial flow, the operator may choose to let the bottom of the pipe sand up to reduce the effective cross-sectional area of the pipe. This principle has been successfully applied to pipelines in a variety of countries. Saltation can eventually lead to blockage of a pipe. It may result in a number of problems, such as water hammer, wear, and freezing in cold climates. Most engineering specifications require that the pipeline be designed to operate at speeds higher than those associated with saltation. 4-1-2 Flow With a Moving Bed When the speed of the flow is low and there are a large number of coarse particles, the bed moves like desert sand dunes. The top particles are entrained in the moving fluid

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above the bed. Consequently, the upper layers of the bed move faster than the lower layers in a horizontal pipe. If the mixture were composed of a wide range of particles with different sizes and settling velocities, the bed would be composed of the particles with the highest settling speed. Particles with a moderate settling speed are maintained in an asymmetric suspension, with most particles concentrated in the lower half of the pipe, whereas the particles with the lowest settling speed move as a symmetric suspension

4-1-3 Suspension Maintained by Turbulence As the flow speed increases, turbulence is sufficient to lift more solids. All particles move in an asymmetric pattern with the coarsest at the bottom of a horizontal pipe covered with superimposed layers of medium- and fine-sized particles. Many particles may strike the bottom of the pipe and rebound. Wear on the bottom of the pipe must be taken into account in maintenance schedules and the pipes must be rotated at intervals suggested by the slurry engineer in order to maintain an even wear pattern of the internal wall of the pipe. Although the flow is not symmetric, from the point of view of power consumption, this regime may be the most economical for transporting a certain mass of solids. Wilson (1991) calls all flows below V3 fully stratified flows and all flows above V3 fully suspended flows. The transition from fully stratified to fully suspended flows is considered by this author to be fairly complex and should be represented by sigmoid or ogee curves. It is a transition over a range of the speed and not an abrupt transition at a single value of the speed. The work of Wilson and colleagues will be examined in Section 4-4-5.

4-1-4 Symmetric Flow at High Speed At speeds in excess of 3.3 m/s (10 ft/s), all solids may move in a symmetric pattern (but not necessarily uniformly). Sometimes this flow is called pseudohomogeneous because of its symmetry around the pipe axis. Power consumption is a linear relationship of the static head multiplied by the velocity, but is proportional to the cube of velocity needed to overcome friction losses. Power consumption in pseudohomogeneous mixtures of coarse and fine particles may be excessive for long pipelines. Pseudohomogeneous mixtures of fine or ultrafine particles may occur at speeds as low as 1.52 m/s (5 ft/s). One definition of fine and coarse particles was explained Govier and Aziz (1972), who proposed the following: 앫 Ultrafine particles: dp < 10 m (mesh 1250), where gravitational forces are negligible. 앫 Fine particles: 10 m < dp < 100 m (mesh 1250 < dp < mesh 140), usually carried fully suspended but subject to concentration gradients and gravitational forces. 앫 Medium sized particles: 100 m < dp < 1000 m (mesh 240 < dp < mesh 15), will move with a deposit at the bottom of the pipe and with a concentration gradient. 앫 Coarse particles: 1000 m < dp < 10,000 m (0.039 in < dp < 0.394 in). These are seldom fully suspended and form deposits on the bottom of the pipe. 앫 Ultracoarse particles are larger than 10 mm (0.4 in). These particles are transported as a moving bed on the bottom of the pipe. Considering particle sizes while ignoring their density is meaningless. Practical engineers do shift the boundaries between different sizes based on the density of the particles. There is no question that beads of high-density polyethylene will behave differently than

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sand particles with the same average diameter because the former is lighter than water while the latter is 2.65 times heavier than water.

4-2 HOLD-UP The previous section describes how different layers of solids move with different speeds, from the bottom, coarser particles, to the finer particles at the top of the horizontal pipe. The theory of hold-ups complicates this process, however. Hold-ups are due to velocity slip of layers of particles of larger sizes, particularly in the moving bed flow pattern. Newitt et al. (1962) conducted speed measurements of a slurry mixture in a horizontal pipe. In the case of light Plexiglas pipe, zircon or fine sand did not result in local slip; particles and water moved at the same speed. However, for coarse sand and gravel, they observed asymmetric suspension and a sliding bed. They also observed that in the upper layers of the horizontal pipe, the concentrations of larger particles were the same as for finer solids, but were marked by differences in the magnitude of the discharge rate of the lower layers.

4.3 TRANSITIONAL VELOCITIES The four regimes of flow described in Section 4-1 can be represented by a plot of the pressure gradient versus the average speed of the mixture (Figure 4-5). The transitional velocities are defined as 앫 V1: velocity at or above which the bed in the lower half of the pipe is stationary. In the upper half of the pipe, some solids may move by saltation or suspension.

1.0

Ratio distanc e from bottom of pipe to the inner diameter (y/D) I

0.8 0.6 0.4

C

0.0 3

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v 7 0.0

0.10

4 0.1

0 0

0.05

0.1

0.15

0.20

Discharge solids concentration C y FIGURE 4-3 Distribution of concentration of solids in a pipe versus average volumetric concentration.

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앫 V2: velocity at or above which the mixture flows as an asymmetric mixture with the coarser particles forming a moving bed. 앫 V3 or VD: velocity at or above which all particles move as an asymmetric suspension and below which the solids start to settle and form a moving bed. 앫 V4: velocity at or above which all solids move as a symmetric suspension.

Velocity (ft/se c)

Volumetric Concentration (%)

0

5

10

15

20

direction of flow 30 20 10 0 0

1

2

3

4

5

6

Velocity (m/s ) FIGURE 4-4 Simplified concept of particle distribution in a pipe as a function of volumetric concentration and speed.

1

at w asymmetric flow

V2 V 3

stationary bed

V1

3

V4

symmetric flow

2

er

4

moving bed

Pressure drop per unit of length

slurry

Speed of flow

FIGURE 4-5 Velocity regimes for heterogeneous slurry flows.

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

V3 is effectively the deposition velocity, often called in the past the Durand velocity for uniformly sized coarse particles. It is no longer recommended that it be called the Durand velocity, as tests in the last 20 years have led to new equations that include the effects of particle size and composition of the slurry. The magnitude of the velocity depends on the volumetric concentration (Figure 4-7).

4-3-1 Transitional Velocities V1 and V2 The transitional velocity V1 is obviously not used for the operation of slurry lines. It may be of interest in lab research, instrumentation, and monitor of start-up. The transitional velocity V2 is determined individually from pressure measurements of the pressure gradient. The main focus of the tests is to determine the height of the bed and to derive a stratification ratio. Wilson (1970) developed a model for the incipient motion of granular solids at V2. He assumed a hydrostatic pressure exerted by the solids on the wall and proposed the following equation: 1 ⌬P ᎏ ᎏ L L

– sin cos

s

+ ᎏ 冢sin – ᎏ 冣冥 冢 冣冤 ᎏᎏ D tan 4 Rw i

r

s(S – 1) Cvb(sin – cos )g = ᎏᎏᎏ 2

(4-1)

where (⌬P/L)2 = pressure gradient at 2 = half the angle subtended at the pipe center due to the upper surface of the bed, in radians s = coefficient of static friction of the solid particles against the wall of the pipe Rw = cross-sectional area of the bed divided by the bed width r = angle of repose of the solid particles

100

10

0

0

10

100

d 50

Cumulative passing (%)

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1000

Particle mesh diameter ( m) FIGURE 4-6 Concept of d50 by cumulative passing percentage versus particle size.

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S = ratio of density of solids to density of liquid Cvb = volume fraction solids in the bed (When USCS units are used, express density in slugs/ft3 rather than lbm/ft3). For 0.7 mm (mesh 24) sand with water in a 90 mm (3.5 in) pipe, Wilson measured = 0.35 and concluded that the assumptions of hydrostatic distribution of the granular pressure were correct.

4-3-2 The Transitional Velocity V3 or Speed for Minimum Pressure Gradient The transitional velocity V3 is extremely important because it is the speed at which the pressure gradient is at a minimum. Although there is evidence that solids start to settle at slower speeds in complex mixtures, operators and engineers often referred to transitional velocity as the speed of deposition. Durand and Condolios (1952) derived the following equation for uniformly sized sand and gravel: VD = V3 = FL{2 · g · Di[(s – L)/L]}1/2

(4-2)

where FL = is the Durand factor based on grain size and volume concentration V3 = the critical transition velocity between flow with a stationary bed and a heterogeneous flow Di = pipe inner diameter (in m) g = acceleration due to gravity (9.81 m/s) s = density of solids in a mixture (kg/m3) L = density of liquid carrier The Durand factor FL is typically represented in a graph for single or narrow graded particles, as in Figure 4-7 after the work of Durand (1953). However, since most slurries

Durand Velocity Factor FD

2.0

1.0

For single or narrow graded slurries

CV = 15% CV = 10%

CV = 5%

C = 2% V

Based on Schiller equation using d50

CV = 15% CV = 5%

0 0

1.0

2.0

3.0

Particle diameter (mm)

FIGURE 4-7 Durand velocity factor versus particle size—comparison between the conventional values for single graded slurries and Schiller’s equation using d50 for wide graded slurry.

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

are mixtures of particles of different sizes, this plot is considered too conservative. The Durand velocity factor has been refined by a number of authors. In an effort to represent more diluted concentrations, Wasp et al. (1970) proposed including a ratio between the particle diameter and the pipeline diameter. Wasp proposed the use of a modified factor F⬘ L so that

S – L VD = V3 = FL⬘ 2gDi ᎏ L

冤

冥 冤ᎏ D 冥 1/2

dp

1/6

(4-3)

i

Schiller and Herbich (1991) proposed the following equation for the Durand velocity factor based on the d50 of the particles: )[1 – exp (–6.9 d50)]} FL = {(1.3 × C 0.125 v

(4-4)

where d50 is expressed in mm. Some reference books define a Froude number as Fr = FL · 兹2苶. The particle size d50 is the statistically determined particle size below which half (or 50%) would be equal or smaller to that set size. The following example illustrates the concept of d50. Example 4-1 A sample of slurry is sieved for particle size. The data is collected in the laboratory (see Table 4-1). Plot the data on a logarithmic graph and determine the d50. Solution The data is plotted in Figure 4-6; the d50 is determined to be 145 m. Example 4-2 A slurry mixture has a d50 of 300 m. The slurry is pumped in a 30 in pipe with an ID of 28.28⬙. The volumetric concentration is 0.27. Using Equations 4-4 and 4-2, determine the speed of deposition for a sand–water mixture if the specific gravity of sand is 2.65. Solution in SI Units From Equation 4-4: FL = (1.3 × 0.270.125)(1-exp (–6.9 × 0.3)) FL = 1.1 × 0.8738 FL = 0.964 From Equation 4-2 the deposition velocity is V3 = 0.964 (2 × 9.81 × 28.25 × 0.0254 × 1.65)1/2 V3 = 4.64 m/s

TABLE 4-1 Data for Example 4-1 Particle size (m)

425

300

212

150

106

75

53

45

38

–38

Cumulative passing (%)

97.2

87.1

68.3

51.3

35.9

20.5

14.5

11.8

10.8

—

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Solution in USCS Units From Equation 4-4: FL = (1.3 × 0.270.125)(1 – exp (–6.9 × 0.3)) FL = 1.1 × 0.8738 FL = 0.964 From Equation 4-2 the deposition velocity is V3 = 0.964 (2 × 32.2 × 28.25/12 × 1.65)1/2 V3 = 15.25 ft/sec Various curves have been published for the magnitude of FL. They are often plotted for a single graded size and use difficult to read logarithmic scales. For the sake of accuracy, Table 4-2 tabulates the magnitude of FL between 0.08 mm < d50 < 5mm on the basis

TABLE 4-2 The Coefficient FL Based on Schiller’s Equation Using the d50 of the Particles for Particles Between 0.080 and 5 mm for Volumetric Concentration up to 30%. FL = {(1.3 × C v0.125)[1 – exp(–6.9 d50)]} d50 (mm)

CV = 0.05

CV = 0.10

CV = 0.15

CV = 0.20

CV = 0.25

CV = 0.30

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 1.5 2 2.5 3.0 3.5 4.0 5.0

0.379 0.446 0.503 0.554 0.598 0.636 0.669 0.735 0.781 0.814 0.837 0.854 0.866 0.874 0.880 0.884 0.887 0.889 0.890 0.891 0.892 0.893 0.8939 0.8940 0.8940 0.8940 0.8940 0.8940 0.8940

0.414 0.486 0.549 0.604 0.652 0.693 0.730 0.801 0.852 0.888 0.913 0.931 0.944 0.953 0.959 0.964 0.967 0.969 0.971 0.972 0.973 0.974 0.9748 0.9749 0.9749 0.9749 0.9749 0.9749 0.9749

0.435 0.511 0.577 0.635 0.686 0.729 0.768 0.843 0.896 0.934 0.961 0.980 0.993 1.002 1.009 1.014 1.017 1.020 1.021 1.023 1.023 1.0245 1.0255 1.0255 1.0255 1.0255 1.0255 1.0255 1.0255

0.451 0.530 0.599 0.658 0.711 0.756 0.796 0.874 0.929 0.968 0.996 1.015 1.029 1.039 1.046 1.051 1.055 1.057 1.059 1.060 1.061 1.062 1.063 1.063 1.063 1.063 1.063 1.063 1.063

0.464 0.545 0.616 0.677 0.731 0.777 0.818 0.898 0.955 0.995 1.024 1.044 1.058 1.069 1.076 1.081 1.084 1.087 1.089 1.090 1.091 1.092 1.0931 1.0932 1.0932 1.0932 1.0932 1.0932 1.0932

0.474 0.557 0.630 0.693 0.748 0.795 0.837 0.919 0.977 1.018 1.048 1.068 1.083 1.093 1.101 1.106 1.109 1.112 1.114 1.115 1.116 1.1172 1.1183 1.1184 1.1184 1.1184 1.1184 1.1184 1.1184

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of Schiller’s equation. The magnitude of FL based on d50 is smaller than values published in the literature for single graded slurry mixtures (lab mixtures using a uniform size of particles). A number of authors have confirmed that this is the case (Warman International Inc., 1990). In order to compare the conventional magnitude of FL based on single and narrow graded particles to the Schiller equation, both ranges of FL are plotted in Figure 4-7. With a more complex approach that takes into account the actual viscosity of the slurry mixture and the density of the particles, Gillies et al. (1999) developed an equation for the Froude number F in terms of the Archimedean number (which we will discuss in Section 4-4-5 for stratified coarse flows): 4 Ar = ᎏ d 3 ( – )g 3L2 p L s L

(4-5)

To estimate the deposition velocity V3, Gilles et al. (1999) developed an equation for the Froude number based on the Archimedean number: Fr = aArb

(4-6)

where Fr = FL · 兹2苶 For Ar > 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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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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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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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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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 = Km 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 V2L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 2L兹苶 C苶 D

fDm – fDL ᎏ CV fDL

gDi (s – L) ᎏᎏ V 2L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 CdVL

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)

Average particle size (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 3LV 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

Average particle size (mm)

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

Iterated concentration C/CA

–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)

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

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

Average particle size (mm)

Drag coefficient for a particle with shape factor of 1

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

Iterated concentration C/CA

–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 “ <<<<<<<<<<<<<<<<<<<” IF rt$ = “Y” OR rt$ = “y” THEN PRINT “ <<<<<<<<<<<<<<<<<<<” IF rt$ = “Y” OR rt$ = “y” THEN PRINT “ calculations for another flow rate” IF ch = 1 THEN INPUT “state the flow rate in m3/HR”; qM IF ch = 1 THEN q1m = qM/3600 IF ch = 2 THEN INPUT “state the flow rate in US gpm”; qus IF ch = 2 THEN q1m = qus * 3.785/60000 IF rt$ = “Y” OR rt$ = “y” THEN GOTO 14 INPUT “specific gravity of carrier liquid”; sgl INPUT “specific gravity of solids”; sgs PRINT PRINT “ please choose between input of weight or volume concentration” PRINT “ 1- weight concentration” PRINT “ 2- volume concentration” PRINT 12 INPUT “ 1 or 2”; cwe

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IF cwe = 1 THEN INPUT “weight concentration in percent”; cwin IF cwe = 2 THEN INPUT “volume concentration in percent”; cvin IF cwe = 0 OR cwe > 2 THEN GOTO 12 PRINT IF cwe = 1 THEN cw = cwin/100 IF cwe = 1 THEN sgm = sgl/(1 - cw * (1 - sgl/sgs)) IF cwe = 1 THEN cv = (sgm - sgl)/(sgs - sgl) IF cwe = 1 THEN PRINT USING “specific gravity of mixture = ##.##, cv #.###”; sgm; cv IF cwe = 2 THEN cv = cvin/100 IF cwe = 2 THEN sgm = cv * (sgs - sgl) + sgl IF cwe = 2 THEN cw = cv * sgs/sgm IF cwe = 2 THEN PRINT USING “specific gravity of mixture = ##.##, cw = #.###”; sgm; cw INPUT “hit any key to continue”; jk$ CLS PRINT 22 INPUT “pipe outside diameter in inches”; d0 IF d0 = 0 THEN GOTO 22 INPUT “wall thickness in inches”; tw INPUT “liner thickness in inches”; tl D1 = d0 - 2 * (tw + tl) PRINT “inside pipe diameter in inches”; D1 d1m = D1 * .0254 a1 = pi * d1m ^ 2/4 14 v1 = q1m/a1 v1us = v1/.3048 IF rt$ = “Y” OR rt$ = “y” THEN GOTO 18 INPUT “viscosity of slurry in cPoise or mPa-s”; viscp visc = viscp/1000 18 ReL = sgl * 1000 * d1m * v1/visc PRINT “Reynolds Number of carrier liquid”; ReL IF rt$ = “Y” OR rt$ = “y” THEN GOTO 20 IF ch = 1 THEN INPUT “pipe roughness in meters”; em IF ch = 2 THEN INPUT “pipe roughness in feet”; ef IF ch = 2 THEN em = ef * .3048 edi = em/d1m PRINT “relative roughness”; edi 20 a = (edi/3.7 + 5.7/ReL ^ .9) b = fnlog10(a) fd = .25/b ^ 2 PRINT “darcy factor for carrier liquid”; fd fan = fd/4 dpl = fd * v1 ^ 2 * sgm * 1000/(2 * d1m) slopliq = 2 * fan * v1 ^ 2/(9.81 * d1m) PRINT USING “head loss per length = ####.##### = “; slopliq PRINT USING “press drop due to carrier liquid = #####.## Pa/m”; dpl PRINT Ub = 9.81 * d1m * (sgm/sgl - 1)/v1 ^ 2 PRINT “Ub”; Ub INPUT “hit any key to continue”; jk$ INPUT “hit any key to continue”; jk$ PRINT “ starting from the top size you are asked to input particle size for “ PRINT “ each fraction and its volumetric concentration as part of the solids” FOR i = 1 TO 50 IF i = 1 THEN PRINT “top fraction” PRINT “size”; i

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IF rt$ = “Y” OR rt$ = “y” THEN GOTO 140 95 INPUT “particle size (in microns) and cumulative volume conc.(%)”; dp(i), cvdp(i) 140 IF cvdp(i) > 100 THEN GOTO 95 IF i = 1 THEN GOTO 85 cvind(i) = –cvdo + cvdp(i) dpav(i) = (dpo + dp(i))/2 GOTO 90 85 cvind(i) = cvdp(i) dpav(i) = dp(i) 90 PRINT USING “average particle size = ###### micron,av.volume conc = ###.#### %”; dpav(i); cvind(i) cvdo = cvdp(i) dpo = dp(i) rep(i) = dpav(i) * 10 ^ -3 * v1 * sgm/visc PRINT “particle reynolds number”; rep(i) IF (rep(i) > 2) AND (rep(i) < 500) THEN cd(i) = 18.5 * rep(i) ^ -.6 IF rep(i) < 2 THEN cd(i) = 27 * rep(i) ^ (–.84) IF (rep(i) > = 500) AND (rep(i) < 200000) THEN cd(i) = .44 vt(i) = (4 * 9.81 * dpav(i) * .000001 * (sgs – sgl)/(3 * cd(i))) ^ .5 IF vt(i) >(v1/2) THEN PRINT “warning deposit velocity is higher than half the average flow velocity” PRINT USING “drag coef for av.diam = #.###;terminal velocity = #.#### m/s”; cd(i); vt(i) dfbed(i) = 82 * cvind(i) * cv * (Ub ^ 1.5) * (cd(i) ^ -.75)/100 PRINT “ correction to friction factor”; dfbed(i) ‘PRINT “pressure drop due this fraction”; dpbed(i) hm = hm + dfbed(i) PRINT USING “ TOTAL correction to friction factor = ###.###”; hm IF dp(i) = 0 THEN GOTO 130 IF cvdp(i) = 100 THEN GOTO 130 ‘INPUT “DO YOU WANT TO CONTINUE (Y/N)”; LKJ$ ‘IF LKJ$ = “n” OR LKJ$ = “N” THEN np = i ‘IF LKJ$ = “n” OR LKJ$ = “N” THEN GOTO 130 120 np = i NEXT i 130 fannew = fan * (1 + hm) PRINT “total friction factor for slurry”; fannew pressure = fannew * 2 * sgm * 1000 * v1 ^ 2/d1m slope = pressure/(9810 * sgm) PRINT USING “pressure = #####.## Pa/m”; pressure PRINT “slope of slurry or head per unit length”; slope 150 INPUT “DO YOU WANT TO DO ITERATION BASED ON CONCENTRATION C/CA (y/n)”; HT$ IF HT$ = “N” OR HT$ = “n” THEN GOTO 325 DFANNEW = fannew uf = v1 * SQR(DFANNEW/2) PRINT USING “ FRICTION VELOCITY = ##.### m/s”; uf FOR i = 1 TO np IF i = 1 THEN dhm = 0 z(i) = –1.8 * vt(i)/(.38 * uf) cca(i) = 10 ^ z(i) PRINT “size,av diam and c/ca “; i, dpav(i), cca(i) dpnew(i) = dfbed(i) * cca(i) dhm = dpnew(i) + dhm NEXT i DFANNEW = fan * (dhm + 1) PRINT “REVISED FANNING FACTOR = “; DFANNEW pressureit = DFANNEW * 2 * sgm * 1000 * v1 ^ 2/d1m

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PRINT USING “iterated pressure = #######.## Pa/m = “; pressureit slopeit = pressureit/(9810 * sgm) PRINT “revised slope “; slopeit 325 INPUT “do you want to repeat the calculation for another flow rate (Y/N)”; rt$ IF rt$ = “Y” OR rt$ = “y” THEN GOTO 10 LPRINT “REVISED FANNING FACTOR = “; DFANNEW LPRINT USING “iterated pressure = #######.## Pa/m = “; pressureit LPRINT “revised slope “; slopeit 525 RETURN

When it was developed in the early 1970s, the Wasp method was considered state of the art. It does, however, ignore or minimize one important parameter, namely the shear stress between the different superimposed layers. This is a topic that the two-layer method attempts to tackle, as we shall see in Section 4-10. It does, therefore, tend to predict pressure losses higher than those from stratified flows in certain circumstances of bimodal (fine and coarse) distribution. Nevertheless, the Wasp method remains a very useful method to this day for the design of pipelines, particularly when it is supported by lab tests, as we showed in Example 4-11.

4-7 SALTATION AND BLOCKAGE Most modern engineering specifications for the design of slurry pipelines categorically forbid flow at speeds below V3. However, the instrumentation engineer needs to know the pressure rise in saltation or at blockage. Herbich (1991) argued that motors should be sized to handle the flow in saltation, and there are incidences where it may be economical to reduce the cross-sectional area of the pipe by allowing flow over a stationary bed. 4.7.1 Pressure Drop Due to Saltation Flows In saltation, there is a bed at the bottom part of the horizontal pipe (Figure 4-11) and different approaches are use to evaluate the pressure losses.

Fines in suspension

Saltation bed FIGURE 4-11

Concept of Saltation.

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Newitt et al. (1955) derived the following equation for saltation flow with d50 > 0.025 mm (0.001 in) and for pipes smaller than 25.4 mm (1 in): im – i Z = ᎏ = 66{[s/L – 1]gDi/Vm2} Cvi

(4-39)

Babcock (1968) conducted tests on water suspensions of coarse sand and steel shot. He expressed a nondimensional ratio of the square of the speed to the product of the pipe inner diameter and the acceleration of gravity. His tests indicated that in the range of 5 < (V 2/gDi) < 40 the friction losses could be expressed as gDi s Z = 60.6 ᎏ – 1 ᎏ L Vm2

冢

冣

(4-40)

But for a water suspension of taconite, in the range of 1 < (V2/gDi) < 12: gDi s Z = 66 ᎏ – 1 ᎏ L Vm2

冢

冣

(4-41)

Newitt’s equation (Equation 4-39) is the most commonly used for saltation flows. Example 4-12 The slurry described in Example 4-7 is in saltation at 1.5 m/s. Determine the resultant friction factor. Solution in SI Units 1.5 × 1,000 × 0.5778 Re = ᎏᎏᎏ = 671,860 0.00129 0.25 fd1 = ᎏᎏᎏᎏᎏ = 0.01572 [log10{(0.0002596/3.7) + (5.74/671,8600.9)}]2 Using the Newitt equation (4-39), which was derived for small pipes, would have given

冤

冥

im – iL 1.65 × 9.81 × 0.5778 = 274 Z = ᎏ = 66 ᎏᎏᎏ CviL 1.52 im ᎏ – 1 = 60.35 iL im = 61.35i Calculating the density of the mixture as:

m = Cv(s – L) + L = 0.22 (1,650) + 1,000 m = 1363 kg/m3 or S.G. = 1.363 Using the Newitt approach with fL = 0.01572

冤

冥

0.01572 × 1.52 im = ᎏᎏ 61.35 = 0.1914 m/m 2 × 0.5778 × 9.81

␦P ᎏ = m gim = 1363 · 9.81 · 0.914 = 2560 Pa/m ␦z

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

A more advanced but less known method to estimate the friction gradient for saltation flows is the Graf–Acaroglu method that is presented in Chapter 6, which we will also discuss in the next section with a worked example.

4.7.2 Restarting Pipelines after Shut-Down or Blockage The loss of power, a water hammer situation, or the freezing of a pipeline are occurrences that require adequate understanding of the power and pressure needed to restart a pipeline. The concept of the hydraulic radius is defined in Chapter 6 and is essential to understand blockage. Vallentine (1955) studied the blockage of a mixture of 0.53 mm (No. 1) and 1.05 mm (No. 2) sand in 50 mm (2 inch) and 150 mm (6 inch) pipe. He proposed to write the Darcy equation in terms of a function for blockage: fD · V 2L fD · Q2L fD · Q2L H = ᎏᎏ = ᎏᎏ2 = ᎏ (B) (4 · RH) 2 · g (8 · g · RH) A 8gD 5I

(4-42)

where B is the blocked area of pipe and (B) is a function of blockage. Vallentine proposed that the blocked area B is a function of the total flow, pipe diameter, flow rate of solids, and flow rate of mixture: 1/3 Q 1/2 L Qm ᎏᎏ B = fn ᎏ 1/2 1/3 2.5 D I [s – L) Q s

冤

冥

(4-43)

Herbich (1991) plotted these functions. They are presented in Figures 4-12 and 4-13. In Chapter 6, the work of Graf and Acaroglu is examined in Section 6.5.4. Schiller (1991) proposed to apply their equation (Equation 6.66b) to the problem of flow over a

1.0 0.9

BLOCKAGE B

abul-4.qxd

0.8

No. 1 Nonuniform (Vallentine)

0.7

No. 2

0.6

Uniform Sands (Craven)

''

(

“

)

0.5 0.4 0.3 0.2 0.1 0.0

0

1

2

3 Q ᎏ d2.5

4

5

6

7

8

冣 冪莦 ␥ ␥ 冢 Qs ᎏ ᎏ – Q s w

–1/3

FIGURE 4-12 Blockage factor B. (From Herbich, 1991, reprinted with permission from McGraw-Hill, Inc.)

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4.46

10 % 10 0. 5% 0.0

BLOCKAGE Q (cfs)

9 8 7 6 5

2.75 50% 2.50 % 0 4 2.25 30% 2.00 20% % 1.75 10 5% 1.50 2% 1.25 1.00

4 3 2 1

PIPE DIAMETER d (ft)

% 3.00 60 0. 25 %

Qs ᎏ Q

1.0 0.7 % 5% 0. 5%

CHAPTER FOUR

2.0 %

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Qs ᎏ = Relative Rate of Sediment Q Transport B = Average Pipe Area Blocked

0.75 0.50

0

FIGURE 4-13 Blockage chart. (From Herbich, 1991, reprinted with permission from McGraw-Hill, Inc.)

stationary bed. Schiller established the hydraulic radius of the bed in terms of the ratio of the mean velocity of flow to the deposition velocity V3 as 苶苶·苶g苶·苶 R苶) 苶苶 g苶 V/兹(4 H苶 = V3/兹(D I ·苶) or RH = 0.25 [(V/V3)2]DI Combining these two equations with the Graf–Acaroglu equation (6.64), Schiller proposed to replace the slope by head losses per unit length. Schiller proposed to replace dp by d50 for a mixture of particles of different sizes: {0.25 [(V/V3)2]DI} (s – L)d50] = 10.39 ᎏᎏᎏ CV · V · ᎏᎏ 3 L(h/L) [0.25 [(V/V3)2]DI] 兹[( 苶苶苶 苶L苶–苶苶)g 1苶苶 d 50 苶]苶 s/

冤

冥

–2.52

(4-44)

Example 4-13 Considering that the slurry of Example 4.1 is partially blocked at a speed of 12 ft/sec. Determine the resultant head per unit length to maintain flow: V3 = 15.25 ft/sec Solution in SI Units V = 3.66 m/s DI = 0.718 m The hydraulic radius is therefore [0.25 [(.787)2]0.718] = 0.111 m.

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冦

1.65(3 × 10–4) 0.27 · 3.66 · 0.111 ᎏᎏᎏ = 10.39 ᎏᎏ –4 3 (h/L) 0.111 兹苶 [(苶1苶 .6苶5苶 )9苶 .8苶1苶(苶3苶 ×苶1苶0苶苶 ) 苶]

冧

–2.52

5247 = 10.39[(h/L)2.52 [839090.5] h/L = 0.0527 Solution in USCS Units V = 12 ft/sec DI = 2.357 ft The hydraulic radius is therefore [0.25 [(12/15.25)2]2.357] = 0.365 ft. d50 = 0.000984 ft

冦

1.65(0.000984) 0.27 · 12 · 0.365 ᎏᎏᎏ = 10.39 ᎏᎏ 3 (h/L) 0.365 兹苶 [(苶1苶 .6苶5苶 )3苶2苶 .2苶 (0苶 .0苶0苶0苶9苶8苶4苶 ) 苶]

冧

–2.52

5256 = 10.39[(h/L)2.52 [844444] h/L = 0.0526 Wood 1979 attempted to simplify this complex problem by proposing that the pressure gradient between the incipient motion velocity V2 (when particles start to leave the bed) and the actual deposition velocity V3 (when the particles start to move as a heterogeneous flow) is essentially composed of three components: 1. A component required to overcome the boundary shear stress and turbulence 2. A component required to accelerate the slurry due to changes in mean velocity of the slurry 3. A component required to accelerate the particles that leave the bed The third component is the principle source of increase in the pressure gradient. The particles are assumed to be eroded from the bed at a rate based on flow conditions. If this rate exceeds the capability of the flow to suspend the solids, they tend to fall back into the bed, until the process is restarted. The analysis of Wood is based on open channel flow, which is the topic of Chapter 6. Wood treats the stationary bed by considering its hydraulic diameter, and proceeds to apply a modified Zandi and Govatos approach.

4-8 PSEUDOHOMOGENEOUS OR SYMMETRIC FLOWS Above V4, the flow is pseudohomogeneous or symmetric (but not necessarily uniform). With negligible hold-up, Govier and Aziz (1972) proposed to express the pressure loss in terms of the ratio of the friction fanning factor for the slurry mixture and the liquid mixture or as im – i fNmm – fNLL ᎏ = ᎏᎏ Cv iL fNLL

(4-45)

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CHAPTER FOUR

where fNm = fanning factor of mixtures fNL = fanning factor of liquid at equivalent volume In Chapter 1, the increase in dynamic viscosity due to the volumetric concentration of solids was discussed; it can be expressed as the Einstein–Thomas equation:

m = L[1 + ACv + BC v2 + C exp (DCv)] where A, B, C, and D are constants. Since the fanning friction factor is a function of the Reynolds number, the ratio of the Reynolds numbers of the mixture and liquid would be

mVmDi Rem = ᎏ m LVmDi ReL = ᎏ m The ratio of the Reynolds number helps to establish a ratio of friction factors: Rem L Cv(s – L) ᎏ = ᎏ = {1 + ACv + BC v2 + C exp(DCv)} ᎏᎏ + 1 ReL m L

冢

冣

(4-46)

The next step consists of using the Blasius equation for transition flows: fm ᎏ = {ReL/Rem)0.25 fL

(4-47)

Although Govier and Aziz (1972) did not discuss turbulent flow, their analysis can be extended. For turbulent flows up to Reynolds numbers from 5000 to 100,000,000, which is well outside the range used for mining, the Swamee–Jain equation (Equation 2.19) can be used to obtain the ratio of friction factors: {log10(0.27 /Di + 5.74/ReL0.9}2 fm/fL = ᎏᎏᎏᎏ {log10(0.27 /Di + 5.74/Re m0.9}2

(4-49)

4-9 STRATIFIED FLOWS In Section 4.6, it was clearly indicated that Wasp et al. (1977) extended the Durand and Zandi approach to a concept of multiple and superimposed layers of particles of different sizes and volumetric concentration, with a logarithmic concentration distribution. This method uses the drag coefficient of the particles as an important parameter. Another school of researchers, particularly lead by Shook, Wilson, and Gilles in Canada, focused on refining the original models of Newitt, which were based on the terminal velocity. These authors proposed that the compound mixture of coarse and fine particles can be simplified to what they called “stratified flows,” in which the fines slide over a moving bed of coarse solids. Newitt, as expressed by Equation 4-5, had proposed that the deposition velocity was a mere factor of 17 times the terminal velocity, Wilson (1991) indicated that this approach was not confirmed by tests for large pipes. The concept he proposed was that the flow was going through a gradual transition from a fully stratified to a fully suspended flow, with a gradual change in the pressure gradient.

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

4.49

Defining a parameter in terms of the pressure gradient: im – iL = ᎏᎏ {m/L – 1}

(4-50)

Moreover, plotting versus the speed, as in Figure 4-14, allows one to find a speed V50 halfway between a fully stratified flow and fully suspended flow. The slope on this plot is defined as M. In the region of partially stratified flow, the logarithmic plot confirms that the pressure gradient is a function of the velocity raised to the power (–M). M is calculated from the slope at V50 considered to be the point at which half-stratified flow occurs, as in Figure 4-14. Using V50 as a reference velocity, and considering that the value of at V50 is effectively half the mechanical sliding coefficient p, Wilson (1992) proposed that = fn(V, V50, p, M) Determining V50 is no easy task. In this partially stratified model, it is argued that the lifting of particles by turbulence is strongly influenced by their diameter. Thus, the fines tend to be transported better by the fluid than the coarse solids at the bottom of the pipe. These particles are transported by eddies, and the largest eddy would be equal to the pipe diameter. The pipe diameter was therefore proposed as a reference. The resistance to motion of the solid particles is proposed to be the result of a contact load associated with mechanical sliding friction in the coarser bed and fluid friction associated with the friction velocity of the carrier fluid. In this complex environment, Wilson et al. (1992) defined a new settling velocity as [{s – L}gL]1/3 Vs = 0.9 Vt + 2.7 ᎏᎏ L2/3

冤

冥

(4-51)

fully stratified flow

gradient M

p

m

L

-1)

partially stratified

m

L

log (i - i )/(

abul-4.qxd

Fully suspended flow V 50

Speed

FIGURE 4-14 Concept of V50 for stratified flows.

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4.50

CHAPTER FOUR

The two constants 0.9 and 2.7 were obtained from test data on sand. The velocity V50 is then expressed in terms of this settling velocity: 苶/f 苶dL 苶)苶 cosh (60 dp/DI) V50 = Vs兹(8

(4-52)

where cosh = the hyperbolic trigonometric function fdL = Darcy friction factor for an equal volume of water In tests conducted for sand in water at 20° C, is 0.1 m/s at a particle size of 0.40 mm, indicated that the magnitude of p, the mechanical friction coefficient, was 0.44. From Figure 4-14:

= 0.5 p(V/V50)–M = 0.22(V/V50)–M

(4-53)

For a mixture of particles of different diameters, a settling velocity Vs85 at d85 and Vs50 at d50 are used to calculate a standard deviation s:

冤

Vs85 cosh (60 d85/DI) s = log ᎏᎏᎏ Vs50 cosh (60 d50/DI)

冥

(4-54)

The coefficient M is expressed as M = (0.25 + 13s2)–1/2. The magnitude of Vm/V50 or ratio of mean velocity of the mixture to V50 is defined as the stratification ratio. The friction losses may be expressed in terms of the stratification ratio. When d85/d50 < 2, the slurry is considered to be narrowly graded, and M is set at 1.7. For 2 < d85/d50 < 5, 1.7 > M > 0.4. There is no question that the approach of Wilson et al. (1992) is extremely interesting, but it is more complex than the methods proposed by Equations 4-3 to 4-7. It requires the support of testing, a database, a computer software, and a personal computer.

4.10 TWO-LAYER MODELS Khan and Richardson (1996) explained that Shook and Roco (1991) developed a two-layer model for stratified slurry flows, which may be summarized as follows: 앫 The slurry flow of heterogeneous mixtures is considered to consist of two layers, each with its own velocity of motion and volumetric concentration; but it is assumed that there is no slip between liquid and solid phases. 앫 The solids in the upper layer are fully suspended, are at a volumetric concentration CVu, and move at a velocityVU. 앫 The coarser solids in the lower layer are considered to be packed. However, because of their irregular shape, there is a certain void fraction between the particles. For sand, the lower layer is considered to be at a maximum volumetric concentration CVU of 60% and move at a velocityVB. The total area of flow for the upper layer of fines and the lower layer of coarser particles is A = AB + AU For the mixture, the mass balance VA = VB AB + VU AU For the liquid phase, (1 – CV)VA = (1 – CVU) VU AU + (1 – CVB)VB AB For the solid phase, CVVA = CVBVB AB + CVUVU AU = CVU AV + (CVB – CVU)VB AB

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4.51

Referring to Figure 4-15: AB = 0.25 D 2I ( – sin cos ) AU = 0.25 D 2I ( – + sin cos ) The upper wet perimeter is WPU = DI ( – ) WPB = DI At the interface WPi = DI sin The momentum and force balance is expressed as dP –AU ᎏ = UWPU + iWPi dx

for the upper layer

dP –AB ᎏ = LWPB – iWPi + fc⌺F dx

for the lower layer

dP –A ᎏ = UWPU + LWPB + fc⌺F dx

for the entire pipe

fc is the friction factor due to the Coulombic friction between the particles and the pipe and ⌺F is the total force per unit length exerted normal to the pipe (e.g., weight of the bed per unit length, etc.) At low to moderate concentrations B = 0.5LL fNBV B2. Certain solids are at a concentration CVB – CVU. They are considered to be of a buoyant weight that is supported at the wall as a result of interparticle contacts. Averaged over the entire cross-sectional area of the pipe, these particles define the “contact load” CC, which is calculated as follows: AB CC = [CVB – CVU] ᎏ A

(4-55)

where AB = cross-sectional area of the lower layer A = cross-sectional area of the entire pipe The upper layer volumetric concentration CVU is obtained from the mean in-situ concentration CX and the total contact load CC as CVU = Cr – CC. A parameter CX is defined as the in-situ concentration in the x-direction, as CX = CVB + CVU

(4-56)

The relationship between CC and CX is established as an experimental correlation: CC ᎏ = e–⌫ CX

(4-57a)

⌫ = 0.124 Ar–0.061[(V 2/(gdp))0.28][(dp/Di)–0.431](S/L– 1)–0.272

(4-57b)

where CVB = 0.60. And for sand slurries, according to SAC (2000): ⌫ = –0.122 Ar–0.12[(V/VD)0.30][(dp/Di)–0.51](S/L– 1)–0.255 The Archimedean number Ar is defined in Equation 4-5.

(4-57c)

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CHAPTER FOUR

The density of the suspended concentration CVU and the density of the carrier liquid L are then used to compute the shear stresses in the two layers (Figure 4-15). A fanning friction factor is then obtained for each layer based on some average velocity of the flow in the pipe, density of the carrier liquid, and dynamic viscosity of the carrier liquid, and using the relative roughness of the pipe /Di. The Churchill (1977) equation is then used to calculate the friction factor as fn = 2[8Re–12 + (A + B)–1.5)1/12

(4-58)

where A = [–2.457 ln[(7/Re)0.9 + 0.27 /Di)]16 B = (37530/Re)16 The Reynolds number is calculated on the basis of the average pipe flow speed, since the speed of flow in each layer is not known at this point. An equivalent “sand roughness” is then defined as the ratio of particle size to pipe diameter (dp/DI). At the interface between the bottom and top layer, a friction factor is derived by modifying the Colebrook equation (Equation 2-17):

WPU WPUB AU DI

2

WPB AB

VB

FIGURE 4-15

VU Speed

above bottom quadrant

Vertical distance (y)

above bottom quadrant

Vertical distance (y)

typical real distribution

C VB C VU Solids volumetric concentration

Two-layer modeling of coarse and fine particle mixtures.

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

2(1 + Y) fNi = ᎏᎏᎏ2 [4 log10(dp/Di) + 3.36]

(4-59)

where Y=0 for dp/Di < 0.0015 Y = 4 + 1.42 log10 (dp/Di) for 0.0015 < dp/Di < 0.15 and Ar (the Archimedean number) < 3 × 105 For bimodal mixtures (mixtures of fine and coarse particles), the Saskatchewan Research Council (SRC) (2000) suggested that the effects of the fines on the viscosity of the carrier liquid should be used to calculate the apparent viscosity f. However, the actual density of the liquid should be used without corrections for volumetric concentration of fines. Khan and Richardson (1996) pointed out that the concept of an equivalent sand roughness is very speculative as the transition between one layer and the other. By subsequent iterations, the friction factor and velocity for each of the two layers are obtained from the shear stress as the product of the friction factor and the dynamic pressure:

U = 0.5fNUUVU2 B = 0.5fNBBV B2 fNBBV B2WPB D 2i gfc(s – L)(CVL – CVU)(1 – CVB)(sin – cos ) BWPB = ᎏᎏ + ᎏᎏᎏᎏᎏᎏ (4-60) 2(1 + CVU – CVB) 2 where fc is the coefficient of kinematic friction between the particles and the pipe. The total normal force per unit length was defined by Shook as D 2i g(s – L)(sin – cos ) (CVB – CVU) (1 – CVB) ⌺F = ᎏᎏᎏ ᎏᎏᎏ 1 – (CVB – CVU) 2

冦

冧

(4-61)

where is the half angle formed by the bottom layer with respect to the center of the pipe. To appreciate the complexity of this approach, SRC (2000) indicated that this approach yielded six unknowns (VB, VU, , Cr, Cc, and –dP/dx). The numbers of iterations that are required are better dealt with on a computer. The two-layer models have gained wide acceptance in the oil–sand industry. The following example is an illustration at the first level of iteration. Example 4-14 Sand slurry in a pipe is flowing at 6.5 m/s (21.3 ft/sec). The pipe diameter is 717 mm (28.35⬙) pipe and the sand particle diameter dp = 360 m (0.0142⬙).The volumetric concentration was presented to be 0.27. Upon review of the composition of the sand, it was noticed that 15% of the solids were fines smaller than 74 m. If the lower bed is packed at 60%, the contact load Cr = 0.30, and the Columbian friction factor fc is 0.50, determine the pressure gradient (assume water dynamic viscosity 1 cP, and sand S.G. 2.65). Volumetric concentration in the upper layer consists essentially of fines: CVU = 0.15 · 0.27 = 0.0405 CVB = 0.85 · 0.27 = 0.23 By the Einstein–Thomas equation, the dynamic viscosity of the carrier liquid needs to be corrected for a concentration of 0.0405 in the upper layer:

= L(1 + (2.5 · 0.0405) + (10.05 · 0.04052) + 0.00273 exp (16.6 · 0.0405)] = 1.123 L = 1.123cP

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CHAPTER FOUR

4 × 9.81 (3.6 × 10–4)3 × 1000 (1650) Ar = ᎏᎏᎏᎏ = 666 3 (1.123 × 10–3)2 ⌫ = 0.124 Ar–0.061 [(V2/(gdp))0.28][(dp/Di)–0.431](S/L – 1)–0.272 = 0.124 (666–0.061) (6.52/(9.81 · 3.6 × 10–4)0.28][0.0005–0.431]1.65–0.272 = 395.52 CC ᎏ = e–395.52 = 0 CX Density of the liquid and fines in the upper layer is U = .0401 · 2650 + (1 – .0401) · 1000 = 1066 kg/m3 If it is assumed that, due to the void fractions, the lower layer is full at 60%, and since the volumetric concentration is 0.23, then the area used is 0.23/0.6 = 0.383. area = 2[0.25D I2( – 0.5 sin cos )] = 0.383 × 0.25DI2

⬇ 80 degrees ⬇ 0.444 AB = 0.25

D I2(

– sin cos ) = 0.25 × 0.7172x(1.396 – 0.171) = 0.1574 m2

AU = 0.25 D I2( – + sin cos ) = 0.25 × 0.7172x(0.556 + 0.171) = 0.246 m2 WPU = DI( – ) = 1.252 m WPB = DI = 1 m The total normal force per unit length was defined by Shook as

冦

0.7172 9.81 (2650 – 1066)(0.985 – 1.39 × 0.1736) (0.23 – 0.0405) (1 – 0.23) ⌺F = ᎏᎏᎏᎏᎏᎏ ᎏᎏᎏ 2 1 – (0.23 – 0.0405) = 632 N/m

冧

fNBBV B2WPB BWPB = ᎏᎏ + fc ⌺F 2 Since WPB = 1 m, 2 (1) fNL 1066V LL B(1) = ᎏᎏ + 0.5 · 632 = 533 fNBV B2 + 316 2

dp/DI = 0.00035/0.717 = 0.000488, so Y = 0 fNI = 2/[4 log10(0.717/0.00035) +3.36]2 = 0.00725 To determine the friction factors for the upper and lower layers it is required to determine the speed in each layer. To obtain the difference between the velocity in the upper and lower layers, it is essential to obtain the shear stress i between the two layers, or to make some assumptions and to proceed with further iterations. In a first iteration, it shall be assumed that VU = 1.1 VLL. AV = ABVB + AUVU 0.25 · · 0.717 · 6.5 = 0.1574 m2(VL)+ 0.246 m2(1.1VL) 2

VB = 6.14 m/s VU = 6.75 m/s

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HETEROGENEOUS FLOWS OF SETTLING SLURRIES

The following calculations require the reader to review some of the concepts of Chapter 6. The hydraulic diameter (Equation 6-2) for the upper layer is 4AU/WPU = 4 × 0.246/1.252 = 0.786 m. The Reynolds Number for the upper layer is ReU = 1066 · 6.75 · 0.786/(1.123 · 10–3) = 5,036,209. The pipe is rubber coated to a roughness of 0.00015 m, /Di = 0.0002. Applying Churchill’s equation to the upper layer: A = {–2.457 ln[(7/Re)0.9 + 0.27 /Di)}16 = 1.192 · 1022 B = (37530/Re)16 = 9.045 · 10–35 fnu = 2[8Re–12 + (A + B)–1.5)1/12 = 2[3 · 10–80 + 7.684 · 10–34]1/12 = 0.0035

U = 0.5fNU UV U2 = 0.5 · 0.0035 · 1066 · 6.752 = 85 Pa For the lower layer the hydraulic diameter is 4ABWPB = 0.6296 m. The Reynolds number for the lower layer is ReU = 1066 · 6.14 · 0.6296/(1.123 · 10–3) = 3,669,531. The pipe is rubber coated to a roughness of 0.00015 m, /Di = 0.000238. Applying Churchill’s equation to the upper layer: A = {–2.457 ln[(7/Re)0.9 + 0.27 /Di)]}16 = 1.006 · 1021 B = (37530/Re)16 = 1.433 · 10–32 fnu = 2[8Re–12 + (A + B)–1.5)1/12 = 2[1.34 · 10–78 + 3.134 · 10–32]1/12 = 0.0047

B = 0.5fNBBV B2 = 0.5 · 0.0047 · 1066 · 6.142 = 94 Pa At the interface i, 0.5fNiU (V U2 – V B2) = 0.5 · 0.0035 · 1066 · (6.742 – 6.142) = 14.68 Pa. dP –AU ᎏ = UWPU + iWPi = 85 · 1.252 + 14.68 · 14.68 · 0.717 · 0.985 = 117 Pa/m dx Since AU = 0.246 m2, then dP/dx = 475.6 Pa/m. Equation 4-60 is not applicable at high concentration of fines, as the slurry starts to behave as a non-Newtonian mixture. For particles with d50 finer than 74 m, the method does not give very reliable results. For flows above a deposited bed (flows with saltation), SRC (2000) proposed to treat the upper layer as a noncircular flow. This means that the hydraulic diameter must be determined from the wetted area. The difference in roughness between these two surfaces (upper surface of the bed) and pipe roughness is not well discussed. The hydraulic diameter is calculated as DHB = 4AB/(WPB). It is considered that for pipe flows, the deposit velocity is a function of the pipe diameter raised to the power of 0.4 for noncircular channels and the friction loss gradient is a function of the ratio V2/DH. By defining VU as the velocity of the upper layer and V3 the critical velocity at which the bed deposits, the following equation is established: 0.4 VU DHB ᎏ=ᎏ 0.4 V3 D

(4-62)

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CHAPTER FOUR

d

WP

U

WP

UL

AU

Deposited solids

Friction loss per unit length

deposited bed

upper layer

i1

Flow rate FIGURE 4-16

Flow above a deposited bed—two-layer model.

The ratio of friction gradients is D im V U2 D ᎏ = ᎏ2 ᎏ = ᎏ i3 V 3 DHB DHB

冢

冣

0.2

(4-63)

where i1 is pressure gradient at V1. At deposition, the flow rate in the upper layer QU is lower than the flow rate through the deposited bed QB, and from the ratio of velocities as per Equation 4-57: QU = AUVU Q3 = AV3 QU DHB ᎏ= ᎏ QB D

冢

冣

0.2

AU ᎏ A

(4-64)

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4.57

4.11 VERTICAL FLOW OF COARSE PARTICLES Newitt et al. (1961) conducted tests for flows of solids in vertical pipes. For fine solids they derived the following empirical equation: Di

S

p

L

冢 冣 冢 ᎏd 冣冢 ᎏ 冣

im – iL gDi ᎏ = 0.0037 ᎏ iLCv V2

1/2

2

(4-65)

In a vertical flow, it would not be possible to develop dunes, a bed, or saltation. There is no concentration gradient and the flow may be treated as pseudohomogeneous for friction loss calculations, as discussed in Section 4-4-3. Since the flow in a vertical pipe is pseudohomogeneous, a simple instrument to measure flow rate of slurry is the inverted U column, which consists of 4 elbows and 2 vertical pipe spools (Figure 4-17). The first vertical branch must be sufficiently high to eliminate entrance effects (previously discussed in Chapter 2). Toward the top of the pipe, a pressure tap measures the static pressure. Pressure loss occurs through the two elbows. Another pressure tap is added on the downward section of the pipe. The inverted U column is calibrated on water and the pressure loss is a function of the flow rate as well as the density of the slurry:

V 2 ⌬P = K ᎏ 2 VmD 2i Q = ᎏ Cd 4 where Cd is the discharge coefficient, or 苶苶 ⌬P 苶苶 /m 苶)苶 Q = CdDi 兹(2

The density of the mixture may be calculated from the input data or measured using a nuclear radiation density gage.

2 3 zA zB 1 4

flow

FIGURE 4-17

Inverted U tube piping for measuring flow of a slurry mixture.

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In a vertical flow, in addition to the friction losses as discussed for a pseudohomogeneous flow, there is a hydrostatic pressure gradient, so that the total pressure drop is: ⌬P = mg[⌬z + fmV m2 ⌬z/(2gDi)] ⌬P = pressure loss between two points ⌬z = height difference between two points For further reading, see the work of Einstein and Graf (1966) who described such a flow and a concentration meter for water–sand mixtures.

4-12 INCLINED HETEROGENEOUS FLOWS Between the two types of horizontal and vertical flows that we discussed in this chapter, there is a range of inclined flows that are important but are not the subject of extensive studies. In fact, it may be argued that in many plants, most flows are either horizontal or vertical, with elbows and fittings between them. An understanding of inclined heterogeneous flows is essential for certain long overland pipelines, thickener feed systems, pumpbox feed systems, and, in some respects, to shed light on open channel flows. Until very recently, attention was focused on uniformly graded slurries. Worster and Denny (1955) indicated that if the pipe is inclined from the horizontal, the fraction loss is i␥ = i + (im – i) cos ␥

(4.65)

where i␥ = pressure gradient at ␥ im = pressure gradient of the mixture in the horizontal pipe This equation suggests that the pressure loss is lower in an inclined pipe than in a horizontal pipe. It also suggests that the pressure gradient is the same upward or downward. The experimental work of Kao and Hawang (1979) indicates that this is not correct. In fact, they noticed that the friction losses for upflows increased up to a certain magnitude of the angle of inclination and then decreased. In the case of downflows, they measured a reduction of friction losses from the values of the horizontal pipe. Wilson et al. (1992) have discussed the effect of pipe inclination on their V50, and suggest that Worster and Denny’s equation be modified by using (cos ␥)1.85 instead of cos ␥. They also published data on certain particles with a diameter between 1 mm and 6 mm. The test data indicated that the Durand factor for deposition velocity FL (see Equation 4-2) increased up to an angle of inclination of 30 degrees and by as much as 38%. They also noticed that the deposition velocity V3 increased by 50% at 30 degrees pipe inclination, but then they noticed a drop at an angle of 40 degrees. They did not conduct further tests. Interestingly, they noticed a reduction of the Durand factor FL by 0.3 at a negative inclination of 20 degrees. There is a dearth of information on flow in inclined pipes, and as overconservative as it may be, Worster and Denny’s equation continues to be used. This approach should change, particularly when the angle of inclination is less than 30 degrees or up to –20 degrees.

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4.12.1 Critical Slope of Inclined Pipes Pipeline have to go through dunes and hills. Certain slurry pipelines may follow the topography (Figure 4-18) but others require pipe bridges to avoid sedimentation of the slurry after a shutdown or power failure (Figure 4-19). A question often raised in designing a pipeline is determining the critical slope of the pipe to avoid areas of blockage once the flow is shut down. To avoid blockage of the line, it is necessary to eliminate areas of steep slopes. A commonly used design restriction of 10–16% (5.7–9°) is often adopted when there is no knowledge of the critical slope. The idea is that the slurry would settle without segregation to a “soft” consistency and not migrate down to the steepest slope in the pipeline during shutdown. Kao and Hwang (1979) criticized this approach as being extremely stringent because it adds to construction costs and capital expenses. They implied that it would be better to properly understand the critical slope rather than to use a rule of thumb. Shook et al. (1974) measured the maximum inclination of a 50 mm (2 in) ID Perspex pipe on a sand–water mixture. They reported that: 앫 The maximum rising angle is 14°, or slope of 24%, before the solids start to slide back. 앫 The sliding bed was at the interface of the solid bed and the pipe wall rather than within the settling bed.

FIGURE 4-18 Tailings pipelines follow the slope of the hills and use soil friction produced by partial burial as an anchor.

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FIGURE 4-19 Pipeline carrying taconite tailings with an important portion of coarse particles required pipe bridges to avoid blockage after shutdown or power failure.

앫 The critical angles for the sand bed increases from 22–26°(slope of 40–48.7%) with the decrease of the size of particles. The critical angle is neither affected by the ratio dp/DI nor by the concentration of the solids. Durand and Gilbert (1960) derived the following equation for inclined pipe: im␥ = iL + iB(cos ␥) where im = energy gradient for mixture iL = energy gradient for liquid iB = energy gradient for solid bed Kao and Hwang (1979) observed that the critical slope for glass beads and for sand occurred at 23°(42% slope) from the horizontal. For other substances such as coal and walnut shells, the initial motion appeared to occur at the interface between the particle bed and the pipe wall. This suggested that the internal friction between irregularly shaped coarse particles was higher than the friction at the wall of the pipe. The critical slope Kao and Hwang (1979) defined was the value for initial particle motion. For sand or glass beads it was 27° ± 2°, and for coal and walnut shells it was 37° ± 2°. Craven and Ambrose (1953) investigated the effect of tube inclination on the head loss for a pipe partially blocked with sediments and for a pipe flowing full. They found that at

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a given average speed, an adverse slope in excess of 10% increased the pressure losses by 25% or more by comparison with a horizontal pipe.

4-12-2 Two-Layer Model for Inclined Flows Matsouk (1996) developed a two-layer model for inclined pipes. His tests were conducted in a 150 mm (6⬙) pipe at angles of inclination between –35 and +35 degrees. The fundamental equations for this model for the upper layer are

UWPU + UBWPUB d(P – Ugh) ᎏᎏ = ᎏᎏ AU dx

(4-67)

For the lower layer they are

BWPB – iWPi + fc⌺FN cos ␥ + FW sin ␥ d(P – Bgh) ᎏᎏ = ᎏᎏᎏᎏᎏ AL dx

(4-68)

where FW is the submerged weight of the sediments in the lower layer. The force balance for the whole pipe is then

BWPB + UWPU + fc⌺FN cos ␥ + FW sin ␥ d(P – Ugh) ᎏᎏ = ᎏᎏᎏᎏᎏ dx A

(4-69)

Equations 4-67 to 4-69 are then solved in a similar manner as presented in Section 4-10. Matsouk pointed out that his approach was different than the models of Shook and Roco (1991) (the SRC model), Lazarus (1989), and Lazarus and Cook (1993), which did not include the pipe axis component of the submerged weight (due to buoyancy) FW sin ␥. The tests of Matsouk did not confirm the Worster and Denny equation. At pipe inclinations close to the angle of internal friction of the transported solids, the behavior of the solids was different for inclining and descending pipes under the same speed and volumetric concentration. The difference was larger with coarse than with fine solids. Matsouk con-

x

P2 submerged lower layer of coarse solids

z

P

1

L

P+ g z FIGURE 4-20

Concept of the two-layer bipolar flow of slurry at an angle of inclination.

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cluded that the deformation of the lower layer was essentially due to the submerged weight of the solids at the bottom of the pipe.

4-13 CONCLUSION In this chapter, we have shown that two principal schools of heterogeneous slurry flows have developed over the last 50 years—one around the Durand–Condolios approach and the other around the Newitt approach. The former evolved gradually until Wasp modified it for multilayer compound systems. The latter gradually evolved to yield the two-layer model. There is no consensus as to which model to use. Some have argued that the Wasp method was more suitable for coal, whereas the two-layer model was better for sand. This is based on the number of papers published on two-layer models for sand slurry mixtures, emanating from the great interest in Canadian oil sands. The science of slurry flows is continuously evolving and needs to be understood in the context of its evolution. It would be a mistake to favor one school of thought over another. Many successful pipelines have been designed using either model. The slurry engineer should appreciate that these models are effective tools to be used with correct empirical coefficients obtained by experimental testing of samples in pumping loops. After reading this chapter, some readers may get the feeling that designing a slurry flow is a combination of science and art. Slurry dynamics may appear to be an exercise of examining each mixture for its properties, much as the physician must examine each patient before administering the cure. The flow of coarse particles in water or mixtures of coarse and fine solids in a liquid is complex. When data is not well accumulated, it is recommended to conduct slurry tests in a pump test loop. The mixture of coarse and fine particles can lead to a concentration gradient of solids in horizontal pipe. The coarse particles tend to flow in the lower layers, whereas the fines flow in the upper layers. Certain authors recommend determining the pressure losses of fine solids separately from coarse solids at the corresponding volumetric concentration and particle diameter of each size. Methods based on concentration ratio for each layer have been developed. A process of iteration is needed to achieve a final estimation of the pressure loss due to the bed. New models for inclined flows are appearing, such as the work of Matsouk (1996) on inclined two-layer models The correct design of inclined flow must be based on empirical data on the critical slope. The principles reviewed in this chapter apply to dredging and transporting sand, gravel, coal, steel shot, and rocks from SAG, rod, or ball mills, cyclone underflows, and tailings, etc., which often have particle sizes larger than 70 m (mesh 200). The practical engineer needs to appreciate the limitations of each method and that models have often been developed for a certain range of particle sizes based on experimental data. It is wise to check the original data. Because the new methods of stratified flows or two-layer models use the actual hydraulic diameter of the bed, whereas the Wasp and Durand methods use the actual pipe diameter, it is easy to get confused. In fact, some of the proponents of the two-layer models leave the impression that Wasp and Durand are using the “wrong pipe diameter.” This is not the approach to take. It is wiser to recognize that the Wasp and Durand methods are useful tools for the range of slurries for which they were developed. This includes concentrations of coarse particles up to a volumetric fraction of 20%. This covers, in fact, most dredged gravels and sands, coal in a certain range of sizes, as well as crushed rocks. It is also important to appreciate that the work of Zandi and Govatos (1967) was based on

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sands up to a volumetric concentration up to 22%, and it would be erroneous to push the envelope of application of their equations beyond such a range. In the last thirty years, considerable progress has been made with stratified flows. The new two-layer models push the envelope of understanding beyond the limits of the models of Durand, Zandi, and Wasp into the range of volumetric concentrations of 30%. But these models have their limitations too. Certainly it would be unwise to use the two-layer model when the d50 is smaller than 74 micrometers. There is still a considerable amount of experimental data associated with these stratified models needed to obtain the Conlombic friction factor and to determine the difference in velocity between the upper and lower layers. When the particles are not too coarse, such a difference is not too large, and approximations such as those proposed by Richardson and Khan are justified, but when the d50 is around 400 micrometers or higher, the difference in velocity and shear between the upper and lower layers become important. In Chapter 6, the analysis presented in this chapter will be extended to include open channel flows. The Acaroglu–Graf equation presented in Chapter 6 was applied here to flows with saltation.

4-14 NOMENCLATURE a A AB Ap Ar AU b B C CA CC CD CE Ct Cv Cvb Cv bed Cvi CVL CVU Cw CX C1 C3 dg dp d85 d50 DH Di dsp

Height of layer A above bottom of conduit Cross-sectional area of the entire pipe Cross-sectional area of the lower layer surface area of particle The Archimedean number Area of upper layer of flow in the two-layer model Factor used to calculate the Archimedean number blocked area of pipe volumetric concentration of the particle diameter under consideration Concentration of solid particles at a reference plane A (usually at 0.08 DI) Contact load in the Shook–Roco two-layer model Drag coefficient Coefficient of discharge In-situ concentration Concentration of solids by volume Volume fraction of solids in the bed Concentration of solids in the moving bed Concentration of solids in the moving bed of fraction i Concentration of solids by volume in the lower layer in the Shook–Rocco model Concentration of solids by volume in the upper layer in the Shook–Rocco model Weight concentration In-situ concentration in the two-layer model Constant Constant Diameter of spherical particle Average diameter of the particle Sieve passage diameter for 85% of the particles Sieve passage diameter for 50% of the particles Hydraulic diameter of a noncircular flow Inner diameter of pipe diameter of equivalent sphere using the sphericity factor

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Es fc fD fDL FL fN fNL fNm fNB fNU Fr g i im i i1 i3 K Ke Kf Kx K2 K3 L Ne m mi mt M P PW Q QB QU Ri r Re Rem ReL RH R1 s Sf Sm U Uf Ufc Uf 0 V VB VD

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The mass transfer coefficient coefficient of kinematic friction between particles and pipe Darcy–Weisbach friction factor Darcy friction factor for equivalent volume of water Durand–Condolios coefficient to determine the deposition velocity Fanning friction factor Fanning friction factor for equivalent volume of water Fanning friction factor for mixture Fanning friction factor for the bottom layer in the two-layer model Fanning friction for the top layer in the two-layer model Froude number Acceleration of falling objects due to gravity (9.78–9.82 m/s2) Equivalent pressure gradient of water at the same volume as the slurry Energy gradient of slurry mixture in equivalent m of water per m of pipe length Pressure gradient for pipe at inclination Pressure gradient at V1 Pressure gradient at V3 Coefficient or constant Experimental factor for the pressure gradient Coefficient in the Durand equation for pressure drop Von Karman constant An experimentally determined constant Coefficient proportional to the mechanical friction factor ␥ Length of conduit Index number Power coefficient in Zandi’s models The mass fraction of solids with particle diameter of dp Total mass of particles Slope of the log scale of pressure gradient versus velocity of a stratified flow Pressure loss Wetted perimeter Flow rate Flow rate in the lower layer of the two-layer model Flow rate in the upper layer of the two-layer model Inner diameter of a pipe Local radius for a point in the flow Reynolds number Reynolds number of the mixture Reynolds number of the liquid carrier Hydraulic radius Cross-sectional area of the bed divided by the bed width Specific gravity of the solids Specific gravity of liquid Specific gravity of slurry mixture Average velocity Friction velocity Critical friction velocity at which the solids start depositing Friction velocity at deposition for limiting case of infinite dilution Velocity Velocity in the bottom layer Deposition velocity (also called V3)

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Vf VL Vm Vmin VR Vs Vt VU V1 V2 V3 V4 V50 WP y Z

4.65

Velocity of carrier fluid Velocity of the lower layer in the two-layer model of Shook and Roco Mean velocity of a mixture Velocity at minimum pressure drop Ratio of solid volume concentration of solids to liquid concentration Settling velocity in the stratified flow model Terminal velocity of the solid particle Velocity of the upper layer in the two-layer model of Shook and Roco Velocity for which flow with a stationary bed is started Velocity for which flow with a moving bed is started Deposition velocity or velocity above which solids start to move Velocity above which all solids move as a pseudohomogeneous mixture Velocity at 50% stratification Wetted perimeter Distance from the lower boundary (pg 33) Nondimensional parameter to express difference between friction losses due to the slurry and an equivalent volume of water

Subscripts B Bottom layer bed Due to the moving bed i Fraction i m Mixture U Upper layer Greek letters  Constant of proportionality Roughness Angle from the vertical starting at the lowest quadrant point ␣ The angle from the horizontal Pressure loss factor r The angle of repose of solid particles L Density of liquid carrier m Density of the slurry mixture s Density of solid sediments U Density of suspended fines and carrier liquid in the upper layer of the two-layer model W Density of water B Shear stress for the lower layer in the two-layer model U Shear stress for the upper layer in the two-layer model ⌫ factor used to compute ratio of concentrations in the two-layer model Factor to determine speed at 50% stratification in the Wilson model L Dynamic viscosity of the carrier liquid m Dynamic viscosity of the slurry mixture p Mechanical friction coefficient s Granular stress of the solid particles Kinematic viscosity of water L Viscosity of carrier liquid at equal volume M Viscosity of slurry mixture Product of the index number and the volumetric concentration s Coefficient of static friction of the solid particles against the wall of the pipe

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4-15 REFERENCES Acaroglu, E. R. 1968. Sediment Transport in Conveyance Systems. Ph.D. dissertation, Cornell University. Acaroglu, E. R., and W. Graf. 1968. Designing conveyance systems for solid–liquid flows. Paper presented at the International Symposium on Solid–Liquid Flow in Pipes, March 3–7, University of Pennsylvania, Philadelphia. Babcock, H. A. 1967. Head losses in pipeline transportation of solids. Paper presented at First World Dredging Conference, WODCON I, the Netherlands. Babcock, H. A. 1968. Heterogeneous flow of heterogeneous solids. Paper presented at International Symposium on Solid Liquid Flow in Pipes, March 3–7, University of Pennsylvania, Philadelphia. Babcock, H. A. 1971. Heterogeneous flow of heterogeneous solids. In I. Zandi (Ed.), Advances in Solid–Liquid Flows in Pipes and its Applications. pp. 125–148. Oxford: Pergamon Press. Bagnold, R. A. 1954. Gravity-free dispersion of large spheres in a Newtonian fluid under shear. Proc. Royal Soc. A, 225, 49–63. Bagnold, R. A. 1955. Some flume experiments on large grains but little denser than the transporting fluid, and their implications. Proc. Inst. Civ. Eng., 4, 3, 174–205. Bagnold, R. A. 1957. The flow of cohesionless grains in fluids. Phil. Trans. Roy. Soc., 249, 235–297. Blatch, N. S. 1906. Water filtration at Washington, DC, discussion trans. Amer. Soc. Civ. Eng. 57, 400–408. Charles, M. E., and G. S. Stevens. 1972. The pipeline flow of slurries—transitional velocities. Paper presented at the Second International Conference on Hydraulic Transport of Solids and Pipes. Second conference of the British Hydromechanic Research Association. Cranfield, England. Churchill, S. W. 1977. Friction factor equation spans all fluid flow regimes. Chemical Engineering 84, no. 7: 91–92. Craven, J. P., and H. H. Ambrose. 1953. The transportation of sand in pipes. Engineering Bulletin (University of Iowa), 34, 67–88. Durand R. 1953. Basic Relationship of the transportation of solids in experimental research. Proc. of the International Association for Hydraulic Research—University of Minnesota, September 1953. Durand, R., and E. Condolios. 1952. Experimental investigation of the transport of solids in pipes. Paper presented at Deuxieme Journée de l’hydraulique, Societé Hydrotechnique de France. Durand, R., and R. Gilbert. 1960. Transport hydraulique et refoulement des mixtures en conduites. Transactions École des Ponts et Chaussees, 130, 3–4. Einstein, H. A., and W. H. Graf. 1966. Loop systems for measuring sand–water mixtures. Journal of Hydraulic Division, Am. Soc. Civ. Eng. 92, HY1, paper 4608, 1–12. Ellis, H. S., and G. F. Round. 1963. Laboratory studies on the flow of nickel–water suspensions. Canadian Mining and Metallurgical Bulletin, 56, 773–781. Faddick R. R. 1982. Ship Loading Coarse Coal Slurries. In The 8th International Conference on Hydraulic Transport of Solids in Pipes, Johannesburg, South Africa. Cranfield, UK: BHRA Group. Gaessler, H. 1967. Experimentelle und Theoretische Untersuchungen uber die Stromungsvorgange Beim Transport von Festoffen in Flassigkeiten durch Horizontale Rohrleitungen. Doctoral thesis. Technische Hochschule, Karlsruhe, Germany. Quoted in G. W. Govier and K. Aziz, The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold Co., 1972, pp. 668 –670. Geller, L. B, and W. M. Gray. 1986. Selected Theoretical Studies Made in Conjunction with the Joint Canada/FRG Research Project on Coarse Slurry, Short Distance Pipeline. CANMET SP 8616 E–Government of Canada Publications Gillies, R. G. J. Schaan, R. J. Sumner, M. J. McKibben, and C. A. Shook. 1999. Deposition velocities for Newtonian slurries in turbulent flows. Paper presented at the Engineering Foundation Conference, Oahu, HI. Submitted for publication in the Canadian J. Chem. Eng. Reference cited by Saskatchewan Research Council (2000). Slurry pipeline course handout. Govier, G. W., and K. Aziz. 1972. The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold.

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Hayden, J. W., and T. E. Stelson. 1968. Hydraulic conveyance of solids in pipes. Paper presented at the International Symposium on Solid–Liquid Flow in Pipes, March 3–7, University of Pennsylvania, Philadelphia. Herbich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill. Hill, R. A., P. E. Snock, and R. L. Gandhi. 1986. Hydraulic transport of solids. In The Pump Handbook, 2nd ed. Edited by I. J. Karassik et al. New York: McGraw-Hill. Hsu, S. T., A. V. Beken, L. Landweber, and J. F. Kennedy. 1971. The distribution of suspended sediment in turbulent flows in circular pipes. Paper presented at the American Institute of Chemical Engeering Conference on Solids Transport in Slurries, Atlantic City, NJ. Hunt, I. N. 1969. Turbulent transport of heterogeneous sediment. Quarterly Journal Mechanics and App. Maths., 22, 234–246. Ippen, A. T. 1971. A new look at sedimentation in turbulent streams. Journal of Boston Soc. Civil Eng., 58, 3, 131–163. Ismail, H. M. 1952. Turbulent transfer mechanism and suspended sediment in closed channels. Trans. Amer. Soc. Chem. Eng., 117, 2500, 409–447. Kao, D. T. Y., and L. Y. Hwang. 1979. Critical slope for slurry pipelines. Paper presented at the Hydrotransport 6 Conference of the British Hydromechanical Research Association, Cranfield, England. Khan, A. R., and J. F. Richardson. 1996. Comparison of coarse slurry pipeline models. In Proceedings of Hydrotransport 13, pp. 259–281. Cranfield, UK: BHR Group. Lazarus, J. H. 1989. Mixed-regime slurries in pipelines. I. Mechanistic model. Journal of Hydraulic Engineering, ASCE, 115, 11, 1496–1509. Lazarus, J. H. & Cooke, R. 1993. Generalised mechanistic model for heterogeneous flow in a nonNewtonian vehicle. In Proceedings of Hydrotransport 12, pp. 671–690. Cranfield, UK: BHR Group. Matsouk V. 1996. Internal structure of slurry flow in inclined pipe—Experiments and mechanistic modeling. In Proceedings of Hydrotransport 13, pp. 187–210. Cranfield, UK: BHR Group. Newitt, D. M., J. F. Richardson, M. Abbott, and R. B. Turtle. 1955. Hydraulic conveying of solids in horizontal pipes. Trans Inst. of Chem. Eng., 33, 93–113. Newitt, D. M., J. R. Richardson, and J. B. Glibbon. 1961. Hydraulic conveying of solids in vertical pipes. Trans. Inst. of Chem. Eng., 39, 93–100. Newitt, D. M., J. R. Richardson, and C. A. Shook (Eds.). 1962. Symposium on Interaction between Fluids and Particles, London. London: Institution of Chemical Engineers. Raj, R. S. 1972. Pressure loss in hydraulic transport of solids in inclined pipes. Paper presented at Hydrotransport 2, Coventry, England. Saskatchewan Research Council. 2000. Slurry Pipeline Course—SRC Pipe Flow Technology Center, Saskatoon, Canada, May 15–16. Schiller, R. E., and P. E. Herbich. 1991. Sediment transport in pipes. In Handbook of Dredging, Edited by P. E. Herbich. New York: McGraw-Hill. Shen, H. W. 1970. Sediment transportation mechanism—Transportation of sediments in pipes. Journal Hydraulics Division Am. Soc. Civ. Eng., 96, 1503–1538. Shook, C. A. 1981. Lead Agency Report for MTCM Cooperative Research Project. Report E-725-6C-81. Report prepared for the Saskatchewan Research Council, Saskatchewan, Canada. Shook, C. A., J. R. Rollins, and G. S. Vassie. 1974. Sliding in Inclined Slurry Pipelines and Shutdown. Report IX. Report prepared for the Saskatchewan Research Council, Saskatchewan, Canada. Shook, C. A., and M. C. Roco. 1991. The two layer model. In Slurry Flow: Principles and Practice. Newton, MA: Butterworth-Heinemann. Spells, K. E. 1955. Correlation for use in transport of aqueous suspensions of fine solids through pipes. Trans Inst. Chem. Eng., 33, 79–84. Thomas, D. G. 1963. Transport characteristics of suspensions: Relation of hindered settling floc characteristics to rheological parameters. Am. Inst. Chem. Eng. Journal, 9, 310–319. Thomas, D. G. 1962. Transport Characteristics of suspensions, Part IV. Am. Inst. Chem Eng. Journal, 8, 373–378. Thomas, D. G. 1964. Transport characteristics of suspensions, Part IX. Am Inst. Chem. Eng. Journal, 10, 303–308. Traynis, V. V. 1970.Parameters and Flow Regimes for Hydraulic Transport of Coal by Pipelines,

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Translated and Edited by W. C. Cooley and R. R. Faddick. Terraspace Inc. report, April 1970, pp 17–19. Turian, R. M., and T. Yuan. 1977. Flow of slurries and pipelines. Am. Inst. Chem. Eng. Journal, 23, 3, 232. Vallentine, H. R.1955. Transportation of Sands in Pipelines. Commonwealth Engineer (Australia), April, 349–355. Warman International Inc. 1990. Slurry Handbook. Madison, WI: Warman International Inc. Wasp, E. J. et al. 1970. Deposition velocities, transition velocities, and spatial distribution of solids in slurry pipelines. Paper read at 1st International Conference on Hydraulic Transportation of Solids in Pipes, Cranfield, England. Wasp, E. J., J. P. Kenny, and R. L. Gandhi. 1977. Solid–Liquid Flow—Slurry Pipeline Transportation. Aedermannsdorf, Switzerland: Trans-Tech Publications. Wicks, M. 1971. Transportation of solids of low concentrations in horizontal pipes. In Advances in Solid–Liquid Flow in Pipes and Its application, edited by I. Zandi. New York: Pergamon Press. Wilson, K. C. 1970. Slip point of beds in solid–liquid pipe flow. Am. Soc. Chem. Eng. Hydraulic Division, no. HY1, paper 6992: 1–12. Wilson, K. C. 1991. Pipeline Design for Settling Slurries. In Slurry Handling, edited by N. P. Brown and N. I. Heywood. New York: Elsevier Applied Sciences. Wilson, K. C., and D. G. Judge. 1976. Paper presented at the International Symposium on Freight Pipelines, Washington, DC Wilson, K. C., G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. New York: Elsevier Applied Sciences. Wilson, W. E. 1942. Mechanics of flow of non-colloidal solids. Trans. Am. Soc. of Chem. Eng., 107, 1576. Wood D. J. 1979. Pressure gradient requirements for re-establishment of slurry flow. In Sixth International Conference on Hydraulic Transport of Solids in Pipes, p. 217. Cranfield, UK: BHRA Group. Worster, R. C., and D. E. Denny. 1955. Hydraulic transport of solid materials in pipes. Proceedings of the Institute of Mechanical Engineers (UK), 38, 230–234. Zandi, I., and G. Govatos. 1967. Heterogeneous flow of solids in pipeline. Proceedings of the Hydraulic Division of Am. Soc. Civ. Eng., 93, no. HY3, paper 5244, 145–159. Zandi, I. 1971. Hydraulic transport of bulky materials. In Advances in Solid–Liquid Flow in Pipes and Its applications, pp. 1–38, I. Zandi (Ed.). Oxford: Pergamon Press.

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CHAPTER 5

HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

5-0 INTRODUCTION The rheology of non-Newtonian flows and homogeneous flows was examined in detail in Chapter 3. With modern methods of grinding, the size of particles can be reduced to values smaller than 70 m (0.0028 in). As shown in Table 3-8, a wide range of metal concentrates and tailings are pumped at a sufficiently high concentration with a small enough particle size for the mixture to behave as a Bingham plastic. There are other clays and slurries that may behave as pseudoplastics. In certain circuits of oil sands processing plants with tar at the level of flotation, the slurry may behave as a thixotropic mixture. With non-Newtonian flows, it is important to take into account the rheology, yield stress, power law exponent, coefficient, and even the time response. Different models have evolved over the years for Bingham and pseudoplastic slurries. Some of these models put more emphasis on the laminar flow regime, in which roughness effects are negligible. Some other models extend to the transition and turbulent regimes. The effect of pipe roughness on friction loss factors in non-Newtonian flows remains a topic worth investigating and researching. For thixotropic slurries, methods are used to predict start-up pressure after a shutdown of the pipeline, and the time required to clean the conduit of gelled material before resuming pumping. The equations for friction factors of non-Newtonian fluids are fairly complex and require iteration and data on the rheology. Throughout the years, different authors have developed equations for “modified” Reynolds numbers, Hedstrom numbers, etc. In this chapter, equations developed by different authors will be reviewed. Through worked examples, the reader will be shown methods of calculating the friction factor. It is the purpose of this chapter to focus on the engineering side of the problem. The reader is strongly advised to read through Chapter 3, to gain the fundamentals for this chapter. For the practical engineer, who is more concerned with the actual design of a pipeline or a pumping system, the numerous and different definitions of the so-called “modified Reynolds number” can be very confusing. Every few years, an author develops a new definition of the “modified Reynolds number” and claims to have found a relationship with the friction factor. The cautious approach for an engineer is to assume that such a universal relationship is illusive, and for every type of slurry there may be a model to use. 5.1

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Sometimes the use of two different methods can yield differences of 25–35% in the estimation of the friction factor. One important difference with the slurries described in Chapter 4 is that most nonNewtonian slurries do not exhibit the stratification of solids, which is usual with coarse particles.

5-1 FRICTION LOSSES FOR BINGHAM PLASTICS Bingham plastics were defined in Chapter 3. They are characterized by a yield stress that must be overcome to start the flow. Examples of Bingham plastics are listed in Table 3-9

5-1-1 Start-up Pressure The start-up pressure for pumping a Bingham plastic is expressed in terms of the true yield stress: 40 L Pst = ᎏ Di

(5-1)

The start-up pressure per unit length is obtained by dividing Equation 5-1 by the length of the pipeline: 40 Pst ᎏ=ᎏ Di L Examples for the starting pressure per unit length for slurries in 3⬙, 6⬙, 12⬙, and 18⬙ pipes are presented in Table 5-1. The Reynolds Number for a Bingham slurry is expressed as: DiVm ReB = ᎏ

(5-2)

The coefficient of rigidity was defined in Equation 3-29 as

0 = ᎏ + ⬁ (d␥/dt)

(3-29)

For Bingham slurries a nondimensional coefficient is defined as the plasticity number:

0Di PL = ᎏ V

(5-3)

The Hedstrom number is the product of the plasticity number and the Reynolds number and is calculated as D 2i m0 He = ᎏ 2

(5-4)

Table 5-2 shows examples of Bingham slurry mixtures and the magnitude of the Hedstrom number for flows in rubber-lined 6⬙ (150 mm NB), 12⬙ (300 mm NB) and 18⬙ (450

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TABLE 5-1 Starting Pressure per Unit Length for Certain Slurries in Pa/m

5.3

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 Aqueous clay suspension III Aqueous clay suspension V 21.4% Bauxite 18% Iron oxide 7.5 % Kaolin 58% Limestone 52.4% Fine liminite 14% Sewage sludge

3⬙ Pipe Sch 40, rubber lined, ID = 2.560⬙ (65 mm)

6⬙ Pipe Sch 40, rubber lined, ID = 5.557⬙ (141.2 mm)

12⬙ Pipe S, rubber lined, ID = 11.500⬙ (292 mm)

18⬙ Pipe S, rubber lined, ID = 16.500⬙ (419 mm)

6.86

234

108

52

37

Particle size, d50

Density, kg/m3

Yield stress, Pa

92% under 74 m

1520

3.8

80% under 1 m

1280

59

13.1

3631

1671

808

567

80% under 1 m

1207

25

6.7

1538

708

342

240

80% under 1 m

1149

7.8

4.0

480

221

107

75

1520 1440 1360 1163 1170 1103 1530 2435 1060

34.5 20 6.65 8.5 0.78 7.5 2.5 30 3.1

44.7 32.8 19.4 4.1 4.5 5 15 16 24.5

2123 1231 409 523 49 462 154 1846 191

977 567 188 241 23 213 71 849 88

473 274 91 116 11 103 34 411 43

332 192 64 82 7.7 72 24 209 30

<200 m <50 m Colloidal <160 m <50 m

*References on rheology data were presented in Table 3-9.

Page 5.3

Slurry

Coefficient of rigidity, mPa · s (cP)

9:17 AM

Starting pressure per unit length (Pa/m)*

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TABLE 5-2 Examples of Hedstrom Numbers for Bingham Slurries in 3⬙, 6⬙, 12⬙, and 18⬙ Rubber-Lined Pipes Hedstrom number*

Density, kg/m3

6⬙ Pipe Sch 40, rubber lined, ID = 5.557⬙ (141.2 mm)

12⬙ Pipe S, rubber lined, ID = 11.500⬙ (292 mm)

18⬙ Pipe S, rubber lined, ID = 16.500⬙ (419 mm)

518,568

2,445,272

1.046 × 107

2.124 × 107

92% under 74 m

1520

80% under 1 m

1280

59

13.1

1,859,285

8,773,821

3.752 × 107

7.616 × 107

80% under 1 m

1207

25

6.7

2,840,039

1.472 × 107

6.078 × 107

1.234 × 108

80% under 1 m

1149

7.8

4.0

2,366,581

1.117 × 107

4.776 × 107

9.694 × 107

1520 1440 1360 1163 1170 1103 1530 2435 1060

34.5 20 6.65 8.5 0.78 7.5 2.5 30 3.1

44.7 32.8 19.4 4.1 4.5 5 15 16 24.5

110,885 113,102 101,528 2,484,607 190,407 1,398,052 71,825 1,205,610 23,129

523,259 533,721 479,100 1.172 × 107 898,514 6,597,299 338,937 5,689,180 109,145

2,237,759 2,282,499 2,048,910 5.014 × 107 3,842,564 2.821 × 107 1,449,488 2.433 × 107 466,768

4,541,865 4,632,671 4,158,568 1.017 × 108 7,799,056 5.726 × 107 2,941,952 4.938 × 107 947,375

<200 m <50 m Colloidal <160 m <50 m

*References on rheology data were presented in Table 3-9.

3.8

6.86

Page 5.4

5.4

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 Aqueous clay suspension III Aqueous clay suspension V 21.4% Bauxite 18% Iron oxide 7.5 % Kaolin clay 58% Limestone 52.4% Fine liminite 14% Sewage sludge

Particle size, d50

3⬙ Pipe Sch 40, rubber lined, ID = 2.560⬙ (65 mm)

9:17 AM

Slurry

Yield stress, Pa

Coefficient of rigidity, mPa · s (cP)

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5.5

mm) pipes. Laminar flows in large pipes are considered for certain slurries at relatively high weight concentration (⬇ 60%), or certain high-energy mixtures (e.g., crude oil with fine and ultrafine coal). Example 5-1 A slurry consists of a clay and water mixture. It is tested and classified as a Bingham mixture with a yield stress of 17 Pa. The pipe inner diameter is 63 mm. The pipe length is 6500 m. Determine the start-up pressure, ignoring any static head. Solution in SI Units Using Equation 5-1: Pst = 40L/Di Pst = 4 × 17 × 6500/0.063 = 7,015,873 (1018 psi) Solution in US units Pst = 40L/Di

0 = 17 Pa/6895 = 2.465 × 10–3 psi L = 6500 m/0.0254 = 255,905 in Di = 63/25.4 = 2.48 in 4 × 2.465 × 10–3 × 255,905 Start-up pressure Pst = ᎏᎏᎏ = 1017.6 psi 2.48 5-1-2 Friction Factor in Laminar Regime Buckingham (1921) was the first to develop an equation for a fully developed laminar flow. This equation has since been modified by Hedstrom (1952) and others to express the friction factor as a function of the Hedstrom and Reynolds numbers: He He4 fNL 1 ᎏ = ᎏ – ᎏ2 + ᎏ 3 ReB 6Re B 3 f NL Re B8 16

(5-5)

or

冤

He4 16 He fNL = ᎏ 1 + ᎏ – ᎏ 3 ReB 6ReB 3 f NL ReB7

冥

(5-6)

This would occur below the critical Reynolds number or in transition between laminar and turbulent flow. The last term between brackets in the equation is often considered second order. Example 5-2 A Bingham slurry with a concentration of 50% by weight is tested in a plastic-lined pipe with an inner diameter of 2.5 in. The tests indicate a yield stress of 1.5 Pa, a slurry mixture specific gravity of 1.54, and a coefficient of rigidity of 0.4 Pa · s. Assuming a flow speed of 4 ft/s in a laminar regime, determine the friction factor by Buckingham’s equation.

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Solution in SI Units Pipe ID = 2.5⬙ or 63.5 mm Speed = 4 ft/s or 1.219 m/s Reynolds number = 0.0635 × 1.219 × 1540/0.4 = 298 Hedstrom number = (0.0635)2 × 1540 × 1.5/(0.42) = 916.8 Using Equation 5-6 and ignoring the higher-order terms: fNL ⬇ [16/ReB][1 + He/(6ReB)] = 0.0812 This a fanning factor, so the Darcy friction factor is 0.3248. Figure 5-1 presents values of the friction factor versus the Reynolds number for a wide range of Hedstrom numbers from 0 to 109. The transition between laminar and turbulent flows is shown by the dotted curve of the critical Reynolds number. From the point of view of engineering, the most practical flows in pipes are in a range of Hedstrom numbers between 105 and 108, as shown in Table 5-2. The transition to turbulent flow will be examined in more detail throughout this chapter. To appreciate the practical magnitude of the laminar friction factor, Table 5-3 presents cases at a speed of 1 m/s (3.3 ft/sec) for rubber-lined pipes in sizes of 3⬙ (80 mm N.B.), 6⬙ (150 mm N.B), 12⬙ (300 mm N.B.), and 18⬙ (450 mm N.B.). The Fanning friction factor based on the Buckingham equation is in the range of 0.001 to 0.15.

FIGURE 5-1 Friction factor versus Reynolds number and Hedstrom number. (From Hill R. A, P. E. Snoek, and R. L. Ghandi, Hydraulic transport of solids, in The Pump Handbook, 2nd ed., I. J. Karassik et al. (Eds.), New York: McGraw-Hill. Reproduced by permission.)

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TABLE 5-3 Friction Factor at a Speed of 1 m/s (3.3 ft/sec) for Bingham Mixtures in Rubber-Lined Pipes*

5.7

54.3% Aqueous suspension of cement rock Flocculated aqueous China clay suspension No. 1 Flocculated aqueous China clay suspension No. 4 Flocculated aqueous China suspension No. 6 Aqueous clay suspension I Aqueous clay suspension III Aqueous clay suspension V 21.4% Bauxite 18% Iron oxide 7.5 % Kaolin clay 58% Limestone 52.4% Fine liminite 14% Sewage sludge

3⬙ Pipe Sch 40, rubber lined, ID = 2.560⬙ (65 mm)

6⬙ Pipe Sch 40, rubber lined, ID = 5.557⬙ (141.2 mm)

12⬙ Pipe S, rubber lined, ID = 11.500⬙ (292 mm)

18⬙ Pipe S, rubber lined, ID = 16.500⬙ (419 mm)

0.00778

0.00718

0.00691

0.00684

Particle size, d50

Density, kg/m3

Yield stress, Pa

92% under 74 m

1520

3.8

80% under 1 m

1280

59

13.1

0.12541

0.12407

0.12348

0.12331

80% under 1 m

1207

25

6.7

0.00566

0.05586

0.05554

0.05545

80% under 1 m

1149

7.8

4.0

0.0186

0.0185

0.0183

0.0182

1520 1440 1360 1163 1170 1103 1530 2435 1060

34.5 20 6.65 8.5 0.78 7.5 2.5 30 3.1

44.7 32.8 19.4 4.1 4.5 5 15 16 24.5

0.0677 0.0426 0.01655 0.0204 0.0272 0.01925 0.0046 0.0345 0.0135

0.0639 0.0396 0.0146 0.0199 0.0221 0.01864 0.00447 0.0336 0.0104

0.0621 0.0382 0.01382 0.0197 0.0199 0.01838 0.00441 0.0332 0.0009

0.0616 0.0379 0.01359 0.0196 0.0193 0.0183 0.00439 0.0331 0.00087

<200 m <50 m Colloidal <160 m <50 m

*References on rheology data were presented in Table 3-9.

6.86

Page 5.7

Slurry

Coefficient of rigidity, mPa · s (cP)

9:17 AM

Fanning friction factor fN at a speed of 1 m/s (3.3 ft/sec)

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CHAPTER FIVE

5-1-3 Transition to Turbulent Flow Regime Hanks and Pratt (1967) analyzed extensive experimental data on critical Reynolds numbers and proposed a relationship between the Reynolds and Hedstrom Numbers at transition as

冢

冣

He 4 1 ReBc = ᎏ 1 – ᎏ xc + ᎏ x 4c xc 3 3

(5-7)

where xc = 0/wc = the ratio of the yield stress to the wall shear stress at the transition from laminar to turbulent flow. At the transition, xc He = 16,800 ᎏ3 (1 – xc)

(5-8)

Figure 5-2 plots the magnitude of critical Reynolds number versus the Hedstrom number for a number of Bingham slurries. Example 5-3 Using Figure 5-2, determine the critical Reynolds number for a clay slurry at a Hedstrom number of 10,000. Solution From Figure 5-2 Rec ⬇ 3700. Wasp et al. (1977) defined the effective pipeline viscosity for laminar flow as

w e = ᎏ 8V/Di

Critical Reynolds Number

10

10

10

(5-9)

5

4

3

10

3

10

4

10

5

10

6

10

8

Hedstrom Number

FIGURE 5-2 The critical Reynolds number versus the Hedstrom number for flow in pipes. (After Hanks, R. W., and D. R. Pratt. 1967.)

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HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

In Chapter 3, in the description of the capillary tube test, the Buckingham equation was derived. When ignoring the fourth-power term, the equation reduces to

w ⬇ 8V/Di + 4/3 0

(3-62)

or 8V/Di + 4/3 0 DI0 e ⬇ ᎏᎏ ⬇ 1 + ᎏ 8V/Di 6V

冤

冥

(5-10)

In many laminar flow pipelines the term Di0/(6V) is much smaller than unity. In such cases Equation 5-10 is then simplified to

e ⬇ Di0/(6V)

(5-11)

At transition, Wasp et al. (1977) point out that this equation can be approximated to

e ⬇ [Di0/(6VTR)] using the effective pipeline viscosity, the critical Reynolds number is expressed as 2 /0 ReBC = 6V TR

At high shear rate for Bingham plastics, e ⬇ ⬁ ⬇ (see Figure 3-10): VTR =

冪莦 ReBC0 ᎏ 6

(5-12)

The Wasp method is based on numerous assumptions, and usually terminates by assuming that the transition Reynolds number is in the range of 2000 to 3000 for numerous Bingham slurries. In some respects, it is a useful tool for hand calculations. A more widely accepted method since the mid 1980s is to compute the transition velocity that was proposed by Wilson and Thomas, to be discussed in Section 5-4-3. It is then assumed that the transition from laminar to turbulent flow occurs when the Wilson–Thomas and the Buckingham equations intersect.

5-1-4 Friction Factor in the Turbulent Flow Regime Hanks and Dadia (1971) developed a semiempirical equation for the turbulent flow of Bingham slurries in closed conduits. These equations were modified by Darby (1981) and Darby et al. (1992) to give a friction factor for the turbulent regime as fNT = 10aReBb

(5-13)

where a = –1.47[1 + 0.146 exp(–2.9 × 10–5He)] b = –0.193 The values of the parameters a and b are based on empirical data for closed conduits. Bingham slurries do not exhibit a sudden change from laminar to turbulent flow. Darby et al. (1992) reviewed the work of previous authors and proposed to combine the laminar and turbulent fanning friction factors into the following equation: m m (1/m) fN = ( f NL + f NT )

(5-14)

m = 1.7 + 40,000/ReB.

(5-15)

where

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CHAPTER FIVE

Equations 5-12 and 5-13 do not account for pipe roughness and are essentially for very smooth pipes such as glass and high-density polyethylene pipes. Studies have been published in the past on flow of Bingham plastics in the laminar regime, where roughness effects are neglected. Thomas and Wilson (1987) have even argued that the non-Newtonian fluids form a viscous sublayer in the boundary layer, that is usually thicker than with Newtonian flows. This viscous sublayer is considered to suppress the contribution of roughness and, in effect, Wilson and Thomas do support the assumptions made by Darby (1981). Many concentrates are pumped at high volumetric concentrations but also at a relatively moderate speed of 2–2.5 m/s (6.6–8.2 ft/s). We will, however, discuss the effects of roughness in Section 5-7. Example 5-4 In Figure 3-9, the relationship between the Bingham plastic apparent viscosity and the shear rate was presented as

= (d␥/d) + ⬁ at high shear rate ⬇ ⬁. Considering an aqueous clay suspension III (Table 3-9) with a mixture density of 1440 kg/m3 (SG = 1.44), a yield stress of 20 Pa, and a coefficient of rigidity of 32.8 mPa · s as reported by Caldwell and Babbitt (see references of Chapter 3), determine the friction factor for flow at 2.5 m/s in a 63 mm ID pipe, using the Darby method. Determine also the pressure drop per unit length. Solution SI Units Reynolds number:

VDI 1440 × 2.5 × 0.063 Re = ᎏ = ᎏᎏ = 6914.6 0.0328 Hedstrom number: (0.063)2 × 1440 × 20 D20 ᎏᎏᎏ He = ᎏ = = 106,249 2 (0.0328)2 m = 1.7 + 40,000/Re m = 1.7 + 40,000/6914.6 = 7.49 a = –1.47[1 + 0.146 exp(–2.9 × 10–5He)] = –1.479 fT = 10aRe–0.193 fT = 10–1.479 × 6914.6–0.193 = 0.006

冢

冣

冢

Re 16 Re4 16 Re ⬇ ᎏ 1+ ᎏ fL = ᎏ 1 + ᎏ – ᎏ 7 3 3f L He Re 6He Re 6He

冢

冣

16 6914.6 fL = ᎏ 1 + ᎏᎏ = 0.00234 6914.6 6 × 106,249 fn = ( f Lm + f Tm)1/m Therefore, fn = (0.002347.49 + 0.0067.49)1/7.49 = 0.006 is the fanning friction factor

冣

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HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

5.11

Darcy factor: fD = 4fN = 0.006 × 4 = 0.024 Pressure drop per unit length: dP/dz = fDV 2/(2Di) dP/dz = 1440 × 0.024 × 2.52/(2 × 0.063) = 1,714 Pa/m (0.816 psi/ft)

5-2 FRICTION LOSSES FOR PSEUDOPLASTICS Pseudoplastic rheology was extensively discussed in Chapter 3, Section 3.4-2. Examples of pseudoplastics are listed in Table 3-10.

5-2-1 Laminar Flow A number of models have been developed for pseudoplastic flows. These treat the fluid as a continuum. 5-2-1-1 The Rabinowitsch–Mooney Relations Herzog and Weissenburg (1928) developed an equation for laminar time-independent, viscous non-Newtonian flows. It was subject to further refinements by Rabinowitsch (1929) and Mooney (1931). For a circular pipe, a relationship is established between the shear stress and the absolute value of the rate of shear ␥ = –du/dr f() = –du/dr Rabinowitsch and Mooney derived a general relationship for the shear rate at the wall:

冢 冣

du – ᎏ dr

w

8V 1 + 3 =ᎏ ᎏ DI 4

冢

冣

(5-16)

where d[ln(Di ⌬P/4L)] = ᎏᎏ d[ln(8V/D)]

(5-17)

5-2-1-2 The Metzner and Reed Approach Metzner and Reed (1955) developed an equation for the Reynolds number in laminar flow as D IV 2– ReMR = ᎏ ␥ where is defined by Equation 5-16 and

␥ = K⬘8(–1) ␥ = gcK⬘8

(–1)

in SI units in USCS units

(5-18)

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CHAPTER FIVE

冢

冣

In SI units

(5-19a)

冢

冣

In USCS units

(5-19b)

1 + 3n K⬘ = K ᎏ 4n 1 + 3n K⬘ = K\gc ᎏ 4n

n

n

The fanning friction factor is then expressed in the laminar flow regime in the conventional manner but using the modified Reynolds number: 16 fNL = ᎏ ReMR

(5-20)

Example 5-5 The pressure drop in a 80 mm ID pipe is to be determined for a slurry with S.G. = 1.37. The power law exponent had been previously determined to be 0.4, and the power law factor K as 16 dynes-spcn/cm2. The speed of the flow is 1.35 m/s. Use the Metzner and Reed approach to calculate the friction factor, assuming a shear rate of 600 s–1. Solution From Equation 5-15: 8V 1 + 3 –600 = ᎏ ᎏ Di 4

冢

冣

8 × 1.35 1 + 3 –600 = ᎏ ᎏ 0.08 4

冢

冣

–4.45 = (1 + 3)/4 –1.11 = 1/ + 3

= 0.529 DV 2– ReMR = ᎏ ␥

冢

1 + 3n K⬘ = K ᎏ 4n

= 18.174 冣 = 16冢 ᎏ 1.6 冣 n

1 + 1.2

n

␥ = K⬘8–1 = 18.174 × 8–0.47 = 6.84 0.080.529 × 1.351.471 × 1370 ReMR = ᎏᎏᎏ 6.84 ReMR = 81.87 16 fn = ᎏ = 0.195 ReMR The Metzner and Reed approach has become a classical method of dealing with time-independent non-Newtonian fluids. It has been extended to Bingham slurries but the opinion of the author is that this approach is fairly difficult to use for Bingham slur-

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HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

5.13

ries, and a more practical method would be the Darby approach, described in Section 51-4. The Metzner and Reed approach requires the engineer to assume a value of the shear stress at the wall w to calculate x. Such assumptions are very difficult to make for engineers outside a research lab. It may be more practical to send samples of the slurry to a rheology lab and to go from plots of yield stress versus weight concentration, as well as from plots of viscosity versus weight concentration and shear rates to a more straightforward computation of friction factors (Figure 5-3). 5.2.1.3 The Tomita Method Tomita (1959) defined a fanning friction factor for power law fluids as 2Di⌬P 1 + 2n fPL = ᎏ2 ᎏ 3LmV 1 + 3n

冢

冢

6 [1/n + 3]1–n RePL = ᎏn ᎏᎏ 2 1/n + 2

冣

D inV 2–nm

冣冢 ᎏᎏ 冣 K

(5-21)

(5-22)

In the laminar flow regimes:

FIGURE 5-3 Friction factor versus Reynolds number for power law factors. [From Hill R. A, P. E. Snoek, and R. L. Ghandi, Hydraulic transport of solids, in The Pump Handbook, 2nd ed., I. J. Karassik et al. (Eds.), New York: McGraw-Hill. Reproduced by permission.]

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CHAPTER FIVE

16 fPL = ᎏ Remod

(5-23)

5-2-1-3 Heywood Method Heywood (1991) proposed to define a modified Reynolds number for pseudoplastics as

mVDi 4n Remod = ᎏ ᎏ K 1 + 3n

冢

冣 冢ᎏ 8V 冣 Di

n

n–1

(5-24)

where K and n are the consistency coefficient and flow behavior indexes for pseudoplastic flows previously defined in Chapter 3. In the laminar flow regime, Heywood (1991) used the conventional method of defining the fanning friction factor in terms of the Reynolds number in the laminar flow regime as previously discussed in Chapter 2, or fNPL = 16/Remod

(5-25)

The effective pipeline viscosity is expressed as

冢

4n e = K ᎏ 1 + 3n

冣 冢ᎏ D 冣 n

8V

n–1

(5-26)

i

5-2-2 Transition Flow Regime Ryan and Johnson (1959) defined a critical Reynolds number for purely viscous pseudoplastics as 6464n (n + 2)(n+2)/(n+1) Rec = ᎏᎏᎏ (1 + 3n)2

(5-27)

The friction factor at the transition from laminar to turbulent, flow called the critical friction factor is 1 (1 + 3n)2 fNc = ᎏ ᎏᎏ 404n (n + 2)(n+2)/(n+1)

(5-28)

Table 5-4 tabulates the critical Reynolds number and fanning friction factor versus the power factor “n.” The minimum friction factor is 0.0067 at n = 0.5. However, Heywood (1991) deducted from various test data that the minimum value for fNc = 0.004, which is even lower than the values indicated by Equation 5-28 (Figure 5.4).

5-2-3 Turbulent Flow Various equations have been developed over the years for turbulent flow of pseudoplastics in smooth pipes. These equations are based on empirical data and semitheoretical models. Using the modified Reynolds number as per Equation 5-17, Dodge and Metzner (1959) developed the following semitheoretical equation for turbulent flow:

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HOMOGENEOUS FLOWS OF NONSETTLING SLURRIES

TABLE 5-4 Critical Reynolds Number and Fanning Friction Factor versus the Flow Behavior Index “n” According to the Ryan and Johnson Method Flow behavior index “n”

Critical Reynolds number

Critical fanning friction factor, fNC

Flow behavior index “n”

Critical Reynolds number

Critical fanning friction factor, fNC

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1577 2143 2345 2396 2381 2337 2280 2219

0.01015 0.00747 0.00682 0.00668 0.00672 0.00685 0.00702 0.0072

0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6

2158 2099 2043 1990 1941 1895 1852 1812

0.00741 0.00762 0.0783 0.00804 0.00824 0.00844 0.00864 0.00883

1 0.4 4 (1– /2) ᎏ=ᎏ log10[Remod f NT ]– ᎏ 0.75 1.2 兹f苶N苶T

(5-29)

2500 f NCR

( from equation 5-28)

0.008

2300

0.006

2100 Re CR

0.004 0.002 0.0 0.0

1900 1700

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1500

(from Equation 5-27)

0.010

Modified nritical Reynolds number

Although Equation 5-29 has been extensively used, it has its own limitations. Measuring the power exponent “n” in laminar flow tests and then trying to apply it to turbulent flows is asking for trouble, particularly for cases when n < 0.5. Heywood and Richardson (1978) showed that pumping flocculated clays yielded higher experimental values of friction coefficient than those predicted by Dodge and Metzner (1959), particularly when the value of “n” had been obtained at low shear stress. Note: Equation 5-20 does not incorporate the effects of roughness. Govier and Aziz (1972) indicated that Equation 5-20 gives excellent agreement between calculated and experimental data in the range of modified Reynolds numbers ReMR of 2900–36,000 and

Critical fanning friction factor

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Flow index "n" FIGURE 5-4 Values of critical Reynolds number and critical fanning factor versus the flow index “n” for pseudoplastic slurries based on Equation 5-27.

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modified power exponent s of 0.36–1.0. Equation 5-20 requires repeated iteration. The use of a personal computer is recommended. For power law fluids, Tomita (1959) extended his laminar flow model (discussed in Section 5.2.1.3) to turbulent flows in smooth pipes by applying Prandtl’s mixing length concept, and developed a different implicit equation: 1 ᎏ = 4 log10(RePL 兹苶 fP苶 LT) – 0.40 兹苶 fP苶 LT

(5-30)

where RePL and fPLT were already expressed for the laminar flow in Equations 5-21 and 522. Tomita’s equation was supported by 40 experimental data points on starch pastes and lime slurries. Irvine (1988) published the following equation: F⬘(n) fn = ᎏᎏ Re[1/(3n+1)] mod

(5-31)

where

冢

冣

8nn 2 ᎏᎏ F⬘(n) = ᎏ n–1 7n 8 7 (1 + 3n)n

1/(3n+1)

(5-32)

Example 5-6 At a volumetric concentration of 30%, a magnetite suspension has a power law coefficient K of 12 dynes-secn/cm2 and a power law exponent of 0.2. If the slurry is homogeneous and nonsettling at a speed of 1.5 m/s, determine the friction factor in a 101 mm ID pipe, at a slurry density of 1600 kg/m3. Solution From Equation 5-24, the modified Reynolds number is calculated as

冢

1600 × 1.5 × 0.101 4 × 0.2 Remod = ᎏᎏ ᎏᎏ 12 × 0.1 1 + 3 × 0.2

冣 冢ᎏ 8 × 1.5 冣 0.2

0.101

–0.8

= 202 × 0.5 × 45,697

Remod = 4615 It is necessary to check if the flow is turbulent. From Equation 5-27: 6464 n(n + 2)(n+2)/(n+1) Rec⬘ = ᎏᎏᎏ (1 + 3n)2 6464 × 0.2(2.2)(2.2/1.2) = 2143 Rec⬘ = ᎏᎏᎏ (1 + 0.6)2 Since Remod > Rec⬘, flow is therefore turbulent. Two different approaches will be used. Irvine Method From Equation 5-32:

冢

2 8 × 0.20.2 ᎏᎏ F⬘(n) = ᎏ –0.8 1.4 8 7 (1 + 0.6)0.2 From Equation 5-31, the fanning friction factor is

冣

1/1.6

= 0.862

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0.862 fn = ᎏ = 0.00442 46151/1.6 Tomita Method From Equation 5-22: RePL = 4615 × 80.8 × 20.2 × 6(1.6/0.2)0.8/2.8 = 316,457 From Equation 5-30: 1 ᎏ = 4 log10(316,457 × 兹苶 fP苶 LT) – 0.40 兹苶 fP苶 LT Iteration 1 starts by assuming at fPLT ⬇ 0.004 correction fPLT = 0.0035 Iteration 2 starts at fPLT ⬇ 0.0042 fPLT = 0.00352 Iteration 3 starts at fPLT ⬇ 0.0045 fPLT = 0.003498 Iteration 4 starts at fPLT ⬇ 0.0035 fPLT = 0.00359 So by theTomita method we obtain a friction factor of 0.0035. The Irvine method yields a friction factor 23% higher than the Tomita method. Both methods do not account for roughness of the pipe wall.

5.3 FRICTION LOSSES FOR YIELD PSEUDOPLASTICS Yield pseudoplastics were described extensively in Chapter 3. Examples were listed in Table 3.11.

5-3-1 The Hanks and Ricks Method In the laminar flow regime, Hanks and Ricks (1978), defined the fanning friction factor in terms of the modified Reynolds number: 16 fNPL = ᎏ Remod

(5-33)

where

冢

(1 – x)2 2x(1 – x) x2 = (1 + 3n)n(1 – x)1+n ᎏ + ᎏ + ᎏ 1 + 3n 1 + 2n 1+n 2 yp yp x = ᎏ = ᎏᎏ2 w fN · · V where yp is the yield stress for pseudoplastic.

冣

n

(5-34)

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For yield-pseudoplastics, Hanks and Ricks (1978), Heywood (1991) proposed to define a modified Hedstrom Number as D i2m yp Hemod = ᎏ ᎏ K K

冢 冣

2–n/n

(5-35)

The critical Reynolds number is established in terms of the modified Reynolds number and Hedstrom number, as in Figure 5-5.

5-3-2 The Heywood Method For the laminar flow regime fanning friction factor, Heywood (1991) modified the Buckingham equation to

冢

16 2Hemod fNLY = ᎏ 1 + ᎏᎏ2 Remod fNLY (Remod)

ᎏᎏᎏ 冣冦1 – (2n + 1) f (Re ) 2Hemod NLY

mod

2

1 + ᎏᎏ 冣冥冧 冤1 + ᎏᎏᎏ (n + 1) f (Re ) 冢 f (Re ) 2nHemod

4nHemod NLY

mod

2

NLY

mod

(5-36)

2

5-3-3 The Torrance Method Defining x = y/w, Torrance (1963) derived the following equation for turbulent friction factor of a yield pseudoplastic:

Critical (Hanks & Ricks) Reynolds Number

3200 2800 2400

4

He = 10 He = 0

2000 1600

6

He = 10 1200 800 400 0

0

0.1 0..2 0.3

0.4 0.5 0.6

0.7 0.8

0.9 1.0

Flow Behavior Index "n"

FIGURE 5-5 Values of the critical value of the “Hanks and Ricks Reynolds number” versus the flow index “n” for yield pseudoplastic slurries.

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1 ᎏᎏ =

兹f苶

– 2.95冥 + ᎏ log 冤冢 ᎏ n 冣 n 2.69

log 冢ᎏ n 4.53

4.53

10RePLC( f

10(1

– x)

冣

0.68 ) + ᎏ (5n – 8) n

[1–n/2]

(5-37)

where RePLC = DnV 2–n/K8n–1. The work of Torrance was essentially an exercise in algebra, and has not been substantiated by experimental data. Nevertheless it has been substantially quoted in the absence of suitable confirming data.

5-4 GENERALIZED METHODS Various models have been developed for complex non-Newtonian flows. The most important ones are listed, but most were derived from empirical data.

5-4-1 The Herschel–Bulkley Model Herschel and Bulkley (1928) developed a model that has been extensively applied to sewage sludge, kaolin slurries, and mine tailings:

= y + j(␥)n

(5-38)

y a = ᎏ = ᎏ + j(␥)n–1 ␥ ␥

(5-39)

The apparent viscosity is

where j is called the Herschel–Bulkley parameter.

5-4-2 The Chilton and Stainsby Method Chilton and Stainsby (1998) indicated that the accuracy of the Herschel–Bulkley model deteriorated at high shear rates. However, this may or may not be significant, depending on the application. At high strain rates the model predicts that the viscosity tends to zero, which is obviously incorrect. In Chapter 3, Section 3-4-2-2 , the Sisko, Cross, Meter, and Bird rheological models were presented. Chilton and Stainsby (1998) stressed the limitations of these models and the shear rates at which they are valid. In an effort to solve the Rabinowitsch and Mooney equations presented in Equations 5-15 and 5-16, Chilton and Stainsby (1998) proposed to express the pressure drop for a Herschel–Bulkley fluid as: ⌬P 4j 8V ᎏ=ᎏ ᎏ L D D

ᎏ ᎏᎏ 冢 冣 冢ᎏ 4n 冣 冢 1 – x 冣冢 1 – ax – bx – cx 冣 n

3n + 1

n

1

1

2

where 4L0 0 x= ᎏ = ᎏ w Di⌬P

3

n

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1 a= ᎏ (2n + 1) 2n b = ᎏᎏ (n + 1)(2n + 1) 2n2 c = ᎏᎏ (n + 1)(2n + 1) For a Bingham slurry, j = or the coefficient of rigidity. For a power law pseudoplastic fluid, 0 = 0, j = p or the power law viscosity. For a Newtonian, Hagen–Poiseuille fluid, 0 = 0, = 1, j = or viscosity. 32V ⌬P 4 8V ᎏ=ᎏ ᎏ =ᎏ L D D D2

冢 冣

Defining an effective viscosity as ᎏ ᎏᎏ 冢 冣 冢ᎏ 4n 冣 冢 1 – x 冣冢 1 – ax – bx – cx 冣

8V * = j ᎏ Di

n–1

3n + 1

1

n

1

2

n

3

(5-40)

Chilton and Stainsby proposed their equation for a generalized Reynolds number as

VDi ReMR = ᎏ *

(5-41)

Using the value (defined by Equation 5-16), the authors indicated that 3 + 1 * = L ᎏ 4

冢

冣

and a modified Reynolds number is defined as 4VDi ReMR = ᎏᎏ L(3 + 1)

(5-42)

if the wall viscosity could be measured. The friction factor in the laminar regime is then expressed as 16 fn = ᎏ ReMR

(5-43)

In the turbulent regime, for Herschel–Bulkley fluids Chilton and Stainsby derived

冢

ReMR fn = 0.079 ᎏᎏ n2(1 – x)4

冣

–0.25

(5-44)

Chilton and Stainsby then proposed another modified Reynolds number: ReMR R⬘ = ᎏᎏ n2(1 – x)4

(5-45)

This indicates that Equation 5-44 is a modified Blasius equation (see Chapter 2): fn = 0.079(R⬘)–0.25

(5-46)

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5.21

Without further proof, they proposed to follow the Prandtl equation of Newtonian fluids as if it could be applied to non-Newtonian fluids: f n–0.5 = 4.0 log10(R⬘f n0.5) – 0.4

(5-47)

Deviations from experimental data were noticed at high Reynolds numbers due to the limitations of the Herschel–Bulkley model on which basis this model was developed. (See Figures 5-6 and 5-7.) Example 5-7 Lab tests are conducted on sewage sludge in a 6⬙ pipe. The density is 1018 kg/m3 (S.G. = 1.018). The yield stress is measured as 1.28 Pa. The slurry is a Herschel–Bulkley fluid mixture with a parameter j = 0.2. The power law coefficient is determined to be 0.74. Calculate the friction factor for a flow of 350 L/s in a 18⬙ pipe of 0.375⬙ thickness, assuming a wall shear stress of 1.6 Pa. Pipe inner diameter = (18 – 2 × 0.375) × 0.0254 = 0.438 m Pipe inner area = 0.25 × × 0.4382 = 0.151m2 Pipe inner speed = 0.35/0.151 = 2.31 m/s

0 1.28 x = ᎏ = ᎏ = 0.8 w 1.6

FIGURE 5-6 The friction factor versus the Chilton–Stainsby Reynolds number for Bingham mixtures. (From A. Chilton and R. Stainsby, 1998. Reproduced with permission of Journal of Hydraulic Engineering, ASCE.)

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FIGURE 5-7 The friction factor versus the Chilton–Stainsby Reynolds number for pseudoplastic mixtures. (From A. Chilton and R. Stainsby, 1998. Reproduced with permission of Journal of Hydraulic Engineering, ASCE.)

冢

8 × 2.31 * = 0.2 ᎏ 0.438

冣

(0.74–1)

ᎏ ᎏᎏ 冢 ᎏᎏ 4 × 0.74 冣 冢 1 – 0.8 冣冢 1 – ax – bx – cx 冣 3 × 0.74 + 1

冢

冢

1

2

1 = 0.366 ᎏᎏ 1 – ax – bx2 – cx3 8 × 2.31 * = 0.2 ᎏ 0.438

1

0.7

3

冣

冣 冢 ᎏᎏ 2.96 冣 –0.26

(2.12 + 1)

0.75

1 5 ᎏᎏᎏᎏᎏᎏ 1 – 0.403 × 0.064 – 0.343 × 0.0642 – 0.254 × 0.0643

冢

冣

* = 0.3725 ReMR = 1018 × 2.31 × 0.438/0.3725 = 2765 This is transition Reynolds Number between laminar and turbulent flow. In laminar regime fn = 16/ReMR = 0.0057.

fnV22 ⌬P Pressure drop ᎏ = ᎏ = 72 Pa/m Di L 5-4-3 The Wilson–Thomas Method The Wilson–Thomas Method was developed in the 1980s for yield pseudoplastic and power law slurries. Wilson (1985) and Thomas and Wilson (1987) assumed that the fluid

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5.23

is a continuum, but that in non-Newtonian cases the viscous sublayer was thicker than with the water. (See Chapter 2 for the definition of viscous sublayer.) Defining a velocity VN for a Newtonian fluid at the same wall shear stress w as for the flow of a non-Newtonian fluid, a bulk velocity V is defined as

冤

冢

冣

冢

1–n n+1 V = VN + Uf 2.5 ln ᎏ + 11.6 ᎏ 2 n+1

冣冥

(5-48)

with VN Di Uf ᎏ = 2.5 loge ᎏ Uf

冢

冣

(5-49)

Since Uf = (w/m)1/2, the Wilson–Thomas model requires the designer to assume a value of wall shear stress. The effective viscosity is defined as

冢

4n eff = ᎏ 3n + 1

冣 冢ᎏ D 冣 n–1

8V

n–1

(5-50)

i

which is slightly different than expressed by Equation 5-26. In the laminar flow regime, the shear rate is expressed as

w 8V 4n ᎏ=ᎏ ᎏ Di 3n + 1 K

冢 冣

1/n

(5-51)

For Bingham fluids, the Wilson–Thomas equation is written as

冤

冢

冣

冥

1–x V = VN + Uf 2.5 ln ᎏ + x (14.1 + 1.25x) 1+x

(5-52)

where x = 0/w ReMR = mVDi(1 – x)/ In the laminar flow regime 8V w 1 4 ᎏ = ᎏ 1 – ᎏ x + ᎏ x4 Di 3 3

冤

冥

(5-53)

(where p = plastic viscosity), which is essentially the Buckingham equation. For Bingham fluids in the laminar regime, if x Ⰶ 0.5 then 4 8V w = ᎏ + ᎏ 0 Di 3 To obtain the transition velocity from laminar to turbulent flow, the following approach based on the Wilson–Thomas method is recommended. In the turbulent regime, a Reynolds number based on the friction velocity Uf is defined as

DIUf Ref = ᎏ p

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The Wilson–Thomas velocity is defined by modifying Equation 5.52 to

冤

冢

冣冥

1–x V = Uf 2.5 ln Re + 2.5 ln ᎏ + x (14.1 + 1.25x) 1+x

(5-54)

The transition occurs at the intersection of Equations 5-53 and 5-54. The Wilson–Thomas method was derived from first principles and does not use empirical correction coefficients. It has proved to be correct for many slurries, but it sometimes overpredicts losses for the Carson slurries (described in Equation 3-52). Example 5-10 shows the method of calculation using the Wilson–Thomas equation. Figure 5-8 compares the Wilson–Thomas method with the Hanks–Dadia friction factor. Figure 5-9 compares it with experimental data.

5-4-4 The Darby Method: Taking into Account Particle Distribution Professor R. Darby (2000) from Texas A&M University recently published a new method to predict the friction factor of power law non-Newtonian slurries. It is, however, much closer to the domain of slurries and takes into account such concepts as the drag coefficient, to which the reader was exposed in Chapter 3. In a method reminiscent of the work of Wasp for compound heterogeneous flows (see Chapter 3), Darby stated that the overall pressure drop for a non-Newtonian fluid is essentially the pressure drop of the liquid phase plus the pressure drop due to the solids: ⌬Pm = ⌬Pf + ⌬Ps

(5-55)

for each fraction i of solids with an average diameter dpi, and with a volumetric concentration Cv, the individual pressure drop must be computed. In a first iteration, the pressure drop for the liquid phase is computed by treating it as homogeneous non-Newtonian liquid by the various methods described in this chapter. A Froude number for the solid particles is defined as

FIGURE 5-8 Comparison between the Wilson–Thomas and the Hanks–Dadia models for friction factor of Bingham slurries. (From A. D. Thomas and K. C. Wilson, 1985. Reproduced by permission of Canadian Journal of Engineering.)

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FIGURE 5-9 Comparison between the Wilson–Thomas and experimental data on pressure losses for a limestone slurry. (From A. D. Thomas and K. C. Wilson, 1985. Reproduced by permission of Canadian Journal of Engineering.)

V 2t V2 Fr2 = ᎏᎏ = ᎏᎏ (s/L – 1)d g (s/L – 1)gDi

(5-56)

where Vt is the terminal velocity of falling particles. At slip velocity Vr between the solids and the liquid phases, with V as the carrier speed of the suspension, a nondimensional pressure drop is defined as Vt ⌬s X = ᎏᎏ ᎏ Cv L(s/L – 1)gL V

冢 冣

2

(5-57)

If Vr = Vf – Vs, W r2 X= ᎏ 1 – Wr

(5-58)

where Wr = Vr /V. For small volume fraction 0 < Cv < 0.25, X = X0. For other concentrations, X = X0 + 0.1F R2(Cv – 0.25)

(5.59)

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Procedures for Newtonian Mixtures: 1. Calculate the Froude number. 2. Determine Vt from the Archimedean number: d 3p f g(s/L – 1) Ar = ᎏᎏ L2 with f being the liquid viscosity. If Ar < 15, the terminal velocity can be obtained from the Stokes equation If Ar > 15, the particle Reynolds number is obtained after rearranging the Dallavalle equation: Rep = [(14.42 + 1.827兹F 苶r苶)0.5 – 3.798]2 = dpVt L/L

(5-60)

3. Calculate the Froude number from Equation 5-56. 4. The value of ratio Wr is then calculated from the Molerus diagram (Figure 5-10). The values of X and ⌬Ps/L are then calculated from Equations 5-59 and 5-60. This step may be repeated for each range of particle size and summed up with other particle sizes to get an overall pressure drop for solids. However for Non-Newtonian mixtures additional procedures are needed.

FIGURE 5-10 Molerus diagram. (From R. Darby, 2000. Reproduced by permission of Chemical Engineering.)

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5.27

Terminal Velocity for Power Law Fluids For a solid particle settling in a non-Newtonian fluid, defining 1.33 + 0.37n Z = ᎏᎏ 1 + 0.7n3.7

(5.61)

as a nondimensional power law parameter and C1 = [(1.88/n)8 + 34]–1/8 as another nondimensional power law parameter, Darby expressed the modified Reynolds number as

冢

4.8 RePL = Z ᎏ 1/2 C d – C1

d nV 2–n t f

冣 =ᎏ K 2

(5.62)

with Cd the drag coefficient: 4g(m/L – 1)d Cd = ᎏᎏ 3V 2t

(5.63)

Friction Factors The transition from laminar to turbulent flow occurs at a critical Reynolds number: RePLC = 2100 + 875(1 – n)

(5.64)

Defining fL as the fanning factor in the laminar regime, and fT as the fanning factor in the turbulent regime, Darby derived the following equations:

␣ fn = (1 – ␣) fL + ᎏᎏ ( f –8 + f L–8)1/8 T

(5.65)

where the laminar flow friction factor is 16 fL = ᎏ RePL

(5.66)

0.0682n–1/2 fT = ᎏᎏ 1/(1.87+2.39n) RePL

(5.67)

At the transition from laminar to turbulent flow, the friction factor is expressed as: (0.414+0.757n) fTR = 1.79 × 10–4 exp(–5.24n)RePL

(5.68)

1 ␣= ᎏ 1 + 4⌬

(5.69)

⌬ = RePL – RePLC

(5.70)

and

Example 5-8 Slurry is required to flow in a 250 mm ID pipe and has the following characteristics:

0 = 15 Pa = 43 mPa · s = 1450 kg/m3

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K = 0.4 Pa · sn n = 0.8 d85 = 65 m V = 1.76 m/s Determine the friction factor. Check on critical Reynolds number: RePLC = 2100 + 875(1 – 0.8) = 2275 Z = 1.13 C1 = 0.4236 Calculations will be done by a series of iterations. Iteration 1. Assume the drag coefficient is Cd = 0.4, then

冢

4.8 RePL = 1.13 ᎏᎏ 0.41/2 – 0.4236

冣 = 597 2

16 16 fL = ᎏ = ᎏ = 0.026 RePL 597 0.0682 × 0.8–1/2 fT = ᎏᎏ = 0.0141 5970.264 ⌬ = 597 – 2275 = –1678 1 ␣= ᎏ 1 + 4–⌬

␣⬵0 ␣ fn = (1 – ␣)fL + ᎏᎏ ( f T–8 + f L–8)1/8 fn = fL = 0.026 By further iteration, the magnitude of the friction factor is refined.

5-5 TIME-DEPENDENT NON-NEWTONIAN SLURRIES Time-dependent non-Newtonian flows are difficult to model. There is a certain lapse of time to overcome before a stable friction factor can be obtained. Govier and Aziz (1972) have published cases in which the lapsing time was as high as 1000 minutes with Prembina crude oil. During this initial lapse of time, the pressure gradient for friction losses dropped from an initial value of 72.5 Pa/m (0.0032 psi/ft) down to 18 Pa/m (0.0008 psi/ft) by the time the flow had stabilized. Such a ratio of 4:1 is certainly not negligible. Govier and Aziz (1972) indicated that once the initial period of stabilization is reached, the general form of pressure loss equations are the same as for time-independent non-Newtonian fluids. At the entry to a pipe, the flow may be laminar, but at a certain distance, once the entrance effects are overcome, the flow can transit to turbulence. The start-up pressures for thixotropic slurries may be quite high, particularly when these slurries coagulate into gels inside the pipeline. During an initial of period of time, gels must be expelled from the pipeline. Crude oils, fuel oils, bentonite clay, and drilling

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5.29

muds are a concern to engineers. Water may be used as an expelling medium to clear up the pipeline. Positive displacement pumps are preferred for thixotropic slurries to overcome the high starting pressures. Various models have been developed for thixotropic fluids. These slurries are sometimes treated as Bingham plastics and sometimes as pseudoplastics, based on the experimental data from test work. The problem of pumping thixotropic froth in oil sand plants was a challenge to manufacturers of pumps, and for a while the belief was that only positive displacement pumps could be used. In 2000, the operators of oil-sand processing plants invited the manufacturers of slurry pumps to develop new appropriate pumps. Research is being conducted at the Saskatchewan Research Institute in Canada and is starting to yield new concepts of pump design.

5-6 EMULSIONS The concept of emulsions was introduced in Chapter 1. In an emulsion, one phase is suspended as droplets rather than particles. Tar or bitumen, for example, can be suspended in water up to a volumetric concentration of 70%. In some of the oil sand extraction processes in Canada, the situation can be further complicated by the addition of flocculants. The Venezuelan corporation, PDVSA, and its research branch, INTEVEP, conducted considerable research on Orimulsion™, a proprietary synthetic fluid composed of surfactants, bitumen droplets, and water. The surfactants keep the bitumen droplets in suspension. Nunez et al. (1996) published a comprehensive paper on the flow characteristics of concentrated emulsions in water with a volumetric concentration in the range of 70–85%.

5-7 ROUGHNESS EFFECTS ON FRICTION COEFFICIENTS Szilas et al. (1981) published the following equation for pseudoplastic oil flow in rough pipes at high Reynolds numbers:

冢

冣

8.03 1 4 1.414 ᎏ = ᎏ log10 (Remod(4fn)1–n/2) + 1.511/n 4.24 + ᎏ – ᎏ – 2.114 兹苶 fn n n n

(5-71)

However, this equation does not include any terms for wall roughness. Torrance (1963) developed a theoretical equation for yield pseudoplastics in fully turbulent flow (at high Reynolds numbers) in rough pipes as

冢

冣

2.65 1 RePLC ᎏ = 4.07 log10 ᎏ + 6.0 – ᎏ 兹苶 fn n

(5-72)

where RePLC = DnV 2–n /K8n–1. This applies to the region where the friction factor is independent of the Reynolds number, or at high Reynolds numbers. Govier and Aziz (1972) suggest the following procedure: 앫 Compute the friction factor for the slurry in a smooth tube using one of the methods described in Sections 5-2, 5-3, and 5-4. 앫 Using the Moody diagram determine the ratio of the friction factor for rough to smooth pipe at the value of the Reynolds number (using ReB, RePLC, or Remod).

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Aral and Kaylon (1994) focused on highly concentrated suspensions and investigated the effects of temperature as well as surface roughness. To take into account the pipe roughness in laminar and turbulent flows, SRC (2000) recommended the use of the equation of Churchill (1977):

冤冢 ᎏ Re 冣

fn = 2

8

12

+ (A + B)–1.5

冥

1/12

(5-73)

where

冦

冤冢 ᎏ Re 冣 7

A = –2.457 ln

冢

0.9

冥冧

+ 0.27 /Di

37,530 B= ᎏ Re

16

冣

16

(5.74) (5.75)

Example 5-9 Assuming turbulent flow and using the Wilson–Thomas model, calculate the bulk velocity for a flow with the following characteristics:

0 = 20 Pa w = 24 Pa = 0.016 Pa · s = 3 × 10–6 m Di = 141 mm = 1350 kg/m3 x = /w = 20/24 = 0.833 Iteration 1. Assume VN = 2.5 m/s, then Re = VDi/

= /(1 – x) = 0.016/(1 – 0.833) = 0.096 Re = VDi/ Re = 1350 × 2.5 × 0.141/0.096 = 4967 Relative roughness /Di = 3 × 10–6/0.141 = 0.000021 From Equations 5-74 and 5-75: A = {–2.457 ln[(7/4967) )0.9 + 0.27 × 0.000021)]16 = 3.86 × 1018 B = 1.1286 × 1014 fn = 0.00949 Iteration 2. Checking the value of VN. From Chapter 2:

w = fnV 2/2 (2 × 24)/(0.00949 × 1350) = 3.75 V = 1.935 m/s The friction velocity from Equation 2-5:

w苶 / = 0.133 m/s Uf = 兹苶

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Using Equation 5-52:

冤

1 – 0.833 V = 1.935 + 0.133 2.5 ln ᎏ + 0.833(14.1 + 1.25 × 0.833) 1.833

冥

2.815 m/s = 1.935 + 0.133[–5.989 + 12.61] Therefore, the equivalent Newtonian fluid velocity is 1.935 m/s, and the mean velocity of the Bingham slurry will effectively be 2.815 m/s. Up until 1990, one author after another tried to treat non-Newtonian slurries as homogeneous mixtures. The arbitrary assumption that the particle diameter was absent in many equations when the diameter was smaller than 44 m or 74 m (depending on the author) is now being challenged. The advent of new experimental methods, such as laser velocimeters, is helping engineers understand the complexity of turbulent non-Newtonian fluids. The work of Park et al. (1989) using laser velocimeters, and Pokryvalio and Grozberg (1995) using electro-diffusion techniques on non-Newtonian slurries was analyzed by Slatter et al. (1996), who confirmed the significance of high-intensity turbulence at the wall. This encouraged them to postulate a theory of particle roughness. Slatter et al. proposed that the d85 particle size be used for particle roughness. (This may be an attempt to correlate with the Nikrudase particle roughness of Newtonian slurries.) For large pipe, when the actual roughness exceeds the value d85, is used. For

< d85

dx = d85

> d85

dx =

For a yield pseudoplastic, Slatter et al. (1996) defined their Reynolds number, Rer, in terms of the friction velocity, consistency factor K, and power coefficient n, as well as roughness dx: 8U 2f Rer = ᎏᎏn 0 + K(8Uf /dx)

(5.76)

If Rer > 3.32, then smooth wall turbulence occurs, and the mean bulk velocity V is expressed as

冢 冣

V Di ᎏ = 2.5 ln ᎏ – 2.5 ln Rer + 1.75 Uf 2dx

(5.77)

If Rer ( 3.32, then fully developed rough turbulent flow occurs, and the mean bulk velocity V is expressed as

冢 冣

Di V ᎏ = 2.5 ln ᎏ + 4.75 Uf 2dx

(5.78)

This correlation produces an abrupt transition from smooth turbulent to fully turbulent flow at the wall of the pipe. We have already explained that the Wilson–Thomas model was based on the assumption that the viscous sublayer in non-Newtonian flows was thicker than with water, thus suppressing the effect of roughness. The work of Slatter, Thorscalden, and Petersen (1996) on mixtures of kaolin clay and sand indicates that this is not very achievable (Figure 5-11). The resultant pressure losses (Figure 5-12) are therefore higher. Figure 5-12 does indicate that the Torrance and the Wilson–Thomas models correlate well. Both models were based on mathematical assumptions at 20 year intervals.

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FIGURE 5-11 A comparison between the Wilson–Thomas and Slatter, Thorscalden, and Petersen models for the viscous sublayer. (From P. T. Slatter et al., 1996. Reproduced by permission of BHR Group.)

FIGURE 5-12 A comparison of pressure drop per unit length between the Slatter, Thorscalden, and Petersen and Wilson–Thomas models. (From P. T. Slatter et al., 1996. Reproduced by permission of BHR Group.)

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Example 5-10 A sand–kaolin mixture with a volumetric concentration of 19.5% is flowing in a 431 mm ID pipe at an average speed of 1.8 m/s. The yield stress is 5.5 Pa, the coefficient of consistency K = 0.124 Pa · sn, and the power coefficient n = 0.64. The specific gravity of the solids are 2.65 and the d85 = 131 m. Using the Slatter method, determine the friction factor. Solution Density of mixture:

m = Cv(s – L) + L m = 0.19(1265) + 1000 = 1240 kg/m3 Iteration 1. Assume fully turbulent flow:

冢

冣

V 0.5Di ᎏ = 2.5 ln ᎏᎏ = 4.75 = 23.26 Uf 131 × 10–6 fD/8 苶. If V = 1.8 m/s, then Uf = 0.0774 m/s, since Uf = V 兹苶 fD = 0.0147 8 × 1240 × 0.07742 = 1.78 Rer = ᎏᎏᎏᎏᎏ 5.5 + 0.124(8 × 0.0774/0.131 × 10–3)0.64 Iteration 2. Since Rer < 3.32, the equation to use is

冢

冣

V 0.5Di ᎏ = 2.5 ln ᎏ + 2.5 ln Rer + 1.75 Uf d85 V ᎏ = 18.51 + 1.44 + 1.75 Uf V ᎏ = 21.70 Uf 0.046 = (fD/8) fD = 0.01698 Uf = 0.0829 m/s

5-8 WALL SLIPPAGE A phenomenon encountered with non-Newtonian mixtures is a tendency for the low-viscosity constituent to migrate to regions of high shear and to lubricate the flow. One example is the core annular flow of crude oil in water, where the more viscous material is lubricated by the less viscous material. In the case of emulsions and certain non-Newtonian slurries, lubrication occurs by a slip layer of water on the wall. Mathematically, the concept of slip can be treated as a discontinuity. Heywood (1991) proposed to represent slip by the slip velocity Vs. In the laminar flow regime, the total flow in a pipe would be

冢 冣冕

D 2i Di Q = Vs ᎏ + ᎏ ᎏ 4 8 w

d␥ 2 ᎏ d dt

3 w 0

(5-79)

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Heywood (1991) noticed that there are no methods to evaluate slip in turbulent flows. Evaluation of slip in laminar flow is conducted using the coaxial cylinder described in Chapter 3. Nunez et al. (1996) indicate that the migration of viscous droplets from the wall in an emulsion exhibit the Segré–Silberberg effect. However, they pointed out that not all emulsions experience slip and that the phenomena of slip appeared to be characteristic of the crude oil or viscous component. In some respects, slip does reduce the friction factor by lubrication from the least viscous phase. However, Aral and Kaylon (1994) found that increasing the surface roughness tended to reduce or eliminate slip. One particular problem with emulsions is the fracture of the droplets under high shear rates. A form of comminution occurs as large droplets come in contact with other ones. Degradation of non-Newtonian slurries under high shear rates is not well documented. There is evidence that high shear rates occur in centrifugal pumps. Adequate clearance may reduce the degradation of the emulsion or slurry but tends to reduce the efficiency of the pump. Degradation can include a form of coalescence or formation of colloids and larger droplets or particles. Clay ball formation is encountered in dredging operations after passage through the pump.

5-9 PRESSURE LOSS THROUGH PIPE FITTINGS The method of the two K-factor was presented in Chapter 2 in Section 2.8. It was explained that Hooper had established a general relationship

冢

1 K1 K = ᎏ + K⬁ 1 + ᎏ DI-in Re

冣

where K1 is the value of K at a Reynolds number of 1 K⬁ is the value of K at high Reynolds number Di-in is the internal pipe diameter in inches Johnson (1982) reviewed some of the problems of pumping non-Newtonian mixtures. In his assessment of fittings for sewage, he indicated important discrepancies in the laminar regime with losses 2–4 times as much as those for water flows. In the turbulent regime, the losses were either of the order of those for water or higher. He recommended that further studies be conducted for laminar flow, but for turbulent flows, the concept of equivalent length be used. In Chapter 2, concepts of pressure losses for Newtonian liquids were examined. Edwards et al. (1985) reviewed the flow of non-Newtonian slurries in laminar regimes. They recommended that the modified Reynolds number (using ReB or Remod) be used to correlate with the loss factor Kf of Newtonian flows in laminar regimes. In certain fittings such as globe valves, turbulence is enhanced by the geometry of the fitting. Although the flow may be laminar in a straight pipe up to a transition to a Reynolds number of 2000, turbulence in the globe valve may actually start at much lower Reynolds number such as 900. Much more work is needed on loss factors for non-Newtonian flows in transition and turbulent regimes. Govier and Aziz (1972) had noticed the lack of any methodology to compute loss from pipe fittings with non-Newtonian flows. They proposed that the

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5.35

method of equivalent length be used, i.e., that the equivalent length of pipe fittings for Newtonian liquids be added to computations of total length for pressure loss of the nonNewtonian slurry.

5-10 SCALING UP FROM SMALL TO LARGE PIPES Non-Newtonian flows are complex. A lab test in a pumping loop can yield very useful rheology data about the pressure drop. Scaling up to larger pipes is one method of predicting pipeline flows. Heywood et al. (1992) proposed the following methods: 앫 For laminar flow, the plot of the shear rate (8V/Di) against the shear stress (Di⌬P/4L) is independent of the pipe diameter and the pipe roughness, so that experimental data could be converted directly into practical data for pipeline design. 앫 For turbulent flows, the Bowen method, which is essentially a modification of the Blasius method, should be used. Bowen (1961) suggested the following modification to the Blasius equation: Dix = kV w

(5.80)

The shear stress is plotted against the flow rate Q to obtain the magnitude of the exponent w. The result is then used to plot /V w against the diameter Di to obtain the values of k and x. The intersection of the turbulent and laminar flow curves gives the transition point. Kenchington (1972) showed that this method showed great discrepancy when the ratio of diameter between the large field pipe and the lab pipe exceeded 6 to 12 folds, so that large pipe tests may be needed. The Bowen method assumed the same range of roughness between a lab and a field pipe. It is therefore important to apply correction factors for roughness when using this method.

5-11 PRACTICAL CASES OF NON-NEWTONIAN SLURRIES The equations presented in the previous sections of this chapter are fairly complex and are based on so many assumptions that the practical engineer may feel lost. Some real examples are needed to guide the designer of non-Newtonian systems.

5-11-1 Bauxite Residue Want et al. (1982) reviewed the design of a bauxite residue pipeline for Alcoa Australia. The plant disposed 4.75 Mtpy (million tonnes per year) of alumina. Tests conducted on samples confirmed that the rheology of the slurry at concentrations in excess of 45% by weight could be expressed by the Carson equation:

冤 冢 冣冥

du w1/2 = 1/2 0 + ⬁ ᎏ dr

1/2

where du/dr is expressed by Equation 5-15 and Equation 5-16.

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In the case of Kwina red mud, tests indicated that the modified consistency coefficient K⬘ (Equation 5-18) was a complex function of the weight concentration: K⬘ = 3.88 × 10–18 (100 Cw)10.57 Pa/s and that the value of was

= 2.55 × 106(Cw × 100)–4.19 for Cw ⱕ 51% = 6.685 (Cw × 100)–0.926 for Cw > 51% Applying the Dodge–Metzner model Want et al. (1982) applied Equation 5-20 to express the consumed power under turbulent conditions as: (Q)3 150 – 100 Cw ᎏᎏ eT = 1.44 × 10–4 fn ᎏ D 5i (100 Cw)1.5

冤

冥

2

in kW/km, and in laminar conditions as: 64mQ(150 – 100Cw) eL = 0.393 K⬘ ᎏᎏᎏ 3 × 104 Cw

冤

冥 冤ᎏ D 冥 1+

1+3

1

in kW/km

i

Want et al (1982) discussed the thixotropic nature of red mud at high concentration, and the importance of flocculants and dispersants on the rheology of this slurry. Referring to Figure 5-13, it is clear that there is a change in the pressure drop per unit length as the weight concentration is increased and the flow changes from turbulent to laminar. This drop in pressure to a minimum at such a transition is often misunderstood, particularly because it goes against the concepts examined in Chapter 4. The important parameters include the diameter of the pipe, so that there is an optimum diameter, and an optimum weight concentration for a given tonnage of fine solids to be transported. Slurries may therefore be pumped at very high concentrations (Figure 5-14) using positive displacement pumps (Figure 5-15) over long distances, provided that the correct weight concentration is used near the turbulent to laminar transition region. Example 5-11 Using the data obtained by Want, examine pumping red mud bauxite residues at a speed of 1.74 m/s in a 141 mm I.D. pipe at weight concentrations of 45% and 60%. Determine the required power for a horizontal pipeline, 3 km long. Assume a density of 1350 at 45% and 1800 at 60%. Solution At Cw = 45%: K⬘ = 3.88 × 10–18 (100Cw)10.57 K⬘ = 1.157

= 2.55 × 106 (100 Cw)–4.19 = 0.302 Q = AV = × 0.25 × 0.1412 × 1.74 = 0.0272 m3/s

冤

64 × 1350 × 0.0272(150 – 45) eT = 0.393 × 1.157 ᎏᎏᎏᎏ 3 × 104 × 0.45 eT = 188 kW/km For 3 km, this is equivalent to 565 kw or 758 hp.

冥 冤ᎏ 0.141 冥 1.3

1

1.906

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FIGURE 5-13 Pressure drop per unit length for red mud tailings. (From F. M. Want et al., 1982. Reproduced by permission of BHR Group.)

At Cw = 60%: K⬘ = 3.88 × 10–18 (60)10.57 = 24.2

= 6.685(60)–0.926 = 0.151

冤

64 × 1800 × 0.0272 (150 – 60) eL = 0.393 × 24.2 ᎏᎏᎏᎏ 3 × 104 × 0.6 eT = 1041.3 kW/km For 3 km, eL = 3123.9 kW or 4186 hp.

冥 冤ᎏ 0.141 冥 1.3

1

1.906

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FIGURE 5-14 Slurry can be pumped in a non-Newtonian regime at high volumetric concentrations. (Courtesy of Geho Pumps.)

5-11-2 Kaolin Slurries Slatter et al. (1996) reported that Kemblowski and Kolodziejski (1973) found that the Dodge and Metzner model did not well represent the flow of kaolin slurries. They derived the following empirical equation: 0.3164 4fn = ᎏ 0.25 ReMR and more generally: E 1/ReMR 4fn = ᎏ m ReMR

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FIGURE 5-15 The pumping of high concentration slurries and pastes may require positive displacement pumps. (Courtesy of Geho Pumps.)

where E, m, and are empirical parameters and functions of the apparent flow behavior index (defined in Equation 5-16) and the modified Reynolds Number ReMR is defined by Equation 5-17.

5-12 DRAG REDUCTION Ippolito and Sabatino (1984) showed that the addition of 3% salt to water tended to reduce the friction factor of bentonite suspensions. Sauermann (1982) indicated that the addition of 0.2 kg/ton of tripolyphsophate (Na5P3O10) to gold slimes at a weight concentration of 67.7% , and with particles smaller than 50 m, reduced the pressure gradient in laminar flow by as much as 30%. Some viscosity reducing agents were discussed in

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Chapter 3.There are, however, very little data published on other methods for reducing friction losses of non-Newtonian slurries. Schowalter (1977) discussed some aspects of drag reduction in non-Newtonian slurries and reported certain cases of mixtures with a pressure drop actually lower than that of water.

5-13 PULP AND PAPER Pulp and paper pump slurries behave as non-Newtonian slurries. The following equations have been reported by the Cameron Hydraulic Data book of IDP (1995), based on work at the University of Maine. A modified Reynolds number is defined as D 0.205 Vsg I ReMR = ᎏᎏ C1.157

(5-81)

where C = % consistency of the pulp, oven dry g = 32.2 ft/s = density in slugs/ft3 Vs = velocity, defined in ft/sec A modified friction factor is defined as 3.97 f= ᎏ 1.636 ReMR

(5-82)

A special equation for friction losses is therefore defined as fV 2LK0 Hf = ᎏ DI

(5-83)

The correction factor K0 depends on the type of pulp. It is considered to be 1.00 for unbleached sulfite softwood, 0.90 for bleached sulfite softwood, 0.90 for unbleached kraft softwood, 0.90 for soda hardwood, 0.90 for reclaimed fiber, 1.0 for presteamed groundwood–softwood, and 1.42 for stoned groundwood–softwood. (IDP 1995). The following program calculates friction factors. 10 PRINT “program for stock , pulp and paper” PRINT “based on the curves of the University of Maine” PRINT “ which are correlations to the data of Brecht and Heller” pi = 4 * ATN(1) INPUT “name of project”; pr$ INPUT “name of client “; nc$ INPUT “date of calculations”; dat$ INPUT “ type of pulp”; pul$ INPUT “consistency of pulp in percent”; CS C = CS/100 15 INPUT “ choose between (1) SI units (m) and (2) US units (feet) “; NB IF ABS(NB) < 1 THEN GOTO 15 IF ABS(NB) > 2 THEN GOTO 15 IF (NB > 1) AND (NB < 2) THEN GOTO 15 GOSUB conversion 18 IF NB = 1 THEN INPUT “inner diameter (m)”, D1M IF NB = 2 THEN INPUT “inner diameter (ft)”; D1US

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IF NB = 1 THEN d1 = D1M * dl IF NB = 1 THEN D1US = D1M/.3048 IF NB = 2 THEN d1 = D1US * dl IF NB = 1 THEN INPUT “pulp flow rate (m3/hr)”, qm IF NB = 2 THEN INPUT “pulp flow rate in USgpm”; qus IF NB = 1 THEN q = qm/60000 IF NB = 2 THEN q = qus * 3.785/60000 a = .25 * pi * d1 ^ 2 v = q/a PRINT USING “pulp speed = ###.### m/s”; v vus = v/.3048 PRINT USING “PULP SPEED = ##.### ft/s”; vus IF (C > .03) AND (vus > 8) THEN PRINT “speed exceeds 8 ft/s please use larger pipe size” IF C > .03 THEN GOTO 25 IF (C < .02) AND (vus > 10) THEN PRINT “speed exceeds 10 ft/s please use larger” IF C < .02 THEN GOTO 25 IF vus > 9 THEN PRINT “SPEED EXCEEDS 9 FT/S, PLEASE USE LARGER PIPE” 25 INPUT “DO YOU WANT TO USE A LARGER PIPE (Y/N)”; p$ IF p$ = “Y” OR p$ = “y” THEN GOTO 18 IF NB = 1 THEN INPUT “DENSITY OF STOCK IN KG/M3 (OFTEN ASSUMED TO BE 1000 KG/M3 “; DENSM IF NB = 1 THEN DENSUS = DENSM * (62.4/1000) IF NB = 2 THEN INPUT “DENSITY OF STOCK IN LBS/CU.FT (OFTEN ASSUMED TO BE 62.4 LBS/CU.FT”; DENSUS REM0 = D1US ^ .205 * vus * DENSUS/CS ^ 1.157 FM0 = 3.97/REM0 ^ 1.636 PRINT USING “MODIFIED REYNOLDS NUMBER = #######”; REM0 PRINT USING “MODIFIED FRICTION FACTOR = #.####”; FM0 PRINT “ please choose between the following pulps” PRINT “ 1- unbleached sulfite “ END conversion: IF NB = 1 THEN dl = 1 IF NB = 2 THEN dl = .3048 IF NB = 1 THEN ql = 3875/60 IF NB = 2 THEN ql = 60/3875 RETURN

5-14 CONCLUSION The world of non-Newtonian flows is very complex and encompasses very different flows, including pulp and paper, food, plastics, and clays. The use of various equations developed by researchers do not yield the same values of the friction coefficient. The problem is compounded by the fact that many rheological tests are conducted in tubes and in laminar flows, yielding values of consistency factor K and exponent n outside the range of turbulent flows. The practical engineer is often left to use his engineering judgment to use the appropriate equation. Proper tests of the slurry flow at the correct range of shear stresses are essential to avoid errors. Various reference books on the subject have equations similar to Equations 5-20 to 5-25 without emphasizing the limitations to their use. Because many of the equations for friction loss factors are implicit and require iteration methods, the use of personal computers is encouraged. The flow of non-Newtonian slurries is complex and requires a significant energy in-

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put. Methods have been developed over the years to reduce friction losses. These include dilution, reduction of volumetric concentration of solids, removal of all flocculants, provisions for air purging of the pipeline, addition of high-aspect-ratio fibers, addition of deflocculants (soluble ionic compounds), reduction of the angularity of particles, and addition of viscosity reducing agents. Not all these methods are always possible, and some require capital investment. It is hoped that the various worked examples in this book will help the practical engineer to design slurry pipelines. It may be necessary to use more than one method and compare results. There are many advantages to pumping slurries at high concentrations, such as concentrates from process plants, food pastes, and ceramic slurry for the manufacture of new materials.

5-15 NOMENCLATURE a A Ar b B C1 CD Cv Cw d85 dp dP/dz dx Di eT E fD fL fN fNC fNL fNLY fNPL fPLT fT fTR Fr g gc He Hemod j K K⬘

Nondimensional parameter and function of Hedstrom number Factor for friction in the Churchill equation Archimedean number Nondimensional parameter Factor for friction in the Churchill equation Nondimensional power law parameter Drag coefficient Volume fraction of solid particles in the slurry mixture Weight concentration Particle diameter passing 85% (m) Diameter of particle Pressure gradient per unit length Equivalent roughness Conduit inner diameter (m) Consumed energy Empirical coefficient Darcy friction factor Laminar component of fanning friction factor for a Bingham plastic Fanning friction factor Fanning friction at transition between laminar and turbulent flow Laminar component of fanning friction factor Laminar component of fanning friction factor for a yield pseudoplastic Fanning friction factor for a pseudoplastic in a laminar regime Tomita laminar friction factor Turbulent component of fanning friction factor Darby friction factor for transition from laminar to turbulent flows with power law fluids Froude number Acceleration due to gravity (9.8 m/s2) Conversion from slugs to pounds mass in USCS units Hedstrom number for Bingham plastic Modified Hedstrom for yield pseudoplastic Herschel–Bulkley parameter Coefficient of consistency for power law fluids Modified coefficient of consistency for power law fluids

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L m n P PL Pst Q R⬘ Re ReB ReBc Rec Remod ReMR RePL RePLC Rer Rep Uf V VN Vr Vs Vt VTR Wr x xc Z

5.43

Length of conduit or pipe Exponent in Darby’s equation for fanning friction factor calculations Flow behavior index for pseudoplastic flows Pressure Plasticity number Start-up pressure to pump a non-Newtonian slurry Flow rate Chilton and Stainsby proposed modified Reynolds number Reynolds number Reynolds number for a Bingham plastic using the coefficient of rigidity for viscosity Critical transition Reynolds number for a Bingham plastic using the coefficient of rigidity for viscosity Reynolds number at transition Modified Hedstrom number for yield pseudoplastic Modified Reynolds number for a power law fluid Tomita Reynolds number for a power law fluid Tomita Reynolds number for a power law fluid at transition Slatter Reynolds number Particle Reynolds number Friction velocity Speed Newtonian velocity Slip velocity Velocity of solids Terminal velocity of falling particles Transition velocity from laminar to turbulent flows Ratio of slip velocity to slurry speed Ratio of the yield stress to the wall shear stress Ratio of the yield stress to the wall shear stress at the transition from laminar to turbulent flow Settling factor for a non-Newtonian fluid

Subscripts L Liquid m Mixture p Particle Greek symbols ␣ Function for use of laminar and friction factors ⌬ Increment Concentration by volume in decimal points ⌬P Pressure drop ⌬m Density change for the mixture Density L Density of liquid carrier in kg/m3 m Density of slurry mixture in Kg/m3 s Density of solids Modified flow behavior index for pseudoplastic flows ␥ Shear strain d␥/dt Wall shear rate or rate of shear strain with respect to time

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⬁ a e e L p * ⬁ 0 w yp

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CHAPTER FIVE

Linear roughness (m) Coefficient of rigidity of a non-Newtonian fluid, also called Bingham plastic viscosity Coefficient of rigidity at high shear rate Carrier liquid dynamic viscosity Apparent viscosity of a pseudoplastic fluid Effective pipeline viscosity Wilson–Thomas effective viscosity Viscosity of liquid carrier Effective pipeline viscosity for pseudoplastic Effective viscosity for Hagen-Poiseuille fluid Bingham plastic limiting viscosity of slurry mixtures (Poise) Empirical function of the power exponent n Pythagoras number (ratio of circumference of a circle to its diameter) Shear stress at a height y or at a radius r Yield stress for a Bingham plastic Wall shear stress Yield stress for pseudoplastic

5-16 REFERENCES Abulnaga, B. E. 1997. Slurcal—Computer Program for Non-Newtonian Flows. Fluor Daniel Wright Engineers, Vancouver, BC, Canada (unpublished). Aral, B. K., and D. M. Kaylon. 1994. Effects of temperature and surface roughness on time dependent development of wall slip in steady torsional flow of concentrated Suspensions. Journal of Rheology, 38, 957–972. Bowen, R. L. 1961. Chemical Engineering, 143–150. Buckingham, E. 1921. On plastic flow through capillary tubes. ASTM Proceedings, 21, 1154. Chilton, R. A., and R. Stainsby. 1998. Pressure loss equations for laminar and turbulent nonNewtonian pipe flow. Journal of Hydraulic Engineering, 124, 5, 522–529. Clifton, R. A, and R. Stainsby. 1998. Pressure loss for laminar and turbulent non-Newtonian pipe flow. Journal of Hydraulic Engineering, 124, 5, 522–529. Churchill, S. W. 1977. Friction factor equation spans all fluid flow regimes. Chemical Engineering 84, 7, 91–92. Darby, R. 2000. Pressure drop of non-Newtonian slurries, a wider path. Chemical Engineering, 107, 5, 64–67. Darby, R. 1981. How to predict the friction factor for flow of Bingham plastics. Chemical Engineering, 88, 26, 59–61. Darby, R., R. Mun, and V. Boger. 1992. Prediction friction loss in slurry pipes. Chemical Engineering, September. Dodge, D. W., and A. B. Metzner. 1959. Turbulent flow of non-Newtonian systems. Am. Inst. Chem. Engr., 5, 2, 189–204. Edwards, M. F., M. S. M. Jadallah, and R. Smith. 1985. Head losses in pipe fittings at Low Reynolds Number. Chem. Engr, Res. Des., 63, 1, 43–50. Govier, G. W., and K. Aziz. 1972. The Flow of Complex Mixtures in Pipes. New York: Van Nostrand Reinhold. Hanks, R. W. 1962. A Generalized Criterion for Laminar–Turbulent Transition in the Flow of Fluids. Union Carbide report. Hanks, R. W., and D. R. Pratt. 1967. On the flow of Bingham plastic slurries in pipes and between parallel plates. Soc. Petr. Eng. Journal, 7, 342–346. Hanks, R. W., and B. H. Dadia. 1971. Theoretical analysis of the turbulent flow of non-Newtonian slurries in pipes. American Journal of Chemical Engineering, 17, 554–557.

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Hanks, R. W., and B. L. Ricks. 1975. Transitional and turbulent pipeflow of pseudoplastic fluids. Journal of Hydronautics, 9, 39–44. Hedstrom, B. O. A. 1952. Flow of plastic materials in pipes. Ind. Eng. Chem., 44, 651–656. Herschel, W. H., and R. Bulkley. 1928. Measurements of consistency as applied to rubber benzene solution. Proc. ASTM, 26, Part 2, 621–633. Herzog, R. L., and K. Weissenberg. 1928. Kolloid Z, 46, 277. Heywood, N. I. 1991. Pipeline design for non-settling slurries. In Slurry Handling, Brown, N. P., and N. I. Heywood (Eds.). New York: Elsevier Applied Sciences. Heywood, N. I., D. C. H. Cheng, and A. J. Carlton. 1992. Slurry systems. In Piping Design Handbook, McKetta, J. J. (Ed.). New York: Marcel Dekker, pp. 585–622 Heywood, N. I., and J. F. Richardson. 1978. Rheological behavior of flocculated and dispersed kaolin suspensions in pipe flow. Journal of Rheology, 22, 6, 559–613. IDP (now called Flowserve). 1995. Cameron Hydraulic Data. NJ: IDP. Ippolito, M., and C. Sabatino. 1984. Rheological behavior and friction resistance of colloidal aqueous suspensions. In Proceedings of the IXth International Congress on Rheology, Mexico. Mexico: Universidad Nacional Autonoma de Mexico. Irvine. 1988. Experimental measurements of isobaric thermal expansion coefficients of Non-Newtonian fluids. Heat Transfer, 1, 2, 155–163. Johnson, M. 1982. Non-Newtonian fluid system design. Some problems and their solutions. In 8th International Conference on the Hydraulic Transport of Solids in Pipe. Johanesburg, South Africa. Cranfield, UK: BHR Group. Kemblowski, Z., and J. Kolodziejski. 1973. Flow resistances of non-Newtonian fluids in transitional and turbulent flow. Int. Chem. Eng., 13, 265–279. Kenchington, J. M. 1972. In Proceedings of the 2nd International Conference on Hydraulic Transportation of Solids.Cranfield, UK: BHR Group. Metzner, A. B., and J. C. Reed. 1955. Flow of non-Newtonian laminar, transition and turbulent regions. Am. Inst. Chem. Eng. Journal, 1, 4, 434. Molerus, O. 1993. Principles of Flow in Disperse Systems. London: Chapman and Hall. Mooney, M. J. 1931. Explicit formulas for slip and fluidity. Journal of Rheology, 2, 2, 210–222. Nunez, G. A., M. Briceno, C. Mata, and H. Rivas. 1996. Flow characteristics of concentrated emulsions of very viscous oil in water. The Journal of Rheology, 40, 3, 405–423. Park, J. T., R. J. Munnheimer, T. A. Grimley, and T. B. Morrow. 1989. Pipe flow measurements of a transparent Non-Newtonian slurry. Journal of Fluids Engineering, 111, 331–336. Porkryvailo, N. A., and Y. G. Grozberg. 1995. Investigation of structure of turbulent wall flow of clay suspensions in channel with electro diffusion method. In Proceedings of the 8th International Conference on Transport and Sedimentation of Solid Particles, Prague, Czech Republic. Rabinowitsch, B. 1929. Veber die viskositat und elastizitat von solen. Z. Phisik Chem. Ser. A., 145, 1–26. Ryan, N. M., and M. M. Johnson. 1959. Transition from laminar to turbulent flows in pipes. Amer. Inst. of Chem. Engr., 5, 433–435 Sauermann, H. B. 1982. The Influence of particle diameter on the pressure gradients of gold slimes pumping In Proceedings of the 8th International Conference on the Hydraulic Transport of Solids in Pipes. Jahannesburg, South Africa, August 1982, Paper E1, pp. 241–246. Cranfield, UK: BRHA Group. Schowalter, W. R. 1977. Mechanics of Non-Newtonian Fluids. New York: Pergamon Press. Slatter, P. T., G. S. Thorvaldsen, and F. W. Petersen. 1996. Particle roughness turbulence. In Proceedings of the 13th International Hydrotransport Symposium on Slurry Handling and Pipeline Transport, Johannesburg, South Africa. Cranfield, UK: BRHA Group. SRC. 2000. Slurry Pipeline Course, May 15–16, 2000, Saskatchewan Research Centre, Saskatoon, Canada. Szilas, A. P., E. Bobok, and L. Navratil. 1981. Determination of turbulent pressure loss Non-Newtonian oil flow in rough tubes. Rheol Acta, 20, 487–496. Thomas, A. D., and K. C. Wilson. 1987. New analysis of non-Newtonian turbulent flow, yield power law fluids. Canadian Journal of Chemical Engineering, 65, 335–338. Tomita, Y. 1959. On the fundamental formula of non-Newtonian flow. Bulletin of the Japan. Soc. Mech.. Engr., 2, 7, 469–474.

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Torrance, B. McK. 1963. Friction factors for turbulent non-Newtonian fluid flow in circular pipes. South African Mechanical Engineer, 13, 4, 89–91. Want, F. M., P. M. Colombera, Q. D. Nguyen, and D. V. Boger. 1982. Pipeline design for the transport of high-density bauxite residue slurries. In Proceedings of the 8th International Conference on the Hydraulic Transport of Solids in Pipes, Johannesburg, South Africa. Cranfield, UK: BRHA Group. Wasp, E., J. Penny, and R. Ghandi. 1977. Solid-liquid flow slurry pipeline transportation. Clausthal, Germany: Trans Tech Publications. Wilson, K. C. 1985. A new analysis of turbulent flow of non-Newtonian fluids. Canadian Journal of Chemical Engineering, 63, 539–546. Further readings Abulnaga, B. E. 1997. Channel 1.0 Computer Program For an Open Channel Slurry Flow. Fluor Daniel Wright Engineers. Vancouver, BC, Canada. Internal report. Al Fariss, T. and K. L. Pinder. 1987. Flow through porous media of a shear-thinning liquid with yield stress. Canadian Journal of Chemical Engineering, 65, 391–405. Bouzaiene, R., and D. Hassani-Ferri. 1992. A selection of pressure loss predictions based on slurry/backfill characterization and flow conditions. C.I.M. Bulletin, 85, 959, 63–68. Chlabra, R. P., J. F. Richardson, and R. Darby. 2000. Non-Newtonian flow in process industry, fundamentals and engineering application. Chemical Engineering, 107, 4, Draad, A. A., G. D. C. Kuiken, and F. T. M. Nieuwstadt. 1998. Laminar turbulent transition in pipe flow for Newtonian and non-Newtonian fluids. Journal of Fluid Mechanics, 377, D25, 267–312. Hedstrom, B. O. A. 1952. Flow of plastic materials in pipes. Journal of Industrial Engineering Chemistry, 44, 651–656. Sandall, O. C., O. T. Hanna, and K. Amurath. 1986. Experiments on turbulent non-Newtonian mass transfer in a circular tube. Am. Inst. Chem. Eng. Journal, 32, 2095–2098. Steffe, J. F., and R. G. Morgan. 1986. Pipeline design and pump selection for non-Newtonian fluid foods. Food Technology, 40, 78–85. Wheeler, J. A., and E. H. Wissler. 1965. Friction factor: Reynolds number relation for the steady flow of pseudoplastic fluids through rectangular ducts. Am. Inst. Chem. Eng. Journal, 11, 207–216.

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CHAPTER 6

SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

6-0 INTRODUCTION The design of mineral processing plants and tailings disposal systems often includes gravity flows in open channels. Such flows are often called slack flows. They involve a free boundary to the atmosphere. In the past, many launders were designed using rules of thumb; however, the development of large mines requires a rigorous scientific approach. Most of the published papers on sediment transportation in open channels dwell extensively on the geophysics of canals and rivers. The field of open channel hydraulics is so vast and complex that the reader may have to consult various reference books such as Graf (1971). One subject of great interest to civil engineers is the carrying capacity of the channel for sediments. This is often termed the sediment load and is measured as a function of the flow rate, width of the channel, and sediment concentration. Using channels to transport solids has limitations due to the fact that no pumps are used to force the flow. Many papers have been published over the years on the geophysics of rivers and the maximum permissible speeds used to avoid scouring and removal of bed materials. Transferring the knowledge about scouring speeds of canals and rivers into useful information for a designer of a hydrotransport system is not a straightforward process. In fact there, is not a single unified mathematical model to represent slurry flows in open channels. In this chapter, a methodology is presented to estimate the friction losses for slurry flows in open channels, cascades, drop boxes, and distribution boxes. In the last twenty years, new developments in thickeners encouraged various operators to develop the concept of adding flocculants to launders. Tailings and concentrate slurries are thereby allowed to flow at higher and higher concentrations in gravity modes. The engineer must take into account the rheology, particularly certain aspects of high yield stress and nonNewtonian characteristics. Mineral processing plants often divide or combine flows in drop boxes, distribution boxes, and plunge pools. The design principles of such entities are presented at the end of the chapter. 6.1

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6-1 FRICTION FOR SINGLE-PHASE FLOWS IN OPEN CHANNELS The words flume, launder, open channel, and slack flow are often used to express the same thing. In the following discussion, these words will be used interchangeably. Even though launders are crucial to mining, very little research on the subject has been published. Despite the lack of reference material on launders for slurries, it is important to start from basic principles. The analysis will focus initially on water flows. The reader will then be introduced to the complexity of slurry flows. The reader should appreciate that an upper practical limit on these flows is a 65% concentration of solids by weight. Since the flow does not fill the launder or pipe, the hydraulic diameter is the defined as the equivalent diameter of flow for an open channel. The hydraulic radius is defined as the ratio of the area of the flow to the wetted perimeter. It is also called the hydraulic mean depth in certain European books. A RH = ᎏ P

(6-1)

FIGURE 6-1 Large concrete structures offer a method of conveying large quantities of slurry. This structure was built to convey 150,000 tons per day of soft high clay tailings at a Peruvian copper mine.

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And the hydraulic diameter is 4A DH = ᎏ P

(6-2)

Figure 6-2 shows various possible shapes for open launders with methods to estimate wetted area and hydraulic radius. The most common launders in mining and dredging are, however, circular and rectangular in shape. Sometimes a circular pipe is opened and vertical walls are added to produce a U-shape. The friction loss for a closed channel and steady-state single-phase flow was examined in Chapter 2. Using the Darcy factor, the head loss for an open launder can be expressed in terms of the hydraulic radius: fDLU 2 h= ᎏ 2g(4RH)

(6-3)

Since the Darcy friction coefficient fD is usually accepted as four times the fanning friction coefficient fN, Equation 6-3 for a slurry may be rewritten in terms of the fanning factor fN, discussed in Chapter 2. fNLU 2 h= ᎏ 2gRH

(6-4)

For a fully developed and uniform flow, the slope or energy gradient of an open launder is established in terms of the head loss per unit of length (Henderson, 1990): fNU 2 H S= ᎏ = ᎏ 2gRH L

2

2 < A = R2( – sin cos ) P = 2R R( – sin cos ) RH = ᎏᎏᎏ 2

(6-5)

A = BH H P = 2H + B BH RH = ᎏ 2H + B

B

2 = RH = DI/4

2

2 > =– A = R2( – sin  cos ) P = 2R R( – sin  cos ) RH = ᎏᎏᎏ 2

H

if H > R A = R[2(H – R) + R/2] R = 2(H – R) + R R[2(H – R) + R/2] RH = ᎏᎏᎏ 2(H – R) + R

2R

FIGURE 6-2 Hydraulic radius for shapes of open channels.

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or fDU 2 H S= ᎏ = ᎏ 8gRH L

(6-6)

Many models for open channel flows of water are based on the Chezy number and the Manning number. The Chezy number is inversely proportional to the square root of the friction factor: Ch =

ᎏ 冪莦 f

(6-7)

ᎏ 冪莦 f

(6-8)

8g D

or Ch =

2g N

The Manning number is a function of both the hydraulic radius and the friction factor: RH1/6 n= ᎏ Ch

(6-9)

or RH1/6 n= ᎏ 2g ᎏᎏ fN

(6-10)

冪 莦莦

Some experimental values for the Manning number “n” are shown in Table 6-1 as derived for water flows. These values are not correct for transportation of solids, particularly solids that introduce a new roughness factor that we will discuss. This table is presented as a reference for dirty water, very dilute mixtures, or decant water that are present in mining and tailings circuits but do not constitute real slurries. Green et al. (1978) summarized the research activities of the U.S. Army Corps of Engineers who derived the following relationship between the hydraulic radius and effective roughness of the channel (in USCS units): RH0.1667 n = ᎏᎏᎏ 23.85 + 21.95 log(RH/ks)

(6-11)

where RH = hydraulic radius in feet k = effective linear roughness in feet n = Manning number in ft–1/3/sec The linear roughness ks (Table 6-2) is also used to compute the flow of water in open channels. The Ministry of Transport (1969) recommends the following equation for flow of viscous liquids in open channels: 1.225(/) k 苶2苶R 苶H 苶S苶 log ᎏ + ᎏᎏ V = – 兹3 14.8 RH RH 兹3苶2苶R 苶苶S H苶

冤

where = absolute viscosity of fluid = density of fluid S = slope or energy gradient g = acceleration due to gravity

冥

(6-12)

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TABLE 6-1 Typical Values for the Manning Number “n” for Water Flows (Do Not Use for Slurries) Channel surface Glass, plastic, machined metal surface Smooth steel surface Sawn timber, joints uneven Corrugated metal Smooth concrete Cement, plaster Concrete culvert (with connection) Glazed brick Concrete, timber forms, unfinished Untreated gunite Brickwork or dressed masonry Rubble set in cement Earth excavation, clean, no weeds Earth, some stones and weeds Natural stream bed, clean and straight Smooth rock cuts Channels not maintained Winding natural channels with pools and shoals Very weedy, winding, and overgrown natural rivers Clean alluvial channels with sediments

Manning factor “n,” ft–1/3 s–1

Manning factor “n,” m–1/3 s–1

0.011 0.008 0.014 0.016 0.0074 0.011 0.009 0.009 0.014 0.015–0.017 0.014 0.017 0.020 0.025 0.020 0.024 0.034–0.067 0.033–0.040 0.075–0.150 0.031 (d75)1/6 using d75 size in feet

0.016 0.012 0.021 0.024 0.011 0.016 0.013 0.013 0.0208 0.022–0.0252 0.0208 0.0252 0.022 0.037 0.030 0.035 0.050–0.1 0.049–0.059 0.111–0.223 0.0561 (d75)1/6 using d75 size in m

After Manning (1895) and Henderson (1990).

Having read Chapters 1–3, the reader must have become aware that sizing pipe involves a straightforward relationship between the flow rate, the cross-sectional area of the pipe, and the required velocity. In the case of open launders, particularly those of rectangular and U-shape, the main concern is to avoid spills. At certain bends, around certain obstacles, or at a sudden reduction of physical slope, the flow may slow down considerably and even spill out of the launder. For straight runs away from such bends and junctions of launders, open conduits are designed to be one-third full. When pipes are used as open launders, they are typically sized to be 50% full, but in the case of tenacious froth they may be sized to be 25% or 30% full (Figure 6-3). Designers often prefer to have a steep launder rather than to suffer a loss of time unblocking settled slurry. As the above equations indicate, the friction loss factor does depend on the hydraulic radius, and larger launders tend to require less slope than small launders. Excessively steep launders tend to lose their liners through fast wear. Obtaining the correct slope without an excessive margin of safety is the correct approach to engineering. Example 6-1 A slurry of unknown properties is flowing in a half full 457 mm (18 in) pipe with wall thickness of 9.5 mm (0.375 in). The measured flow rate is 0.189 m3/s (3000 US gpm). The launder is inclined at a slope of 2%. Determine the friction factor and the Chezy and Manning numbers.

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TABLE 6-2 Recommended Values of Absolute Roughness in mm

Classification (assumed clean and new unless otherwise stated) Smooth Drawn nonferrous pipes of aluminum, brass, copper, alkathene, glass, Perspex, HDPE Plastic pipe, welded joints Plastic pipe, flanged or coupled joints Fiberglass pipe (FRP), flanged or coupled Metal Asbestos cement Spun bitumen-lined pipe Spun concrete-lined pipe Wrought iron pipe Rusty wrought iron pipe Uncoated steel pipe Coated steel pipe Rubber-lined steel pipe Plastic-lined steel pipe Galvanized iron Coated cast iron Tate relined pipes Riveted steel pipes (untuberculated)—(good = girth, riveted only; normal = full riveted, taper, or cylinder joints; poor = full riveted, butt-strap joints) Riveted steel pipes (untuberculated)—plates < 6 mm Riveted steel pipes (untuberculated)—plates > 6 mm Concrete Class 4—Monolithic construction against oiled steel surface with no surface irregularities, smooth-surfaced precast pipelines with no shoulders or depressions at the joints Class 4a—Monolithic construction in units of 2 m or over with spigot and socket joints, or ogee joints pointed internally Class 3—Monolithic construction against steel, wet-mix, or spun pre-cast pipes, or with cement or asphalt coating Class 2—Monolithic construction against rough texture precast pipe or cement gun surface (for very coarse textures, take = size of aggregate in evidence) Class 1—Precast pipes with mortar squeeze at joints Smooth trowel led surface Lined concrete pipe Unlined concrete pipe

Values of roughness ks, mm (*) Good

Normal

Poor

Average effective roughness of launder mm (**)

0.003 0.146 2.292 2.292

0.03 0.15 0.015 0.03

0.015 0.03 0.03 0.06 0.6 0.03 0.06

0.15 3 0.06 0.15

0.06 0.06 0.15

0.15 0.15 0.3

0.30 0.30 0.6

0.6 1.5

1.5 3

3 6

0.06

0.15

0.15

0.3

0.3

0.6

1.5

0.6

1.5

0.3

3 0.6

6 1.5

0.725 1.35 0.350 0.726

3.63 1.35 0.73 1.35

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TABLE 6-2 Continued

Classification (assumed clean and new unless otherwise stated) Steel construction Welded sections, unlined Rolled sections, unlined Rubber lined Plastic lined Plastic construction, free formed Clayware Pitch fiber Pitch fiber Glazed vitrified clay, very accurately lined joints Glazed vitrified clay in 1 m under 600 mm diameter Clayware, glazed vitrified clay in 1 m under 600 mm diameter Clayware, glazed vitrified clay in 0.6 m under 300 mm diameter Clayware, glazed vitrified clay in 0.6 m over 300 mm diameter Clayware, butt jointed drain tile Clayware, glazed brickwork Clayware, brickwork, well pointed Clayware, old brickwork in need of pointing Mature foul sewers constructed of materials with roughness when new, not exceeding those given for mature sewers Slimed not more than 6 mm Lime incrustations, grease, or slime not more than 25 mm thick, or even layer of fine sludge Gritty solids, lying unevenly in inverts (higher figures relate to shoals of debris at Froude number of order of 0.3 to 0.5) Unlined rock tunnels Granite and other homogeneous rocks Diagonally bedded slates (use values with design diameter) Earth channels Straight uniform artificial channels Straight natural channels, free from shoals, boulders, and weeds

Values of roughness ks, mm (*) Good

Normal

Poor

Average effective roughness of launder mm (**) 1.35 0.73 1.35 0.73 0.73

0.003

0.03

0.06 0.15 0.3

0.3 0.6

0.15

0.3

0.3

0.6

0.6 0.6 1.5

1.5 1.5 3 15

3 3 6 30

0.6 6

1.5 15

3 30

60

150

300

60

150 300

300 600

15 150

60 300

150 600

1.35

*Data from the Ministry of Technology, United Kingdom (1969), Hydraulics Research Paper 4, Tables for the Hydraulic Design of Storm-drains, Sewers and Pipe-Lines. **Data from Green (1978).

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H = Z/3

H = Z/3

H=R Z

Z

H 2R

B

FIGURE 6-3 Recommended degree of fill for open channel slurry flows (does not apply to junction boxes).

Solution in Metric Units Q = 3,000 US gpm × 3.785 = 11,355 L/min = 0.1893 m3/s ID of pipe = 438.15 mm Area for a half full pipe = /8 × 0.438152 = 0.0754 m2 Velocity = 0.1893/0.0754 = 2.51 m/s Wetted perimeter = × 0.43815/2 = 0.688 m Hydraulic radius = A/P = 0.0754/0.688 = 0.1095 Slope = 2% 2

fN(2.51)2 fNV 0.02 = ᎏ ᎏᎏ = 2.93 fN 2gRH 2 × 9.81 × 0.1095 Fanning friction factor, fN = 0.0068 The Darcy factor, fD = fN × 4 = 0.027 The Chezy number, Ch =

ᎏ = ᎏ = 53.71 m 冪莦 冪莦 f 0.0068 2g

2 × 9.81

1/2

/s

N

The Manning number, n = RH1/6/Ch = 0.10951/6/53.714 = 0.0129 m–1/3/s Solution in USCS Units Q = 3000 US gpm = 3000/7.4805 = 401.04 ft3/min ID of a pipe = 17.25 in = 17.25/12 = 1.4375 ft Area of a half full pipe A = (/8)1.43752 = 0.8115 ft2 Velocity = P/A = 401.04/0.8115 = 494.21 ft/min (8.237 ft/s) P = D/2 = × 1.4375/2 = 2.258 ft Hydraulic radius = A/P = 0.8115/2.258 = 0.359 ft

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SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

Slope = 2% fN × 8.2372 fNV 2 0.02 = ᎏ = ᎏᎏ = 2.935 fN 2gRH 2 × 32.2 × 0.359 The fanning friction factor, fN = 0.0068 The Darcy factor, fD = fN × 4 = 0.027 The Chezy number, Ch =

ᎏ = ᎏ = 97.2 ft 冪莦 冪莦 f 0.0068 2g

2 × 32.2

1/2

/s

N

The Manning number, n = RH1/6/Ch = 0.3591/6/97.2 = 0.0087 ft–1/3/s The reader should be careful when using the Manning roughness “n.” Contrary to common belief, it is not a nondimensional number and its value changes from SI units to USCS units by the ratio of conversion from feet to meters to the power of 1/3 or 0.673.

6-2 TRANSPORTATION OF SEDIMENTS IN AN OPEN CHANNEL Determining the Chezy number for slurries is a method of approaching the design of launders. Julian et al. (1921) measured an average Chezy number of 80 ft1/2/s for rectangular launders (with width = twice the depth) of minimum wetted perimeter for carrying slime overflow and average stamp-battery pulp in cyaniding gold and silver circuits. Classical theories of suspended solids in open channels are based on two-dimensional turbulent flow. Consider a two-dimensional turbulent flow with a velocity U in the horizontal x-direction and V in the vertical y-direction. Reynolds (1895) defined the shear stress parallel to x on a plane normal to y as

= – (U⬘V⬘)average

(6.13)

where

= density of fluid and (U⬘V⬘)average = average of the fluctuations of the turbulent velocities Boussinesq (1877) developed an equation in the form of dU = m ᎏ dy

(6-14)

where m = the eddy viscosity, analogous to the dynamic viscosity discussed in Chapter 2. m = the coefficient of exchange of momentum between neighboring streams of the fluid, expressed in m2/s or ft2/sec. Von Karman (1935) developed the following equation: dU = V⬘Lmix ᎏ dy where  = correlation coefficient ⬵ 1.0 (see Section 6.2.3) V⬘ = average of absolute values of fluctuations normal to the main flow Lmix = mixing length

(6-15)

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By substituting Equation 6-15 into Equation 6-13 it is concluded that

m = V⬘Lmix

(6-16)

A rate of transfer of mass of suspended particles per unit area is defined as dC dm ᎏ = –V⬘Lmix ᎏ dt dy

(6-17)

But the values of , V⬘, and Lmix are not necessarily equal in magnitude in Equations 6-14 and Equation 6-17. C = concentration of suspended solids O’Brien (1933) studied the suspension of sediments in an open channel flow. He developed a theory that the rate of transfer of solids upward must be in equilibrium with the downward exchange of momentum due to gravitational forces: dC VtCy = –V⬘Lmix ᎏ dy

(6-18a)

dC VtCy = s ᎏ dy

(6-18b)

where Cy = volume concentration of solids at level y y = distance from the lower boundary s = mass transfer coefficient for sediments, similar to m but not necessarily equal to it. Solving Equation 6-18 yields C loge ᎏ = Ca

冕

dC ᎏ dy

y

a

(6-19)

where Ca is the concentration of solids at an arbitrary reference plane of height “a.” If s is constant over the depth, then s(y) = constant. Equation 6-18 is then solved to give C ᎏ = e–J Ca

(6-20)

where J = (y – a)(Vt/s). The correct procedure consists of establishing a relationship between s and the vertical coordinate y before solving Equation 6-18. In the absence of detailed information about the relationship between s and m, they are assumed to be equal so that

m = –V⬘Lmix = ᎏ dU/dy

(6-21)

Substituting Equation 6-21 into Equation 6-18 yields the concentration of distribution C loge ᎏ = –Vt Ca

冕

y

a

dU/dy ᎏ dy

(6-22)

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6.11

In a uniform open channel with a large ratio of width to depth, the shear stress is expressed as

冢

ym – y = w ᎏ ym

冣

(6-23)

where w = shear stress at the wall ym = distance from the boundary to the liquid surface Substituting Equation 6-23 into Equation 6-21 yields C loge ᎏ = –Vt Ca

冕

y

a

dU/dy ᎏᎏ dy (1 – y/ym)w

(6-24)

The velocity gradient is expressed in the form of the universal defect law as

冢 冣

U – Umax 1 2y ᎏ = ᎏ loge ᎏ 兹苶 (w苶/苶苶) Kx DI

(6-25)

For a pipe, Kx = 0.4 and y is the distance from the internal wall at the bottom of a horizontal pipe. Keulegan (1938) demonstrated that Equation 6-24 applied to open channels. For a wide-open channel, the value of ym, or depth of flow, is used:

冢 冣

U – Umax 1 y ᎏ = ᎏ loge ᎏ 兹苶 苶 苶 (苶 / ) K y w x m

(6-26)

In Chapter 2, the friction velocity Uf was defined as Uf =

w

ᎏ =U ᎏ 冪莦 冪莦2 fN

Substituting Equation 6-6 yields 苶R 苶H 苶S 苶)苶 Uf = 兹(g

(6-27)

w = gRHS

(6-28)

where fN = the fanning friction factor U = the mean velocity of the flow S = slope Equation (6-28) is called the DuBoys equation (Wood, 1980). It clearly establishes that for a slurry to move at a density , a minimum level of shear stress must be available: ᎏ 冤冢 ᎏ y 冣y –a冥

C Vt loge ᎏ = ᎏ loge Ca KxUf

(6-29)

m

冢 冣

C h ᎏ= ᎏ Ca ha

a

ym – 1

Z

(6-30)

where Vt Z = ᎏᎏ Kx兹g苶y苶S m苶

(6-31)

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ym h= ᎏ –1 y

(6-32)

ym – a ha = ᎏ a

(6-33)

Equation 6-30 establishes that the relative concentration of solids depends on their vertical position and on the factor Z, which is a function of the ratio of the terminal velocity of the particles Vt to the group KxUf. It is therefore a measure of the intensity of turbulence.

6-2-1 Measurements of the Concentration of Sediments Determining the magnitude of the plane “a” is the starting point to solve Equation 6-28 and 6-29. Rouse (1937) suggested the height “a” be equal to the height of the roughness elements ks. He suggested using Equation 6-19, assuming an interval from y = 0 to y = ks, and by assuming that (y) = (ks) = constant. Rouse indicated that at y = 0 the solids concentration corresponding to the bed of sediments should be used. Richardson (1937) reported test results for a 305 mm × 305 mm × 1830 mm (1 ft × 1 ft × 6 ft) flume and indicated that in the boundary region the concentration of sediments was inversely proportional to the vertical coordinate y, but that in open streams it conformed better with Equation 6-30. Vanoni (1946) conducted a series of tests in an 838 mm (33 in) wide by 18.29 m (60 ft) long flume to validate Equation 6-29 (Figure 6-4). Average velocity was noticed to occur at 0.368 ym or the depth of the liquid. The velocity profile followed a logarithmic function of depth. The following results were obtained by Vanoni (1946). 앫 The sediment concentration profile followed the pattern set by Equation 6-30. However, the exponent was smaller than the value of Z expressed by Equation 6-31 when the sediments became coarser. 앫 A random turbulence was observed and slip between fluid and sediment was suspected as the sediment accelerated. Thus, the assumption that mass and momentum transfer were equal was not satisfied, as the theory did not account for slip and random turbulence. 앫 For fine materials, the coefficient of sediment mass transfer was smaller than the coefficient of momentum transfer. 앫 For coarse materials, the coefficient of sediment mass transfer was larger than the coefficient of momentum transfer. 앫 Suspended load decreased the coefficient of mass transfer. The reduction was more important with fine solids than with coarse solids. 앫 Suspended load reduced the value of the Von Karman constant. Values of Kx between 0.314 and 0.342 were measured (by comparison with 0.4 for full pipes). The reduction of the Von Karman coefficient indicated a reduced level of mixing and a tendency by the sediments to suppress turbulence. 앫 The suspended load tended to reduce the resistance to flow. Sediment-laden water moved faster than clear water and the Manning roughness number decreased with the sediment load, as shown in Figure 6-5.

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FIGURE 6-4 Velocity profile for the flow of a sand–water mixture in a rectangular open channel. (From Vanoni, 1946, by permission of ASCE.) 6.13

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FIGURE 6-5 Variation of the equivalent roughness, Von Karman coefficient, and Z1 with the weight concentration of the sand–water mixture. (From Vanoni, 1946, by permission of ASCE.)

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앫 Suspended load tended to cause a flow to become unevenly distributed. Unsymmetrical sediment distribution within the flow caused secondary circulation. 앫 The velocity distribution near the center of the flume followed Von Karman universal defect law. Example 6-2 Iron sand is flowing in a rectangular open channel that is 300 mm wide × 120 mm high at a speed of 3 m/s. Assume the Von Karman coefficient K = 0.33. The average particle size is 0.3 mm. Using Einstein’s approaches, it is assumed that the reference layer “a” is to be twice the particle size diameter. The flume is one-third full (i.e., ym = 40 mm). The slurry weight concentration is 45%, the dynamic viscosity is 1 cP, and the specific gravity of sand is 4.1. Calculate C/Ca if the slope is 3% at depth with 2% increments of depth. Ignore any dunes. Assume  = 1.0. Solution in Metric Units The first approach is to determine the terminal velocity of the sand. The Particle Reynolds number is: dpVm/ = (0.3 × 10–3 × 3 × 1000/1 × 10–3) = 900 This is turbulent flow. By Newton’s law (Equation 3-13): Vt = 1.74[g(s – L/L)]0.5dp0.5 Vt = 1.74[9.81 × 3.1 × 0.3 × 10–3]0.5 Vt = 0.167 m/s Using Equation 6-31: Z = 0.167/[0.33 × (9.81 × 0.04 × 0.03)0.5] Z = 4.664 Applying Equation 6-29: C/Ca = (h/ha)4.664 As it will be explained later in this chapter, Einstein proposed that the value of “a” be equal to twice the grain diameter, or in this case 0.6 mm = twice the particle size of sand. Let us calculate the concentration of solids at 2% of the depth of the flume. 2% of depth = 0.02 × 40 = 0.8 mm: ym h = ᎏ – 1 = h = (1/0.02) – 1 = 49 y ym – a ha = ᎏ a ha = (40/0.6) – 1 = 65.67 h ᎏ = 49/65.67 = 0.735 ha C ᎏ = 0.7354.664 = 0.2378 Ca

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4% of depth: h = (1/0.04) – 1 = 24 h ᎏ = 24/65.67 = 0.3655 ha C ᎏ = 0.36554.664 = 9.15 × 10–3 Ca So most of the solids will be in the bottom 4% of the launder. Example 6-3 Iron sand (SG = 4.1) is flowing in a rectangular channel 600 mm wide × 120 mm high. The liquid level is 60 mm high. The slope is 1% and the liquid is flowing at 1.5 m/s. The particle average diameter is 0.5 mm. The specific gravity of the solids is 4.1 and the dynamic viscosity of the mixture is 1.5 cP. Calculate C/Ca at 4% intervals. Assume Von Karman Kx = 0.33. Ignore any dunes. Assume  = 1.0. Solution The particle Reynolds number is : 0.5 × 10–3 × 1.5 m/s × 1000/1.5 × 10–3 = 500 This is transition flow. From Chapter 3, Allen’s law would apply:

s/L Vt = 0.20 g ᎏ L

冢

冣

0.72

d 1.8 ᎏ (/)0.45

(0.5 × 10–3)1.8 ᎏ Vt = 0.20(9.81 × 3.1)0.72 ᎏᎏ (1.5 × 10–6)0.45 Vt = 1.116 mm/s Using Equation 6-30: Z = 1.116 × 10–3/[0.33 (9.81 × 60 × 10–3 × 0.01)0.5] Z = 3.3822 × 10–3/0.0767 = 0.044 The magnitude of “a” is assumed to be twice the particle’s diameter: a = 2dp = 2 × 0.5 mm = 1 mm C ᎏ = (h/ha)0.044 Ca 60 mm ha = ᎏᎏ = 59 1 mm – 1 Let us calculate the concentration at 2% depth: y = 0.02 × 60mm = 1.2 mm 1 h = ᎏ = 49 0.02 – 1

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h ᎏ = 49/59 = 0.83 ha C ᎏ = 0.830.044 = 0.99 Ca 4% depth: y = 2.4 mm 1 h = ᎏ = 24 0.04 – 1 h ᎏ = 24/59 = 0.4067 ha C ᎏ = 0.40670.044 = 0.96 Ca 8% depth: y = 4.8 mm 1 h = ᎏ = 11.5 0.08 – 1 C ᎏ = (11.5/59)0.044 = 0.931 Ca 12% depth: y = 7.2 mm C ᎏ = (7.33/59)0.044 = 0.912 Ca 16% depth: y = 9.6 mm C ᎏ = (5.25/59)0.044 = 0.899 Ca 20% depth: y = 12 mm C ᎏ = (4/59)0.044 = 0.888 Ca 24% depth: y = 14.4 mm C ᎏ = (3.17/59)0.044 = 0.879 Ca

6.17

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28% depth: y = 16.8 mm C ᎏ = 0.871 Ca 32% depth: y = 19.2 mm C ᎏ = 0.864 Ca Examples 6-2 and 6-3 show the effect of slope on the distribution of solids. In the case of Example 6-2, the slope is higher at 3% and most of the solids move at the bottom of the channel. In Example 6-3, the slope is low at 1% and all the sand is mixed with the water. Graf (1971) reviewed the experiments conducted by various researchers and a tendency developed to measure an empirical Z1 as a substitute for Z. He summarized the work of Einstein and Chien (1955) who developed an approximate relationship between Z and Z1: Z Z1 = ᎏᎏᎏᎏᎏ LZ兹(2 苶 苶)苶 2 Z L exp(–L2Z 2/) + ᎏᎏ exp(–x2/2)dx (2苶 兹苶 苶) 0

冕

(6-34)

where x = loge y L = loge(1 + RKx) The best fit occurs when RKx = 0.3. Einstein (1950) called the flow layer right on top of the bed the “bed layer,” and indicated that it would be impossible to have suspension of the solids there. He measured a thickness of layer t = 2dp. The material within this layer was the source of the suspended load and established the lower limit for Ca. Einstein then proceeded to derive very complex equations that require numerical integration. It would be beyond the scope of this book to dwell on such equations.

6-2-2 Mean Concentrations for Dilute Mixtures (Cv < 0.1) Celik and Rodi (1991) published data suggesting that Einstein’s equations were overestimating the concentration at a = 2d. For fairly dilute suspensions with a volumetric concentration of less than 10%, which is common in a lot of applications, they proposed a more simplified approach, defining the suspended sediment load qbs as qbs = qbCT =

冕

RH

␦a

where qbs = flow rate of sediment per unit width qb = total flow rate of mixture per unit width C = time averaged (mean) concentration CT = mean transport capacity concentration U = time averaged velocity in x-direction RH = hydraulic radius or mean depth of liquid

LUCdy

(6-35)

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␦a = depth of bed load layer y = vertical coordinate above bottom of channel Defining parameter Cm as the depth-averaged concentration, Celik and Rodi (1991) used their results from their previous paper (1984) to conclude that Cm/CT ⬇ 1.13, and derived the following equation:

w Um CT = ᎏᎏ ᎏ (s – L)gym Vt

(6-36)

where is a constant of proportionality (⯝ 0.034 from tests on sand). The exercise consists of calculating CT. For this purpose, these two authors discussed the problem of the effective shear stress. Reviewing the work of other authors and their own, Celik and Rodi (1991) pointed out that there are certain important factors in the turbulent regime, such as: 앫 When there are elements causing increased roughness, the flow separates, and only a part of the shear stress determined from the energy slope is effective in moving the particles in suspension. The work of gravity is then used partly to overcome friction at the bed. 앫 The turbulent energy, which is produced in a turbulent regime, is related to the total shear stress, but the drag force is then not effective in maintaining the particles in suspension. 앫 The turbulence energy of the upper layers needs to be transferred by convection and diffusion first to the region above the bed. Smaller quantities of energy are then available to suspend the bed in the presence of large amounts of roughness. 앫 The mean velocities and the wall shear stresses in separated regions are much smaller than in the areas where the flow is still attached. 앫 The presence of separated regions in the flow and stagnant areas cause the particles to settle in dead water zones and it becomes very difficult to resuspend them. 앫 The permeability of a bed increases the resistance of the bed. (Zippe and Graf, 1983). 앫 For rivers, a typical value of the ratio of friction velocity to average velocity (Uf /U) is 0.05. Tests conducted by Van Rijn (1981) and by Apmann and Rumer (1967) indicate that the flow over an approximately flat bed of loose sand shows great similarity to the characteristics of flow over a rough surface. To take in to account all these effects in the turbulent regime, Celik and Rodi (1991) proposed an equation for the effective shear rate:

e = [1 – (ks/ym)]w

(6-37)

where ks = the equivalent resistance parameter (in most cases the roughness height or absolute roughness) = empirical constant ( = 0.06 in tests obtained by these authors) ym = average depth of liquid in flume In conclusion, Celik and Rodi (1984) established a simplified relationship between roughness and friction velocity for dilute mixtures as

冤

ks Um ᎏ = Er exp –1 – Kx ᎏ ym Uf

冥

(6-37)

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where Er ⬇ 30 Kx = the Von Karman constant Substituting Equation 6-37 into Equation 6-36 and using the effective shear stress yields

w Um CT = [1 – (ks/ym)] ᎏᎏ ᎏ (s – L)gym Vt

(6-39)

Equation 6-39 is plotted in Figure 6-6. From test data, Celik and Rodi (1991) obtained a value where = 0.034 and = 0.06 for flow over a flat bed of loose sand without large dunes or antidunes. The particle diameter was between 0.005 mm and 0.6 mm and volumetric concentration was limited to 10%. On a logarithmic scale, the slope he obtained was 0.034 in the range of CT from

ks F= 1– ᎏ ym

Um2

Um

ᎏ 冤 冢 冣 冥 ᎏᎏ ( Ⲑ – 1)gy V s

L

m

t

FIGURE 6-6 The volumetric capacity CT from Celik and Rodi (1991). (Reprinted by permission of ASCE.)

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0.00001 to 0.10. Nevertheless the authors pointed that the value of is very empirical and strongly dependent on the friction velocity Uf. Unfortunately, the did not provide the correlation, and this leaves the engineer facing a situation of measuring such a value or iterating from case to case.

Example 6-4 Fine sand with an average particle diameter dp of 0.3 mm (0.000984 ft) and SG = 2.625 is transported at a volumetric concentration CV = 7.91%. The volume flow rate is 1200 m3/hr (42,378 ft3/hr). The launder is 600 mm (1.97 ft) wide and the height of the liquid is 200 mm. Determine the slope of the launder using the Celik–Rodi method. Assume Kx = 0.33, = 0.06, and = 1.5 cP (or 3.13 × 10–5 lbf-sec/ft2). Solution in SI Units Cv Ct = ᎏ = 0.07 1.3 The average speed is Q Um = ᎏ = (1200/3600)/(0.2 × 0.6) = 2.79 m/s A Assuming the roughness of the bed to be equal to twice the average particle diameter, ks = 0.6 mm From equation 6-38:

冤

ks Um ᎏ = Er exp –1 – Kx ᎏ ym Uf

冥

冤

2.79 = 30 exp –1 – 0.33 ᎏ Uf

冥

0.9207 ln(0.001) = 1 – ᎏᎏ Uf Uf = 0.112 m/s Using Equation 6-27: Uf = 兹g 苶R 苶苶S H苶 The hydraulic radius for this rectangular channel is (0.6 · 0.2) RH = ᎏᎏ = 0.12 m (0.6+0.2+0.2) Uf = 0.1395 m/s = (9.81 × 0.12 × S)1/2 S = 0.0107 or 1.07%

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Solution in USCS Units Cv Ct = ᎏ = 0.07 1.3 Area of flow = 1.97 × 0.656 = 1.293 ft2 The average speed is Q Um = ᎏ = [(42378 ft3/hr)/3600]/1.293 = 9.1 ft/sec A Assuming the roughness of the bed to be equal to twice the average particle diameter, ks = 0.0236 in If the depth is 8 in: ks ᎏ = 0.0236/8 = 0.003 ym From Equation 6-38:

冤

ks Um ᎏ = Er exp –1 – Kx ᎏ ym Uf

冥

冤

9.1 = 30 exp –1 – 0.33 ᎏ Uf

冥

3.003 ln(0.001) = 1 – ᎏ Uf Uf = 0.38 ft/sec Using Equation 6-27: Uf = 兹g 苶R 苶H 苶S苶 The hydraulic radius for this rectangular channel is RH = (1.987 · 0.65)/(1.987 + 0.65 + 0.65) = 0.391 ft Uf = 0.38 ft/sec = (32.2 × 0.391 × S)1/2 S = 0.011 or 1.1%

6-2-3 Magnitude of  Carstens (1952) demonstrated that  never exceeds unity (1.0). For fine particles,  ⬇ 1.0 or s ⬇ . For coarse particles,  < 1.0 or s < .

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Brush et al. (1962), Matyukhin an Prokofyev (1966) and Majumdar and Carstens (1967) have confirmed that  = 1.0 or s ⯝ for fine particles. For coarse particles,  < 1.0 or .s < .

6-3 CRITICAL VELOCITY AND CRITICAL SHEAR STRESS Graf (1971) established a relationship between the forces for the incipient movement of a set of loose, cohesionless solid particles and the angle of repose as FT tan = ᎏ FN where FN = force normal to angle of repose FT = force tangential to angle of repose These two forces are the resultants of the lift and drag forces (discussed in Chapter 3) referred to in Figure 6-7: FN = W cos ␣ – L FT = W cos ␣ + D W cos ␣ + D tan = ᎏᎏ W cos ␣ – L

(6-40)

The surface area resisting motion is expressed in terms of a shape factor 1 and the particle diameter d. The surface area associated with lift is expressed in terms of a shape factor 2 and the particle diameter d: L = 0.5CLU b2 2d 2

(6-41)

D = 0.5CDU b2 1d 2

(6-42)

where Ub is the bed velocity. The submerged weight of the particle is expressed as a shape factor and the diameter of the particle: W = 3gd 2(s – )

(6-43)

Substituting Equations 6-40, 6-41, and 6-42 into 6-43 establishes the relationship between the critical velocity and the actual shape and density of the particle: 2 23(tan cos ␣ – sin ␣) Ubc ᎏᎏ = ᎏᎏᎏ (s/L – 1) CD1 + CL 2 tan

(6-44)

where Ubc is the the critical bed velocity to start the motion of the particles. Graf (1971) defined the right-hand part of Equation 6-44 as the sediment coefficient . Fortier and Scobey (1926) conducted extensive experiments on permissible canal velocities to understand the erosion and transportation of sediments. Their results are presented in Table 6-3. Their main conclusions which are still valid today, were

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ity

loc

Ve

Lift

Drag Weight

FIGURE 6-7 Lift and drag forces on sediments in an open channel.

앫 The laws governing the transport of silt and detritus in open channels are very distantly related to the laws governing scouring of the canal bed and are not directly applicable. 앫 The material of seasoned canal beds consists of solids of different shapes and sizes. When the fines fill the interstices between the coarser solids, they form a dense and stable mass that is more resistant to the erosion of water. 앫 The velocity required to scour the bedded canal is much higher than the velocity required to suspend particles outside the bed. 앫 Colloids tend to cement clay, sand, and gravel in such a way that the compound mixture resists erosion. 앫 The grading of materials ranging from fine to coarse, coupled with the adhesion between colloids and these solids, makes it possible to operate at high velocities without appreciable scouring effects. Neill (1967) derived the following equation to estimate the critical velocity for coarse particles in launders: 2 dP U bc ᎏᎏ = 2.50 ᎏ (s/ – 1)gdP ym

冤 冥

–0.20

(6-45)

The term dP/ym is sometimes called “relative sand roughness.” In Chapters 3 and 4 we examined definitions of velocity during all periods of movement from settling to deposition, etc. Similarly, in open channels the critical scour velocity is well above the sedimentation velocity (or equal to the terminal velocity in full pipe flow). Sometimes engineers confuse these two terms, although they are quite different in magnitude. The critical shear stress is at the point of the incipient motion, or at which the motion starts, and is expressed as

cr ᎏᎏ = (s – )d

(6-46)

where is the the sediment coefficient. Thomas (1979) argued that sand particles finer than 0.15 mm would be completely enveloped in the viscous sublayer so that the critical shear stress could be simplified to

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cr = 1.21 m[g (s/L – 1)]2/3

(6-47)

With = kinematic viscosity. Wilson (1980) indicated that this correlated well with the work of Ambrose (1953), who had indicated a change of resistance as the grains of sand became of the order of magnitude of the roughness of the flume. Substituting Equation 6-47 into the DuBoys equation (6-28), the critical slope to initiate motion of a bed of small sand particles in an open channel is

w = mgRH S = 1.21 m[g (s/L – 1)]2/3

TABLE 6-3 Permissible Canal Velocities (after Fortier et al., 1925) Velocity, after aging, of canals at a depth of 910 mm (3 ft) or less

Original material excavated for the canal Fine sand (noncolloidal) Sandy loam (noncolloidal) Silt loam (noncolloidal) Alluvial silts when noncolloidal Ordinary firm loam Volcanic ash Fine gravel Stiff clay (very colloidal) Graded loam to cobbles, when noncolloidal Alluvial silts when colloidal Graded, silt to cobbles, when colloidal Coarse gravel (noncolloidal) Cobbles and shingles Shales and hard pans

Clear water, no detritus ____________ m/s ft/s 0.45 0.54 0.61 0.61 0.84 0.84 0.84 1.15 1.15 1.15 1.22 1.22 1.5 1.83

1.5 1.75 2.0 2.0 2.5 2.5 2.5 3.75 3.75 3.75 4.0 4.0 5.0 6.0

Water transporting colloidal silts ____________ m/s ft/s 0.84 0.84 0.91 1.07 1.07 1.07 1.52 1.52 1.52 1.52 1.67 1.83 1.67 1.83

2.5 2.5 3.0 3.5 3.5 3.5 5.0 5.0 5.0 5.0 5.5 6.0 5.5 6.0

Water transporting noncolloidal silts, sands, gravel, or rock fragments _____________ m/s ft/s 0.45 0.61 0.61 0.61 0.69 0.61 1.15 0.91 1.52 0.91 1.52 1.98 1.98 1.5

TABLE 6-4 Effects of Sediment Load on Equivalent Roughness (after Vanoni, 1946) Average sediment load in grams/liter

Ratio of equivalent roughness to size of bottom sand

0 0.17 3.21 7.36 16.2

0.328 0.282 0.190 0.110 0.072

1.5 2.0 2.0 2.0 2.25 2.00 3.75 3.0 5.0 3.0 5.0 6.5 6.5 5.0

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The critical slope to start motion is 1.21[(s/L – 1)]2/3 Scrit = ᎏᎏᎏ RHg1/3

(6-48)

Equation (6-47) confirms the correlation between the average depth of the slurry, the weight of solids, the viscosity and density of the mixture, and the minimum slope. Example 6-5 A slurry mixture of fine particles and water has a specific gravity of solids 4.1 and is flowing in a partially filled rectangular launder 600 mm wide. Determine the critical slope if the launder is to flow one-third full. The density of the mixture is 1250 kg/m3 (2.42 slugs/ft3) and the dynamic viscosity is 20 mPa · s (4.17 × 10–4 lbf-sec/ft2). Solution in Metric Units The kinematic viscosity of the mixture is 20 × 10–3/1250 = 0.000016 m2/s. If the launder is one-third full, the hydraulic radius is 0.6 × 0.2 RH = ᎏ = 0.12 m 0.6 + 0.4 From Equation 6-47: 1.21[ (s/L – 1)]2/3 1.21[16 × 10–6(4.1 – 1)]2/3 ᎏᎏᎏ = 0.0063 or 0.63% Scrit = ᎏᎏ 0.12 × 9.811/3 RHg1/3 is the minimum slope to start motion of the slurry in these conditions. Solution in USCS Units The width of the launder is 1.97 ft, the height of the liquid would be 0.66 ft, and the hydraulic radius is 1.97 × 0.66 RH = ᎏᎏ = 0.395 ft 1.97 + 0.66 The kinematic viscosity of the mixture is 4.17 × 10–4 lbf-sec/ft2 ᎏᎏᎏ = 1.723 × 10–4 lbf-sec-ft/slugs 2.42 slugs/ft3 From Equation 6-47 1.21[(s/L – 1)]2/3 1.21[1.723 × 10–4 (4.1 – 1)]2/3 ᎏᎏᎏᎏ Scrit = ᎏᎏᎏ = = 0.0063 or 0.63% RHg1/3 0.395 × 32.21/3 is the minimum slope to start motion of the slurry in these conditions. This analysis was extended by Wilson (1980) for sand flowing in partially filled pipes. Wilson defined three regimes for sand flowing in open launders with particle diameters in the range of 0.02 mm to 4 mm (mesh 625–5):

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1. Homogeneous flow occurs at a slope S < 0.0006/DI 2. Optimum slope for heterogeneous flow S = 0.10(dp/1000)1.5D I–0.7 3. Fully blocked pipe occurs at S > 0.42 For rocks with solid specific gravity between 1.5 and 6.0, Wilson (1980) extended the analysis and stated, for homogeneous flow: 0.6 [ (s/L – 1)]0.6 S1 ⱕ ᎏᎏᎏ × 10–3 DI 1650

(6-49)

31.6 × 10–3 d p1.5[(s/L – 1)]1.2 S1 < S2 ⱕ ᎏᎏᎏᎏ D I0.7 1.651.2

(6-50)

0.42[(s/L – 1)]0.35 S3 ⱖ ᎏᎏ 1.650.35

(6-51)

or heterogeneous flow:

for blocked flow:

Example 6-6 A slurry mixture of fine particles and water of d85 = 0.08 mm with a density of solids 4100 kg/m3 is flowing in a partially filled circular pipe with an inner diameter of 438 mm (17.25 in). The pipe inner diameter is 438 mm (17.25 in). Determine the minimum slope for flow as a heterogeneous mixture and the slope for a blocked pipe. Solution from Equation 6-50 For heterogeneous flow: 31623d p1.5[(s/L – 1)]1.2 S2 ⱕ ᎏᎏᎏ D I0.7 1.651.2 316230.000081.5 [(4.1- 1)]1.2 S2 = ᎏᎏᎏ 0.4380.7 1.651.2 S2 = 0.086 or 8.6% The slope for a blocked pipe is determined from Equation 6-51 as 52.4%.

6-4 DEPOSITION VELOCITY The terrains over which tailing flumes are built do not always have an appropriate slope. If the flow operates in a subcritical flow regime, the engineer must calculate a realistic estimation of the deposition velocity. Dominguez et al. (1996) published an equation based on experimental data measured at Codelco and the Chilean Research Center of Mining and Metallurgy. For cases where the viscosity effects are negligible

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8gRH (S – m) VD = 1.833 ᎏᎏ m

冤

冥 冢ᎏ R 冣 1/2

d85

0.158

(6-52)

H

However, in cases where the dynamic viscosity of the carrier liquid is instrumental, such as with alkaline water, Dominguez et al. (1996) derived the following equation: 8gRH (S – m) VD = 1.833 ᎏᎏ m

冤

冥 冢ᎏ R 冣 1/2

d85

0.158

1.2(3,100/J)

(6-53)

H

where J = RH(gRH)1/2/m m = the absolute viscosity of the mixture A comparison between the deposition velocity as calculated by Equation 6-52 and experimental data is presented in Figure 6-8. Example 6-7 A slurry mixture of coarse particles of d85 = 12 mm with Cw = 40%, density of solids 4100 kg/m3, and a specific gravity 4.1 is flowing in a circular pipe with an inner diameter of 438 mm (17.25 in). Assuming that the pipe is to be half full, determine the deposition velocity. Solution RH = D/4 = 0.438/4 = 0.1105 m (4.35 in)

m = 1000/[1 – 0.4(4100 – 1000)/4100] = 1433kg/m3 Using Equation 6-51:

FIGURE 6-8 Correlation between calculated and experimental measurements of the deposition velocity of coarse slurries. (From Dominguez et al., 1996, reprinted by permission of BHR Group.)

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VD = 1.833[8 × 9.81 × 0.1105 (4100 – 1433)/1433]0.5 (0.012/0.1105)0.158 VD = 1.29 × 4.0174 = 5.183 m/s Green et al. (1978) and the ASCE and WPCF (1977) proposed that Camp’s equation for the self-cleaning speed of sewers be used in the design of slurry launders: 8Ke s – m Vsc = ᎏ dpg ᎏ fD m

冤

冢

冣冥

1/2

(6-54)

where Ke = an experimental constant (the ASCE recommends a constant of 0.06 for grit chambers, whereas Green recommends a constant of 0.8 for sewers) Vsc is expressed in m/s (ft/s) d is expressed in m (ft) g = 9.81 m2/s (32 ft/s) To use this equation, one must first determine the Darcy friction factor and then solve by iteration. Because the average speed is a logarithmic distribution of depth, it would be wise to design launders with a mean speed equal to twice the self-cleaning speed. Example 6-8 Calculate the self-cleaning speed of Example 6-1, assuming a sewer flow where Ke = 0.8, solids at a SG of 3.1, Cw = 20%, and dp = 2 mm. Solution in Metric Units In Example 6-1, the Darcy factor was calculated to equal fD = fN × 4 = 0.027.

m = 1000/[1 – 0.2(3100 – 1000)/3100] = 1157 kg/m3 Vsc = [8 × 0.8 × 2 × 10–3 × 9.81(3100 – 1157)/(1157 × 0.027)]1/2 Vsc = 2.79 m/s Solution in USCS Units In Example 6-1, the Darcy factor was calculated to equal fD = fN × 4 = 0.027. dp = 6.56 × 10–3 ft Sm = 1/[1 – 0.2(3.1 – 1)/3.1] = 1.157 Vsc = [8 × 0.8 × 6.56 × 10–3 × 32.2(3.1 – 1.157)/(1.157 × 0.027)]0.5 Vsc = 9.2 ft/s

6-5 FLOW RESISTANCE AND FRICTION FACTOR FOR HETEROGENEOUS SLURRY FLOWS Equations 6-6 to 6-10 established the parameter for friction loss of single-phase Newtonian flows in open channels. Equation 2-25 established the correlation between friction factor and friction velocity. Two approaches are often used by designers of launders; one is based on the friction factor and the other on the velocity.

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6-5-1 Flow Resistances in Terms of Friction Velocity Liu (1957) and Acaroglu (1968) indicated that the ratio of the friction velocity to the settling speed of a particle were implicitly related and proposed the following function:

冤

Uf dUf ᎏ = fct ᎏ Vt

冥

(6.55)

Graf (1971) proposed including the Froude number to reflect the degree of turbulence:

冤

Uf Uav dUf ᎏ = fct ᎏ ’ ᎏ Vt 兹(g 苶y苶 苶 m)

冥

where ym is the average depth of the fluid. This is confirmed in studies by Garde and Dattari (1963) and Bogardi (1965). The presence of sand dunes at the bottom of a channel with a typical wavelength is a function of the Froude number (Kennedy, 1963). As shown in Figure 6-9, antidunes are formed in the regime of critical flow (0.8 < Fr < 1.5). The presence of dunes increases the effective wall shear stress in the form of profile drag. Graf (1971) proposed to establish a hydraulic radius RH⬘ due to the grain roughness and a separate value RH⬘⬘ based on the bed forms so that:

w = gS(RH⬘ + RH⬘⬘) where S is the physical slope. He defined a friction velocity Uf as a combination of the component due to grain roughness Uf⬘ and due to the bed form as Uf⬘⬘: Uf2 = Uf⬘2 + Uf⬘⬘2

2.8 2.4

Froude Froude numbe numberFr F

2.0 antidunes 1.6 1.2 dunes 0.8 0.4 0.0

0

2

4

6

8

10

= wavelength

12

14

16

18

2 * * D/

FIGURE 6-9 Wavelength of dunes and dunes versus the Froude number in open channel flows. (After Kennedy, 1963.)

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6-5-2 Friction Factors The previous paragraph demonstrates that the presence of the bed forms (dunes and antidunes) tends to increase the friction velocity and therefore the friction factors. 6-5-2-1 Effect of Roughness There is a dearth of information on the effect of roughness factors on slurry flow in closed or open conduits. The Ministry of Technology of the United Kingdom (1969) recommended modifying the Colebrook equation by using the hydraulic radius for the single-phase fluids. Green et al. (1978) concurred with this approach and proposed that the Darcy friction factor be obtained from the following equation by replacing the diameter with four times the hydraulic radius and using the Reynolds number based on the hydraulic radius: ks 1 2.51 ᎏ = –2 log10 ᎏ + ᎏ 14.8R 兹f苶 Re 兹f苶 H D D

冤

冥

(6-56)

where ks = the linear roughness (measured in the same units as the hydraulic radius, e.g., meters) Re = 4RHV/, the Reynolds number expressed in terms of the hydraulic radius This definition of the linear roughness is difficult to calculate. In a fast flow, the roughness of the pipe or channel wall may be used. Attempts have been made to define a (Nikuradse) sand roughness for closed conduits, such as the ratio of the particle diameter to the inner diameter of the conduit (dp/DI) but very little has been published for open channels. The problem is far from simple, and the roughness is often taken as twice the grain diameter. The presence of dunes at low Froude number tends to complicate the picture by introducing another parameter for roughness. Example 6-9 Considering Example 6-1, assume that the roughness is 0.0045 mm. Reiterate the friction factor using Equation 6-56. Assume m = 1350 kg/m3 and = 2.8 cP. fD = 0.027 RH = 0.1095 m V = 2.51 m/s Re = V × 4RH/ = 1350 × 2.5 × 4 × 0.1095/2.8 × 10–3 Re = 131,987 Iteration 1 0.027–0.5 = –2log10[0.0045/(14.8 × 0.1095) + (2.51/(131,987 · 0.0270.5)] 6.085 ⫽ 5.077 Iteration 2 Assume fD = 0.038, then 0.038–0.5 = –2log10(2.777 × 10–3 + 3.707 × 10–6) 5.13 ⬇ 5.11 Therefore the Darcy factor is iterated to 0.038. 6-5-2-2 Effect of Particle Concentration on Slurry Viscosity Green et al. (1978) proposed to incorporate the effect of particle concentration in the form of increased effective viscosity by using the Einstein–Thomas equation (Equation 1-9) to

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define an effective viscosity. This effective viscosity is then used to compute the Darcy factor by using the hydraulic radius in the Colebrook equation. This approach is, however, essentially limited to Newtonian slurries in pseudohomogeneous flow well above the deposition velocity. This approach has been covered by Equation 6-53. It does not take in account any dunes or partial deposition at the bottom of the bed . It is, however, a useful and straightforward approach for flows at supercritical Froude number. 6-5-2-3 Effects of Particle Sizes on the Chezy Coefficient Richardson et al. (1967) derived the following equations. For a plane with little or no sediment transportation: Ch ym ᎏ = 5.9 log ᎏ + 5.44 兹苶 g d85

冢 冣

(6-57)

For a plane with appreciable sediment transport: Ch ym ᎏ = 7.4 log ᎏ 兹苶 g d85

(6-58)

冣 冢 冣

(6-59)

冢 冣

For ripples (in English units):

冢

Ch 0.3 ym 0.13 ᎏ = 7.66 – ᎏ log ᎏ + ᎏ + 11 兹苶 g Uf d85 Uf For dunes and antidunes: ⌬RHS

– ᎏ冣 冢 冣冪冢莦1莦莦莦 R S 莦

Ch ym ᎏ = 7.4 log ᎏ 兹苶 g d85

(6-60)

H

where ⌬RHS is the increase of RHS due to the form roughness. Example 6-10 A launder is designed to transport appreciable coarse sediments over a plane with d85 = 4 mm. The height of the slurry must be limited to 150 mm. Using Equation 6-60, determine the required slope if the flow rate is 850 m3/hr and the width of the channel is 450 mm. Solution Ch/兹g苶 = 7.4 log(0.15/0.004) = 11.65 Ch = 36.5 m1/2/s Area of flow = 0.15 × 0.45 = 0.0675 m2 Q = 850 m3/hr = 0.236 m3/s V = 3.498 m/s fD = 8g/Ch2 = 8 × 9.81/36.52 = 0.0589 Slope = fDV 2/8gRH RH = A/P = (0.15 × 0.450)/(0.450 + 0.15 + 0.15) RH = 0.793 m S = 0.0589 × 3.4982/[8 × 9.81 × 0.793] = 0.0116 or 1.16%

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6-5-2-4 Effect of Bed Form on the Friction Graf (1971) discussed the importance of Equation 6-52 and proposed to write the total friction factor for flow in a channel in the presence of dunes or bed forms. He suggested the following equation for the overall friction factor: fD = f D⬘ + f D⬘⬘

(6-61)

where f D⬘ = Darcy friction factor for the channel without bed forms f D⬘⬘ = Darcy friction factor due to bed forms In concordance with Silberman (1963) and Vanoni and Hwang (1967), Graf (1971) indicated that f D⬘ can be estimated from conventional pipe equations by substituting the diameter of the pipe with four times the hydraulic radius. This has already been presented in Equation 6-54. A similar approach was developed by Lovera and Kennedy (1969). From lab tests, Vanoni and Hwang (1967) derived the following equation for f D⬘⬘: 1 RH ᎏ = 3.5 log ᎏ – 2.3 兹苶 f D苶 ⬘⬘ e⌬Hav

(6-62)

where ⌬Hav = mean height of the bed form e = A/Ab = (where A = total area and Ab = the horizontal projection of the lee face of the bed forms) When the magnitude of “e” cannot be determined, Equation 6-62 can be written in terms of the wavelength of the bed form as:

RH 1 ᎏ = 3.3 log ᎏ2 – 2.3 兹苶 f D苶 ⬘⬘ (⌬Hav)

(6-63)

where is the length of the dune. In this section, the importance of dunes was well emphasized. The designer of a slurry flume should avoid these troublesome regimes by designing for supercritical flows wherever the topography allows it.

6-5-3 The Graf–Acaroglu Relation Starting from basic principles of lift and drag forces and buoyancy and weight on a solid particle, Acaroglu (1968), Graf and Acaroglu (1968) proceeded to develop a methodology that applies equally well to both closed conduits and open channels. They considered that the drag force was a function of the Reynolds number based on the bed velocity Ub and the shape factor 1: D = 0.5CDLUb21d 2 However, they chose a different shape factor D for the drag coefficient:

冢

冣

Ubdp CD = f1 ᎏ , D

(6-64)

The bed velocity for the solids-water mixture is expressed as

冢

Uf ks y Ub = Uf f2 ᎏ , CV, ᎏ ks

冣

(6-65)

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where ks = the Nikuradse’s equivalent sand roughness (d/DI) CV = the volumetric concentration of solids The shear friction velocity is expressed as Uf =

w

ᎏ= 冪莦

兹R 苶H 苶S 苶g 苶

(6-66)

By assuming that the absolute roughness of the bed is equal to the particle diameter, Acaroglu and Graff proceeded to define the shear intensity parameter as (s – L)d ⌿A = ᎏᎏ LSRH

(6-67)

At a critical value of the shear intensity parameter ⌿Acr, the shear stress is equal to the critical shear stress previously discussed. When ⌿A > ⌿Acr or w < cr, no movement of sediments occurs. When ⌿A < ⌿Acr or w > cr, movement of sediments occurs. The power consumed with friction or head losses in the open channel is expressed in terms of the energy slope (head loss per unit length) and a nondimensional transport parameter is derived as CVUavRH A = ᎏᎏ3 兹苶 (苶 L苶苶–苶1苶 )g苶 d 苶p s/苶

(6-68)

By examining data from various authors and by regression analysis, Graf and Acaroglu extrapolated the following relationship:

A = 10.39 (⌿A)–2.52

(6-69a)

or CVUavRH (s – L)d ᎏᎏ3 = 10.39 ᎏᎏ 兹苶 (苶 L苶苶–苶1苶 )g苶 d 苶 LSRH s/苶

冤

冥

–2.52

(6-69b)

Equation 6-66 was obtained for finely graded sand with a particle diameter between 0.091 mm and 2.70 mm (0.0036 – 0.1063 in) and was studied in rivers and open channel flumes. This equation applied to both closed conduits and open channels (Figure 6-10). Graf (1971) pointed out that this equation was based on extensive data that was often difficult to analyze. For some unknown reasons, there has not been much research since 1970 to refine the Graf–Acaroglu equation. This is probably due to the fact that practically all research on slurry flows tends to limit itself to full pipes. Example 6-11 A mine is located at high altitude. The tailings need to be transported by gravity over a long distance. The estimated particle size is 6 mm. The volumetric concentration is set at 25%. The specific gravity of the solids is 3.1. The carrier liquid is water. The channel is rectangular in shape with a width of 750 mm. Assuming an average speed of 2 m/s, determine the minimum slope for a flow rate of 1100 m3/hr using the Graf–Acaroglu method. Solution in Metric Units Q = 1100 m3/hr or (1100/3600) = 0.3055 m3/s For a speed of 2 m/s, the height of the liquid would be: 0.3055/(2 × 0.75) = 0.204 m

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SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

S

L

L

SRH

A

dp

100.0

10.0

1.00

0.10 0.01

0.1

1.0

10

100

1000

CV U RH A S

L

gd

3 p

FIGURE 6-10 The Graf–Acaroglu relationship for flow of sand in open and closed channels, assuming a particle roughness equal to the grain size. Adapted from Graf and Acaroglu, 1968.

The hydraulic radius is RH = (0.75 × 0.204)/(0.75 + 2 × 0.204) = 0.132 m From Equation 6-68: 0.0662 0.25 × 2 × 0.132 A = ᎏᎏᎏ = ᎏ = 0.188 兹苶 [(苶3苶 .1苶–苶1苶 ) 苶9苶 .8苶1苶 ×苶6苶 ×苶1苶0–3 苶苶] 0.3515 From Equation 6-69a:

A = 10.39 ⌿ A–2.52 or ⌿A = 4.911 From Equation 6-67: 4.911 × 0.132 = 3.1 × 6 × 10–3s–1 S = 0.0287 or 2.87%. The Graf-Acaroglu method is very useful to determine the bed depth of a full pipe with saltation. Equation 4-43 of Chapter 4 uses this method.

6-5-4 Slip of Coarse Materials Kuhn (1980) conducted velocity measurements on transportation of coarse coal in trapezoidal flumes under controlled laboratory conditions. He reported slip between the coarse solids and liquid. At the inclination of 2.5°, the speed of the coarse material was of 10% slower than the liquid speed of 4.5 m/s, but it decreased gradually to a slip of 8.5% of the speed of 6 m/s at an inclination of 6°. Such a degree of slip is reminiscent

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of the two-layer models of heterogeneous flow extensively discussed in Chapter 4 for full pipe flows.

6-5-5 Comparison between Different Models Blench et al. (1980) conducted a comparison between the different equations to calculate the sediment discharge versus the water discharge in open channels and plotted them as in Figure 6-11. It is obvious that different equations yield different results. The sediments are transmitted in different patterns. When the flow is tranquil (Froude number Fr < 1), two kinds of sand waves may develop: dunes and ripples. They are similar in shape, with an upstream surface and a gentle and gradually varying slope, finishing with an abrupt downstream slope (Figure 6-12). Although similar in shape, ripples are independent of the magnitude of flow, whereas dunes are strongly dependent on flow. At Froude number larger than unity (sometimes referred to as supercritical flows), the flow suppresses the formation of bed forms. Anti-dunes, which are more symmetrical than ripples, form. They move in the same direction as the flow or even opposite to the flow.

FIGURE 6-11 Comparison between the different models for transport of sediments in open channels. (From ASCE, 1975. Reprinted by permission of ASCE.)

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liquid level Flow direction

ym

wavelength FIGURE 6-12 Sand dunes at low Froude number (Fr < 1).

Anti-dunes accentuate the deformation of the free surface (Figure 6-13). They do not occur in closed conduits and their motion is at a lower speed than the fluid. Tournier and Judd (1945) reported that the specific gravity of the ore is an important factor to consider. Heavier ores require more slope to be transported in an open channel, as shown in Figure 6-14. Tournier and Judd (1945) reported that the size of the particles play an important role, and that larger particles require more slope, as shown in Figure 6-15. Figures 6-12 and 6-13 clearly demonstrate that it may be erroneous to use conventional Manning formulae for water flow depending on the roughness of the pipe, as these ignore the resultant roughness due to sand dunes and anti-dunes. Figures 6-14 and 6-15 clearly in-

liquid level ym

Flow direction

wavelength FIGURE 6-13 Sand anti-dunes at high Froude number (Fr > 1).

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2. 7 =

cg ra vi ty

sp ec ifi

sp ec ifi

cg ra vi ty

=

3. 8

slope of launder (%)

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0

20 40 60 80 Weight concentration (%)

slope of launder (%)

FIGURE 6-14 The slope is a function of the specific gravity of the ore as well as the weight concentration (after Tournier and Judd, 1945). The magnitude of the slope is not shown here as it depends also on the hydraulic radius or shape of the launder.

0

4

8 12 16 particle size (mm)

18

20

FIGURE 6-15 Larger particles require more slope (after Tournier and Judd, 1945). The magnitude of the slope is not shown here as it depends also on the hydraulic radius or shape of the launder.

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dicate that the density of the ore as well as the size of the solids do increase the slope. These important factors are unfortunately too often ignored by some engineers who rely on the Manning equation.

6-6 FRICTION LOSSES AND SLOPE FOR HOMOGENEOUS SLURRY FLOWS Green et al. (1978) attempted to estimate the effect of particle sizes and concentration in the form of increased effective viscosity. In this approach, however, they essentially limited their studies to Newtonian slurries and did not take into account the effects of yield stress and plasticity, effects often encountered at high concentrations of fine particles. Geophysicists prefer to talk about cohesive bed forms when exploring the movement of clays in rivers. For certain soils, cohesive forces develop between the solid particles. Graf (1971) indicated that Equation 6-43 should be modified to include X0, a coefficient of cohesion of the material. For a Bingham slurry, the yield stress is added to Equation 6-45 to express the critical shear stress at the point of the incipient motion:

cr ᎏᎏ = + X0 (s – )gd

(6-70)

Cohesive (soil) materials include clay-sized (colloidal) particles, silt-sized particles, and sometimes sand-sized particles. Graf (1971) classified clays into the following three main categories: 1. Kaolinites 2. Montmorillonites 3. Illites Other more minor clays include halloysites, chlorites, and vermiculites. Clay materials have residual electrostatic forces that attract cations and anions. This is measured as a cation exchange capacity in milliequivalents per 100 grams. Grim (1962) stated that Kaolinites have an exchange capacity of 3–15 milliequivalent per 100 grams, whereas illites rated higher at 10–40, and montmorillonites at 80–150. In very simple terms, Grim explained that there are two main structures for clays. One structure consists of two close sheets of packed hydroxyl molecules, in which aluminum, iron, and magnesium atoms are embedded in an octahedral coordination equidistant from the six oxygen atoms in the hydroxyls. The second structure consists of silica tetrahedrons. In the tetrahedron, the silicon atom is equidistant from the hydroxyls. The silica tetrahedral groups are arranged to form a hexagonal network, which is repeated from sheet to sheet. It would be beyond the scope of this book to discuss the physical aspects of clays. The slurry engineer should appreciate that these electrostatic forces and arrangement of chemical groups influence the yield stress at certain concentrations that may be at either the lower end or the higher end. Cohesive soils have the ability to absorb water and develop plasticity; they also have limit liquid. All three characteristics were briefly defined in Chapter 1. Cohesive colloids can form lumps. Forces on such lumps are shown in Figure 6-16. Graf (1971) summarized the test work of Karasev (1964) who proposed that the erosion of cohesive material beds in rivers is due to aggregate-to-aggregate rather than by

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CHAPTER SIX

Lift Force

Cohesion Force

Drag + Cohesion forces

Weight Force FIGURE 6-16

Forces on a colloidal lump moving in an open channel.

particle-to-particle contact. Karasev (1964) derived the following equation for the average scouring velocity: ym 2d50(s – L) + 3⌳ Ucr = 0.142 Ch ᎏᎏᎏ 1.2 + 8 ᎏ L Patm

冤

冢

冣冥

0.5

(6-71)

where ⌳ = information about cohesion Patm = atmospheric pressure A comparison between the computed and measured values of a scouring velocity is presented in Table 6-5. This value of critical speed should be considered the speed for minimum transportation of clay by hydrotransport. The flow of water in canals containing sand, cohesive soils, and cohesive banks was examined by Graf (1971) on the basis of the work of Simons et al. (1963). The graph in Figure 6-17 suggests that the hydraulic radius can be correlated to the flow rate by the following equation: RH = 0.43 · Q0.361 6-6-1 Bingham Plastics Whipple (1997) developed numerical models for open channel flow of Bingham fluids but did not provide a methodology to calculate friction losses. This paper is important for a geophysicist but provides no useful tools for the slurry engineer. For a homogeneous slurry, there are two important numbers to calculate: the Reynolds number and the plasticity number (defined in Chapter 5). In the absence of well-defined models for friction losses of Bingham slurries, Abulnaga (1997) proposed a methodology to modify some of the equations of full-pipe flows by expressing the Reynolds and Hedstrom numbers in terms of the hydraulic radius. At high shear rate, the coefficient of rigidity is taken as the viscosity for a Bingham plastic: Re = 4RHV/

(6-72)

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TABLE 6-5 Scouring Velocity of Clays (after Karasev, 1964) Scouring velocity in streams ______________________________ Diameter of Karasev’s aggregate Adhesion ⍀ Experimental formula ______________ _____________ _______________ _____________ mm in kPa psi m/s ft/sec m/s ft/sec

#

Material

1 2 3 4 5 6 7 8 9 10 11

Aggregate materials Medium loam Rammed clay Rammed clay Heavy loam Heavy loam Clay Clay Clay Clay Compacted clay Heavy loam

2.2 3.0 4.0 2.7 4.0 4.0 4.0 6.0 4.0 0.8 1.0

0.087 0.118 0.157 0.106 0.157 0.157 0.157 0.24 0.157 0.03 0.04

225.6 550 202 373 225.6 245 285 196 285 510 304

12 13 14 15 16 17

Dispersed materials Medium loam Clay Clay Clay Clay Clay

4.0 1.5 3.2 4.0 0.8 4.0

0.157 0.059 0.126 0.157 0.03 0.157

746 706 785 432 785 549

32.7 80 29.3 54 32.7 35.5 41.3 28.4 41.3 73.9 44.1

1.74 2.16 2.06 2.04 1.20 2.20 1.30 1.43 1.54 2.23 2.14

3.28 7.09 6.76 6.69 3.94 7.22 4.26 4.69 5.05 7.32 7.02

1.83 2.82 2.36 2.22 1.70 1.75 1.86 1.45 1.86 3.20 2.35

6 9.3 7.74 7.28 5.58 5.74 6.1 4.76 6.10 1.05 7.71

108 102 114 62.6 114 79.6

1.91 3.06 2.35 2.87 2.40 1.54

6.27 10.04 7.71 9.42 7.87 5.05

2.88 2.68 2.4 1.93 2.42 2.50

9.45 8.79 7.87 6.33 7.94 8.2

FIGURE 6-17 Correlation between the hydraulic radius and the discharge flow rate. (From Graf, 1971, reprinted by permission of McGraw-Hill.)

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And the Plasticity number is written in terms of the hydraulic radius as

04RH PL = ᎏ V

(6-73)

The Hedstrom number is the product of the Plasticity number and the Reynolds number and is calculated as 16RH20 He = ᎏ 2

(6-74)

In Chapter 3, the different categories of non-Newtonian flows were reviewed. Methods to calculate friction losses for Bingham slurries and power law slurries were presented in Chapter 5. Modified Reynolds and Hedstrom numbers for pseudoplastics were also introduced in Chapter 5. In Chapter 5, the Buckingham equation was introduced as He He4 fNL 1 ᎏ = ᎏ – ᎏ2 + ᎏ 3 ReB 6 ReB 3 f NL ReB8 16

(5-5)

Modifying it for an open channel in laminar flow yields

16RH20/2 fNL ᎏ ⬇ ᎏ – ᎏᎏ2 4RHVm 6(4RHVm/) 16 0 fNL ᎏ⬇ᎏ–ᎏ 4RHV 6V 2m 16

(6-75)

The Darby equation for the friction factor in turbulent regime is fNT = 10aReBb where a = –1.47[1 + 0.146 exp(–2.9 × 10–5He)] b = –0.193 The values of the parameters “a” and “b” were based on empirical data for closed conduits. They may be tentatively modified to open channels to yield 4RHVm fNT = 10a ᎏ

冢

冣

b

(5-10)

where a = –1.47 [1 + 0.146 exp{–2.9 × 10–5(16RH2m0/2)}] b = –0.193 Equations for the empirical parameters “a” and “b” should be confirmed by extensive testing in open channels. It is unfortunate that very little research is conducted in this extremely important field of fluid dynamics. Darby et al. (1992) proposed to combine the laminar and turbulent fanning friction factors into the following equation: m (1/m) fN = (f mNL + f NT )

(5-11)

40,000 m = 1.7 + ᎏ ReB

(5-12)

where

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For an open channel this becomes 10,000 m = 1.7 + ᎏ RHVm

(6-76)

Example 6-12 The soft high clay tailing from a copper concentrator develops a Bingham viscosity of 400 mPa · s at a weight concentration of 45% as well as a yield stress of 5 Pa. The slurry density is 1350 kg/m3. The tailings flow rate is 1600 m3/hr. If the maximum allowed slope is 1.5%, determine the suitable size for a half-full smooth HDPE pipe (i.e., ignore roughness factor). Solution in SI Units Iteration 1 Assume a speed of 1.8 m/s. Required area for flow is A = (1600/3600)/1.8 = 0.247 m2 Pipe ID = [(8/)A]0.5 = 0.789 m (31.09 in)

= 0.4 Pa · s RH = DI/4 = 0.789/4 = 0.19725 ⬇ 0.198 m Re = 4RHmV/ = 4 × 0.198 × 1350 × 1.8/0.4 = 2673 He = 16 × 0.1982 × 1350 × 5/0.42 = 26,463

冤

冥

冤

26,463 16 He 16 fL ⬇ ᎏ 1 + ᎏ ⬇ ᎏ 1 + ᎏ Re 6Re 2673 6 × 2673

冥

fL ⬇ 0.0158 a = –1.47[1 + 0.146 exp(– 2.9 × 10–5Re)] a = –1.7 a

fT = 10 Re0.193 = 0.00435 m = 1.7 + 40,000/Re = 16.67 fN = ( f Lm + f Tm)1/m = 0.016 Using Equation 6-5: fNV 2 S= ᎏ 2gRH S = (0.016 × 1.82)/(2 × 9.81 × 0.198) = 0.0132 S = 1.32% This approach was used by Abulnaga (1997) at Fluor Daniel Wright Engineers to design a tailings launder (see Figure 6-1) to transport tailings rich in soft high clay for a Peruvian copper mine. The tailings system functioned well. The Wilson–Thomas method for full flow in closed channels does not rely on empirical coefficients such as the Darby method, but is based on the assumption that a thick sublayer lubricates the wall surface of the pipe. It has not been modified yet for open channel flows. This is a topic well worth further research.

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6-7 FLOCCULATION LAUNDERS Hydraulic flocculation is sometimes conducted in launders feeding a thickener. The G gradient is a measure of the average shear rate for a flocculation tank. For a tank flocculated using a mixer, Camp (1955) defined the G gradient as: G=

ᎏ 冪莦 V P

(6-77)

ol

where P = power applied by the mixer Vol = volume of the liquid in the tank = average viscosity of the solution Camp (1955) as well as the American Society of Civil Engineers (1977) modified this equation for launders, conduits, and flocculation basins by proposing that the power term P be replaced by the power due to friction in the launder: P = Qg⌬H

(6-78)

where = density of the mixture g = acceleration due to gravity Q = flow rate ⌬H = head loss due to friction As a measure to control ⌬H, ASCE proposed to install a baffle system in the flume. They defined this process as hydraulic flocculation: G=

Qg⌬H

ᎏ= ᎏ 冪莦 V 冪莦 V P

(6-79)

ol

Assuming that no baffles are installed in the launder, it would be possible to express the volume V as the product of the wetted area by the length of the launder: Vol = AL Assuming that the slope S of the launder is equal to the energy gradient (as per Equation 6-5), or friction loss per unit length (or S = ⌬H/L), then G=

Qg⌬H

QgS

ᎏ= ᎏ 冪莦 冪莦 VA A

(6-80)

Equation 6-80 is valid only if no baffles or other artificial means are added to the launder to increase friction losses. Example 6-13 A tailing is flowing to a thickener. In-line flocculation is applied to raise the viscosity to 20 mPa · s. The flow rate is 1800 m3/hr. The density of the mixture is 1420 kg/m3. The launder is rectangular with a slope of 1.2%. The chemical process requires that the G gradient be smaller than 100. Determine a suitable size for the launder. Solution From Equation 6-79: 100 = [1420(1800/3600) 9.81 × 0.012/(A × 0.02)]0.5

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100 = 64.66 (1/A)0.5 A0.5 = 0.646 m A = 0.4179 m2 If the depth of the liquid is one-third the width of the launder, A = w2/3 W = 1.12 m Depth = 0.373 m Velocity = Q/A = 0.5 m3/s/0.4179 m2 = 1.196 m/s

6-8 FROUDE NUMBER AND STABILITY OF SLURRY FLOWS A measure of the stability of a flow in an open channel can be expressed in the form of the Froude number (Fr), a ratio of inertia to gravity forces: V Fr = ᎏ 兹苶 gy苶 m

(6-81)

In the case of slurry flows, it is important to avoid subcritical flows (Fr < 0.8), as they often cause settling problems. Critical flows (0.8 < Fr < 1.5) may be associated with a degree of instability and wavy motion, leading in some cases to working problems and overflows. Kennedy (1963) reported test work on sand and suggested that antidunes occur at a Froude number of 0.8 to 1.4 (critical regime). Green et al. (1978) recommended that slurry launders be designed for Froude numbers in excess of 1.5, to avoid regimes of instability of flow. On the other hand, excessively high Froude numbers (Fr > 5) are associated with steep slopes. Steep slope instability was discussed by Niepelt and Locher (1989). Slug flow is reported at high Froude numbers, causing working instability in the form of roll waves. Although economics may often dictate maintaining gentle slopes of < 2% on many long-distance tailings projects, the design of plants must too often accommodate tight spaces. The use of steep slopes (> 8%) may often cause high speeds, in excess of 6 m/s (20 ft/s). To avoid premature wear of liners or pipes, excessive slopes in plants should be avoided. If it is not possible to avoid steep slopes, drop boxes, pressurized boxes, chokes, or full-flow closed conduits should be used wherever wear is a major concern. There are certain conditions in an open channel that may lead to a sudden change from a supercritical regime to a subcritical regime. This is often associated with the so-called “hydraulic jump,” an increase in the depth of the liquid due to the lower speed of motion.

6-9 METHODOLOGY OF DESIGN The design of open launders is complex, with many implicit functions. Abulnaga (1997) suggested starting the calculations assuming a speed of 2 m/s (6.56 ft/s). For many types of slurries, this is a good starting point. The flow rate is divided by the assumed speed to

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obtain an area of the flow A. If a pipe is selected, the flow area A is taken as ⬇ 45–55% of the cross-sectional area of commercial pipes to obtain a pipe diameter. 1. If a rectangular launder is to be manufactured, the width is used to compute the height of the liquid. For these launders, the height of the liquid is assumed to be 1/3 the width. 2. For the area of the flow, the perimeter and hydraulic radius are then obtained. 3. The Froude number is then calculated. If it appears that the flow will be in a subcritical or critical regime, the speed should be increased. 4. Repeat steps 1 to 4 until the Froude number is larger than 1.5. 5. Input the rheology of the slurry to obtain the plasticity, Reynolds, or Hedstrom number based on the hydraulic radius of the flow. 6. Compute the friction factor in accordance with one of the equations listed. 7. Obtain the friction loss per unit length and equate it to the slope, as per Equation 6-5. 8. If the energy gradient or slope from Step 9 exceeds the physical contour of the terrain where the launder is to be installed, reiterate assuming a slower flow. 9. Check on the deposition velocity or self-cleaning abilities of the solids. If the deposition velocity is more than 50% of the average velocity, speed up the flow by changing the cross section of the launder or the physical slope. The following computer program, “Non-Newt-Channel,” uses the Darby method to design open channel flow for non-Newtonian fluids on the basis of modifications to the Darby method. Computer Program “Non-Newt Channels” CLS PRINT “CHANNEL FLOW PROGRAM FOR NON-NEWTONIAN FLOWS PRINT “****************************************” PRINT c = 0 pi = 4 * ATN(1) DEF fnlog10 (x) = LOG(x) * .43242944# DEF FNASN (x) = x + x ^ 3/6 + 3 * x ^ 5/(2 * 4 * 5) + 15 * x ^ 7/(2 * 4 * 6 * 7) + (15 * 7)/(48 * 7 * 8) * x ^ 9 DEF fnacos (x) = pi/2 – (x + x ^ 3/6 + (3/(2 * 4 * 5)) * x ^ 5 + 15/(8 * 6 * 7) * x ^ 7 + 15 * 7 * x ^ 9/(48 * 7 * 8)) g = 9.81 INPUT “PROJECT “; proj$ PRINT INPUT “DATE “; d$ CLS INPUT “NAME OF ENGINEER “; e$ ‘e$ CLS 5 INPUT “AREA”; a$ INPUT “LINE NUMBER “; li$ CLS INPUT “DO YOU INTEND TO USE US UNITS (y/n)”; U$ GOSUB CONVERSION INPUT “INITIAL FLOW RATE (m3/s)”; q ‘INPUT “INITIAL FLOW RATE (m3/HR)”; QH ‘q = QH/3600

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q = q * qul PRINT USING “flow rate = ##.### m3/s”; q PRINT INPUT “DESIGN FACTOR “; SF c = 0 6 IF c = 1 THEN GOTO 19 PRINT INPUT “specific gravity of carrier liquid”; sgl INPUT “specific gravity of solids”; sgs PRINT PRINT “ please choose between input of weight or volume concentration” PRINT “ 1- weight concentration” PRINT “ 2- volume concentration” PRINT 12 INPUT “ 1 or 2”; cwe IF cwe = 1 THEN INPUT “weight concentration in percent”; cwin IF cwe = 2 THEN INPUT “volume concentration in percent”; cvin IF cwe = 0 OR cwe > 2 THEN GOTO 12 PRINT IF cwe = 1 THEN cw = cwin/100 IF cwe = 1 THEN sgm = sgl/(1 - cw * (1 - sgl/sgs)) IF cwe = 1 THEN cv = (sgm - sgl)/(sgs - sgl) IF cwe = 1 THEN cvin = 100 * cv IF cwe = 1 THEN PRINT USING “specific gravity of mixture = ##.##, cv = #.###”; sgm; cv IF cwe = 2 THEN cv = cvin/100 IF cwe = 2 THEN sgm = cv * (sgs - sgl) + sgl IF cwe = 2 THEN cw = cv * sgs/sgm IF cwe = 2 THEN cwin = cw * 100 IF cwe = 2 THEN PRINT USING “specific gravity of mixture = ##.##, cwin = ##.##%”; sgm; cw dens = sgm * 1000 PRINT INPUT “DO YOU KNOW THE VISCOSITY (y/n)”; ZS$ IF ZS$ = “Y” OR ZS$ = “y” THEN INPUT “VISCOSITY (mPa.s)”; vu1 CLS IF ZS$ = “N” OR ZS$ = “n” THEN KRAT = 1 + 2.5 * cv + 10.05 * cv ^ 2 + .00273 * EXP(16.6 * cv) ‘ASSUMED VISCOSITY OF WATER 1 mPa · s IF ZS$ = “N” OR ZS$ = “n” THEN vu = KRAT * .001 IF ZS$ = “Y” OR ZS$ = “y” THEN vu = vu1/1000 CLS PRINT USING “VISCOSITY = ##.##### Pa.s”; vu INPUT “hit any key”; t$ CLS INPUT “yield stress in dynes/cm2 “; y1 y = y1/10 CLS PRINT “you can let the program iterate for itself or you can input an initial speed” PRINT v1 = 2 PRINT “iteration starts at 2 m/s (6.6 ft/s)” INPUT “do you prefer to input an initial speed (Y/N)”; b$

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IF b$ = “N” OR b$ = “n” THEN GOTO 18 17 PRINT USING “ current speed = ##.## m/s “; v1 IF us = 1 THEN INPUT “ initial speed in ft/s “; v1 IF us = 2 THEN INPUT “ initial speed in m/s “; v1 v1 = v1 * leg 18 INPUT “maximum allowed slope in percent “; s1 19 IF c = 1 THEN q = q * SF IF c = 1 THEN GOTO 50 21 PRINT “choose shape of launder from following list “ PRINT PRINT “1- rectangular “ PRINT “2- circular” PRINT “3- U shape” INPUT “which choice (1,2,3 etc.....)”; ck 50 IF ck = 1 THEN GOSUB rect IF ck = 2 THEN GOSUB circ IF ck = 3 THEN GOSUB ushape PRINT “v1” re = v1 * 4 * rh * dens/vu he = (4 * rh/vu) ^ 2 * dens * y PRINT USING “Reynolds No = #########, Hedstrom No = ######## “; re; he GOSUB friction IF c = 0 THEN GOSUB settling IF v2m > (v1/2) THEN PRINT “to avoid settling, flow should be speeded up” IF v2m > (v1/2) THEN GOSUB increase IF sm > s1 THEN PRINT “slope exceeds maximum allowed slope” IF sm > s1 THEN GOTO 17 ‘INPUT “do you want to print out these results as a minimum slope”; min$ ‘IF min$ = “Y” OR min$ = “y” THEN min = 1 ‘IF min = 1 THEN GOSUB print1 ‘IF min = 1 THEN GOTO 999999 GOSUB gradient INPUT “do you want a hard copy (Y/N)”; mv$ IF mv$ = “y” OR mv$ = “Y” THEN GOSUB print1 999999 IF SF > 1 THEN c = c + 1 IF c = 1 THEN GOTO 19 END increase: v1 = v1 * 1.03 PRINT USING “velocity = ##.## m/s”; v1 area = q/v1 RETURN decrease: v1 = v1 * .97 area = q/v1 RETURN are: area = q/v1 PRINT “flow area “; area RETURN

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CONVERSION: IF U$ = “Y” OR U$ = “y” THEN us = 1 IF U$ = “n” OR U$ = “N” THEN us = 2 IF us = 1 THEN PRINT “for us units use foot for length, US gallon for flow” IF us = 1 THEN PRINT “speed in ft/s” ft = .3048 gal = 3.785 inch = .0254 IF us = 2 THEN GOTO 888 leg = ft qul = gal/60000 GOTO 890 888 leg = 1 qul = 1 890 RETURN rect: CLS IF c = 1 THEN GOTO 1600 PRINT “you have chosen a rectangular launder - do you want to continue (Y/N)”; mre$ IF mre$ = “n” OR mre$ = “N” THEN GOTO 21 14 IF us = 1 THEN INPUT “width of channel (ft) “; w IF us = 2 THEN INPUT “width of channel (m) “; w IF w = 0 THEN GOTO 14 w = w * leg PRINT INPUT “do you want the program to calculate height of walls (Y/N)”; hr$ IF hr$ = “N” OR hr$ = “n” THEN INPUT “height of walls “; hl IF hr$ = “y” OR hr$ = “Y” THEN hl = .5 * w hl = hl * leg 1600 area = q/v1 dep = area/w IF c = 1 THEN GOTO 1606 IF p$ = “y” OR p$ = “Y” THEN GOTO 1606 INPUT “what ratio of fill is acceptable (0.333, 0.5, 0.75)”; fill 1605 IF hr$ = “N” OR hr$ = “n” THEN hl = dep/fill hfill = hl * fill 1606 IF (dep > hfill) THEN PRINT “depth of liquid exceeds preferred fill ratio” IF dep > hfill THEN v1 = v1 * 1.01 IF dep > hfill THEN area = q/v1 IF dep > hfill THEN dep = area/w IF dep > hfill THEN GOTO 1605 ‘calculation of hydraulic radius rh = area/(w + 2 * dep) mhd = dep GOSUB froude IF nf < = 1.5 THEN PRINT “froude number too low at “; nf IF nf < = 1.5 THEN INPUT “flow is unstable do you want to stabilize the flow “; p$ IF p$ = “N” OR p$ = “n” THEN GOTO 1612 IF nf > 1.5 THEN GOTO 1612 v1 = v1 * 1.01

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area = q/v1 GOTO 1600 1612 RETURN circ: rf = 0 PRINT “calculation for a circular launder” ‘RINT “ITERATION STARTS FOR 1/2 FULL PIPE” INPUT “degree of fullness as a ratio of area of flow to pipe area (.5,.6 etc..)”; full1 1999 PRINT “speed (m/s)”; v1 INPUT “do you want to change speed (Y/N)”; lk$ IF lk$ = “y” OR lk$ = “Y” THEN INPUT “new speed in m/s “; v1 ‘ IF lk$ = “n” OR lk$ = “N” THEN GOTO 2001 2095 area = q/v1 IF c = 1 THEN GOTO 456 dia = SQR(area * 4/(full1 * pi)) IF us = 1 THEN diapus = dia/.0254 IF us = 1 THEN PRINT USING “recommended inner pipe diameter = ##.## in “; diapus IF us = 2 THEN PRINT USING “recommended inner pipe diameter = ###.####m”; dia 2001 IF us = 2 THEN GOTO 2100 IF nff = 0 THEN GOTO 2002 PRINT USING “present pipe i.d = ##.### m”; id IF us = 1 THEN idus = id/.0254 IF us = 1 THEN PRINT “present pipe id = ###.##inches”; idus 2002 INPUT “ pipe outer diameter in inches”; dout IF rf > 0 THEN GOTO 2004 INPUT “pipe thickness in inches “; thickus INPUT “pipe liner thickness in inches “; linus 2004 idin = dout - 2 * thickus - 2 * linus PRINT USING “pipe i.d = ###.### in “; idin id = idin/12 PRINT USING “pipe id = ##.## ft”; id GOTO 2105 2100 INPUT “pipe outer diameter in mm”; d2m IF nff = 1 THEN GOTO 2101 IF rf > 1 THEN GOTO 2101 INPUT “pipe thickness in mm”; thick INPUT “pipe liner thickness in mm”; lin 2101 idm = d2m - 2 * thick - 2 * lin id = idm/1000 2105 id = id * leg r1 = id/2 a2 = pi * r1 ^ 2 456 IF area < a2 THEN GOSUB depth1 IF area > a2 THEN GOSUB depth2 IF a2 = area THEN PRINT “DEPTH = RADIUS” RATIO1 = dep/(2 * r1) INPUT “HIT ANY KEY TO CONTINUE”; l$ RATIO1 = dep/id 457 PRINT “RATIO OF LIQUID DEPTH TO DIAMETER”; RATIO1 IF RATIO1 < 1.05 * full1 AND RATIO1 > .95 * full1 THEN GOTO 470 IF RATIO1 < .2 THEN GOTO 490

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IF RATIO1 > .85 THEN GOTO 492 IF RATIO1 < .48 THEN INPUT “ DO YOU WANT TO INCREASE THE DEPTH OF LIQUID TO REDUCE SLOPE (y/n)”; n$ IF RATIO1 > .52 THEN INPUT “do you want to decrease liquid depth “; dp$ IF RATIO1 < .48 AND n$ = “N” OR n$ = “n” THEN GOTO 470 IF RATIO1 < .48 AND n$ = “y” OR n$ = “y” THEN GOTO 465 IF RATIO1 > .52 AND dp$ = “N” OR dp$ = “n” THEN GOTO 470 IF RATIO1 > .52 AND dp$ = “Y” OR dp$ = “y” THEN GOTO 467 GOTO 470 490 PRINT “ please reduce pipe diameter as depth is less than 20% of diameter” INPUT “do you want to decrease the pipe diameter “; kjl$ IF kjl$ = “n” OR kjl$ = “N” THEN GOTO 456 IF kjl$ = “Y” OR kjl$ = “y” THEN GOTO 2001 492 PRINT “please increase pipe diameter as depth is more than 90% of diameter” INPUT “do you want to change pipe diameter (Y/N)”; qg$ IF qg$ = “Y” OR qg$ = “y” THEN GOTO 465 IF qg$ = “n” OR qg$ = “N” THEN GOTO 470 465 GOSUB decrease GOSUB are GOSUB depth1 GOTO 456 467 GOSUB increase GOSUB are GOSUB depth2 GOTO 456 470 PRINT “speed (m/s)”; v1 RATIO1 = dep/id INPUT “hit any key to continue”; l$ GOSUB angle 2120 mhd = dep mhd = id * (fnacos(1 - 2 * RATIO1) - (2 - 4 * RATIO1) * SQR(ABS (RATIO1 - RATIO1 ^ 2)))/(8 * SQR(ABS(RATIO1 - RATIO1 ^ 2))) mhds = mhd/.0254 PRINT USING “mean hydraulic depth = ##.##m ##.### in”; mhd; mhds GOSUB froude PRINT “FROUDE NUMBER”, nf INPUT “HIT ANY KEY TO CONTINUE”; lk$ IF nf < = .8 THEN PRINT “a new diameter is recommended” IF nf < = 1.4 THEN GOTO 1999 IF nf < = .8 THEN GOSUB increase IF nf < = .8 THEN GOSUB are IF nf < = .8 THEN PRINT “flow is subcritical” 3000 IF (nf > .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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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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SLURRY FLOW IN OPEN CHANNELS AND DROP BOXES

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

d

2

L

d

r L >L r b

L

FIGURE 6-26

3

Geometry of the nappe from a waterfall.

subcritical flow

Lj

ventilation Dd df

d1 d d3

Lp Ld

d < d3

Lr

Ld < Lj

Lr < Lb Case (A)

FIGURE 6-27 1955.)

Patterns of flow with free fall with entry in a subcritical regime. (After Rand, 6.62

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In Case C (Figure 6-28), the nappe is submerged, and the depth of the tail water is higher than d2. Compared with the previous two cases, this turbulent under-nappe region is the deepest of the three cases. The turbulent roller extends much further and is less intense than the hydraulic jump. Referring to Figure 6-29, if Dd is the depth of the fall from the bottom to the brink of the bottom of the drop box, a drop number Dr can be defined as q b2 Dr = ᎏ3 gD d

(6-89)

In real life there is always a sort of churning area of liquid under the nappe, but from a theoretical point of view, which ignores this pool of liquid, the location of the centerline of the nappe intersecting with the bottom of the apron or the drop box would be expressed as Lp ᎏ = 1.98[Dr1/3 + 0.357 Dr2/3]1/2 Dd

(6-90)

This point is also called the toe of the nappe. The hydraulic jump occurs at a distance Ld, which may be smaller, equal to, or even larger than Lp (Figure 6-26) The ratio of Ld/Lp is maximum at 1.87 when Dr = 1. Tests reported by Rand (1955) indicate that the value of Ld ᎏ = 4.30Dr0.27 Dd

(6-91)

In cases where the hydraulic jump starts at the toe of the nappe, the experimental work of Rand (1955) indicates that the reference depth d2 for the tail water can be expressed as a constant: Ld ᎏ = 2.60 d2

(6-92)

ventilation Dd d2

df d1 d = d2 Ld

Lr

Ld = Lj

Lr = Lb Case (B)

FIGURE 6-28 Rand, 1955.)

Geometry of nappe from a free fall with entry in a critical regime. (After

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ventilation Dd df

d

d

Lj Ld

Lr

d > 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 22

(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 + 22 = 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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-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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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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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 <1% by volume 2 Wet grinding with the addition of >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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hydrocyclone

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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7.21

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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TABLE 7-4 Selection Guide for Grinding Mills

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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 (<350 mm, 10⬙) Large feed (<25 mm, 1⬙) Intermediate feed (<12 mm, 0.5⬙) Fine feed (<1 mm, 14 mesh) Coarse product (<3.4 mm, 6 mesh) Fine product (0.4 mm, 35 mesh) Maximum production of fines Minimum production of fines Production of cubical particles Primary mill of two-stage circuit Secondary mill of two-stage circuit Operation in open or closed circuit Operation in closed circuit only

2/28/02

Mineral

Rod __________________ Autogenous Ball Vertical _________________ Peripheral ______________________ _____________ Primary Secondary Overflow Discharge Overflow Grate Pebble Spindle Tower Vibrating Hammer

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7.25

A special chamber on the tip is installed to feed the material and the grinding medium. The liners on the inside are designed to be ribbed either along the length of the cylinder or in spiral shaped ribs. Ni-hard is the most commonly used alloy for liners (Figure 7-19), but manganese steel and chrome steel are also used. In some designs, rubber is used as the liner material (Figure 7-18) It is important to cut down maintenance costs and periods of outage to replace liners. Many mines open their mills once a month for maintenance and to replace the worn liners. The grinding medium is steel rods in the case of rod mills and steel balls in the case of ball mills. As the cylinder or tumbler rotates, the heavy rods and balls are lifted by the ribs of the liners. The rods and balls fall by gravity after a certain angle of rotation is reached. The impact in turn fractures and grinds the rocks into smaller stones. The spacing between the ribs of the liner is critical. Too narrowly spaced ribs may jam the coarser rocks and delay their fracture. Speed of rotation is extremely important. At a certain speed, the material, which is lifted by friction against the liner, starts to fall down. The cascading effects of stone against stone causes grinding by attrition. The material output is fine but wear is high. As the speed of the mill is increased, grinding takes place by impact of the rods or balls against the rocks. As the speed increases even further, centrifugal forces become sufficient for the material to centrifuge. This speed is called “the critical speed of the mill.” Mills are designed to operate at 75% of their critical speed. The diameter of the rods is often 50 mm (2 in) but can be set by the designer of the mill. It is, however, important to separate the rods or balls from the slurry at the discharge of the mill before they enter the slurry pump. The successful separation of steel balls from the slurry involves proper design of trommels, a mechanism to catch the balls, and screens on top of the pump box. Ideally, the balls should be recycled back to the feed of the milling unit.

FIGURE 7-19

Worn-out metal liners removed during monthly maintenance of a SAG mill.

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7-4-1 Rod Mills Rod mills are a type of fine crusher and can reduce the size of rocks down to 1 mm (0.04 in). They perform better than a fine crusher in less than optimum conditions when the feed is damp or contains clay. Typically, the length to diameter ratio of the rod mills is 1.5 to 2.5. Milling occurs by impact of rod against rod. The stones are trapped between the rods and disintegrate. The coarser stones are the first to break. The finer escape milling. Rod mills are not used on closed circuits. In the last 20 years, the mining industry has tended to replace rod mills with large autogenous and semiautogenous mills 7-4-2 Ball Mills In ball mills, metal balls are used as the grinding media. The balls are made of a variety of materials. Steel balls are forged. High chrome balls are cast with 28% chrome and are available from special foundries. About 1 kg of balls is used per ton of stone. Small balls with a diameter of about 25 mm (1 in) are preferred to larger ones in order to maximize the area of contact between balls and stones. The slurry weight concentration in a ball mill is 65–80%. Excessive concentration will cause the particles to stick to the balls and will decrease the effectiveness of grinding. The ball mill may then “freeze” and spill out its contents, causing costly downtime to empty the mill. For this reason, the weight concentration should not be allowed to exceed 80%. A trunnion at the discharge of the ball mill separates balls from slurry. The balls are then conveyed back to the feed. Balls gradually wear out through repeated feeding to the mill and must be replaced. Ball mills are built in different diameters up to a maximum of 6.5 m (21 ft), and in power drives up to 9650 kW (13,000 hp). Their shape is determined by the type of output (Figure 7-20). 7-4-3 Autogenous and Semiautogenous Mills Autogenous and semiautogenous (AG and SAG) mills are extremely large mills with a maximum diameter of 12.2 m (40 ft). In the last few years, Siemens and ABB have devel-

feed

feed

discharge

balls

slurry

Cascade mills (wet and dry grinding) - Used for autogneous and semiautogeneous milling in closed circuit - Primary Grinding with minimum retention time for very fine output - Diameter to length ratio 2:1

balls

discharge

slurry

(b) Conical shape mill - Suitable for fine discharge

FIGURE 7-20 The shape of ball grinding mills is determined by the type of discharge and ore.

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7.27

FIGURE 7-21 SAG mill with wrap-around or ring motor. [Courtesy of Metso Minerals (formerly known as the companies Nordberg and Svedala).]

oped and installed “wraparound” motors. In this design, the outside diameter of the tumble on one side is part of the rotor of the motor (Figure 7-21). These motors are manufactured up to a power size of 7000 kW (9400 hp). The large diameter of these mills maximizes the impact forces. Although the feed is typically 150–180 mm (6–7 in) in diameter, the output can be as fine as 0.3 mm (0.012 in). Particles tend to cleave along their natural grain boundaries. Six to ten percent of steel balls are added on a continuous basis to the feed to assist grinding through a separate entry. Wet milling and grinding is less dusty and less noisy than dry grinding. The feed and output trunnions are on opposite sides. The trommel on one side catches the steel or high chrome balls to prevent them from falling into the pump box.

7-5 AGITATED GRINDING Agitation is another method of grinding. The whole body of the mill may sit on springs and be agitated by crankshafts or an eccentric mechanism driven by a motor. Another ap-

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proach is to have a rotor, an agitator, and a rotating hammer inside the mill to impart energy.

7-5-1 Vertical Tower Mills A vertical mill was developed in Japan for grinding fine minerals. The tower mill is a combination of vertical tanks, a screw mixer, a mill, and a classifier. From a chute on the side of the mill, the rocks, steel balls, and liquid are introduced. The vertical screw rotates around a vertical shaft and creates an upward vertical counterflow. The finer materials float to the top and are led to a side chamber for classification, while the heavier and coarser solids sink with the steel balls (Figure 7-22). The diameter of the balls is 6–32 mm (0.25–1.25 in). Size reduction of the ore is limited to 5 mm (0.197 in) due to limited grinding. Vertical tower mills are manufactured for a maximum output of 100 tph. They require limited floor space and have a low consumption of power.

7-5-2 Vertical Spindle Mills The vertical spindle mill (also known as SAM) uses a central vertical multistage mixer (Figure 7-23). Each stage consists of a number of wolfram carbide pins fixed on a hollow shaft. They provide horizontal stirring. This machine operates with feed smaller than 1 mm (16 mesh) for fine and ultrafine wet or dry grinding. The units are small and compact and can be relocated within the plant. Maximum power is 75 kW per unit.

7-5-3 Roller Mills Roller mills are used for soft grinding of industrial minerals in a dry state. The mill consists of a rotating table on a vertical axis. Two rollers rotate around their own shafts at an angle with respect to each other. The rollers are spring loaded. The output is diverted to dry cyclones and the oversized material is fed back to the roller mill. A new generation of high-pressure roller mills has appeared on the market since the 1980s. A very high level of torque is transmitted to the rollers to maximizing the crushing loads. High-pressure rollers are mainly used in cement plants, diamond processing (when the extraction is from rocks, as it is in Canada), and to a certain extent in the field of metalliferrous minerals.

7-5-4 Vibrating Ball Mills The body of the mill consists of a central feed chamber and two side chambers. The feed is from the top and the discharge from the central chamber is at the opposite end. The whole body of the mill sits on four strong springs. Two electric motors synchronized by V-belts rotate an eccentric mechanism linked to each of the side chambers (Figure 7-24). This machines uses fine feed smaller than 5 mm (mesh 4) and is particularly suited for difficult material with an energy index Wi > 30 kWh/sh.ton. These are essentially small machines with maximum motor size rated at 55 kW or 75 hp. However, they are often chosen over tumbling mills for lower installation cost, lower operating cost, less floor space, increased grinding flexibility, and improved product control within the limitation of their size. Rods or balls may be used as grinding media within open or closed circuits with these machines.

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classifier box

vertical bars for wall protection and help to grinding

7.29

recirculation of coarse material helix for upwards pumping while mixing and grinding slurry pump

FIGURE 7-22

Slurry circuit of vertical grinding tower mill for solids with a maximum diameter of 6.4 mm (1/4⬙).

Page 7.29

launder for fines (output)

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Page 7.30

vertical bars for wall protection and help to grinding 7.30

recirculation of coarse material helix for upwards pumping while mixing and grinding slurry pump 5

K

M

A Z D A

A

FIGURE 7-23

Vertical spindle mill slurry circuit.

P

U -

2

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two motors in parallel at each end

7.31

one side chamber at each end feed

discharge

FIGURE 7-24

Vibrating mill.

7-5-5 Hammer Mills In the hammer mill, a central rotor arm is fitted with rings of arms that crush and mill the feed against the wall of the mill. It is essentially used for dry milling of low-abrasive and friable minerals such as cement, coal, gypsum, and limestone. It is considered by some engineers a crusher rather than a mill.

7-6 SCREENING DEVICES In Chapters 1 and 3, the concept of d50 was introduced as the particle size diameter below and at which 50% of the particles can pass through the opening of a sieve. The same concept applies to the definition of screen size. The screen size aperture is equal to d50. An ideal screen would let all particles equal to or smaller than d50 pass through. This is not always the case, as the performance of the screen depends on a variety of factors: 앫 Screen deck size. In order for all particles to use the screen effectively, the layer of solids above the screen needs to be very thin. This means a large deck size for a given mass of solids. For economical reasons, this is not possible and a thick layer of solids forms on the smaller screens. 앫 Vibration. To move away the coarse particles that block the passage of the finer ones, it is essential to oscillate the screen. The amplitude of the oscillation must match the specifics of the solids. Too much vibration could cause the solids to float as a cloud without passing through the screens. 앫 Presentation angle. Ideally, the solids should be fed as normal as possible to the screen. This means that the solids should come in at a 90° angle. Unfortunately, this is not always possible. 앫 Screen material. Screens are manufactured of metal, rubber, and even fiberglass. Metal screens have a wider aperture than rubber, which is more flexible and less prone to particle binding. 앫 Moisture content. Sprays are sometimes added to screens to improve their efficiency and flush the solids. Sprays suppress clouds of fine particles.

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7-6-1 Trommel Screens Trommel screens are essentially rotating cylindrical mesh. These trommels rotate at a slight angle of inclination to facilitate the removal of material. Trommel screens can be supplied as a set of concentric screens of different aperture. The finest screens are the quickest to show wear. 7-6-2 Shaking Screens Shaking screens move in a horizontal reciprocal motion along the length of the screen. Solids are fed in a horizontal circular movement. The discharge moves in the direction of the horizontal movement of the screen. The motion of the solids changes from circular at the feed to eccentric, and finally to horizontal shaking. 7-6-3 Vibrating Screens These screens are set at an angle with respect to the horizontal. The vibration occurs at a right angle to the screen by the rotation of unbalanced counterweights on a shaft above the screen. Vibration can also be induced by electromagnets and oscillating currents. Vibration levels are high and the screens must be mounted on vibration isolation rubber pads. These screens are extremely noisy and exceed 100 dBA levels of noise, nevertheless, they are the most widely used. 7-6-4 Banana Screens Banana screens are essentially stationary screens. The sieve is bent around a curved screen. The top of the screen is vertical and solids are fed from the top. Particles pass successive wedge bars and solids are removed between them based on the trigonometric opening normal to the fall. To avoid clogging, the bars are pneumatically tilted at regular intervals. These screens can be designed to sieve particles as fine as 50 m.

7-7 SLURRY CLASSIFIERS Classification is the process that separates coarse from fine. Various methods use the effect of size, density, and magnetic and electrostatic properties of the solids. When the weight concentration is smaller than 15%, particles settle in a “free settling” mode. When the weight concentration increases, turbulence promotes settling of the heavier particles faster than the lighter particles. Two families of density classifiers are available: 1. Classifiers that use the principle of free settling to achieve size separation 2. Classifiers that use the particles’ hindered settling speed for density separation and for concentration of a particular mineral 7-7-1 Hydraulic Classifiers In a hydraulic classifier, solids are fed at the top through a chamber that leads into a column. Water is pumped from the bottom of the column. The counterflow moves into a

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number of successive columns or stages. Products settle in accordance with the principles described in Chapter 3. Solids are removed at the bottom through a restriction such as a spigot. The spigot valve opens and closes in accordance with the applied pressure from the accumulation of solids. This is the principle of operation of the hydrosizer, which is often connected to other gravity and magnetic separators.

7-7-2 Mechanical Classifiers A mechanical classifier is a combination of an open channel flow, a weir, and a mechanical device to remove solids. The trough to which the slurry is directed is inclined with respect to the horizontal. On one side, a weir is constructed opposite to the direction of the flow. The heavier solids deposit upstream of the weir, whereas the finer solids pass over it. This system operates on the principle of the sliding bed described in Chapters 4 and 6. The weir is designed to minimize turbulence. A mechanical rake or rotating Archimedean screw removes the coarse solids. Water drains away as the solids are removed. The speed of the rake or rotating shaft is critical to the efficiency of separation. The height of the weir must be adjustable to change the depth of the pool, the rising velocity, and the cut point between coarse and fine. The addition of water controls the density of the slurry. Dilution may be required for effective separation, but excessive water dilution may have to be followed by thickening after classification. Mechanical classifiers are expensive to install but in some applications they are selected for high-density valuable minerals as they assist in their immediate recovery without requiring further complicated flotation circuits. In some respects, flotation circuits use some of the principles of mechanical classifiers by using a circular internal weir, a mixer, an underflow pump, and a separate froth pump.

7-7-3 Hydrocyclones Hydrocyclones (Figure 7-25) are the most common classifiers in the mining industry. They require little space and operate on the pressure from the mill discharge pumps (typically 104–152 kPa, 15–22 psi). They are typically used to classify solids from a size of 40 to 400 m (mesh 325 to 35). The principle of the cyclone relies on creation of a vortex; sometimes primary and secondary vortexes are created by feeding the material tangentially. In a vortex, a certain pressure field is created to counterbalance centrifugal forces. In the cyclone, the applied pressure is converted into a swirling motion. The intensity of the swirl is measured as the swirl number: angular momentum S = ᎏᎏᎏ axial momentum If the swirl flow number exceeds 0.5, the swirl is classified as a strong swirl. Strong swirl is associated with a low-pressure zone at the core. A strong swirl is associated with an important pressure drop. The cyclone feed gauge pressure at the inlet flange is often of the order of 70–100 kPa (10 to 15 psi). The swirling chamber of the cyclone is where the separation starts. The inlet area to the cyclone is often of the order of 5–7% of the chamber area, or the inlet diameter is be-

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FIGURE 7-25 A number of hydrocyclones may be used in parallel to classify slurry flow in a grinding circuit.

tween 20% and 25% of the swirling chamber diameter. The inlet nozzle is rectangular in shape in some sheet metal fabricated cyclones, or circular in fiberglass cyclones. Inside the swirling chamber, a pipe section protrudes from the top of the cyclone. It is called the vortex finder. It must extend below the feed entrance to avoid shortcuts of unclassified slurry to the top discharge or overflow. The diameter of the vortex finder is typically 32% to 36% of the swirling chamber diameter. The finer and lighter particles flow out of the hydrocyclone through the vortex finder (Figure 7-26). In some of the earlier metal fabricated designs, the swirling chamber consisted of a single cylinder. In fiberglass designs, it is split into two halves, which are individually lined with removable rubber liners. The cyclone chamber is followed by a cylindrical chamber with a depth approximately equal to its diameter. This chamber provides some retention time. The cylindrical transition chamber is followed by a conical chamber, often designed with an included angle between 10 and 20 degrees. It provides further retention time. At the bottom of the cyclone, the apex is installed. It acts as a sort of nozzle or orifice. For different applications, different orifice diameters may be used, and for different apex diameters, different pressures are required. The apex is therefore a sort of controlling element to the cyclone. The minimum apex orifice diameter is on the order of 10% of the swirling chamber diameter, and the largest orifice diameter is on the order of 35%. In either case, the apex must allow the flow of the coarse materials. At the bottom of the apex, the discharge is called the cyclone underflow. At the top of the vortex finder, the discharge, which consists of fines, is called the cyclone overflow. For primary grinding circuits, the underflow typically contains 50 to 53% by volume

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COMPONENTS OF SLURRY PLANTS

discharge of finer particles (cyclone overflow)

Involute design vortex finder

inlet pipe

cylindrical feed chamber (top half)

cylindrical chamber (bottom half)

fiberglass body Conical section angle from 10 to 20 degrees discharge pipe for coarse solids (cyclone underflow)

Rubber liner Apex

FIGURE 7-26 A cross-sectional representation of a rubber-lined cyclone. (Courtesy of Mazdak International Inc.)

of solids, whereas for regrinding circuit, the underflow typically delivers 40 to 45% solids by volume. Hydrocyclones can be manufactured from dough-molded compound fiberglass, cast iron, or sheet metal lined with polyurethane. Metal and fiberglass cyclones are lined with rubber or with hard metal (Ni-hard or 28% chrome white iron). Burgess and Abulnaga (1991) presented a finite element analysis of fiberglass cyclones. The performance of the hydrocyclone is calculated by using a partition curve similar to a screen curve. This gives the d50 size, or 50% probability at the cut point. This cut point is defined as the condition for which 50% of the feed will be discharged as coarse particles in the cyclone underflow and 50% as fines or cyclone overflow. For every cyclone design, there is a base d50C or cut-off for the recovery (Figure 7-27). Cyclones are usually operated in a steady mode with constant pressure. Surges can lead to unfavorable air entrainment. To maintain constant pressure from the pumps, the pump box must have a constant level of slurry. To adjust the slurry level, the sump must be provided with a water addition mechanism. Normal feed to cyclones consists of a slurry at 30% solids concentration by weight. Some mines operate with slurry weight concentrations as high as 35%. Higher concentration by weight imposes higher pressures of operation, which can cause a reduction in efficiency of operation of the hydrocyclone while coarsening the cut point. Vortex finders are changed in accordance with the required cut. A larger diameter vortex finder tends to coarsen the overflow while increasing its discharge flow rate at a constant pressure.

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CHAPTER SEVEN

100 ACTUAL RECOVERY

CORRECTED RECOVERY

50

Recovery to underflow, %

d 50c

0 Diameter of particles in micrometers FIGURE 7-27

Typical particle recovery curve for the underflow from a hydrocyclone.

The size of the discharge spigot is selected in order to maximize the flow of solids at high density and to reduce the flow of water in the underflow. Too small a spigot tends to dewater the underflow and to break the air core while reducing the overall efficiency. Because of the wide range of slurries with different particle sizes, the cutoff d50C is used to normalize the particle size (Arterburn, 1982). The actual particle diameter from a recovery is divided by the d50C size and a parameter X is defined as: X = particle diameter/d50C particle diameter The recovery to the underflow (Arterburn, 19xx) on a corrected basis is defined as: e4X – 1 Rr = ᎏᎏ 4X e + e4 – 2

(7-15)

Using the base d50C, Arterburn (1982) proposed to use three correction factors for an application, C1, C2, C3, or d50C(application) = d50C(base) × C1 × C2 × C3

(7-16)

The base d50C is defined as a polynomial function of the cyclone swirling chamber diameter. Arterburn proposed d50C = 2.84(D/100)0.66

(7-17)

where D is the cyclone chamber diameter in meters. The correction factor C1 is based on the volumetric concentration of solids fed to the cyclone:

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COMPONENTS OF SLURRY PLANTS

冢

53 – 100CV C1 = ᎏᎏ 53

冣

–1.43

(7-18)

Equation 7-18 applies in a range CV < 0.4. The pressure drop between the feed nozzle and the cyclone overflow ⌬P is used to compute the second correction factor C2: C2 = 3.27(⌬P)–0.28

(7-19)

where ⌬P is expressed in kPa. To minimize energy losses, ⌬P should be in the range of 40 to 70 kPa, Arterburn (1982), particularly for coarse separation in grinding circuits. The final correction factor C3 is based on the density of the solids with respect to the liquid density: C3 =

ᎏ 冪莦 – 1650 S

(7-20)

L

Tables 7-5 and 7-6 show the typical size ranges for cyclones. Example 7-2 A hydrocyclone with a diameter of 250 mm is selected for a flow rate of 15 L/s at a pressure of 140 kPa. The specific gravity of the solids is 4.8 and the volumetric concentration is 0.3. If the pressure drop between the feed and the overflow is maintained at 50 kPa, determine the corrected d50C. Assuming a discharge coefficient of 0.5 and a remaining pressure of 20 kPa at the apex, determine the underflow capacity for an apex diameter of 80 mm if the underflow density is 2000 kg/m3. From Equation 7-17 the base d50C = 2.84 (D/100)0.66 = 23.77 m. From Equation 7-18 the correction factor C1 is

冢

53 – 30 C1 = ᎏ 53

冣

–1.43

= 3.3

From Equation 7-19 the correction factor C2 is C2 = 3.27(⌬P)–0.28 = 3.27 (50)–0.28 = 1.0935

TABLE 7-5 Typical Range of Sizes for Cyclones Operating at Pressures from 20 to 500 kPa (3–72 psi) Diameter (of swirling chamber) in mm 100 150 250 380 510 660 760

Capacity in L/s

Diameter (of swirling chamber) in inches

Capacity in USgpm

1 L/s @ 20 kPa–6 L/s @500 kPa 3 L/s @ 20 kPa–15 L/s @500 kPa 7 L/s @ 20 kPa–35 L/s @500 kPa 12 L/s @ 20 kPa–60 L/s @500 kPa 26 L/s @ 20 kPa–140 L/s @500 kPa 50 L/s @ 20 kPa–250 L/s @500 kPa 85 L/s @ 20 kPa–450 L/s @500 kPa

4 6 10 15 20 26 30

16 [email protected] 3psi–96 gpm @ 72 psi 48 gpm @ 3psi–240 gpm @ 72 psi 110 [email protected] 3psi–555 gpm @ 72 psi 190 [email protected] 3psi–950 gpm @ 72 psi 410 [email protected] 3psi–2200 gpm @ 72 psi 793 [email protected] 3psi–3963 gpm @ 72 psi 1350 [email protected] 3psi–7100 gpm @ 72 psi

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TABLE 7-6 Typical Cyclone Size versus Particle Cut Diameter of hydrocyclone

Discharge cut size

25 mm (1⬙) 100 mm (4⬙) 250 mm (10⬙) 500 mm (20⬙)

5 m (mesh 2500) 40 m (mesh 380) 75 m (mesh 200) 150 m (mesh 100)

The final correction factor C3 is based on the density of the solids with respect to the liquid density: C3 =

ᎏᎏ = 0.66 冪莦 4800 – 1000 1650

d50C application = 23.77 × 3.3 × 1.0935 × 0.66 = 57 m The flow rate in the underflow is determined from nozzle equations: 苶P 苶苶 / = 0.5 · 0.00503 兹2 苶0苶 = 0.01124 m3/s = 11.24 L/s = 178 USgpm Q = CD · A兹2苶⌬

7-7-4 Magnetic Separators In beach and mineral sand plants as well as in taconite processing plants, minerals have magnetic properties. The presence of a magnet would attract the ferrous ores and separate them from other solids. This is the principle of magnetic separation. Magnetic separators work on two principles. 1. An electromagnetic drum set in a stream 2. A belt driven by an electromagnetic drum on which solids in a dry state or slurry form are allowed to pass to separate the ferrous ores

7-8 FLOTATION CIRCUITS Flotation is a method of separating solids from streams by creating a froth to which they are attracted. Thus in a slurry circuit, flocculants are added to create a froth rich with the metal concentrate. The trick is to make mineral particles hydrophobic, or water repellant. Flotation involves the selected “adsorption” of hydrocarbons (e.g., ethyl xanthate) on liberated minerals (e.g., chalcopyrite), which can then be attached to and transported by air bubbles in the slurry to a so-called froth layer and then separated from the hydrophilic (wetted) particles. For flotation to be efficient, it must be repeated a few times in a circuit that includes a rougher, a scavenger, and a cleaner as shown in Figure 7-28. The collector in a flotation circuit consists of a hydrophobic hydrocarbon chain of melecules (grease or wax) that repels the mineral-laden water and causes it to attach itself to the passing air bubble. The surface chemistry is divided into three categories:

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COMPONENTS OF SLURRY PLANTS

Air bubble

Wa ter

particles

circulating concentrate

feed

rougher

tails

scavenger

tails

concentrate circulating tails

cleaner concentrate

FIGURE 7-28

Fig 7-28

Flow chart for flotation circuit with rougher, scavenger, and cleaner.

1. Physical adsorption with a free energy of the collector smaller than 5 kcal/mol 2. Chemisorption with a free energy of the collector larger than 30 kcal/mol 3. Intermediary stages between adsorption and chemisorption Sulfide minerals are relatively easier to separate by chemisorption because they can use the major collectors such as xanthates and dithiosphophates. Certain special additives with high surface energy capabilities can also be added to separate different grades of sulfides (e.g., to sink pyrite while floating chalcopyrite). Oxide minerals (e.g., hematite, apatite. etc.) and silicate minerals are more difficult to separate by flotation than sulfide minerals. For oxides and silicate minerals, flotation is difficult because it is done by adsorption with minimal free surface energy using anionic fatty acids and cationic amines, which operate essentially by electrostatic forces. When various ores are present, flotation may be done in stages using tanks in series. In each tank, a different pH level may be set or different collectors may be added, with the output from each tank going to a different circuit for further treatment. Depressants are chemicals that make the particle surface hydrophilic and nonfloatable. Typical depressants include bichromate, cyanide, zinc sulphate, and lime. Activators are chemicals that make the surface of nonfloating particles active for collector attachment. Typical activators are copper sulphate and sodium sulphide. The pH value is a determining factor in many flotation circuits. It is adjusted by using various chemicals such as lime, caustic soda, sulfuric acid, etc. Frothers are chemicals that are used to decrease the surface tension of water in order to 앫 Develop improved stability in the pulp 앫 Achieve smaller and better bubble size

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앫 Create a suitable froth layer 앫 Help destroy froth, after which it is removed Typical frothers include alcohols and pine oil. The size of the flotation tank is based on the required flow rate as well as the required retention time, an aeration factor, and a scale factor: Q · tr · sc volume = ᎏ a

(7-21)

where sc = scale factor = 0.85 for plants = 1.0 for pilot plants = 1.7 for lab batch Q = flow rate tr = retention time a = aeration factor = 0.85 Example 7-3 A flow rate of 500 m3/hr requires a retention time of 20 minutes. Size the tank assuming four tanks. volume = (500/60) · 20 · 1/0.85 = 196 m3 196/4 = 49 m3 per tank The number of cells in a flotation circuit is determined by the degree of metallurgical control and the concern for short-circuiting. It used to be believed that the correct approach would consist of small cells and longer banks. However, with the advent of special mixers and good aeration techniques, it is now possible to use larger tanks (Figure 7-29). Flotation circuit can be very simple or very complex. A simple circuit such as used with coal, achieves floatation in a single step and does not involve cleaning of the froth. In a more complex circuit, an initial stage, called the rougher, is added; it acts as a preconcentrator. The flocculated output goes then to a second stage, called cleaning, that is done at higher dilution and is sometimes associated with regrinding at various stages. When the ore grade is fairly low but the mineral is of high value, a scavenger is used for additional preconcentration. Froth is a real challenge in the design of pumps. This will be reviewed in Chapter 8.

7-9 MIXERS AND AGITATORS Mixers or agitators (Figure 7-30) are very important components of mineral and chemical process plants. They are used in various stages such as flotation circuits, leaching circuits, gold adsorption on carbon, preparation of special chemicals such as milk of lime, preparation of feed for pipelines, and final storage where sedimentation is likely to occur. Mixers are used in the gold leaching processes. Special tanks for carbon in pulp (CIP) or carbon in leach (CIL) are built with mixers. The largest diameter of these tanks is approximately 17 m (56 ft). For large plants, the process of flotation or leaching in a single tank is not very efficient. To increase productivity, tanks are installed in series (up to five stages or five tanks in a series), thus eliminating possible short-circuiting.

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COMPONENTS OF SLURRY PLANTS

concentrate air

feed

agitator impeller gangue FIGURE 7-29

FIGURE 7-30

Simplified flotation circuit.

Top-entry agitator. (Courtesy of Hayward Gordon, Canada.)

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There are processes that require a single mixing tank with one or two agitator mixers; an example is the preparation milk of lime, where solids are simply mixed with water and the mixers are used to prevent sedimentation. Special processes in solvent extraction plants require gentle mixing while pumping the organic solution over a weir. For example, in a copper SX-EW plant the organic solution is kerosene-based; it absorbs the copper sulfates and separates them from other solids. For these applications, manufacturers have developed special pump mixers, which are essentially large, open shrouded pump impellers with a large number of vanes. Most tanks used in mining are built with a height to diameter ratio around unity and use a single-stage mixer. The mixer is of a vertical shaft design (Figure 7-30). The impeller diameter is in the range of 30% to 45% of the tank diameter. The impeller is usually situated at about 30% of the depth of the tank. Baffles at the wall of the tank break the vortex that is formed by the agitator. These baffles have a width of 8% of the tank diameter. Certain processes use tall, concentric tanks (sometimes called Pechuka tanks) with two agitators in a series on a single shaft. These are more common in South Africa than in North America. Horizontal agitators are installed on the periphery of very large tanks, particularly in the pulp and paper industry. They have not been popular in slurry mixing tanks in mineral processes as they are difficult to maintain. A vertical mixing tank (Figure 7-31) is fit with baffles at the walls to break the vortex generated by the agitator. A certain gap is left between the baffles and the wall of the tank. In some respects, the propeller-type mixer causes continuous mass flow against a stationary flat bottom. If the speed of rotation is not sufficient, a stagnation area develops. The levels of turbulence in the tank, as well as the shape of the bottom of the tank,

b = DT/12 to D T /10 feed

H = DT

gap = (1/72) tank diameter

C = D A = DT /3

D A = 0.3 D T to 0.45 D T DT

FIGURE 7-31

Typical dimensions for the design of mixing tanks.

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are important parameters. Sometimes a conical deflector is installed at the bottom of the tank. Mixing can be done by using an outside pump. The tank must then have a conical bottom. The discharge of the tank at the bottom flows to a pump. The discharge of the pump is returned to the tank and passes through a jet mixer. For tanks with a flat bottom, the discharge may be from the bottom or through a pipe on the side (Figure 7-32). For very viscous mixtures, anchor agitators are recommended. The blades are vertical and rotate fairly close to the wall surface of the tank. For some difficult and frothy pulps in biological slurry treatment, helicoidal mixers are installed (Figure 7-33). For some complex mixtures, the agitator may incorporate a hollow shaft to sparge oxygen, an impeller to break up the froth at a high level, and one at the bottom of the shaft to mix the slurry (Figure 7-34). The propeller-type agitator is the most common in the mining industry. Its design can be examined from various angles: mechanical strength, speed of operation, hydrofoil shape of the blades, etc. The shaft is designed for the “jamming” condition or “start-up” in a settled tank. The main force is taken at 75% of the maximum radius blade span meas-

feed

bottom and side discharge

feed

Side discharge

feed

bottom and central discharge

feed

Top discharge

FIGURE 7-32 Various patterns of discharge from the mixing tank in accordance with the required degree of agitation.

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Anchor mixer for very viscous mixtures FIGURE 7-33

Helicoidal mixer for very Helicoidal viscousmixer mixtures for v

Anchor and helicoidal mixers are used for particularly viscous mixtures.

hollow shaft hollow shaft vertical verticalmotor motor Hollow shaftfor for Ho llow shaft central oxygen oxygen flow flow

to su support SSole o lepplate late to pport mixer Topoof structure Top f vvessel essel stru cture

threaded threaded coupling coupling Open housingfor for Op en housing breaker foam break er

oTTop p column column (flanged) (flanged)

foambreaker break er foam Grease lubricatedsleeve sleeve Gre ase lubricated bearing & packing bearing & pack ing

Bo ttom column Bottom column(flanged) (flanged) Mixer Mixer

oxygendischarge discharge oxygen

FIGURE 7-34 Complex mixer for biological slurries with central injection of oxygen and with a baffle to skim the froth. 7.44

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ured from the center of the shaft. For severe duty application, the shaft is designed for 2.5 times the rated motor torque without exceeding the yield stress (or 0.2% proof stress) of the shaft material. For light-duty mixers, the shaft is designed for 1.5 times the rated motor torque without exceeding the yield stress of the shaft material. The shaft must not operate at speeds higher then 70% of the first critical speed if the impeller is not dynamically balanced, or higher than 85% of the first critical speed if the impeller is dynamically balanced. The deflection of the shaft should be limited, particularly in the case of anchor agitators, so that the blades do not hit the walls or baffles. The shaft is supported between two bearings in a cantilever arrangement. The bearings may be part of a vertical gearbox or an independent bearing assembly. For large agitators, the gearbox and bearing assembly are integral. In Figure 7-35, the flow between the two levels z1 and z2 contracts across the propeller, which induces a velocity Vi. Since the flow at the free surface as well as at the bottom of the tank is negligible, this flow resembles the ground effect of a hovering helicopter. From airscrew theory, the thrust across the propeller is: T = 2AV 2i

(7-22)

where Vi = the induced velocity A = area of flow across the propeller 苶/2 苶A 苶 苶 Vi = 兹T

(7-23)

power = TVi = 2AVi3

(7-24)

where T = thrust Another important theory used to calculate thrust is the blade theory. In Chapter 3, the concept of lift and drag around an aerofoil or an aircraft wing was introduced. The blade of a propeller mixer is essentially a rotating wing exposed to a flow velocity V and a rotating speed in rpm. Due to the contraction of the stream across the propeller, the flow velocity is half the induced velocity. Since the blades are set at a certain pitch angle (quite often 40–45°), they are at an angle of attack with respect to the relative speed. The relative speed is the vectorial addition of these two perpendicular speeds (Figure 7-35). 2 2 苶V 苶2苶 苶苶r苶 苶 W= 兹¼ i +

(7-25)

where the angular speed = 2N/60 and N is rotations per minute. When the flow approaches a blade at a relative speed W and an angle of incidence ␣ with respect to the chord of the blade, a certain pressure distribution develops around the blade. The result is a lift force L perpendicular to W and a resistance drag force D tangential to it. A good designer keeps the angle of attack at a value that corresponds to the maximum lift-to-drag ratio. For every airfoil, a plot of lift-to-drag curve (Figure 7-36) is obtained. If the blade is pitched at an angle , then the vertical force is Y = lift cos – drag sin = L cos – D sin

(7-26)

and the horizontal force X = L sin + D cos

(7-27)

The vertical force Y is opposite to thrust, whereas the horizontal force X multiplied by the radius gives the resistant torque. Near the tips of the propeller, the flow degrades due to the presence of tip vortices.

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b = DT /12 to D T /10 feed

Y = L cos – D sin

7.46

X = L sin + D cos angle of incidence

z1

V /2

i

V

Lift

W

pitch angle of blade Drag

i

z2

W

V /2

i

U=r FIGURE 7-35

Induced velocity and hydrodynamic forces for a propeller-type mixer.

Page 7.46

W

gap

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angle of incidence Lift coefficient C L 1.3

FIGURE 7-36 flow.

Lift W

Drag

Drag coefficient C D 0.1

stall

stall

10 o Angle of attack (or incidence )

10o Angle of attack (or incidence )

Lift and drag forces as a function of the angle of incidence with respect to the

The load distribution for many propellers is a maximum at 75% of the radius (Figure 737) so that the effective torque is measured at this region. In order to predict the performance of a full-scale mixer, a test may be conducted on a reduced scale model under laboratory conditions. Performance is scaled up to large units using nondimensional factors such as the power factor from the theory of rotating equipment. In basic terms, it means that two mixers of the same geometrical design (but differ-

radial distribution of load

FIGURE 7-37

mixer propeller blade

Distribution of the total force as a function of the span of the blade.

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ent sizes) will behave in similar ways with the same fluid and tank conditions. They would have the same power number. The power number is defined as P Cp = ᎏ N3D5

(7-28)

where D = tip diameter of the mixer N = rotational speed (rpm) = density of the slurry P = power The mixer Reynolds number is defined as ND2 Rem = ᎏ 2

(7-29)

The relationship between the power number and the Reynolds number is shown in Figure 7-38. Examples of the power number are presented in table Table 7-7. The ability of the mixer impeller to pump or induce flow is measured and defined by a nondimensional flow factor: Q CQ = ᎏ3 ND

(7-30)

It is also a function of the Reynolds number, as shown in Figure 7-39. Gates et al. (1976) examined the use of mixerss to maintain solids in suspension. An equivalent volume Voleq is defined is defined as

FIGURE 7-38 Power coefficient versus Reynolds number. The top curve is typical of flatblade mixers with wide blades. The middle curve is typical of flat-blade mixers with narrow blades. The bottom curve is typical of pitched-blade mixers. (Reproduced by permission of Hayward Gordon.)

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TABLE 7-7 Typical Power Number for Mixers in Turbulent Flows Type

3 Blades

4 Blades

1.6 0.7 0.2

1.7 0.8 0.5

45° flat-pitched blades Propeller (marine) Hydrofoil

Voleq = Sm Vol

(7-31)

where Sm is the specific gravity of the slurry mixture. The terminal velocity for spheres was discussed in Section 3-1-3-1. A design settling velocity Vd is correlated to Vt by a correction factor fw: Vd = fwVt

(7-32)

where Vd = design settling velocity Vt = the terminal (or free settling) speed The correction factor fw is presented in Table 7-8 as a function of weight concentration. This empirical coefficient was developed by Chemineer Inc., based on experimental work. It is often difficult to predict the nature of the flow or the drag coefficient near an impeller blade. At weight concentrations in excess of 15%, the solids start to interact, hindering settling so that the settling velocity must be adjusted. The level of agitation is very important to the mechanics of suspension. Chemineer Inc. developed a scale of agitation from 1 to 10, summarized by Gates and al. (1976) as in Table 7-9. Figure 7-40 shows the level of suspension of solids in correlation with the Chemineer scale. Often, manufactures define mixing as simple, mild, medium, vigorous, or violent (Figure 7-40). The science of mixing and keeping solids in suspension is highly empirical. The engineer should take into account existing similar installations as well as lab work results. Because of wear associated with slurries, a simple flat blade system is used to design

Flow Coefficient

D/H = 0.40 D/H = 0.45 D/H = 0.50 Ratio of Impeller diameter to tank height

C = Q 3 Q ND

Turbulent Laminar

Transition

Reynolds Number

2

Re = ND /(2

)

FIGURE 7-39 Flow coefficient versus Reynolds number.

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TABLE 7-8 The Correction Factor fw Presented as a Function of Weight Concentration Solids weight concentration (%)

Factor fw

2 5 10 15 20 25 30 35 40 45 50

0.8 0.84 0.91 1.0 1.1 1.2 1.3 1.42 1.55 1.70 1.85

From Gates et al., 1976, reprinted by permission from Chemical Engineering.

TABLE 7-9 Chemineer Scale for Agitation of Solids in Suspension Scale of agitation 1–2

3–5

6–8

9–10

Description At levels 1–2, agitation is required for minimal suspension of solids. Agitators capable of working at an agitation level of 1–2 will: 앫 Produce motion of all of the solids of the design-settling velocity in the vessel 앫 Permit moving fillets of solids on the bottom, which are periodically suspended Agitation levels 3–5 characterize most chemical process industries solids suspension applications. This scale range is typically used for dissolving solids. Agitators capable of working at an agitation level of 3–5 will: 앫 Suspend all of the solids of design velocity completely off the vessel bottom 앫 Provide slurry uniformity to at least one-third of the fluid batch height 앫 Be suitable for slurry draw-off at low exit-nozzle elevations Agitation levels 6–8 characterize applications where the solids suspension level approaches uniformity. Agitators capable of scale level 6 will: 앫 Provide concentration uniformity of solids to 95% of the fluid batch height 앫 Be suitable for slurry draw-off up to 80% of the fluid batch height Agitation levels 9–10 characterize applications where the solids suspension uniformity is the maximum practical. Agitators capable of scale 9 will: 앫 Provide concentration uniformity of solids to 98% of the fluid batch height 앫 Be suitable for slurry draw-off by means of overflow

From Gates et al., 1976. Reprinted by permission from Chemical Engineering.

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a. Unstable particles are on vessel bottom (Scale of agitation = 1)

b. Particles swept off vessel bottom (Scale of agitation = 3)

c. Solids are homogeneously distributed (Scale of agitation = 9)

FIGURE 7-40 Intensity of agitation yields different patterns of solid suspension. (From Gates et al., 1976. Reproduced by permission from Chemical Engineering.)

the mixer. The blades are often pitched at an angle of 45° with respect to the horizontal plane. To size such a flat-bladed propeller in a mixing tank, with baffles at 90° to each other, baffle width of 1/12th of the tank diameter, and offset from the wall with a gap of 1/72nd of the tank diameter, Gates et al. (1976) proposed the following empirical equation in USCS units:

冢

HP Din = 394 ᎏ SmN 3n

冣

0.2

(7-33a)

where Din = diameter in inches HP = power in hp n = number of impellers N = speed in rev/min Sm = specific gravity of the slurry mixture To express Equation 7-33a in SI units:

冢

P Dimp = 37.57 ᎏ SmN3n

冣

0.2

(7-33b)

where Dimp = diameter in meters P = power in Watts To correlate between the levels of agitation in a tank, the speed of rotation, and diameter of the impeller, Gates et al. (1976) plotted the scale of agitation versus , a factor defined in U.S. units as:

= N3.75Dimp2.81/Vd with Vd expressed in ft/min, D in inches, and N in rev/min (Figure 7-41).

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FIGURE 7-41 Chart to determine speed and diameter of a mixer versus the solids suspension scale. [From Gates et al. (1976). Reprinted by permission of Chemical Engineering.]

Example 7-4 A tank contains 600 ft3 of a slurry mixture. The specific gravity of the solids is 4.1 and the average particle size d50 is 0.01 ft. It is required to be designed for overflow output at an agitation scale of 9. The weight concentration of the mixture is 20%. Size the mixer, assuming a single impeller with an impeller-to-tank diameter of 0.4, baffles 1/12th the tank diameter and baffle gap of 1/72nd the tank diameter; use Figure 7-34. Assume viscosity of 1 cP. Compare the power consumption with a mixer operating at a Chemineer scale of 5. Solution in USCS Units Since most tank have a diameter equal to the height, and assuming a volume of 90% the tank volume, the effective volume occupied by the slurry is Vol = 0.9 × 0.25 × DT3 = 600 ft3 Hence, DT = 9.47 ft T = 9.47/0.9 = 10.52 Dimp = 0.4 × 10.52⬘ = 4.21⬘ or approximately 50.5 inch For a scale of 9 and D/T = 0.4, = 30 × 1010, so

= N3.75D2.81/Vd We must determine Vd. The particles are coarse enough to assume a drag coefficient CD = 0.44. Substituting into Equation 3-7 4(3.1) × 32.2 × 0.01 4(S – L)gdg = ᎏᎏᎏ CD = ᎏᎏ 3 × V 2t 3LV 2t Vt = 1.5 ft/s = 90 ft/min From Table 7-8, the correction factor fw = 1.0 and Vd = VT, or

=30 × 1010 = N3.75D2.81/Vd = N3.7550.52.81/90 = 679.5 N3.75 N = 202 rev/min To determine the required horsepower, Equation 7-33a is used:

冢

HP Din = 394 ᎏ SmN 3n

冣

0.2

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The specific gravity of the mixture is obtained by using Equation 1-4: 100 m = ᎏᎏᎏ = 1128 kg/m3 15/4100 + (100 – 15)/1000 or Sm = 1.128. [50.5/394] × (1.128 × 2023)0.2 = HP0.2 or HP = 322 hp. For a scale of 5, and D/T = 0.4, = 5 × 1010, so

= 5 × 1010 = N 3.75D2.81/Vd = N 3.7550.52.81/90 = 679.5 N 3.75 N = 125 rev/min To determine the required horsepower, Equation 7-33a is used: [50.5/394] × (1.128 × 1253)0.2 = HP0.2 or HP = 76 hp. Going from a mild level of agitation at a 5 on the Chemineer scale to level 9 increases the power consumption 4.24 fold. The shear rate was previously defined in Chapter 2 as the rate of change of the velocity with respect to height above the wall. More generally for a mixer, the shear rate is the change of velocity with respect to depth. The induced velocity is essentially created by the impeller, and the maximum shear rate is experienced at the tip of the blade. There is, however, an average shear rate estimated for the impeller zone, and an average in bulk of the tank. For a radial impeller, all the flow is discharged at the tip of the impeller, and all the solids are subject to the same speed and head. In the case of the propeller or axial turbine, the velocity distribution is proportional to the radius. All solids pass through a higher shear zone in the case of the radial machine. The closer the impeller is to the bottom of the tank, the more the induced velocity is suppressed. A proximity factor hc/D is defined as the ratio of the gap of the impeller to the diameter. This principle is similar to those in the world of aeronautics. The ground effect is essentially the pressure exerted by a helicopter or airplane. The closer it is to the ground, the more pronounced is the effect and, eventually, the induced drag is reduced when the machine flies very close to the ground. In the case of the mixers, the flow is restricted as the impeller gets closer to the bottom of the tank. This affects power consumption. A radial mixer tends to induce flow tangentially, whereas an axial machine tends to induce it vertically. Propeller and radial mixers tend to create different patterns of recirculation around their blades. Mounting more than one impeller on the same shaft gives different patterns of performance based on the mutual interference that the different inpellers exert on each other. The power from two axial turbines is not twice the power from a single propeller, as the induced velocity of the second propeller is not twice the induced velocity of the first propeller. In fact, in some applications, the first propeller is smaller in diameter than the bottom propeller. However, two radial impellers closely spaced consume more than twice the power of a single unit. In sewage treatment plant processes as well as in chemical and mining processes, gas is sparged in the liquid. The impeller of the mixer is then used to provide dispersion of the gas and circulation of the tank contents. For radial machines, a small radial impeller is installed at the bottom to mix the gas. In the case of hydrofoils, different approaches are used. If the shaft and blades are hollow, gas may be pumped through the shaft and blades. In other applications, the impeller is contained within a cylinder that is within the tank. This prevents flooding the impeller with gas.

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A hydrofoil type of mixer would have the highest pumping capacity but would develop the lowest head or shear at constant horsepower or torque. Hydrofoil mixers are often chosen for high-flow applications. For heat transfer, solid suspension, blending, or solid dissolving, bulk pumping is critical. A hydrofoil mixer would be the preferred machine. However, for gas–liquid contacting, molecular mixing, solid dispersion (reduction of agglomerates), an axial flow, flatblade machine or radial mixer are preferred options. The mixing of non-Newtonian slurries is fairly complex and must rely on experimental data. Wasp et al. (1977) proposed that the power consumption for mixing non-Newtonian slurries is a function of the Reynolds number, the Hedstrom number, and the Froude number: P 2 0 N2Dimp ᎏ = fn ᎏ ,ᎏ ,ᎏ 3 5 2 2 2 N D imp ND imp N D imp g

冢

冣

(7-34)

In other words, the power consumption is based on the Reynolds number, Hedstrom number, and Froude number based on the impeller diameter. The mixing of non-Newtonian fluids is common in the manufacture of polymers and considerable data is available. It is, however, not very wise to extrapolate to non-Newtonian slurry mixtures. In the last 25 years of the 20th century, larger and larger mixers have been built. Certain shaft failures and blade failures have occurred on some large agitators. Some were due to corrosion of the bolts holding the blades and others due to jamming of the shaft. It is important to understand that starting an agitator from fully settled conditions can be very stressful to the machine. The reader should consult appropriate reference books on machine design and gearbox design. A service factor of 1.5 is a minimum for sizing the gearbox. It would be beyond the scope of this book to explore the selection of gearboxes for agitators. However, the designer of slurry mixers should be aware of certain important mechanical criteria, such as critical speed. To examine the distribution of loads on the blade, the program “Agitblade” may be used. Program “Agitblade” for Propeller Blade Loads CLS REM propeller design for AGITATOR PI = 3.1415 DIM R(20), PITCH(20), V(20), W(20), ALPHA(20), BETA(20), BLPHA(20) DIM C(20), CL(20), CD(20), CT(20), CU(20), T(20), INCIA(20) DIM P(20), TK(20), VV(20), VH(20), VS(20), PK(20) INPUT “FLUID DENSITY”; DENS INPUT “radius at hub “; RHUB INPUT “ RADIUS AT TIP”; RTIP AP = PI * RTIP ^ 2 100 PRINT “OPTIONS FOR COMPUTATION” PRINT “1-CALCULATIONS ASSUMING KNOWLEDGE OF INDUCED VELOCITY” PRINT “ USING MOMENTUM THEORY” PRINT “2- BLADE THEORY CALCULATIONS FOR BLADEWISE DISTRIBUTION OF FORCES” PRINT “ AND FLOW CHARACTERISTICS” PRINT “3- VORTEX FLOW CALCULATIONS”

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INPUT “YOUR OPTION “; OPT IF OPT = 1 THEN 300 IF OPT = 2 THEN 2000 IF OPT = 3 THEN 6000 300 INPUT “NUMBER OF PROPELLERS ON THE SAME SHAFT”; NU IF NU > 1 THEN PRINT “CALCULATIONS FOR FIRST PROPELLER” INPUT “VELOCITY UPSTREAM THE PROPELLER “; VS IF VS = 0 THEN 305 INPUT “IS THIS VELOCITY PARALLEL TO SHAFT (Y/N)”; V$ IF V$ = “Y” OR V$ = “y” THEN 305 INPUT “INCLINATION OF U/STREAM VELOCITY WRT SHAFT AXIS IN DEGREES”; INC INCI = INC * PI/180 305 FOR M = 1 TO NU VS(M) = VS PRINT PRINT PRINT “CALCULATION FOR PROPELLER “; M INCIA(M) = INCI * 180/PI VV(M) = VS * COS(INCI) VH(M) = VS * SIN(INCI) GOTO 320 310 VV(M) = VS 320 IF VV(M) = 0 THEN PRINT “COMPONENT OF U/S VEL PARALLEL TO PROP IS NIL” 321 INPUT “IS THE HYDROSTATIC PRESSURE EQUAL ON BOTH SIDES OF THE PROP (Y/N)”; PH$ IF PH$ = “Y” OR PH$ = “y” THEN 340 INPUT “HYDROSTATIC PRES UPSTREAM PROP “; PHUS INPUT “HYDROSTATIC PRES DOWNSTREAM PROP “; PHDS 340 INPUT “DO YOU KNOW THE INDUCED VELOCITY (Y/N)”; IV$ IF IV$ = “N” OR IV$ = “n” THEN 600 INPUT “INDUCED VELOCITY “; VU(M) VU = VU(M) T(M) = DENS * AP * SQR((VV(M) + VU) ^ 2 + VH(M) ^ 2) * VU ^ 2 + (PHUS – PHDS) * AP P(M) = T(M) * VV(M) + .5 * T(M) * VU(M) TK(M) = T(M)/1000 PRINT USING “ THRUST = ###########.#### kN”; TK(M) PK(M) = P(M)/1000 PRINT USING “POWER = #########.### kW”; PK(M) VH = VH(M) VS = SQR((VV(M) + VU) ^ 2 + VH(M) ^ 2) INCI = ATN(VH/(VV(M) + VU(M))) NEXT M LPRINT “CALCULATIONS BASED ON MOMENTUM THEORY” FOR M = 1 TO NU

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LPRINT LPRINT LPRINT “CALCULATION FOR PROPELLER “; M LPRINT USING “VELOCITY UPSTREAM SHAFT = ###.## m/s”; VS(M) LPRINT USING “ITS COMPONENT PARRALLEL TO SHAFT= ###.## m/s “; VV(M) LPRINT USING “ITS COMPONENT PERPANDICULAR TO SHAFT = ###.## m/s “; VH(M) LPRINT USING “ITS INCLINATION WRT TO SHAFT = ###.# deg “; INCIA(M) LPRINT USING “INDUCED VELOCITY = ###.## m/s “; VU(M) LPRINT USING “ RESULTANT THRUST = #####.## KN”; TK(M) LPRINT USING “ INDUCED POWER CONSUMPTION = #####.## KW”; PK(M) NEXT M GOTO 8000 600 PRINT “YOU MAY HAVE TO CALCULATE THE INDUCED VELOCITY BY ASSUMING A CERTAIN THRUST MAGNITUDE” 2000 INPUT “VERTICAL SPEED”; V INPUT “PITCH AT HUB”; PITCHR INPUT “PITCH AT TIP”; PITCT INPUT “REQUIRED TIP SPEED “; VTIP RPS = VTIP/RTIP PRINT USING “THE ROTATIONAL SPEED IN ######.## RAD/S “; RPS RDIV = (RTIP - RHUB)/10 PITDIV = (PITCHR - PITCT)/10 INPUT “chord at the root”; CR INPUT “TIP CHORD “; CT REM CALCULATE THE ADVANCE RATIO OF THE PROPELLER J = V/(2 * PI * RTIP) PRINT “ADVANCE RATIO OF PROP “; J REM CALCULATE BLADE AREA AND HENCE ASPECT RATIO SB = (RTIP - RHUB) * .5 * (CT + CR) AR = (RTIP - RHUB) ^ 2/SB PRINT “BLADE ASPECT RATIO”; AR PRINT “BLADE AREA “; SB CD0 = .015 K = .1/(1 + 2/AR) KD = K ^ 2/3 * AR REM K IS THE LIFT COEFFICIENT SLOPE DCL/DALPHA IN THE LINEAR RANGE PRINT “LIFT SLOPE PER DEGREE IN THE LINEAR RANGE “; K ACR = (CR - CT)/(RTIP - RHUB) CDIV = (CR - CT)/10 DTIP = 2 * RTIP INPUT “POWER NUMBER “; CP P = CP * DENS * RPS ^ 3 * DTIP ^ 5 * CP/2 * PI PK = P/1000 TOR = PK/RPS PRINT USING “REQUIRED TORQUE = #####.## KNm”; TOR PRINT USING “REQUIRED POWER #####.## KW”; PK FOR N = 1 TO 11 R(N) = RHUB + (N - 1) * RDIV PITCH(N) = PITCHR - (N - 1) * PITDIV

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V(N) = R(N) * RPS W(N) = SQR(V(N) ^ 2 + V ^ 2) BLPHA(N) = ATN(V/V(N)) ALPHA(N) = BLPHA(N) * 180/(2 * PI) BETA(N) = PITCH(N) - ALPHA(N) C(N) = CR - ACR * (R(N) - RHUB) CL(N) = K * BETA(N) CD(N) = CD0 + KD * CL(N) CT(N) = CL(N) * COS(BLPHA(N)) - CD(N) * SIN(BLPHA(N)) CU(N) = CL(N) * SIN(BLPHA(N)) + CD(N) * COS(BLPHA(N)) T(N) = CT(N) * .5 * W(N) ^ 2 * DENS * C(N) * RDIV NEXT N RPM = RPS * 60/(2 * PI) LPRINT “AGITATOR PROPELLER” LPRINT USING “VERTICAL SPEED = #####.### m/s”; V LPRINT USING “HUB RADIUS = ###.#### m TIP Radius = ####.#### m “; RHUB, RTIP LPRINT USING “TIP SPEED = ####.### M/S”; VTIP LPRINT USING “ANGULAR SPEED = ######.## RPM”; RPM LPRINT USING “VERTICAL SPEED = ####.## m/s “; V LPRINT USING “PITCH AT TIP= ####.## ; PITCH AT HUB = ####.##”; PITCT, PITCHR LPRINT “BLADE AREA”; SB LPRINT “BLADE ASPECT RATIO”; AR LPRINT “APPROX LIFT SLOPE COEF IN LINEAR RANGE”; K LPRINT “APPROX LIFT/DRAG POLAR RATIO “; KD LPRINT USING “PROPELLER AREA ######.## m^2”; AP LPRINT “LOCAL RADIUS PITCH ANGLE ANGLE OF INCIDENCE TOTAL VELOCITY ALPHA” FOR N = 1 TO 11 LPRINT R(N), PITCH(N), BETA(N), W(N), ALPHA(N) NEXT N LPRINT “ LOCAL RADIUS,CHORD, LIFT COEF, DRAG COEF, THRUST COEF, TANG FORCE COEF” FOR N = 1 TO 11 LPRINT R(N), C(N), CL(N), CD(N), CT(N), CU(N) LPRINT NEXT N 6000 8000 INPUT “DO YOU WANT TO PROCEED WITH OTHER THEORIES OF DESIGN (Y/N) “; D$ IF D$ = “Y” OR D$ = “y” THEN 100 END REM CT=THRUST COEFFICIENT PARALLEL TO PROPELLER SHAFT REM CU= FORCE COEFFICIENT PERPANDICULAR TO SHAFT AND CAUSING RESISTANCE

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The designer of mixers with hydrofoils should be aware that the center of pressure does not always correspond with the center of gravity of the blade. The center of pressure could be as far as 25% of the blade chord on high-aspect-ratio blades or those with a high ratio of blade length to blade chord. A pitching–bending moment results; it must be considered when sizing the bolts of the blade. It is important to check the stress on the shaft of an agitator. The following program, written in QuickBasic, allows one to check for the case of a shaft with two impellers in series. Program “Dblagit.Bas” for Two Agitators Mounted on a Shaft PRINT “double agit” pi = 4 * ATN(1) ‘INPUT “are you using SI units (Y/N)”; l$ l$ = “n” IF l$ = “n” OR l$ = “N” THEN conv = .0254 IF l$ = “y” OR l$ = “Y” THEN conv = 1! ‘INPUT “distance from bottom bearing to first coupling”; ab ab = 9.41 ‘INPUT “shaft O.D. for first section of shaft ab “; dab0 ‘INPUT “shaft I.D. for first section of shaft ab “; dabi dab0 = 5 jab = (pi/32) * (dab0 ^ 4 - dabi ^ 4) * conv ^ 4 ‘INPUT “distance from first coupling to second coupling”; bc bc = 135.64 ‘INPUT “shaft O.D. for second section of shaft ab “; dbc0 ‘INPUT “shaft I.D. for second section of shaft ab “; dbci dbc0 = 8.625 dbci = 7.625 ‘sch 80 jbc = (pi/32) * (dbc0 ^ 4 - dbci ^ 4) * conv ^ 4 ‘INPUT “distance from second coupling to first impeller”; cd cd = 48 ‘INPUT “shaft O.D. for third section of shaft cd “; dcd0 ‘INPUT “shaft I.D. for first section of shaft cd “; dcdi ‘dcd0 = 6.625 ‘dcdi = 6.065 dcd0 = dbc0 dcdi = dbci jcd = (pi/32) * (dcd0 ^ 4 – dcdi ^ 4) * conv ^ 4 ‘INPUT “distance from first to second impeller”; de de = 150 dde0 = dcd0 ddei = dcdi jed = jcd ‘INPUT “ power on top impeller”; p1 p1 = 21 * 746 ‘INPUT “ power on bottom impeller”; p2 p2 = 35 * 746 INPUT “rpm”; rpm

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7.59

w = rpm * 2 * pi/60 ‘assume start up torque factor 250% ts = 2.5 t1 = p1/w t2 = p2/w PRINT “torque from top impeller”; t1 ‘INPUT “diameter of first impeller”; dimp1 dimp1 = 84 ‘INPUT “diameter of second impeller”; dimp2 dimp2 = 66 r1 = dimp1 * conv/2 r2 = dimp2 * conv/2 f1 = t1/(.75 * r1) PRINT “ tangential force on top impeller”; f1 f2 = t2/(.75 * r2) PRINT “tangential force on bottom impeller”; f2 ma = (f1 * conv * (ab + bc + cd) + f2 * conv * (ab + bc + cd + de))/10 ‘assume unsymmetry of forces at 10% radius may = SQR(ma ^ 2 + .75 * (t1 + t2) ^ 2) mayn = ma/1000000! PRINT USING “total bending moment at a = ####.## MNm”; mayn syab = mayn/jab PRINT USING “shaft stress syab = ###.## MPa”; syab IF syab > 100 THEN PRINT “warning the required stress limit is 100 MPa” mb = (f1 * conv * (bc + cd) + f2 * conv * (bc + cd + de))/10 mby = SQR(mb ^ 2 + .75 * (t1 + t2) ^ 2) mbyn = mby/1000000! PRINT USING “total bending moment at b = ####.## MPa”; mbyn sybc = mbyn/jbc PRINT USING “shaft stress syab = ###.## MPa”; sybc IF sybc > 100 THEN PRINT “warning the required stress limit is 100 MPa” mcd = (f1 * conv * cd + f2 * conv * (cd + de))/10 mcd = SQR(mcd ^ 2 + .75 * t2 ^ 2) mcdn = mcd/1000000! PRINT USING “total bending moment at c = ####.## MPa”; mcdn sycd = mcdn/jab PRINT USING “shaft stress sycd = ###.## MPa”; sycd IF sycd > 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 = mR2

(7-38)

where = 2N/60 R = radius of rotation The ratio of the centrifugal force to the weight is called the centrifugal number Nc: Nc = mR2/mg = R2/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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FIGURE 7-44

7.63

Centrifugal separator. (Courtesy of Knelson Concentrators.)

冪莦

R Ks = 2N ᎏ g

(7-40)

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

2RNdp 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

冣

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 = 2R0Hus = CvQ Separation will occur when us > CvQ/2R0H.

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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

7.65

Liquid Mixture

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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

앫 앫 앫 앫 앫

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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water connection packings rings

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bearings cartridge

shaft sleeve gland plate

8.3

adjustment bolt

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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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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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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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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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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8.9

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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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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8.17

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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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

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

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

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CHAPTER EIGHT

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

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

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

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

190 280 420 760 874

231 462 693 924 1063

297 594 891 1188 1366

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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

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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CHAPTER EIGHT

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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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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CHAPTER EIGHT

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 = 2RN/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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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

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

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

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

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

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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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 222 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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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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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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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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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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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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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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8.41

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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CHAPTER EIGHT

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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8.43

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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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

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 = 0.25d 2SLPA On the suction side, there is suction pressure Ps, so the thrust force is: TS = 0.25d 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 – 0.1252imp{[R 22 – R 2b] – [(1 + tb/xb)2 · (R 22 – R 2b)]}

(8-40)

where P2 = 0.75g(TDH) + PS.

Ps

FIGURE 8-31

R1

P1 dA

d sl

PA

Axial loads on an impeller with plain shrouds.

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CHAPTER EIGHT

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 – 0.1252imp{[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 = PAR 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 – 0.1252imp{[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 = PsRS2

(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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8.47

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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CHAPTER EIGHT

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) <= 35 THEN nr(i) = n/1.354 Rem radial load equation is in SI unit Fr(i)=k(i) * sg * 1000 * 9.81 * d2 * b2 * h(i) u1 = nr(i) * pi * (dr/304.5)/60 u2 = nr(i) * pi * (d2/304.5)/60 u3 = nr(i) * pi * (dsl/304.5)/60 u4 = nr(i) * pi * (ds/304.5)/60 u5 = nr(i) * pi * (db/304.5)/60 PRINT USING “u5 = ####.## ft/s u3= ####.## ft/s”; u5; u3 h1 = h2 - (nr(i)/1000) ^ 2/13.55 * ((1 + tfv/sf) ^ 2 * (d2 ^ 2 - dr ^ 2))/c ^ 2 PRINT “h1= “; h1 INPUT “hit any key t continue “; l$ h5 = h2 - (nr(i)/1000) ^ 2/13.55 * ((1 + tbv/sb) ^ 2 * (d2 ^ 2 - db ^ 2))/c ^ 2 PRINT “h5= “; h5 INPUT “hit any key t continue “; l$ h3 = h5 - (u5 ^ 2 - u3 ^ 2)/(8 * 32.2) PRINT “h3= “; h3 INPUT “hit any key t continue “; l$ t21 = .5 * (a2 - a1) * (h2 + h1) * (sg/2.31) PRINT “t21 = “; t21 INPUT “hit any key t continue “; l$ t14 = .5 * (a1 - a4) * (h4 + h1) * (sg/2.31) PRINT “t14= “; t14 INPUT “hit any key t continue “; l$ t25 = .5 * (a2 - a5) * (h2 + h5) * (sg/2.31) PRINT “t25= “; t25 INPUT “hit any key t continue “; l$ t53 = .5 * (a5 - a3) * (h5 + h3) * (sg/2.31) PRINT “t53= “; t53 INPUT “hit any key t continue “; l$ fa(i) = t21 + t14 - t25 - t53 fan(i) = fa(i)/(9.81 * 2.2) f(i)=fr(i)/(9.81 * 2.2) PRINT USING “flow= ##### L/s head = #### m speed = ##### rpm”; Q(i); h(i); nr(i) PRINT USING “axial load = ####### lbs ####### N “; fa(i); fan(i) 998 END

8-5-3 Thread Pull Force Most modern slurry pumps feature a threaded shaft assembly. The torque from the operation of the pump is ultimately transmitted to the pump assembly through the shaft. The

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8.49

TABLE 8-13 Limiting Dimensions of American National Standard General Purpose Single Start ACME Threads. External Threads (for Shafts), Class 2G Nominal diameter (inch)

Threads per inch

Major diameter, min/max, in inch

Minor diameter, min/max, in inch

Pitch diameter, min/max, in inch

1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 4.50 5.00

4 4 4 3 3 3 2 2 2 2 2

1.4875–1.5000 1.7375–1.7500 1.9875–2.0000 2.2333–2.2500 2.4833–2.5000 2.7333–2.7500 2.9750–3.0000 3.4750–3.5000 3.9750–4.0000 4.4750–4.5000 4.9750–5.0000

1.1965–1.2300 1.4456–1.4800 1.6948–1.7300 1.8572–1.8967 2.1065–2.1467 2.3558–2.3967 2.4326–2.4800 2.9314–2.9800 3.4302–3.4800 3.9291–3.9800 4.4281–4.4800

1.3429–1.3652 1.5916–1.6145 1.8402–1.8637 2.0450–2.0713 2.2939–2.3207 2.5427–2.5700 2.7044–2.7360 3.2026–3.2350 3.7008–3.7340 4.1991–4.2330 4.6973–4.7319

For more information consult ANSI standard B1.5-1977.

ACME external thread (Table 8-13) is used for the shaft and the ACME internal thread (Table 8-14) is used for the impeller. Because the impeller thread is cast, particularly with hard metals, Class 2G is suggested because it has a wider range of tolerances than the 3G, 4G, and 5G Classes. BSW shaft threads are used on the smallest sizes. Figure 8-32 represents a typical ACME shaft thread. In order to determine the shaft stresses and the axial pull due to torque, the first step is to assess the torque due power: Tq = 60PW/(2N)

(8-51)

Example 8-5 A pump is sized for 200 m3/hr, at a TDH of 36 m and specific gravity of 1.4. The pump speed is 600 rpm and the hydraulic efficiency is 67%. Determine the power and the torque.

TABLE 8-14 Limiting Dimensions of American National Standard General Purpose Single Start ACME Threads, Internal Threads (for Impellers), Class 2G Nominal diameter (inch)

Threads per inch

Major diameter, min/max, in inch

Minor diameter, min/max, in inch

Pitch diameter, min/max, in inch

1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.50 4.00 4.50 5.00

4 4 4 3 3 3 2 2 2 2 2

1.5200–1.5400 1.7700–1.7900 2.0200–2.0400 2.2700–2.2900 2.5200–2.5400 2.7700–2.7900 3.0200–3.0400 3.5200–3.5400 4.0200–4.0400 4.5200–4.5400 5.0200–5.0400

1.2500–1.2625 1.5000–1.5125 1.7500–1.7625 1.9167–1.9334 2.1667–2.1834 2.4167–2.4334 2.5000–2.5250 3.0000–3.0250 3.5000–3.5250 4.0000–4.0250 4.5000–4.5250

1.3750–1.3973 1.6250–1.6479 1.8750–1.8985 2.0833–2.1096 2.3333–2.3601 2.5833–2.6106 2.7500–2.7816 3.2500–3.2824 3.7500–3.7832 4.2500–4.2839 4.7500–4.7846

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CHAPTER EIGHT

pitch p

pitch dia

2 = 29˚

minor dia

major dia

p/2

h = p/2

bt

FIGURE 8-32 ACME thread for pump shafts.

Solution in SI Units power = (200/3600) · 1.4 · 9810 · 36/0.67 = 40,997 W torque = power/rotational speed = 40,997/(2 · · N/60) = 652.5 N-m The helix angle of the thread is defined as

冤

L tan = ᎏ dm

冥

(8-52)

where L = length of a full turn = pitch for single-start threads L = 2 × pitch for double start threads pitch = distance between two consecutive threads measured at the thread diameter dm = pitch diameter The axial load transmitted through the thread from the torque is expressed as

dm cos ␣n – fL Fth = 2 · Tq ᎏᎏᎏ dm( fdm + L cos ␣n)

冤

冥

(8-53)

tan ␣n = tan ␣ cos For ACME threads it equals 14.5°. For square threads it is nil. For modified square it is 5°. For buttress threads it is 7°. So for an ACME thread: tan ␣n = 0.968 cos The coefficient of friction f is measured between the shaft and the impeller. In some pumps, the shaft is of steel but the impeller may be of bronze. Slurry pumps are essentially steel against iron and the coefficient of friction is considered to be in the range of 0.14 to 0.15:

dm cos ␣n – fL Fth = 2 · Tq ᎏᎏᎏ dm( fdm + L cos ␣n)

冤

冥

(8-54)

If n is the number of engaged threads, the axial load from this thread pull force creates a bending stress Sb and a shear stress Ss at the root of the shaft thread:

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8.51

冤

冥

(8-55)

冤

冥

(8-56)

3Fthh Sb = ᎏ2 dmnb t Fth Ss = ᎏ dmnb t

where h = height of the thread tooth = (major diameter – minor diameter)/2; in the case of ACME threads h = p/2 bt = thread width at the root 8-5-4 Radial Force on the Drive End When pulleys are installed to drive the pump, the torque transmitted through a pulley diameter Dp results in a force. Different equations are available, but the simplest expresses the resultant pulley force as:

冤 冥

4Tq Fp = ᎏ Dp

(8-57)

8-5-5 Total Forces from the Wet End The total radial force transmitted by the impeller to the shaft is due to the combination of the hydraulic radial thrust and the weight of the impeller: F1 = FR + Wimp

(8-58)

It is assumed that F1 is acting on the center of gravity of the impeller. The total axial load is: F2 = ±FA as the axial force may change direction as the number of stages exceeds two pumps in series. The torque, a source of torsion stress, was defined in Equation 8-51. On the drive side, the pulley force is upward for overhead-mounted motors or sideways for sidemounted motors. Calculations are often made on the assumption of overhead-mounted motors: F3 = Fp – Wp

(8-59)

On this basis, the shaft of the pump is designed. Due to fatigue considerations, the maximum stress should be smaller than the lesser of 18% yield, or 30% ultimate tensile strength. Referring to Figure 8-33, the equilibrium of forces shows that the reaction force at the wet end is RW and at the drive end it is RD: RW – F1 – RD + F3 = 0 Taking moments at the point of contact load of the wet end bearing: –A · F1 + RD · B – F3(B + C) = 0

(8-60)

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FR + Wimp Fp - Wpulley RD (with belt drive)

dA

Torque

d

d WE

FA

d L th

k

RW B

A

d D.E

C

FIGURE 8-33 Loads on the shaft of a horizontal slurry pump.

冤

冥

F3(B + C) A · F1 RD = ᎏᎏ + ᎏ B B

冤

冥

F1 · (A + B) F3C RW = ᎏ + ᎏᎏ B B

(8-61)

(8-62)

The reader should refer to specialized books on machine design that detail all aspects of the design of shafts, stress concentration, and bearing life calculations from the reaction forces at both the drive end (outboard) and wet end (inboard) bearings. The manufacturers of bearings have their own detailed factors for type of lubricant and ratio of axial to radial force. Some manufacturers of slurry pumps offer grease lubricated bearing assemblies and reserve the oil version for high-speed and high-thrust loads (as in pumps in series), whereas some use oil all across their range of pumps. 8-5-6 Flange Loads A common misconception is that the flanges of slurry pumps can take the same loads as water pumps. The fact that the discharge flange is split radially to allow access to rubber or metal liners by itself is an indication that this is not the case at all. The casing of a slurry pump can be distorted by excessive pipe loads on the flange. The consultant engineer is therefore well advised to contact the manufacturer for allowed flange loads. It is also necessary to provide proper pipe supports at the discharge of the slurry pump, and not to use the pump by itself as an anchor block to piping. The common error is to apply a large expansion at the discharge of the pump, such as from a 4⬙ pump discharge to an 8⬙ pipe. Doubling the diameter is effectively multiplying by four the area exposed to the full pressure, of which a quarter is absorbed by the pump, leaving three quarters to be balanced by a pipe fitting such as a properly supported dead end bulkhead, an anchor block, or, whenever possible, by soil friction, as is the case with pipeline pumps.

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THE DESIGN OF CENTRIFUGAL SLURRY PUMPS

8-6 ADJUSTMENT OF THE WET END The wear of the impeller front vanes, the throatbush, is believed to cause a drop in efficiency. The impeller must be readjusted by moving it forward. To move the impeller relative to the casing, the shaft assembly must be moved relative to the pump frame, as the latter is bolted to the pump casing. Two different methods are available: 1. A special bolt under the bearing cartridge (Figure 8-34) 2. Push and pull bolts at the drive end (Figure 8-35) Once the bearing cartridge is moved, it is fixed in place by clamping bolts that are tightened against the frame.

8-7 VERTICAL SLURRY PUMPS The vertical sump (Figure 8-36) complements the horizontal pump. The vertical pump is particularly suitable for floor sumps in mill discharge areas and in dealing with flotation circuits. The vertical sump pump may be supplied as: 앫 A stand-alone pump with double suction impeller (Figure 8-37) to be installed in a concrete or metal sump, particularly with flotation columns

bearing cartridge

pump frame

clamping bolts for bearing cartridge

adjustment bolt for bearing cartridge

FIGURE 8-34 Adjustment of the pump impeller by a special bolt between the bearing cartridge and the frame.

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clamping bolts bearing cartridge

push bolts pump frame FIGURE 8-35 Bearing assembly of the ZJ slurry pump (made in China) with adjusting push and pull bolts. (Courtesy of AJP Services Inc. The distributor for Canada.)

FIGURE 8-36 Inc.)

Sump pump with double suction impeller. (Courtesy of Mazdak International

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(a)

(b) Rubber-Lined, Acid-Proof Pumps with Double Suction Impellers Pump Size Frame Units

A

SP2 BV inch 26 mm 660 SP3 CV inch 32 mm 813 SP4 DV inch 37 mm 940 SP6 EV inch 48 mm 1220 SP8 FV inch 52 mm 1321

B

C

D

E

F

11 280 14 356 14 356 14 356 14 356

32 813 36 915 48 1220 60 1524 60 1524

36 915 48 1220 60 1524 72 1829 84 2987

12 305 16 406 20 508 24 610 14 356

20 508 20 508 26 660 35 889 35 889

G

H

J

K

L

6 16 20 20 20 152 406 508 508 508 8 22 22 26 26 203 559 559 660 660 10 26 26 30 30 254 660 660 762 762 14 34 34 38 38 356 864 864 965 965 16 47 47 52 52 406 1194 1194 1321 1321

N

P

40 1016 60 1524 72 1829 84 2134 96 2438

8 200 12.6 320 14.8 376 19.7 500 23 584

*D is the standard depth—other shaft length are available in 12⬙ increments—consult the plant for critical speed *E is the minimum priming level *C and N are typical sump dimensions for the sump

FIGURE 8-37 Dimensions for sump pump and corresponding sump. (Courtesy of Mazdak International Inc.) 8.55

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앫 A single suction impeller with an auger or agitator below the impeller to agitate settled solids in floor sumps (Figure 8-38) 앫 A top suction pump supplied integrally with a metal conical tank, called a “tank pump” (Figure 8-39) The vertical slurry pump is designed to have all its bearings above the baseplate so as to be well protected from slurry ingestion (Figure 8-40). Due to the depth of the sump, the design engineer must pay particular attention to the critical speed of the pump. For this reason, the shaft of these pumps can be as large as 200 mm (8⬙) to offer the necessary rigidity. Vertical slurry pumps are particularly popular in froth handling circuits. To handle the combination of solids, air, and liquids, a double suction impeller is often recom-

motor & pulleys

bearing assembly baseplate 2" discharge eyebolt

column wearplate casing

shaft

impeller screen

agitator

fig 8-38

FIGURE 8-38 Sump pump with single suction impeller and auger to agitate settled solids particularly suited for mill discharge floor. (Courtesy of Mazdak International Inc.)

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motor & pulleys

bearing assembly

baseplate eyebolt

inlet tank shaft impeller casing discharge

93 FIGURE 8-39 Tank sump pump with top suction impeller and integral tank for particularly difficult frothy slurries.

mended with vertical sump pumps. As an alternative, tank pumps with top suction (Figure 8-39) are used. In either configuration, the impeller must be designed to resist air biting. A special type of process used to extract gold is based on cyanide leaching. Leached gold is then separated by adsorption, the property of certain materials such as carbon to fix gold on their surface. Carbon spheres are used as an adsorption material. This process is done in special “carbon in leach” or “carbon in pulp” circuits with mixing tanks. The transfer of these solutions requires recessed or vortex impellers that can pump without breaking the carbon lumps. The impeller is recessed out of the flow as shown in Figure 8-41. A design that is gaining popularity in plants for recycling newspaper is the vertical pump with a recessed impeller and a chopper blade. It is not uncommon that the recycling bins for paper now found in every suburb of North America end up containing milk cartons, plastic bottles, toys, and even pieces of wood. These materials are not very good for conventional stainless steel pumps with mechanical seals. The cantilever sump pump (Figure 8-41) with the chopper offers the ability to pump long fibers while chopping them and eliminating the maintenance problems of mechanical seals.

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FIGURE 8-40 Components of a vertical slurry pump showing that the bearings are above the baseplate. 1. Shaft sea. 2. Top bearing cover. 3. Top bearing. 4. Bearing assembly. 5. Crease nipple. 6. Bearing locknut. 7. Bearing washer. 8. Bottom bearing—spherical roller for heavy duty. 9. Discharge pipe. 10. Baseplate. 11. Shaft seal. 12. Bottom hub. 13. Pedestal— Open structure. 14. Top suction strainer. 15. Wear plate. 16. Shaft. 17. Shaft sleeve. 18. Double suction impeller for minimum thrust loads. 19. Pump casing. 20. Lower suction strainer. (Courtesy of Mazdak International Inc.) 8.58

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stator

Rotor

Wet end with cutter

FIGURE 8-41 Vertical slurry pump with a recessed impeller. This pump is suitable for carbon transfer in gold cyanide circuits and for wastewater treatment applications. The addition of a cutter (rotor and stator) renders this pump particularly suitable for certain applications such as sewage treatment and newspaper recycling plants.

8-8 GRAVEL AND DREDGE PUMPS Hard metal pumps play an important role in dredging lakes and ports. Some sizes are presented in Table 8-12. A typical construction of a dredge pump is presented in Figure 8-42. Dredge pumps are designed to handle particularly large boulders and lumps of clay. Some of the largest dredge pumps are designed to handle 6.3 m3/s or 100,000 US gpm. A special low-pressure, high-flow pump called the ladder pump is designed to be mounted at the tip of the suction arm. Its purpose is essentially to move the material up to the boat hopper or up to a booster pump on a hopper. The booster pump is designed for higher discharge head. A particular type of pump is the phosphate-matrix-handling pump. It does resemble in many aspects a sort of dredge pump, but is built of materials to handle both corrosion and

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One piece shell and engine side door/liner for minimum parts replacement

Integral stuffing box shell mount

Solid impeller hub eliminates problems with threaded or bolted inserts

OPTIONAL ONE PIECE DOOR/SIDE LINER DESIGN AVAILABLE

Precision machined heavy duty shell Adjusting bolt allows for easy adjustment of impeller for proper operating clearances

Separate thrust bearing

Heavy duty bearing assembly for high power & loading conditions

Advanced design impeller. Good hydraulic performance without sacrificing spherical clearance

FIGURE 8-42 Components of the Marathon dredge pump. (Courtesy of Mobile Pulley and Machine Works.)

wear. It is often driven by a diesel engine through a gearbox. The complete baseplate with driver and pump are relocated from one area to another as mining is done.

8-9 AFFINITY LAWS Affinity laws are used to predict the effects of changing the speed of a pump, trimming an impeller, and extrapolating the performance of a pump from case (A) to case (B). They state that: HA/HB = N A2/N B2

(8-63)

HA/HB = D A2/DB2

(8-64)

HA/HB = N A2/N B2

(8-65)

QA/QB = DA/DB

(8-66)

QA/QB = NA/NB

(8-67)

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8.61

8-10 PERFORMANCE CORRECTIONS FOR SLURRY PUMPS Slurry pumps are designed and tested for hydraulics using water as a reference fluid. However, they are designed to handle large spherical rocks. Often, four-vane impellers are less efficient but outlast other types, particularly on abrasive slurries. Attempts have been made to quantify and qualify the spacing of vanes on the performance of dredge slurry pumps. As early as 1932, Fischer and Thoma conducted tests on a pump built of a transparent material. Although the flow was observed to be close to the designed value at best efficiency point, it quickly deviated at other values. They observed a large area of flow separation on the trailing edge of the vane, with reverse flow in certain instances. Understanding the effects of solids on centrifugal pumps has been a slow process. Fairbanks (1941) developed a theory to correlate the head developed by a pump for a slurry mixture with the volumetric concentration and specific gravity of the solids. He explained that the fundamental Euler equation could be modified to account for the density and flow rate of the mixture as: T = mQm(r2Vt2m – r1Vt1m) The power needed to pump the mixture by an ideal pump (at 100% hydraulic efficiency) is then expressed as: T = mQmgHm where is the angular speed . The mixture head is then expressed as two components for solids and carrier fluid: Hm = (/gm) · [s · Cv · (r2Vt2s – r1Vt1s) + (1 – Cv) · (r2Vt2L – r1Vt1L)

(8-68)

where Cv = volumetric concentration of solids m = density of mixture s = density of solids Fairbanks concluded from his tests on a single pump that : 앫 The drop in the head-capacity curve varies not only as the concentration increases, but also as the particle size of the material in suspension increases. 앫 The fall velocity of the suspended material is the most important parameter for predicting the effect of solids on pump performance. 앫 The power input is a linear relationship of the apparent specific gravity of the solids in suspension

8-10-1 Corrections for Viscosity and Slip Viscosity must be taken in account when pumping viscous slurries. Viscosity reduces the efficiency of pumping and the head developed by a pump (Figure 8-43). The Hydraulic Institute Standards provides correction curves for viscous fluids pumping, but warns against extrapolating to other pumps or fluids. The Institute does not publish curves for viscous slurries. Duchham and Aboutaleb (1976) derived equations to predict the effects of viscosity

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1.4 1.2

Head (Wate

1.0

Ef fic Ef ie nc fic y i (v en (w isc cy at ou er sf ) lu id )

1.0 0.8 H/HN

r)

us fluid)

0.6 0.4

0.8

N

Head (Visco

0.6 0.4

0.2

0.2

0.0 0.0

0.5 Q/Q

Fig 8 43 FIGURE 8-43

0.0 1.5

1.0 N

Effect of viscosity on the performance of centrifugal pumps.

and density on the flow rate, head, and power consumption by comparing particle Reynolds number and power factor. Their analysis did not present a definite appreciation of the effects of viscosity. Sheth et al. (1987) investigated slip factors for slurry pumps by conducting tests on a Wilfley pump. The pump had a 267 mm (10.5 in) diameter, 27 mm (1.06 in) blade width, and a discharge angle of 31°. The following equation was derived by Sheth et al. (1987) to account for the effects of the slurry mixture carrier densities:

s ᎏᎏ2 L · N · D

冢

冣

0.12

m ND2 = 0.0989 – 0.00157 ᎏ ᎏ L Q

where s = slip factor = dynamic (absolute viscosity) of liquid carrier Dimp = impeller diameter N = rotating speed of impeller m = density of slurry mixture L = density of liquid carrier Q = flow rate

冢

冣

0.5

(8-69)

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The above equation is empirical and the exponents and coefficients may change for different pump designs. More research work on different designs would have to be published before a universal formula is adopted. Example 8-6 A slurry pump is to be designed to pump slurry under the following conditions: maximum speed at intake 4 m/s (13 ft/s) flow rate 120 L/s (1858 USGPM) head 40 m (131 ft) slurry density 1470 kg/m30 (SGm = 1.47) water carrier slurry viscosity 100 mPa · s max solid particle size 25 mm (1 in) Using the Sheth formula, determine the geometry of the impeller. Solution 0.120 m3/s suction area = ᎏᎏ = 0.04 m2 3 suction diameter = 0.225 m (8.85 in) suction area at 4 m/s = 0.03 m2 suction diameter = 0.195m (7.7 in)

s ᎏᎏ2 L · N · D

冢

冣

0.12

m ND2 = 0.0989 – 0.00157 ᎏ ᎏ L Q

冢

Or, calling a = N · D2:

s · 0.331 = 0.0989a0.12 – 0.0067a0.62 s = 0.298a0.12 – 0.020a0.62 If A = 30, then:

s = 0.298 × 1.5 – 0.0202 × 8.23 s = 0.28 If a = 40, then:

s = 0.464 – 0.198 = 0.265 If a = 20, then:

s = 0.427 – 0.129 = 0.297 If a = 15, then:

s = 0.4124 – 0.108 = 0.304 If a = 10, then:

s = 0.393 – 0.084 = 0.31

冣

0.5

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Let us assume a = 10, then: 10 = N · D2 Since the particle size passage is 25 mm (⬇ 1⬙), assume discharge width = 30 mm or 0.03 m (1.18 in). The head ratio = 2gH/u2 (in the United States) is:

= 2H[1 – (cm/u2) cot 2] If we assume D = 0.4m (⬵ 16u), then: 10 = N × 0.16 ⇒ N = 62.5 rev/s U = 78.5 m/s Cm = Q/ADis ⇒ discharge area = × 0.4??(1 – zt/sin 2) If b = 30 mm (1.181⬙) then: A = × 0.4 × 0.03 (1 – zt/sin 2) = 0.037 (1 – zt/sin 2) If Z = 4 vanes and t = 30 mm then: A = 0.037 (1 – 0.12/sin 2) If 2 = 15° then: A = 0.0198 m2 Cm = 0.12/0.0198 = 6.05 m/s Cm/U2 = 0.077

= 2Hs(1 – (Cm/U2) cot 2) = 0.44H 2gH 2 × 9.81 × 40 = ᎏ = ᎏᎏ = 0.127 2 U2 78.52 0.127 = 0.44 H ⇒ H = 0.289 This is not a very efficient pump due to the combination of viscosity and solid density: consumed power = gQH/H = 9.81 × 470 × 0.120 × 40/0.289 ⬵ 240 kw or 327 hp It is recommended to install a 400 hp motor. 8-10-2 Concepts of Head Ratio and Efficiency Ratio When Pumping Solids Stepanoff (1969) explained that when pumping solids in suspension, a pump impeller imparts energy to the carrier liquid. For a homogeneous mixture, he explained, the impeller will be able to impart as many feet of mixture as it would have been able to impart head of water. The performance of the impeller is not impaired but the power consumption increases linearly with the specific gravity of the mixture. In reality, at best efficiency point, the presence of solids tends to reduce the hydraulic head by the energy wasted to move them through the impeller passageways. Similarly, the efficiency of the pump when handling the mixture will be reduced by the presence of solids. Two factors can be defined head ratio and efficiency ratio.

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8.65

The head ratio is HR = Hm/Hw

(8-70)

or the ratio of head developed when pumping slurry to the head developed when pumping water. The efficiency ratio is ER = Em/Ew

(8-71)

or the ratio of efficiency developed when pumping slurry to the efficiency developed when pumping water. Stepanoff (1969) indicated that at best efficiency: HR = ER Stepanoff (1969) reported work by Japanese investigators who indicated that tests on carbide slurries tended to show that the head–capacity ratio may increase or decrease depending on whether the solids concentration tended to cause the slurry to behave as a Newtonian or non-Newtonian mixture. Reviewing published data between 1941 and 1971, Hunt and Faddick (1971) reported that various tests in different labs and field applications confirmed that: 앫 The drop in head (in feet of mixture flowing) developed for a given volumetric discharge rate decreased as the concentration of the solids in suspension increased. 앫 The required brake horsepower for a given pump operating at a given capacity increased as the concentration of solid material in suspension increased. 앫 The efficiency at a given capacity decreased as the concentration of the solid material in suspension increased. Hunt and Faddick (1971) simulated the performance of centrifugal pumps pumping wood chips by tests using rectangular plastic parts with an average specific gravity of 1.02. They used four different impeller designs in two different volute designs. There was no consistency in the extent of head drop or efficiency with solid concentration, and the results indicated that the actual design of the pump was a very important factor. A difference of head and efficiency of 5 to 7% was noticed for the different designs. The authors therefore discouraged applying head and efficiency ratio factors for one pump to another pump of a different geometry, but encouraged further research into the mechanisms of flow through the rotating passages of these pumps. It is important to appreciate the work of Hunt and Faddick. Often, a pump vendor will produce a chart or curve to obtain the head and efficiency ratio. The limitations of such curves are that they apply only to pumps of similar geometrical design. The discrepancy of 5–10% between one design and another may have to be absorbed by the motor. McElvain (1974) published data on the effects of solids on pump performance. He worked on the concept of the head and efficiency reduction factors defined as: RH = 1 – HR

(8-72)

R = 1 – ER

(8-73)

He tested impellers up to a diameter of 35 cm (13.78 in) and on various concentrations of silica and one grade of heavy mineral. He developed a set of curves and established a relationship between volumetric concentration and the head and efficiency reduction factors as: RH = R = 5 · K · CV

(8-74)

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1.2 Head (Slurry)

Head (Water) 1.0

Ef fic Ef ie nc fic y( ie w nc at y er (s ) lu rry )

1.0

H/H N

0.8 0.6 0.4

0.8

N

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0.6 0.4

0.2

0.2

0.0 0.0

0.5

1.0 Q/Q

Fig 8-44 FIGURE 8-44

0.0 1.5

N

Effect of solids on the performance of centrifugal pumps.

The K factor was then plotted against the d50 and for solids of various specific gravity (see Figure 8-45). The assumption that RH = R was accepted to hold true for slurry volumetric concentrations smaller than 20%. This covers a substantial number of pump applications. Example 8-7 Heavy metal oxide slurry is to be pumped at a volumetric concentration of 18%. The specific gravity of the solids is 5.0 and the d50 is 400 m. The calculated head on slurry is 35 m. Determine the head ratio and the equivalent water head on the pump performance curve. Solution Using the McElvain equation, the value K is determined from the lower curve at 0.38. Substituting in Equation 8-74, at a volumetric concentration of 18% (less than 20%): RH = R = 5 · K · CV = 5 · 0.38 · 0.18 = 0.342 Substituting into Equation 8-72: HR = 1 – RH = 0.658 Since the calculated head for friction, the equivalent value on water is Hw = Hm/HR = 35 m/0.658 = 53.2 m The engineer must therefore select the appropriate pump speed from the pump curve that would develop 53.2 m.

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0.1 K-Factor

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1.5

0.2 2.65

0.3

4.0

0.4

S = 5.0 s

0.5 10

100

1000

10000

Particle Size d 50 ( m) FIGURE 8-45 Correction to the head K factor of centrifugal pumps on the basis of the specific gravity and particle size Ss = specific gravity of solids. (After McElvain, 1974.)

Sellgren and Vappling (1986) reported that at high volumetric concentration the efficiency ratio was smaller than the head ratio, thus indicating a more pronounced loss of efficiency. Sellgen and Addie (1993) reported losses as low as half of those predicted by McElvain. The curves of McElvain do not take into account another important factor, namely the ratio of particle size to impeller diameter. Burgess and Reizes (1976) proposed that the head ratio and efficiency ratio were a function of three parameters: 1. Weight concentration 2. Ratio of d50 to impeller diameter 3. Specific gravity of the solid particles Sellgen and Addie (1993) indicated that there is a size effect and that head and efficiency losses were less drastic in large pumps than in small pumps (Figure 8-46). This clearly demonstrates the importance of pump design on performance. The importance of pump design on the head and efficiency ratio was confirmed by Czarnota et al. (1996) through tests on ITT-Flygt submersible pumps. Their work confirmed that high-efficiency pumps suffered from less degradation of performance than less efficient pumps. Head reduction was confirmed to be a linear function of the volumetric concentration of solids. Larger particles were found to slip more than smaller particles. An important factor they reported is that settling or separation can occur due to centrifugal forces. These forces are proportional to the square of the radius, and in the presence of large particles can lead to partial blockage, higher water velocity, and more slip between solids and liquid. A well-mixed particle distribution tended to decrease derating of pumps. Russian engineers developed a very advanced mathematical model based on full screen analysis instead of the average d50, which has been the focus of most equations in

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FIGURE 8-46 Effect of the size of the pump impeller on the correction factor for head RH for slurry at a weight concentration of 42%. (From Sellgren and Addie, 1992.).

Australia, Europe, and North America. The work of Kuznetsov and Samoilovich (1985, 1986) was summarized by Angle et al. (1997). These advanced mathematical models permit corrections based on the number of vanes, discharge angle of the vanes, and volumetric concentration of each range of diameter of solids in the slurry. It would be very appropriate to explore these models; however, when examining a worn-out impeller, as in Figure 8-47, the reader may wonder how practical such models may be. 8-10-3 Concepts of Head Ratio and Efficiency Ratio Due to Pumping Froth Flotation froth is complex slurry and may contain an important amount of air and gases (Figure 8-48). The industry uses the froth factor as a measure. Basically, it is determined by filling a column of flotation slurry and measuring the height H0. It is then left for 24 hr to rest. The height of the slurry H⬘ is then measured. The froth factor F is defined as: F = H0/H⬘

(8-75)

Using this concept of froth factor to size pumps must be done very carefully. Different grades of froth leads to different levels of entrained gases, as shown in Table 8-15. Conventional centrifugal pumps can not handle excessive amounts of entrained gases. A very common misunderstanding in the industry is that the flow rate of slurry must be multiplied by the froth factor to size the pump. This violates a very fundamental principle that gases or air are compressible fluids. In other words, as the bubbles pass through the impeller they are compressed and reduced in size. In fact, the proper sizing of slurry pumps to handle froth must be based on a full examination of the system. For example, if

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8.69

FIGURE 8-47 Worn-out impeller, showing gradual degradation of the impeller tip diameter and vanes. Deterioration of hydraulics and head efficiency may occur throughout the wear life of the impeller and pump.

FIGURE 8-48 Flotation slurry froth contains sufficient air bubbles to degrade the performance of centrifugal pumps. (Courtesy of EOMCO Process Equipment Co.)

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TABLE 8-15 Correlation between Froth Factor and Percentage of Entrained Air Froth factor

% entrained air

1.5 2.0 2.5 3.0

2–3% 3–5% 5–7% > 7%

Example Normal flotation tailings Flotation tailings with minimum retention time Tenacious flotation tailings with minimum retention time Froth with very fine particles

the flotation cells are away from the pump and the froth is transported by gravity in launders, it may be argued that the surface area of the launders acts as a deaerator for air removal. In that case, does a 24 hr tube test apply well? The correct approach is in fact to remove as much of the air as possible before the froth enters the pump feed sump. This sump may also be designed in a conical shape to maximum surface area at the top. Certain forms of froth are very difficult to pump, such as the type associated with tar sands, in which viscosity plays a major role. Efficiency as low as 10% was reported with conventional pumps. Cappelino et al. (1992) presented a very thorough study on the performance of centrifugal pumps with open impellers with emphasis on pulp and paper flotation circuits and deinking cells. High-consistency stock (12%) can have as much as 20–28% entrained air. At the inlet to the impeller, the pressure drop tends to cause an expansion of the air and gases, and this indicates well that the concept of the froth factor can be misleading Example 8-8 The height of the liquid in a froth cell is 30 m above the pump. The depression at the inlet to the pump is about 6 m. Determine the expansion of gases, assuming a barometric pressure of atmospheric air at 9.5 m. The pump is designed to deliver a total head of 54 m. Determine the final volume of the gases. Solution The effective absolute head in the sump is: 30 + 9.5 = 39.5 m Due to the depression of 6 m, the absolute pressure is then: 39.5 – 6 = 33.5 m The expansion ratio at constant temperature is: 39.5/33.5 = 1.179 The absolute discharge head is: suction head + TDH + atmospheric barometric height = 30 + 54 + 9.5 = 93.5 m Ratio of discharge to suction absolute head is: 93.5/39.5 = 2.37 The size of the air or gas bubbles will then shrink by the inverse of this ratio, or 42.2%. Since the laws of thermodynamics apply, the concept of a constant froth factor is illusive. It would be a grave error to size piping and equipment based on the suction froth factor.

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FIGURE 8-49 Head and power correction factors for entrained gas due to flotation circuits. (From Cappellino, Roll, and Wilson, 1992. Reproduced by permission of Texas A&M University.)

The performance of slurry pumps deteriorates in the presence of entrained gases. Cappelino et al. (1992) have therefore proposed to define appropriate head and power correction factors as: head measured with entrained gas HF = ᎏᎏᎏᎏ head measured without entrained gas

(8-76)

power measured with entrained gas PF = ᎏᎏᎏᎏ power measured without entrained gas

(8-77)

or

Special pumps are available for handling froth and entrained gases. One interesting design is the Sulzer–Ahlstrom ART pump. It features holes through the impeller leading

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straight to an expeller at the back of the impeller. This expeller discharges the air through a separate discharge flange at the back of the casing. In some other designs, the stuffing box is connected to an external liquid ring vacuum pump that can remove any entrained air in the slurry. The manufacturers of pulp and paper pumps have also designed special impeller with protruding vanes that extend into the suction pipe to break up any large air particles. This concept is gaining popularity in some oil sand applications to handle particularly thixotropic and viscous froth. Dredge pumps sometimes face a similar problem. Gases are disturbed or released (particularly methane) during certain phases of dredging and end up accumulating in the ladder pump. Herbich and Miller (1970) conducted extensive test work on the effect of air on the development of head. Herbich (1992) proposed that special air removal systems be installed on the suction side of the pump with an ejector, as this is better than a vacuum pump.

8-11 CONCLUSION In this chapter, some of the important parameters that give slurry pumps their final shape were examined. It is obvious that slurry pumps are different from water pumps and that considerable research should be undertaken in fields such as pump-out vanes, expeller design, and effects of wide impellers on performance. The successful performance of these pumps depends on their resistance to wear. The slurry—in terms of its composition and concentration, and in terms of any froth-induced gases—is the determining factor for power consumption and the final hydraulics across the impeller and casing. These parameters are extremely important for the successful installation of these pumps.

8-12 NOMENCLATURE A Ap bt b2 B2 BHP C Cm Cv Cw Cp D d2 dSL dm ER Fth FA Fp

Factor to calculate slip, depending on the use of volute or diffuser Equivalent casing area for stress calculations Thread width at the root Width of the impeller at the impeller tip diameter Width of the casing at the impeller tip diameter Power in bhp Constant Meridional velocity across the impeller volumetric concentration of solids Concentration by weight of the solid particles in percent Heat capacity Equivalent diameter of casing The tip diameter of the impeller vanes Diameter of shaft sleeve Pitch diameter Efficiency ratio Force due to thread pull Axial thrust on shaft due to hydraulic forces Force on casing due to design pressure

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Fp FR G H H1 H2 H30 HE HIMP SV HR HVOL SV HSV HV K Kx Ky L Lth m M N NPSH Nq NUS NSS PA Pb Pe Ps PD PW PV Q R1 R2 R3 R4 Rb RC RH R RMD Rr Rs Rv0 R1 R2 Rv0 RD

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Force due to belts at the drive end of the pump Radial thrust Acceleration due to gravity (9.78 to 9.81 m/s2 or 32.2 ft/sec) Height of the thread tooth = (major diameter – minor diameter)/2; in the case of ACME threads h = p/2 Head at the medium diameter of the eye of the impeller Head at the tip diameter of vane of the impeller Head at 30% of best efficiency capacity Euler ideal head for an impeller Shut-off head due to the impeller Head ratio Shut-off head due to the volute Total shut-off head Vapor head Correction factor for the head ratio Coefficient to determine Xv Coefficient to determine Yv length of a full turn in a shaft thread = pitch for single start threads The length of the shaft thread Exponent in vortex equation Number of pumps in series Rotational speed of the pump in rev/min Net Positive Suction Head Specific speed in SI units Specific speed in US units Suction specific speed Atmospheric pressure Pressure at the root of the pump-out vanes Rb Pressure at the surface of the liquid in absolute terms on the suction side Pressure at the suction diameter d1 Pressure losses between the surface of the liquid and the pump, due to friction, valves, etc. Pump power in Watts Vapor pressure Flow rate Root radius of the vanes of the impeller Tip radius of the vanes of the impeller Radius of the smaller circle of a twin circle volute Radius of the larger circle of a twin circle volute Radius at the root of the pump out vanes Cutwater radius Head correction factor Efficiency correction factor Meridional radius of the volute at the throat Tip radius of the pump-out vanes Tip radius of the shaft sleeve Local radius of vanes Radius of the root of the impeller vane Tip radius of the vanes of an impeller Radius of vortex Reaction force at the drive end bearing

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RW s S Sb Ss Tq t tC tL TS TSL TDH U V W Wimp Wp X XV YV Z Z1 Ze

CHAPTER EIGHT

Reaction force at the wet end bearing Gap between the edge of the pump-out vanes and the back wear plate of the casing Static moment, obtained by graphical integration along the meridional plane of the vanes Bending stress at the root of the shaft thread Shear stress at the root of the shaft thread Torque Depth of the pump-out vanes Thickness of the pump casing Thickness of the liner Thrust on suction side Thrust on shaft sleeve Total Dynamic Head Tip speed Absolute velocity across the impeller Relative velocity across an impeller Weight of impeller Weight of pulleys Total gap between the pump-out vanes’ impeller surface and the pump back plate Width of the volute in the x-direction Width of the volute in the y-direction Number of vanes Geodetic elevation of liquid surface above the centerline of the pump impeller Geodetic elevation of the centerline of the pump impeller

Greek Letters  Angular inclination of the vane with respect to the tangent Slip factor Density of liquid Cavitations parameter Angular velocity liq Angular velocity of the liquid in the gap between the pump-out vanes and the pump back plate imp Rotational velocity of the impeller or expeller SI Head coefficient to SI convention US Head coefficient to US convention Subscripts 1 At the root of the vane 2 At the tip of the vane E Eye of the impeller F Front shroud R Rear shroud Imp Impeller Liq Liquid Sl Sleeve

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8-13 REFERENCES Abulnaga, B. E. 2001. Recommendations for the design of mill discharge slurry pumps. Mazdak International Inc. Internal Report 02/2001 (unpublished). Addie, G. R. and F. W. Helmly. 1989. Recent improvements in dredge pump efficiencies and suction performances. Europort Dredging Seminar, Central Dredging Association, Delft, Netherlands. Anderson, H. H. 1938. Mine pumps. J. Mining Soc. Durham, United Kingdom. Anderson, H. H. 1977. Statistical records of pump and water turbine effectiveness. International Mechanical Engineers Conference on Scaling for Performance Prediction in Rotodynamic Pumps. September, pp. 1–6. Anderson, H. H. 1980. Centrifugal Pumps. Trade and Technical Press: UK. Anderson, H. H. 1984. The area ratio system. World Pumps, 201. Angle, T. and J. Crisswell (Editors). 1977. Slurry Pump Manual. Salt Lake City, Utah: Envirotech. Angle, T. and A. Rudonov. 1999. Slurry Pump Manual. Salt Lake City, Utah: Envirotech. ANSI/ASME B106.1. 1985. Design of Transmission Shafting. American Society of Mechanical Engineers, New York. Burgess, K. E. and B. E. Abulnaga. 1991. The application of finite element methods to Warman pumps and process Equipment. Paper presented to the Fifth International Conference on Finite Element Analysis in Australia, University of Sydney, Australia, July 1991. Burgess, K. E. and J. A. Reizes. 1976. The effect of sizing, specific gravity and concentration on the performance of centrifugal slurry pumps. Proc. Inst. Mech. Eng., 190, 36. Cappellino, C. A., D. Roll, and G. Wilson. 1992. Design considerations and application guidelines for pumping liquids with entrained gas using open impeller centrifugal pumps. Proceedings of the Ninth International Pump Users Symposium, Texas A&M University. Czarnota, Z., M. Fahlgren, M. Grainger, and S. Saunders. 1996. The effects of slurries on the performance of submersible pumps. BHR Group Hydrotranport, 13, 643–655. Duchham C. D. and Y. K. A. Aboutaleb. 1976. Some tests in a single stage semi-open impeller centrifugal pump handling coal dust slurries. In Proceedings Pumps and Turbine Conferences, Vol 1. Fairbanks, L. C. Jr. 1941. Effects on the characteristics of centrifugal pumps. Solids in Suspension Symposium, Proc. Am. Soc. Civ. Eng., 129, 129. Fischer, K. and D. Thoma. 1932. Investigation of the flow conditions in a centrifugal pump. Transactions ASME, 54. Frost, T. H. and E. Nielsen. 1991. Shut-off head of centrifugal pumps and fans. Proc. Inst. Mech. Eng., 205, 217–223. Herbrich, J. 1991. Handbook of Dredging Engineering. New York: McGraw-Hill. Herbich, J. B. and R. J. Christopher. 1963. Use of high speed photography to analyze particle motion in a model dredge pump. In Proceedings of the International Association for Hydraulic Research, London England. Herbich, J. B. and R. E. Miller. 1970. Effect of air content on performance of a dredge pump. In Proceedings of the World Dredging Conference, Wodcon 70, Singapore. Hunt, A. W and R. F. Faddick. 1971. The effects of solids on centrifugal pump characteristics. In Advances in Solid–Liquid Flow in Pipes and Its Application, I. Zandi (Ed.), New York: Pergamon Press. Jekat, W. K. 1992. Centrifugal pump theory. Section 2.1 in Pump Handbook, J. Karassik et al. (Eds.), New York: McGraw Hill. Kuznetsov, O. V. and D. C. Samoilovich. 1986. Increase of Reliability of Slurry Pumps in Service (in Russian). Moscow: CINTIchimneftemash, ser.XM-4. McElvain, R. E. 1974. High pressure pumping. Skillings Mining Review, 63, 4, 1–14. Pfeiderer, C. 1961. Die Kreiselpumpen. Berlin: Springler-Verlag. Samoilovich, D. C. 1986. Experimental Study of Slurry Pumps Performances (in Russian). Moscow: CINTIchimneftemash, ser.XM-4. Sellgren, A. and L. Vappling. 1986. Effects of highly concentrated slurries on the performance of centrifugal pumps. Proceedings of the International Symposium on Slurry Flows, FED Vol 38, ASME, USA, pp. 143–148. Sellgren, A. and G. R. Addie. 1992. Effects of solids on the performance of centrifugal slurry pumps.

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Paper presented at the 10th Colloquium: Massenguttransport durch Rohrietungen in Meschede, Germany, May 20–22. Sellgren, A. and G. R. Addie. 1993. Solids effect on the characteristics of centrifugal slurry pumps. Paper presented at the 12th International Conference on Slurry Handling and Pipeline Transport, Brugge, Belgium. Sheth, K. K., G. L. Morrison, and W. W. Peng. 1987. Slip factors of centrifugal slurry pumps. A.S.M.E. Journal of Fluids Engineering, 109, 313–318. Stepanoff, A. J. 1969. Gravity flow of bulk solids and transportation of solids in suspension. New York: Wiley. Stepanoff, A. J. 1993. Centrifugal and Axial Flow Pumps. Melbourne, FL: Krieger. Sulzer Pumps. 1998. Centrifugal Pump Handbook. New York: Elsevier. Turton, R. K. 1994. Rotodynamic Pump Design. Cambridge: Cambridge University Press. K. C. Wilson, G. R. Addie, and R. Clift. 1992. Slurry Transport Using Centrifugal Pumps. London: Elsevier Applied Sciences. Wilson, G. 1976. Construction of solids-handling centrifugal pumps. In Pump Handbook, J. Karassik et al. (Eds.) New York: McGraw Hill. Worster, R. C. 1963. The flow in volutes and effect on centrifugal pump performance. Proc. Inst. Mech. Eng., 177, 843. Further Reading Kazim, K. A. and B. Maiti. 1997. A correlation to predict the performance characteristics of centrifugal pumps handling slurries. In Proceedings of the Institution of Mechanical Engineers. Part A. Journal of Power and Energy, 211, A2, 147–157. Cader, T., O. Masbernat, and M. C. Rocco. 1994. Two phase velocity distributions and overall performance of a centrifugal slurry pump. Journal of Fluid Engineering, 116, 316–323. Gandhi, B. K., S. N. Singh, and V. Seshadri. 2000. Improvements in the prediction of performance of centrifugal slurry pumps handling slurries. Proceedings of the Institution of Mechanical Engineers. Part A. Journal of Power and Energy, 214, 5, 473–486.

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CHAPTER 9

POSITIVE DISPLACEMENT PUMPS

9-0 INTRODUCTION Positive displacement slurry pumps and mud transfer pumps play a major role in a number of industries such as mining and metallurgical processes, chemicals, power generation, porcelain and ceramics, and sugar refining. These pumps are versatile, efficient, and suitable for pressures up to 17.3 MPa (2500 psi). Positive displacement pumps have gained acceptance on long-distance mineral concentrate pipelines as their high capital cost is recuperated through lower installation cost of electric systems, elimination of booster stations, and high hydraulic efficiency, which is superior to centrifugal pumps. Plunger or diaphragm pumps do not handle large flow rates in excess of 100 m3/hr (4400 US gpm), but they are suitable for a wide range of applications at higher volumetric concentrations than centrifugal pumps. Positive displacement pumps can pump slurries with a weight concentration of 70%.

9-1 SOLID PISTON PUMPS Positive displacement slurry pumps are used in a number of industries (Table 9-1). Solid piston pumps are reserved for the pumping of slurries of a low to medium abrasiveness (Miller Number <50) such as chalk slurry, fine coal, flotation material, and drilling mud sludge. In these pumps, the slurry comes into contact with the piston and with the packings. In a duplex pump, the flow splits into two cylinders inside the pump, whereas in a triplex it splits into three cylinders. Table 9-2 compares both designs. A special feature of the duplex pumps is that they can be built in a single- or doubleacting configuration. In the case of the double-acting pump, the slurry flows on both sides of the piston. On one side, there is a piston rod that goes through the packing before connecting to the connecting rod and crankshaft (Figure 9-1). Because the flow is divided in a duplex double-acting piston pump to the two sides of

9.1

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TABLE 9-1 Applications of Positive Displacement Pumps Industry

Application

Mining

Coal transportation (e.g., Novo Siberski pipeline, Black Mesa Pipeline) Flotation material Washery refuse Deep mine dewatering of water with solid particles Limestone, milk of lime Potash rock salt, phosphate, iron ore, nickel ore concentrate Bauxite, red mud, gold mud Sand, pyrite, REA gypsum Backfilling Underground drainage Filter press feed Autoclave feed

Chemicals

Salt slurry Porcelain slurry Pastes Detergent slurry Combustion furnace feed Filter press feed

Power generation

Coal and coal slurry Flue Pressurized fluidized bed combustion Wet ash removal Ship loading Long-distance pipelines

Construction

Bentonite, clay mash, cement

Porcelain

Clay slurry Filter press feed

Sugar

Carbonation slurry Sugar beet washing

Information provided by courtesy of Wirth-Maschinen and Bohrgerate, Germany.

TABLE 9-2 Comparison between Duplex and Triplex Pumps Duplex Single Acting 2 cylinder liners 2 piston gaskets 2 cylinders 4 valves Slurry does not come in contact with packing

Duplex Double Acting 2 cylinder liners 4 piston gaskets 2 cylinders 2 piston rods with packing 8 valves Slurry does come in contact with packing

Triplex Single Acting 3 cylinder liners 3 piston gaskets 3 cylinders 6 valves Slurry does not come in contact with packing

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FIGURE 9-1 Concept of the double-acting duplex piston pump. (Courtesy of Wirth Pumps.)

the piston, these pumps operate at lower speeds than single-acting duplex and triplex pumps (Figure 9-2). The single-acting duplex pump seems to have disappeared from the world of manufacturing. For proper balancing, the pistons of duplex pumps are 180° out of phase (Figure 9-3), but for triplex pumps they are 120° out of phase with each other (Figure 9-4). Triplex pumps have a lower degree of oscillation than duplex units. The degree of

FIGURE 9-2 Concept of the triplex piston pump. (Courtesy of Wirth Pumps.)

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FIGURE 9-3 Concept of gear mechanism for duplex piston pumps. (Courtesy of Wirth Pumps.)

FIGURE 9-4 Pumps.)

Concept of gear mechanism for triplex piston pumps. (Courtesy of Wirth

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FIGURE 9-5 Pulsation diagram for triplex pump. (Courtesy of Wirth Pumps.)

variation of flow in the former is 23% (Figure 9-5) compared to 46% in the latter (Wallrafen, 1983; Figure 9-6). Duplex and triplex slurry pumps are manufactured to a power frame of approximately 1500 kW (2000 bhp). The Black Mesa Pipeline featured 13 duplex pumps, each with a driving power of 1250 kW (1675 bhp) to transport 4.8 million tons of coal over a distance of 440 km (275 miles) (Wallrafen, 1983). Some of these pumps were manufactured by Wilson-Snyder in the United States. Piston slurry pumps are used extensively as mud transfer pumps. Gardner-Denver in the United States offers a range of duplex pumps in the power range of 12–76 kW (16–102 hp).

FIGURE 9-6 Pulsation diagram for duplex pump. (Courtesy of Wirth Pumps.)

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FIGURE 9-7 Triplex pump TPK 7⬙ × 12⬙/1600. Driving power 1200 kW (1600 hp). (Courtesy of Wirth Pumps.)

9-2 PLUNGER PUMPS For a long time, plunger pumps were not considered to be suitable for slurry transportation, but in the late 1970s, manufacturers developed a suitable flushing system to minimize wear of the plunger. Plunger pumps are single acting. They use plungers instead of pistons (Figure 9-8) with valves. They operate at 80–120 cycles per minute. Plunger pumps are prone to wear. They are less expensive to purchase than diaphragm pumps but have a higher maintenance cost. These pumps use three types of valves: 1. Free-floating valves 2. Spring-loaded spherical (Rollo) valves 3. Spring-loaded elastomer-seal (mud) valves These valves are shown in Figure 9-9. It is important to minimize packing wear with piston pumps. Smith (1985) proposed four methods: 1. Use of conventional packing of plungers at low speed with slurries of low abrasiveness 2. Provision of a clean, slurry-free environment for the packing rubbing surface (by synchronized or continuous injection of water or cleaning fluid) 3. Separation of the slurry from the pumping element 4. Total isolation of the slurry from the packing (by providing a separate diaphragm chamber) The SAMARCO pipeline in Brazil used 14 plunger pumps with a driver power of 920 kW each to deliver 12 million metric tons of iron oxide ore concentrate over a distance of

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FIGURE 9-8 Schematic representation of a plunger pump.

400 km (250 mi) (Wallrafen, 1983). These triplex plunger pumps were manufactured by Wilson-Snyder. The Wilson-Snyder line of plunger slurry pumps features 21 different sizes from 45–1250 kW (60–1700 hp). They have also been used in Georgia, U.S.A. on a kaolin pipeline. Kaolin is not very abrasive. These pumps have also been used for mine dewatering from a depth of 1036 m (3400 ft). The water contained solids.

FIGURE 9-9 Categories of valves for slurry pumps. (a) Free floating; (b) spring-loaded spherical (Rollo); (c) spring-loaded elastomer seal. (From Smith, 1985. Reprinted by permission of McGraw-Hill.)

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Some of the triplex plunger pumps are rated at 41.4 MPa (6000 psi) (Wilson-Snyder, 1977), such as the model 85-25. The volume capacity is rated from 363–2941 L/min (96–777 US gpm).

9-3 PISTON DIAPHRAGM PUMPS To handle abrasive slurries that piston pumps would find difficult, manufacturers such as Geho Pumps (Netherlands), Wirth (Germany), and Gorman-Rupp (United States) have developed pumps to use a diaphragm or a sort of flexible piston that comes in contact with the slurry or sludge. Feluwa of Germany added a hose so that there is an isolating bath of oil between the hose and the diaphragm. The pumps from Geho, Wirth, and Feluwa feature a crankshaft mechanism to move the diaphragm but the Gorman-Rupp pump uses an air cylinder to actuate the diaphragm. Diaphragm piston pumps use a sort of oil chamber between a reciprocating piston (operated by a crankshaft) and the diaphragm (Figure 9-10). Provided that no puncture occurs in the diaphragm, slurry does not come in contact with the piston. A special control system is installed to detect diaphragm rupture. Wallrafen (1983) reported that Wirth manufactured its first piston diaphragm pump in 1969 to pump sand slurries. The first unit lasted 1000 hrs without having to replace worn parts. By the early 1980s, wear life of 6000 hrs was achieved with rubber materials and

FIGURE 9-10 Pumps.)

Concept of double-acting piston duplex diaphragm pump. (Courtesy of Wirth

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9.9

proper design of the diaphragm. Diaphragm piston pumps are designed as duplex doubleacting or as triplex single-acting pumps in a similar concept rather to solid piston pumps (Figures 9-10 and 9-11). Piston diaphragm pumps (Figure 9-12) are more expensive than plunger pumps. For autoclave feed pumps, Geho developed a special design to handle slurries as hot as 200°C (392°F) at a high flow rate. Solids concentrations can be as high as75% and pumps can operate at slurry temperatures up to 200°C. Typical uses in the mining industry include: 앫 앫 앫 앫 앫 앫 앫

Long-distance slurry (mineral concentrate) pipelines (up to 300 km long) Clean and efficient tailings disposal High-pressure bauxite digester feed Autoclave and reactor feed Mine backfilling Mine dewatering (single stage) Hydraulic ore hoisting

With more than 400 piston diaphragm pumps installed on some of the world’s most demanding long-distance pipeline applications, Geho Pumps has taken the opportunity, through a significant research and development effort, to constantly improve piston diaphragm pump design. This has resulted in numerous proprietary design improvements and technical innovations relevant to severe slurry pumping.

FIGURE 9-11 Pumps.)

Concept of single-acting piston triplex diaphragm pump. (Courtesy of Wirth

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FIGURE 9-12

Geho pump at Freeport. (Courtesy of Geho.)

Geho Piston diaphragm are used for tailings disposal and pumping at very high concentration so that: 앫 Amount of free water is virtually eliminated, allowing slurry to be stacked or distributed in layers 앫 Dry stacking requires less storage space 앫 Prevents contamination of the environment by leakage 앫 Rain and wind do not affect the solidified tailings 앫 Mechanical stability of tailings allows a high stack with rehabilitation possibilities after use Typical tailings applications of Geho pumps include: 앫 앫 앫 앫 앫 앫

Bayswater Power Station—fly ash disposal (Australia) Nabalco, Gove Refinery—red mud disposal (Australia) Ledvice Power Station—fly ash disposal (Czech Republic) Pingguo Aluminium Company—red mud disposal (Peoples Republic of China) Khaperkheda Ash Handling Plant—fly ash disposal (India) National Aluminum Company—red mud disposal (India)

TABLE 9-3 Examples of Installation of Piston Diaphragm Pumps on Slurry Pipeline and Tailings Applications Location

Manufacturer

Installation application flow rate stated for each pump

Alsen Zementwerke

Geho

Antamina, Peru

Wirth

Ashanti Goldfields Ghana

Wirth

3 piston diaphragm pumps to pump limestone slurry over 10 km (6.3 mi) 4 piston pumps, 100 m3/hr (440 gpm), 25.2 MPa (3650 psi), copper concentrate 1 pump, 215 m3/hr (947 gpm), 6 MPa (870 psi) backfill slime (gold tailings)

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9.11

TABLE 9-3 (continued) Location

Manufacturer

Installation application flow rate stated for each pump

Bajo Alumbrera, Argentina

Geho

Cameco, Canada

Wirth

Cia Minera Disputada de Las Condes, Chile Codelco, Chile

Wirth

Course Nickel , Australia

Wirth

Doña Ines Collahuasi, Chile

Geho

ECPSA, Cuba

Wirth

Empresa Minera Yauliyacu, Peru Eskay Creek, Canada

Wirth Wirth

Freeport, Indonesia

Geho

Goldmine, South Africa

Wirth

ISCOR, Hillendake Mine, South Africa Jian Shan

Wirth

Nabalco, Australia

Geho

Norilsk Nickel Combinat

Geho

Pasminco, Australia

Wirth

Los Pelambres, Chile

Geho

Sicartsa, Mexico

Geho

J. R. Simplot, USA

Geho

Batu Hijau, Indonesia

Geho

Cockburn Cement, Australia

Geho

New Zealand Steel

Geho

Rio Capim, Brazil

Geho

Minera Escondid, Chile

Geho

OEMK, Ukraine

Geho

6 piston diaphragm pumps in 3 booster stations to transport copper concentrate over a distance of 320 km (225 mi) in a 150 mm (6 in) line, 91 m3/h at 217 bar 2 pumps, 80 m3/hr (352 gpm), 12.5 MPa (1810 psi), uranium ore 3 pumps 115 m3/hr (510 gpm), 2.5 MPa (360 psi) copper tailings 3 pumps, 140 m3/hr (616 gpm), 3.5 MPa (507 psi), c opper tailings 2 pumps, 193 m3/hr (850 gpm), 5 MPa (725 psi), lateritic nickel ore 2 piston diaphragm pumps for 203 km of copper concentrate transport, 117 m3/h at 217 bar 10 pumps, 200 m3/hr (800gpm), 6.4 MPa (920 psi), iron–nickel slurry 2 pumps, 90 m3/hr (400 gpm), 14.4 MPa (2080 psi), tailings 1 pump, 25 m3/hr (110 gpm), 11.7 MPa (1700 psi) for a 6.5 km (4 mi) pipeline 2 piston diaphragm pumps to transport copper concentrate over 120 km (75mi), 159 m3/h at 40 bar 1 pump, 12 m3/hr (66 gpm), 12 MPa (1714 psi), backfilling 2 pumps 450 m3/hr (1980 gpm), 7.4 MPa (1075 psi), heavy mineral tailings 100 kms iron ore concentrate transport (PR of China), 2 piston diaphragm pumps, 216 m3/h at 153 bar 3 piston diaphragm pumps for a highly concentrated red mud slurry to a disposal area, 200 m3/h at 160 bar 9 piston diaphragm pumps, 55 km of multimetallic ore transport (North Siberia, Russia), 400 m3/h at 80 bar 3 pumps, 161 m3/hr (710 gpm), 12.5 MPa (1810 psi) zinc and lead concentrate 2 piston diaphragm pumps for 120 km pipeline transportation of copper concentrate slurry, 165 m3/h at 150 bar 1 piston diaphram pump for iron ore concentrate slurry, 380 m3/h at 110 bar 4 piston diaphragm pumps for 100 km transportation of phosphate slurry, 97 m3/h at 228 bar 2 piston diaphragm pumps for 120 km copper concentrate slurry transportation, 123 m3/h at 228 bar 3 piston diaphragm pumps for shell and slur