Third Edition
© 2006 by Taylor & Francis Group, LLC
Third Edition edited by Hiroaki Masuda Ko Higashitani Hideto Yoshida
Boca Raton London New York
A CRC title, part of the Taylor & Francis imprint, a member of the Taylor & Francis Group, the academic division of T&F Informa plc.
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Published in 2006 by CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2006 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number-10: 1-57444-782-3 (Hardcover) International Standard Book Number-13: 978-1-57444-782-8 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Catalog record is available from the Library of Congress
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Preface to the Third Edition Powders and particles are extending their applications in recent developing fields such as the information market, including mobile phones, copy machines, and electronic displays, as well as pharmaceuticals, cosmetics, foods, chemicals, and many other fundamental engineering fields. Nanoparticles are also promising research subjects for the development of more effective applications of various particles. They need powder technology in their preparation, characterization, measurement, and functional treatment and for totally systematic handling. Since the first edition published in 1988 and the second edition in 1997, the Powder Technology Handbook has been accepted in many fields of engineering. Because of this acceptance and the encouragement of Russell Dekker, we agreed to update the book and present this third edition. This new edition has been substantially revised and expanded. The content has been updated, not only by adding newly developed areas such as simulations, surface analysis, and nanoparticles, but also by introducing new young authors. Some sections are combined, so that the readers can easily refer to the subject. We hope this handbook will serve as a strong guide to the field of powder and particle technology. Special acknowledgment is given to all contributors and also to all the original authors whose valuable work is cited in this handbook. We also extend acknowledgment to Dr. Matsusaka, associate professor of Kyoto University, for his collaboration on the editing work. With this, our third edition, we thank our new editorial staff: Kari Budyk, Tao Woolfe, and others at Taylor & Francis Books (formerly Marcel Dekker) for their laborious editing and production work. HIROAKI MASUDA KO HIGASHITANI HIDETO YOSHIDA
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Preface to the Second Edition Since first published in 1988, the Powder Technology Handbook has been accepted as a reference in many areas of engineering. Because of this acceptance and the support of Russell Dekker, the Chief Publishing Officer of our publisher, we agreed to update the book and present this second edition. This entire book has been substantially revised. American and European authors have been invited to join the Japanese authors who participated in writing the first edition, giving the book a wider viewpoint. In response to reviews for the first edition, the contents have been reorganized. New sections have been added to treat the recent advances in computer simulation (V.19) and working atmospheres and hazards (VII.1–VII.3). This second edition contains a more detailed table of contents so that readers can understand the whole range of the handbook. It is hoped that this handbook will serve as a strong guide to the field of particle technology. Special acknowledgment is given to Dr. Koichi Iinoya, professor emeritus of Kyoto University, for his advisory work. This second edition is dedicated to him on the occasion of his 80th year of age. We are indebted to Joseph Stubenrauch and others at Marcel Dekker, Inc., for their laborious editing work. KEISHI GOTOH HIROAKI MASUDA KO HIGASHITANI
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Preface to the First Edition Particulate or powder technology deals with a variety of particles, from submicrometer to large grains and gravels and from liquid mist or droplets to bubbles, as well as solid particles and aggregates. It also deals with two-phase (solid–gas, solid–liquid, and liquid–gas) and three-phase (solid– liquid–gas) mixtures and is related to work in the medical, biological, pharmaceutical, chemical, mining, metallurgical, mechanical, civil, food and agricultural science, and engineering fields. Many natural and artificial phenomena encountered in our daily lives may be explained by knowledge of powder technology. The first book on particle technology in general was probably Micromeritics by J. M. Dalavalle, published in 1943; the second may be Particulate Technology by C. Orr, published in 1966. There are also various books on specific fields in powder technology. In Japan, the first handbook was published in 1965 and the second appeared in 1986. Particulate or powder technology is a fundamental engineering field involving newly developed high or evolutionary technology. Submicrometer particles, for example, are a focus of practical applications. This handbook has been developed from the second Japanese handbook, although the contents have been revised and updated. It provides a comprehensive understanding of powder technology. Although many investigators have contributed, the various manuscripts have, as much possible, been edited in a unified fashion, so as not to become a mere collection of monographs: Keishi Gotoh has edited the sections on dry powders and Ko Higashitani those on wet powders. Koichi Iinoya assumes the entire responsibility as editor in chief. Special acknowledgment is given to all contributors who prepared their manuscripts on time. The editors also acknowledge all original authors whose valuable work is cited in this handbook. We are indebted to Nikkan Kougyo Shimbunsha Publishers, who kindly supplied us with the original drawings of all figures used in the Japanese edition. The work of a number of students in relettering the artwork is gratefully acknowledged. Once again, this handbook provides a comprehensive understanding of powder technology as a sort of encyclopedia and it is hoped that it will be valuable as a guide to this invaluable engineering field. We are indebted to Ms. Lila Harris and others at Marcel Dekker, Inc., for their laborious editing work. KOICHI IINOYA KEISHI GOTOH KO HIGASHITANI
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Editors Hiroaki Masuda is a professor in the Department of Chemical Engineering at Kyoto University, Japan. He is a member of the Society of Chemical Engineers, Japan and the Institute of Electrostatics (Japan), among other organizations, and is the president of the Society of Powder Technology, Japan. His research interests include electrostatic characterization, adhesion and reentrainment, dry dispersion of powder, and fine particle classification. Dr. Masuda received his Ph.D. degree (1973) in chemical engineering from Kyoto University, Japan. Ko Higashitani graduated from the Department of Chemical Engineering, Kyoto University, Japan in 1968. He worked on hole pressure errors of viscoelastic fluids as a Ph.D. student under the supervision of Professor A. S. Lodge in the Department of Chemical Engineering, University of Wisconsin– Madison, USA. After he received his Ph.D. degree in 1973, he moved to the Department of Applied Chemistry, Kyushu Institute of Technology, Japan, as an assistant professor, and then became a full professor in 1983. He joined the Department of Chemical Engineering, Kyoto University, in 1992. His major research interests now are the kinetic stability of colloidal particles in solutions, such as coagulation, breakup, adhesion, detachment of particles in fluids, and slurry kinetics. In particular, he is interested in measurements of particle surfaces in solution by the atomic force microscope and how the surface microstructure is correlated with interaction forces between particles and macroscopic behavior of particles and suspensions. Hideto Yoshida is a professor in the Department of Chemical Engineering at Hiroshima University, Japan. He is a member of the Society of Chemical Engineers, Japan and the Society of Powder Technology, Japan. His research interests include fine particle classification by use of high-performance dry and wet cyclones, standard reference particles, particle size measurement by the automatic-type sedimentation balance method, and the recycling process of fly-ash particles. Dr. Yoshida received his Ph.D. degree (1979) in chemical engineering from Kyoto University, Japan.
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Contributors Motoaki Adachi Department of Chemical Engineering Frontier Science Innovation Center Osaka Prefecture University Sakai, Osaka, Japan
Toyohisa Fujita Department of Geosystem Engineering Graduate School of Engineering University of Tokyo Tokyo, Japan
Charles S. Campbell School of Engineering, Aerospace and Mechanical Engineering University of Southern California Los Angeles, California, USA
Yoshinobu Fukumori Faculty of Pharmaceutical Sciences Kobe Gakuin University Kobe, Japan
Masatoshi Chikazawa Division of Applied Chemistry Graduate School of Engineering Tokyo Metropolitan University Hachioji, Tokyo, Japan
Mojtaba Ghadiri Institute of Particle Science and Engineering School of Process, Environmental and Materials Engineering University of Leeds Leeds, United Kingdom
Reg Davies E. I. Du Pont de Nemours, Inc. Wilmington, Delaware, USA Shigehisa Endoh National Institute of Advanced Industrial Science and Technology Tsukuba, Japan Yoshiyuki Endo Process and Production Technology Center Sumitomo Chemical Co., Ltd. Osaka, Japan Richard C. Flagan Division of Chemistry and Chemical Engineering California Institute of Technology Pasadena, California, USA Masayoshi Fuji Ceramics Research Laboratory Nagoya Institute of Technology Tajimi, Gifu, Japan
Kuniaki Gotoh Department of Applied Chemistry Okayama University Okayama, Japan Jusuke Hidaka Department of Chemical Engineering and Materials Science Faculty of Engineering Doshisha University Kyotanabe, Kyoto, Japan Ko Higashitani Department of Chemical Engineering Kyoto University Katsura, Kyoto, Japan Hajime Hori Department of Environmental Health Engineering University of Occupational and Environmental Health Kitakyushu, Fukuoka, Japan
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Hideki Ichikawa Faculty of Pharmaceutical Sciences Kobe Gakuin University Kobe, Japan
Matsuoka Masakuni Department of Chemical Engineering Tokyo University of Agriculture and Technology Koganei, Tokyo, Japan
Kengo Ichiki Department of Mechanical Engineering The Johns Hopkins University Baltimore, Maryland, USA
Hiroaki Masuda Department of Chemical Engineering Kyoto University Katsura, Kyoto, Japan
Hironobu Imakoma Department of Chemical Science and Engineering Kobe University Kobe, Japan
Shuji Matsusaka Department of Chemical Engineering Kyoto University Katsura, Kyoto, Japan
Eiji Iritani Department of Chemical Engineering Nagoya University Nagoya, Chikusa-ku, Japan
Minoru Miyahara Surface Control Engineering Laboratory Department of Chemical Engineering Kyoto University Katsura, Kyoto, Japan
Chikao Kanaoka Ishikawa National College of Technology Tsubata, Ishikawa, Japan Yoichi Kanda Department of Chemical Engineering Kyoto University Katsura, Kyoto, Japan Yoshiteru Kanda Department of Chemistry and Chemical Engineering Yamagata University Yonezawa, Yamagata, Japan Yoshiaki Kawashima Gifu Pharmaceutical University Mitahora-Higashi, Gifu, Japan
Kei Miyanami Department of Chemical Engineering Osaka Prefecture University Sakai, Osaka, Japan Yasushige Mori Department of Chemical Engineering and Materials Science Doshisha University Kyotanabe, Kyoto, Japan Makio Naito Joining and Welding Research Institute Osaka University Ibaraki, Osaka, Japan
Yasuo Kousaka Department of Chemical Engineering Osaka Prefecture University Sakai, Osaka, Japan
Yoshio Ohtani Department of Chemistry and Chemical Engineering Kanazawa University Kanazawa, Ishikawa, Japan
Ryoichi Kurose Central Research Institute of Electric Power Industry Yokosuka, Kanagawa, Japan
Morio Okazaki Department of Chemical Engineering Kyoto University Kyoto, Japan
Hisao Makino Central Research Institute of Electric Power Industry Yokosuka, Kanagawa, Japan
Kikuo Okuyama Department of Chemical Engineering Hiroshima University Higashi-Hiroshima, Japan
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Jun Oshitani Department of Applied Chemistry Okayama University Okayama, Japan Yasufumi Otsubo Center of Cooperative Reserch Chiba University Image-ku, Chiba, Japan Isao Sekiguchi Department of Applied Chemistry Chuo University Bunkyo-ku, Tokyo, Japan Mamoru Senna Faculty of Science and Technology Keio University Yokohama, Kanagawa, Japan Manabu Shimada Department of Chemical Engineering Division of Chemistry and Chemical Engineering Hiroshima University Higashi-Hiroshima, Japan Kunio Shinohara Materials Chemical Engineering, Lab. Division of Chemical Process Engineering Hokkaido University Sapporo, Japan Yoshiyuki Shirakawa Department of Chemical Engineering and Materials Science Doshisha University Kyotanabe, Kyoto, Japan Minoru Sugita Ohsaki Research Institute, Inc. Tokyo, Japan Michitaka Suzuki Department of Mechanical System Engineering University of Hyogo Himeji, Japan
Hiroshi Takahashi Department of Mechanical Systems Engineering Muroran Institute of Technology Muroran, Hokkaido, Japan Minoru Takahashi Ceramics Research Laboratory Nagoya Institute of Technology Tajimi, Aichi, Japan Takashi Takei Faculty of Urban Environmental Science Tokyo Metropolitan University Hachioji, Tokyo, Japan Isamu Tanaka Department of Environmental Health Engineering University of Occupational and Environmental Health Kitakyushu, Fukuoka, Japan Tatsuo Tanaka Hokkaido University Sapporo, Japan Toshitsugu Tanaka Department of Mechanical Engineering Osaka University Suita, Osaka, Japan Ken-ichiro Tanoue Yamaguchi University Ube, Yamaguchi, Japan Yuji Tomita Department of Mechanical and Control Engineering Kyushu Institute of Technology Kitakyushu, Fukuoka, Japan Shigeki Toyama Nagoya University Nagoya, Japan JunIchiro Tsubaki Department of Molecular Design and Engineering Nagoya University, Nagoya, Japan
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Hirofumi Tsuji Central Research Institute of Electric Power Industry Yokosuka, Kanagawa, Japan Yutaka Tsuji Department of Mechanical Engineering Osaka University Suita, Osaka, Japan Hiromoto Usui Department of Chemical Science and Engineering Kobe University Kobe, Japan Satoru Watano Department of Chemical Engineering Osaka Prefecture University Sakai, Osaka, Japan Richard A. Williams Institute of Particle Science and Engineering School of Process, Environmental, and Materials Engineering University of Leeds Leeds, United Kingdom
Hiroshi Yamato Department of Environmental Health Engineering University of Occupational and Environmental Health Kitakyushu, Fukuoka, Japan Toyokazu Yokoyama Hosokawa Powder Technology Research Institute Hirakata, Osaka, Japan Hideto Yoshida Department of Chemical Engineering Hiroshima University Higashi-Hiroshima, Japan Hiroki Yotsumoto National Institute of Advanced Industrial Science and Technology Tsukuba, Ibaraki, Japan Shinichi Yuu Ootake R. & D. Consultant Office Fukuoka, Japan
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Contents PART I Chapter 1.1
Chapter 1.2
Chapter 1.3
Chapter 1.4
Chapter 1.5
Chapter 1.6
Particle Characterization and Measurement Particle Size
3
1.1.1 1.1.2 1.1.3
3 3 6
Definition of Particle Diameter Particle Size Distribution Average Particle Size
Size Measurement
13
1.2.1 1.2.2 1.2.3
13 14
Introduction The Approach Particle Size Analysis Methods and Instrumentation
15
Particle Shape Characterization
33
1.3.1 1.3.2 1.3.3 1.3.4 1.3.5
33 34 35 45 47
Introduction Representative Size Geometrical Shape Descriptors Dynamic Equivalent Shape Concluding Remarks
Particle Density
49
1.4.1 1.4.2
49 50
Definitions Measurement Method for Particle Density
Hardness, Stiffness and Toughness of Particles
53
1.5.1 1.5.2 1.5.3 1.5.4
53 56 59 60
Indentation Hardness Measurement of Hardness Measurement of Stiffness Measurement of Toughness
Surface Properties and Analysis
67
1.6.1 1.6.2 1.6.3
67 76 93
Surface Structures and Properties Surface Characterization Atomic Force Microscopy
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PART II Chapter 2.1
Chapter 2.2
Chapter 2.3
Chapter 2.4
Chapter 2.5
Chapter 2.6
Chapter 2.7
Chapter 2.8
Fundamental Properties of Particles Diffusion of Particles
103
2.1.1 2.1.2
103 109
Thermal Diffusion Turbulent Diffusion
Optical Properties
115
2.2.1 2.2.2 2.2.3 2.2.4 2.2.5
115 116 119 122 123
Definitions Light Scattering Light Extinction Dynamic Light Scattering Photophoresis
Particle Motion in Fluid
125
2.3.1 2.3.2 2.3.3
125 125 130
Introduction Motion of a Single Particle Particle Motion in Shear Fields
Particle Sedimentation
133
2.4.1 2.4.2 2.4.3 2.4.4
133 133 138
Introduction Terminal Settling Velocity Settling of Two Spherical Particles Rate of Sedimentation in Concentrated Suspension
139
Particle Electrification and Electrophoresis
143
2.5.1 2.5.2
In Gaseous State In Liquid State
143 147
Adhesive Force of a Single Particle
157
2.6.1 2.6.2 2.6.3 2.6.4
157 159 162 163
Van der Waals Force In Gaseous State In Liquid State Measurement of Adhesive Force
Particle Deposition and Reentrainment
171
2.7.1 2.7.2
171 174
Particle Deposition Particle Reentrainment
Agglomeration (Coagulation)
183
2.8.1 2.8.2
183 190
In Gaseous State In Liquid State
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Chapter 2.9
Viscosity of Slurry
199
2.9.1 2.9.2 2.9.3 2.9.4
199 199 200
2.9.5 2.9.6 2.9.7 Chapter 2.10
Chapter 2.13
Chapter 2.14
202 202 203 205
2.10.1 2.10.2 2.10.3
205 206
Impact Force Mode of Breakage Analysis of Breakage for the Semibrittle Failure Mode Analysis of Breakage of Agglomerates
Sintering 2.11.1 2.11.2 2.11.3
Chapter 2.12
200
Particle Impact Breakage
2.10.4 Chapter 2.11
Introduction Basic Flow Characteristics Time-Dependent Flow Characteristics Viscosity Equations for Suspensions of Spherical Particles of Narrow Particle Size Distribution Effect of Particle Size Distribution on Slurry Viscosity Measurement of Slurry Viscosity by a Capillary Viscometer Measurement of Slurry Viscosity by a Rotating Viscometer
208 210 213
Mechanisms of Solid-Phase Sintering Modeling of Sintering of Agglomerates Sintering Process of Packed Powder
213 214 216
Ignition and Combustion Reaction
219
2.12.1 2.12.2 2.12.3 2.12.4
219 219 221 222
Combustion Profile Devolatilization and Ignition Gaseous Combustion Solid Combustion
Solubility and Dissolution Rate
225
2.13.1 2.13.2 2.13.3 2.13.4 2.13.5
225 225 226 230 230
Solubility of Fine Particles Factors to Increase Solubility Theories of Dissolution Measurement of Dissolution Rate Methods to Increase the Dissolution Rate
Mechanochemistry
239
2.14.1 2.14.2 2.14.3 2.14.4
239 239 240
2.14.5
Terminology and Concept Phenomenology of Mechanochemistry Theoretical Background Structural Change of Solids under Mechanical Stress Mechanochemical Solid-State Reaction and Mechanical Alloying
240 242
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2.14.6 2.14.7
PART III Chapter 3.1
Chapter 3.2
Chapter 3.3
Chapter 3.4
Chapter 3.5
Chapter 3.6
Soft Mechanochemical Processes and Their Application Recent Developments and Future Outlook
242 244
Fundamental Properties of Powder Beds Adsorption Characteristics
249
3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6
249 249 250 255 256
Introduction Adsorption Measurement Theory of Adsorption Isotherms Adsorption Velocity Adsorbed State of Adsorbate Estimation of Surface Properties by Adsorption Method
259
Moisture Content
265
3.2.1 3.2.2
265
Bound Water and Free Water Methods for Determining Moisture Content in a Particulate System
267
Electrical Properties
269
3.3.1 3.3.2
269 275
In Gaseous State In Nonaqueous Solution
Magnetic Properties
283
3.4.1 3.4.2 3.4.3
283 286 288
Magnetic Force on a Particle Ferromagnetic Properties of Small Particles Magnetism of Various Materials
Packing Properties
293
3.5.1 3.5.2
293 299
Packing of Equal Spheres Packing of Multisized Particles
Capillarity of Porous Media
309
3.6.1 3.6.2 3.6.3
309 310
3.6.4
Common Phenomenon: Young-Laplace Effect Nitrogen Adsorption Method Mercury Intrusion Method (Mercury Porosimetry) Other Techniques of Interest
313 314
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Chapter 3.7
Chapter 3.8
Chapter 3.9
Chapter 3.10
Chapter 3.11
Chapter 3.12
Chapter 3.13
Permeation (Flow through Porous Medium)
317
3.7.1 3.7.2
317 321
Resistance to Flow through a Porous Medium Pressure Drop across a Fibrous Mat
Specific Surface Area
325
3.8.1 3.8.2 3.8.3 3.8.4
325 326 330 332
Definition of Specific Surface Area Adsorption Method Heat of Immersion Permeametry
Mechanical Properties of a Powder Bed
335
3.9.1 3.9.2 3.9.3
335 338 341
Shearing Strength of a Powder Bed Adhesion of a Powder Bed Yielding Characteristics of a Powder Bed
Fluidity of Powder
349
3.10.1 3.10.2 3.10.3 3.10.4
349 349 357 357
Definition of Fluidity Measurement of Fluidity Factors Affecting Fluidity Improvement of Fluidity
Blockage of Storage Vessels
361
3.11.1 3.11.2 3.11.3
361 361 368
Phenomena and Factors Mechanisms and Flow Criteria Methods of Preventing Blockage
Segregation of Particles
371
3.12.1 3.12.2 3.12.3 3.12.4 3.12.5
371 371 372 376 380
Definition and Importance Related Operations Fundamental Mechanisms Patterns and Degrees Minimizing Methods
Vibrational and Acoustic Characteristics
383
3.13.1 3.13.2 3.13.3
383 385
3.13.4 3.13.5 3.13.6
Behavior of a Particle on a Vibrating Plate Behavior of a Vibrating Particle Bed Generating Mechanism of Impact Sound between Two Particles Frictional Sound from a Granular Bed Vibration of a Small Particle in a Sound Wave Attenuation of Sound in a Suspension of Particles
386 390 394 396
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PART IV Chapter 4.1
Chapter 4.2
Chapter 4.3
Chapter 4.4
Particle Generation and Fundamentals Aerosol Particle Generation
401
4.1.1 4.1.2 4.1.3 4.1.4
401 404 407 408
Generation of Particles by Reaction
413
4.2.1 4.2.2
413 418 425
4.3.1 4.3.2
425 428
436 443 445 449
4.5.1 4.5.2
449 456
Particle Dispersion in Gaseous State Particle Dispersion in Liquid State
Electrical Charge Control
465
4.6.1 4.6.2
465
In Gaseous State Charge Control by Contact Electrification in Gaseous State In Liquid State
473 474
Surface Modification
485
4.7.1 4.7.2 4.7.3
485 485
4.7.4 4.7.5 4.7.6 4.7.7 4.7.8 Chapter 4.8
Composite Particle Formation in the Fluid Bed Process Particle Design in the Emulsifying Process Summary
435
Dispersion of Particles
4.6.3 Chapter 4.7
Crystallization Phenomena and Kinetics Operation and Design of Crystallizers
Design and Formation of Composite Particles
4.4.2 4.4.3
Chapter 4.6
Gas-Phase Techniques Liquid-Phase Techniques
Crystallization
4.4.1
Chapter 4.5
Condensation Methods Liquid Atomization Powder Dispersion Generation of Monodisperse Particles
Purpose of Surface Modification Methods of Surface Modification Conventional Treatments with Surfactants, Coupling Agents, and Simple Heating Microencapsulation and Nanocoating Polymerization and Precipitation In Situ Mechanical Routes and Apparatus Characterization of Coated Particles Remarks and Recent Developments
Standard Powders and Particles
486 486 488 488 488 490 493
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PART V Chapter 5.1
Chapter 5.2
Chapter 5.3
Chapter 5.4
Powder Handling and Operations Crushing and Grinding
503
5.1.1 5.1.2 5.1.3 5.1.4 5.1.5 5.1.6
503 503 504 512 515 517
Classification
527
5.2.1 5.2.2 5.2.3 5.2.4
527 530 532 546
Chapter 5.6
551
5.3.1 5.3.2 5.3.3 5.3.4 5.3.5 5.3.6 5.3.7
551 551 552 554 554 556 559
General Characteristics of Silos Classification of Silos Planning Silos Design Load Load due to Bulk Materials Calculation of Static Powder Pressure Design Pressures
Feeding
561 Introduction Various Feeders
561 562
Transportation
567
5.5.1 5.5.2
567 573
Transportation in the Gaseous State Transportation in the Liquid State
Mixing 5.6.1 5.6.2 5.6.3 5.6.4 5.6.5
Chapter 5.7
Basis of Classification Dry Classification Wet Classification Screening
Storage (Silo)
5.4.1 5.4.2 Chapter 5.5
Introduction Comminution Energy Crushing of Single Particles Kinetics of Comminution Grinding Operations Crushing and Grinding Equipment
577 Introduction Powder Mixers Mixing Mechanisms Power Requirement for Mixing Selection of Mixers
577 577 581 584 587
Slurry Conditioning
591
5.7.1 5.7.2
591 594
Slurry Characterization Slurry Preparation
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Chapter 5.8
Chapter 5.9
Chapter 5.10
Granulation
599
5.8.1 5.8.2
599 607
Kneading and Plastic Forming
615
5.9.1 5.9.2
615 618
5.10.2
Chapter 5.12
Chapter 5.13
Chapter 5.14
Chapter 5.15
Kneading Plastic Forming
Drying 5.10.1
Chapter 5.11
Granulation Mechanisms Granulators
621 Drying Characteristics of Wet Particulate and Powdered Materials Dryer Selection and Design
621 623
Combustion
629
5.11.1 5.11.2 5.11.3 5.11.4
629 629 629 633
Introduction Control of the Combustion Process Combustion Burner Furnace and Kiln
Dust Collection
637
5.12.1 5.12.2 5.12.3 5.12.4
637 644 649 651
Flow-Through-Type Dust Collectors Obstacle-Type Dust Collectors Barrier-Type Dust Collectors Miscellaneous
Electrostatic Separation
653
5.13.1 5.13.2
653 656
Separation Mechanism Separation Machines
Magnetic Separation
661
5.14.1 5.14.2 5.14.3 5.14.4
661 663 668
Classification of Magnetic Separators Static Magnetic Field Separators Magnetohydrostatic Separation Electromagnetic-Induction-Type Separation
670
Gravity Thickening
673
5.15.1 5.15.2 5.15.3 5.15.4 5.15.5
673 673 675 676 678
Pretreatment Ideal Settling Basin Settling Curve Kynch Theory Design of a Continuous Thickener
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Chapter 5.16
Filtration 5.16.1 5.16.2 5.16.3 5.16.4 5.16.5
Chapter 5.17
Chapter 5.18
Chapter 5.21
684 688 690 692 693 697
5.17.1 5.17.2 5.17.3 5.17.4 5.17.5
697 697 701 703 703
Basis of Expression Modified Terzaghi Model Secondary Consolidation Simplified Analysis Expression Equipment
Flotation
5.18.3 5.18.4 5.18.5 5.18.6
Chapter 5.20
Basis of Cake Filtration Theory Constant-Pressure and Constant-Rate Filtration Internal Structure of Filter Cake Non-Newtonian Filtration Filtration Equipment
Expression
5.18.1 5.18.2
Chapter 5.19
683
705 Principles of Flotation Classification of Minerals according to Their Flotation Behavior Flotation Reagents Flotation Machines Differential Flotation Plant Practice of Differential Flotation
705 707 708 710 714 714
Electrostatic Powder Coating
717
5.19.1 5.19.2 5.19.3 5.19.4
717 719 720
Coating Machines Powder Feeding Machine Powder Coating Booth Numerical Simulation for Electrostatic Powder Coating
721
Multipurpose Equipment
725
5.20.1 5.20.2 5.20.3
725 727 731
Fluidized Beds Moving Beds Rotary Kiln
Simulation
737
5.21.1 5.21.2 5.21.3 5.21.4 5.21.5 5.21.6
737 748 754 756 760 764
Computer Simulation of Powder Flows Breakage of Aggregates Particle Motion in Fluids Particle Methods in Powder Beds Transport Properties Electrical Properties of Powder Beds
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PART VI Chapter 6.1
Chapter 6.2
Chapter 6.3
Chapter 6.4
Chapter 6.5
Chapter 6.6
Chapter 6.7
PART VII Chapter 7.1
Process Instrumentation Powder Sampling
771
6.1.1 6.1.2
771 774
Sampling Equipment Analysis of Sampling
Particle Sampling in Gas Flow
779
6.2.1 6.2.2 6.2.3
779 780 783
Anisokinetic Sampling Error Sampling in Stationary Air Practical Applications of Particle Sampling
Concentration and Flow Rate Measurement
795
6.3.1 6.3.2
795 799
Particle Concentration in Suspensions Powder Flow Rate
Level Measurement of a Powder Bed
805
6.4.1 6.4.2 6.4.3 6.4.4 6.4.5 6.4.6
805 806 808 809 809 810
Level Meters and Level Switches Mechanical Method Electrical Method Ultrasonic Wave Level Meters Radiometric Method Pneumatic Method and Others
Temperature Measurement of Powder
813
6.5.1 6.5.2
813 814
Thermal Contact Thermometers Radiation Thermometers
On-Line Measurement of Moisture Content
817
6.6.1 6.6.2 6.6.3 6.6.4
817 817 821
Introduction Electrical methods Infrared Moisture Sensor Application of Moisture Control to Powder-Handling Processes
823
Tomography
827
6.7.1 6.7.2 6.7.3
827 829 830
Introduction Sensor Selection and Specification Examples of Powder-Processing Applications
Working Atmospheres and Hazards Health Effects Due to Particle Matter
837
7.1.1
837
Introduction
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7.1.2 7.1.3 7.1.4 7.1.5 7.1.6 Chapter 7.2
Chapter 7.3
Respiratory System Penetration and Deposition of Particles in the Respiratory Tract Fate of Deposited Particles Health Effects of Inhaled Particles Threshold Limit Value
837 838 839 840 841
Respiratory Protective Devices for Particulate Matter
843
7.2.1 7.2.2 7.2.3 7.2.4 7.2.5 7.2.6
843 843 843 845 846 847
Introduction Types of Respirators Air-Purifying Respirators Atmosphere-Supplying Respirators Protection Factor Notes for Using Respirators
Spontaneous Ignition and Dust Explosion
849
7.3.1 7.3.2
849 861
Spontaneous Ignition of Powder Deposits Dust Explosion Mechanism and Prevention
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Part I Particle Characterization and Measurement
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1.1
Particle Size JunIchiro Tsubaki Nagoya University, Nagoya, Japan
1.1.1
DEFINITION OF PARTICLE DIAMETER
Particle size data are essential to anyone treating powders. Expressing the size of a single particle is not a simple task when the particle is nonspherical. Expressions of individual particle size and a hypothetical equivalent sphere with regard to some properties. Table 1.1 lists the physical meaning of variously defined characteristic diameters. When a particle is circumscribed by a rectangular prism with length l, width w, and height t, its size is expressed by the diameter, obtained from the three dimensions. l, w, and t are measured with a microscope. Feret and Martin diameter are statistical diameters, which are affected by the particle orientation or measuring direction. The mean values of them are often defined as characteristic diameters. Unrolled diameter is the mean value of a statistical diameter. The equivalent diameters are the diameters of spheres having the same geometric or physical properties as those of nonspherical particles.
1.1.2
PARTICLE SIZE DISTRIBUTION
Size Distribution When a certain characteristic diameter, shown in Table 1.1, is measured for N particles, and the number of particles, dN, having diameters between x and x + dx, is counted, the density size distribution q 0 (x) is defined as dN 1 N dx
(1.1)
q ( x ) dx 1
(1.2)
qo ( x ) where
∫
∞
0
0
The cumulative distribution Q0(x) is given as Q0 ( x ) ∫ q0 ( x ) dx x
0
(1.3)
Therefore, dQ0 ( x ) dx
q0 ( x )
(1.4)
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TABLE 1.1
Powder Technology Handbook
Expression of Particle Size Definition of characteristic diameters Geometric size
Physical meaning and corresponding measuring method
Breadth: b Length: l Thickness: t b 1 b l t ⎛ b ⎞ , ,⎜ ⎟ ⎝t⎠ 2 3
1/ 3
,
3 ⎛ 2lb 2bt 2lt ⎞ , lb ⎜ ⎟⎠ ⎝ 1 / l 1 / b 6
1/ 2
Feret diameter
Distance between pairs of parallel tangents to the particle silhouette in some fixed direction
Martin diameter
Length of a chord dividing the particle silhouette into two equal areas in some fixed direction
Unrolled diameter
Chord length through the centroid of the particle silhouette
Sieve diameter
a1, a2: Openings of sieves
1 (a a2 ) or a1a2 2 1 Volume: v
Equivalent diameter
Surface: s
Equivalent projection area diameter (Heywood diameter)
Diameter of the circle having same area as projection area of particle, corresponding to diameter obtained by light extinction
Equivalent surface area diameter (specific surface diameter) (s/p)1/2
Diameter of the sphere having the same surface as that of a particle
Equivalent volume diameter (6v/p)1/3
Diameter of the sphere having the same volume as that of a particle, corresponding to diameter obtained by (electrical sensing zone method)
Stokes diameter
Diameter of the sphere having the same gravitational settling velocity as that of particle obtained by gravitational or centrifugal sedimentation and impactor
Aerodynamic diameter
Diameter of the sphere having unity in specific gravity and the same gravitational settling velocity as that of a particle obtained by the same methods as above
Equivalent light-scattering diameter
Diameter of the sphere giving the same intensity of light scattering as that of a particle, obtained by the light-scattering method
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Particle Size
5
The size distributions thus defined are on a number basis. In the case of mass or volume basis, total mass M and fractional mass dM are used instead of N and dN, respectively, and also the subscript value is changed from 0 to 3, so that q3(x), Q3(x) should be used. In general the subscript is described as r, and r 0, 1, 2, 3 corresponds to number, length, area, and mass or volume basis, respectively. Density distribution qr (x) can be transformed to another basis distribution qr(x) by x sr qr ( x ) dx
qs ( x )
∫
0
x sr qr ( x ) dx
(1.5)
When particle size distributes widely, the distributions are plotted versus ln x instead of x and defined as Qr ( ln x ) Qr ( x ) q∗r (ln x )
dQr ( x ) xqr ( x ) d ln x
(1.6) (1.7)
The density distribution in a representation with a logarithmic abscissa is distinguished by superscript *. The discrete expression, which gives the size distribution histogram, q–r,i, q–*r,i, becomes q r,i
∗
q r ,i
Q r , i xi
Qr ,i ln xi
Qr ( xi ) Qr ( xi1 )
xi xi1 Qr ( xi ) Qr ( xi1 ) ln( xi xi1 )
(1.8)
(1.9)
Normal Distribution The normal or Gaussian distribution function is defined as
qr ( x )
⎡ ( x x )2 ⎤ 50 ⎥ exp ⎢ 2 s 2 s 2p ⎢⎣ ⎥⎦ 1
(1.10)
where x 50 is the 50% or median diameter defined as Qr (x50 ) 0.5 and is the standard deviation expressing the dispersion of the distribution. s x84.13 x50 x50 x15.87
(1.11)
Log-Normal Distribution The log-normal distribution function is given by substituting ln x and ln g, respectively, for x and in Equation 1.10 as follows: ⎡ ( ln x ln x )2 ⎤ 1 50 ⎥ q r (ln x ) exp ⎢ 2 ln 2 s g ln s g ⋅ 2p ⎢⎣ ⎥⎦ ∗
(1.12)
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where g is the geometric standard deviation given as sg
x84.13 x 50 x50 x15.87
(1.13)
If a size distribution obeys the log-normal form, the other distribution converted by Equation 1.5 also obeys the log-normal form. The geometric standard deviation of any distribution is the same value; meanwhile, the median diameters are different but can convert each other. In the case of the volume and number distribution the median diameters, x50,3 , x50,0, , can convert each other by Hatch’s equation. ln x50,3 ln x50,0 3 ln 2 s g
(1.14)
Rosin–Rammler (Weibull) Distribution The Rosin–Rammler or Weibull distribution function is written as
(
Qr ( x ) 1 exp bx
n
)
or
⎡ ⎛ x ⎞n⎤ Qr ( x ) 1 exp ⎢ ⎜ ⎟ ⎥ ⎢⎣ ⎝ xe ⎠ ⎥⎦
(1.15)
where b is a constant equal to xe–n and x e is an absolute size constant defined as x e x 63.2 , that is, xe x63.2 . n is the distribution constant expressing the dispersion of particle sizes. The density distribution is written as qr ( x ) nbx
n1
n ⎡ ⎛ x ⎞n⎤ 1⎛ x ⎞ exp(bx ) or qr ( x ) ⎜ ⎟ exp ⎢ ⎜ ⎟ ⎥ x ⎝ xe ⎠ ⎢⎣ ⎝ xe ⎠ ⎥⎦ n
(1.16)
Graphical Representation As an example, a set of data obtained by a sieving test is illustrated in Table 1.2, and the size distributions are illustrated in Figure 1.1 with a general abscissa and Figure 1.2 with a logarithmic abscissa. Q3,i in Table 1.2 is plotted on log-normal and Rosin–Rammler probability paper, as shown in Figure 1.3. Since the plots are on a straight line on a log-normal probability graph, the particle size distribution obeys the log-normal function of which x50 1.0 mm and x84.13 2.2 mm. If we read the two values of x15.87, x50 or x84.13, g can be calculated. From Figure 1.3 x50 1.0 mm and x84.13 2.2 mm, then g 2.2. The dotted line in Figure 1.3 is the number distribution converted from measured volume distribution by Hatch’s equation. ISO 9276-1 (JIS Z 8819-1) standardizes the graphical representation of particle size analysis data.
1.1.3 AVERAGE PARTICLE SIZE All average particle diameters except the geometric mean diameter are defined by x k ,r k M k ,r k
M kr ,0 Mr ,0
k
M kr3,3 Mr3,3
(1.17)
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Particle Size
7
TABLE 1.2 1 i
Calculation of the Histogram and the Cumulative Distribution 3 Mi ( 10–3kg)
0
2 xi (mm) 0.063
4 Q3,i (%) 0
1
0.09
0.1
0.1
2
0.125
0.27
0.37
3
0.18
1.03
4
0.25
5
0.355
6 7
5 xi (mm)
6 q–3,i (%mm–1)
7 lnxi
8 (q–3,i) (%)
0.027
3.7
0.357
0.3
0.035
7.7
0.329
0.8
1.4
0.055
18.7
0.365
2.8
2.2
3.6
0.07
31.4
0.329
6.7
5.4
9
0.105
51.4
0.351
15.4
0.5
9.3
18.3
0.145
64.1
0.342
27.2
0.71
13.7
32
0.21
65.2
0.351
39.1
8
1
17
49
0.29
58.6
0.342
49.6
9
1.4
17
66
0.4
42.5
0.336
50.5
10
2
14.6
80.6
0.6
24.3
0.357
40.9
11
2.8
9.9
90.5
0.8
12.4
0.336
29.4
12
4
5.6
96.1
1.2
4.7
0.357
15.7
13
5.6
2.4
98.5
1.6
1.5
0.336
7.1
14
8
1.1
99.6
2.4
0.5
0.357
3.1
15
11.2
0.3
99.9
3.2
0.1
0.336
0.9
16
16
0.1
4.8
0.0
0.357
0.3
100
Where Mk,r is complete kth moment of a qr (x) -distribution. ∞
M k ,r ∫ x k qr ( x ) dx
(1.18)
0
Geometric mean diameter is defined as ∞
ln x geo, r ∫ ln x ⋅ qr ( x ) dx
(1.19)
0
Although a lot of average diameters can be defined, the several listed in Table 1.3 are generally used. If a density distribution is given as a histogram, Mk,r is calculated by the following equations. Equation 1.18 is rewritten as follows if k –1.
m
M k ,r ∑ q r ,i i1
xi
∫ x dx k 1 ∑ q ( x k
1
m
r ,i
xi1
i1
⎛ xik1 xik11 ⎞ 1 m Δ Q ∑ r ,i ⎜⎝ x x ⎟⎠ k 1 i1 i i1
k1 i
xik11
) (1.20)
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100
Q3
3
q* [%], q* [%], Q [%]
80
60
3
q
3
3
40
q
20
0
3
0.1
1
x FIGURE 1.1
10
[mm]
A graphical representation of particle size distribution with a linear abscissa.
If k –1, x ln i m xi1 xi M1,r ∑ q r ,i ln ∑ Qr ,i xi xi1 xi1 i1 i1 m
.
(1.21)
Equation 1.19 is rewritten as follows:
m
ln x geo, r ∑ q r ,i i1
xi
m
xi1
i1
i
Qr ,i
m
q r ,i
∫ ln xdx ∑ x x
i1
∑ i1
( xi xi1 )2
(1.22)
The values of the average diameters listed in Table 1.3 can be calculated from the data in Table 1.2. The calculation results are illustrated in Table 1.3. The spread of a size distribution is represented by its variance, which represents the square of the standard deviation, . The variance, 2 , of a qr (x) -distribution is defined as 2 ∫
0
( x x ) q ( x)dx M 2
1,r
r
2 ,r
(
M1,r
)
2
(1.23)
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Particle Size
9
100 Q3
q3
60
3
3
q* [%mm–1], q* [%mm–1], Q3 [%]
80
40
20 q3
0
FIGURE 1.2 abscissa.
0
2
4 6 x [mm]
8
10
A graphical representation of particle size distribution with a logarithmic
From a histogram, 2
⎤ 1⎡ m ⎤ 1⎡ m ⎢ ∑ q r ,i x 3i x 3i1 ⎥ ⎢ ∑ q r ,i x i2 x i21 ⎥ 3 ⎣ i1 ⎦ 4 ⎣ i1 ⎦ 2 3 3 ⎛ x x i1 ⎞ ⎤ 1 ⎡ m ⎤ 1⎡ m ⎥ ⎢ ∑ Qr ,i ⎜ i Q x x − ⎟ i1 )⎥ ⎢ ∑ r ,i ( i 3 ⎢ i1 ⎦ ⎝ xi xi1 ⎠ ⎥⎦ 4 ⎣ i1 ⎣ 2
(
)
(
)
(1.24)
The standard deviation of the particle sizes illustrated in Table 1.2 is calculated as 1.29 mm by Equation 1.24. ISO 9276–2 (JIS Z 8819-2) standardizes the calculation of average particle sizes.
Notation i k m Mk,r n
number of the size class with upper particle size xi power of x total number of size classes complete kth moment of a qr (x) -distribution distribution constant
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99.9 99
90 70 50
90 70 50
30 20
30 20
10
10
5
5
1
1
0.1 0.01
0.1
Q3 [%]
Q0.Q3 [%]
99.9 99
0.1 10
1 x [mm]
FIGURE 1.3 graph.
Cumulative distributions on a log-normal and Rosin–Rammler distribution
TABLE 1.3 Average Diameters Average diameters
Symbols
Definition
Examples of
q0(x)
q3(x)
from
from
Table 1.2 [mm]
Arithmetic average length diameter Weighted average number diameter
x–1,0
0.228
Arithmetic average surface diameter
x–2,0 x–
0.303
x–1,1 x–
0.402
1.40
Harmonic mean diameter
x–1,3 x–
Geometric mean diameter
x–geo,r
Arithmetic average volume diameter Weighted average length diameter Weighted average surface diameter, Sauter-diameter Weighted average volume diameter
3,0
1,2
1,r
0.410
0.749
0.147 (r0) 0.749 (r3) 3.98 (r3)
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Particle Size
N qr(x) qr qr*(ln x) qr* q–r,i q–r,i Qr(x) Qr ΔQr,i x xe xi xi1 Δ xi x15.87, x50, x63.2, x84.13 x50,3 x50,0 r, s s sg
11
particle number density distribution qr(x) density distribution in a representation with a logarithmic abscissa qr*(x) average density distribution of the class Δ xi , histogram average density distribution of the class Δ In xi, histogram cumulative distribution Qr(x) Qr(xi)Qr(xi1) particle size, diameter of a sphere absolute size constant upper size of a particle size interval lower size of a particle size interval xixi1, width of the particle size interval defined as Qr(x15.87) 0.1587,Qr(x50) 0.5 ,Qr(x63.2) 0.632 Qr(x84.13) 0.8413 median particle size of a cumulative volume distribution median particle size of a cumulative number distribution type of quantity of a distribution, r, s 0, 1, 2, 3 standard deviation geometrical standard deviation of log-normal distribution
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1.2
Size Measurement Reg Davies E. I. Du Pont de Nemours, Inc., Wilmington, Delaware, USA
1.2.1
INTRODUCTION
The particle size distribution has been usefully defined as the state of subdivision of a material. For a smooth dense sphere, size (X) is equivalent to diameter (D). In most industrial applications, however, size (X) is equivalent to diameter (D) only through the use of an adjustment factor (F), where Dρ = FX F can be a shape factor or, in the case of optical methods, a combined shape and extinction coefficient. Unless F is known, accurate particle size measurement of irregular particles is impossible. In these instances, size will be a function of the direction of linear measurement. It will be a function of the physical principle by which it is detected and a function of the instrument by which it is measured. Hence, an average size calculated from data obtained via an envelope—volume diameter of a powder in an electrolyte, a light-scattering measurement of the powder in suspension, or a permeability measurement of dry powder in air—will not be the same. The resulting differences will provide a valuable fingerprint of shape/optical variations and can be used advantageously for this purpose. Table 2.1 shows many common methods of experiencing the size of irregular particles. The first six of these are obtained from microscopy and image analysis: the sieve diameter from sieving methods, specific surface diameter from permeability measurements, surface diameter from gas adsorption; Stokes’ diameter from sedimentation methods, volume diameter from electrical sensing zone methods, and optical diameters from instruments using optical sensing detectors. Again, for most industrial powders, all these diameters will be different. This is not necessarily a disadvantage if one is aware of the reasons for the differences and is able to convert one equivalent diameter to another. It is used to advantage when one wishes to increase the sensitivity of detection of a change in size distribution of a powder during a processing step. For example, if one is concerned with breakage and the reduction of mass, then a volume-sensitive (e.g., electrical sensing zone instrument or X-ray sensing) instrument will be most useful. Whereas in attrition, where volume is conserved but large numbers of very fine particles are abraded from the surfaces of large particles, number counting or surface methods will have the highest sensitivity to the change. Such considerations lead one to conclude that one should always choose the most suitable method and instrument for the job in hand. No sample should ever be submitted to or analyzed by a physical resting laboratory simply with a request for particle size analysis. Neither should one use an instrument because it is conveniently located near one’s workplace. Several hundred instruments are documented in the literature for the measurement of particle size distribution and, their range of application spans six orders of magnitude. The most comprehensive treatise that discusses size measurement and the instruments available is given by Allen.1 At the time of writing this chapter, Allen’s book had just been submitted for its fifth edition. Allen’s treatment combines historical size measurement with in-depth theory. Many of the older techniques are described, but many of these are now almost obsolete. In this chapter, theory will be minimal and only modern instruments will be discussed. For information on other methods, theory and detail, simply read the work by Allen. 13 © 2006 by Taylor & Francis Group, LLC
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TABLE 2.1
Several Methods of Expressing the Size of Irregular Particles
Average thickness
The average diameter between the upper and lower surfaces of a particle in its most stable position of rest
Average length
The average diameter of the longest chords measured along the upper surface of a particle in the position of rest
Average breadth
The average diameter at right angles to the diameter of average length along the upper surface of a particle in its position of rest
Feret’s diameter
The diameter between the tangents at right angles to the direction of scan, which touch the two extremities of the particle profile in its position of rest
Martin’s diameter
The diameter which divides the particles profile into two equal areas measured in the direction of scan when the particle is in a position of rest
Projected area diameter
The diameter of the sphere having the same projected area as the particle profile in the position of rest
Particle sieve diameter
The width of the minimum square aperture through which the particle will pass
Specific surface diameter
The diameter of the sphere having the same ratio of external surface area to volume as the particle
Surface diameter
The diameter of the sphere having the same surface area as the particle
Stokes’ diameter
The diameter of the sphere having the same terminal velocity as the particle
Volume diameter
The diameter of the sphere having the same volume as the particle
Optical diameter
The diameter of the sphere having the same optical (e.g., backscattering or forward scattering) cross section as the particle; these will be different.
1.2.2 THE APPROACH There are several basic steps that might have to be followed from the initial thought that size distribution data are required. Some of these could be the following: 1. Define the particle-processing problem. What is the hypothesis? What are you expecting to find? Why are you measuring size? 2. Define the type and extent of the size distribution data required—if any! 3. Acquire a reproducible sample of the test material. 4. Prepare (disperse) the sample for analysis. 5. Check the approximate upper and lower sizes and the state of dispersion in the sample by microscopy. 6. Select the most relevant (sensitive) physical principle to measure this range of size present. In some instances, this could require more than one principle. 7. Select an instrument that uses this principle. Again, in some instances this could require more than one instrument. 8. Calibrate the instrument(s) against international standards. 9. Conduct the analysis as per international (ISO) or national standards. 10. Select the most relevant data-handling method, calculate the most relevant distribution, and display it as per ISO or national standards. This chapter will provide information to assist the reader with steps 6 and 7 only—selection of a method/instrument for size measurement. Information on other steps is provided elsewhere. © 2006 by Taylor & Francis Group, LLC
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Size Measurement
15
1.2.3 PARTICLE SIZE ANALYSIS METHODS AND INSTRUMENTATION For simplicity, the methods and instruments have been loosely classified into six groups: 1. Visual methods (e.g., optical, electron, and scanning electron microscopy and image analysis) 2. Separation methods (e.g., sieving, classification, impaction, electrostatic differential mobility) 3. Stream scanning methods (e.g., electrical resistance zone, and optical sensing zone measurements) 4. Field scanning methods (e.g., laser diffraction, acoustic attenuation, photon correlation spectroscopy) 5. Sedimentation 6. Surface methods (e.g., permeability, adsorption) Discussion will not be uniform across these classes. Class 6 will have minimum discussion, as the methods are not widely used for measuring distributions; classes 1, 2, and 5 will have moderate discussion, as they receive widespread use in laboratories where high-cost instruments are not common; classes 3 and 4 will receive the most discussion, as many of the instruments in these classes are being used for on-line, in-process measurement and in today’s industrial environment are being implemented into process measurement and control schemes. Use for powders, suspensions, and aerosol will be treated simultaneously; that is, aerosol will not be treated separately, as in previous versions.
Visual Methods: Microscopy Use of a microscope should always accompany size analysis by any method because it permits an estimate to be made of the range of sizes present and the degree of dispersion. In most instances, particle size measurement rarely follows, as other procedures are faster and less stressful to the operator. Microscope size analysis is used primarily as an absolute method of size analysis, as it is the only one in which individual particles are seen. It is used more widely in aerosol analysis, as airborne particles are more often deposited onto surfaces [e.g., by impactors, thermal precipitators (i.e., in a state more conducive to visual examination)]. It is used perhaps more for shape/morphology analysis than size analysis and for beneficiation studies of minerals when combined with x-rays analysis. It is also used when other methods are not possible (e.g., inclusions in steel, porosity in ceramics). Clearly, the range of sizes and their degree of dispersion strongly influence the ease and reliability of size measurement. For instances where particles are not already deposited on surfaces, sample preparation is a critical step. This is discussed in detail by Allen1. Typically, samples can be extracted from an agitated well-dispersed suspension, but dry, temporary, or permanent mounts also possible. For dry mounts, particles can be dusted onto a surface; for temporary slides, powder can be dispersed-held in viscous media. For permanent slides, a 2% solution of collodion in amylacetate or other similar systems can be used to fix the particles permanently in place for later examination. In transmission electron microscopy (TEM), particles are deposited on a very thin film that is transparent to the electron beam. This film is supported on metal grids or frames. For scanning electron microscopy (SEM) backscatter measurements, the powder is dispersed onto a metal substrate and made conductive by coating with a thin layer of carbon from a vacuum evaporator. It cannot be stressed enough that preparation of the sample for visual examination is one critical step and the fact that it receives little space in this chapter does not reflect on its importance. When a satisfactory dispersion of particles on a relevant substrate has been achieved, particle size analysis can follow. Points to consider are resolution, the total number of particles to be counted, the choice of size distribution (whether number or mass distribution is required), and the © 2006 by Taylor & Francis Group, LLC
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effect of material properties. The limit of resolution (i.e., the distance at which two particles in close proximity appear as a single particle) is proportional to the wavelength of the illuminating source and inversely proportional to both the refractive index of the immersion medium and the sine of the angular aperture of the objective. Although absolute limits of resolution exceed the following, the more common ranges of measurement are 3-1000m for optical microscopy, 2 nm to 1m for TEM, and 20 nm to 1000m for SEM. For manual counting and operator comfort, the total number of particles should be typically below 800, and procedures have been developed to attain acceptable statistical accuracy with this constraint. Counting particles by number is easier than by mass, as statistical reliability does not rely on the omission of one particle, whereas counting by mass (e.g., of a distribution containing 999 particles of diameter 1µm and one 10 mm particle), the omission of the one 10mm particle removes approximately 50% of the mass. Hence, very accurate assessment of the largest particles in the sample is essential for mass distribution measurement. Allen 1 shows the expected standard error S(Pr ) of the percentage Pr by number in each size class, out of the total number in all size classes, to be ⎛ P (100 ⫺ Pr ) ⎞ S ( Pr ) ⫽ ⎜ r ⎟⎠ Smr ⎝
(2.1)
where ⌺mr is the total number of particles of all size classes. The standard error is a maximum when Pr=50; hence, S(Pr) will always be less than 2%—an acceptable error for most instances—if Mrⱖ 625 particles. For mass counting, if S(Mg) is the standard deviation experienced as a percentage of the total by weight, Mg is the percentage by weight in a given size range, and Mr is the number of particles counted in the size range, then S(Mg ) ⫽
Mg
(2.2)
Mr
So if 10% by weight of particles lie in the upper or top size range for a similar acceptable accuracy of 2%, it is absolutely necessary to count 25 particles in the top size range. Without special procedures, the total number count to achieve this accuracy would be many thousands and beyond the endurance of an operator. Two approaches can be taken: count the thousands using automatic microscopes/image analyses or maintain the count below 800 by reducing the area examined for the smaller sizes. This procedure is beyond the scope of this chapter, but it is described in British Standard 3406 2 and by Allen1. Furthermore, for most purpose, a choice is made to count on a projected area basis rather than a linear basis. Although this requires an estimate of a circle of equivalent area to the irregular particle, the law of compensating errors generally results in lower errors than by manipulation of any of the possible linear measurements. Figure 2.1 shows a typical reticule given in British Standard 3406 showing seven circles in a root-2 progression of sizes and five different geometric areas. The importance of a well-dispersed and nonsegregated sample becomes clear when the geometric areas of this reticule are examined. Spatial variations can be minimized by using opposite quadrants. Materials properties affect the procedure depending on particle strength, wettability, and particle refractive index, to name three factors. Others can be similarly important. Strength and wettability will certainly influence the method of dispersion and the mounting of the sample. The refractive index will influence resolution. With image analysis, gray value discrimination and edge definition is an acute problem for automatic counting. These latter instruments vary widely in cost, and cost generally correlates with the ability to enhance the image electronically. Allen1 discusses the © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.1 3406 (1963).
17
The British Standard Graticule (Graticules Ltd). From British Standard
commercially available instruments today. Typically, manufacturers are sensitive to cost and offer lines of expensive instruments with full image enhancement capability and color representation, moderately priced instruments for general–purpose analysis and lower-cost instruments for applications where resolution, gray value discrimination, and edge detaching are not a problem. Software packages are also appearing whereby images can be analyzed by a personal computer with software interactive capability on a routine basis. Size analysis by microscopy is performed in our laboratory in less than 2% of size measurement cases, but it is significantly higher when shape information is required.
Separation Methods These methods are typified by procedures which permit the extraction of a certain size class from the feed sample followed by a measurement of the percentage by number or mass of the removed size class relative to the feed concentration. Sieving is perhaps the oldest of these methods, although classification by winnowing was widely used to separate grain in ancient Egypt. Sieving is very well known, but perhaps at the same time the least well known for potential errors in the analysis. Because it appears to be a simple and low-cost process, errors are commonplace. Sieving consists of placing a powder on a surface (e.g., a plate with numerous openings of a fixed size) or a mesh of intersecting wires with numerous openings of a fixed size, and agitating the surface (sieve) so that particles of size smaller than the holes pass through. To speed up the analysis, several sieves are placed on top of each other, the coarsest at the top and the rest shaken until the residue on each sieve contains particles which cannot pass through the lower sieve. This is a well-known © 2006 by Taylor & Francis Group, LLC
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procedure; what is not always appreciated are the errors involved in the process. For the sake of this discussion, only wire-woven sieves will be discussed, as these generally pose the greatest problems and are the most commonly used. Separation of sizes on the sieve depends on the maximum breath and maximum thickness of a particle. Length affects the separation only when it becomes excessive. The sieve surface contains apertures with a range of sizes and a range of shapes. They are classified internationally by their mesh sizes and percentage of open area. The first error arises in the assumption that this classification is accurate for all sieves both new and old. Apertures vary widely and the standard deviation of apertures is smaller (e.g., 3–5% for large aperture sieves of nominal aperture widths of 630 µm and larger, 10% for small aperture widths of 40 µm 3). Leschonski 4 examined eight 50-µm sieves and the median aperture varied from 47.3 to 63.2 µm. Hence, sieve calibration is necessary at frequent intervals of use if size analysis is to be accurate. Several approaches have been used, including microscopy and sieving with standard powders. The latter is more tedious, but for woven sieves, it gives a better measure of three-dimensional aperture sizes and calibrates the sieve in terms of volume diameter using the actual procedure used for size measurement. Leschonski recommends a counting and weighing technique similar to that of Andreasen 5 applied to the fraction of particles passing the sieve just prior to the completion of the analysis. These are usually of narrow size range and are typical of the cut size of the sieve. As sieving is a rate process and the passage of a particle through an aperture a statistical procedure, sieving accuracy (i.e., the time of sieving) depends also on the size distribution spread of sizes, the number and mass of particles on the sieve surface, the particle shape, the method of shaking, and the percentage of open area. Agglomeration and strength of particles affect the outcome. For highly agglomerated dry powders, agglomeration effects can be overcome by wet sieving; hence, wet sieving is recommended often for particle sizes less than 200 mesh. Friability strongly influences the choice of the sieving procedure and, again, wet sieving is less aggressive. Sieve wear with time influences aperture size and requires recalibration at frequent intervals. Sieve cleaning is sometimes the primary source of aperture distortion. Sieving machines are numerous and use different methods of agitating the particles on the surface, thus assisting them to pass through the sieve. A Ro-tap machine with horizontal rotary and vertical tapping motion is aggressive, but for strong particles, it is highly effective. Air-swept sieves (e.g., Hosokawa Alpine and Allan Bradley sonic sifters) can be equally aggressive on the particle, whereas wet sieving is gentler (e.g., Retsch, Hosokawa, Alpine). Automatic versions such as the Gradex Size Analyzer simultaneously reduce analytical times and operator error. The sieving process is simple but probably experiences the most inaccuracy of all sizing methods. Classifications in liquid and air can be used to separate size classes but is not commonly used for size analysis. Online analyzers have been developed using air classification (e.g., the Humboldt size analyzer), but perhaps the more recent applications have been in classification/fractionation methods using hydrodynamic chromatography (HDC) and field flow fractionation (FFF) for colloidal suspensions. Figures 2.2 and 2.3 shows the essential differences between the two. In HDC, colloidal particles in a dilute suspension are separated by injecting them into a nonporous packing of large particles in a laminar flow of clean fluid . Particle separation occurs due to the hydrodynamic interaction of the laminar flow profile and the cross section of the particle. Large particles whose diameters interact with the higher-velocity central stream lines of the parabolic laminar flow profile move faster through the column, whereas small particles whose diameters interact only with the slower profiles near the packing surfaces move slower. Size analysis is obtained by measuring the concentration of particles existing the column with time using an ultraviolet detector. Size resolution and size discrimination are not generally as good as FFF. Here, a similar injection of particles is made into a long channel under laminar flow, but this time a field is applied across the channel. In centrifugal fields typified by the DuPont SFFF, shear-induced hydrodynamic lift forces oppose the driving force of the field. Because of diffusional effects, finer particles diffuse nearer the higher velocity central portion of the laminar flow field and are eluted first, whereas larger particles pushed by the © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.2
FIGURE 2.3
Capillary model of hydrodynamic chromatography separation.
Centrifugal sedimentation field flow fractionation.
centrifugal field nearer to the outer wall and away from the higher-velocity portions of the flow profile are retarded. Again, concentration exiting the channel is measured by an ultraviolet (UV) detector and size distribution calculated. In the DuPont SFFF, a time-delayed exponential centrifugal field is used. Although a constant force field provides high resolution of particles, an exponential force field in which the centrifugal field is allowed to decay exponentially speeds up the analysis, particularly for wide size distributions. Alternate types of fields have been reported in the literature. These include magnetic, thermal, and steric designs. For aerosols, typical devices that fractionate feed distributions in gases include impactors and electrostatic differential mobility analyzers. In impactors, the particles in gas are passed through a nozzle and made to interact normally with a plane surface. The resulting particle trajectories can be calculated using the equation of motion of a particle in a flow field. Above a certain size of particle termed the cut size, particles will strike the surface with the potential for collection. Particles smaller than the cut size will follow the flow of gas, not touch the surface, and so have low potential for collection. The collection efficiency at the impaction surface is expressed in terms of the Stokes number (Stk) where Stk ⫽
rpCc D p2U o
(2.3)
18m(W 12)
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Where pp density of the particle and Dp its diameter, Cc is the Cunningham correction, Uo is the velocity of the gas, µ is the gas viscosity, and W the diameter/width of the nozzle. Impaction surfaces are calibrated by determining the collection effiency as a function of the Stokes number. For a given impactor geometry and operating condition, the Stokes number at the 50% collection efficiency is estimated and used to determine the cut size Dp for those conditions. Thus, by reducing the value of W, the nozzle diameter for the identical gas flow, the impaction velocity. Up increases and the cut size is reduced. In this way, multiple impaction stages can be designed within an instrument to collect a variety of sizes. These instruments are termed cascade impactors. By weighing or otherwise estimating (microscopy) the weight of particles on each surface, a size distribution can be measured. Impactors suffer from errors in that there is some interaction between neighboring surfaces, deposition occurs inside the impactor by other mechanisms (e.g., eddy diffusion, thermal, or electrostatic). More importantly, particles rebound from the collection surfaces and are recollected on later stages. To minimize rebound, surfaces can be made tacky. Agglomeration is another concern in that it broadens the size distribution and the collected mass on any impactor stage. The mass of the agglomerate will determine its collection efficiency, and agglomerates will be collected as if they were large particles. Upon impact however, they may mechanically disperse into their smaller components on the collection surface. The mass loading of any stage influences the collection efficiency and impactors are used with low concentrations of particles in gas and are operated for short sampling times. Solids concentrations and sampling times can be increased by redesign of the collection surface (e.g., in the Lundgren impactor, the collection stage is held on a slowly rotating drum). The lower size of collection can be extended by other designs (i.e., high-flow-rate systems, the use of microjets, and low pressure). Impactors are generally used for particles sizes in excess of 1 µm with cyclone or sedimentation pre-stages for sizes greater than 10 µm. They have found widespread use in health and biological studies, including respiratory/lung deposition studies. Typical of these is the Andersen impactor, Figure. 2.4. For fine particles in gases (e.g., 1µm and smaller), electrical mobility analyzers have become common. Figure 2.5 shows two types of systems in which the velocity Ve of a charged spherical particle in an electric field is given by
Ve ⫽
peCc E ⫽ Be E 3pmD p
(2.4)
where E is the strength of the electric field, e the elementary charge, P the number of elementary charges carried by the particle, Cc the Cunningham correction, µ the gas viscosity, and Dp the particle diameter. Be, the electrical mobility, is inversely proportional to diameter and can be varied by changing the electrical voltage applied to the central collection rod and other parameter values remaining unchanged. Hence, particles are collected on the center rod according to their mobility. Particles larger than a critical size determined by a critical electrical mobility do not reach the rod and pass through the analyzer. Their concentration is detected by either an electrometer or a condensation nuclei counter. Thermosystems manufacture both types of instrument: TSI 3071 is a differential mobility analyzer (DMA) fitted with an impactor to preclassify particles coarser than 1 µm, and a condensation nuclei counter to measure number concentration. Two DMA are sometimes used in tandem to study kinetics and growth, where the first acts as a generation and classification stage to present a narrow size distribution to the second. A conditioner between the DMAs can apply humidity or gases (e.g.. SO3, NH3, etc.) for interaction studies. The TSI 3030 electrical aerosol analyzer uses an electrometer and employs 11 voltage steps which span the range 0.003–1 µm. Automatic and manual modes are available at an aerosol flow rate of 4 L/min. © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.4
Anderson impactor.
Stream Scanning Methods Stream scanning methods are those in which particles are examined one at a time, and their interaction with an external field is taken as a measure of their size. Such methods are usually limited to measurements at low particle concentrations and are most suitable for particle number counting. They have found widespread use in air quality and contamination monitoring. For the determination of mass distributions, many thousands of particles typically have to be counted. They are used for both particles in liquid and particles in gases. The Coulter principle is used for particles in liquids only. Patented in 1949 and published in 1956 for blood cell counting, the principle has found widespread use in all branches of particle science. Figure 2.6 shows the basic principle. Particles are dispersed in an electrolyte and a tube containing an aperture of known size is inserted into the suspension. By placing an electrode both inside and outside the tube and initiating the flow of an electrical current through the orifice, an electric field of known characteristics becomes a sensing zone. When suspension flow is initiated by applying vacuum on the immersed tube, particles in suspension flow singly through the orifice, and momentary changes in impedance give rise to voltage pulses, the heights of which are proportional to particle volume. Theses pulses are amplified, sized, and counted and expressed as a size distribution. Size ranges from 0.6 to 1200 µm are typical, but several apertures/tubes have to be used to achieve this. This is because © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.5
FIGURE 2.6
Electrical mobility analyzer.
Coulter counter.
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each orifice can sense particles from 2% to 60% of the orifice diameter. Problems can arise due to the necessity to use and match multiple aperture tubes, due to aperture blockage by large particles, by the passage of more than one particle through the sensing zone at a time (coincidence), by the nondetection of particles smaller than the lower detection limit of an aperture, and by the evolution of erroneous pulses due to abnormal interactions with the electric field. The first problem is easily overcome by trial and error and by calibration. The second is simply overcome by preclassifying the sample. Coincidence errors are routinely overcome by the application of simple formulas supplied by the manufacturer. Abnormal pulses arise due to the fact that the electric field is not uniform and that the field is dense near the inlet and outlet edges of the orifice. Passage of particles through these regions or interaction of extreme particle shapes with these regions can generate nonuniform electrical pulses. These can be eliminated by the use of mechanical focusing of particles through the orifice center or by electronic editing of abnormal pulses by the electronic data analyzer. The latter is more common today. The nondetection of particles due to each orifice having a lower detection limit of 2% of its diameter is overcome by the performance of a mass balance. This is discussed at length by Allen1. Accurate particle size analysis is best performed on dilute suspensions in which particles pass through the center of the orifice and mass balances are performed. In addition to Coulter, the principle is also applied in the Elzone electrozone counter manufactured by Particle Data. One special use of the Coulter principle is for the determination of oversize counts in coating slurries. As there are very few, a precision transparent sieve with a ±2% tolerance is used in combination with a Coulter mass balance procedure to prefill the few oversize particles and examine them by microscopy to determine their nature, count them by, size and recombine the data into the total mass balance. Allen describes this procedure on precision sieves manufactured by Collimated Holes Inc. The primary value of the Coulter principle is that it can be used to measure both mass and population distributions accurately, and as volume diameter is one typical size representation, this agrees closely with most sedimentation analysis and with sieve analyses when calibrated, as discussed earlier in this section. Of more widespread use are optical sensing zones which operate according to the following principles: 1. Collecting and measuring the light intensity in a forward direction 2. Collecting and measuring the light intensity in other directions (e.g., 90° or backscatter) 3. Light blockage or geometric shadowing 4. The measurement of phase shift The most common types of instruments fall under class 1. They are available in various degrees of sophistication and wide variations in design. Particle size response is a function of the size, the shape, the orientation of the particle, the flow rate, and the relative refractive index between the particle and its surroundings. Instrumentation cost and lower measurement limit are determined by the light source, the light collection system, and the efficiency of the detector. White-light sources are cheaper but give rise to a lower detection limit of 0.5 µm. Extension below this limit demands the use of lasers or laser diodes and highly efficient light collection systems. It is a balance among the intensity of the light source, the particle response, and the collection optics. The relationship between particle size and scattered light intensity at any angle can be obtained for spheres with Mie theory. For a monotonic increase in scattering cross section with size, forward scattering at small angles is used. This has been dependence on particle refractive index, but extraneous light scattered from instrument internals is more of a problem than with
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large-angle scattering. This is why class 1. instruments are numerous and class 2. not. Typical manufacturers include HIAC-Royco, Particle pleasuring Systems, Climet, Kratel, and Polytec among many others. Each manufacturer has a relatively broad choice of instruments for gas or liquid applications. These are discussed more extensively by Allen1. Geometric shadowing or light blockage is typified by the HIAC-Royco light-blocking principle. When particles in liquid pass singly through the sensing zone, an amount of light equivalent to the effective cross section/shadow of the particle is cut from the incident beam. Sensors are available for the detection of sizes 1 µm to 3mm. They are used particularly for sizes larger than 1 µm, and because they are less sensitive to refractive index and to the nature of the surrounding liquid, they have found use in the contamination monitoring of hydraulic oils. Geometric shadowing has also been use d to measure particles in molten polymer. The flowvision analyzer (Figure.2.7) uses fiber optics both to pass light across and detect particles in pipelines containing polymer. As particles pass through the beam, strobe-actuated cross sections of the particles are analyzed by image analysis. Particles of size 2–1000µm are typically counted and Figure 2.8 shows a series of gel events in a polymer line monitered on-line by the device6. The Partec 200 (Figure 2.9) offers on-line measurement capabilities at higher concentration in liquids, typically 5–30% volume percent, and so finds applications in the measurement and control of crystallizers, precipitators, and reactors. A light source is focused as a spot in the dispersion by a scanning focusing lens. This spot moves in a circular path. Light intersecting a particle as a chord is backscattered via the identical optics used to focus the light, and the chord length is recorded and scaled. Software to transform chord length distribution into particle diameter is used for some applications. Figure 2.10 shows typical results of a heating and cooling cycle on an organic system. Clearly, nucleation and dissolution are followed as a function of temperature.
Field Scanning Field scanning methods measure the interaction of an assembly of particles and interpret the signal in terms of the size distribution of the assembly. Low-angle laser light-scattering instruments (LALLS) collect light scattered from particles in a collimated laser beam by an array of detectors in the focal plane of the collecting lens
FIGURE 2.7
Flowvision analyzer.
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FIGURE 2.8
25
Output of flow vision.
FIGURE 2.9
Partec 200.
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FIGURE 2.10 Typical output of Partec 200 (heating and cooling cycle on a crystal system).
(Figure 2.11). The light-intensity distribution for a single opaque spherical particle as given by the Fraunhofer equation falls off rapidly as particle size is reduced. This results in a diffraction pattern of circular light and dark rings, the intensity due to coarse particles being near the center and fine particles at the periphery. When an array of sizes are illuminated, a similar pattern emerges, but particles contribute to the intensities of more than one ring; hence, the light-scattering pattern is a matrix which has to be deconvoluted in order to determine the size distribution from the scatter pattern measurement. Many commercial instruments are available that use this principle. They are manufactured by Leeds & Northrup, Cilas, Coulter, Seishin, Shimadzu, Sympatec, Malvern, Fritsch, Insitec, and Horiba & Nitto. Instruments differ by the way they design the sensor/array to measure the pattern of their scattering algorithm, their mathematical deconvolution routine, their method of extending the lower limit of 0.1 µm, and their method of extending the upper limit to 3000 µm. A full discussion of these commercial instruments, their similarities and their differences are given by Allen1. The devices are limited to a few percent mass concentration and have found extensive use in off-line and on-line applications. The instruments are easy to operate and yield highly reproducible data. However, their general tendency is to oversize the coarse end of a distribution and assign an excess of particles to the fine end of the distribution, thus broadening the size distribution. For online applications in dense slurries, they require a dilution pre-analysis step, although multiple scattering studies have been made but not widely applied. They are calibrated using standard powders or by photomask reticules7.
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FIGURE 2.11 Low-angle laser light-scattering instrument.
Ultrasonic attenuation systems measure the attenuation of a range of ultrasonic frequencies passed through a concentrated dispersion of particles in liquid. A series of relationships among particle size, mass concentration, and wavelength are obtained, which can be deconvoluted to produce a size distribution. A single-point sizing device manufactured by ARMCO-Autometrics found widespread use in the mining industry in the 1970s and 1980s, whereas later developments by Herbst and Alba8 developed a multipoint sizing option. This was extended to develop the Alba-DuPont UltraspecTM technology. This was embodied in an instrument to measure the size distribution of TiO2 slurries at 70% by weight.9 Reibel and Loeffler10 reported on an on-line ultrasonic device for particle size distribution that is now available from Sympatec. Figure 2.12 shows the concept. Data on narrow and broad size ranges of glass beads showed good correlation with LALLS for particles greater than 15 µm at volume concentration up to 10%. Pendse and Sharma11 developed a laboratory acoustic analyzer for dense slurries up to 70% by weight. This is marketed by Pen-Kem. Acoustic sensing is becoming more widely used due to the simplicity of the sensor/sample configuration and for its lower sensitivity to volume concentration. The systems are compact and robust, needing no optical benches as with LALLS. Gas bubbles in the suspension create problems however. For very small particles in liquids, photon correlation spectroscopy is used. Figure 2.13 shows the configuration in which a laser beam is passed into a suspension under Brownian motion and the scattered light fluctuations measured at 90° to the incident beam. The autocorrelation function of the scattered light fluctuations is calculated and related to the diffusion coefficient. This then yields an average size together with a polydispersity factor—some measure of the spread of sizes around the average value. Commercial instruments are available from Amtec. Wyatt, Malvern, NiComp, Brookhaven, Coulter, and Munhall. Different designs, some using fiber optics and some multiangle systems, are available. These are discussed more fully by Allen.
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FIGURE 2.12 On-line ultrasonic device.
FIGURE 2.13 Photon correlation spectroscopy.
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Sedimentation Methods Sedimentation methods are based on the settling of a single sphere under gravitational or centrifugal fields of infinite extent. Sedimentation techniques have been classified by Allen according to the following principles: Class
Suspension Type
Measurement Principle
Force Field
1
Homogeneous
Incremental
Gravitational
2
Homogeneous
Incremental
Centrifugal
3
Line start
Incremental
Centrifugal
4
Homogeneous
Cumulative
Gravitational
5
Line Start
Cumulative
Gravitational
6
Homogeneous
Cumulative
Centrifugal
7
Line Start
Incremental
Gravitational
8
Line Start
Cumulative
Centrifugal
Most modern instruments in use today fall into classes 1 and 2. Only examples from these will be discussed here. For all others, see the work by Allen1. In the homogeneous incremental gravitational class of methods, the solids concentration is measured at a known depth below the surface of an initial homogeneous suspension. The correlation will remain constant until the largest particle present has fallen below the measurement zone, when the particle concentration begins to fall. In consequence, the concentration will be that of particles smaller than the Stokes diameter, and a plot of concentration against the Stokes diameter will be the mass undersize distribution. In class 2, the same principles apply, except that the particles now fall in radial paths, hence, the number of particles entering the measuring zone is less than the number leaving, so the concentration of these particles is less than the initial concentration. Corrections have to be applied for the following: • Reynolds number, if that of the particle exceeds 0.2 • Radial dilution • Extinction coefficient, if optical incremental sensors are employed An example from class1 is the Micromeretics Sedigraph 5000 and 5100 (Figure 2.14) in which the measuring principle is x-ray attenuation, which is directly proportional to the atomic mass of the suspended particles (the mass undersize). In this unit, the measuring zone is changed with time by permitting the measurement cell to scan through the x-ray beam, resulting in smaller and smaller sedimentation heights. In this way, analytical measurement is speeded up, and size measurement as low as 0.2 µm is routinely recorded. For other instruments in this class, see the work by Allen1. Examples from class 2 include the following: • The DuPont/Brookhaven Scanning X-ray Disc Centrifugal Sedimentometer (Figure 2.15). which is used for the accurate measurement of particles less than 0.5 µm. In this device, the suspension is measured in a hollow x-ray transparent disk which generally contains 20 ml of suspension at 0.2% by volume. Speed ranges from 750 to 6000 rpm. The source and detector remain stationery for a preset time and then scan toward the surface. The commercial version (i.e., BI-XDC) has a stationary measurement disk and a scanning x-ray source and detector module. Total analysis time is under 8 min for most inorganic pigments. • The Horiba Cuvet Photocentrifuge is typical of several Japanese-manufactured instruments. Here, the disk is replaced with a rectangular spectrophotometric cell containing a homogeneous suspension. Extinction coefficient and radial dilution corrections © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.14 Sedigraph.
FIGURE 2.15 Scanning X ray disk centrifugal sedimentometer (DuPont/Brookhaven).
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FIGURE 2.16 Centrifugal sedimentometer (Horiba).
must be made. Figure 2.16 shows the Horiba configuration. Clearly, smaller amounts of sample can be used, and as Allen1 shows, a wide range of operating conditions are available by choice. Sedimentation methods measure the Stokes diameter which for many particle systems closely resembles the volume diameter of the distribution. Hence, sedimentation and Coulter methods often agree closely, except for extreme shapes.
Surface Methods For completeness, surface methods can be used to measure the average size of a powder. Generally, this is done dry and by two approaches. For dense, smooth particles of insignificant internal porosity, permeability methods give average specific surface diameters comparable to other methods, providing the lowest size present is greater than 2 µm. Allen1 describes these. Gas adsorption methods, which measure the internal and external surface area, have been used but significantly undersize powders with internal surface area. These methods are rarely used today.
1.2.4 SUMMARY This chapter has outlined the more prevalent methods of size analysis in the 1990s. It does not provide detail of each method or the range of choices that are available even with one manufacturer’s instrument. Today, most instruments are modular and have options that can be selected for specific needs. Allen provides a detailed list of manufacturers, and they should be contacted in order to understand their options and thereby provide a basis for instrument selection and purchase.
REFERENCES 1. Allen, T., Particle Size Measurement, 4th ed., Chapman & Hall, New York, 1990. 2. British Standard 3406, Methods for PSD of Powders, Part 4., Optical Methods, 1963. 3. Ilantzis, M. A., Ann. Inst. Tech. Taim. Trav. Publ., 14 (161):165 and 484, 1961. © 2006 by Taylor & Francis Group, LLC
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Powder Technology Handbook 4. Leschonski, K. in Proc. Particle Size Analysis Conference, M. J. Groves and J. L. Wyatt-Sargent. eds., Society Analytical Chemistry, 165, 1970. 5. Andreasen, A. H. M., Sprechsaal, 60:515, 1927. 6. Joche, E. C., Proc. Control of Particulate Processes, 1995. 7. Hirrleman et al., PARTEC 5th European Symp., Part. Characterization, pp.412, 655–671, 1992. 8. Herbst, J. A., and Alba, F., Particle and Multiple Processes, Volume 3, Colloidal and Interfacial Phenomenon, pp. 297–311, 1989. 9. Alba, F., U.S. Patent 5.121, 629, 416,1992. 10. Riebel, U. and Loeffler, F., Eur. Symp. Particle Characterization, p. 416, 1989. 11. Pendse, H. P., Sharma A., Proc. PTF Forum, pp. 136–147, 1994.
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1.3
Particle Shape Characterization Shigehisa Endoh National Institute of Advanced Industrial Science and Technology, Tsukuba, Japan
1.3.1
INTRODUCTION
Particles have various shapes depending on their manufacturing method and mechanical properties. Unlike a sphere or a rectangular parallelepiped, which are objects with clear geometrical definitions, powder particles are very complicated objects with no definite form. It is generally very difficult to describe their shapes. They have therefore been conventionally described and classified by the use of various terms. On the other hand, along with recent rapid advances in computer and information technologies, image processing technology has remarkably progressed in both software and hardware, making it easy to obtain image information and geometrical features of particles. Accordingly, various quantitative methods for expressing particle shape have been proposed and adopted. The particle shape description or expression is classified by several criteria. In the most primitive manner, the descriptions are classified into quantitative methods and qualitative ones. In qualitative description, the shape is expressed by several terms such as “spherical,” “granular,” “blocky,” “flaky,” “platy,” “prismodal,” “rodlike,” “acicular,” “fibrous,” “irregular,” and so on. 1,2 The verbal description is sometimes convenient to express irregular shape and makes it easy to understand the shape visually. But how are a platy shape and a flaky shape distinguished? Under present conditions, this distinction must depend on human visual judgments in many cases. Therefore, quantitative descriptions of particle shape will be necessary. Quantitative shape descriptors can be calculated from two- or three-dimensional (2D or 3D) geometrical properties and can be calculated by comparing with physical properties of the reference shape. The following are required for the shape descriptor3: • • • • • •
Rotation invariance: values of the descriptor should be the same in any orientation. Scale invariance: values of the descriptor should be the same for identical shapes of different size. Reflection invariance. Independence: if the elements of the descriptors are independent, some can be discarded without the need to recalculate the others. Uniqueness: one shape always should produce the same set of descriptors, and one set of descriptors should describe only one shape. Parsimony: it is desirable that the descriptors are thrifty in the number of terms used to describe a shape.
The form of a particle is essentially a 3D property, but the shape of 2D images is mainly treated in this paper.
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1.3.2
REPRESENTATIVE SIZE
If a given particle has the k-dimensional geometric property quantity Qk , perimeter P for onedimension, surface area S and projected area A for two-dimensions, volume V for three-dimensions, the equivalent diameter x of a sphere having the same quantity can generally be expressed by the following equation:
(
)
x ⫽ Qk f
k
(3.1)
where the coefficient fb is called the geometrical shape factor. From the mathematical relations for a sphere, the equivalent diameters and the factors are given as follows2: •
•
•
volume equivalent diameter: xV ⫽ (6V / p )1 / 3 , f V ⫽ p / 6
(3.2)
xS ⫽ S / p , fS ⫽ p
(3.3)
surface equivalent diameter:
projected are equivalent diameter (Heywood diameter): xA ⫽ 4 A / p , fA ⫽ p 4
•
(3.4)
perimeter equivalent diameter: xP ⫽ P / p
(3.5)
When particles have similar shapes, the averages of volume and surface area are given by the following expressions, respectively: V ⫽ f  M3,0
(3.6)
S ⫽ f␥ M 2,0
(3.7)
where Mk, 0 is the complete kth moment of number density distribution q0(x) of x defined in Equation 8: M k ,0 ⫽ ∫ xk q0 ( x )dx
(3.8)
Accordingly, a volume-related specific surface is as follows: SV ⫽
S V
⫽
f␥ M 2,0 f M3,0
⫽
f23 M1,2
(3.9)
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where f23 is a specific surface area shape factor given by the ratio between fγ and f. M1,2, which is a ratio of M3,0 to M2,0, is called the Sauter diameter –x vs(⫽–x 1,2) (an average weighted surface diameter):
(
( )
)
xVS ⫽ M1,2 ⫽ M3,0 M 2,0 ⫽ ∫ x q2 x dx
(3.10)
where q2(x) is area density distribution. The specific surface area shape factor of spheres is given as f23 ⫽ 6 and the specific surface area is SV ⫽ 6/x. By applying a correction coefficient fc, that is, Carman’s shape factor, to a general particle, SV ⫽ 6/fcx is given. A 2D image of a particle gives the Feret diameter x F , which is a distance between parallel tangents as shown in Figure 3.1. Since the Feret diameter changes with the angle of the tangents xF(u), that is, this means statistical diameter, the average value –x F can be employed as a representative diameter, given by p
1 xF (u)du p ∫0
xF ⫽
(3.11)
The maximum and minimum values of xF give the “length” and “breadth,” respectively. The unrolled diameter, xR(u), is also defined as a diameter passing through the center of gravity of the image, as shown in Figure 3.1. The average is given by p
1 xR (u)du p ∫0
xR ⫽
1.3.3
(3.12)
GEOMETRICAL SHAPE DESCRIPTORS
Quantitative descriptors for the geometry of a particle image can be also classified into three categories based on the scale of the inspection of form: 1. Macroscopic descriptor, expressing overall form and referring to “proportion” or “elongation” of the particle image. The Fourier descriptors of lower-order harmonic number also give overall form features.
Df
)
(Q
)
Da
(Q
Q
Q)
r( C. G.
F (s)
Po s
FIGURE 3.1 function.
Statistical diameter and angular
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2. Mesoscopic descriptor, expressing details of form and profile outline and defining space filling feature (or bulkiness), concavity, and robustness of the particle. 3. Microscopic descriptor, expressing surface structure and texture, giving roundness and fractal dimension of the image. The Fourier descriptors of higher order also express surface properties.
Macroscopic Descriptors The definition and classification of shape descriptors expressing macroscopic shape properties are calculated from macroscopic geometrical features such as representative diameter, axis lengths, thickness, and so on.
Proportion This parameter describes the anisotropy or elongation of a particle. It is given by the ratio of lengths of axes (length L, breadth B, thickness T ): elongation ⫽ L/B or ⫽ L⬘/B⬘
(3.13)
flatness ⫽ B/T or L/T
(3.14)
where L and B⬘ are maximum and minimum Feret diameters, respectively, and B and L⬘ are Feret diameters perpendicular to L and B⬘, respectively.4 The axial ratio of the radius equivalent ellipse of the image that has the same geometrical moments up to the second order as the original particle silhouette (Figure 3.2) is referred to as “anisometry” and also gives the proportional property4 K ⫽ I1 / I 2 ⫽ a / b (I1 ⬎ I 2 )
(3.15)
where I1 and I2 are the major and minor principle inertia moments of the particle silhouette, and the axis lengths are a ⫽ I1 / A and b ⫽ I 2 / A , respectively.
FIGURE 3.2 Radius equivalent inertia ellipse and its principle axis length. © 2006 by Taylor & Francis Group, LLC
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Index Given by the Ratio of Characteristic Particle Diameters Circularity cAP is defined by the ratio of the perimeter PC of a circle with the same projected area and that of the particle P, and gives the degree of similarity to a circle. The circularity is also given the ratio of the Heywood diameter and the perimeter equivalent diameter, as follows: cAP ⫽ PC ⲐP ⫽ xH ⲐxP
(3.16)
For 3D inspection, Wadell’s sphericity cS defined by the ratio of the surface area SS of a sphere with the same volume and that of the particle S gives the degree of similarity to a sphere, as follows5: cS ⫽ SS / S
(3.17)
cS is approximately obtained by the ratio xA/xmin where xmin is the diameter of circumscribed minimum circle as a projected image (Figure 3.3). The several kinds of shape descriptors given by the ratios of characteristic particle diameters are6 cAF ⫽ xA Ⲑ xF , cRA ⫽ xR ⲐxA , xRP ⫽ xR ⲐxP ,…
(3.18)
The coefficient of variation of the statistical diameter is thought of as a shape descriptor giving deviation from a circle with mean size, defined as si ⫽
1 xi
p
1 2 xi (u) ⫺ xi ) du , for i ⫽ F or R ( ∫ p0
(3.19)
This descriptor is related to the circularity.
FIGURE 3.3 Circumscribed and inscribed circles and their diameters. © 2006 by Taylor & Francis Group, LLC
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Mesoscopic Descriptors Space Filling Factor This factor is called “bulkiness,” “bulkiness factor,” and so on and expresses the bulky space occupied by a particle. The descriptor also indicates the degree of compactness. Bulkiness is defined as a ratio between the area of a circumscribed rectangle or circumscribed circle of the image and that of the particle (Figure 3.3), as follows7: fbR ⫽ A ⲐLB,
(3.20)
2 fbC ⫽ 4 A Ⲑpxmin
(3.21)
The bulkiness corresponds to the solidity defined as an area ratio between the circumscribed convex hull Ah and the image, as follows8: Solidity ⫽ A ⲐAh
(3.22)
Also, the projected area relative to the area of the equivalent ellipse of inertia can be defined as the space filling factor, as follows 9 : fBE ⫽ A Ⲑpab
(3.23)
where a and b are the major and minor radii of ellipse of gyration, respectively. In 3D inspection, a circumscribed polyhedron (rectangular prism) or sphere is employed and the factor is given by the ratio between the volumes. Irregularity index xmax/xmin, defined by the ratio of diameters of the inscribed maximum circle to that of the circumscribed minimum circle, gives the bulky property, as shown in Figure 3.3. For an elongated particle, “curl” is defined by the following equation10: Curl ⫽ L (⫽ xF max ) ⲐLG ,
(3.24)
where LG is the geodesic length of a fibrous particle, as shown in Figure 3.4.
FIGURE 3.4
Polygonal convex hull.
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Concavity and Robustness This descriptor is based on mathematical relation. The descriptor cFP is defined by the ratio of the average Feret diameter and equivalent perimeter one: cFP ⫽ xF Ⲑxp
(3.25)
cFP for a shape without any convex portions on the surface is unity, thus the descriptor cFP can be used to assess the surface concavity.7 The descriptors indicating the degree of convexity or concavity on the surface of the image are proposed as follows: Convexity ⫽ P ⲐPh ,
(3.26)
Concavity ⫽ (Ah ⫺ A)ⲐAh .
(3.27)
where Ph is the perimeter of the convex circumscribed polygon.9 The image processing technique gives the morphological descriptors. v1 and v2 are, respectively, the numbers of erosions (shrinking) in image processing necessary to eliminate completely the image and its residual set with respect to the convex hull of area Ah (Figure 3.5). The robustness factor is V 1 ⫽ 2v1 Ⲑ A
(3.28)
V 2 ⫽ 2 v2 / A
(3.29)
The largest concavity is
Robustness is correlated to the elongation. The plot of V1 versus V2 is used for shape screening.10,11
FIGURE 3.5 particle.
Elongated concave
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Surface Roughness Hausner’s surface index is given by a reciprocal of circularity cAP8: 2 C ⫽ 1 / c AP ⫽ P 2 / 4p A
(3.30)
If the convex portion of the silhouette outline of particle is an arc, roundness is defined by the ratio of the average value of the curvature radii of the corners and edges to the diameter of the maximum inscribed circle Rmax (Figure 3.6)6: Rn ⫽
∑r
i
NRmax
(3.31)
where ρi is the curvature radius of the corner i and N is the number of corners.
Application of Fourier Analysis: The Fourier Descriptor The morphology of a particle can be expressed by the Fourier descriptor. The Fourier descriptor is a set of Fourier coefficient Ak (⫽ 2|Ck|), or pairs of the amplitude Ak and phase angle αk of the Fourier coefficient that can be obtained by the Fourier transformation of the projected image function F(Q):
Ck ⫽
2p ak ⫺ jbk 1 ⫽ F (Q)exp( ⫺jkQ)dQ, Ak ⫽ ak2 ⫹ bk2 2 2p ∫0
ak ⫽ tan⫺1 (bk / ak )
(3.31)
(3.32)
for the harmonic order k ⫽ 0, ⫾1, ⫾2,…., j ⫽ ⫺1.
FIGURE 3.6
Surface structure and roundness.
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The composite values of the Fourier coefficients and their statistical values are referred to as the Fourier descriptor, also. The values of Ak for lower harmonic orders give the macroscopic form feature,12–22 that is, A2 corresponds to the elongation (i.e., aspect ratio), and A3 and A4 mean triangularity and squareness, respectively. The Fourier descriptor is distinguished from other shape descriptors in its representation of the measured image. The representation of original form F(Q) can be guaranteed within the number of coordinate points N measured or interpolated for the Fourier transformation, as follows: F (⌰) ⫽
a0 N / 2⫺1 ⫹ ∑ Ak cos(k⌰ ⫺ ak ) 2 k⫽1
(3.33)
The following methods for various image function of outline are proposed (Figure 3.1 and Figure 3.7).
FIGURE 3.7
Angular function.
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The Radial Fourier Descriptor: Polar Coordinate Method In the polar coordinate method, the radius vector r(u) from the center of gravity to the boundary of the image as a function of an angle u of counterclockwise rotation about the center of gravity is transformed into the Fourier space (Figure 3.1).12–15 The radial length r(u)(⫽|r(u)|) is given by the Fourier series, as follows: r (u) ⫽ r0 ⫹
N / 2⫺1
∑
Ak cos(ku ⫺ ak )
(3.34)
k⫽1
r0 ⫽
1 2p
2p
∫ r (u)du
(3.35)
0
Some of the geometrical quantities, such as area and mean unrolled diameter, are given by the Fourier coefficient in polar coordinates. The mean radial length is r 0(⫽x—R /2), given by Equation 3.35, and the area A is given by N / 2⫺1 ⎤ ⎡ A ⫽ p ⎢r02 ⫹ ∑ Ak2 ⎥ k⫽1 ⎦ ⎣
(3.36)
The Fourier descriptors of the polar coordinate method can be defined as follows: IRR ⫽ (DA / 2r0 )2 ⫽ cRA⫺2
(3.37)
Randance ⫽ m2 /r 0 2 ⫽ ∑ Ak *2
(3.38)
Skewness ⫽ m3 /r0 3
(3.39)
where Ak* is the normalized amplitude Ak/A0, and mm is the mth moment around the mean radius r0, as follows: mm ⫽ ∫ [r (u) ⫺ r0 ] du/2p m
(3.40)
IRR corresponds to the equivalent dimensionless area diameter and is a measure for irregularity. Dimensionless moments m2/r02 and m3/r03 are called randance and skewness. This method, however, cannot be applied unless the radius vector r(u) becomes a single-valued function of u (Figure 3.7a). The Angular Bend Function Method (f-l Method): ZR Fourier Descriptor The ZR Fourier descriptors are derived by the Fourier transformation of the angular function f(s), which is the direction change (angle of deviation) of the outline tangent on the arc length s from the starting point P0 (Figure 3.7).16–19 The normalized angular function f *(t) is meaningful in comparison between different size particles by defining ,
f* (t ) ⫽ f ( s ) ⫺ t ⫽ f ( Pt / 2p ) ⫺ t
(3.41)
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where t is a normalized argument with the perimeter P, as follows: t ⫽ 2π(s/P)
(3.42)
f*(t) is subjected to the Fourier transformation: ⬁
f* (t ) ⫽ m0 ⫹ ∑ [ ak cos(kt ) ⫹ bk sin(kt )]
(3.43)
k⫽1
The profile of the particle image can be approximated by a polygon (Figure 3.7b), then the Fourier descriptors are derived: m0 ⫽ p ⫹
1 m ∑ s j ⌬fj P j⫽1
(3.44)
ak ⫽
1 m ∑ ⌬fj cos(kt j ) kp j⫽1
(3.45)
bk ⫽
1 m ∑ ⌬fj sin(kt j ) kp j⫽1
(3.46)
where sj is the arc length of jth vertex from the starting point, Δfj is the change of slope at vertex j, and m is the total number of sides of the polygon. The angular function method can be applied to every shape, even if the radial function r(u) is not unique with u. In some cases, however, the starting point and the endpoint of the represented profile do not coincide. 18 The Radius Vector Method In the case where r(u) does not become a single-valued function, the modified function by considering the rotational direction of radius vector is introduced and subjected to the Fourier transformation.20 The modified radial function is defined as R(u⬘) ⫽ r (u⬘)fD(u⬘)
(3.47)
where fD(u⬘) indicates the rotational direction of the radius vector. The function gives fD(u⬘) ⫽ ⫹1 for the counterclockwise rotational direction and fD(u⬘) ⫽ –1 for the clockwise rotational direction, respectively. u⬘ is a cumulative value of angle variations at u from the starting point, such as u⬘ ⫽ ⌺ⱍΔuⱍ. The (x, y) coordinates can be transformed into the Fourier spectrum.21,22
Fractal Dimensions Fractal dimensions 23 are applied to the quantification of particle shape. 24 Generally, the fractal dimension is defined as follows. The geometrical quantity Q k of a k-dimensional pattern (topological dimension k) measured by a certain scale h is proportional to the number of elements for k-dimensional volume N. Qk ⫽ ηΝ
(3.48)
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If the scale h and N is given by the following relation: N ⫽ h⫺Df , Df ⫽ k
(3.49)
the pattern is fractal, which means statistical self similarity, and Df is called a fractal dimension. As shown in Figure 3.8, let us follow the profile by a divider (linear element) of constant width h, and plot the perimeter estimated by Equation 3.48 or number of linear elements N plotted against h on an amphi-logarithmic sheet. The relation shown in Figure 3.9 is generally obtained with changing h. This relation is expressed in Equation 3.49, and the fractal dimension Df is obtained by its slope: log(P) ⫽ (1 – Df)log h ⫹ c
(3.50)
FIGURE 3.8 Estimation of perimeter with changing liner element length h. © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.9
Log N versus log h plot.
With the influence of the concavity and convexity on the surface, D f for the perimeter of the outline (k ⫽ 1) becomes 1 ⬍ Df ⬍ 2. The Df value of a particle with a smooth surface is close to unity, whereas the fractal dimension is higher for complicated curves.10,26,27 Therefore, it is believed that the fractal dimension quantitatively expresses the complexity of the surface and structure of the surface. Usually, the logN-logh plot for the profile with a fractal structure shows the relation as shown in Figure 3.8, that is, the linear relation divided into two regions or more. The region of scale larger than some break point refers to the macroscopic morphology, on the other hand, the smaller region corresponds to the microscopic surface properties, such as surface texture. When a 2D pattern of a binary image (k ⫽ 2) seems to be fractal, the correlation function C(r) of density between the patterns with a distance r satisfies the following relation, 28,29 C (r ) ⫽⬍ f (r ⫹ r⬘) ⋅ f (r⬘) ⬎ ∞ r − b
(3.51)
where ⬍⬎ expresses an average operation and b is the slope of the logC(r)-logr plot. The fractal dimension is given by the following expression: Df ⫽ k ⫺ b
(3.52)
where f (r⬘) is an image function at a point r⬘, which is 0 or 1.
1.3.4
DYNAMIC EQUIVALENT SHAPE
Drag Force Shape Factor Stokes’ fluid drag force FD, which acts on an irregular-shaped particle moving at a relative velocity v in a fluid with a viscosity m, is given by the following expression when the characteristic particle diameter is x: FD ⫽ 3pmvx K
(3.53)
where the coefficient K is called the drag force shape factor. The FD value of an irregular-shaped particle varies depending on the orientation to fluid flow, and thus the drag shape factor K depends on the orientation of the particle. Also, the value of K depends on the characteristic particle diameter employed in the equation. The value for equivalent volume diameter xV differs from that for the equivalent surface diameter xS, respectively. © 2006 by Taylor & Francis Group, LLC
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Stokes diameter xst is generally given using the volume shape factor as follows: xst2 ⫽
6 f3, b 2 xb p Kb
(3.54)
If the drag shape factors with the axes of an ellipsoid orthogonal and parallel to the flow are Ka and Kc, respectively, the factor for random orientation can be given by the following equation30: 3 / K b ⫽ (1 / K ba ⫹ 2 / K bc )
.
(3.55)
Dynamic Shape Factor The dynamic shape factor κ is defined as follows31: k⫽
fluid drag force acting on a particle drag force to a spheere with equal volume
(3.56)
and is equal to the drag force shape factor of the equivalent volume diameter KV. It is also given by the ratio of the volume equivalent diameter and the Stokes diameter, as follows: k ⫽ ( xV / xst )2
(3.57)
Also, according to (fV /k)x2V ⫽ (fA/KA)x2A, the following relation is obtained: 2
k⫽
⎛ p ⎞ p ⎛ xV ⎞ KA ⫽ ⎜ ⎜ ⎟ 6fA ⎝ xA ⎠ ⎝ 6fA ⎟⎠
1/ 3
KA
(3.58)
κ of a spheroid is shown in Figure 3.10.2,32
FIGURE 3.10 Dynamic shape factor of spheroid. © 2006 by Taylor & Francis Group, LLC
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The Wadell sphericity is related with the dynamic shape factor in the following expression: cS ⫽ (K A / k)2 ⫽ (6fA / p )2 / 3
(3.59)
The dynamic shape factor of agglomerates with porosity is given by the volume shape factor fV,agst based on the Stokes diameter of aggregate xagst , as follows33: k⫽
(xV / xagst )2 (1 ⫺ )
⎛ 6⎞ ⫽⎜ ⎟ ⎝ p⎠
2/3
2 fV,agst
(1 ⫺ )
(3.60)
For the blocky and chainlike aggregate composed by n particles, empirical relationships for κ are obtained, as follows34: k ≈ 1.232, (blocky, 5 ⱕ n ⱕ 23), k ≈ 0.863n1/3 , (chainlike, n ⱕ 8)
1.3.5
(3.61)
CONCLUDING REMARKS
The shape descriptors presented here are mainly classified into two categories based on the scale of inspection. The descriptors derived from geometric quantities represent overall or mesoscopic features of shape. Those descriptors convenient to treat the shapes, however, are not unique with the shapes. This means that the form cannot be reconstructed using the overall or mesoscopic descriptor. On the other hand, harmonic analysis, such as the Fourier transformation, for several images of a particle is useful. The harmonic spectrum extracted from the image function describes not only the overall shape feature but the microscopic shape. Further, the shape reconstruction can be done using the Fourier coefficients. However, the Fourier descriptor includes too much information on the shape to select appropriate parameters for characterization or distinction of particles. In this paper, classification and distinction of particles based on shape have not been treated, but these are significant in practical powder processing. The statistical methods and artificial neural networks are effective tools for particle characterization.35,36,19 The form of a particle is essentially a 3D property. A few descriptors for 3D shape have been proposed and used in restricted fields. The 3D morphological analysis including measuring techniques in three dimensions should be developed.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
Allen, T., in Particle Measurement, Chapman and Hall, London, 1981, p. 107. Endoh, S., J. Powder Technol. Japan, 29, 854, 1992; 35, 383, 1998. Chan, L. C. L. and Page, N. W., Part. Part. Syst. Charact., 14, 67, 1997. Heywood, H., Trans. Inst. Chem. Eng., 25S, 14, 1947. Wadell, H., J. Geo., 40, 443, 1932; 43, 250, 1935; J. Franklin Inst., 217, 459, 1934. Tsubaki, J. and Jimbo, G., Powder Technol., 22, 161, 1979. Hausner, H. H., Planseeber. Pulvermetall., 14, 75, 1966. Faris, N., Pons, M. N., et al., Powder Technol., 133, 54, 2003. Medaria, A. I., Powder Technol., 4, 117, 1970/71. Pons, M. N., Vivier, H., et al., Powder Technol., 103, 44, 1999. Vocak, M., Pons, M. N., et al., Powder Technol., 97, 1, 1998. Schwarcz, H. P. and Shane, K. C., Sedimentology, 13, 213, 1969,
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Meloy, T. P., Powder Technol., 16, 233, 1977. Luerkins, D. W. et al., Powder Technol., 31, 209, 1982. Beddow, J. K. et al., Powder Technol., 18, 19, 1977. Zahn, C. T. et al., IEEE Trans. Compt., C-21, 269, 1972. Fong, S. T. et al., Powder Technol., 22, 17, 1979. Koga, J. et al., Report IPCR, 56, 63, 1980. Hundal, H. S. et al., Powder Technol., 91, 217, 1997. Gotoh, K., Powder Technol., 23, 131, 1979. Shibata, T. et al., J. Powder Technol. Jpn., 24, 217, 1987. Browman, E. T. et al., Geotechnique, 51, 545, 2001. Mandelbrot, B. B., Fractal, Form, Chance and Dimension, Freeman, San Francisco, 1977. Kaye, B. H., Powder Technol., 21, 1, 1978. Schwarz, H. et al., Powder Technol., 27, 207, 1980. Kaye, B. H., Part. Charact., 1, 14, 1984. Suzuki, M. et al., J. Powder Technol. Jpn., 25, 287, 1988. Meakin, P., J. Colloid Interface Sci., 96, 415, 1983. Meakin, P., Phys. Rev. A, 29, 997, 1984. Blumberg, P. N. et al., AIChE J., 14, 331, 1968. Fuchs, N. A., in The Mechanics of Aerosol, Pergamon Press, New York, 1964, p. 40. Gans, R., Sitzber math-physk Klasse Akad. Wiss. München, 41, 191, 1911. Kosaka, Y. et al., J. Colloid Interface Sci., 84, 91, 1981. Stober, W., in Fine Particles, Liu, B. Y. H., Ed., Academic Press, New York, 1970, p. 363. Chien, Y. T., Interactive Pattern Recognition, Marcel Dekker, New York, 1978. Endoh, S. et al., Kagaku Kogaku Ronbunshu, 8, 476, 1982.
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1.4
Particle Density Yasuo Kousaka Osaka Prefecture University, Sakai, Osaka, Japan
Yoshiyuki Endo Sumitomo Chemical Co. Ltd., Osaka, Japan
1.4.1 DEFINITIONS The density of particles and powder is a physical property as important as the particle size. Although the density is defined as the ratio of mass to volume, the density of the particles does not necessarily agree with that of the particle material, as a particle occasionally includes pores, as shown in Figure 4.1. The density defined for particles and powder is as follows.
True Density, rs This density is defined as the ratio of the mass of the particle to its actual volume excluding inside pores. If the particles are ground so fine as to make the pores disappear, the true density can be measured by the method described later.
Particle Density, rp This is defined as the particle mass divided by the particle volume, including the inside closed pores. The volume can be obtained by the method mentioned below.
Bulk Density, rB The bulk density is defined for powder or particle beds. It is given as the ratio of the mass of the powder bed in a vessel to the volume of the bed, including the voids among primary particles. If the values of rp and rB are measured, the porosity or void fraction can be determined by ⎛ rB ⎞ ⫽1⫺ ⎜ ⎟ ⎝ rp ⎠
(4.1)
Closed pore
Particle FIGURE 4.1
Illustration of a cross section of a particle. 49
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1.4.2 MEASUREMENT METHOD FOR PARTICLE DENSITY The measurement of the particle’s density consists of measuring its mass and volume. The mass can be accurately measured with an electronic balance. However, it is difficult to measure the volume of particles. As described below, the volume of a particle is determined from the liquid volume increase upon adding the particle to a liquid, or in the case of gaseous medium, it is determined from the increased amount of volume or pressure.
Pycnometer Method This is the most representative method using a liquid medium. Figure 4.2 shows the Wadon-type pycnometer, whose volume is generally about 2 ⫻ 10⫺5 m⫺3 (20 ml). In this method, the following masses are measured: empty pycnometer mo, pycnometer containing liquid ml, pycnometer including sample particles msl. The particle density rp is given by rp ⫽
rl (ms ⫺ m0 )
( ml ⫺ m0 ) ⫺ (msl ⫺ ms )
(4.2)
where rl is the liquid density. If rl is not known, it can be measured with this pycnometer, whose volume is determined by filling it with water. In the present method, one must use a nonvolatile liquid in which the particles are wettable and insoluble. Air bubbles that adhere onto the particle surface can be effectively removed by heating or boiling the liquid under diminished pressure.
Constant Gas Volume Method In this method, gas such as air or helium is used instead of liquid. The density of almost any particle can be measured because, in contrast with the pycnometer method, there is no problem of particle solubility in the fluid medium.
FIGURE 4.2
Pycnometer.
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Figure 4.3 shows the basic principle of this method.1 The well-known Beckmann method is based on a principle similar to that of the present method. First, without any particle sample in the cylinder, the piston is moved from position A to position B. In the compression process the volume and the pressure of gas in the cylinder change as follows: V ⫹ v0 → V Pa → Pa ⫹ ΔP1
(4.3)
where v0 is the cylinder volume between planes A and B, and V is the volume between the plane B and the bottom of the cylinder. According to Boyle’s law, the following relation holds. Pa(V ⫹ v0) ⫽ (Pa ⫹ ΔP1)V
(4.4)
⎛ P ⎞ V ⫽ v0 ⎜ a ⎟ ⎝ ⌬P1 ⎠
(4.5)
This may be rewritten as
Second, after introducing a known weight of particle sample into the cylinder, the above procedure is repeated. In this case, the change in volume and pressure is V ⫹ v0 ⫺ Vs → V ⫺ Vs Pa → Pa ⫹ ΔP2
(4.6)
Therefore, the following equation is obtained: ⎛ P ⎞ V ⫺ Vs ⫽ v0 ⎜ a ⎟ ⎝ ⌬P2 ⎠
FIGURE 4.3
(4.7)
Constant gas volume method.
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From Equation 4.5 and Equation 4.7, the particle volume Vs becomes ⎛ P P ⎞ Vs ⫽ v0 ⎜ a ⫺ a ⎟ ⎝ ⌬P1 ⌬P2 ⎠
(4.8)
Although some other methods are available,2⫺4 their principles are almost the same as that described here.
REFERENCES 1. Mular, A. L., Hockings, W. A., and Fuerstenau, D. W., AIME Trans. Soc. Mining Eng., 226, 404–406, 1963. 2. von Neumann, B., Gross, W., Kremser, L., and Schmidt, J., Brennstoff Chemie, 15, 161–180, 1934. 3. Poisson, R., Peintures Pigments Vernis, 42, 767–769, 1966. 4. von Rennhack, R., Gas und Wasserfach, 109, 1209–1213, 1968.
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1.5
Hardness, Stiffness, and Toughness of Particles Mojtaba Ghadiri University of Leeds, Leeds, United Kingdom
Hardness, stiffness, and toughness represent the resistance of solid materials to plastic and elastic deformations and crack propagation, respectively. They are important in processes such as comminution, tableting, polishing, abrasive and erosive wear, and attrition, as they define the mode and pattern of mechanical failure. Their characterization is highly desirable for product and process improvement and optimization.
1.5.1
INDENTATION HARDNESS
The stress required for plastic deformation can be quantified directly only for a number of well-defined geometric shapes such as spherical, conical, and pyramidal objects1,2 by, for example, compression of a particle between two platens. For nonspherical particles the indentation hardness is easier to measure (as compared to the deformation stress), from which the resistance to plastic deformation can be inferred. Hardness is related to a number of fundamental properties of solids, such as the yield stress, work-hardening rate, and anisotropy, and is influenced by the geometry of the deformed region. Indentation hardness is commonly defined as the resistive pressure, H, when a permanent impression is made. It is measured after a certain amount of plastic strain has been induced, typically around 8%,3 and is therefore affected by anisotropy in the structure and the work-hardening process. However, it is still useful to relate hardness to yield stress, and the ratio of hardness over yield stress is commonly defined as the constraint factor. 4 This factor is usually much greater than unity, because the plastic deformation in a hardness test is constrained by the surrounding region, which is under elastic deformation. The indentation hardness and constraint factor depend on the properties of the deformed material, the indenter geometry, 5 and the coefficient of friction. 6,7 . The material properties of concern here are the elastic modulus, yield stress, rate of work-hardening, and anisotropy. The yield stress is a function of temperature and strain rate.8 As temperature is decreased, the dislocation mobility is reduced, and hence the yield stress increases. A similar behavior is observed when the strain rate is increased. The hardness is related to the yield stress and work-hardening by considering the characteristics of the uniaxial stress–strain curve for several types of material failure. Figure 5.1A and B represent materials that do not work-harden; elastic deformation is included in Figure 5.1B, while ideal plastic behavior is shown in Figure 5.1A. The effect of work-hardening is included with and without an elastic deformation in Figure 5.1D and C, respectively. The elastic deformation and rate of work-hardening have a very strong influence on the value of the constraint factor, and this is analyzed by the theoretical models of indentation hardness as outlined below.
Rigid–Perfectly Plastic Indentation A general treatment of the indentation process for this case has been described by Hill,9 based on the wedge-cutting mechanism. Here, the deformation process is represented by Figure 5.1A. The deformation for a blunt indenter has also been analyzed by approximating the process to the deformation by a rigid flat punch penetrating a semi-infinite rigid–perfectly plastic medium.6 53 © 2006 by Taylor & Francis Group, LLC
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(A)
(B)
(C)
(D)
FIGURE 5.1 Characteristic uniaxial stress-strain curves: (A) rigid, perfectly plastic, (B) elastic, perfectly plastic, (C) rigid, plastic with work-hardening, and (D) elastic, plastic with work-hardening.
According to the wedge-cutting mechanism, the indentation is treated as cutting by a smooth wedge, where it is assumed that slip occurs along planes of maximum shear. In this case, a simple relationship exists between the yield stress, Y, and hardness, H: f⫽ H ⁄ Y ⬵3
(5.1)
Tabor3 tested various ductile metals such as aluminum, copper, and mild steel and found that the experimental results are in good agreement with the above theory. However, the above simple relationship does not apply to anisotropic and work-hardening materials. Furthermore, the experimental observations made by Samuels and Mulhearn10 have shown that the wedge-cutting mechanism operates only when the wedge angle is less than 30°. For wedge angles greater than 30°, the plastic deformation occurs by radial compression, producing hemispherical surfaces of constant strain, where Hill’s theory of expansion of a spherical cavity is more applicable (see below).
Elastic–Perfectly Plastic Indentation The deformation characteristics are shown in Figure 5.1B. The expansion of a spherical cavity in an infinite elastic–plastic medium has been used to describe the process of indentation, where the mean pressure, p, at which the cavity expands, is given by Hill9: ⎛ ⎞ ⎪⎫ p 2 ⎪⎧ E ⫽ ⎨1 ⫹ ln ⎜ ⎟⎬ Y 3 ⎪⎩ ⎝ 3 (1 ⫺ v ) Y ⎠ ⎪⎭
(5.2)
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Where E is Young’s modulus and v is Poisson’s ratio. The contact pressure, p, is equivalent to the hardness of the flat surface, H, and the above model has been widely used to interpret the problems associated with elastic–plastic indentation. Following Hill’s approach, Marsh11 has proposed a similar expression for the expansion of a hemispherical cavity, where the constraint around the plastic zone is less than that around a spherical cavity. It can be seen from Equation 5.2 that pⲐY (or HⲐY) is a function of EⲐY. Tabor has presented the hardness of a wide range of materials on such a coordinate system.7 The onset of plastic flow occurs at a ratio of pⲐY of about 1, and the rigid–perfectly plastic theory predicts pⲐY ⫽ 3. Indentation is often made with a conical or pyramidal shape indenter where the angle of indenter affects the indentation pressure. A model of indentation hardness for this case has been presented by Johnson5 based on the spherical cavity expansion as H 2⎛ E cotu ⎞ ⫽ ⎜ 1 ⫹ ln ⎟ Y 3⎝ 3Y ⎠
(5.3)
where u is the indenter semiangle. For a spherical indenter, cot u may be replaced by aⲐR, where a is the radius of the impression and R is the indenter radius. This is a simple correlation between the indentation hardness and the indenter angle. However, Equation 5.3 is not valid for very sharp indenters having a narrow apex angle (e.g., wedges12 and cones13).
Work-Hardening Model For materials which exhibit significant work-hardening, the hardness is much larger than the yield stress. This is because hardness is conventionally measured at a strain of about 8%, 4 where significant work-hardening can take place. It is possible to use the analysis of expansion of a spherical cavity and to incorporate the effect of work-hardening, whose characteristics are shown in Figure 5.1C and D. Here, the stress after yielding can be expressed in the form of s ⫽ Y ⫹P
(5.4)
where P is the augmented stress associated with work-hardening, and it is a function of strain . P is normally approximated by a linear relationship of the form P ⫽ P′ ⫻ ( ⫺ o), ⬎ o
(5.5)
where P′ is the rate constant of hardening, and o is the strain corresponding to the onset of workhardening. For an incompressible material, in other words, no change of volume (v ⫽ 0.5), a simple relationship between the resistive pressure and the rate of work-hardening has been given by Bishop et al.14: 2 2 ⎧ ⎛ 2E ⎞ ⎫ 2p H ⫽ p ⫽ Y ⎨1 ⫹ ln ⎜ ⎟ ⎬ ⫹ P′ ⎝ 3Y ⎠ ⎭ 3 ⎩ 27
(5.6)
An alternative approach to determine the constraint factor for work-hardening materials has been suggested by Tabor,4 where the constraint factor is described as the ratio of the indentation hardness to the representative uniaxial flow stress, YR, at an effective strain R. f ⫽ H/YR
(5.7)
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FIGURE 5.2
Relationship between hardness and yield stress.
For a Vickers indenter, the average effective strain produced by the indenter is between 8% and 10%. 3 The flow stress at this value of strain is taken to calculate the constraint factor from Equation 5.7. This approach produces significantly different values for f and does not include the yield stress and work-hardening in an explicit way. In summary, the relationship between the hardness and yield stress is illustrated in Figure 5.2. It is clear that the material is subjected to more constraint as its morphology changes from the amorphous state to the crystalline state.
1.5.2
MEASUREMENT OF HARDNESS
Measurement and theory of hardness testing have been developed in the field of material testing, where it has been possible to carry out hardness tests on large specimens. Until very recently, it has been difficult to characterize the hardness of small particles. However, with rapid progress in the past two decades in the field of nanotesting and atomic force microscopy, detailed characterization of surfaces of even micrometer-size particles is now possible. The measurement of hardness started with scratch testing of flat surfaces by various particles having varying degrees of hardness, for example, the Mohs hardness test.15 This method provided a qualitative indication of resistance to plastic flow, but it was too complex for analysis to obtain quantitative results. The method of indentation hardness testing evolved by the use of well-defined indenter geometry, such as spherical, pyramidal, or conical shapes. Tabor’s pioneering work3 established the basis of understanding of the deformation processes involved in the indentation. A number of common test methods are available using different indenter shapes: Brinell, using a 10 mm sphere of steel or tungsten carbide; Vickers, using a diamond pyramid; Knoop, using also a diamond pyramid; and Rockwell, using a diamond cone with a spherical tip and steel spheres.7 In all these tests, a load is applied to the surface under testing by slowly penetrating the indenter at the right angle into the specimen. The hardness number is calculated from the applied load, the projected area of the impression, and by taking account of the shape of the indenter. The relevant formula for hardness numbers may be found, for example, from Tabor.7 The above techniques cannot be easily applied to the measurement of hardness of a particle unless the particle is large, that is, above a few millimeters. However, the recent development of nanotest methods has enabled a full characterization of mechanical properties as well as topography of fine particulate solids. 16 Because of the small size of impression, which can be submicrometer, almost all nanoindentation methods measure the resistance of material to penetration (i.e., the applied load) as a function of depth of penetration, from which an effective hardness can be inferred. © 2006 by Taylor & Francis Group, LLC
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FIGURE 5.3 Schematic diagram of the nanoindenter device. [From Arteaga, P. A. et al. Tribol. Int., 26 (Suppl 5), 305–310, 1993. With permission.]
A schematic diagram of a typical nanotest device is shown in Figure 5.3. There are several designs.16⫺18 The design shown here was originally developed by Pollock et al.19 and is now manufactured by Micro Materials, Ltd., Wrexham, U.K. The device measures the movement of a calibrated diamond indenter penetrating into a specimen surface at a controlled loading rate. It uses a pendulum pivoted on bearings that are essentially frictionless. A coil is mounted at the top of the pendulum so that when a coil current is present the coil is attracted toward a permanent magnet, producing motion of the indenter toward the specimen and into the specimen surface. The displacement of the diamond is measured by means of a parallel plate capacitor, one plate of which is attached to the diamond holder; when the diamond moves, the capacitance changes, and this change is measured by means of a capacitance bridge. The most commonly used indenter is a Berkovitch diamond, a three-sided pyramid, which is particularly suitable for nanoindentation work because it can be machined down very accurately to a very sharp tip. However, 90° trigonal and spherical indenters have also been used. The above device is capable of measuring load and displacement with typical resolutions around 25 mN and 20 nm, respectively. In a nanoindentation test, the specimen is first brought in contact with the indenter; then the load is increased at a prescribed rate until the desired maximum load is reached, and then it is decreased back to zero at the same rate. Therefore a continuous recording of the penetration depth and the applied load is made, from which an effective hardness modulus may be inferred at a specified strain rate by invoking a relation between the penetration depth and the indentation area. A typical depthⲐload cycle is shown in Figure 5.4. During the unloading part of the cycle, the elastic deformation of the sample will recover, and therefore the unloading curve is not horizontal. The maximum depth and characteristics of the unloading curve are used to calculate the hardness. However, it is also possible to evaluate the elastic modulus from the unloading curve. Two distinct regions can be observed here: the initial unloading stage bearing all the features of an elastic unloading, and © 2006 by Taylor & Francis Group, LLC
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FIGURE 5.4
Typical loadingⲐunloading curve in a nanoindentation test.
a highly nonlinear stage toward the end of the unloading cycle. Reasons for this nonlinearity are complex; changes in the contact area arising from an increase in the apical angle of the impression are thought to be responsible.20–22 There are several methods for the evaluation of hardness from such data.17,20 The method described below is based on the analysis of Oliver and Pharr17 and is one of the most widely used methods. At the initiation of unloading (Figure 5.4), the elastic component of the deformation starts to recover so that contact between the material surface and the indenter is still maintained. As the load is decreased the elastic displacement continues to recover until at zero loads the displacement reaches a final value, h f. At the peak load, Fmax, the displacement is hmax. A schematic diagram depicting a cross-sectional view of an indentation is shown in Figure 5.5. The total displacement, h, can be described at any time during loading as h ⫽ hc ⫹ hs
(5.8)
where hc is the contact depth, that is, the vertical distance along which contact is made, corresponding to the contact size, a, and hs is the elastic displacement of the surface at the perimeter of the contact. In this analysis, hardness, H, can be determined by H=
Fmax A
(5.9)
where A is the projected area of the impression. Now, from the geometry of the indenter 2 A ⫽ khc,max
(5.10)
with k being the indenter shape factor, and hc,max the contact depth at Fmax. Now, to determine the contact depth from the experimental data, it can be noted that hc,max ⫽ hmax ⫺ hs,max
(5.11)
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FIGURE 5.5
59
A schematic diagram depicting a cross-sectional view on an indentation.
which follows directly from Equation 5.8. hmax is experimentally measured, and hs,max can be ascertained from the unloading data by the use of an expression derived by Oliver and Pharr17 from Sneddon’s solution23 for the elastic contact of indenters of different shapes. This expression is given by hs,max ⫽
Fmax S
(5.12)
where S is the stiffness at load Fmax (as shown in Figure 5.4), and is a constant that depends upon indenter geometry. The value of for a Berkovitch indenter cannot be determined analytically; however, it has been shown that the Berkovitch indenter behaves very much like a paraboloid of revolution for which a solution for the parameter exists, that is, ⬇ 0.75.24 In practice, the stiffness, S, is obtained by fitting the unloading data to a power law relation of the form F ⫽ a(h ⫺ hf)m
(5.13)
where (h ⫺ hf) is the elastic displacement of the indentation during unloading; a and m are constants. The values of a, m, and hf are all determined by a nonlinear curve fitting procedure. Equation 5.13 can then be differentiated analytically with respect to (h ⫺ hf) to obtain the value of stiffness S⫽
dF d ( h − hf )
(5.14)
which can then be evaluated at the peak load, Fmax. Then, by first substituting the value of stiffness obtained from Equation 5.14 into Equation 5.12, the value of hs,max is calculated. Equation 5.11, Equation 5.10, and Equation 5.9 are then used to determine the hardness.
1.5.3
MEASUREMENT OF STIFFNESS
Particles subjected to mechanical stresses below the critical level for plastic yielding undergo elastic deformation. The resistance to elastic deformation is represented by the ratio of load over the extent of deformation, termed stiffness. As in the case of plastic deformation, for nonspherical particles the contact geometry is too complex for characterizing the stiffness by standard techniques. The approach © 2006 by Taylor & Francis Group, LLC
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outlined above for indentation hardness measurement can also be used to evaluate the stiffness. For materials obeying Hooke’s law, the stiffness can be calculated from Young’s modulus, and this in turn can be determined from the unloading portion of the depth–load data (as shown in Figure 5.4) by the use of Sneddon’s analysis.23 This analysis is independent of indenter geometry24 and is therefore applicable to results obtained with the Berkovitch indenter p S 2 A
Er ⫽
(5.15)
where S is calculated from Equation 5.14, A is the contact area given by Equation 5.10, and Er is the reduced modulus defined by
(
) (
1 ⫺ v2 1 ⫺ vi2 1 ⫽ ⫹ Er E Ei
)
(5.16)
where E and v are Young’s modulus and Poisson’s ratio of the specimen, and Ei and vi are the same parameters for the diamond indenter. Since Young’s modulus of the indenter is two orders of magnitude greater than that of the test materials, the reduced modulus is effectively Young’s modulus of the specimen. In conclusion, the above approach enables the determination of hardness and Young’s modulus at the micro- and nanoscale levels. However, great care is needed for the experimental procedure and interpretation of the data because of the scale of operation.
1.5.4
MEASUREMENT OF TOUGHNESS
Fracture toughness represents the resistance of solid materials to crack propagation. The basis of its measurement lies in the principles of linear elastic fracture mechanics and the Griffith energy balance for crack propagation.25 For large specimens, the measurement of fracture toughness is made using a uniform stress field. A common method is the three-point bend test as shown in Figure 5.6, where a crack of length c is made much shorter than the specimen’s depth, d, providing uniform
F/2
F/2
b
c d
F FIGURE 5.6
Three-point bend test.
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tension at the crack tip. In this case the fracture toughness is calculated based on the simple beam theory and is given by25 K ⫽ m(3FbⲐd 2)(pc)1Ⲑ2
(5.17)
where m is a factor which depends on cⲐd and has a limiting value of 1.12. For small particles the indentation method is used to measure the fracture toughness as described below.
Indentation Fracture Toughness In indentation fracture by a sharp indenter causing plastic deformation, there is a critical size of indentation, rc, above which fracture is initiated26,27: rc ⬵ a
EG Y2
(5.18)
where a is a constant, E is Young’s modulus, G is the fracture surface energy, and Y is the yield stress in a uniaxial compression test. This critical size reflects the influence of the mechanical properties. The yield stress is related to the hardness, as Y ⫽ HⲐf, where f is the constraint factor. The fracture surface energy and Young’s modulus are related to the fracture toughness based on linear elastic fracture mechanics. For the case of plane strain, E G ⫽ K c2 (1⫺ n 2)
(5.19)
where v is Poisson’s ratio. Considering that v2 is much smaller than unity, the critical size can be written approximately in the following form: rc ⬇ a⬘(
Kc 2 ) H
(5.20)
FIGURE 5.7 Indentation crack geometry: (A) Berkovitch impression; cross-section view of (B) radial crack and (C) median crack systems. © 2006 by Taylor & Francis Group, LLC
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FIGURE 5.8 Scanning electron micrograph of a nanoindentation in the (001) face of lab-produced paracetamol. Applied force: 50 mN. Indenter: Berkovitch pyramid. [From Arteaga, P. A. et al., Tripartite Programme in Particle Technology, Final Report, 1997. With permission.]
FIGURE 5.9 Scanning electron micrograph of a nanoindentation in the (100) face of lab-produced lactose. Applied force: 100 mN. Indenter: Berkovitch pyramid. [From Arteaga, P. A. et al., Tripartite Programme in Particle Technology, Final Report, 1997. With permission.]
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FIGURE 5.10 Relationship between crack length and load from nanoindentation experiments on paracetamol and lactose crystals. [From Arteaga, P. A. et al., Tripartite Programme in Particle Technology, Final Report, 1997. With permission.]
When the surface is indented by a sharp indenter, the plastic zone size increases until a critical size is reached, above which cracks propagate. From the indentation fracture mechanics, it can be shown that the size of the stress field scales with √r c. Therefore the crack length is controlled by rc, which in turn is related to Kc. The fracture toughness can then be determined from the relationship of the crack length with applied load by the use of an expression which has the following general form28: k
n
a ⎞⎟ ⎛⎜ E ⎞⎟ P ⎟ ⎟ ⎜ *3/ 2 ⎜⎝ l ⎟⎠ ⎜⎝ H ⎟⎠ c ⎛
K c ⫽ x v ⎜⎜
(5.21)
where xv is a calibration factor, aⲐl is the ratio of crack length to impression size as defined in Figure 5.7, c* (⫽ a ⫹ l) is the crack size as measured from the center of the impression, P is the load, and k and n are power indices. It is clear that this method relies on the factor PⲐc*3Ⲑ2 being a constant (for a given material). There are several correlations for fracture toughness cited in the literature, all of which suggest different values for xv, k, and n. A review of these correlations is given by Ponton and Rawlings.28 The main advantage of measuring fracture toughness by indentation is that no special specimen geometry is required. However, since it relies upon the indentations producing a radial or median crack system with well-formed surface traces, its applicability is limited to materials that exhibit brittle and semibrittle fracture during indentation. In a typical test a number of different loads are applied, and the indent size and crack length, as shown in Figure 5.7, are measured. A plot of crack length, c, as a function of P2Ⲑ3 should produce a straight line, the slope of which would give Kc. As an example, the results of measurement of fracture toughness of paracetamol and lactose crystals by nanoindentation are shown below. Figure 5.8 and Figure 5.9 show indentations on the (001) face of paracetamol and (100) face of lactose crystals, respectively. It can be seen clearly that surface traces of well-defined cracks are produced on nanoindentation of these crystals. © 2006 by Taylor & Francis Group, LLC
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The relationships between crack length and load are shown in Figure 5.10 for indentations in lab-produced crystals of paracetamol and lactose. Error bars of one standard deviation of the mean are also included in the figure. The load–crack length relationship given previously, specifically, PⲐc*3Ⲑ2 ⬇ constant, has been added for comparison. The values of the intercept, b, as indicated in the figure, have been chosen to be slightly different from the values of the intercept for the best fits of the experimental data to show the close agreement in the slopes of the lines. It is clear that the load–crack length data for both materials approximate reasonably well the expected relationship. The results of fracture toughness for both paracetamol and lactose are shown in Figure 5.11 as plots of fracture toughness against load. Here, the equation proposed by Laugier29 has been used to calculate Kc, as this correlation has been developed for materials which are well behaved in their indentation response, and it takes into account the role of plastic deformation in driving the crack growth. This correlation, which is based on the modification of the equation proposed by Anstis et al.30 to include the effect of the plastic driving force, gives the constants of the equation as xv ⫽ 0.010, k ⫽ 0, and n ⫽ 2Ⲑ3. Care needs to be taken to account for the effect of indenter geometry. Most correlations reported in the literature are based on the Vickers indentation with four associated radial cracks, while with Berkovitch (three-sided pyramid) indentations a smaller number of cracks is produced. The work of Dukino and Swain31 can be used to account for this difference. These workers compared measurements of fracture toughness obtained from Berkovitch and Vickers indentations and proposed a method to relate the two measurements. An issue that is very important in predicting the comminution behavior of powders is the determination of the critical particle size for breakage transition. Several models have been proposed.32–35 All these models, although based on different considerations and assumptions, follow the same general form. Therefore, with the values of fracture toughness and hardness of paracetamol and lactose as measured
FIGURE 5.11 Plots of fracture toughness against load for paracetamol and lactose crystals. © 2006 by Taylor & Francis Group, LLC
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by nanoindentation, the values of rc for paracetamol and lactose are calculated as 0.5 mm and 2.4 mm, respectively.36 These values seem reasonable. However, further work is needed to establish a comparison with the actual sizes of the particles obtained in comminution experiments.
REFERENCES 1. Chaudhri, M. M., J. Mater. Sci., 19, 3028–3042, 1984. 2. Ghadiri, M., Arteaga, P., and Cheung, W., Proceedings of Second World Congress in Particle Technology, Kyoto, 1990, pp. 438–446. 3. Tabor, D., Proc. Roy. Soc., A192, 247, 1948. 4. Tabor, D., Microindentation Techniques in Material Science and Engineering, ASTM STP889, Blau, P. J. and Lawn, B. R., Eds., ASTM, Philadelphia, 1986, pp. 129–159. 5. Johnson, K. L., J. Mech. Phys. Solids, 18, 115–126, 1970. 6. Gilman, J. J., The Science of Hardness Testing and Its Research Application, Westbrook, J. H. and Conrad, H., Eds., American Society for Metals, Materials Park, Ohio, 1971, pp. 54–58. 7. Tabor, D., Rev. Phys. Tech., 1, 145–155, 1970. 8. Hull, D. and Bacon, D. J., Introduction to Dislocations, Pergamon Press, Oxford, 1984. 9. Hill, R., The Mathematical Theory of Plasticity, Clarendon Press, Oxford, 1950. 10. Samuels, L. E. and Mulhearn, T. O., J. Mech. Phys. Solids, 5, 125–134, 1957. 11. Marsh, D. W., Proc. Roy. Soc. A, 279, 420, 1964. 12. Hirst, W. and Howse, M., Proc. Roy. Soc. Lond. A, 311, 429–444, 1969. 13. Atkins, A. G. and Tabor, D., J. Mech. Phys. Solids, 13, 149–164, 1965. 14. Bishop, R. F., Hill, R., and Mott, N. F., Proc. Phys. Soc., 57, 147–159, 1945. 15. Tabor, D., Proc, Phys, Soc. B, 67, 249–257, 1954. 16. Arteaga, P. A., Ghadiri, M., Lawson, N. S., and Pollock, H. M., Tribol. Int., 26 (Suppl. 5), 305–310, 1993. 17. Oliver, W. C. and Pharr, G. M., J. Mater. Res., 7 (Suppl. 6), 1564–1583, 1992. 18. Field, J. S. and Swain, M. V., J. Mater. Res., 10 (Suppl. 1), 101–112, 1995. 19. Pollock, H. M., Maugis, D., and Barquins, M., Microindentation Techniques in Materials Science and Engineering, ASTM STP 889, Blau, P. J. and Lawn, B. R., Eds., ASTM, Philadelphia, 1986, pp. 47–71. 20. Doerner, M. F. and Nix, W. D., J. Mater. Res., 1 (Suppl. 4), 601–609, 1986. 21. Loubet, J. L., Georges, J. M., and Meille, G., Microindentation Techniques in Materials Science and Engineering, ASTM STP 889, ASM, Blau, P. J. and Lawn, B. R., Eds., ASTM, Philadelphia, 1986, pp. 72–89. 22. Ion, R. H., Pollock, H. M., and Roques-Carmes, C., J. Mater. Sci., 25, 1444–1454, 1990. 23. Sneddon, I. N., Int. J. Eng. Sci., 3, 47–57, 1965. 24. Pharr, G. M., Oliver, W. C., and Brotzen, F. R., J. Mater. Res., 7 (Suppl. 3), 613–617, 1992. 25. Lawn, B. R. and Wilshaw, T. R., Eds., Fracture of Brittle Solids, Cambridge University Press, Cambridge, 1975. 26. Puttick, K. E., J. Phys. D Appl. Phys., 11, 595–604, 1978. 27. Puttick, K. E., Energy Scaling in Elastic and Plastic–Elastic Fracture, paper presented at the Griffith Centenary Meeting, U.K., 1993. 28. Ponton, C. B. and Rawlings, R. D., Mater. Sci. Technol., 5, 865–872, 1989. 29. Laugier, M. T., J. Mater. Sci. Lett., 4, 1539–1541, 1987. 30. Anstis, G. R., Chantikul, P., Lawn, B. R., and Marshall, D. B., J. Am. Ceram. Soc., 64, 533–538, 1981. 31. Dukino, R. D. and Swain, M. V., J. Am. Ceram. Soc., 75 (Suppl. 12), 3299–3304, 1992. 32. Lawn, B. R. and Evans, A. G., J. Mater. Sci., 12, 2195–2199, 1977. 33. Kendall, K., Nature, 272, 710–711, 1978. 34. Hagan, J. T., J. Mater. Sci., 16, 2909–2911, 1979. 35. Puttick, K. E., J. Phys. D Appl. Phys., 13, 2249–2262, 1980. 36. Arteaga, P. A., Bentham, A. C., and Ghadiri, M., Tripartite Programme in Particle Technology, Final Report, University of Surrey, Guildford, U.K., 1997.
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1.6
Surface Properties and Analysis Masayoshi Fuji, Ko Higashitani, and Yoichi Kanda Nagoya Institute of Technology, Tajimi, Gifu, Japan
1.6.1
SURFACE STRUCTURES AND PROPERTIES
Particle Surface and Character A material that does not have flowability at temperatures below its melting point can exist as a solid in various shapes. The significant feature is the degree of the unsaturation of the chemical bond to which the internal structure terminates when the surface of the solid is looked at on the microscopic scale. It is not possible to diffuse because the activation energy of the surface diffusivity is generally high, when the adjacent potential energy is different by atoms, ions, or molecules that compose the surface. Therefore, a feature of the solid surface is that it is not able to produce a uniform energy like the liquid surface. Various characteristics of a solid surface are strongly influenced by these two features. The population of atoms, ions, and molecules that compose the fine particle surface increases in comparison with the particulate inside. In this case, a lot of peculiar characteristics begin to appear in fine particles. 1 Especially, the property concerning handling the powder begins to strongly depend on the character of the particulate surface.2 Therefore, how one can know the character of the particulate surface and control it are the major technologies in all fields where the powder has an effect. Here, we describe in detail an important matter for understanding the character of the particulate surface.
Surface Relaxation Various surface relaxations appear for the stabilization because the solid surface has broken chemical bonds and exists in an unstable state. These relaxations are different depending on the kind of chemical bond, such as covalent bond, ionic bond, and metallic bond, and the kind of material. In this section we separately describe chemical relaxation with a chemical change and physical relaxation with a structural change. Moreover, because the solid is usually handled in the atmosphere, we also describe the stabilization of the surface by water adsorption. Physical Surface Relaxation An atom, ion, or molecule on the surface receives an unsymmetrical interaction force, because chemical bonds have been cut on the crystal surface. Therefore, these positions change from the position forecast from the internal crystal lattice. In the case of a surface of alkali halides, which are ionic crystals, the anion or the cation is displaced to a position that is positive or negative in the vertical direction against the surface from the position expected from the lattice location. Figure 6.1 shows such an example. In the surface layer of the (100) face of NaCl, Cl–, which is the anion, is polarized with the Na+ ion on the circumference and, as a result, displaces from the surface to the outside. The cations internally displace. 67 © 2006 by Taylor & Francis Group, LLC
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The electrical double layer is formed as a result and the surface has been stabilized. That is, a center position for both ions does not exist on the same face. Moreover, the distance between the position of the first layer as the average of both ions and the position of the second layer is compressed neater than interlayer distances in the crystal. It is assumed that such displacement is caused from the first layer to the fifth layer in NaCl. A similar phenomenon in other materials is listed in Table 6.1. In the (100) face of MgO, whose bond energy is larger than the alkali halide though it is classified into the same ionic crystal system, the interionic distance of the first surface layer and the second layer is compressed to about 85% of the interlayer distance in the crystal. The physical surface relaxation phenomenon is presumed to be different depending on the difference in the ion species, the coordination number, the crystal face, and the bond energy for the crystal with the same ionic bond. A similar relaxation is also caused on the surface of a metallic crystal. The surface relaxation is high in the crystal face; therefore, the atomic density is small, as shown in Table 6.1. The change region is assumed to be from the surface to
1st (Surface) 2nd 3rd 4th 5th 6th 7th FIGURE 6.1 Physical relaxation model for (100) face of alkali halide alkali. Open circle is anion. Closed circle is cation. Dotted line indicates the position of inner crystal lattice.
TABLE 6.1
Physical Surface Relaxation
Crystal structure
(110)
(001)
(111)
Cu
face-centered cubic
0.804
0.871
0.944
Al
face-centered cubic
0.900
1.00
1.15
Fe
body-centered cubic
1.00
0.986
0.85
NaF
rock-salt
—
0.972
—
NaCl
rock-salt
—
0.970
—
NaBr
rock-salt
—
0.972
—
MgO
rock-salt
—
0.85
—
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about two layers. The physical relaxation phenomenon also includes the relaxation phenomenon by the two-dimensional rearrangement of the surface atoms. In the (111) face of the Si crystal formed with the covalent bond, one of four sp3 covalent bonds is cut and is unstable. Therefore, the surface atom arrangement is caused by rearrangement in two dimensions, and changes into another regular surface structure (2 ⫻ 2). Moreover, the hybrid orbital of the Si atoms, which are adjacent to the surface, changes, and regular relief structures are formed. In the latter case, the bonding orbital of the Si atom alternately changes from the hybrid orbital of sp3 into sp2 ⫹ p and 3p ⫹ s. In the former case, because the sp2 orbit is contained, the bond angle in the Si interatomic is large, and the height of the Si atom is less than for the sp3 cases. On the other hand, because the bond angle in the Si interatomic is small when three p3 orbitals are contained, the height of the Si atom is greater than for the sp3 case. Therefore, the height of the Si atom alternately changes, and the formation of a regular relief structure occurs. The (2 ⫻ 1) structure in the Si(001) face and the (7 ⫻ 7) structure are well known.6,7 Chemical Surface Relaxation Because a chemical bond is unsaturated in the solid surface as previously described, the reactiveness is high. When the solid is exposed to an atmosphere, the surface, where the reactivity is high, chemically reacts with the reactive gas in the atmosphere, that is, oxygen, carbon dioxide,or water vapor, and forms a surface composition different from the chemical composition of the bulk. The metallic surface is oxidized by atmospheric oxygen and water vapor, and corrosion is a daily phenomenon. Moreover, oxygen in the air is chemically adsorbed and an oxide is also formed on the surface of the nitride. This oxide surface further chemically adsorbs the water vapor, and a hydroxyl group is formed on the surface. Examples of the chemisorption of oxygen on a metallic surface and the nitride surface are as follows. 2Fe + O2 → 2FeO
(6.1)
4AlN + 3O2 → 2Al2O3 + N2
(6.2)
Si3N4 + 3O2 →3 SiO2 + 4N2
(6.3)
2TiB2 + 6H2O → 2TiO2 + 2B2H6 + O2
(6.4)
The material chemically adsorbs the atmospheric carbon dioxide and forms a carbonic acid salt. For instance, carbonate and a surface hydroxyl group are formed with the carbon dioxide and water vapor in air on MgO, CaO, and the surface of the MgO–ZnO complex oxide. Water vapor is easily chemically adsorbed in general, and the surface becomes covered by hydroxyl groups on the oxide surface. Figure 6.2 shows the chemisorptions of water vapor on the surface of MgO having a simple crystal structure as one example. The chemisorption of water vapor has the ability to absorb only on the surface layer and absorb into a layer in the solid. Not only the first surface layer but even the inside of the solid MgO can absorb. The water vapor is chemically adsorbed, and the oxide is also formed on
Surface
FIGURE 6.2
Chemical adsorption of water molecules on MgO (100).
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the surface of nitride and boride, as shown by the following equations. In addition, the oxide surface changes into a surface of hydroxyl groups by chemisorption of the water vapor. ≡Si–O–Si≡ + H2O → 2≡Si–OH
(6.5)
Si3N4 + 6H2O → 3SiO2 + 4NH3 ≡Si–O–Si≡ + H2O → 2≡Si–OH
(6.6)
The surface hydroxyl group is desorbed by heating. The desorption behavior not only differs depending on the oxide type, but also changes depending on synthetic method and the thermal hysteresis of the sample. Figure 6.3 shows the change in the amount of the surface hydroxyl groups by heating various oxides.8–11 The physical adsorption of the water vapor occurs on the polar surface when the solid has been left in the atmosphere and the surface free energy has decreased. The adsorption layer is formed from about two molecular layers at the relative humidity about 60% on the hydrophilic surface such as glass. Figure 6.4 shows water vapor adsorption isotherms for various glass surfaces. The solid surface is stabilized by the existence of these adsorbed water molecules, and the dynamic, chemical and electrical properties of the material are significantly influenced. Moreover, it is known
Surface density of hydroxyl (nm–2)
12 n-Al203 aFe2O3 SiO2 TiO2 ZnO
10 8 6 4 2 0
600 200 400 800 Temperature (°C)
0
FIGURE 6.3 treatment.
Changes of surface density of hydroxyl on several oxides with heat-
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that the adhesive force between particles changes and produces a remarkable influence on the aggregation and the solidifying properties of the powder.12–14
Surface Acidity and Basicity Influence of the Element For the chemical property of the oxide surface, one of the important characteristics for practical use might be the solid acid and base. The acidity and the basicity of the oxide surface are predictable to some degree based on the chemical property of the atom composition. Figure 6.5 shows the relation to electronegativity (χi) of the partial charge and the metal ion on the oxide oxygen.17 It is understood that the negative charge density of oxygen becomes smaller when the electronegativity of the metal ion is high and shows acidity. The electronegativity of the metal ion is shown here by the following equation: χi = (1+2z) χ0
(6.7)
where z is the charge, and χ0 is the electronegativity of the metal atom. Auroux et al. found a linear relation, as shown in Figure 6.6, from the heat of adsorption measurement of CO2 by some oxides and the ratios of the charge/metal ion radius. When the ratio of the charge/metal ion radius is low, and the ionicity of chemical bond is high, the basicity is strong.18 Influence of Partially Chemical Structure
Adsorption amount (ml-STP.m–2)
The strength of the acid and the base changes depending on the chemical structural difference, even if it is a single chemical composition. As for the free surface hydroxyl group observed on the Al2O3 surface, two or more structures are possible based on the relation of the coordination number. If the net charge of the hydroxyl of each structure is requested, one can use it becomes as Table 6.2. It is expected that the higher the amount of the positive charge, the stronger the acidity. The complex oxide, which consists of the oxide and contains more than two kinds of metals, occasionally shows an acidity which does not
0.3
0.2 180 °C, 4hr. 800 °C, 4hr.
0.1
0 0
FIGURE 6.4
0.1
0.2 0.3 Relative Pressure (–)
0.4
0.5
Water vapor adsorption isotherms on porous silica glass.
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Cs+ K+
Partial charge
-0.8
Rb+
Na+
Li+ Ba2+ 2+ Sr
-0.6
Ca2+
Y3+ Mg2+ 2 + Be Ag2+ Sc2+ 2+ Cu Cu+
-0.4
Ti+
-0.2
Zn2+ Cd2+
Al3+ In3+
Zr4+ Ti4+
Hg2+ Pb2+
Sn4+
Pb4+
0 0 FIGURE 6.5 metal ion.
2
4
6 10 8 Electronegativity
12
14
Relationship between partial charge of oxide surface and electronegativity of
appear in a single oxide. Tanabe et al. reported that the acid appearance of the complex oxide can be explained according to the coordination number and each electric charge of the metal ion and oxygen.19 Moreover, the acid strength of the complex oxide is found to correlate to the average of the electronegativity of the composition metal ion.20 Recently, the acid and the base on the surface were also explained by the quantum chemical calculation using the cluster model, by which an oxide part is cut out.21 Influence of Surface Acidity and Basicity on Dispersion The surface hydroxyl group receives the proton or dissociates and positively electrifies the particle surface or negatively in water. M–OH + H.+ + OH– → M–OH2+ + OH–
(6.8)
M–OH + H.+ + OH– → M–O– + H2O + H+
(6.9)
The charge of the particulate surface strongly influences the dispersibility of the particle in a liquid. The character of the proton receipt or the dissociation of the hydroxyl depends on the acidity and basicity. Therefore, the difference in the positive and negative charges of the surface and the amount of the charge is caused by the oxide type. The typical oxide isoelectric point is shown in Table 6.3.22 When the oxide powder is dispersed in the medium, it is influenced by dissociation according to which type of the above equation by the character of the H+ or OH– acceptor. That is, the state of the interfacial interaction is determined by the relative intensity of an acid, the base character of the powder, and the acid and base characteristics of the medium. The principle of this interfacial phenomenon is shown in Figure 6.7. For instance, when the acidity of the powder surface is strong and the medium is basic, the particle gives the medium H+ and the surface is charged negatively. On the other hand, when the basicity of the particulate surface is strong and the medium is acidic, the © 2006 by Taylor & Francis Group, LLC
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Mo Si V
8
Charge per radius
73
Ta Nb
6
Al Ti
Ba Zr
Cr
Ga Pt
4
Th Nd
Mg Zn La Ca
2
0
20
0
40
60
80
100
120
Adsorption heat of CO2 (kJ.mol-1) FIGURE 6.6 Relationship between adsorption heat of CO2 and ratio of surface charge and radius of ion.
TABLE 6.2
Net Charges of Hydroxl Group on Alumina Surface
Note: O, octahedral coordination; T, tetrahedral coordination.
particulate surface receives H+ from the medium, and the surface is charged positively. When acidity and the basicity of the particulate surface and the dispersion medium are the same strength, the particle is not charged.
Surface Roughness and Porosity The surface geometry, which influences the particle interaction predicted from the size of the particle when the particle size is submicron, is a nanoscale structure, because a bigger surface geometry than © 2006 by Taylor & Francis Group, LLC
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TABLE 6.3 Material WO3
Isoelectric point >0.5
Material ␥-Al2O3
Isoelectric point 7.4–8.6
SiO2
1.8
Y2O3
9.0
SiC
3.4
␣-Fe2O3
9.04
4.3
␣-A12O3
9.1
ZnO
9.3
Au Al(OH)2
5.0–5.2 6.6
CuO
9.4
␣-FeO(OH)
6.7
BeO
10.2
TiO2
6.7
La2O3
10.4
CeO2
6.75
ZrO2
10–11
Cr2O3
7.0
Ni(OH)2
11.1
␥-FeO(OH)
7.4
Co(OH)2
11.4
Zn(OH)2
7.8
MgO
12.4 ± 0.3
Basic
SnO2
Particle surface Negative charge
Neutral
(+)
(–) Particle surface Positive charge
Acidic
Property of dispersion medium
Isoelectric Point of Various Powders
Acidic FIGURE 6.7 medium.
Basic Neutral Surface property of particle
Acidic and basic properties of particle surface in nonaqueous dispersion
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this is roughly thought to be a category of particle shapes including macropores (above 50 nm). The surface geometry of this large scale can also be observed using an electron microscope, and it allows one to observe the relation between the particle shape and the particle interaction. Surface geometry structure can be roughly divided into the four classes shown in Figure 6.8, according to these ideas and the relative sizes of the adsorbed molecule and the particle. The first structure is a smooth surface on an atomic scale, as shown in Figure 6.8A. The second is a surface structure with roughness on an atomic level, as shown in Figure 6.8B. The third structure is a surface structure, shown in Figure 6.8C, which has micropores. The fourth structure, Figure 6.8D, is a surface structure that has mesopores. The studies concerning the quantitative relationship of these surface structures and powders and their various physical properties are few. An example concerning the surface structure, the physical adsorption of water, and the particle–particle adhesive force is introduced here.23,24 The adhesive force between particles in the presence of water vapor is influenced by the state of adsorption of the water molecule on the particulate surface and the geometric structure of the particulate surface. The main cause of the adhesive force due to the humidity before the liquid bridge forms among particles is a hydrogen bond between particles by mediation of the water molecule. The main cause of the adhesive force in the humidity after the particle–particle capillary condensation occurs is a liquid bridge force. When the particle has pores, these basic mechanisms are the same. Therefore, the adhesive force because of the humidity before the liquid bridge forms among particles is the force of the hydrogen bond among particles by the mediation of the water molecule. However, the adhesive forces become small for a nonporous particle, according to the decrease in the interfacial area. Moreover, only a partial liquid bridge force is formed in the humidity before the capillary condensation in the pore is completed, and the adhesive forces are smaller than for when the capillary condensation in the pore is completed. On the other hand, the liquid bridge force is equal for the nonporous particle that is caused when neither the capillary condensation among particles nor the capillary condensation in the pore is completed. For instance, when the granulation operation is done in humidity, the capillary condensation in the pore is not completed, and the granulation will be not able to be done enough. On the other hand, to cause aggregation, one only has to set the opposite condition. As mentioned above, controlling various physical properties of powder by understanding the surface structure will become possible.
Actual Surface Various surface relaxations and the physical adsorption of water occur, and the stabilization of the surface energy is achieved as on an actual solid surface. The surface inhomogeneity may exist as a step, a kink, a defect, or a dislocation, plus the adsorption impurities; therefore, the state of the actual surface is more complex. The exposure ratio of the crystal face and the number of heterogeneity sites on the surface differ and depend on the preparation conditions of the solid. The surface
FIGURE 6.8 Conceptual classification of surface structure: (A) smooth surface on molecular level, (B) surface with roughness on molecular level, (C) surface with micropore, (D) surface with mesopore. © 2006 by Taylor & Francis Group, LLC
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Water C
A R r
Position D
B D
E
Detachment force
D
FIGURE 6.9
D
Bubble
=
R- D R
Ratio of edge length of cubic sodium chloride crystal and number of atoms.
structures on the top and the edge increase when the particle becomes a fine particle. Many of these unsaturated bonds are skillfully used as active sites, as with solid catalysts. The ratio of the surface of the atom to the entire atom that composes the particle increases when the particle of a salt crystal with six coordinations becomes small, as shown in Figure 6.9. For instance, when particle size or atomic diameter becomes 50 or less, it is understood that the ratio of the surface of the atom rapidly increases. Moreover, when the solid is used as a material, it is necessary to process the solid. With respect to processing factors such as generated heat and chemical reaction, the surface layer of the particle is different from the inside; that is, a processing-altered layer is formed on the surface, and defects and dislocations are generated. On the other hand, it is thought that the amorphous solid of glass, for instance, is solidified without making the supercooled liquid a crystal. Therefore, there is no regular array of atoms, ions, or molecules observed in the crystal. The surface of melted glass is a smooth plane because of the surface tension as well as the liquid. When the cooling solidification is done, the surface smoothness is maintained. Because the mobility of the material is high in the melt, the element by which the surface energy is decreased is concentrated at the surface. Especially, alkaline components easily gather on the surface. Moreover, because the melt is at a high temperature, it seems that the vaporization of the element and the alteration of the element along with the disappearance, by the reaction with the reactive gas in the atmosphere, have occurred on the surface. As mentioned above, the actual surface is complex. However, with an understanding of the major characteristics it is possible to predict how it will react by carefully considering the generation process of the powder and the matters explained here.
1.6.2
SURFACE CHARACTERIZATION
Classification of Surface Characterization The characterization methods for particulate surfaces are shown in Table 6.4. The characterization methods, separate for the dry and the wet processes, are roughly classified; a strict classification is difficult. Many solid surface analyses based on the spectrum can be used to determine composition and chemical conditions. The methods as shown in Table 6.4 are a part of all measurements. For a powder sample, various physical properties are generally measured as the powder sample undergoes compression molding as a disk. When the powder sample is measured using these methods, it is necessary to note the relative size difference between the particle size and the equipment, such as the beam diameter of the probe, penetration depth of the probe, escape depth of the response signal, and diffusion effects of the response signal. It cannot be judged whether the information is for the powder surface or the entire powder. Moreover, the surface roughness of the powder sample that undergoes compression molding influences the detection efficiency. The level of qualitative © 2006 by Taylor & Francis Group, LLC
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TABLE 6.4 Classification of Particle Surface Analysis For Wet Powder and/or Wet Process For Dry Powder and/or Dry Process Composition chemical condition
Magic angle spinning nuclear magnetic resonance: MAS-NMR Fluorescent X-ray photoelectron spectroscopy: AES Secondary ion mass spectrometry: SIMS Electron spin resonance: ESR
Inductively couple plasma: CP Atomic absorption Atomic absorption spectrometry: AAS
Functional groups
Chemical reaction method Infrared Spectroscopy: IR Chemical reaction method Zeta Thermogravimetric and Differential thermal analysis: potential measurement TG/DTA
Acidity and basicity
Gas adsorption, adsorption heat: IR Temperature-programmed desorption: TPD Thermal desorption spectroscopy: TDS
Wettability
Contact angle measurement by direct observation of Heat of immersion, preferential droplet, wet velocity, or AFM, gas adsorption, heat dispersion test of adsorption Inverted gas chromatography
Roughness and porosity
Gas adsorption Scanning electron microscope: SEM Transmission electron microscope: TEM Atomic force microscope: AFM X-ray Computerized tomography: X-CT
Adsorption (CV, titration etc.), ultraviolet and visible spectrophotometry: UV, titration, heat of immersion
Adsorption AFM Magnetic resonance imaging: MRI
measurement is generally appropriate to this method. It is easy for the composition analysis in the wet process to analyze the chemical composition of the entire particle. On the other hand, various instruments are necessary, as only the surface is dissolved to determine information on the surface. However, this method has enough ability to qualitatively analyze the ion, which undergoes segregation to the particulate surface. The analysis of surface functional groups, surface acidity and basicity, wettability, and roughness and porosity, and their measurements are described below.
Surface Functional Groups The analysis method of a surface functional group is chiefly accomplished with a method based on a chemical reaction, spectroscopy methods such as infrared (IR) and nuclear magnetic resonance (NMR) spectroscopy, and methods that combine both approaches. Here, the measurement method of the surface hydroxyl group is explained. A similar method can be applied for other functional groups. The ignition loss method, Grignard reagent method,25 alkyl silane reagent method,16,26–28 and so forth, are chemical reaction methods for surface hydroxyl group determination. The D2O displacement method, which uses IR spectroscopy, is a kind of chemical reaction method.29,30 The chemical reaction method and the analysis methods that use IR spectroscopy are discussed below. The basis of the ignition loss method for an oxide powder is a chemical reaction due to dehydration between the surface hydroxyl groups by heating. The chemical reactions for silica are as follows: ⬅Si– OH ⫹ HO–Si⬅ → ⬅Si–O–Si⬅ ⫹ H2O
(6.10)
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Therefore, if the water molecule generated along with heating is measured by a suitable method, the amount of the hydroxyl can be estimated. The weight change according to heating or the water capacity generated along with heating is then measured. In the method of applying gas chromatography, the generated water is measured with a thermal conductivity detector. It is possible to measure not only the hydroxyl, which exists in the powder surface, but also the water, which exists inside the particle, and this method is very simple and easy.16 It explains the characterization of the surface hydroxyl group on the oxide that uses the chemical reaction method. The chemical reaction method is classified based on the name of the molecule used for the evaluation, such as the Grignard reagent method and the silane method. The Grignard reagent method measures CH4 generated by the reaction shown in Equation 6.11. ⬅Si–OH + CH3MgI → ⬅Si–O–MgI + CH4
(6.11)
The DDS silane method can be used to distinguish the iso-type hydroxyl, which does not form the hydrogen bond, and the hydroxyl of other types according to the difference in the reaction, as shown in the following equations.16 ⬅
Si – OH +
Me
Cl
⬅
= Si ⬅
+
Me
⬅
(6.12)
Me
Si – O Si
Si
OH
Cl
Me
⬅
Si – O
Me
Si – OH
Cl
Me
⬅
Si – O
Me
Si ⬅
Si – OH
Cl
HCI
Me
Cl
Me
Cl
OH
+
Si
Si Cl
Me
Si – O
Si Me
⬅
Si – O
+ 2HCI
(6.13)
+ 2HCI
(6.14)
Me
However, distinction between the gem-type hydroxyl and the H-bonded-type hydroxyl is difficult. The iso-type hydroxyl discharges one molecule of HCl reacting with one molecule of silane. On the other hand, the gem-type hydroxyl or the H-bond-type hydroxyl discharges two molecules of HCl reacting with one DDS molecule of silane. Therefore, the hydroxyl of each type can be separately measured by measuring the amount of consumption of the DDS silane and the amount of HCl generated. The reagents that can be used for the chemical reaction method are shown in Table 6.5. The surface hydroxyl group gives clear IR spectroscopy results. Therefore, IR spectroscopy analysis is an extremely effective evaluation method of the surface hydroxyl group to observe the O–H stretching vibration (3000 to 3800 cm–1). The O–H stretching vibration of the physisorbed water exists in this wave-number region. Therefore, a cell that can do the degassing operation and the heating operation in the system, as shown in Figure 6.10, is used.31,32 The type of surface hydroxyl group that can be identified by IR analysis, as shown in Figure 6.11A, is the H-bonded hydroxyl, which forms the hydrogen bond and the free-type hydroxyl, which does not form the hydrogen bond. There is much research about surface hydroxyl groups. It is believed that three types of hydroxyls with different chemical structures, as shown in Figure 6.11B, exist on the surface of SiO2.33 The iso-type hydroxyl and the gem-type hydroxyl have a narrow absorption at about 3750 cm –1 . It is difficult to distinguish them by IR spectroscopy.34,35 These can be distinguished using29 Si solid-state NMR spectroscopy.36–38 Reports concerning the existence of the tri-type hydroxyl are very few.39,40 As for the Hbonded hydroxyl, wide peak, which centers on about 3600 cm–1, has been observed. In addition, the © 2006 by Taylor & Francis Group, LLC
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TABLE 6.5 Hydroxyls
79
Reagents Used for Determination of Surface
Reagent Diazomethane
Molecular formula Ch3N2
Detected molecule N2
Lithium aluminium hydride
LiAlH4
H2
Trimethylchlorosilane
CH3ClSi
HCl
FIGURE 6.10 An optical cell of in situ measurement for IR spectra.
FIGURE 6.11 Classification of surface silanol (A) according to chemical state, (B) according to chemical structure. © 2006 by Taylor & Francis Group, LLC
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H-bonded hydroxyl marked by the arrow in Figure 6.11A is called the terminal hydroxyl and has an absorption at about 3720 cm–1.41 The wave number of the O–H stretching vibration changes when the kind of atom that bonds the hydroxyls changes. The relation between the electronegativity of the atom that unites the hydroxyls and the OH stretching vibration is shown in Figure 6.12.42 In addition, two or more infrared absorption spectra appear when the oxygen coordination number is different, even if the chemical composition is the same. Figure 6.13 shows the structure and absorption spectrum of the surface hydroxyl group observed on the surface of Al2O3.43,44 Moreover, the dehydration behavior caused by heating of the surface hydroxyl group differs depending not only on the oxide type but also on the preparation method and the hysteresis of the sample. IR spectroscopy is used to measure of strength of the adsorption site and to identify the adsorption site. The history of the research on the observations of the adsorbed molecular state, which uses the infrared spectroscopy and interaction with the surface, is long. Especially, many of the interactions of the surface hydroxyl group and the adsorbed molecule have been determined for the oxide.45 Kiseleve et al.46 found the relationship between the heat of formation of the hydrogen bond between the organic molecule and the hydroxyl, and wave number shifts of the O–H stretching vibration by adsorption from the heat of adsorption measurement of the organic molecule and the IR spectroscopy measurement for silica. These results are shown in Figure 6.14. Because the solvent molecule, in addition to the adsorbed molecule, exists in the liquid-phase adsorption, this adsorption behavior is more complex than gas-phase adsorption. However, the interaction between the surface hydroxyl group and the adsorbed molecule can be examined by IR spectroscopy and calorimetry, as well as by gas-phase adsorption.47,48 The cleavage reaction of (M–O–M) on the oxide surface in addition to the reaction of the surface hydroxyl group is an important characteristic of the powder surface. Because the reactiveness to water is the same as the mechanism of surface hydroxyl group generation, it is considered an index of the chemical stability of the water molecule on the surface. Moreover, some reactivities with other molecules have been reported. Morrow et al.49 observed the dissociation of NH3 to the siloxane linkage (⬅Si–O–Si⬅) and adsorption from the change in the IR absorption spectrum of the NH3 adsorption on silica gel. Bunker et al.50 found the structure of reactivity siloxane that tetrahedron SiO4 share the edge from the analysis of the IR absorption spectrum. SiO2 is a coupled structure of the SiO4 tetrahedron. The ring structure with three types of distortion shown in Figure 6.15 is reported in IR spectroscopy50,51 and Raman spectroscopy.52 Especially important, the structure that the SiO4 tetrahedron shares the edge and the D2 structure has a big distortion. Therefore, it is confirmed that various molecules easily
FIGURE 6.12 Relation between electronegativity of atom with hydroxyl and stretching vibration of free hydroxyl. © 2006 by Taylor & Francis Group, LLC
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Self supported sample
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Vacuum
Cock
CaF 2 window
Sample holder Thermocouple FIGURE 6.13 Structures and stretching frequencies of hydroxyl observed in alumina surface: (A) free hydroxyl, (B) hydrogen-bonded hydroxyl. O, octahedral coordination; T, tetrahedral coordination. 81
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(a) H O
H O
H O
H O
H O
Si
Si
H O
H O
Terminal
Si
Si
H-bond type
free type
(b)
H O
Si
H O
H O
HHH OOO
Si
Si
Si
iso. type
gem. type
tri. type
FIGURE 6.14 Relation between binding formation heat of various molecules and silanols and peak shift values of surface hydroxyl by adsorption.
have a dissociation adsorption on the surface with these structures.53–55 The wave number corresponding to these structures and the calculation value of the strain energy are shown in Table 6.6.56 It is expected that the amount of siloxane of such reactiveness differs according to the production method and the hysteresis of the silica.
Surface Acidity and Basicity The strength of the acid–base of the solid surface applies to the definition of the H 0 function and H⫺ function, which shows the strength of the acidity/basicity of a uniform solvent. That is, the acid strength of the solid surface is defined by H0, and the base strength is defined by H⫺. For instance, when indicator B of basicity is adsorbed on the acid site of the solid surface, a part of the indicator is made a proton with the acid site. The strength H0 of these acid sites is shown in Equation 6.15 by the value of pKBH+ of BH+. which is the conjugate acid of indicator B when the concentration CBH+ of indicator BH+ made a proton is equal to the concentration CB of indicator B not made a proton. H0 ⫽ pKBH+ – log(CBH+/CB)
(6.15)
Therefore, the acid strength indicated by the H0 value shows the ability to change into the conjugate acid by the acid sites which give the protons half of the base indicator B of the neutral which the acid site on the surface adsorbs. In Lewis’ definition , the H0 value shows the ability that the electron pair can be received from half of the adsorbed base indicator B. The investigation of base strength is similar to the investigation of acid strength. When indicator BH, which defines base strength, © 2006 by Taylor & Francis Group, LLC
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83
3900 Al
Stretching vibraction (cm-1)
3800
Mg
Si B
Cu 3700
Ge P
S
3600
Cl N
3500
3400 1
2 Electronegativity
3
: hydroxyl on oxide surface : hydroxyl of oxyacid FIGURE 6.15 Distortion ring structures into silica.
TABLE 6.6 Infrared and Raman Absorption and Strain Energy of the Structure of [SiO4] in Silica Ring structure
Absorption Spectrum
Spectroscopy
Strain energy
4-membered
888/908 cm⫺1
Infrared
58 kJ/mol (Si–O)
6-membered
605 cm⫺1 (D2 )
Raman
17 kJ/mol (Si–O)
8-membered
490 cm⫺1 (D1)
Raman
—
adsorbs on the solid surface, H+ is pulled out by the basic site, and a part of indicator BH becomes the state of B–. The strength H– of the basic site is shown by the pK BH value of indicator BH when concentration CB– of B– is equal to the concentration CBH of BH from which H+ is not pulled out. However, because the measurement of the ratio of the concentration (CBH+/CB) and (CBH/CB–) of the indicator on the solid surface is difficult, the following techniques are actually used. Because it is © 2006 by Taylor & Francis Group, LLC
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possible to apply it to both the acid and the base, the evaluation method of the acid site is described here. Many indicators of basicity in which the charge neutrality becomes a conjugate acid by receiving H+ are known as Hammett indicators. Here, each pKa value of these conjugate acids is assumed to be b, c, d, and e (b < c < d < e). The indicator pKa ⫽ b is also assumed not to be changed by the acid structure at the acid site, nor is the indicator pKa ⫽ c assumed to have been changed by the acid structure. At this time, it is understood that the acid strength H0 value of the surface acid site is between b and c. The indicator causes a discoloration, and the change occurs in the UV spectra when the indicator is changed into the acid structure by the solid acid site. Therefore, the strength and the amount of the acid site are measurable by using these phenomena. An example of the determination of the acid site by the discoloration of the indicator is shown as follows. p-dimethylaminoazobenzene (yellow), which is the indicator, is adsorbed on the solid acid site and becomes red with the following acid structures:
(6.16)
n-butylamine, which is a stronger basicity molecule than the indicator, is dropped by the following procedure. Then, n-butylamine causes the exchange adsorption as the indicator of the acid structure which has been adsorbed according to the next equation, and the indicator returns to the former yellow color.
(6.17)
Acid strength is understood from the difference in the base strength of the indicator used. Moreover, the amount of the acid site can be obtained according to the amount of n-butylamine necessary to erase the color of the acid structure. The reacted indicator is made the acid structure, and discolor, even if the acid sites are a Brønsted acid site or a Lewis acid site. It is possible to do the same for the evaluation of basicity. Indicators to examine acidity and basicity are shown in Tables 6.7 and 6.8, respectively.57 In this procedure, it is necessary to leave it for a long time at the time of each n-butylamine dropping to achieve an adsorption equilibrium and confirm that the color of the acid structure disappears. Therefore, a long time is required for the measurement. The Benesi method in which this fault is improved is widely used. In the Benesi method, many sample solutions, which have different dosages of n-butylamine, are prepared. After the sorption equilibrium isotherm of n-butylamine, various indicators are added for each sample. The discoloration of these indicators to the acid structure is examined, and the acid strength distribution is measured. The intensity distribution of the basic site in the solid surface is obtained by the same principle. That is, one only has to use an acid molecule such as benzoic acid as a titration reagent instead of n-butylamine, and the indicator for the H– measurement is used. Adsorption of the basic gaseous molecule is caused on the acid site, and the absorbed amount corresponds to the acidity. The adsorptive interaction of the basicity molecule that adsorbs on a © 2006 by Taylor & Francis Group, LLC
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TABLE 6.7
85
Indicators Used for Acid Strength Measurements Indicator
Neutral red
Basic Color Colorless
Acid Color Red
pK3H+ +6.8
H2SO4 wt. % 8 x 10-8
Methyl red
Yellow
Red
+4.8
—
Phenylazonaphthylamine
Yellow
Red
+4.0
5 x 10-5
p-Dimethlaminoazobenzene
Yellow
Red
+3.3
3 x 10-4
2-Amino-5-azotoluene
Yellow
Red
+2.0
0.005
Benzeneazodiphenylamine
Yellow
Purple
+1.5
0.02
4-Dimethylaminoazo-1-naphthalene
Yellow
Red
+1.2
0.03
Crystal violet
Blue
Purple
+0.8
0.1
diphenylamine
Yellow
Purple
+0.473
—
Dicinnamalacetone
Yellow
Red
–3.0
48
Benzalacetophenone
Colorless
Yellow
–5.6
71
Anthraquinone
Colorless
Yellow
–8.2
90
p−Νitrotoluene
290 μm
350 μm
−10.5
99.9
2,4-Dinitrotoluene
255 µm
320 µm
–12.8
106
p-Nitrobenzeneazo-(p⬘-nitro)-
TABLE 6.8
Indicators Used for Base Strength Measurements
Indicator Bromothymol blue
Acid Color Yellow
Basic Color Blue
pKBH 7.2
Phenolphthalein
Colorless
Pink
9.3
2,4,6-Trinitroaniline
Yellow
Tango
12.2
2,4-Dinitroaniline
Yellow
Purple
15.0
4-Chloro-2-nitroaniline
Yellow
Orange
17.2
4-Nitroaniline
Yellow
Orange
18.4
4-Chloroaniline
Colorless
Pink
26.5
strong acid site is strong, and desorption is difficult at low temperature. Therefore, the desorption temperature of the adsorbed base molecule is closely related to the strength of the acid site. Several measurement examples are given as follows. The base molecules that adsorb onto the surface, and then the physisorbed and hydrogen-bonded molecules are removed at a suitable temperature when basicity gases such as ammonia, pyridine, and alkylamines are used. The amount of the base molecules, which chemically adsorb on the acid site, is measured as follows. The temperature gradually rises while removing the desorption gas, and the amount of chemisorption at each temperature is measured; or the temperature continuously rises while removing the desorption gas, and the intensity distribution of the acid site is obtained as the desorption spectrum of the base molecule. The amount of the desorption base as the amount of chemisorptions is measured by the weight method with a balance or by gas chromatography. That is, the kind of site where the adsorption activity is different is distinguished from the number of desorption peaks in the temperature-programmed desorption method. Moreover, the chemical bonding state and strength of the acid site are obtained from © 2006 by Taylor & Francis Group, LLC
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the difference in the desorption temperature, and the amount of the acid site is obtained from the amount of each desorption. The activation energy of desorption assumes it is almost equal to the heat of adsorption, and it is used as an index of the strength of the acid. The next equation, for the temperature-programmed desorption method, shows a relation between the programming rate (Vt) and desorption peak temperature (Tm), by which the maximum elimination kinetics is given when the gas, in which the strength of the acid site is uniform, and when detached is not adsorbed again. 2log Tm – log Vt = Ed/(2.303RTm) + log(Ed/AR)
(6.18)
A is a frequency factor, Ed is the activation energy, and R is the gas constant. The sample with the same absorbed amount of the base probe molecule is prepared, the programming rate is changed, and the desorption peak temperature Tm corresponding to the changes is obtained. The activation energy, Ed, is obtained from the slope of the straight line of the plot of 2log Tm – log Vt to 1/Tm. As for the features of the gas absorption method, the solvent has no influence, and the surface acidity at various temperatures can be measured. Moreover, as another advantage, the amount of the initial adsorption can be easily controlled, and a catalytic reaction mechanism and kinetics data might be able to be comparatively facilitated. On the other hand, strong physisorption and chemisorption on powder surfaces cannot be distinguished, which is a disadvantage. In addition, because the diffusion process of the gas cannot be disregarded and re-adsorption of desorption gas occurs, obtaining an accurate kinetic parameter might be difficult. Moreover, when the amount of the acid site is measured using ammonia and n-butylamine is considered, as the adsorption mechanism to the acid site is the same, the amount obtained with ammonia might be large. It is considered that the pore distribution and the distance between acid sites are closely related to this difference. The evaluation of the acid–base site where IR spectroscopy was used is subsequently introduced. The amounts of the Brønsted acid site and the Lewis acid site are measured from the strength of IR spectroscopy of the chemically adsorbed base molecules. The IR absorption spectrum of pyridine, which adsorbs on the Brønsted acid site and the Lewis acid site, is shown in Figure 6.16.58 The infrared absorption spectrum in the 1400 to 1700 cm–1 region is derived from the in-plane vibration of the pyridine ring. The wave number is remarkably different according to whether the adsorbed pyridine forms a hydrogen bond, a coordinate bond, or an ionic bond. The infrared absorption spectrum features of the adsorbed pyridine are shown in Table 6.9. Moreover, the acid site type can be identified according to the IR absorption spectrum of the adsorption ammonia. The reason for this is that the IR absorption of the deformation vibration of H–N–H changes depending on the kind of adsorption site. As for the ammonium ion (NH4+) and ammonia, which formed the coordinate bond, absorption appears at about 1400 cm–1 and at about 1620cm–1, respectively. The former is a Brønsted acid site, and the latter is a Lewis acid site.59,60 There is a method of measuring the thermal transformation when the probe molecule directly adsorbs on the acid–base site. Because a suitable basic or acidic molecule can be used, the adsorbate and the temperature of the heat of adsorption measurement can be arbitrarily selected according to the calorimeter; acid strength can be obtained under various conditions. For the heat of adsorption measurement, a suitable amount of sample is first placed in the cell, and the heating exhaust processing is done under optimum conditions. The sample is placed in the calorimeter with the heating exhaust processing done. The adsorbate is introduced after it reaches the thermal equilibrium state and the generated heat of adsorption is measured. The absorbed amount is obtained at the same time. In general, because the change in the heat of adsorption is large in the region where the absorbed amount is low, a little probe gas is introduced. The heat of adsorption is converted into the amount of 1 mol adsorbate, and a graph of absorbed amount (V) versus heat of adsorption (q) is obtained. If ΔV/Δq is obtained from the graph based on the V–q curve, and ΔV/Δq is plotted for q, the intensity distribution curve of the acid–base is obtained. © 2006 by Taylor & Francis Group, LLC
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(a)
87
H
H
H
H
H
O
O
O
O
O
AI
AI
AI AI
AI AI
AI AI AI
[O]
[T]
[O] [O]
[T] [O]
[O][O][O]
3750 Wavenumber (cm-1)
3800 (b)
3600
H
H
H
O
O
O
AI
AI
AI AI
3400 Wavenumber (cm-1)
3700
3200
FIGURE 6.16 Infrared absorption spectrum of pyridine: (A) chloroform solution of pyridine, (B) chloroform solution of pyridine: BH3, (C) chloroform solution of (pyridine:H) + Cl–. The absorption band at 1520 is wide and is an absorption peak of the chloroform solvent.
TABLE 6.9 IR Absorption Band Assigned Pyridine Molecules Adsorbed on Acidic Surface Hydrogen Bonded
Coordinately Bonded
Ionically Bonded
1400–1447 (VS)
14.7–1465 (VS)
1480–1490 (W)
1471–1503 (W-S) 1525–1542 (S)
1562–1580 (W)
1580–1600(S)
1592–1633 (S)
1600–1620 (S)
1478–1500(VS)
1634–1640 (S) Note: Absorbance strength: VS, very strong; S, strong; W, weak.
Wettability For the large particle shown in Figure 6.17, the contact angle can be directly measured with a microscope.61 Moreover, the contact angle is directly obtained by attaching the particle to the AFM probe, as shown in Figure 6.18. 62 However, it is difficult to directly measure the contact angle in a general powder. It assumes as follows, and the contact angle is calculated. It is considered that a minute space in the packed bed of the powder shown in Figure 6.19 is formed with the capillary having a © 2006 by Taylor & Francis Group, LLC
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Heat of formation of hydrogen bond (kcal/mol)
88
12 17
10 14
8 13
9
15 16
6
11
12
10
4
: Ar 2 : n-C H 3 : 4: N 6 : CCl : C H 7 : CH NO 8 : O 9 : CH COOC H 10 : CH CN 11 : CO 12 : O O O 15 : 13 : CH COCH 14 : (C H ) O 16 : N 17 : (C H ) N 1
6
5
8
4
14
6
2
6
2
3
3
5
2
3
7
2 4
3
3 5
6
2
5
3
2
2
5
3
1 2
0 0
200
400
600
800
1000
1200
Peak shift (cm -1 ) FIGURE 6.17 Contact angle measurement for large particle.
uniform cylindrical shape. The contact angle can be measured according to the velocity of the liquid as it infiltrates into this capillary. The following equation holds when the radius of the cylindrical capillary in the powder-packed bed is assumed to be r. dx rg cos u r 2 rg ⫽ ⫺ dt 4hx 8h
(6.19)
Here, x is the height to which the capillary rises, t is the time, g is the liquid surface tension, h is the viscosity of the liquid, r is the density, g is gravitational acceleration, and u is the contact angle. The first term on the right in Equation 6.19 is the contribution of the capillary force. The second term is the contribution of gravity. When the second term can be neglected, the following equation is obtained by integrating Equation 6.19. x2 ⫽
rg cos u t 2h
(6.20)
Therefore, the capillary radius r in the powder-packed bed is obtained from the straight line relation between x2 and t beforehand by using the liquid which has contact angle u = 0 where the powder is wet. The contact angle u is then obtained by measuring the relation between x2 and t for the powder with a target liquid. Moreover, the wettability evaluation of the powder is determined from the preference dispersing test of the powder in various liquids or mixed solvents. Additionally, the wettability can be judged according to the analysis of the adsorption isotherm, heat of adsorption, and heat of immersion.63 © 2006 by Taylor & Francis Group, LLC
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Si
Si
Si
Si
O O
O
O Si
4-membered ring
6-membered ring (D2)
Si
O
O Si
Si O
O
Surface Properties and Analysis
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O
Si
8-membered ring (D1)
FIGURE 6.18 Schematic of a force–position curve between a spherical particle and a bubble. The position is the height position of the bubble. It was assumed that no long-range forces are active between the particle and bubble before contact and that the particle surface is not completely wetted by the liquid. At large distances the cantilever is not deflected (A). When the particle comes into contact with the air–water interface, it jumps into the bubble and a TPC line is formed (B). The reason for the jump-in is the capillary force. Moving the particle further toward the bubble shifts the TPC (three points contact) line over the particle surface (C). The important factor is the receding contact angle θ . When retracing the particle again (D), at some point the force is high enough to draw the air–water interface (E). The drawing of right side is a particle at equilibrium in the air–water interface. θ is the equilibrium contact angle.
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1440 1583
Absorbance
1480 ~1520
1600 (a)
1465
1490
1630
~ 1520 1585
1485 1540
(b)
16101640 (c)
1400
1500 1700 1600 Wavenumber (cm)
FIGURE 6.19 Contact angle estimated from wet velocity of powder bed.
Surface Roughness and Porosity In gas absorption, the absorbed amount changes according to the adsorbing gas pressure change observed at a constant temperature. A lot of information can be obtained by analyzing this isothermal line. For instance, the specific surface area, which uses the BET theory, is the most popular analytical result. The adsorption isotherm can be classified according to type. Here, the classifi cation by which the isothermal line of the step is added to the BDDT classification is shown in Figure 6.20. 64 Type I is called the Langmuir type, and much chemisorption and adsorption into the micropores is observed. Type II is adsorption of the multilayers in the isothermal line, which is called the BET type © 2006 by Taylor & Francis Group, LLC
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91
particle liquid contact angle
center
FIGURE 6.20 Classification of adsorption isotherms.
and is often observed because of nitrogen adsorption on a solid without micropores and mesopores. Type III is observed when the interaction of the solid surface and the adsorbed molecule is weak; it is often observed in water vapor adsorption on a hydrophobic surface.65 Type IV and type V are shapes when hysteresis is observed in the isothermal line. When the solid, which has the adsorption isotherm of type II or type III, has mesopores, the adsorption isotherm becomes type IV and type V, respectively. However, the porous silica with mesopores from which hysteresis in the adsorption isotherm is not observed has been synthesized in recent years,66 and the disappearance of hysteresis has been discussed.67 When the adsorption isotherm is on an extremely uniform surface and the phase transition occurs in the adsorption phase, a step is observed. For instance, there is Kr adsorption on a graphite surface. 68 The classification of such an isothermal line is extremely qualitative information. However, it is useful as a means to know the outline of the structure and the characteristics of the powder surface. The roughness factor, r, is used as an index which shows the complexity of the solid surface.
r⫽
SN 2
(6.21)
Sd
SN2 is the surface area measured by N2 adsorption, and Sd is the apparent surface area estimated from particle diameter. When r is larger than 1, it is presumed that roughness on a molecular order exists in the surface. Recently, an attempt was made to describe the complex structure of the solid surface by fractal dimensions.69 Some methods of obtaining the fractal dimension of the solid surface have been suggested.70–72 Here, we show an example for obtaining it from the adsorption of the molecule with a different size. The following equation occurs when the surface fractal dimension, Da, occupation area, s, of the molecule used for adsorption, and n, a monomolecular absorbed amount are used. log n ⫽⫺ ( Da 2) log s ⫹ const.
(6.22)
Therefore, Da is obtained from the plot of log n and log s. Figure 6.21 shows the result of obtaining the fractal dimension of the activated carbon surface.73 As a result, Da of the activated carbon is 3.03. In general, the fractal dimension of the solid surface indicates the value of 2 < Da < 3. Da ⫽ 2 is a surface as smooth as a graphite surface. On the other hand, Da approaches 3 as the surface structure becomes complex. Silica gels and active carbon are examples of Da approaching 3. It is only recently that a solid surface was described by fractal dimensions. Therefore, the relation between a © 2006 by Taylor & Francis Group, LLC
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92
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Water C
A R r
Position D
B D
E
Detachment force
FIGURE 6.21 Monolayer volume, Vm, as a function of the cross-sectional area, σx.
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Bubble =
R-D R
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D
D
Surface Properties and Analysis
93
fractal dimension and various surface physical properties has not yet been clarified. However, there are some advance reports concerning the relation between catalytic kinetics and the fractal dimensions,74 the relations between the fractal surface and wettability,75 and structure predictions of the surface modification group. 76 Future developments are expected in the form of a new index which shows the complexity of the surface structure of a solid.
1. 6.3 ATOMIC FORCE MICROSCOPY These days, various advanced materials have been produced from nanoscale functional particles. This implies that the molecular-scale information of particle properties, such as the shape and surface property of particles and the interaction forces between them in the medium, is extremely important to control the quality of particulate products. The microscopic shape and size of particles have been evaluated, using electron microscopes (SEM and TEM), but they are observed only under the high vacuum state, so the information obtained there is not necessarily appropriate for the real production processes of materials. In 1982, the Scanning Tunneling Microscope (STM) was invented to observe the atomic image of surfaces by measuring the tunneling current between the cantilever tip and conductive surfaces. This invention of STM stimulated the further invention of the atomic force microscope (AFM) in 1985, which enables us to image the atomic-scale properties of nonconductive surfaces either in vacuum or in any medium. For engineers who want to know what is happening in the real processes, the information given by AFM has been extremely valuable, because of the following features of AFM: 1. Molecular-scale information of local surface properties and interaction forces between surfaces in any medium is obtainable. 2. In situ measurements are possible. 3. The apparatus is easy to handle. The detailed explanation of AFM is given elsewhere, examples of the data obtained are given below.
77,78
but the main principle of AFM and
Apparatus of Atomic Force Microscope The AFM apparatus is composed of the measuring, controller and data-treatment units. As illustrated in Figure 6.22, the measuring unit is composed of a laser beam generator and a photo detector, the fluid cell with a cantilever and a sealing o-ring and the piezo system whose atomic-scale movement of the horizontal x/y direction and vertical z-direction is controlled by the controller unit. The probe popularly used is made of Si or Si3N4. The tip radius of curvature is 10~60 nm, the cantilever length is 100~500 nm, and the nominal spring constant is 10-2~102 N/m. The sample plate glued on the metal plate is attached magnetically on the piezoelectric crystal, which can move to the vertical direction with a constant speed by controlling the voltage of the piezo scanner. The cantilever deflection due to the interaction between the tip and sample surfaces is detected as the voltage change of the photo detector onto which a laser beam reflected from the rear of the cantilever is focused. This deflection is converted into the interaction force, applying Hooke’s law to the cantilever. The zero separation is determined from the onset of the “constant compliance” region, where the cantilever deflection is linear with the surface displacement. The separation distance between surfaces is evaluated by the displacement of the sample plate from the constant compliance region. This principle of AFM can be used for two different purposes.
Evaluation of Surface Property by AFM The AFM was originally constructed to evaluate the surface roughness on the atomic scale by using a so-called contact-mode procedure. In this case, the surface is scanned to x/y direction, keeping the separation between the tip and sample surfaces constant by the feedback control to the © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.22 Schematic illustration of the principle of Atomic Force Microscope (AFM) and the measuring modes.
piezo scanner. Then the feedback data are converted to the so-called height image of the surface. It is known that the height of the surface roughness obtained by this method is accurate enough on the molecular scale, but the width depends on the radius of curvature of the tip used. The similar image of roughness is also obtainable by using the data of cantilever deflection accumulated during the scanning without the feedback adjustment. This image is called the deflection image, which is not as accurate as the height image, because the contacting force varies during the scan. Hence, the height mode is popularly employed. These contact-mode procedures can not be used with soft surfaces, because the tip may scratch off the part of the soft surface. For soft surfaces, the so-called tapping mode is usually employed. In this mode the cantilever piezo system oscillates the cantilever at a given frequency. When the tip interacts with the sample surface, the amplitude of the cantilever damps. Hence, if the sample plate is scanned, keeping the amplitude of the cantilever constant by the feedback control, the height image of the surface can be obtained. Because the possibility of scratching off the surface is very small in this case, the tappingmode procedure is widely used to gain the image of soft surfaces, especially in the case of the surface with adsorbed materials, such as surfactants and polymers. Figure 6.23 shows an example of AFM image of the polymer chains adsorbed on the mica surface, which was measured in situ in water.79,80 This success to take the AFM image of single polymers was followed by the AFM observation of the nuclei formation of latex particles.81 If the phase difference exists between the input given by the cantilever piezo system and the detected output during scanning, which is caused by the interaction between the tip and sample surfaces, the phase image of the surface is obtainable. This image is often used to gain the measure of the rigidity of surfaces. If the data of amplitude are taken by fixing the height constant, the amplitude image can be obtained.
Usage of AFM for Other Surface Properties In the case of usual AFM measurements, the surface properties are measured by using the intermolecular forces which exist commonly between surfaces. If there exist some other forces between © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.23 An example of a height image of high cationic polymers (poly[2-(acryloy loxy)ethyl(trimethyl)ammoniumchloride], molecular weight ⫽ 1.25⫻107 g/mol) adsorbed on a mica surface measured in situ in water.
surfaces, the AFM can to be used to detect the force by employing the probe tip, which interacts with surface: 1. Magnetic Force Microscope (MFM) If the probe tip is made of magnetic material, the magnetic interaction between the tip and sample surfaces is detected, so that the magnetic map of the surface can to be obtained even if the surface is completely flat. 2. Lateral (Friction) Force Microscope (LFM) When a probe (usually colloidal probe) is pressed against the substrate at a constant applied force and slides horizontally, the friction force between surfaces can be detected from the torsion of the cantilever. The magnitude of lateral force is determined by using a four divided photo detector. It is found that the lateral force is much more sensitive to the roughness of the sample surface, compared with the normal force measurements.82
Evaluation of Interaction Forces between Surfaces in Media As explained above, the essential principle of AFM is to measure the interaction force between surfaces. By using this principle without scanning, one can evaluate the interaction forces between surfaces in any medium. The interaction between the probe tip and the sample surface is usually very small because of the small radius of curvature of the tip. This difficulty was overcome by gluing a micron-size colloidal particle on the top to the cantilever, shown as a colloid probe in Figure 6.22.83 A typical force curve given by AFM force measurements is schematically illustrated in Figure 6.24. © 2006 by Taylor & Francis Group, LLC
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From the one cycle of measurement, one can measure the long-range interaction force, the shortrange interaction force, and the adhesive force between surfaces simultaneously. The long-range interaction force is usually consistent with the force curve predicted by the DLVO theory described in 2.8.2, but the short-range interaction force depends on the adsorbed layer of water molecules, ions, and so on.84-86 An interesting example of the short-range interaction force is given in Figure 6.25, where a step-wise force curve was obtained for a 98 wt% alcohol solution. This indicates that the
FIGURE 6.24 Schematic illustration of the force curve (the interaction forces vs. the surface separation) obtained by AFM.
FIGURE 6.25 An interesting example of the short-range interaction, the step-wise force curve found for the interaction between a probe tip and a mica surfaces in an alcohol solution with 2wt% water, and the corresponding mechanism. © 2006 by Taylor & Francis Group, LLC
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alcohol molecules are adsorbed on the mica surface standing, and the step-wise force curves arise when the structured alcohol layers on both surfaces are broken, as illustrated in the figure. 87,88 Another important contribution by AFM force measurements is to clarify the fact that the longrange attractive force between hydrophobic surfaces in aqueous solutions, as shown in Figure 6.26, which was a debated issue for a long time, originated from the coalescence of nanobubbles attached to the surfaces.89,90 As described above, the AFM is a very powerful tool for investigating the surface microstructure and interactions between surfaces in media, mainly because the molecular-scale information is obtainable for any kind of the combination of surfaces and media without any high vacuum as in the case of SEM and TEM. Hence, now a huge number of data obtained by AFM have been reported in the wide range of research fields. Some AFM data overturned mechanisms predicted by macroscopic experiments that have been believed for a long time. However, there exists an inevitable problem of AFM; one cannot know the genuine separation distance between surfaces, because the zero separation is defined at the point where the piezo movement coincides with that of the cantilever. If there exists an adsorbed layer into which the probe tip can not penetrate, the separation between the tip surface and the outer surface of the adsorbed layer is regarded as the zero separation.
FIGURE 6.26 An interesting example of the long-range attractive force due to nanobubbles on hydrophobic surfaces in water. © 2006 by Taylor & Francis Group, LLC
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Kubo, R., Phys. Lett., 1, 49–50, 1962. Chikazawa, M. and Takei, T., Gypsum Lime, 228, 255–267, 1990. Benson, G. C. and Claxton, T. A., J. Chem. Phys., 48, 1356–1364, 1968. Taroni, A. and Haneman, D., Surf. Sci., 10, 215–221, 1968. Nakamatsu, H., Sudo, A., and Kawai, S., Surf. Sci., 194, 265–271, 1988. Schlier, R. E. and Farnsword, H. E., J. Chem. Phys., 30, 917–925, 1959. Pandey, K. C., Phys. Rev. Lett., 49, 223–229, 1982. Morimoto, T., Nagao, M., and Tokuda, F., Bull. Chem. Soc. Jpn., 41, 1533–1539, 1968. Morimoto, T. and Naono, H., Bull. Chem. Soc. Jpn., 46, 2000–2006, 1973. Naono, H., Kodama, T., and Morimoto, T., Bull. Chem. Soc. Jpn., 48, 1123–1129, 1975. Zhuravlve, L. T., Langmuir, 3, 316–322, 1987. Chikazawa, M. and Kanazawa, T., J. Soc. Powder Technol. Jpn., 14, 18–25, 1977. Harnb, N., Hawkins, A. E., and Opalinski, I., Trans. IchemE, 74, 605–626, 1996. Chikazawa, M., Kanazawa, T., and Yamaguchi, T., KONA (Powder Particle), 2, 54–61, 1984. Chikazawa, M. and Tajima, K., Kaimenkagaku, Maruzen, Tokyo, 2001, pp. 14–14. Kanazawa, T., Chikazawa, M., Takei, T., and Mukasa, K., J. Ceram. Soc. Jpn., 92, 11, 654–660, 1984. Ozaki, A., Shokubai-kougaku-kouza 10, Chijinsyoten, Tokyo, 1972, p.778. Auroux, A., Gervasini, A., J. Phys. Chem., 94, 6371–6376, 1990. Tanabe, T., Sumiyoshi, T., Shibata, K., Kiyoura, T., and Kitagawa, J., Bull. Chem. Soc. Jpn., 47, 1064–1070, 1974. Shibata, K., Kiyoura, T., Kitagawa, J., Sumiyoshi, T., and Tanabe, T., Bull. Chem. Soc. Jpn., 46, 2985–2991, 1973. Yoshida, S. and Kawakami, H., Hyomen, 21, 737–743, 1983. Yoon, R. H., Salman, T., and Donnay, G., J. Colloid Interface Sci., 70, 483–489, 1979. Fuji, M., Machida, K., Takei, T., Watanabe, T., and Chikazawa, M., J. Phys. Chem. B, 102, 8782–8787, 1998. Fuji, M., Machida, K., Takei, T., Watanabe, T., and Chikazawa, M., Langmuir, 15, 4584–4589, 1999. Fripiat, J. J. and Uytterhoeven, J., J. Phys. Chem., 66, 800–806, 1962. Hair, M. L. and Hertl, W., J. Phys. Chem., 73, 2372–2378, 1969. Van Der Voort, P., Gillis-D’hamers, I., and Vansant, E. F., J. Chem. Soc. Faraday Trans., 86, 3751– 2757, 1990. Yoshinaga, K., Yoshida, H., Yamamoto, Y., Takakura, K., and Komatsu, M., J. Colloid Interface Sci., 153, 207–213, 1992. Madeley, J. D., Richmond, R. C., and Anorg, Z., Allg. Chem., 389, 92–98, 1972. Davydov, V. Ya., Kiserev, A. V., Kiserev, S. A., and Polotnyuk, V. O. V., J. Colloid Interface Sci., 74, 378–386, 1980. Yates, Jr., J. T. and Madey, T. E., Vibrational Spectroscopy of Molecules on Surfaces. Plenum, New York, 1987, p. 116. Kondo, J. and Domen, K., Shokubai (Catalysts Catalysis), 32, 206–212, 1990. Iler, R. K., The Chemistry of Silica, John Wiley & Sons, New York, 1979, p. 622. Hoffmann, P. and Knozinger, E., Surf. Sci., 188, 181–187, 1987. Ferrari, A. M., Ugliengo, P., and Garrone, E., J. Phys. Chem., 97, 2671–2677, 1993. Sindrof, D. W. and Maciel, G. E., J. Am. Chem. Soc., 105, 1487–1493, 1983. Leonardelli, S., Facchini, L., Fretigny, C., Tougne, P., and Legrand, A. P., J. Am. Chem. Soc., 114, 6412–6418, 1992. Hayashi, S., J. Soc. Powder Technol. Jpn., 32, 573–579, 1995. Takei, T., J. Jpn. Soc. Colour Mater., 69, 623–631, 1996. Hauuka, S. and Root, A., Langmuir, 98, 1695–1701, 1994. Morrow, B. A., Cody, I. A., and Lee, L. S. M., J. Phys. Chem., 80, 2761–2767, 1976. Smirnov, E. P. and Tsyganenko, A. A., React. Kinet. Catal. Lett., 7, 425–431, 1977. Knozinger, H. and Ratnasamy, P., Catal. Rev. Sci. Eng., 17, 31–37, 1978. Ballinger, T. H. and Yates, Jr., J. T., Langmuir, 7, 3041–3047, 1991. Little, L. H., in Infrared Spectra of Adsorbed Species, Kagaku-doujin, Kyoto, 1971, pp. 246–308.
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46. Curthoys, G., Davydov, V. Y., Kiselev, A. V., and Kiselev, S. A., J. Colloid Interface Sci., 48, 58–64, 1974. 47. Korn, M., Killmann, E., and Eisenlauer, J., J. Colloid Interface Sci., 76, 7–13,1980. 48. Zhao, Z., Zhang, L., and Lin, Y., J. Colloid Interface Sci., 166, 23–29, 1994. 49. Morrow, B. A. and Cody, I. A., J. Phys. Chem., 80, 1998–2004, 1976. 50. Bunker, B. C., Haaland, D. M., Ward, K. J., Michalske, T. A., Smith, W. L., and Balfe, C. A., Surf. Sci., 210, 406–421, 1989. 51. Chiang, C., Zegarski, B. R., and Dubois, L. H., J. Phys. Chem., 97, 6948–6956, 1993. 52. Krol, D. M. and van Lierop, J. G., J. Non-Cryst. Solids, 63, 131–137, 1984. 53. Bunker, B. C., Haaland, D. M., Michalske, T. A., and Smith, W. L., Surf. Sci., 222, 95–101, 1989. 54. Grabbe, A., Michalske, T. A., and Smith, W. L., J. Phys. Chem., 99, 4648–4656, 1995. 55. Wallance, S., West, J. K., and Hench, L. L., J. Non-Cryst. Solids, 152, 101–107, 1993. 56. O’Keeffe, M. and Gibbs, G. V., J. Chem. Phys., 81, 876–884, 1984. 57. 58. Little, L. H., in Infrared Spectra of Adsorbed Species, Kagaku-doujin, Kyoto, 1971, pp. 202–203. 59. Mapes, J. E. and Eischens, R. P., J. Phys. Chem., 58, 809–815, 1954. 60. Pliskin, W. A., Eischens, R. P., J. Phys. Chem., 59, 1156–1162, 1955. 61. Fuji, M., Fujimori, H., Takei, T., Watanabe, T., and Chikazawa, M., J. Phys. Chem. B, 102, 10498– 10504, 1998. 62. Preuss, M. and Butt, H.-J., J. Colloid Interface Sci., 208, 468–477, 1998. 63. Fuji, M., Takei, T., Watanabe, T., and Chikazawa, M., Colloid Surf. A, 154, 13–24, 1999. 64. Burnauer, S., Deming, L. S., Deming, W. E., and Teller, E., J. Am. Chem. Soc., 62, 1723–1729, 1940. 65. Iwaki, T. and Jellinek, H. H. G., J. Colloid Interface Sci., 69, 17–23, 1979. 66. Kresge, C. T., Leonowicz, M. E., Roth, W. J., Vartuli, J. C., and Beck, J. S., Nature, 359, 710–716, 1992. 67. Ravikovitch, P. I., Domohnaill, S. C. O., Neimark, A. V., Schuth, F., and Unger, K. K. Langmuir, 11, 4765–4771, 1995. 68. Amberg, C. H., Spencer, W. B., and Beebe, R. A., Can. J. Chem., 33, 305–311, 1995. 69. Pfeifer, P. and Avnir, D., J. Chem. Phys., 79, 3558–3566, 1983. 70. Avnir, D., Farin, D., and Pfeifer, P., J. Colloid Interface Sci., 103, 112–118, 1985. 71. Williams, J. M. and Beebe, Jr., T. P., J. Phys. Chem., 97, 6249, 6255–6261, 1993. 72. Rojanski, D., Huppert, D., Bale, H. D., Xie Dacai, Schmidt, P. W., Farin, D., Seri-Levy, A., and Avnir, D., Phys. Rev. Lett., 56, 2505–2511, 1986. 73. Avnir, D., Farin, D., and Pfeifer, P., J. Chem. Phys., 79, 3566–2572, 1983. 74. Farin, D. and Avnir, D., J. Phys. Chem., 91, 5517–5523, 1987. 75. Hazlett, R. D., J. Colloid Interface Sci., 137, 527–533, 1987. 76. Fuji, M., Ueno, S., Takei, T., Watanabe, T., and Chikazawa, M., Colloid Poly. Sci., 278, 30–36, 2000. 77. Sarid, D., Scanning Force Microscopy, Oxford Univ. Press, New York, 1991. 78. Bhushan, B., Fuchs, H., Hosaka, S., Applied Scanning Probe Methods, Springer, Berlin, 2004. 79. Arita, T., Kanda, Y., Hamabe, H., Ueno, T., Watanabe, Y., Higashitani, K., Langmuir, 19, 6723–6729, 2003. 80. Arita, T., Kanda, Y., Higashitani, K., J. Colloid Interface Sci., 273, 102–105, 2004. 81. Yamamoto, T., Kanda, Y., Higashitani, K., Langmuir, 20, 4400–4405, 2004. 82. Donose, B. C., Vakarelski, I. U., Higashitani, K., Langmuir, 21, 1834–1839, 2005. 83. Ducker, W.A., Senden, T. J., and Pasheley, R. M., Langmuir 8, 1831–1836, 1992 84. Israelachvili, J.N., Intermolecular and Surface Forces, 2nd ed. Academic Press, New York, 1992. 85. Vakarelski, I. U., Ishimura, K., Higashitani, K., J. Colloid Interface Sci., 227, 111–118, 2000. 86. Vakarelski, I. U., Higashitani, K., J. Colloid Interface Sci., 242, 110–120, 2001. 87. Kanda, Y., Nakamura, T., Higashitani, K., Colloids Surf., A, 139, 55–62, 1998. 88. Kanda, Y., Iwasaki, S., Higashitani, K., J. Colloid Interface Sci., 216, 394–400, 1999. 89. Ishida, N., Sakamoto, M., Miyahara, M., Higashitani, K., Langmuir, 16, 5681–5687, 2000. 90. Sakamoto, M., Kanda, Y., Miyahara, M., Higashitani, K., Langmuir, 18, 5713–5719, 2002.
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Part II Fundamental Properties of Particles
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2.1
Diffusion of Particles Kikuo Okuyama Hiroshima University, Higashi-Hiroshima , Japan
ShinichiYuu Kyusyhu Institute of Technology, Kitakyushu, Fukuoka, Japan
2.1.1 THERMAL DIFFUSION When a particle is small, Brownian motion is caused by random variations in the incessant bombardment of molecules against the particle. As the result of Brownian motion, aerosol particles appear to diffuse in a manner analogous to the diffusion of gas molecules. The Brownian motion of a particle having mass mp is expressed by the equation of motion of a single particle:
mp
3pmDp dv v mp a ( t ) dt Cc
(1.1)
where v is the velocity of the particle, a(t) the random acceleration resulting from the thermal motion of the background molecules and Stokes drag, and Cc the Cunningham correction factor accounting for noncontinuum effects. Cc is equal to unity in liquid, and in air it is given by ⎛ l ⎞ ⎛ l ⎞ ⎡ ⎛ Dp ⎞ ⎤ Cc 1 2.514 ⎜ ⎟ 0.80 ⎜ ⎟ exp ⎢ 0.55 ⎜ ⎟ ⎥ ⎝ l ⎠ ⎥⎦ ⎝ Dp ⎠ ⎝ Dp ⎠ ⎢⎣
(1.2)
where l is the mean free path of gas molecules; l = 6.5 × 10–6 cm at 25°C under atmospheric pressure. Figure 1.1 shows a particle trajectory in a stationary fluid. In this case, the average square of the — — particle displacement x2 in a time interval t and the average displacement | x|, by Brownian motion, are given as follows: x 2 2 Dt ,
x
4 Dt p
(1.3)
In the above equations, D (cm 2 /s) is the Brownian diffusion coefficient of particles with diameter Dp given by D
Cc kT 3pmDp
(1.4)
where k is the Boltzmann constant (1.38 × 1023 J/K) and T is the temperature (K). Figure 1.2 shows the Brownian diffusion coefficient of particles in air and water at 20 ° C . 103 © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.1
Particle movement by Brownian motion.
FIGURE 1.2 Diffusion coefficient of particle in air and water.
For the agglomerate particles, the Brownian diffusion coefficient can be expressed by replacing the solid particle diameter by the collision diameter, Dc, of agglomerate in the Fuchs interpolation expression for Brownian coagulation in the free molecule and continuum regimes1,2 D
kT ⎛ 5 4 Kn 6 Kn 2 18 Kn3 ⎞ 3pmDc ⎜⎝ 5 Kn (8 p )Kn2 ⎟⎠
(1.5)
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where Kn (2l/Dc) is the Knudsen number of the agglomerate. The collision diameter of an agglomerate is proposed by2,3 ⎛ v⎞ Dc dp ⎜ ⎟ ⎝ vp ⎠
1 / Df
( )
dp np
1 / Df
(1.6)
where dp and vp are the diameter and the volume of the primary particle, respectively, np is the number of primary particles in an agglomerate, and Df is the mass fractal dimension. The number concentration of particles of a uniform size undergoing Brownian diffusion in a flow with gas velocity u can be determined by solving the following equation of convective diffusion of particles with an allowance for external forces: t∑ F n ∇ ⋅ un D∇2 n ∇ ⋅ vn, v t mp
(1.7)
where n is the particle number concentration, v is the particle velocity caused solely by external forces F, u is the velocity of the surrounding gas or liquid, and τ is the relaxation time of the particle (ρpCcDp2/ 18μ, rp = particle density, m = viscosity of fluid). If particles are polydisperse with a number distribution function n(Dp, t) at time t, the corresponding diffusion equation can be given by substituting n(Dp, t) into n in Equation 1.7. The local deposition rate of particles by Brownian diffusion onto a unit surface area, the deposition flux j (number of deposited particles per unit time and surface area), is given by j D∇n vn un
(1.8)
If the flow is turbulent, the value of the deposition flux depends on the strength of the flow and the Brownian diffusion coefficient. Table 1.1 indicates the representative solutions for steady-state particle deposition in laminar flow. The solutions in cases (b) and (c) are applicable to size analysis of small aerosol particles based on Brownian diffusion. Figure 1.3 shows the change in number concentration of aerosols in laminar flow through a horizontal tube as a function of the dimensionless parameter m [Dx/(uxavR2); uxav = average velocity of fluid, x = pipe length, R = pipe radius]. 8 n 0 and n are the inlet and outlet particle number concentrations, respectively. The value of s (utR/D; ut = gravitational settling velocity) indicates the relative effect of gravitational sedimentation to the Brownian diffusion, and s = 0 means the case of Brownian diffusion only, case (b) in Table 1.1. With increasing s, the effect of gravitational sedimentation increases, and the curve of s = 50 agrees with that of the gravitational sedimentation without any Brownian diffusion9:
(
n 2 1 2ab a1 3 b arcsin a1 3 n0 p
)
(1.9)
23 where a = (3/8)sm and b 1 a . When small particles are enclosed in a chamber and its number concentration is uniform except in the vicinity of the chamber walls, the change in the number concentration of particles with time is given by
n n0 exp ( bt )
(1.10)
where n0 is the initial particle number concentration, b the deposition rate coefficient, and t the elapsed time. In the case of a cylindrical chamber having an inner surface area S and volume VT , © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.3 Decrease in number concentration of aerosol due to Brownian diffusion and gravitational settling in a horizontal pipe flow. [From Taulbee, D. B., J. Aerosol Sci., 9, 17–23, 1978. Reprinted with permission from J. Aerosol Sci., copyright (1978), Pergamon Press, Ltd.] 107
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b corresponds to DS/δVT , and the thickness δ of the concentration boundary layer can be given as 2.884 D –1/3 for natural convection flow. 10 If the flow is turbulent, b becomes a function of the strength of the flow field and the Brownian diffusion coefficient.10,11 The transport rate of particles suspended in turbulent flow onto a wall surface over which the fluid stream is fl owing as shown in Figure 1.4 can be described by N ( D DE )
dn nut cos u dy
(1.11)
where D and DE are the Brownian and turbulent diffusion coefficients, ut the gravitational settling velocity, y the distance from the surface, and u the angle from the direction of y to the direction of the gravity. By solving Equation 1.11 analytically, Crump and Seinfeld11 derived an expression for the deposition velocity V(u) [(deposition flux)/(particle number concentration above the surface)] for a wall with the angle u approximating the turbulent diffusion coefficient DE to Ke y m as follows: ⎤ ⎡ ⎛ ⎞ p ut cos u 1⎥ V (u) ut cos u ⎢exp ⎜ ⎟ ⎜⎝ m sin (p m ) m K D m 1 ⎟⎠ ⎥ ⎢ e ⎦ ⎣
1
(1.12)
In the turbulent diffusion coefficient D E, Ke is considered to be proportional to the velocity gradient of fluid as k|du/dx|, where k is a constant depending on the flow condition of fluid. For stirred turbulent flow field, k = 0.4 and m = 2.7 could explain the experimental values reasonably. When the
FIGURE 1.4
Schematic diagram of particle deposition onto an inclined flat surface.
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surface area of walls of the chamber with the angle u is S(u), the deposition rate coefficient b can be related to the deposition velocity V(u) as ⎛ 2p ⎞ b ⎜ ∑ V (u) S (u)⎟ VT 1 ⎝ u0 ⎠
(1.13)
where VT is a total volume of the chamber. When external forces such as electrostatic force have influence on particle transport, the deposition velocity can be predicted by considering the particle velocity induced by the forces in Equation 1.12.12 In such a case, utcos u in Equation 1.12 is replaced by utcos u vext, where vext is the component of this velocity in the direction normal to the wall. Deposition velocity is also influenced by wall roughness. Because the concentration boundary layer of gas-borne particles is generally much thinner than the momentum boundary layer, even a wall with very small roughness that does not affect the momentum boundary layer (i.e., a hydraulically smooth wall) can enhance particle deposition rate if the height of the roughness is comparable to the thickness of the concentration boundary layer.13
2.1.2 TURBULENT DIFFUSION Turbulent diffusion is defined as the transportation of physical properties such as mass, heat, and momentum by the turbulent random motion of a fluid. Because knowledge of statistical functions describing turbulent motions is limited compared with that of molecular motions, any solution of turbulent transport problems is incomplete and only approximate. In comparing turbulent mass flux with the corresponding flux caused by molecular motion, it is usually assumed that the turbulent mass flux is similar in description to molecular diffusion; that is, the flux is directly proportional to the concentration gradient (gradient transport). Although the solution based on the gradient transport cannot be correct in all details, the phenomenological theory (the gradient transport theory) is most useful for practical engineering problems. The present section is devoted to a description of the turbulent diffusion, particularly the particle turbulent diffusion based on the gradient transport. The treatment of transport phenomena in the various flow fields and these applications are omitted in this section, and fundamental descriptions about the turbulent diffusion (i.e., turbulent diffusion equation, turbulence properties, and diffusion coefficient) are presented.
Turbulent Diffusion Equation The law of conservation of particle mass gives n ∇ ⋅ n v D∇2 n G t
(1.14)
where n, v, D, and T are particle concentration, particle velocity vector, Brownian diffusion coefficient, and source, respectively. is the nabla operator. The instantaneous values can be written as n n n ′,
v v v′
(1.15)
where the overbar and prime denote the average and the fluctuating values, respectively. Some rearrangement, after substituting Equation 1.15 into Equation 1.14 gives n ∇ ⋅ n v ∇ ⋅ n ′ v ′ D∇ 2 n G t
(1.16)
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The first term on the left-hand side represents change with time, the second is a convection term, and the third is a turbulent transport term. The first term on the right-hand side represents mass flux due to Brownian motion and the second is a source or reaction term. Only the third term on the left-hand side has turbulence fluctuating values. If the turbulent transport term n––v– is expressed by the time-averaged particle concentration n–, Equation 1.16 can be solved because then the only unknown in Equation 1.16 is n–. The introduction of the gradient transport, which shows an analogy to the Brownian diffusion, gives n ′ vi′ p
n xi
(1.17)
where p is the particle turbulence diffusion coefficient. If the diffusion coefficient is a tensor of the second order, the relation between n ′v ′ and ∂n / ∂x j should be n ′ vi′ pij
n x j
(1.18)
In this section the summation convention with respect to repeated indices is used. Substitution of Equation 1.18 into Equation 1.16 gives n ∇⋅n v t xi
⎛ ∂n ⎞ 2 ⎜ pi j ∂ x ⎟ D∇ n G ⎝ j⎠
(1.19)
Equation 1.19 is a general form of a particle diffusion equation. If we consider the transport by the turbulent motions without molecular diffusion and source, Equation 1.19 reduces to n ⎛ n ⎞ ∇ ⋅ nv pi j ⎜ t xi ⎝ x j ⎟⎠
(1.20)
When the time-averaged particle velocity –v and the turbulent diffusion tensor p i j are known, Equation 1.20 can be solved and the solution with the boundary conditions gives a particle concentration distribution. If the time-averaged particle velocity is equal to that of fluid (i.e., the particle inertia is negligibly small), substitution of the fluid continuity equation into Equation 1.20 gives ∂n ∂ v ⋅ ∇n ∂t ∂ xi
⎛ ∂n ⎞ ⎜ pij ∂x ⎟ ⎝ j⎠
(1.21)
where the fluid is assumed to be incompressible.
Turbulent Diffusion Coefficient The basic problem now is to derive an expression for p i j. Batchelor14 showed that when the probability distribution of the displacement of particles has a Gaussian form, the diffusion tensor can be interpreted as ij ji
(
)(
d X i X j X j dt i
)
(1.22)
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where Xi and Xj are i and j components of the Lagrangian position of the particle. When i = j = 2, Equation 1.22 reduces to 22
(
)
2
2
2
)
2
(1.23)
2
The mean square value X 2 X 2
(X X )
(
1 d X X2 2 dt 2
= 2∫
t
0
of the Lagrangian displacement can be obtained by
∫
t
0
n’2 (t)n2 (t − t )d tdt 2n22 ∫ (t − t ) RL (t )d t t
(1.24)
0
The Lagrangian correlation coefficient RL(t) is defined as n2 (t )n2 (t t )
RL (t ) =
1
⎡n (t )2 n (t t )2 ⎤ 2 2 ⎣ 2 ⎦
(1.25)
Now consider some simple cases. The Lagrangian correlation RL(t) is nearly equal to unity for a very small t (i.e., for a very short time diffusion). Substitution of RL(t) 1 into Equation 1.24 gives
(X
X 2 ) v2′2 t 2 2
2
(1.26)
Substituting Equation 1.26 into Equation 1.23, we obtain 22 v2′ 2 t
(1.27)
Hence, the diffusivity for very short time diffusion depends linearly on the time and the mean square value of fluctuating velocity (i.e., turbulent intensity). On the other hand, we now consider a long diffusion time. When diffusion time t is much longer than the time t0 for which RL(t0) 0, Equation 1.24 is reduced to 2 v2′2 ∫ ( t t ) RL ( t ) d t 2 v2′2 t ∫ RL ( t )d t t
t0
0
0
∞
2 v2′2 ∫ RL ( t )d t
(1.28)
0
2 v2′2 t TL where TL, defined as TL ∫ 0 RL (t )d t , is the Lagrangian integral time scale. Substitution of Equation 1.28 into Equation 1.23 gives 22 v2′2 TL
(1.29)
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Solid particles, mist, and small bubbles cannot follow the fluid motion completely due to their inertia. Hinze15 derived the relation between p and f based on Equation 3.14 as follows: p f
(
1 B2 exp (At ) exp t Tf L A2 Tf2L 1 1 exp t Tf L
1
(
)
)
(1.30)
Equation 1.30 shows that p f for a long diffusion time. This is reasonable for homogeneous turbulence, even though for a long diffusion time p is not equal to f in an inhomogeneous turbulence in which the turbulence characteristics change spatially. The assumption that the same fluid will surround the particle as it moves (no overshooting) causes an unreasonable result. However, it is difficult to take the effect of the overshooting rigorously into account. Hence, the Lagrangian turbulent trajectories of particles and the turbulent diffusivity are calculated by a simple model. Yuu et al.,16 Brown and Hutchinson,17 and Gosman and Ioannides18 present models. The model of Yuu et al.16 is explained here. The particle moves, surrounded by an eddy that has a lifetime TL defined as ∞
TL ∫ RL ( t )d t
(1.31)
0
where RL(t) is the Lagrangian time correlation coefficient and TL is the averaged longest time during which a fluid particle persists in moving in a given direction. In the model, RL is 1 when the diffusion time t TL , and RL becomes zero when t TL .In other words, an independent eddy surrounds the particle and affects the particle motion during the next TL. The fluid fluctuating velocity is obtained by using a normal random function in which the average value is zero and the variance is equal to the fluid intensity. Substituting the fluid fluctuating velocity into the equation of particle motion, one can calculate the Lagrangian particle trajectory. Similarly, the particle trajectory during the next TL is calculated, and the process is repeated until the diffusion time passes. Since the early work by Taylor,19 the relation between the turbulent diffusivity and Lagrangian displacements has been studied by many investigators to obtain values for the turbulent flux of transferable quantities. Most of them, however, did not completely consider the effect of inhomogeneity of the turbulence field nor the effect of mean velocity distribution in an inhomogeneous turbulent flow. The ensemble average of the Eulerian fluctuating velocities at the Lagrangian positions is not zero in an inhomogeneous turbulent flow. For example, Batchelor’s description (Equation 1.22) of the turbulent diffusivity for homogeneous turbulence would not be strictly applied to real turbulent flows, as these usually have mean velocity distributions. Yuu20 derived the relation (Equation 1.32) between turbulent diffusivities and the Lagrangian displacement of particles in a flow field with a mean velocity distribution based on the gradient transport: i j
1 ∂ 2 ∂t
( x x )( x i
i
j
xj
) { x
i
u j x iu j
}
(1.32)
in which indicates an ensemble averaged values. Figure 1.5 shows the calculated results of %yy in two interacting plane parallel jets where the distance between nozzles is 24D. The experimental data of Yuu et al.21 indicates that the strong interactions between two jets exist primarily in the region where 20 x/D 80 when the distance between the two nozzles is 24D. As shown in Figure 1.5, the effect of the mean velocity distribution on %yy in this region is very large. If Equation 1.22, which assumes that there is homogeneous turbulence, is used for the calculation, we find that this gives about a fivefold overestimation of the diffusivity in this region. This is the curve shown by the open circles. © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.5 Calculated results of the dimensionless lateral turbulent diffusivity in two interacting plane parallel jets (the distance between the two nozzle centers is 24D). z, {, and indicates the dimensionless values of the first and second terms of Equation 1.32, respectively. % U D. yy
yy
0
REFERENCES 1. Seinfeld, J. H. and Pandis, S. N., Atmospheric Chemistry and Physics from Air Pollution to Climate Change, John Wiley & Sons, New York, 1998, pp. 656–664. 2. Kruis, F. E., Kusters, K. A., and Pratsinis, S. E., Aerosol Sci. Technol. 19, 514–526, 1993. 3. Matsoukas, T. and Friedlander, S. K., J. Colloid Interface Sci. 146, 495–506, 1991. 4. Levich, V. G., Physicochemical Hydrodynamics, Prentice-Hall, Englewood Cliffs, N.J., 1962. 5. Gormely, P. G. and Kennedy, M., Proc. Roy. Irish Acad. A 52, 163–169, 1948. 6. De Marcus, W. and Thomas, J., U.S. Atomic Energy Commission Report ORN 1–1413, 1952. 7. Stechkina, I. and Fuchs, N. A., Ann. Occup. Hyg., 9, 59–64, 1966. 8. Taulbee, D. B., J. Aerosol Sci., 9, 17–23, 1978. 9. Fuchs, N. A., The Mechanics of Aerosols, Dover, New York, 1964, pp. 110–112. 10. Okuyama, K., Kousaka, Y., Yamamoto, S., and Hosokawa, T., J. Colloid Interface Sci., 110, 214–223, 1986. 11. Crump, J. G. and Seinfeld, J. H. J. Aerosol Sci. 12, 405–415, 1981. 12. Shimada, M. and Okuyama, K., J. Colloid Interface Sci., 154, 255–263, 1992. 13. Shimada, M., Okuyama, K., Kousaka, Y., and Seinfeld, J. H., J. Colloid Interface Sci., 125, 198–211, 1988. 14. Batchelor, G. K., Aust. J. Sci. Res., A2, 437–451, 1949. 15. Hinze, J. O., Turbulence, 2nd Ed., McGraw-Hill, NewYork, 1975, pp. 460–471. 16. Yuu, S., Yasukouchi, N., Hirosawa, Y., and Jotaki, T., AIChE J., 24, 509–519, 1978. 17. Brown, D.J. and Hutchinson P., J. Fluids Eng., 101, 265–269, 1979. 18. Gosman, A. D. and Ioannides, E., AIAA Paper 81–0323, 1981. 19. Taylor, G. I., Proc. London Math. Soc. Ser. 2, 20, 196–215, 1921. 20. Yuu, S., Phys. Fluids, 28, 466–472, 1985. 21. Yuu, S., Shimoda, F., and Jotaki, T., AIChE J., 25, 676–685, 1979.
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2.2
Optical Properties Yasushige Mori Doshisha University, Kyotanabe, Kyoto, Japan
2.2.1
DEFINITIONS
Light is electromagnetic radiation in the wavelength range from 3 nm to 30 μm. Visible light is the part of electromagnetic radiation to which the human eye is sensitive, and the wavelength of it ranges from 400 nm to 750 nm. When a light beam illuminates a particle having a dielectric constant different from unity, a part of the light will be reflected on the surface of particle, and the rest of the light will pass into the particle, as shown in Figure 2.1. The part of the light passed into the particle will go out as refraction, and the rest will be absorbed. When the particle size is much larger than the wavelength of the incident light, the light beam is diffracted near the particle. The scattering phenomenon of light includes diffraction, deflection, and refraction of light. The net results of the absorption and scattering caused by the particle are known as the extinction of incident light. In describing such optical phenomena, the size parameter α defined by the following expression may be important: a ⫽ p Dp l
(2.1)
where Dp is the particle diameter and l is the wavelength of the incident light. Another important definition is the refractive index. The refractive index of the medium is given by the ratio of the velocity of light in a vacuum, c, to that in the medium, v: m0 ⫽ c v
FIGURE 2.1
(2.2)
Scattering geometry. 115
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This value is always larger than unity. For an absorbing material having appreciable electrical conductivity, the refractive index is expressed as a complex number:
mm ⫽ n1 ⫺ i n2
(i ⫽
⫺1
)
(2.3)
The imaginary part n2 is zero for nonabsorbing particles, as n2 expresses the property of absorption of light. m is defined as the ratio of the complex refractive index of the medium, m m, and the refractive index of the absorbing material, m0, via m ⫽ mm m0
2.2.2
(2.4)
LIGHT SCATTERING
Mie used electromagnetic theory to obtain the rigorous solution of light scattering by a uniform and spherical particle.1 When unipolarized light of intensity I0 illuminates a particle as shown in Figure 2.2, the scattering intensity at a distance r in the direction u from the particle is given as
I⫽
l2 (i1 ⫹ i2 ) I 0 8p 2 r 2
(2.5)
where i1 and i2 are the Mie intensity parameters for scattered light with perpendicular and parallel polarization against the observed plain, respectively. These parameters are expressed in the following equations as functions of m, α, and u.2
i1 ⫽
i2 ⫽
an ⫽
bn ⫽
∞
2 n ⫹1 {an pn ⫹ bn t n } ∑ n⫽1 n ( n ⫹ 1)
∞
2n ⫹1 {bn pn ⫹ an t n } ∑ n n ⫽ 1 ( n ⫹ 1)
2
(2.6)
2
n (a) n′ (m a) ⫺ m n′ (a) n (m a) zn (a) n′ (m a) ⫺ m zn′ (a) n (m a)
m n (a) n′ (m a) ⫺ n′ (a) n (m a) m zn (a) n′ (m a) ⫺ zn′ (a) n (m a)
(2.7)
(2.8)
(2.9)
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FIGURE 2.2
117
Scattering and diffraction of light.
where c⬘n and z⬘n means the first derivative of cn and zn, respectively. The functions cn and zn are related to the Bessel (Jn) and second-order Hankel (Hn(2)) functions via cn ( x ) ⫽
pa J 1 ( x) 2 n⫹ 2
⎛ xp⎞ zn ( x ) ⫽ ⎜ ⎝ 2 ⎟⎠
12
(2.10)
H (2) 1 ( x ) n⫹
(2.11)
2
where x is equal to α or mα. Figure 2.3 shows the angular distribution of scattered intensity calculated for uniform spheres with mm = 1.5.3 The intensity of scattered light becomes a minimum at a scattering angle between 90° and 150°. Figure 2.4 and Figure 2.5 show the changes in the intensity of scattered light with size parameter α as a function of the scattering angle u and the refractive index mm of the particle material.3 For absorbing particles having the imaginary part of refractive index n2, the fluctuation of the scattered intensity decreases with the value of n2.4 If a particle is much smaller than the wavelength of light (mα ⬍ 0.5), the contribution of the nth term rapidly decreases as n increases in Equation 2.6 and Equation 2.7. When the values of an and bn are adopted to a sum of n = 1 and 2, and a value only of n ⫽1, respectively, and the value for higher than α3 is neglected, the Rayleigh scattering equation is obtained:
I⫽
p 4 Dp6 ⎛ m 2 ⫺ 1 ⎞ 8 r 2 l4 ⎜⎝ m 2 ⫹ 2 ⎟⎠
2
(1 ⫹ cos ) I 2
0
(2.12)
In the Rayleigh scattering regime (α ⬍ 2), the scattered intensity is proportional to the sixth power of the particle size, as shown as Equation 2.12. In the Mie scattering regime (α ⫽ 2–10), the scattered intensity increases with particle size while fluctuating. In the geometrical optics regime (α ⬎ 10), the intensity increases as the square of particle size, and the intensity fluctuation is more drastic than in the Mie scattering regime. When the particle size is much larger than the wavelength (α ⬎ 300), the light beam is diffracted near the particle, as shown in Figure 2.1 or Figure 2.6. For a low diffraction angle (u ⬍ 10°), the angular distribution of the intensity of diffracted beam I(u) is given by the Fraunhofer formula: I (u) ⫽
2
p 2 Dp4 ⎧ J1 (a sin u) ⎫ ⎨ ⎬ I0 4 l2 ⎩ a sin u ⎭
(2.13)
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FIGURE 2.3 Angular distribution of scattered intensity for uniform sphere with mm = 1.5. [From Kanagawa, A., Kagaku Kogaku, 34, 521, 1970. With permission. ]
where J1(α sin u) is the first-order Bessel function. As the intensity of the Fraunhofer diffraction depends on the particle size and is independent of the refractive index of the particle material, this fact is often useful in particle-sizing applications. The light scattering technique is nowadays one of the typical particle size analysis methods.5 However it is difficult to apply to nonuniform and nonspherical particles for the above theory. The analysis for nonuniform6 and nonspherical particles7 was made. Mishchenko et al. recently published a book on light scattering by nonspherical particles in which theoretical and numerical techniques are also included.8
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FIGURE 2.4 Changes in the intensity of scattered light with size parameter α as the parameter of the scattering angle u. [From Kanagawa, A., Kagaku Kogaku, 34, 521, 1970. With permission.]
2.2.3
LIGHT EXTINCTION
Aerosol particles or colloidal particles illuminated by a light beam absorb some light, as well as scatter. Thereby, the intensity of the light beam diminishes, as shown in Figure 2.6. According to the Lambert– Beer law, the light extinction can be expressed by I ⫽ I 0 exp (⫺ K L )
(2.14)
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FIGURE 2.5 Changes in the intensity of scattered light with size parameter α as the refractive index mm. [From Kanagawa, A., Kagaku Kogaku, 34, 521, 1970. With permission.]
where I0 and I are respectively the intensity of incident light and the intensity at penetrated distance L. K is the turbidity given as K ⫽ N Qext ( Dp )
(2.15)
where N is the particle number concentration, and Qext(Dp) is the extinction cross section of one particle, whose projected area equivalent diameter is Dp. As the extinction cross section of a particle is the sum of its scattering cross section and its absorption cross section, the following relation for monodisperse particles holds: Qext ⫽ Qscat ⫹ Qabs
(2.16)
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FIGURE 2.6
121
Setup for light-extinction measurements.
where Qscat and Qabs are respectively the scattering and absorption cross sections. For a spherical particle, Qext and Qscat can be calculated when Equation 2.5 is integrated on all solid angle.9
)
(2.17)
l2 ∞ ∑ (2 n ⫹1) Re {an ⫹ bn } 2 p n ⫽1
(2.18)
Qscat ⫽
Qext ⫽
(
l2 ∞ 2 2 ∑ (2 n ⫹1) an ⫹ bn 2 p n⫽1
(
)
where Re{ } means the calculation only of the real part. As Qscat, Qabs, and Qext have the dimension of area, the nondimensional values are obtained from the division by the projected area of a particle and named the scattering, absorption, and extinction efficiencies, respectively. When the sample of particles has a size distribution, the turbidity can be expressed as follows: ∞
K ⫽ N T ∫ Qext ( Dp ) q0 ( Dp ) dDp 0
(2.19)
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where NT is the total particle number concentration, and q0(Dp) is the density distribution of particles on a number basis. If the particle number concentration is sufficiently high, the light scattered by each particle illuminates other particles. When such multiscattering occurs, the coefficient Q ext should be corrected.10
2.2.4
DYNAMIC LIGHT SCATTERING
Dynamic light scattering has been used for the size measurement of particles between about 3 nm and 5 μm in diameter.11 When fine particles are illuminated by monochromatic light, a fl uctuating scattering intensity occurs due to the Brownian motion of the particles. The fluctuation is correlated to the diffusion coefficient D of the particles.12
D=
k T Cc 3p m Dp
(2.20)
where k is Boltzmann’s constant, T is temperature, Cc is the slip correction factor, and μ is viscosity of the medium. From this relationship, the particle size Dp can be determined. The advantage of this measurement is that it is independent of the particle refractive index. Several techniques have been developed for dynamic light scattering. These techniques can be classified in two ways: by the difference in data analysis (photon correlation spectroscopy and frequency analysis method) and by the difference in optical setup (homodyne and heterodyne detection optics). The scattered light from each particle is mixed with either light scattered from the rest of the particles (homodyne) or with light from the incident beam (heterodyne). The detector signal has two components: a constant level, representing the average intensity of the collected light, and a time-varying component, representing the dynamic light scattering effect. The time-dependent component is analyzed by following two methods: time-based correlation function (photon correlation spectroscopy) or frequency-based power spectrum (frequency analysis method). The normalized intensity autocorrelation function for photon correlation spectroscopy is defined by g(2) (t ) ⫽
I (t ) I (t ⫹ t ) I (t )
2
(2.21)
where I(t) and I(t + τ) are the scattered intensities at time t and t + τ, respectively. The autocorrelation function of the scattered intensity is related to the electric field correlation function, g1(τ): g ( 2 ) ( t ) ⫽ 1 ⫹ B g1 ( t )
2
(2.22)
where B is an instrumental factor. g1(τ) is also expressed as follows: g1 ( t ) ⫽ exp (⫺G t )
(2.23)
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q⫽
G ⫽ D q2
(2.24)
4p n sin (u 2 )
(2.25)
l0
where n is the refractive index of the dispersion medium, λ0 is the wavelength of the laser in vacuum, and u is the scattering angle. In the frequency analysis method, the power spectrum for homodyne and heterodyne detection optics are given by p ( v) ⫽
2G v ⫹ (2 G )
p ( v) ⫽
2
2
for homodyne detection optic
2G for heterodyne detection optic v ⫹ G2 2
(2.26)
(2.27)
For sample has the size distribution, g1(τ) and p(ω) are related to the normalized distribution function of decay rates G(G), which is a function of particle diameter as shown as Equation 2.20 and Equation 2.24. ∞
g1 ( t ) ⫽ ∫ G ( G ) exp (⫺G t ) d G 0
∞
p ( v) ⫽ ∫ G ( G ) 0
∞
2G v ⫹ (2 G )
p ( v) ⫽ ∫ G ( G ) 0
2
2
d G for homodyne detection optic
2G d G for heterodyne detection optic v ⫹ G2 2
(2.28)
(2.29)
(2.30)
The average diameter of sample is calculated by ∞1 1 ⎛ 1⎞ ⫽∫ G(G) d ⎜ ⎟ 0 x ⎝ x⎠ Dp
2.2.5
(2.31)
PHOTOPHORESIS
The mechanism of photophoresis is similar to that of thermophoresis. When a very small particle is illuminated from one side, a temperature gradient is induced in the particle. The difference of strength of the gas molecule impact on the particle causes the particle to move in the direction opposite to the © 2006 by Taylor & Francis Group, LLC
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illuminated side. Because photophoresis is a result of an interaction between the particle and the surrounding gas molecules, it cannot occur in the cases of nonabsorbing particles or in a vacuum. The photophoresis force depends on the light intensity and wavelength, the size, shape, and material of the particle, and the gas pressure. Although the evaluation of the force is rather difficult, some analysis has been proposed,13 and measurement data have been obtained.14
REFERENCES 1. Mie, G., Ann. Phys., 25, 377, 1908. 2. Kerker, M., The Scattering of Light and Other Electromagnetic Radiation. Academic Press, New York, 1969. 3. Kanagawa, A., Kagaku Kogaku, 34, 521, 1970. 4. Kanagawa. A., Kagaku Kogaku Ronbunshu, 2, 325, 1976. 5. Xu, R., Particle Characterization: Light Scattering Methods. Kluwer Academic, Dordrecht, 2000. 6. Druger, S. D., Kerker, M., Wang, D. S., Cooke, and D. D., Appl. Opt., 18, 3888, 1979. 7. Schuerman, D. W., Light Scattering by Irregularly Shaped Particles. Plenum Press, New York, 1980. 8. Mishchenko, M. I., Hovenier, J. W., and Travis, L. D., Eds. Light Scattering by Nonspherical Particles: Theory, Measurements, and Applications. Academic Press, San Diego, 2000. 9. Hodkinson, J. R., In Aerosol Science, Davies, C. N., Ed. Academic Press, New York, 1966, pp. 287 and 324. 10. Born, M. and Wolf. E., Principles of Optics, Pergamon Press, Elmsford, NY, 1970. 11. Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge university Press, Cambridge, 1989, p. 441. 12. Berne, B. and Pecora, R.. Dynamic Light Scattering. John Wiley and Sons, New York, 1976. 13. Kerker, M. and Cooke, D. D., Opt. Soc. Am., 72, 1267, 1982. 14. Lin, H. B. and Campillo, A. J. Appl. Opt., 24, 422, 1985.
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2.3
Particle Motion in Fluid Shinichi Yuu Kyushu Institute of Technology, Kitakyushu, Fukuoka, Japan
Yoshio Ohtani Kanazawa University, Kanazawa, Ishikawa, Japan
2.3.1
INTRODUCTION
Motion of particles in fluid is divided into two classes: rectilinear or curvilinear motion, and random Brownian motion. For particles with diameters larger than about 0.5 μm, inertia or external force acting on the particles and the fluid resistance dominate the motion of particles. In such a case, tracing the motion of single particles is applied to picture the motion of the whole dispersed system of particles in fluid. Motion of single particles can be further divided into three classes according to the extent of particle–fluid interaction. The simplest case is one in which the particles have the same density as the fluid at a low particle concentration so that the particles exactly follow the motion of fluid. The second case is such that the particle concentration is fairly low and the fluid induces the relative motion of particles but the particle motion does not affect the motion of fluid. The third case is such that the particle motion and fluid motion interact with each other by exchanging momentum. In this section, motions of single particles in the second case without particle–particle interaction are described.
2.3.2
MOTION OF A SINGLE PARTICLE
Resistance Force on a Spherical Particle Particle motion relative to a fluid is dominated by resistance forces acting on the particle and external forces, such as gravitational, centrifugal, and electrostatic forces. Therefore, we should first know the resistance force on a particle for the description of particle motion. Because FD depends on Dp, u, r, and m, FD would be written as FD k Dp u r m a
b
c
e
(3.1)
where FD, DP, u, and r are resistance force, particle diameter, fluid velocity, and fluid density, respectively, and m is fluid viscosity. Dimensional analysis of Equation 3.1 gives
FD = k Dp u 2
2
⎛ D ur ⎞ r⎜ ⎝ m ⎟⎠ p
−e
(3.2)
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Substituting the cross-sectional area of spherical particle A 14 pDp2 into Equation 3.2 gives
FD
1 2
⎛ D ur ⎞ ⎝ m ⎟⎠
r u A k′ ⎜ 2
−e
p
1
2
2
r u A CD
(3.3)
where CD is a resistance coefficient. Equation 3.3 shows that CD depends on only the particle Reynolds number; that is, C D f (Re p ) k ′ Re p
−e
(3.4)
Experimental results indicate that the exponent e in Equation 3.4 also depends only on Rep. In other words, the value of e is changed by the fluid moving conditions (i.e., whether the fluid flow is viscous, transitional, or turbulent). As the particle size in powder technology is usually small, in most cases the particle Reynolds numbers are less than unity. When Rep < 1 (from the engineering point of view, Rep < 2), the fluid flow is dominated by the viscous force, and then the Navier–Stokes equation of fluid motion reduces to the linear equation, which is the creeping motion equation, by neglecting the inertia terms. Stokes 1 first solved the creeping motion equation and obtained the fluid velocity and stress distributions around a spherical particle settling in a uniform flow. He further obtained the resistance force acting on a spherical particle in a uniform viscous flow by integrating the stress distribution over the spherical surface of the particle. The resistance force is FD 3p m D p u
(3.5)
Equation 3.5 is the well-known Stokes resistance law. Substitution of Equation 3.5 into Equation 3.3 gives the resistance coefficient CD for the Stokes law: CD =
24
(3.6)
Re p
As mentioned earlier, the Stokes law applies to the creeping flow only. As the inertia effect on flow surrounding a particle increases with increasing Rep, the Stokes law cannot apply to the flow field where Rep >2. Resistance forces for flow fields where Rep >2 are given by the following experimental formulas: 2 < Re p < 500
FD
5 4
(
p mr Dp u
CD
500 < Re p < 10
5
(Allen’s region)
)
1.5
(3.7)
10 Re p
(Newton’s region)
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(
FD 0.055p r Dp u
)
2
(3.8)
C D ≅ 0.44
Recently, the Schiller–Naumann experimental formula 2 has become more popular: 0 < Re p < 800
CD
(
24 1 0.15 Re 0p.687 Re p
)
(3.9)
Figure 3.1 shows CD of these equations, where the bold line indicates the experimental result. When the particle diameter is comparable to or less than the mean free path of the air molecule, the effect of the slip around the particle should be considered. Thus, FD is FD =
3p m u Dp
(3.10)
Cc
where Cc is the slip correction factor. Kennard (3) formulized the slip correction factor using Millikan’s experimental data4: ⎡
⎛ ⎝
C c = 1 ⎢ 2.46 0.82 exp ⎜
⎣
0.44 Dp ⎞ ⎤ l l
⎟⎠ ⎥ ⎦D
(3.11) p
Neglecting the exponential term in Equation 3.11 gives the following approximate equation for a particle suspended in air of normal conditions Cc 1
0.165
(3.12)
Dp
where the unit of Dp is in μm.
Equation of Particle Motion Tchen5 derived the Lagrange equation of motion for a particle suspended in an unsteady velocity field by extending the Basset–Buossinesq–Oseen equation 6 :
p 3 dn D r 3pm Dp (u n ) 6 p p dt p 3 du p 3 ⎛ du dn ⎞ D r D r 6 p dt 12 p ⎜⎝ dt dt ⎟⎠ t du / dt ′ dn / dt ′ 3 dt ′ Fe Dp2 prm ∫ t0 2 t t′
(3.13)
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FIGURE 3.1 Resistance coefficient versus particle Reynolds number.
Particle Motion in Fluid
129
The term on the left-hand side of Equation 3.13 is the force required to accelerate the particle. The first term on the right-hand side of Equation 3.13 is the Stokes resistance force. The second term is the force due to the pressure gradient of the fluid surrounding the particle. The third term is the force to accelerate the virtual added mass of the particle relative to the fluid. The fourth term is the Basset term describing the effect of the deviation in flow pattern from steady state. Fe is the external force. The condition |(uv)/u|<<1 reduces Equation 3.13 to t du / dt dn / dt dn du A (n u ) B C∫ dt ′ t dt dt t t′ 0
6 Fe 0 pDp3 rp
(3.14)
where
A
(
36
)
2 rp r Dp
2
B
,
3r 2 rp r
,
C
(
18
rm
)
2 rp r Dp
p
As rp>>r and m<<1 in an airflow, B and C in Equation 3.14 are much smaller than A. Therefore, the third and fourth terms on the left-hand side of Equation 3.14 can be neglected in an airflow. When the resistance force is expressed with the resistance coefficient CD, the equation of particle motion is
m
dn dt
Cd A c r
2 Cc
⎛
n u (n u ) m ⎜ 1
⎝
r⎞
rp ⎟⎠
g
(3.15)
where the third and fourth terms in Equation 3.14 are neglected, and the particle mass m (p/6)D3p rp and the cross-sectional area Ac (p/4)D2p for the spherical particle. Two-dimensional normalized equations of Equation 3.15 are
2
C c rp u 0 Dp 18 m D
24
C D Re p d t
2
C c rp u 0 Dp 18 m D
24
2
n u
x
2
d y
C D Re p d t
24C c
⎛ dx ⎞ u ⎟ 0 ⎜⎝ ⎠ dt
2
d x
2
2
rp D p
C D Re p 18 m u 0
n u
⎛ dy ⎜⎝ dt
⎛ r⎞ ⎜⎝ 1 r ⎟⎠ g 0
⎞ ⎠
uy ⎟
(3.16)
p
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where x
(
C D 24 m / Dv v u u 0 r
x D
, y
y
, t
D
u0 t
, u
D
x
ux u0
, uy
uy u0
(3.17)
) for the Stokes law reduces Equation 3.16 to the following equations: 2
c
d x dt
2
dx dt
2
ux 0,
c
d y dt
2
dy dt
uy G 0
(3.18)
where c and G are the inertia parameter and normalized settling velocity, respectively, defined as 2
c=
C c rp Dp u 0
,
18 m D
G
)
(
C c rp r Dp g 2
18 m u 0
(3.19)
The solution of Equation 3.18 based on the initial conditions gives the particle trajectory in the airflow with the velocity distribution (ux,uy).
2.3.3
PARTICLE MOTION IN SHEAR FIELDS
When a particle moves in a shear field, lift force (lateral force) is exerted on a particle due to the local velocity gradient of fluid flow and the rotation of particle. The lift force plays an important role in the motion of particles in pipe flow, boundary layers, and other shear fields. Saffman7 theoretically derived the lift force for spherical particle at a low Reynolds number:
FL 1.62u Dp
2
⎛ du ⎞ ⎜⎝ rm ⎟⎠ dy
1/ 2
(3.20)
where u is the local relative velocity between the particle and the fluid. The lift force exerts on particle in the direction of increasing the fluid velocity. In the shear field, the particle rotates due to the viscous force. At a steady state, the angular velocity is given by
v
1 du
(3.21)
2 dy
The lift force occurs even in uniform flow when the particle is rotating. At a low Re the lift force due to the rotation of particle is given by
FL
p 8
3
u Dp r v
(3.22)
The above expression was theoretically derived by Rubinow and Keller.8 © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.2 Lift coefficient as a function of dimensionless angular velocity. [From Tsuji et al., J. Fluids Eng., 107, 484–491, 1985. With permission.]
In general, particles are not spherical in shape. Even a small deviation of particle shape from sphere causes a large lift force. Since the lift force always acts on particles in the direction perpendicular to the fluid flow, it makes the particles concentration nonuniform in the direction perpendicular to the fluid flow. Equation 3.20 and Equation 3.21 are applicable only at a low Re less than unity. At high Re the formulae for the lift force are only empirical. The lift force at high Re is expressed in a similar manner to the drag force by introducing the lift coefficient:
FL C L
⎛p D ⎞ ⎜⎝ ⎟⎠ 4 2
p
2
rf n r 2
(3.23)
Unfortunately, experimental data are still limited for the lift force due to the shear at high Re. Some data of the lift force were measured for a particle near the wall so that a high shear layer was developed and the wall effects were included in the data. Data by Yamamoto9 even show that the lift force exerts in the direction of decreasing the velocity, which is completely opposite to the lift force direction predicted by Equation 3.20. Experimental data of the lift due to the rotation are also very much limited, and the relation between the lift coefficient and the dimensionless angular velocity shown in Figure 3.2 is often used to estimate the lift due the particle rotation at high Re.
REFERENCES 1. 2. 3. 4. 5.
Stokes, G. G., Trans. Cambridge Philos. Soc., 8, 287–318, 1845. Schiller, L. and Naumann, A., Z. Ver. Dtsch. Ing., 77: 318–321, 1933. Kennard, E. H., Kinetic Theory of Gases, McGraw-Hill, New York, 1938, p. 83. Millikan, R. A., Phys. Rev., 22: 1–18, 1923. Tchen, C. M., Dissertation, Delft, Martinus Nijihof, The Hague, 1947.
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132 6. 7. 8. 9. 10.
Powder Technology Handbook Basset, A. B., Philos. Trans. Roy. Soc., 179, 43–71, 1888. Saffman, P. G., J. Fluid Mech. 22, 385–403, 1965. Rubinow, S. and Keller, J. B., J. Fluid Mech., 11, 447–457, 1961. Yamamoto, F., Trans. JSME, 57, 3414–3421, 1992. Tsuji, Y., Morikawa, Y., and Mizono, O., J. Fluids Eng., 107, 484–491, 1985.
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2.4
Particle Sedimentation Shinichi Yuu Kyushu Institute of Technology, Kitakyushu, Fukuoka, Japan
2.4.1
INTRODUCTION
Sedimentation is the most important phenomenon not only in measuring fundamental characteristics of particles but also in understanding mechanisms of unit operations such as dust and mist collection, classification, and pneumatic transport. In this chapter, we describe the resistance force on a spherical particle moving through a fluid, the Lagrangian equation of a particle in motion, and the sedimentation velocities of particles in various situations. Particles moving through a viscous fluid interact with each other; thus, interactions between two or more spherical particles falling through a viscous liquid are also discussed.
2.4.2 TERMINAL SETTLING VELOCITY Substituting u 0 into Equation 3.15 from 2.3 gives the equation of particle motion in a quiescent fluid:
(
)
rp r g dn 3 CD n 2 r dt 4 rp Dp rp
(4.1)
When the first term (the hydrodynamic resistance force) on the right-hand side of Equation 3.15 from 2.3 is counterbalanced by the second term (the hydrodynamic resistance force), the spherical particle settling under the gravity attains a uniform velocity. It is called the terminal settling velocity. Substituting dv/dt0 and Equation 3.6, Equation 3.7, or Equation 3.8 from 2.3 into Equation 4.1 gives the equation of terminal settling velocity for each region:
(r r)D u
2 p
p
t
g
, Re p 2
18 m
(
)
(
)
⎛ 4 r r 2 g2 ⎞ p ⎟ ut ⎜ 225mr ⎟ ⎜⎝ ⎠ ⎛ 3g rp r Dp ⎞ ut ⎜ ⎟ ⎜⎝ ⎟⎠ r
(4.2)
1/ 3
Dp
, 2 Re p 500
(4.3)
1/ 2
,
500 Re p 10 5
(4.4)
Resistance Force Acting on a Spherical Particle Settling Relative to Cylindrical and Plane Walls Lorentz1 derived the theoretical equation of the resistance force acting on a spherical particle settling near a wall, as shown in Figure 4.1: ⎤ ⎡ 9 ⎛ Dp ⎞ FD 3pm Dp u ⎢1 ⎜ ⋅⋅⋅⎥ , Re p 1 ⎟ ⎥⎦ ⎢⎣ 16 ⎝ 2 H ⎠
(4.5) 133
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FIGURE 4.1
Spherical particle settling near a wall.
Faxen2 derived theoretically the following equation, which describes the torque acting on a particle settling near a wall: 5 ⎤ ⎡ 3 ⎛ Dp ⎞ 4 9 ⎛ Dp ⎞ T 2pm Dp2 u ⎢ ⎜ ⋅⋅⋅⎥ , Re p 1 ⎟ ⎜ ⎟ 256 ⎝ 2 H ⎠ ⎥⎦ ⎢⎣ 32 ⎝ 2 H ⎠
(4.6)
Both FD and T increase with increasing Dp/2H. This means that the settling and rotational velocities decrease with increasing Dp/2H. Bohlin3 derived the theoretical equation of the resistance force acting on a spherical particle settling on a centerline of a cylindrical pipe as shown in Figure 4.2: FD 3pm D p uK1
(4.7)
3 5 ⎡ ⎛ Dp ⎞ ⎛ Dp ⎞ ⎛ Dp ⎞ K1 ⎢1 2.10443 ⎜ 2.08877 ⎜ 6.94813 ⎜ ⎝ 2 R0 ⎟⎠ ⎝ 2 R0 ⎟⎠ ⎝ 2 R0 ⎟⎠ ⎢⎣ 6 8 10 ⎤ ⎛ Dp ⎞ ⎛ Dp ⎞ ⎛ Dp ⎞ 1.372 ⎜ 3 . 87 4 . 19 ⋅⋅⋅⎥ ⎜ ⎟ ⎜ ⎟ ⎟ ⎝ 2 R0 ⎠ ⎝ 2 R0 ⎠ ⎝ 2 R0 ⎠ ⎥⎦
−1
(4.8)
, Re p 1
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FIGURE 4.2 Spherical particle settling along a centerline of a cylindrical pipe.
Bohlin3 also presented the theoretical equation of the resistance force acting on a spherical particle moving with velocity u in an axisymmetrical Poiseuille flow whose centerline velocity is u 0.
(
FD 3pm Dp uK1 u 0 K 2
)
(4.9)
3 7 ⎡ ⎛ Dp ⎞ ⎛ Dp ⎞ 2 ⎛ Dp ⎞ ⎢ K 2 K1 1 ⎜ 0.1628 ⎜ 0.4059 ⎜ 3 ⎝ 2 R0 ⎟⎠ ⎝ 2 R0 ⎟⎠ ⎝ 2 R0 ⎟⎠ ⎢⎣ 9 10 ⎤ ⎛ Dp ⎞ ⎛ Dp ⎞ 0.5236 ⎜ 1 51 ⋅⋅⋅⎥ , . Re p 1 ⎟ ⎜ ⎟ ⎝ 2 R0 ⎠ ⎝ 2 R0 ⎠ ⎥⎦
(4.10)
Fayon and Happel 4 presented the following equation, which represents the resistance force for flows of higher Reynolds number than the creeping flow: ⎡ 1 FD 3pm Dp u ⎢ ⎢1 2.104 D / 2 R 2.088 D / 2 R 0 0 p p ⎣ ⎞⎤ ⎛C Re p 100 ⎜ A 1⎟ ⎥ , ⎠ ⎥⎦ ⎝ CS
(
)
(
)
8
(4.11)
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where CA is a real resistance coefficient of a spherical particle settling in an unbounded fluid, and CS is the Stokes resistance coefficient (i.e., CS24/Rep). Therefore, CA/CS1 represents a deviation from Stokes law. Brenner5 obtained solution of the creeping motion equation for the steady motion of a spherical particle toward or away from a plane surface of infinite extent, as shown in Figure 4.3. His equation of the resistance force is FD 3pm Dp u b
(4.12)
∞ n( n 1) 4 b sinh a∑ 3 n1 ( 2 n 1)( 2 n 3)
⎛ ⎞ ⎜ 2 sinh ( 2 n 1) a ( 2 n 1) sinh 2 a ⎟ ⎜ 1⎟ ⎜ 4 sinh 2 ⎛ n 1 ⎞ a ( 2 n 1)2 sinh 2 a ⎟ ⎜⎝ ⎟ ⎜⎝ ⎟⎠ 2⎠
(4.13)
FIGURE 4.3 Spherical particle settling toward a plane surface of infinite extent. © 2006 by Taylor & Francis Group, LLC
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137
where a cosh1 (2h/Dp). The approximate equation of Equation 4.13 presented by Honig et al.6 is 2
b
6 h 13h 2 2
6h 4h
,
h
2h Dp
(4.14)
Equation 4.12, Equation 4.13, and Equation 4.14 show that in a continuum, the resistance force of a spherical particle moving toward a plane surface (or another particle) increases with decreasing separation distance between them and finally becomes infinite just before touching. However, there are effects of discontinuity in discontinuous media. The atmospheric air in the minimum separation between a particle and a plane surface or two particles whose diameters are less than several tens of microns creates the discontinuum. Therefore, a fine particle collides with another particle or a plane surface more easily. Umekage and Yuu7 measured resistance forces of a 6.3-mm foam styrene ball under reduced air pressure by using a high-speed camera. At 10 Pa, the mean free path of air is 670 μm, and the Knudsen number of a 6.3-mm foam styrene ball is equivalent to that of a 0.6-μm particle in atmospheric air. The experimental data in Figure 4.4 show that the increment of the resistance force acting on a 6.3-mm foam styrene ball near a plane surface under reduced pressure decreases with decreasing air pressure. Therefore, it is concluded that the resistance force of a fine particle moving toward a plane surface or another particle in the air is much less than that acting in a continuum.
Settling Velocity of a Nonspherical Particle The dynamic shape factor K defined by Equation 4.15 is used to estimate the difference between the settling velocities of nonspherical and spherical particles: K
Transport velocity of equispherical particle Transport velocityy of nonspherical particle
(4.15)
FIGURE 4.4 Resistance force acting on a 6.3-mm foam styrene ball near a plane surface under reduced pressure. © 2006 by Taylor & Francis Group, LLC
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where the transport velocity usually corresponds to the settling velocity. The Stokes resistance law reduces Equation 4.15 to K
Equispherical diameter Stokes diameter of nonspherical particlee
(4.16)
Stober8 presented the equations of K for ellipsoids: K1
K11
(
)
(8 / 3) (q 2 1) q 1 / 3
)
for q 1
(4.17)
)
for q 1
(4.18)
(
⎡ 2 q 2 3 / q 2 1 ⎤ ln q q 2 1 q ⎣ ⎦
(
)
( 4 / 3) (q 2 1)q1 / 3
(
⎡ 2 q 2 1 / q 2 1 ⎤ ln q q 2 1 q ⎣ ⎦
K1
K11
(
(8 / 3) (q 2 1)q1 / 3
)
⎡ 2 q 2 3 / 1 q 2 ⎤ cos1 q q ⎣ ⎦
(
( 4 / 3) (q 2 1)q1 / 3
)
⎡ 2 q 2 1 / q 2 1 ⎤ cos1 q q ⎣ ⎦
for q 1
(4.19)
for q 1
(4.20)
where q polar diameter of ellipsoid/equator diameter of ellipsoid, and K1 and K11 are dynamic shape factors of nonspherical particles whose settling directions are perpendicular and horizontal to the polar axis, respectively. Stober8 measured the settling velocities of polystyrene latex aggregates and obtained the following experimental formulas for dynamic shape factors of aggregate particles: K ⬇ 1.233 K ⬇ 0.862n1/3
for massive aggregates for chainlike aggregates
(4.21) (4.22)
where n is the number of single particles composing the aggregate.
2.4.3
SETTLING OF TWO SPHERICAL PARTICLES
The exact solution of Stokes equation for the motion of a viscous fluid around two spherical particles settling with equal small constant velocities along their line of centers was presented by Stimson and Jeffery.9 The resistance force for one of two equal spherical particles is FD 3pm Dp b n( n 1) 4 b sinh a∑ 3 n1 ( 2 n 1)( 2 n 3)
(4.23)
⎛ ⎞ 1⎞ 2 ⎛ 4 sinh 2 ⎜ n ⎟ a 2 ( n 1) sinh 2 a ⎟ ⎜ ⎝ ⎠ 2 ⎜1 ⎟ sinh n 2 2 1) a ( 2 n 1) sinh 2a ⎟ ( ⎜ ⎜⎝ ⎟⎠ © 2006 by Taylor & Francis Group, LLC
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where 1/ 2 ⎧ ⎡⎛ l ⎞ 2 ⎤ ⎫ ⎪ l0 ⎪ 0 a ln ⎨ ⎢⎜ ⎟ 1⎥ ⎬ ⎥ ⎪ ⎪ Dp ⎢⎣⎝ Dp ⎠ ⎦ ⎭ ⎩
(4.24)
When the distance between the centers of two spherical particles l0 is 1.54 DP, b is 0.7025; that is, the interaction of two particles gives about a 30% reduction in resistance. Wakiya10 gave the solutions of the Stokes equation for the motion of two spherical particles moving in an arbitrary direction: ⎧⎪ ⎡ ⎤ 9 ab 3 ⎛ a 3 b 27 a 2 b 2 ab3 ⎞ Fax 6pm a ⎨uax ⎢1 2 3 ⋅⋅⋅ ⋅⎥ ⎜ 2 4 4 4 ⎟ 4 l0 4⎝ 4 l0 l0 l0 ⎠ ⎥⎦ ⎩⎪ ⎢⎣ ⎡ 3 b 1 ⎛ a 2 b 27 ab 2 b3 ⎞ ubx ⎢ ⎜ 3 3⎟ 4 l 30 l0 ⎠ ⎢⎣ 2 l 0 2 ⎝ l 0
(4.25)
⎤ ⎫⎪ 9 ⎛ a 3 b 2 27 a 2 b3 ab 4 ⎞ ⎜ 5 ⋅⋅⋅ ⎥⎬ ⎟ 4 ⎝ l0 8 l 50 l 50 ⎠ ⎥⎦ ⎭⎪ ⎧⎪ ⎡ ⎤ 9 ab 3 ⎛ a 3 b 27 a 2 b 2 ab3 ⎞ Fay 6pm a ⎨uay ⎢1 ⎜ 4 4 ⎟ ⋅⋅⋅⋅⎥ 2 4 32 l 0 l0 ⎠ ⎥⎦ ⎪⎩ ⎢⎣ 16 l 0 8 ⎝ l 0 ⎡ 3 b 1 ⎛ a 2 b 27 ab 2 b3 ⎞ uby ⎢ ⎜ 3 3⎟ 16 l 30 l0 ⎠ ⎣⎢ 4 l 0 4 ⎝ l 0
⎤ ⎫⎪ 27 ⎛ a 3 b 2 9 a 2 b3 ab 4 ⎞ 5 ⎟ ⋅⋅⋅⎥ ⎬ ⎜ 5 5 64 ⎝ l 0 16 l 0 l0 ⎠ ⎥⎦ ⎪⎭
⎧⎪ ⎡ ⎤ 9 ab 3 ⎛ a 3 b 27 a 2 b 2 ab3 ⎞ ⎜ 4 4 ⎟ ⋅⋅⋅⎥ ⎨uay ⎢1 2 4 32 l 0 l0 ⎠ ⎥⎦ ⎪⎩ ⎢⎣ 16 l 0 8 ⎝ l 0 ⎤ ⎫⎪ ⎡ 3 a 1 ⎛ a 3 27 a 2 b ⎞ uby ⎢ ⎜ 3 ⋅ ⋅ ⋅⎥ ⎬ 3 ⎟ ⎥⎦ ⎪ ⎢⎣ 4 l 0 4 ⎝ l 0 16 l 0 ⎠ ⎭
va
(4.26)
3 bx 0 4 l 30
(4.27)
where |x0| l0.
2.4.4 RATE OF SEDIMENTATION IN CONCENTRATED SUSPENSION The rate of sedimentation of the particles in a suspension which are well distributed throughout the fluid in a vessel is slower than the velocity given by the Stokes resistance law. Steinour 11 considered © 2006 by Taylor & Francis Group, LLC
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that the resistance force on a particle in suspension can be represented by the Stokes resistance law with a correction term that is a function of the concentration of particles only, as follows: FD
3pm Dp u
(4.28)
f ( )
where u is the relative velocity between a particle and a fluid. is the function of the volume occupied by liquid, which is analogous to the porosity in powder beds. When 1 at infinite dilution, f() is equal to unity. The force balance between FD given by Equation 4.28, and the gravitational force of 1 the particle given by p ( rp rm ) Dp3 g provides the terminal relative velocity: 6
u
(
)
p rp r Dp2 f ( ) 18m
ut f ( )
(4.29)
where the density difference between the particle and suspension is
(
)
rp rm rp ⎡⎣(1 ) rp r ⎤⎦ rp r
(4.30)
The measured rate of sedimentation uc is not a relative velocity between a particle and a fluid in suspension, but a settling velocity of the particle relative to a fixed cross section. The continuity condition states that the volume of settling particles is equal to the ascending fluid volume; that is, (1)uc (u uc ). Hence, u c u
(4.31)
Substituting Equation 4.29 into Equation 4.31, both from 2.3, gives u c ut 2 f ( )
(4.32)
Steinour11 measured the rate of sedimentation for tapioca particles and glass beads and obtained the experimental formula of f (): f ( ) 101.82(1 ) for 0.6
(4.33)
Notation Cc CD Dp FD Fe F∞ g H h K l
Slip correction factor Resistance coefficient Particle diameter (μm) Resistance force (N) External force (N) Resistance force of a single particle in an infinite medium (N) Gravitational acceleration (m/s2) Distance between particle center and wall (m) Minimum distance between particle surface and wall (m) Dynamic shape factor Mean free path (μm)
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R0 r Re, Rep t uv ut r, rp c
141
Radius of pipe (m) Radial coordinate (m) Reynolds number and particle Reynolds number Time (s) Fluid and particle velocity (m/s) Terminal settling velocity of single particle (m/s) Fraction of volume occupied by liquid Fluid and particle density (kg/m3) Inertia parameter
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Lorentz, H. A., Abh. Theor. Phys. Leipzig, 1, 23–36, 1907. Faxen, H., Ann. Phys. IV Folge, 68, 89–98, 1922. Bohlin, T., Trans. Roy. Inst. Technol. Stockholm, No. 155, 1960. Fayon, A. M. and Happel, J., AIChE J., 6, 55–63, 1960. Brenner, H., Chem. Eng. Sci., 16, 242–248, 1961. Honig, E. P., Roebersen, G. J., and Wiersema, P. H., J. Colloid Interface Sci., 36, 97–103, 1971. Umekage, T. and Yuu, S., Nihon Kikai Gatsukai Ronbunshu Ser. B, 59, 1559–1565, 1993. Stober, W., in Fine Particles, Liu, B. Y. H., Ed., Academic Press, NewYork, 1976. Stimson, M. and Jeffery, G. B., Proc. Roy. Soc. London, A111, 110–122, 1926 Wakiya, S., Niigata Univ. Kogakubu Kenkyu Hokoku, 5, 155–160, 1957. Steinour, H. H., Ind. Eng. Chem. 36, 618–626, 1944.
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2.5
Particle Electrification and Electrophoresis Hiroaki Masuda, Shuji Matsusaka, and Ko Higashitani Kyoto University, Katsura, Kyoto, Japan
2.5.1
IN GASEOUS STATE
Contact Electrification When two different metallic materials come into contact, electrons in the metal having the higher Fermi level move into the other metal so as to equalize the Fermi levels of the two materials. Then the number of electrons in the material having a lower Fermi level increases, resulting in electrification to negative charge. As a matter of course, the other material is electrified to positive charge. Contact electrification of semiconductive materials is explained in a similar way. The contact region acts as an electrical capacitor. The amount of charge stored in the capacitor can be obtained by solving Poisson’s equation. Electrification of a nonconductive material is, however, more complicated. For metal⫺insulator contact, the charge density is estimated by the following equation1: s⫽
fs ⫺ fm ez 1 + 0 0 eN s ⫹ N b
(5.1)
where ⫽ charge density (C/m2) s ⫽ work function of nonconductive solid material (eV = 1.602 × 10⫺19 J) m ⫽ work function of metal (eV) Ns ⫽ density of surface states (l/(m2 eV)) Nb ⫽ density of traps of acceptor type (l/(m3 eV)) e ⫽ elementary charge (1.602 × 10⫺19 C) e ⫽ dielectric constant of solid (F/m) e0 ⫽ dielectric constant of vacuum (8.85 × 10⫺12 F/m) z0 ⫽ gap between solid and metal (4Å = 0.4 nm) A similar equation is derived for insulator⫺insulator contact by assuming surface states for both materials.2 The difference of work functions divided by elemental charge e is called the contact potential difference. If the surface states have negligible effects (z0 ⫽ 0, Ns ⫽ 0), Equation 5.1 is simplified to s ⫽ Vc z
(5.2a) 143
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where Vc is the contact potential difference and z is the effective Debye length ( / N b / e). If the density of traps is negligible (Nb ⫽ 0) and the density of surface states is sufficiently high (Ns >>0/e2 z0), Equation 5.1 is simplified to s=
0 Vc z0
(5.2b)
When the contact bodies are separated, charges are partly neutralized or discharged through the surrounding medium. The process is called charge relaxation.3 The maximum charge of a spherical particle is controlled by discharge from the surface of the particle. The breakdown condition in air is given by an electric field strength of 3 MV/m for particles larger than 200 µm and a surface potential of 300 V for those less than 200 µm.4
Electrification of a Particle by Impact Charge transfer between impacted materials proceeds during the very short time of collision. Such a high-speed electrification will be represented by5: ⎡ ⎛ ⌬t ⎞ ⎤ ⌬q ⫽ CVc ⎢1 ⫺ exp ⎜⫺ ⎟ ⎥ ⎝ t ⎠⎦ ⎣
(5.3)
where C is the capacitance of the contact region, t the time constant for electrification, and ⌬ t the effective duration time of contact. The capacitance C is given by C=
S S or C = 0 z z0
(5.4)
corresponding to Equation 5.2a or 5.2b, where S is the contact area. Equation 5.3 shows that metallic particles (t << ⌬t ) may be highly electrified by impact as long as the back discharge is negligible. 6 For nonconductive particles, the time constant t is given by t ⫽ rd
(5.5)
where rd is the specific resistance of the particle. The charge transferred is given by ⌬q ⫽
SVc ⌬t z rd
(5.6)
Equation 5.6 shows that the perfect insulator (rd → ∞) does not acquire charge, which coincides with the experimental results obtained by the use of solid rare gases.7 The contact area S depends on the mode of impact. It is nearly proportional to impact velocity v for inelastic collision and proportional to v0.8 for elastic collision. In both cases it is proportional to the square of the particle size.5 If particles collide with the wall several times, they are all electrified and produce a strong electric field. Also, the contact potential difference Vc is affected by both the electric field and the image charge of the particle. The electrification process, including the electric-field effect and the imagecharge effect, is represented by the following equation8,9: ⎡ ⎛ N⎞ ⎛ N ⎞⎤ Δq ⫽ q0 exp ⎜⫺ ⎟ ⫹ q∞ ⎢1⫺ exp ⎜⫺ ⎟ ⎥ ⎝ N0 ⎠ ⎝ N0 ⎠ ⎦ ⎣
(5.7)
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where N is the number of collisions of a particle, N0 the relaxation number of collisions for charge transfer, q0 the initial charge of a particle, and q the final charge attainable in the process. In gas⫺solids pipe flow, particle charging depends on other factors such as the number of particle collisions, initial charge on particles, and the state of impact charging. The value of each factor is spread, and therefore, the particles have a distribution after passing through the pipe. The charge distribution on particles can be estimated with the distribution of the factors.10
Electrification of a Particle through Breakage Electric charge of particles after breakage is represented by a normal distribution with zero mean. The cause of electrification through breakage is the unequal partition of positive or negative charge. Electrification of particles in a crusher is caused by both the contact between different materials (particles and the wall) and the breakage. Therefore, the mean value of charge is no longer zero. Further, smaller particles produced through breakage are apt to take a negative charge because the work function will be higher.11 The charge distribution of mist produced through atomization of water solution is symmetrical. The absolute mean is given by12
ne ⫽
2 N i Dp1.5 3
(5.8)
where ⫺ ne is the average number of elementary charge, Ni the number density of ions, and Dp the particle diameter.
Motion of a Charged Particle When a charged particle is in an electric field, the following force (Coulomb force) will act on it: F ⫽ qE
(5.9)
where F is the force (N), q the charge (C), and E the strength of the electric field (V/m). The movement of charged particles produces an electric current. This is called the convective current. The convective current is affected by a magnetic field, and the force acting on a charged particle is given by F ⫽ qE + qv × B
(5.10)
where v is the particle velocity (m/s), and B is the magnetic flux density (T). The electrostatic force between the particle and a body having electric charge dq is given by dF =
q dq 4p0 r 2
(5.11)
where r is the distance between the particle and the charged body. When the electric charges are distributed on various surrounding bodies, integration of Equation 5.11 will give the total force acting on the particle. Velocity and trajectory of a charged particle in a uniform electric field can be obtained by solving an equation of motion, including the Coulomb force term. For example, the terminal velocity v in an electric field is obtained as v⫽
qECc 3pmDp
(5.12)
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where Cc is Cunningham slip correction factor and µ is the viscosity of fluid. The particle velocity under unit strength of the electric field v/E is called the electric mobility. Electrically charged particles produce an electric field, which is calculated by the following Poisson’s equation: div E ⫽
r «0
(5.13)
where r is the space-charge density (C/m3). Table 5.1 lists the strength of electric fields for one-dimensional cases. Unipolar charged particles repulse each other and gradually disperse. This phenomenon is called electrostatic dispersion or diffusion. Particle deposition on a wall is enhanced by the electrostatic effect as estimated approximately by the following Wilson equation13 both for a circular tube and a parallel-plate channel14: h⫽
t / t0 1+ t / t0
(5.14)
where h is the deposition fraction, t the residence time, and t0 the half-life time. The half-life time is given by t0 ⫽
3pmDp0
(5.15)
n0 q 2Cc
where n0 is the number concentration (m⫺3) and Dp is the equivalent volume diameter. A charged particle near the equipment wall will be affected by the so-called image force, which is always attractive. If the particle is assumed to be a point charge, the force is given by F =⫺
q2 16p0r 2
(5.16)
_ where r is the distance from the wall. For a nonconductive wall of specific dielectric constant , the force is given by F ⫽⫺
TABLE 5.1
Inside
Outside
q2 ⫺1 16p0r 2 ⫹ 1
(5.17)
Electric Field Produced by Electrically Charged Particles In Cylindrical Tube [Er (V/m)]
Between Parallel Plates [Ex (V/m)]
In spherical vessel [Er (V/m)]
1 r r 2 0
r x 0
1 r r 3 0
1 r r0 2 2 0 r
r x 0 0
1 r r0 3 3 0 r 2
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If the assumption of point charge does not hold, infinite set of images should be adopted. 15 A particle in an electric field attracts other particles even if the particle is not charged. This is because the electric field is distorted by the particle. The force acting on a particle in a distorted electric field is given by16 1 ⫺1 2 F ⫽ pDp30 grad E 4 ⫹2
2.5.2
(5.18)
IN LIQUID STATE
Particles in solutions are more or less charged. This charge generates the electrostatic interaction between particles when their surfaces are sufficiently close to each other, and also their migration toward the electrode of the opposite sign when particles are exposed to an external electric field. In this section, the electrical characteristics of particles in electrolyte solutions, which are understood at present, are discussed.
Surface Charge17 When an interface is formed between solid and liquid, the solid surface is usually charged because of the difference of the affinity of electrons to the surfaces. The charging mechanism depends on both the properties of the solid and the medium. The following mechanisms of charging are known for particles in electrolyte solutions. Nernst-Type Charging When crystal particles such as AgI crystals are dispersed in water, an equilibrium will be established ⫺ between Ag+ and I ions adsorbed on the particle surface and those in the bulk solution. If the con⫺ ⫺ centration differs between adsorbed ions of Ag+ and I , the excess amount of either Ag⫹ or I results in the corresponding charge of the particle surface. These ions are called potential-determining ions. In this case the surface potential c0 is able to be calculated by the so-called Nernst equation: ⎛ 2.3kT ⎞ c0 ⫽⫺ ⎜ (pAg ⫺ pAg 0 ) ⎝ e ⎟⎠
(5.19)
where k is the Boltzmann constant, T is the temperature, e is the elementary charge, pAg ⫽ ⫺logaAg+, and aAg+ is the activity of silver ions. The point of zero charge (p.z.c.), pAg0, is given in Table 5.2. A few crystal particles, such as AgBr, AgCl, Ag2S, AgCNS, and BaSO4, are known to be charged by the same mechanism. Charging by Dissociable Groups Particles with groups, such as ⫺OH, ⫺COOH, and ⫺NH3, will be charged by their dissociation in an aqueous solution, and the degree of dissociation depends on the pH of the solution. For example, ⫺ + ⫺COOH dissociates as ⫺COOH ← → ⫺COO + H , and so the surface potential is always negative but the magnitude increases with the pH, as illustrated by curve I in Figure 5.1. When a surface is zwitterionic because of ⫺COOH and ⫺NH3 groups on the surface, the particle will be charged positively or negatively depending on the pH as known from Equation 5.20, and the p.z.c. exists in between. The potential in this case varies with the pH, as shown schematically by curve II in Figure 5.1. +
⫺
H OH ⎯ ⎯⎯⎯ ⎯⎯ → NH 3 ⫺ R ⫺ COO⫺ NH⫹4 ⫺ R ⫺ COOH ← ⎯⎯⎯ ⎯⎯ → NH 3⫺ R ⫺ COOH ← ⎯
(5.20)
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TABLE 5.2
Points of Zero Charge
Oxides SiO2 (precipitated)
pH0 2–3
SiO2 (quartz)
3.7
␦-MnO2
2.8
-MnO2
7.2
SnO2 (cassiterite)
5–6
␣-Al(OH)3
5.0
␣-Al2O3
9.1
␥-Al(OH)3 (gibbsite)
8–9
␥-AlOOH
8.2 5.7
RuO2 TiO2 (rutile)
5.7–5.8
TiO2 (antase)
6.2
␣-FeOOH (goethite)
8.4–9.4
␣-Fe2O3 (haematite)
8.5–9.5
ZnO
8.5–9.5
CuO
9.5
MgO
12.4
Other materials AgBr
pAg0 ⫽ 4.9–5.3
AgI
pAg0 ⫽ 5.64–5.66
Montmorillonite
pH0 ⫽ 2.5
Kaolinite
pH0 ⫽ 4.6
Clay platelets, sides
pH0 ⫽ 6–7
When oxide particles, such as SiO2, TiO2, and Al2O3, are dispersed in water, water molecules adsorb and form ⫺OH groups on the particle surface. Then the surface will be charged by the adsorption or desorption of H+, as illustrated by Equation 5.21; the surface potential of particles varies with the pH in the similar way as that of zwitterionic particles: +
⫺
H OH ⎯ ⎯⎯⎯ ⎯⎯ → M ⫺ O⫺ M ⫺ OH 2+ ← ⎯ ⎯⎯⎯ ⎯⎯ → M ⫺ OH ←
(5.21)
where M indicates a metal atom. The values of p.z.c. for various oxide particles are given in Table 5.2. Charging by Isomorphic Substitution This charging mechanism is found particularly in clay minerals. When there are defects in the crystal lattice of particles in which Si 4+ is substituted by Al 3+, the deficit of positive charge results in charging particles. This charge is not affected by the pH value. Charging by Other Mechanisms When ions, ionic surfactants, or ionic polymers are adsorbed on the particle surface by the covalent force, the van der Waals force, or electrostatic force, the particle will be charged because of the © 2006 by Taylor & Francis Group, LLC
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Surface Potential
o
+ p.z.c
( II )
pH
0
(I) –
FIGURE 5.1 with pH.
Typical variation of surface potential
charge of adsorbed molecules. The adsorption of impurities with charge in the solution also results in the charge of particles. In these cases, the charging mechanism is generally hard to predict.
Electrical Double Layer around Spherical Particles and Their Charge As described earlier, particles in solutions are usually charged. Then, because of the electroneutrality principle, ions which are of the same amount but different sign are attracted toward the particle surface. A part of the ions is adsorbed directly on the particle surface to form the so called Stern layer. The rest of the ions are in thermal motion, balancing the electrical attractive force to form a diffuse layer. The potential at the outside of the Stern layer is called the Stern potential, cd. Because the structure within the Stern layer is not well known, the Stern potential is often regarded as the surface potential of particles for many engineering purposes. The potential, c, in the diffuse layer around a particle of radius a is expressed by the Poisson⫺Boltzmann equation, which cannot be solved analytically. However, when the Debye⫺Hückel approximation that c is small everywhere (i.e., zec << kT ) holds, and the electrolyte is symmetric with valency z, c is given explicitly as follows: ⎛ c a⎞ y = ⎜ d ⎟ exp[ ⫺ k(r ⫺ a)] ⎝ r ⎠ ⎛ 2n z 2 e2 ⎞ =⎜ 0 ⎝ kT ⎟⎠
(5.22)
0.5
(5.23)
where r is the distance from the center of the particle, n0 is the ionic concentration of the bulk solution, and is the permittivity of the medium. 1/k is a measure of the thickness of the diffuse layer. In the case of particles in an aqueous solution at 25°C, the thickness of the diffuse layer is given by
(
1 ⫽ 3 ⫻ 10 −10 z C k
)
−1
(m)
(5.24)
where C is the electrolyte concentration (mol/dm3). It is clear that the diffuse layer becomes thinner as values of z and C increase. At z ⫽ 1 and C ⫽ 10⫺3 mol/dm3, 1/k ~ 10⫺8 m ( = 10 nm). © 2006 by Taylor & Francis Group, LLC
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The total charge, Q, of the particle is given by Q ⫽ 4 pa (1 + ka)cd
(5.25)
Electrical Interaction between Spherical Particles18 When two charged particles are so close that their double layers overlap each other, the ionic concentration between the particle surfaces becomes much higher than that in the bulk solution. This difference in ionic concentration generates an osmotic pressure between the bulk solution and the solution between the surfaces and, therefore, the corresponding interaction force between the particles. The interaction is normally expressed in terms of the potential energy, VR, which is correlated with the interaction force, F, by F = ⫺dVR /dr. The electrostatic interaction potential between dissimilar spherical particles is given by
VR =
(
)
2 2 ⎡ 2c c pa1 a2 cd1 ⫹ cd2 ⎛ 1⫹ exp (⫺kh ) ⎞ ⎢ 2 d1 d22 ln ⎜ ( a1 ⫹ a2 ) ⎢⎣ cd1 ⫹ cd2 ⎝ 1⫺ exp (⫺kh) ⎟⎠
(
)
(5.26)
⎤ ⫹ j ln [1⫺ exp(⫺2 kh)]⎥⎥ ⎥⎦ where the subscripts 1 and 2 indicate particles 1 and 2, respectively, and h is the distance between the particle surfaces. j = 1 when the surface potential is regarded as constant, and j = ⫺1 when the surface charge density is regarded as constant. This equation is valid only when the Debye⫺Hückel approximation holds and the value of ka is reasonably large (say ka > 10). Equation 5.26 indicates that the interaction is not only repulsive but also attractive, depending on the values of cd1 and cd2. For particles of equal size and potential, the equation is simplified as VR ⫽ j 2 pa cd2 ln[1 ⫹ j exp( ⫺ kh)]
(5.27)
It is clear that the interaction is always repulsive between similar particles. The values of VR for j ⫽ ±1 coincide well when h is reasonably large (say kh > 3), but the value of VR for j ⫽ ⫺1 becomes greater than that for j ⫽ 1 when h is small (say kh < 1). It is known that the potential curve of real colloids is often found between the potential curves of j ⫽ ±1.
Electrophoresis When an external electric field is applied to a suspension, charged particles will migrate toward the electrode of the opposite sign, as illustrated in Figure 5.2. This phenomenon is called electrophoresis. Measurements of the electrophoretic velocity of particles give information about their surface potential. When a particle undergoes electrophoresis, a thin layer of liquid around the Stern layer moves with the particles as if it is fixed on the particle surface. The outer surface of this layer is called the slipping plane, and the potential there is called the zeta potential. Hence, the zeta potential can be correlated with the electrophoretic velocity of particles. According to Henry,19 the electrophoretic velocity, uE , of spherical and cylindrical particles is given as a function of the zeta potential z as follows: uE ⫽
u∞ 2zf ( ka, K ) ⫽ E 3m
(5.28)
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U +
+ + + + +
+
+
+ +
+
Particle
+ + +
+
+
+
+
+
+
+
+
+ + + + + +
+ +
+
Stern layer
+ +
E
Slipping plane d
Potential curve
Surface potential
Diffusion layer
0
potential
FIGURE 5.2 Electrochemical double layer around particle surface in an applied electric field.
where u∞ is the velocity of particles, E is the intensity of the external electric field, µ is the viscosity of the medium, a is the particle or cylinder radius, K is lp/l0, where lp and l0 are the electric conductivities of the particle and the medium, respectively, and f(ka, K) is the Henry function drawn by the solid lines in Figure 5.3. When the particle is a nonconducting sphere (K ⫽ 0) and ka >> 1, f(ka, K) ⫽ 3/2 and Equation 5.28 results in the so called Smoluchowski equation.20 This equation is applicable to nonspherical particles also: uE ⫽
z m
(5.29)
If K ⫽ 0 and ka << 1, Equation 5.28 coincides with the Huckel equation,21 where f(ka, K) ⫽ 1, uE =
2z 3m
(5.30)
When the double layer around a particle is relatively thick (i.e., 0.1
2zf ( ka, z ) 3m
(5.31)
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1.5
Cylindrical particle (parallel to electric field, K=0)
Spherical particle (K=0)
1.0
0.75
=t
Cylindrical particle (perpendicular to electric field, K=0)
0.01
FIGURE 5.3
0.1
=5
1
10
100
1000
Henry and Overbeek functions.
where f(ka, z) is given elsewhere,24 and two examples are drawn by the dashed lines in Figure 5.3. Equation 5.31 is valid only at relatively low zeta potential. A more widely applicable method to estimate uE was developed by O’Brien and White.25 uE for a spherical particle in a KCl solution is shown in Figure 5.4. The electrophoretic velocity for particles of arbitrary shape is given by the Smoluchowski equation if ka >> 1, and by the following equation in the case of ka << 1: uE ⫽
Q fh
(5.32)
where fh is the hydrodynamic friction factor. Stigter26,27 calculated the electrophoretic velocity of cylindrical particles in which the relaxation effect is taken into account.
Measuring Methods of Electrophoretic Velocity and Zeta Potential The measuring methods of electrophoretic velocity have been developed extensively because the zeta potential can be evaluated by measuring the value of uE. The methods of microelectrophoresis and electrophoretic light scattering are widely employed. Microelectrophoresis This method is employed when the movement of particles is detectable by microscope or ultramicroscope. The electrophoretic velocity of colloidal particles is determined by direct observation © 2006 by Taylor & Francis Group, LLC
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7
7 0
6
250 150
0.01 0.1
100 70 60 50 40
6 0.2
5
0.3 0.4 0.5
5
4
0.75 1.0 2.0
4
3
2.75
3
30
6
20 15 10 8 4
3
2
2
1
1
0
FIGURE 5.4
5
0
5
10
Electrophoretic velocity of spherical particle in KCl solution.
of particle movement in a closed capillary of circular or rectangular cross section. The medium flows with respect to the cell wall in an electric field because of the electroosmosis, and at the same time the Poisuille flow occurs toward the opposite direction because the cell is closed at both ends. Hence, there exist stationary levels where the net velocity of the medium is zero. The velocity of particles at these levels is their electrophoretic velocity. Measurement at an accurate stationary level is required to minimize the error because the velocity gradient of the medium is quite large around the stationary levels. Convenient methods have been developed to detect the electrophoretic velocity of particles in the capillary tube. In the rotary prism method, the migrating particles look stationary when their velocity synchronizes with the rotational velocity of the prism; thus, the density distribution of uE can be evaluated by changing the velocity of the prism. In the rotary grating method, the image of particles is detected by photomultiplier tubes through a rotary grating plate, and the frequency of the detected light is analyzed by a spectrum analyzer with a computer to determine the entire frequency of mobility. Laser Doppler Method28 When colloidal particles in the electrophoretic cell are in Brownian motion, a power spectrum of the laser scattered from the particles is obtained, because of the Doppler broadening of the frequency. Measurement of a small shift of the power spectrum due to the particle displacement in an electric field offers a method to evaluate the electrophoretic velocity. The method is computerized and a prompt measurement for nanosize particles is possible. Ultrasonic Method When colloidal particles are placed under the ultrasonic field, both the particles and counter ions oscillate, but their mobility is different. Hence, when electrodes are placed at the distance of half of © 2006 by Taylor & Francis Group, LLC
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the sonic wavelength, the potential difference proportional to the zeta potential is detectable. The advantage of this method is that it is applicable to suspensions of high particle concentration, and a prompt measurement for nanosize particles is possible. Electrokinetic Sonic Amplitude Method By applying the electric field of high frequency to colloidal suspensions, the ultrasound will be generated from the movement of particles. The zeta potential can be calculated from the strength of the ultrasound. The advantage of this method is that it is applicable to suspensions of high concentration and to a wide range of particle size. It is even applicable to flowing suspensions. Moving Boundary Method When a clear boundary between a suspension and a particle-free medium is able to be formed, measurement of the boundary displacement in an electric field offers a method to determine the electrophoretic velocity of particles in the suspension. The Tiselius method24 and Ottewill method29 are applications of this principle. Tracer Electrophoresis30 The electrophoretic cell is composed of a central horizontal tube filled with a colloidal solution and has electrodes at both ends. Concentration of particles in the central tube varies with time in an electric field. The electrophoretic velocity is determined by measuring the concentration change of particles with water-insoluble dyes. Mass Transport Method31 The electrophoretic cell is composed of a reservoir and a small collection chamber, which are joined together. When an electric field is applied between the electrodes in the reservoir and in the chamber, particles migrate between them. The electrophoretic velocity is determined by measuring the difference in the net weight of solid in the chamber at the beginning and end of an experiment. This method is suited for highly concentrated suspensions.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Kasai, A., Proc. Inst. Electrostat. Jpn., 1, 46⫺51, 1977. Gutman, E. J. and Hartmann, G. C., J. Imaging Sci. Technol., 36, 335⫺349, 1992. Matsuyama, T. and Yamamoto H., J. Phys. D Appl. Phys. 28, 2418⫺2423, 1995. Crowley, J. M., Fundamentals of Applied Electrostatics, John Wiley & Sons, New York, 1986, pp. 26⫺30. Masuda, H. and Iinoya, K., AIChE J., 24, 950⫺956, 1978. John, W., Reischl, G., and Devor, W., J. Aerosol Sci., 11, 115⫺138, 1980. Cottrell, G. A., Reed, C., and Rose-Innes, A. C., Inst. Phys. Conf. Ser., 48, 249⫺356, 1979. Masuda, H., Komatsu, T., and Iinoya, K., AIChE J., 22, 558⫺564, 1976. Matsusaka, S. and Masuda, H., Adv. Powder Technol., 14, 143⫺166, 2003. Matsusaka, S., Umemoto, H., Nishitani, M., and Masuda, H., J. Electrostat., 55, 81⫺96, 2002. Gallo, C. F. and Lama, W. L., J. Electrostat., 2, 145⫺150, 1976. Smoluchowski, M., Phys. Z., 13, 1069⫺1080, 1912. Wilson, I. B., J. Colloid Sci., 2, 271⫺276, 1947. Masuda, H., Ikumi, S., and Ito, T., KONA, 3, 17⫺25, 1985. Smythe, W. R., Static and Dynamic Electricity, McGraw-Hill, New York, 1950, pp. 118⫺122. Pohl, H. A., Dielectrophoresis, Cambridge Univ. Press, Cambridge, U.K., 1978, chap. 4.
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Particle Electrification and Electrophoresis 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.
155
Hunter, R. J., Foundations of Colloid Science, Vol. 1, Clarendon Press, Oxford, U.K., 1987. Usui, S., J. Colloid Interface Sci., 44,107, 1973. Henry, D. C., Proc. Roy. Soc. London, A133,106, 1931. Smoluchowski, M., Phys. Z., 5, 529, 1905. Hückel, E., Phys. Z., 25, 204, 1924. Overbeek, J. T. G., Kolloid-Beih., 54, 287, 1943. Wiersema, P. H., Loeb, A. L., and Overbeek, J. T. G., J. Colloid Sci., 22, 78, 1966. Kruyt, H. R., Colloid Science, Elsevier, Amsterdam, 1952, p. 215. O’Brien, R. W. and White, L. R., J. Chem. Soc. Faraday Trans., 74, 1607, 1978. Stigter, D., J. Phys. Chem., 82, 1417, 1978. Stigter, D., J. Phys. Chem., 82, 1670, 1979. Uzgiris, E. E., Rev. Sci. Instrum., 45, 117, 1974. Ottewill, R. H. and Shaw, J. N., Kolloid-Z., 218, 34, 1967. Stigter, D. and Mysels, K. J., J. Phys. Chem., 59, 45, 1955. Oliver, J. P. and Sennett, P., paper presented at the Third Annual Meeting of the Clay Mineral Society, 1966.
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2.6
Adhesive Force of a Single Particle Hiroaki Masuda Kyoto University, Katsura, Kyoto, Japan
Kuniaki Gotoh Okayama University, Okayama, Japan
Ko Higashitani and Shuji Matsusaka Kyoto University, Katsura, Kyoto, Japan The adhesive force between particles or a particle and a solid surface plays an important role in powderhandling processes such as dry dispersion, transportation, and classification of particles. The van der Waals force, the electrostatic force, and the liquid bridge are the main sources of the adhesive force.
2.6.1 VAN DER WAALS FORCE The van der Waals force is a short-range electromagnetic force interacting between two molecules (or atoms). However, the force also acts between two macroscopic bodies such as particle–particle and particle–wall. The van der Waals force can be determined by London–van der Waals theory (microscopic theory) or Lifshitz–van der Waals theory (macroscopic theory).1,2 In London–van der Waals theory, the so-called dispersion force acts between two symmetrical and electrically neutral molecules (or atoms). The potential energy is approximated by EA b11 r 6
(6.1)
where r is the distance between center points of two molecules (or atoms) and b11 is a constant that depends on the molecular (or atom) characteristics. Hamaker3 assumed that the interaction between molecules is expressed by the London–van der Waals potential energy (Equation 6.1), and the interaction between two macrobodies can be calculated by the integration of all interactions of molecules that exist in the bodies. Therefore, the fundamental equation for the calculation of van der Waals potential energy is E =∫
q12 b11 dv1 dv2 v1 ∫ v 2 r 6
(6.2)
where q1 is a number of molecules per unit volume in the body and v1 and v2 are the volumes of the two bodies. Representative expressions of van der Waals potential energy based on the above equation are summarized in Table 6.1.4–7 In the table, the equations in which the retardation effect is taken into consideration are also listed.8,9 The retardation effect should be taken into account when the distance between surfaces of two bodies is beyond 100 nm ( l/2p, where l is the wavelength of light). The van der Waals force can be obtained through differentiation of the potential energy; (dE/dr), also 157 © 2006 by Taylor & Francis Group, LLC
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Table 6.1 Representative expression of van der Waals potential energy and ven der Waals force van der Waals Potential Energy (E)
van der Waals Force (F)
Without retardation effect:
– R1
R2
AR1 R2 = Esphere , z << R 1 6(R1 + R2 )z
–
AR1 R2 = Fsphere 6(R1 + R2 )z 2
–
AR1 R2 R 6(R1 + R2 ) z 2
–
AR 6z 2
–
A R 6 z2
–
z (B1 + B2 ) E [z + 0.5(B1 + B2 )]2 sphere
(1)
With retardation effect: z
R
–
AR1 R2 zb 1– ln 1 + λ λ 6(R1 + R2 )z zb
Without retardation effect: AR – , z << R1 6z
z
b
–λ
b zb + λ
–
1 z
(2)
(3)
With retardation effect:
–
Without retardation effect: B1
R1
R2 z+
B2
B1 + B2 =S 2
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z Esphere z + 0.5(B1 + B2 )
+
–
b b 1 – λ zb + λ z
z Fsphere z + 0.5(B1 + B2 )
(4)
(5)
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AR zb 1 – ln 1 + λ λ 6z zb
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summarized in Table 6.1. The constant A in these equations is called the Hamaker constant, which is given by A = p 2 q12 b11
(6.3)
– ,10,11 and the Hamaker The Hamaker constant can be related to the Lifshitz–van der Waals constant ––h v constant between two different materials in air can be approximately expressed by12 A12 A11 A22
(6.4)
where A11 and A22 are the Hamaker constants for material 1 and material 2, respectively, in a free space. In the presence of a third material between the two bodies, the Hamaker constant is given by A132 ( A11 A33 ) ( A22 A33 )
(6.5)
where A33 is the Hamaker constant of the third material. The values of the Hamaker’s constants in air for carbon hydrate, oxides or halides, and metals are (4 ~ 10) × 10–20 J, (6 ~ 15) × 10–20 J, and (15 ~ 50) × 10–20 J, respectively.13 The separation distance z in the equations in Table 6.1 is determined by Born’s repulsion force2 and is usually taken as 0.4 nm in air. On the other hand, if the particle or wall is soft, the elastic deformation occurs at the contact point. In Johnson–Kendall–Roberts (JKR) theory,14 the van der Waals force for the soft material can be expressed by a linear function of the particle diameter, as well as for the hard material. According to Dahneke,15 the van der Waals force for a soft and large particle can be approximated by a quadratic function of the particle diameter. Derjaguin et al.16 proposed an equation (Derjaguin–Muller–Toporov [DMT] theory) that is a linear function of the particle diameter. The value calculated by DMT theory is 4/3 times the value calculated by JKR theory. Tabor17 and Muller et al.18,19 studied the difference between JKR theory and DMT theory, showing JKR theory and DMT theory are the special case of their model. Tsai et al.20 proposed a new model introducing a new adhesive parameter that is similar to the model of Muller et al. In their model, when particle deformation is negligible, the van der Waals force is proportional to the diameter, and the value is 0.4 times that calculated by JKR theory, and the force is proportional to the 4/3-th power of the diameter when deformation is large enough.
2.6.2
IN GASEOUS STATE
Electrostatic Force Electrostatic force in the gas phase arises from (1) particle–charge interaction, (2) image–charge effect, and (3) electrostatic contact potential difference. Between two charged particles, coulombic force Fec acts, and the force is approximately calculated by the following equation (cf. 2.5.1): Fec
1 q1q2 4p 0 r 2
(6.6)
where q1 and q2 are the charge of the particle, r is the distance between centers of the particles, and 0 is a dielectric constant of the medium. If two particles are in contact and the gap between particle surfaces is extremely smaller than the diameter, Equation 6.6 can be expressed as follows: Fec
ps1s 2 2 Dp 0
(6.7)
where s1 and s2 are the surface charge densities of particles. © 2006 by Taylor & Francis Group, LLC
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The image force Fei acts between a charged particle and a neutral surface: Fec
1 0 q2 4p 0 0 ( 2r )2
(6.8)
where q is the particle charge and is the dielectric constant of the wall material. When two different materials are brought into contact, both of them are electrified because of the contact potential difference (cf. 2.5.1). The induced electrostatic force Fed between a particle and a wall is estimated by the following equation through adopting Dahneke’s deformation equation 15 as a first approximation: 2 V 2 ⎧⎪ AkDp 1 Fed p 0 c2 ⎨ 2 z ⎩⎪ 32 z 2
k
⎛ A2 k 2 Dp ⎞ ⎫⎪ 1 ⎜ ⎟⎬ 108 z 7 ⎠ ⎭⎪ ⎝
(6.9)
1 n12 1 n22 E1 E2
where, E1, E2 are Young’s modulus and n1, n2 are Poisson ratio for particle and wall, respectively.
Liquid Bridge Force When the relative humidity of atmosphere is relatively high (>65%21), the liquid bridge is formed at the contact point of two particles, as shown in Figure 6.1. For the completely wettable surface of particles, adhesive force caused by the liquid bridge can be obtained as the sum of the capillary force and the force caused by the surface tension of the liquid as follows: FL pr22 PL 2psr2
(6.10)
where r2 is a radius of liquid bridge as shown in Figure 6.1, s is the surface tension of liquid, and PL is the capillary pressure inside the liquid bridge. If the cross section of the liquid bridge is approximated by a circular arc, capillary pressure PL is expressed by ⎛ 1 1⎞ PL = s ⎜ ⎟ ⎝ r1 r2 ⎠
(6.11)
Then, the liquid bridge force can be calculated by ⎛ 1 1⎞ FL pr22s ⎜ ⎟ 2psr2 ⎝ r1 r2 ⎠
(6.12)
If it is assumed that r1 is very much smaller than r2, the geometric relations among particle diameter Dp, r1, and r2 [r1 1– Dp(seca1), r2 1– Dp(1tan asec a), a→0] gives the following equation for 2 2 contacting spheres of the same size21: FL ≅ psDp
(6.13)
For a spherical particle on a plane wall, it becomes FL ≅ 2psDp
(6.14)
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FIGURE 6.1
161
Liquid bridge formed between two particles.
For a nonwetting surface, the right-hand sides of Equation 6.13 and Equation 6.14 must be multiplied by cos u, where u is a contact angle between the liquid and the surface. On the other hand, r1 and r2 can be correlated with a vapor pressure Pd at the vicinity of the surface by Kelvin’s equation as follows22: ⎧⎪ M s cos u ⎛ 1 1 ⎞ ⎪⎫ Pd exp ⎨ ⎬ Ps0 RT r L ⎜⎝ r1 r2 ⎟⎠ ⎭⎪ ⎩⎪
(6.15)
where Ps0 saturation vapor pressure M molecular weight R gas constant T temperature r L density of liquid u contact angle If the liquid contains a solute, vapor pressure Pd decreases with the increase in the number of solute molecules. The pressure can be expressed by22,23 Pd (1 g ) Pkcl Ps0 g≡
(6.16)
ins ( nw ins )
where Pkcl ( Pd in Equation 6.15) is the vapor pressure without solute, i is the van’t Hoff factor, and ns and nw are the number of solute and solvent molecules, respectively. © 2006 by Taylor & Francis Group, LLC
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Comparison of Adhesive Force Adhesive forces calculated by the above equations are shown in Figure 6.2. The liquid bridge force (Equation 6.14) is the dominating adhesive force, as long as the liquid bridge is formed. Without the liquid bridge, the van der Waals force (Equation 1 in Table 6.1) dominates. For a charged particle such as toner particles,24 the coulombic force becomes important. When the surface charge density σ is assumed to be 26.5 μC/m 2, which is the maximum value decided by the electric field limit for discharging, the coulombic force (Equation 6.7) dominates for the particles over 200 μm in diameter. For the particles smaller than 200 μm, the coulombic force may be larger than the van der Waals force because the maximum surface charge density is determined by the voltage limit rather than the field limit. 25 In the above discussion, the effects of the atmospheric conditions are not taken into consideration. The humidity of the atmosphere changes the shape of the liquid bridge, which results in the alternation of the adhesive force.26,27 The humidity also changes the adsorbed water layer thickness,28 and it also affects the adhesive force.29,30 The effect of the humidity strongly depends on the roughness of a surface.31 The adhesive force will also be affected by temperature.32,33
2.6.3
IN LIQUID STATE
The adhesive force between particles is often assumed to be given by the van der Waals force, even in the case of particles in liquids, without checking the adequacy. It is known that two or three layers of water molecules, ions, and hydrated ions are adsorbed on the solid–liquid interface.34 Recent measurements of adhesive forces between a plate and a particle with an atomic force microscope indicate that the strength of adhesive force depends greatly on the microstructure of the adsorbed layer, even in
FIGURE 6.2 Comparison of adhesive forces. A 1019 J (in air), z 0.4, s 0.072 N/m, rp 103 Kg/m3, g 9.8 m/s2, s1 s2 26.5 mC/m2, 0 8.8 1012 F/m. © 2006 by Taylor & Francis Group, LLC
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simple solutions like electrolyte solutions of monovalent cations. The characteristics are summarized as follows, and the details are given elsewhere.35,36 (1) The adhesive force Fad depends on how many particles can break the adsorbed layer to contact directly during the contact time tc. (2) The magnitude of Fad depends on the hydration enthalpy of ions ΔH, as well as the electrolyte concentration Ce; Fad decreases with increasing Ce and decreasing value of ΔH. It is especially important to know that the dependence of Fad on tc varies greatly with ΔH in concentrated solutions, as shown in Figure 6.3. This is explained as follows. Since highly hydrated cations like Li+ form a thick but weak adsorbed layer, surfaces can contact directly to have a strong adhesion by destroying the adsorbed layer. On the other hand, because poorly hydrated cations like Cs+ form a thin but strong adsorbed layer, the gap between surfaces at contact reduces the strength of adhesive force greatly. The mechanism is schematically drawn in Figure 6.4.
2.6.4
MEASUREMENT OF ADHESIVE FORCE
When a solid particle comes into contact with a wall or another particle, interaction forces occur between the surfaces. The forces depend on the physical and chemical properties, particle shape, roughness, electrostatic charge, and interaction media. In addition, the forces vary according to the distance between the surfaces. The term “adhesive force” is defined as the maximum attractive interaction force during which the particle is removed in the normal direction; thus, it is called the “pull-off force.” In this section, basic techniques for the measurement of the adhesive force are summarized, and the analysis based on the force balance and moment balance is explained. Furthermore, atomic force microscopy, which became a powerful tool for the measurement of a force–distance curve, is described.
Methods of Measurement The techniques for the measurement of the adhesive force are classified into several categories. 37–39 The principles of the techniques are schematically shown in Figure 6.5. They are applicable to the measurement of particle–particle interaction as well as particle–wall interaction.
FIGURE 6.3 Difference between the adhesive forces at tc 0.1 and 50 s in various 1 M electrolyte solutions. © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.4 Mechanism for the difference of adhesive force between cations of low and high hydration enthalpies.
Gravity method (pendulum type): A vertical plate is brought up to a freely hanging particle until contact is made, and then shifted in a direction perpendicular to the contact area. The gravity component that tends to separate the particle is increased with the inclination angle of the plate. From the force balance, the adhesive force Fa can be calculated using the following equation: Fa Mp g sin a
(6.17)
where Mp is the mass of the particle, g is the gravitational acceleration, and a is the inclination angle. To separate the particle, the gravity component must be larger than the adhesive force. Therefore, this method is used for comparatively large particles of the order of 1 mm in diameter. © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.5
165
Principles of measurement of adhesive force.
Spring balance method: The separation force is supplied by a spring. Since the spring force can be made larger than the gravitational force, this method is applicable for micro-sized particles. There are several types of springs, including helical and cantilever springs. Atomic force microscopy is an application using the cantilever spring, which is described later. Centrifuge method40–42: This method is widely used for the measurement of the adhesive forces of micron-sized particles. The important advantages consist in the sensitivity and statistical accuracy. Since the adhesive forces of many particles on a surface can be measured in a single experiment, the adhesive-force distribution is also obtained. The centrifugal force Fcen is calculated using the following equation: Fcen Mp acen
(6.18)
where acen is the centrifugal acceleration ( lv2), in which l is the distance of the object from the axis, and v is the angular velocity. To prevent the effect of the aerodynamic drag caused by the rotation, the object should be in a closed box or under vacuum. If the particles are sensitive to temperature changes, the frictional heat generated by rotation should be removed. © 2006 by Taylor & Francis Group, LLC
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Vibration method43: The apparatus is rather small and simple since there is no mechanical rotation unit. The separation force can be estimated in the way similar to the centrifugal force, namely, Fvib Mp avib
(6.19)
where avib is the vibration acceleration ( yv f2), in which y is the amplitude, and v f is the angular frequency. If the vibration waveform is not sinusoidal or contains noise, the maximum acceleration must be measured. To separate fine particles, high vibration acceleration is needed. Since such vibration raises the temperature of the vibrating plate, a cooling system is required. Impact method44: Particles are attached on one side of a wall, and an impact is applied to the reverse side. The impact force does not act directly on the particles, but the wall pushes the particles forward. This method is very simple, and the separation force can be calculated from the product of the mass of the particle and the effective acceleration. However, since the accurate determination of the acceleration is not so easy, this method is often used for a qualitative evaluation of the adhesive force. Fluid dynamic method 45–48 : Particles can be removed by an air jet or a shear flow. This method is suitable for the evaluation of the particle adhesion to a wall in a particle–fluid system. The adhesive strength distribution can also be obtained from the relationship between the particle reentrainment efficiency and the fluid dynamic separation force.
Analysis of Forces Exerted on a Single Particle Adhesive force is greatly affected by the contact states such as deformation and multiple-point contact, which also depend on the viscoelasticity, geometry, and surface roughness. Figure 6.6 illustrates several contact states between a particle and a wall. In addition to the contact states, the points of application of force and their directions are important to determine the adhesive force. Figure 6.7 shows two cases of particle–wall contacts in a centrifugal field. The moment balance for the particle adhering to a horizontal wall is expressed as42
(F M g) a ≈ M a
p
p
l v 12
Dp 2
(6.20)
where a is the distance from the point of application to the fulcrum. The adhesive force Fa becomes Fa ≈ M p (l v 12
FIGURE 6.6
Dp 2a
g)
(6.21)
Contact states between a particle and a wall.
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FIGURE 6.7
167
Particle-wall contacts in a centrifugal field..
For the particle adhering to a vertical wall, the adhesive force Fa can be expressed as Fa ≈ M p (l v 22 g
Dp 2a
)
(6.22)
If lv 22 >> gDp/(2a), Fa is simply equal to Mplv 22. When the gravitational component cannot be neglected, the value of a is important to estimate the adhesive force. However, from Equation 6.21 and Equation 6.22, the term of Dp/(2a) can be eliminated, and the adhesive force is calculated using the following equation:
Fa M p
(l v 1 v 2 ) 2 g 2 l v 12 g
(6.23)
Also, the value of a can be obtained by
a≈
Dp l v 12 g 2
l v 22 g
(6.24)
Atomic Force Microscopy The research on the force–distance curve, as well as the adhesive force, has attracted much attention since the atomic force microscope was invented.49 In particular, replacing the tip of a cantilever by a spherical particle (i.e. the colloidal probe technique) has accelerated the study of particle–wall interaction.50–53 Figure 6.8 shows the principle of the atomic force microscope with a colloidal probe. The sample is moved up and down by applying voltage to a piezoelectric translator. When a force acts on the probe, the cantilever bends. The deflection of the cantilever is measured using the optical lever technique. Figure 6.9 shows a force–distance curve. (1) The colloidal probe is approaching but still far from the surface, and the interaction force is negligible. (2) When the probe gets close to the surface, the interaction force will occur; this is supposed to be attractive here. When the attractive force exceeds the spring force of the cantilever, the probe will jump © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.8
Atomic force microscope with a colloidal probe.
FIGURE 6.9
A force-distance curve.
into contact with the surface. (3) The probe and the surface move in parallel, which is called the “constant compliance region.” (4) During retraction, the probe will usually adhere to the surface, causing the cantilever to bend downward. (5) When the spring force of the cantilever becomes larger than the adhesive force, the cantilever will jump out of contact into its equilibrium position. Using this method, interaction forces can be analyzed in detail and compared with theoretical models on van der Waals, liquid bridge, electrostatic forces, and so forth.
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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
Israelachvili, J. N., Intermolecular and Surface Forces with Applications to Colloidal and Biological Systems, Academic Press, New York, 1985. Krupp, H., Adv. Colloid Interface Sci., 1, 111–239, 1967. Hamaker, H. C., Physica, 4, 1058–1072, 1937. Vold, M. J., J. Colloid Sci., 16, 1–12, 1961. Vincent, B., J. Colloid Interface Sci., 42, 270–285, 1973. Czarnecki, J. and Dabros, T., J. Colloid Interface Sci., 78, 25–30, 1980. Göner, P. and Pich, J., J. Aerosol Sci., 20, 735–747, 1989. Gregory, J., J. Colloid Interface Sci., 83, 138–145, 1981. Clayfield, E. J., Lumb, E. C., and Markley, P. H., J. Colloid Interface Sci., 37, 382–389, 1971. Lifshitz, E. M., Soviet Phys-JETP, 2, 73–83, 1956. Langbein, D., Physical Review B, 2, 3371–3383, 1970. Israerlachvili, J. N., Proc. Roy. Soc. Ser. A, 331, 35–55, 1972. Visser, J., Adv. Colloid Interface Sci., 3, 331–363, 1970. Johnson, K. L., Kendall, K., and Roberts, A. D., Proc. Roy. Soc. Ser. A, 324, 301–313, 1971. Dahneke, B., J. Colloid Interface Sci., 40, 1–13, 1972. Derjaguin, B. V., Muller, V. M., and Toporov, Y. P., J. Colloid Interface Sci., 53, 314–326, 1975. Tabor, D., J. Colloid Interface Sci., 58, 2–13, 1977. Muller, V. M., Yushchenko, V. S., and Derjaguin, B. V., J. Colloid Interface Sci., 77, 91–101, 1980. Muller, V. M., Yushchenko, V. S., and Derjaguin, B. V., J. Colloid Interface Sci., 92, 92–101, 1983. Tsai, C. J., Pui, D. Y. H., and Liu, B. Y. H., Aerosol Sci. Technol., 15, 239–255, 1991. Zimon, A. D., Adhesion of Dust and Powder, 2nd Ed., Consultant Bureau, New York, 1982, p.69, p.114. Carman, P. C., J. Phys. Chem., 57, 56–64, 1953. Endo, Y., Kousaka, Y., and Nishie, Y., Kagaku Kogaku Ronbunshu, 18, 950–955, 1992. Lee, M. H. and Ayala, J., J. Imaging Technol., 11, 279–284, 1985. Masuda, H. and Matsusaka, S., J. Soc. Powder Technol. Jpn., 30, 713–716, 1993. Endo, Y., Kousaka, Y., and Nishie, Y., Kagaku Kogaku Ronbunshu, 19, 55–61, 1993. Chikazawa, M., Nakajima, W., and Kanazawa, T., J. Res. Assoc. Powder Technol., Japan, 14, 18–25, 1977. Chikazawa, M., Yamaguchi, T., and Kanazawa, T., Proceedings of the International Symposium on Powder Technology 1981, Hemisphere Publ., pp. 202–207, 1981. Danjo, K. and Otsuka, A., Chem. Pharm. Bull., 26, 2705–2709, 1978. Gotoh, K., Takebe, S., Masuda, H., and Banba, Y., Kagaku Kogaku Ronbunshu, 20, 205–212, 1994. Gotoh, K., Takebe, S., and Masuda, H., Kagaku Kogaku Ronbunshu, 20, 685–692, 1994. Nishino, M., Arakawa, M., and Suito, E., Zairyo, 1, 535–540, 1969. Otsuka, A., Iida, K., Danjo, K., and Sunada, H., Chem. Pharm. Bull., 31, 4483–4488, 1983. Israelachvili, J. N., Intermolecular and Surfaces Forces, 2nd Ed., Academic Press, New York, 1992. Vakarelski, I. U., Ishimura, K., and Higashitani, K., J. Colloid Interface Sci., 227, 111–118, 2000. Vakarelski, I. U., Higashitani, K., J. Colloid Interface Sci., 242(1), 110–120, 2001. Krupp, H., Adv. Colloid Interface Sci., 1, 79–110, 1967. Zimon, A. D., Adhesion of Dust and Powder, 2nd Ed., Consultants Bureau, New York, 1982, pp. 69–91. Israelachvili, J. N., Intermolec. Surface Forces, 2nd Ed., Academic Press, London, 1991, pp. 165–175. Asakawa, S. and Jimbo, G., J. Soc. Mater. Sci., Japan, 16, 358–363, 1967. Emi, H., Endo, S., Kanaoka, C., and Kawai, S., Kagaku Kogaku Ronbunshu, 3, 580–585, 1977. Matsusaka, S., Koumura, M., and Masuda, H., Kagaku Kogaku Ronbunshu, 23, 561–568, 1997. Mullins, M. E., Michaels, L. P., Menon, V., Locke, B., and Ranade, M. B., Aerosol Sci. Technol., 17, 105–118, 1992. Otsuka, A., Iida, K., Danjo, K., and Sunada, H., Chem. Pharm. Bull., 31, 4483–4488, 1983. Visser, J., J. Colloid Interface Sci., 34, 26–31, 1970. Masuda, H., Gotoh, K., Fukada, H., and Banba, Y., Adv. Powder Technol., 5, 205–217, 1994. Masuda, H., Matsusaka, S., and Imamura, K., KONA, 12, 133–143, 1994. Matsusaka, S., Mizumoto, K., Koumura, M., and Masuda, H., J. Soc. Powder Technol. Jpn., 31, 719–725, 1994.
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49. Binnig, G., Quate, C. F., and Gerber, Ch., Phys. Rev. Lett., 56, 930–933, 1986. 50. Ducker ,W. A., Senden, T. J., and Pashley, R. M., Nature, 353, 239–241, 1991. 51. Claesson, P. M., Ederth, T., Bergeron, V., and Rutland, M. W., Adv. Colloid Interface Sci., 67, 119 –183, 1996. 52. Cappella, B. and Dietler, G., Surf. Sci. Rep., 34, 1–104, 1999. 53. Kappl, M. and Butt, H.-J., Part. Part. Syst. Char., 19, 129–143, 2002.
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2.7
Particle Deposition and Reentrainment Manabu Shimada Hiroshima University, Higashi-Hiroshima, Japan
Shuji Matsusaka and Hiroaki Masuda Kyoto University, Katsura, Kyoto, Japan Particle deposition is a phenomenon in which particles, suspended in a fluid, are transported to a wall and make permanent or temporary contact with the surface of the wall. Particle deposition is an important phenomenon in many fields: the fouling of channel walls exposed to dusty gas, microcontamination problems in advanced material processing, the dry deposition of particulate matter in an atmospheric environment, the development of gas filtration devices, the designed deposition of particles for the fabrication of functional devices, and so forth. Particle reentrainment means resuspension of deposited particles. In general, small primary particles are hard to reentrain from a wall; however, aggregate particles are readily reentrained from a particle deposition layer. The aggregate reentrainment affects various engineering applications, such as powder dispersion, particles size classification, dust collection, particle synthesis, and aerosol sampling.
2.7.1
PARTICLE DEPOSITION
To predict and evaluate particle deposition phenomena, mechanisms that govern the transport process of particles must be better understood. Important mechanisms include transport by fluid flow, transport by Brownian diffusion and turbulent eddies, and motion induced by inertial forces and external forces such as gravitational sedimentation, electrostatic migration, thermophoresis, diffusiophoresis, and photophoresis.
Fundamental Concepts The transport of submicron-sized particles suspended in a laminar flow gas is generally dominated by fluid flow motion, Brownian diffusive motion, and migration by external forces. In such a case, the spatial and temporal change of particle number concentration per unit gas volume, n, is described by the following convection–diffusion equation:
{
}
dn ⋅ rf D ( n / rf ) ( u v ) n Q dt
(7.1)
where t is the time, rf the density of the medium gas, D the Brownian diffusion coefficient, u the medium gas velocity, v the particle velocity induced by the sum of external forces (ΣF) and equals BΣF (mobility B CC /3pmDp; CC, Cunningham correction factor; m, viscosity of the medium gas; Dp, particle diameter), and Q the rate of generation or consumption of particles. The gas velocity u 171 © 2006 by Taylor & Francis Group, LLC
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can usually be determined by solving the governing equation for fluid flow. The energy equation also needs to be solved in order to incorporate the temperature dependence of rf and D into Equation 7.1 when deposition in a nonisothermal system is under consideration. Sections 2.3–2.5 and literaturee.g.,1,2 will be referred to for the expression of particle velocity v induced by various external forces F. To obtain the rate of deposition, Equation 7.1 can be solved with appropriate boundary conditions that are relevant to the state of the particles on the surface. Strictly speaking, a particle having apparently arrived at a surface is separated from the surface by a very short distance. However, small, submicron particles for which Equation 7.1 holds can be regarded as being held at the surface immediately and permanently, once the surface is reached. Since the separation between a particle and surface is much smaller than the particle diameter, the particle concentration n is usually set to zero at a distance of half the particle diameter from the surface. The number of particles deposited per unit surface area and time, deposition flux j, can then be calculated from j = D
dn dz z = Dp / 2
(7.2)
where z is the distance from the surface. Since j is generally proportional to the particle concentration sufficiently far from the surface, ( n 0 ), v d j / n 0 gives the deposition flux per unit particle concentration and is denoted as the deposition velocity.
Deposition in Laminar Flow Analytical or approximate solutions have already been obtained for some geometrically simple systems. As an example, the deposition velocity of particles is approximated as
vd
1.08 D ⎛ rf u0 dd ⎞ dd ⎜⎝ m ⎟⎠
12
⎛ m ⎞ ⎜⎝ r D ⎟⎠ f
13
2D Ex Dp
(7.3)
for particles suspended in a laminar flow impinging against a disc with diameter d d and, at the same time, experiencing a constant external force in the direction normal to the wall.3 u0 and Ex in the above equation are the flow velocity and a parameter that indicates the magnitude of the external force, respectively. Ex equals EqCC /(6pmD) (E is electric field strength, q is particle charge) for Coulomb force in an electric field, and (r p rf)Dp3gCC /(36 mD) (rp is density of the particle, g is acceleration of gravity) for gravitational force. Figure 7.1 shows examples of some calculated results for vd for particles flowing above a horizontally oriented disc. Examples of analytical or approximated solutions of vd for several other systems are also found in Section 2.1 and in a report by Adamczyc et al.4 Although systems for which such analytical or approximated solutions can be directly applicable are limited, such solutions are often very useful for a rough estimation of deposition with the actual systems being simplified. However, if a detailed distribution of the deposition velocity must be known, a rigorous calculation of the flow field and particle concentration distribution is necessary. Distributions of electric potential and temperature must be analyzed in detail when a nonuniform electrostatic or thermophoretic force influences the deposition. 5,6
Deposition in Turbulent Flow In addition to Brownian diffusion, the mixing and transport of particles suspended in a turbulent flow field are enhanced by turbulent eddies in the medium gas. To evaluate the rate of deposition in a turbulent flow, the detailed temporal change (fluctuation) of particle concentration induced by turbulent eddies is not usually analyzed. In most cases, a time-averaged rate of deposition is derived © 2006 by Taylor & Francis Group, LLC
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Deposition velocity,vd [m/s]
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10-2 3 10-3
2
10-4
4 5 1
10-5 0.01
0.1
1
Particle diameter [ m] FIGURE 7.1 Deposition velocity of airborne particles on the upper surface of a horizontally oriented disc 125 mm in diameter; (1) for particles in downward flow at a speed of 0.3 m/s; (2) same conditions as (1) except that pressure is reduced to 1.01 kPa; (3) same conditions as (1) except that each particle possesses a unit elementary charge and that uniform electric field of 1 V/m exists; (4) same conditions as (1) except that uniform temperature gradient of 1 K/mm exists; (5) for particles suspended in turbulent flow at a certain intensity.
from the basic equations for gas velocity and particle concentration distribution, which have been averaged with respect to time and space (“Reynolds average”). In this case, the transport of particles due to turbulent eddies can be regarded as a diffusion phenomenon, and the magnitude of the diffusion is described by the turbulent diffusion coefficient, Dt. As a result, u and D in Equation 7.1 are – and D + D , respectively. replaced by u t Several notable experimental results and theoretical models for particle deposition in a turbulent flow have been reviewed by Papavergos and Hedley. 7 Figure 7.2 shows representative experimental results for turbulent deposition in a circular pipe. The ordinate and abscissa of the figure are the dimensionless deposition velocity vd+ ( vd /u*, u*, friction velocity) and the dimensionless relaxation time of particles + ( rf rpDp2 u*2CC /(18 m2)), respectively. Turbulent deposition is roughly divided into the following three regimes on the basis of the magnitude of +, which indicate the effect of particle inertia. In the regime in which + is less than about one and thus particles will follow the fluid motion in turbulent eddies (referred to as “turbulent particle diffusion regime”), the magnitude of Dt is of the same order as the eddy kinematic viscosity of the fluid ν t and has a relationship of Dt nt/Sct (Sct, turbulent Schmidt number (~ 0.7–1.0)). νt can be determined by measurement, empirical formula, and numerical simulations of flow based on the turbulence models8–10 When + is in the range of about 1–10 (referred to as the “eddy diffusion–impact regime”), the deposition velocity depends strongly on + , since the motion of particles tends to deviate from that of fluid near a wall, as the result of inertial force. A number of theoretical or semiempirical models have been proposed to explain the measured deposition velocities: the free-flight model (stopping-distance model), in which particles are assumed to “fly” due to inertia in the viscous sublayer on a wall11,12; the effectivediffusivity model, in which Dt is assumed to be enhanced apparently by inertial motion13,14; and so on. On the other hand, the deposition velocity for values of + larger than about 10 (referred to as the “particle inertia moderated regime”) is only slightly dependent on +, since the inertia-induced © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.2 Summary of experimental deposition data in vertical flow system. [From Papavergos, P. G. and Hedley, A. B., Chem. Eng. Res. Des. 62, 275–295, 1984. With permission.]
motion of particles tends to prevail over the entire region of the fluid. Despite many existing models, a method capable of accurately evaluating turbulent deposition has not been fully established. For the present, the following empirical equations will be useful as a rough estimation7:
{
}
vd 0.065 m ( rf D ) vd 3.5 104 2 vd 0.18
−2 / 3
(
(0.2
0.2
(
+
20
)
)
(7.4)
)
(7.5)
20
(7.6)
In addition to the above-mentioned models based on the convection–diffusion equation, recent advances in computer performance have made it possible to more accurately calculate the transport of particles influenced by microscopic fluid motion in turbulent eddies. This type of calculation is based on the equation of motion for each particle and often employs a Monte Carlo simulation. The effects of spatial nonuniformity of turbulent intensity and external force on particle motion and deposition can be analyzed.15–18
2.7.2
PARTICLE REENTRAINMENT
When particles are deposited on a surface, adhesive forces act on the particles. However, if aerodynamic forces or other separation forces based on particle collision and so on are sufficiently large, © 2006 by Taylor & Francis Group, LLC
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the deposited particles will be reentrained into the flow. The reentrainment phenomenon should be distinguished from bounce or saltation of coarse particles, since the contacting time and the state of the interaction are very different. In general, fine particles immersing deeply within the viscous sublayer in a turbulent boundary layer are hard to reentrain; however, when fine particles accumulate on the surface, aggregate particles are readily reentrained.19 The aggregate reentrainment affects various engineering applications, such as powder dispersion, particle size classification, dust collection, particle sythesis, and aerosol sampling.
Concept of Reentrainment of Aggregates The concept of the reentrainment of aggregates from a particle deposition layer is illustrated in Figure 7.3. Since the surface of the particle layer is not smooth, projecting particles experience a relatively large drag; consequently, the aggregate particles can be reentrained. A simple model for the reentrainment of a spherical aggregate is shown in Figure 7.4. When the flow around the aggregate is very slow, namely, creeping flow, the drag on a small segment of the aggregate in the shear flow is approximated by 20,21
dRD ( y)
(
24 w y y* Dag
) y (D
ag
)
y dy
(7.7)
where w is the wall shear stress. The moment of force M is represented by M ∫
Dag y*
( y y ) dR *
D
( y)
(7.8)
The maximum bending stress sb in the break point is given by sb
M Z
(7.9)
FIGURE 7.3 Concept of reentrainment of aggregates from a particle deposition layer in turbulent pipe flow. © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.4
A reentrainment model of an aggregate particle.
where Z is the section modulus: Z
{
}
p d3 p * y (Dag y* ) 32 4
3/2
(7.10)
From Equation 7.7 through Equation 7.10, the following equation is obtained: sb
96 w f (Y * ) p
(7.11)
where f (Y *) is represented by
( )
(
)
3 / 2
f Y * ⎡⎣Y * 1 Y * ⎤⎦
∫ (Y Y ) 1
Y*
* 2
Y (1 Y )dY
(7.12)
where Y*
y* ≈ 0.5 Dag
(7.13)
y Dag
(7.14)
Y
From Equation 7.11 through Equation 7.14, b 3w is obtained; the adhesive strength can be calculated from the wall shear stress. © 2006 by Taylor & Francis Group, LLC
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Particle Reentrainment Efficiency The adhesive strength distribution is related to the reentrainment efficiency. In general, the distribution function is represented by the following log-normal equation20: g (c )
1 2p ln s g
∫
⎛ ( ln ln )2 ⎞ c c50 exp ⎜ ⎟ d ( ln c ) 2 ⎜⎝ ⎟⎠ 2 ln s g
c
0
(7.15)
where c is the critical wall shear stress corresponding to the reentrainment. For turbulent pipe flow: 3 × 103 < Re <105, g (c) is rewritten as a function of the average velocity u–:
()
g u
1 2p ln s g’
∫
u
0
(
⎛ ln u ln u 50 exp ⎜ ⎜ 2 ln 2 s g’ u ⎝ 1
) ⎞⎟ du 2
⎟ ⎠
(7.16)
where s g’ s g4 / 7
(7.17)
When aggregate particles are reentrained from the surface of the particle layer, a new surface of the particle layer is exposed to the flow; the next reentrainment can occur from the newly exposed surface with the same reentrainment efficiency g. This is a hierarchical phenomenon. After renewal of the surface n times, the total reentrainment efficiency G(n) is represented by n
G(n) = ∑ g s s =1
g (1 g n ) 1 g
(7.18)
The mass of particles reentrained per unit area W/Ap is proportional to G(n → ∞ ): W kg k G (n → ∞) Ap 1 g
(7.19)
where k is a constant.
Particle Reentrainment Rate The particle reentrainment is not instantaneous, but takes place over a period of time. This phenomenon cannot be explained solely by the static concept. It should be considered that the reentrainment has a statistical origin associated with flow characteristics such as turbulent bursts. Using both the burst generation probability hB and the adhesive strength distribution (or reentrainment efficiency g), the total reentrainment efficiency G(m) after fluid fluctuation m times is represented by m
n
(m) ∑ ∑ n1 Cs1 (1 hB )n − s (hBg )s
(7.20)
n1 s1
g[1 (1 hB hBg )m ] 1 g
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Using the relation m t /Δ. t, where Δ. t is the interval of the fluctuation, G( m ) is rewritten as a function of elapsed time t:
(t )
⎛ t ⎞ ⎤ g ⎡ ⎢1 exp ⎜ ⎟ ⎥ 1 g ⎢⎣ ⎝ Tc ⎠ ⎥⎦
(7.21)
where Tc is a time constant, which is given by Tc
t ln(1 hB hBg )
(7.22)
As 0 < B < 1 and 0 < g < 1, Tc has a positive value. Also, the normalized total reentrainment efficiency R(t) is represented by R (t )
⎛ t ⎞
(t ) 1 exp ⎜
(t → ∞) ⎝ Tc ⎟⎠
(7.23)
There are actually two types of reentrainment,22,23 namely, short-delay and long-delay reentrainment; therefore, the reentrainment flux J is expressed as J JS J L
(7.24)
When average fluid velocity increases from u– 0 at a constant rate a ( d u– / dt ), the amount of reentrained particles increases with elapsed time. Taking into account the time delay (see Equation 7.23), the reentrainment fluxes JS(t) and JL(t) can be calculated by the convolution, respectively, as follows20: ⎛ t t ⎞ ak a t 1 dg exp ⎜ dt ∫ 2 0 TS (1 g ) du ⎝ TS ⎟⎠
(7.25)
⎛ t t ⎞ 1 (1 a )k a t dg exp ⎜ dt ∫ 2 0 TL (1 g ) du ⎝ TL ⎟⎠
(7.26)
J S (t )
J L (t ) =
where a is the mass ratio of the short-delay reentrainment to total reentrainment. Equation 7.25 and Equation 7.26 can be rewritten as follows: ⎛ u u ⎞ ak u 1 dg exp ⎜ ⎟ du ∫ 2 0 TS (1 g ) du ⎝ aTS ⎠
(7.27)
⎛ u − u ⎞ (1 a )k u 1 dg exp ⎜ du ∫ 2 0 TL (1 g ) du ⎝ aTL ⎟⎠
(7.28)
J S (u)
J L (u)
When the fluid velocity increases at a constant acceleration, the reentrainment flux is calculated from Equation 7.25 and Equation 7.26, or Equation 7.27 and Equation 7.28. When the fluid velocity is maintained at u– u–0 after stopping the fluid acceleration, the reentrainment flux in the steady-state — flow is calculated by substituting zero for dg / du' . © 2006 by Taylor & Francis Group, LLC
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Figure 7.5 shows the experimental and calculated values for the reentrainment flux in accelerated airflow (a 0.01, 0.05, 0.1, and 0.6 m/s 2). The reentrainment flux increases with the airflow acceleration as well as the velocity. Figure 7.6 shows the reentrainment flux in steady-state flow after
FIGURE 7.5 Reentrainment flux of aggregate particles from a particle deposition layer by accelerated air flow ( c50 4.4 Pa, u–50 29 m/s, sg 2.0, k 1.7 × 10–3 kg/m2, a 0.6, TL 250 s, TS 1/a).
FIGURE 7.6 Dimensionless Reentrainment in steady-state flow after stopping air acceleration (c50 4.4 Pa, u–50 29 m/s, sg 2.0, k 1.7 × 10–3 kg/m2, a 0.5~0.7, TL 200~300 s, TS 1/a). © 2006 by Taylor & Francis Group, LLC
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stopping the air acceleration. The average air velocity is increased at a constant acceleration up to u–0 30 or 40 m/s, and maintained for 1000 s. The reentrainment flux is normalized by the flux J0 at t t 0. The dimensionless flux J/J0 decreases rapidly in the early stage and approaches zero gradually. The rate of decrease is larger for higher acceleration, and the time dependence of the reentrainment in the steady-state flow is still affected by the preceding unsteady-state flow (acceleration effect).
Simultaneous Particle Deposition and Reentrainment In gas–solids pipe flow, particle deposition and reentrainment often occur simultaneously. The amount of particles deposited on the surface increases with elapsed time; however, the increasing rate gradually decreases and finally reaches a certain value, as shown in Figure 7.7. This is because aggregate particles are more readily reentrained as the amount of particles on the wall increases.19 The variation of the mass of particles per unit area can be expressed as24 W ⎛W⎞ ⎜ ⎟ A ⎝ A⎠
*
⎧⎪ ⎛ t ⎞ ⎫⎪ ⎨1 exp ⎜ ⎟ ⎬ ⎝ 0 ⎠ ⎭⎪ ⎩⎪
(7.29)
where (W/A)* is the equilibrium mass of particles per unit area, and 0 is a time constant. The calculated lines are also shown in Figure 7.7. The value of (W/A)* depends on the adhesion and separation strengths. Figure 7.8 shows the photographs of typical particle deposition layers formed 24 on the wall. The form of the deposition layer is classified into two categories: a continuous (filmy) deposition layer covering over the wall and a striped-pattern deposition layer; the state depends on the particle diameter and the air velocity, as shown in Figure 7.9. The submicron particles form only a filmy deposition layer, and micron-sized particles form a striped deposition layer as well as a filmy deposition layer. The formation of the striped deposition layer depends on the inertia of the suspended particles.25,26 In addition, other factors such as surface roughness27–30 and external vibration31 should be taken into consideration.
FIGURE 7.7 Mass of particles deposited per unit area as a function of time elapsed (horizontal rectangular channel: 3 mm high, 10 mm wide; particle concentration 0.05kg/m3). © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.8 Photographs of particle deposition layers (horizontal rectangular channel 3 mm high, 10 mm wide, Alumina powder Dp50 1.7 μm).
FIGURE 7.9 State diagram of particle deposition layer as a function of average air velocity and particle diameter. © 2006 by Taylor & Francis Group, LLC
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REFERENCES 1. Rosner, D. E., Mackowski, D. W., Tassopoulos, M., and Castillo, J., Garcia-Ybarra, P., Ind. Eng. Chem. Res., 31, 760–769, 1992. 2. Chen, S. H., Aerosol Sci. Technol., 30, 364–382, 1999. 3. Cooper, D. W., Peters, M. H., and Miller, R. J., Aerosol Sci. Technol., 11, 133–143, 1989. 4. Adamczyc, T., Dabros, T., Czarnecki, J., and van de Ven, T. G. M., Adv. Colloid Interface Sci., 19, 183–252, 1982. 5. Shimada, M., Seto, T., and Okuyama, K., AIChE J., 39, 1859–1869, 1993. 6. Tsai, R., Chang, Y. P., and Lin, T. Y., J. Aerosol Sci., 29, 811–825, 1998. 7. Papavergos, P. G. and Hedley, A. B., Chem. Eng. Res. Des., 62, 275–295, 1984. 8. Shimada, M., Okuyama, K., and Asai, M., AIChE J., 39, 17–26, 1993. 9. Shimada, M., Okuyama, K., Okazaki, S., Asai, T., Matsukura, M., and Ishizu, Y., Aerosol Sci. Technol., 25, 242–255, 1996. 10. Schmidt, F., Gartz, K., and Fissan, H., J. Aerosol Sci., 28, 973–984, 1997. 11. Friedlander, S. K. and Johnstone, H. F., Ind. Eng. Chem., 49, 1151–1156, 1957. 12. Sehmel, G. A., J. Geophys. Res., 75, 1766–1781, 1970. 13. Sehmel, G. A., J. Aerosol Sci., 4, 125–138, 1973. 14. Liu, B. Y. H. and Ilori, T. A., Environ. Sci. Technol., 8:351–356, 1974. 15. Lin, C. H. and Chang, L. F. W., J. Aerosol Sci., 27, 681–694, 1996. 16. Brooke, J. W., Kontomaris, K., Hanratty, T. J., and McLaulin, J. B., Phys. Fluids A, 4, 825–834, 1992. 17. Wang, L. P. and Maxey, M. R., J. Fluid Mech., 256, 27–68, 1993. 18. He, C. and Ahmadi, G., J. Aerosol Sci., 30, 739–758, 1999. 19. Ikumi, S., Wakayama, H., and Masuda, H. Kagaku Kogaku Ronbunshu, 12, 589–594, 1986. 20. Matsusaka, S. and Masuda, H., Aerosol Sci. Technol., 24, 69–84, 1996. 21. Kousaka, Y., Okuyama, K., and Endo, Y., J. Chem. Eng. Jpn., 13, 143–147, 1980. 22. Reeks, M. W., Reed, J., and Hall, D., J. Phys. D Appl. Phys., 21, 574–589, 1988. 23. Wen, H. Y., Kasper, G., and Udischas, R., J. Aerosol Sci., 20, 923–926, 1989. 24. Matsusaka, S., Theerachaisupakij, W., Yoshida, H., and Masuda, H., Powder Technol., 118, 130–135, 2001. 25. Matsusaka, S., Adhiwidjaja, I., Nishio, T., and Masuda, H., Adv. Power Technol., 9, 207–218 , 1998. 26. Theerachaisupakij, W., Matsusaka, S., Akashi, Y., and Masuda, H., J. Aerosol Sci., 34, 261–274, 2003. 27. Adhiwidjaja, I., Matsusaka, S., Tanaka, H., and Masuda, H., Aerosol Sci. Technol., 33, 323–333, 2000. 28. Reeks, M. W. and Hall, D., J. Aerosol Sci., 32, 1–31, 2001. 29. Ziskind, G., Fichman, M., and Gutfinger, C., J. Aerosol Sci., 31, 703–719, 2000. 30. Soltani, M. and Ahmadi, G., J. Adhesion, 51, 105–123, 1995. 31. Theerachaisupakij, W., Matsusaka, S., Kataoka, M., and Masuda, H., Adv. Powder Technol., 13, 287– 300, 2002.
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2.8
Agglomeration (Coagulation) Kikuo Okuyama Hiroshima University, Higashi-Hiroshima, Japan
Ko Higashitani Kyoto University, Katsura, Kyoto, Japan
2.8.1
IN GASEOUS STATE
Coagulation of aerosols, which is used to describe the growing process of aerosol particles in contact with each other, causes a continuous change in number, concentration, and size distribution of agglomerates, keeping the total particle volume constant. Of the various coagulations classified by the kinds of force to cause collision, Brownian coagulation (thermal coagulation) may be the most familiar and fundamental. Others are gradient coagulation, turbulent coagulation, electrostatic coagulation, acoustic coagulation, coagulation due to the velocity difference under gravity or centrifugal force, and so on. When two particles having diameters Dpi and Dpj collide, the number of collisions per unit time per unit volume can be written as
(
)
N ⫽ K Dpi , Dpj ni n j
(8.1)
where ni and nj are the number concentrations of particles i and j, respectively, and K(Dpi, Dpj) is the coagulation rate function or the collision rate between particles of diameters Dpi and Dpj. In the special case of the initial stage of coagulation of a monodisperse aerosol having uniform diameter Dp, the decreasing rate of particle number concentration n can be given from Equation 8.1 as dn ⫽⫺0.5K 0 n 2 dt
(8.2)
where K0 ⫽ K(Dp, Dp). When the coagulation rate function is not a function of time, the decrease in particle number concentration from n0 to n can be obtained from the integration of Equation 8.2 over a time period from zero to t: n⫽
n0 1 ⫹ 0.5K 0 n0 t
(8.3)
Because the particle volume is kept constant during the coagulation process, the average particle diameter Dp based on particle volume after the period of time t can be given by Dp ⫽ Dp 0 (1 ⫹ 0.5K 0 n0 t )
13
(8.4) 183
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However, in the case of coagulation of a polydisperse aerosol, the basic equation that describes the time-dependent change in particle number concentration of volume v at time t, n(v, t)dv, can be given as ⭸n ( v, t ) ⭸t
⫽
1 v K ( v ′, v ⫺ v ′ ) n ( v ′, t ) n ( v ⫺ v ′, t ) dv ′ 2 ∫0 ∞
⫺n ( v, t ) ∫ K ( v, v ′ ) n ( v ′, t ) dv ′
(8.5)
0
The first term on the right-hand side represents the rate of formation of particles of volume v due to coagulation, and the second represents the rate of loss of particles of volume v by coagulation with all other particles. The time-dependent change in number concentration and size distribution of aerosol particles undergoing coagulation can basically be obtained by solving Equation 8.5 for a given initial particle size distribution. In the derivation of Equation 8.5, however, the following assumptions are made: (1) particles are electrically neutral, and (2) particles are spherical and collide with each other to form another spherical particle whose mass is the same as the combined mass of the two smaller particles. Accordingly, Equation 8.5 cannot be applied to evaluate the coagulation of solid particles as particle fusion takes place slowly after the collision. In order to consider the nonspherical effect of agglomerated particles on coagulation, the population balance equation for collision and sintering has recently been derived.1 Although their model predicted the average primary particles sizes of agglomerates, the effect of agglomerate structure on the collision rate was neglected by using the collision rate of spherical particles having an equivalent volume of the agglomerate.2
Brownian Coagulation For aerosol particles of submicron diameter, Brownian coagulation by Brownian motion of particles is essential for the characterization of the behavior of aerosols. In the general case, the rate function of Brownian coagulation is characterized by the Knudsen number Kn ⫽ 2l/Dp, where l is the mean path of the background gas. In the continuum regime, having very small values of Kn, Smoluchowski3 applied the theory of Brownian diffusion to the statistical problem of the collision of particles thermally agitated in a continuum. In the diffusion process where a certain particle (radius r i and diffusion coefficient D i ) among other particles is fixed, the surrounding particles (radius rj and diffusion coefficient D j ) collide with the fixed particle due to their Brownian motion, as shown in Figure 8.1. The number of particle r j diffusing in unit time is ⎛ ⭸n ⎞ N ⫽ 4prij 2 D j ⎜ ⎟ ⎝ ⭸r ⎠ r⫽r
(8.6)
ij
where rij ⫽ ri ⫹ rj is the distance between the centers of the two particles at the moment of contact, and n is the concentration of particles rj at time t as a function of distance r from particle ri. The value of the concentration gradient ⭸n/⭸r can be determined by solving the diffusion equation in spherical coordinates. For submicron particles, this concentration instantaneously shifts to the stationary one by the following equation: rij ⎞ ⎛ n ⫽ n j ⎜1 ⫺ ⎟ r⎠ ⎝
(8.7)
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Agglomeration (Coagulation)
FIGURE 8.1
185
Brownian coagulation by Brownian diffusive deposition.
Substitution of this equation to Equation 8.6 gives N ⫽ 4pD j rij n j
(8.8)
Because the central fixed particle is also in Brownian movement, Equation 8.8 leads to the next equation by adding the Brownian diffusion coefficient D i of particle ri.
(
)(
)
N ⫽ 4p Di ⫹ D j ri ⫹ rj n j
(8.9)
Accordingly, the Brownian coagulation rate function in the continuum regime can be given from its definition as Equation i in Table 8.1, which lists the available coagulation function KB(Dpi, Dpj) for Brownian coagulation.4,5 Figure 8.2 shows the coagulation rate functions of monodispersed particles, 0.5KB(Dp, Dp), as a function of particle diameter in air at ambient conditions. There exists a distinct maxima in the size range from 0.01 to 0.1 μm, depending on particle diameter.6 At the present stage, for Brownian coagulation in the transition regime, Fuchs’ interpolation formula (Equation ii in Table 8.1) is considered to be reasonable due to the better agreement with more rigorous theories. Furthermore, the effect of agglomerate structure on the collision kernel is incorporated by replacing the particle diameter with the collision diameter.2,7 The collision diameter of agglomerate, Dc,i with a primary particle diameter, dp,i and the number of primaries per an agglomerate, np,i is given by ⎛ v ⎞ Dc ,i ⫽ dp,i ⎜ i ⎟ ⎝ vp , i ⎠
1 / Df
( )
⫽ dp,i np,i
1 / Df
(8.10)
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TABLE 8.1 Available Expressions for Brownian Coagulation Rate Function Continum Regime Kn ⬍ 0.1 (i) 2p Di ⫹ D j Dpi ⫹ Dpj
(
)(
)
Transition Regime 0.1 ⱕ Kn ⱕ 10 (ii) 2p Di ⫹ D j Dpi ⫹ Dpj
(
)(
)
⫺1 ⎡ Dpi ⫹ Dpj 8( Di ⫹ D j ) ⎤ ⫻⎢ ⫹ ⎥ ⎢⎣ Dpi ⫹ Dpj ⫹ 2 gij vij ( Dpi ⫹ Dpj ) ⎥⎦ ............(ii)
............(i)
Free-Molecular Regime Kn ⬎ 10 2 p (iii) D ⫹ Dpj vij 4 pi ............(iii)
(
)
k ⫽ Boltzmann constant ( ⫽ 1.38 ⫻ 10⫺23 J/K), T ⫽ temperature Di ⫽
kT 3pmDpi
Di ⫽
(
Cc kT 3pmDpi
vij ⫽ vi2 ⫹ v j2
⎡ ⎛ ⫺1.1⎞ ⎤ Cc ⫽ 1 ⫹ Kn ⎢1.257 ⫹ 0.4 exp ⎜ ⎝ Kn ⎟⎠ ⎥⎦ ⎣
( (D g⫽
) + l ) ⫺ (D
pi
i
li ⫽
FIGURE 8.2
mi ⫽
0.5
gij ⫽ gi2 ⫹ g 2j
3
2 pi
i
⎛ 8kT ⎞ vi ⫽ ⎜ ⎝ pmi ⎟⎠
+ li2
3 D pi li
)
)
0.5
0.5
p 3 D r 6 pi p
1.5
⫺ Dpi
8 Di pvi
Brownian coagulation rate against particle diameter.
and dp,i ⫽
6 vi ai
(8.11)
where vp,i is the volume of a spherical primary particle in agglomerate size i, and Df is the mass fractal dimension. However, the enhancement of coagulation rate due to the influence of van der Waals or the dispersion force between colliding particles was quantitatively explained by Marlow’s theory.8 © 2006 by Taylor & Francis Group, LLC
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In order to evaluate the change in particle size distribution of aerosols, the following three types of solutions have been obtained: (1) analytical solution, (2) asymptotic solution, and (3) numerical solution. Table 8.2 shows representative analytical and approximate solutions. In a series of reports, Friedlander9 developed the similarity theory for the uniformity in shape of observed size spectra of atmospheric aerosols. The asymptotic size distribution obtained is called the self-preserving distribution function (SPDF). This similarity theory evaluates the size spectra only after a sufficiently
TABLE 8.2 Representative Analytical and Approximate Solutions for Brownian Coagulation (1) Kn ⬍ 0.01, initially monodispersed particlesa ⎛ t ⎞ nk ⫽ n0 ⎜ ⎟ ⎝ tB ⎠
k⫺1
⫺1
k +1 ⎡⎛ t ⎞ ⎤ ⎢⎜ 1 ⫹ ⎟ ⎥ tB ⎠ ⎥ ⎢⎣⎝ ⎦
where t B ⫽
(
1 , K = 0.5K B Dp , Dp K B0 n0 B0
)
a
(2) Kn ⬎ 10, initially monodispersed particlesb n ⫽ 0.1624a⫺6 / 5 ( kT )
⫺3 / 5
⎛ Mn0 t 6 ⎞ ⎜ r4 N ⎟ ⎝ p av ⎠
⫺1 5
⎛ kT ⎞ , dv ⫽ 2.274a2 / 5 ⎜ 4 ⎟ ⎝ rp ⎠
1/ 5
⎛ Mn0 t ⎞ ⎜⎝ N ⎟⎠
2/5
av
(3) Kn ⬍ 0.1, initially polydispersed particlesc
(
n 1 vg ⫽ , ⫽ A exp 4.5 ln 2 s g 0 n0 A vg 0
{
(
where A ⫽ 1 ⫹ 1 ⫹ exp ln 2 s g 0
)
(
)
⎛ exp 9 ln 2 s g 0 ⫺ 2 ⎞ ⎟ ⎜2⫹ ⎟⎠ ⎜⎝ A
)} 0.5K
(
⫺1 2
n t , K B0 ⫽ 0.5K B Dp , Dp
B0 0
(
)
exp 9 ln 2 s g 0 ⫺ 2 ⎞ 1 ⎛ , ln 2 s g ⫽ ln ⎜ 2 ⫹ ⎟ ⎟⎠ A 9 ⎜⎝
)
(4) Kn ⬎ 10, initially polydispersed particlesd ⎛ n 1 vg E15 / 2 ⫺ 2 ⎞ , ⫽ ⫽ E 9 / 2 A6 / 5 ⎜ 2 ⫹ n0 A6 5 vg 0 A ⎟⎠ ⎝
⫺3 / 5
⎡2 ⎛ E 15 2 ⫺ 2 ⎞ ⎤ , s g ⫽ exp ⎢ ln ⎜ 2 ⫹ ⎥ 15 A ⎟⎠ ⎦ ⎝ ⎣
12
where A ⫽ 1 ⫹ ( 85 ) H t , t ⫽ (6 kT rp )1 2 rg10/ 2 n0 t , H ⫽ E 1 / 8 ⫹ 2 E 5 / 8 ⫹ E 25 / 8 , E ⫽ exp ( z0 ) , z0 ⫽ ln 2 s g 0 (5) Kn ⬎ 10, for fractal agglomeratese 4⫺Df ⎤ n ⎡ ⎛ 3 Df ⫺ 4 ⎞ ⫽ ⎢1 ⫹ ⎜ H 0 bKvg20Df n0 t ⎥ ⎟ n0 ⎣⎢ ⎝ 2 Df ⎠ ⎥⎦
−
2 Df 3 Df ⫺4
,
⫺ Df
4⫺Df ⎫⎪ ⎛9 ⎞ ⎧⎪ ⎛ 3 D ⫺ 4 ⎞ 2 Df H bKv n0 t ⎬ ⫽ exp ⎜ ln 2 s 0 ⎟ ⎨1 ⫹ ⎜ f 0 g0 ⎟ ⎝ ⎠ vg 0 2 ⎩⎪ ⎝ 2 Df ⎠ ⎭⎪
vg
2 Df 3 Df ⫺4
⎤ 3 Df ⫺4 ⎡ ⎧⎪ 9 (3 Df ⫺ 4 ) 2 ⎫⎪ exp ⎨ ln s 0 ⎬ ⫺ 2 ⎥ ⎢ 2 Df ⎥ ⎢2 ⫹ ⎩⎪ ⎭⎪ , 4⫺Df ⎥ ⎢ ⎛ 3 Df ⫺ 4 ⎞ 2 D f ⎥ ⎢ H bKv n t 1⫹ ⎜ 0 0 g0 ⎥⎦ ⎢⎣ ⎝ 2 Df ⎟⎠
⎤ ⎡ ⎪⎧ 9 (3 Df ⫺ 4 ) 2 ⎫⎪ exp ⎨ ln s 0 ⎬ ⫺ 2 ⎥ ⎢ 2 D D 2 ⎪ ⎪ f ⎥ ⎩ ⎭ f ln 2 s ⫽ ln ⎢2 ⫹ 4⫺Df ⎥ 9 (3 Df ⫺ 4 ) ⎢ 3 Df ⫺ 4 2 Df ⎢ 1⫹ H 0 bKvg 0 n0 t ⎥ ⎥⎦ ⎢⎣ 2 Df (continued) © 2006 by Taylor & Francis Group, LLC
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TABLE 8.2 Representative Analytical and Approximate Solutions for Brownian Coagulation (Continued) 2 1⎞ 1 1⎞ 1⎞ ⎪⎧ 9 ⎛ 4 ⎪⎫ ⎪⎧ 9 ⎛ 2 ⎪⎫ ⎪⎧ 9 ⎛ 4 ⎪⎫ where H ⫽ exp ⎨ ⎜ 2 ⫺ ⫹ ⎟ ln 2 s ⎬ ⫹ 2 exp ⎨ ⎜ 2 ⫺ ⫹ ⎟ ln 2 s ⎬ ⫹ exp ⎨ ⎜ 2 ⫹ ⎟ ln 2 s ⎬ , Df 4 ⎠ Df 4 ⎠ 4⎠ ⎪⎩ 2 ⎝ Df ⎪⎭ ⎪⎩ 2 ⎝ Df ⎪⎭ ⎪⎩ 2 ⎝ Df ⎪⎭ 2
⎛ 3 ⎞ Df K ⫽⎜ ⎟ ⎝ 4p ⎠
⫺
1 2
⎛ 6 kT ⎞ ⎜ r ⎟ ⎝ p ⎠
1/ 2
rp2⫺6 / Df
n, particle number concentration (m−3) rg, geometric mean radius rp, primary particle radius (m) nk, number concentration of particle containing k initial particles vg, geometric mean volume [⫽ pdpg3/6, dpg : geometric mean diameter (m)] sg, geometric standard deviation k, Boltzmann constant (1.38 × 10−23 J/K) Df, mass fractal dimension T, absolute temperature (K) rp, particle density (kg/m3) t, time (s) a, sticking probability between collision Nav, Avagadro’s number M, molecular weight 0, initial value b, values depend on sg0 and Dfe. a Friedlander, S. K., Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, 2nd Ed., Oxford University Press, New York, 2000, pp. 188–196. b Ulrich, G. D., Combust. Sci. Technol., 4, 47–58, 1971. c Lee, K. W., J. Colloid Interface Sci., 92, 315–325, 1983. d Lee, K. W., Curtis, L. A., and Chen, H., Aerosol Sci. Technol., 12, 457–462, 1990. e Park, S. H. and Lee, K. W., J. Colloid Interface Sci, 246, 85–91, 2002.
long time has passed. Figure 8.3 depicts the time-dependent change of particle size distribution obtained numerically solving Equation 8.5.10
Laminar and Turbulent Coagulation When aerosol particles are suspended in a laminar shear flow, particles tend to coagulate due to their relative motion, and the coagulation rate function can be given as follows3:
(
)
(
K L Dpi , Dpj ⫽ 0.17hL Dpi ⫹ Dpj
)
3
du dx
(8.12)
where du/dx is the velocity gradient of the flow and h L is the collision efficiency, which expresses the effect of the aerodynamic interaction between particles. The collisions between particles suspended in a turbulent flow are generally considered to be caused by the two essentially independent mechanisms. In the first mechanism, particles may collide with each other due to the velocity difference between particles caused by the spatial © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.3 coagulation.
189
Time-dependent changes in size distribution of particles undergoing Brownian
nonhomogeneity characteristic of turbulent flow. The theoretical rate function by Saffman and Turner11 is
(
)
(
LT1 Dpi , Dpj ⫽ 0.16hT Dpi ⫹ Dpj
) ⎛⎜⎝ v ⎞⎟⎠ 3
12
0
(8.13)
where 0 is the energy-dissipation per unit mass of fluid, n the kinematic viscosity, and h T the collision efficiency. The second mechanism may be caused by a relative motion of each particle to the local turbulent motion of the fluid, because the inertia of a particle is not the same between unequalsized particles. Saffman and Turner11 also obtained the following equation:
(
)
(
K T2 Dpi , Dpj ⫽ 1.43hT Dpi ⫹ Dpj
⎛ r ⎞⎛ 3⎞ Dpi ⫺ Dpj ⎜ 1⫺ f ⎟ ⎜ 0 ⎟ rp ⎠ ⎝ v ⎠ ⎝
) ( ) ( ) 2
0.25
(8.14)
where τ(Dp) is the particle relaxation time ( ⫽ rpCcDp2/18m). Figure 8.4 shows the turbulent coagulation function. It is seen that the turbulent coagulation plays an important role in the coagulation of aerosol particles larger than a few microns even under the small value of 0.12
Acoustic Coagulation In an acoustic field, particles coagulate mainly by the following three mechanisms: (1) orthokinetic coagulation due to relative oscillating motion, (2) coagulation due to hydrodynamic attractive force between particles, and (3) turbulent coagulation due to acoustically induced turbulence in a highly intensive acoustic field. Shaw 13 has published an excellent review on acoustic coagulation. © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.4 Turbulent coagulation rate against particle diameter. Solid line, Equation 8.13 and Equation 8.14; dotted– dashed line, Equation 8.14; dashed line, Equation 8.13.
2.8.2
IN LIQUID STATE
The fundamental behavior of particles in liquids is very similar to that in gases, except that (1) the viscosity and permitivity of the medium are much greater, (2) the charge of particles is related closely with their stability, and (3) the effect of particle inertia is usually negligible. The motion of a particle in a solution is expressed by .. mr ⫽ fT ⫹ fB ⫹ fF
(8.15)
.. where m and r are the mass and acceleration of the particle; fT, the total static interaction force with other particles; f B , the force due to the Brownian motion; and f F , the force generated by the flow of the medium. Subtraction of equations between an arbitrary pair of particles gives an equation of the relative motion as follows. Here the effect of particle inertia is neglected. FT ⫹ FB ⫹ FF ⫽ 0
(8.16)
where FT, FB, and FF are the corresponding forces acting relatively between particles. FT is the static forces, such as the electrical repulsive force and the van der Waals attractive force. FB and FF are the Brownian and hydrodynamic forces caused by the relative motion of particles respectively, which may be named the dynamic interaction force. If Equation 8.16 is able to be solved, the relative motion of particles will be available to calculate the coagulation rate.
Static Interaction and Coagulation (DLVO Theory) In order to calculate the coagulation process quantitatively, FT , FB and FF must be known. However, as far as the stability of a suspension is concerned, it may be estimated using the static interaction © 2006 by Taylor & Francis Group, LLC
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force F T ⫽ |F T| only. For example, if F T is only attractive, one will find that the particles are unstable and, if the maximum value of FT is very large, the particles must be free from coagulation, even though collisions between particles occur. In this section, the relationship between the static interaction and particle stability is discussed, employing the interparticle potential energy VT, given by FT ⫽ –dVT /dr, where r is the distance between particle centers. The charge of particles in aqueous solutions makes them repulsive to each other as explained in Section 2.5.3. The repulsive force between similar particles of radius a is explicitly given by the following equations, when the surface potential, c d, is sufficiently low (say, c d < 25 mV) 14 :
{
VR ⫽⫾ 2p acd2 ln 1 ⫾ exp ⎡⎣⫺ka ( s ⫺ 2 )⎤⎦
} (ka ⬎ 10)
VR ⫽ 4p acd2 exp[⫺ka(s ⫺ 2)] s ( ka ⬍ 5)
(8.17) (8.18)
where is the permitivity of the medium; s, the dimensionless distance between particle centers normalized by a; k ⫽ (2n0z2e2/kT )0.5; n0, the ionic concentration of bulk solution; z, the ionic valency; e, the elementary charge; k, the Boltzmann constant; and T, the temperature. k is a measure of the electrolyte concentration. The ⫹ sign in Equation 8.17 indicates that the equation is for the constant surface potential, and the – sign indicates that the equation is for the constant surface charge. Equation 8.18 is applicable for both cases. The instantaneous dipole moment generated by the fluctuation of electrons around an atomic nucleus polarizes the neighboring atoms and generates the attractive force between atoms. This interaction force is called the disperson force. Permanent dipoles also generate the interaction force. The sum of these forces is called the van der Waals force. Because the contribution by the permanent dipole is small in most cases, the dispersion force is considered to be the van der Waals force. Integrating the contribution due to all the atoms in particles, the van der Waals force between spherical particles is given by 2 ⎛ s2 ⫺ 4 ⎞ ⎤ ⎛ A⎞ ⎡ 2 VA ⫽ ⎜ ⎟ ⎢ 2 ⫹ 2 ln ⎜ 2 ⎟ ⎥ ⎝ 6 ⎠ ⎣s ⫺ 4 s ⎝ s ⎠⎦
(8.19)
where A is the Hamaker constant. The value of A is evaluated either theoretically or experimentally.15 The value of A between dissimilar particles 1 and 2, in a medium, 3, may be estimated by A132 ⫽
(
A11 ⫺ A33 )( A22 ⫺ A33
)
(8.20)
where Aij is the Hamaker constant of the material, i, in free space. When particle surfaces are so close that the electron clouds overlap, a very strong Born repulsion force appears. It is said that the Born potential, V Born , is infinitely large when the separation distance is a few angstroms (say, 4Å) and zero elsewhere. The total interaction potential, VT, is given by the sum of these potentials: VT ⫽ VR ⫹ VA ⫹ VBorn
(8.21)
A typical potential curve is illustrated in Figure 8.5a. When the energy given by the Brownian or fluid motion overcomes the maximum potential energy, particles will coagulate. Otherwise, particles will be either dispersed or coagulated loosely in the secondary minimum even if particles collide each other. © 2006 by Taylor & Francis Group, LLC
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Vaom
VT kT
4Å
0
S
Secondary minimum Primary minimum VT
C = 10
(a) VT
-3 mol/l
A=
70mV
50mV
10 -2 mol/l 10 -1 mol/l
20mV S
S
VA
VA
(b)
(c)
FIGURE 8.5 Interparticle potential and the dependence on electrolyte concentration and surface potential.
Most parameters in Equation 8.21 are not variable for a given suspension, but the values of k and cd may vary by changing the electrolyte concentration and the pH of the solution, respectively. As illustrated in Figure 8.5b and 8.5c, the maximum value of the potential decreases as the value of κ, that is, the electrolyte concentration, increases and also as the absolute value of cd decreases. At a sufficiently high electrolyte concentration, the peak of the potential curve disappears. This indicates that every collision results in coagulation. This electrolyte concentration is called the critical coagulation concentration (CCC). Coagulation where the electrolyte concentration is higher than the CCC is called rapid coagulation, whereas coagulation where the concentration is lower than the CCC is called slow coagulation. The theoretical value of CCC for particles with a relatively high surface potential in an aqueous solution of 25°C is given by CCC(mol/L) ⫽
87 ⫻ 10⫺40 z 6 A2
(8.22)
This indicates that the CCC is very sensitive to the ionic valency and, therefore, so is the stability. The correlation has been confirmed experimentally and is known as “Shultze–Hardy rule.” The stability of a suspension is controllable also by changing the value of cd. The variation of cd depends on the mechanism of surface charge. cd varies with the pH of the medium in the case of oxide particles, and with the concentration of the potential determining ions in the case of the Nernst-type particles, as described in Section 2.5.3. © 2006 by Taylor & Francis Group, LLC
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Various surfactants and polymers are possibly employed to control the stability of particles, but these methods are discussed in Section 4.5.2.
Dynamic Interaction and Coagulation Rate Particles interact with each other through their Brownian motion and the fluid motion. As explained earlier, the stability of particles can be evaluated roughly by the static interaction, but the dynamic interactions should be taken into account to know the stability quantitatively and to calculate the coagulation rate. Brownian Coagulation and Stability Ratio In the case of rapid coagulation, every collision due to the Brownian motion results in coagulation. The rate of the Brownian coagulation between particles i and j in a stationary medium, Jij , is given by16 ⎛1 ⎛ ∞ exp(VTij / kT ⎞ ⎡ 2 KT ⎤ 1⎞ J ij ⫽ ⎢ ai ⫹ a j ⎜ ⫹ ⎟ ni n j ⎜ ∫ ⎥ ⎟⎠ r 2 dr ⎝ ai ⫹a j ⎝ ai a j ⎠ ⎣ 3m ⎦
(
)
−1
(8.23)
where ai and ni are the radius and number concentration of particle i, and VTij is the interparticle potential between particles i and j. Then the change of particle concentration is calculated by the population balance equation: ⬁ dnk 1 k⫺1 ⫽ ∑ J ij ⫺ ∑ J ik dt 2 i⫽1 i⫽1
(8.24)
i⫹ j⫽k
Under the condition that VTij ⫽ 0 and the particles are initially monodispersed, substitution of Equation 8.23 into Equation 8.24 gives the following equations for total and i-fold particle concentrations at times t, Nt, and ni, respectively: t 1 1 ⫺ ⫽ N t N 0 N 0 t1 / 2
(8.25)
N 0 (t / t1 / 2 )i −1 (1 ⫹ t / t1 / 2 )i +1
(8.26)
ni ⫽
where N0 is the initial total particle concentration and t1/2 is 3m/4kTN0. Equation 8.25 indicates 1/Nt varies linearly with time. It is found that this theory overestimates the rapid coagulation rate by about 40%, and this is because no hydrodynamic interaction is taken into account.17 Because the van der Waals force and the hydrodynamic interaction act even in rapid coagulation, the correction factor representing these effects, aB ( ⫽ αBii), can be calculated by Equation 8.16 as shown in Figure 8.6.19 Then the change of ni can be evaluated numerically as shown in Figure 8.7, and Nt is expressed analytically by dN t ⎛ 4a kT ⎞ ⫽⫺ ⎜ B ⎟ N t2 ⎝ 3m ⎠ dt
(8.27)
In the case of the slow coagulation (VTij ⫽ 0), the prediction of the coagulation rate is not easy, but the stability of suspensions is evaluated by the so-called stability ratio W. W is a measure of the © 2006 by Taylor & Francis Group, LLC
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8 (-)
1.0
Smoluchowski Theory
0.8 0.6 0.4 10
–21
10
–20
10
–19
10
–18
A (J)
FIGURE 8.6 constant A.
Dependence of aB on the Hamaker
deviation from the rapid coagulation at the initial stage of coagulation where the coagulation between particles of the same size is dominant: W⫽
J 0R J 0S
(8.28)
where the superscripts R and S indicate the rapid and slow coagulation, the subscript 0 indicates t ⫽ 0, and J ⫽ J11. Then, the theoretical value of W is derived from Equation 8.23 as W ⫽ 2a∫
⬁
2a
exp(VT 11 / kT ) dr r2
(8.29)
The experimental value of W is determinable using the following equation from the initial change of the turbidity, , of suspensions19: R
⎛ d ⎞ W ⫽⎜ ⎟ ⎝ dt ⎠ 0
S
⎛ d ⎞ ⎜⎝ ⎟⎠ dt 0
(8.30)
It is known that the theoretical and experimental values of W agree qualitatively but not quantitatively. This discrepancy is one of the unsolved problems of colloidal phenomena.16 Coagulation in Shear Flow Particles in flow fields collide with each other because of their relative velocity. It depends on the balance between the energy of particles given by the flow and the interparticle potential energy whether the collision results in coagulation or not. The simplest model for the shear coagulation was proposed by Smoluchowski,20 in which it is assumed that FT ⫽ FB ⫽ FF ⫽ 0, but particles collide because of the geometrical interception. In this case, the collision rate is given by 4 J ij ⫽ gRij3 ni n j 3
(8.31)
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1
10
Particle Size (Fold)
–1
1 2
ni/N0 (—)
4 6 8
10–2
10
Experimental Run 1
10
–3
Run 2 Run 3 Calculated
0
20
40
60
t (min.) FIGURE 8.7 Comparison of ni versus t between theoretical prediction and experimental results of Brownian coagulation.
Where g is the shear rate and Rij ( ⫽ ai ⫹ aj) is the coagulation radius. Then, the total particle concentration is given by dN t ⎛ 4gf ⎞ N ⫽⫺ ⎜ ⎝ p ⎟⎠ t dt
(8.32)
where f is the volume fraction of particles. The trajectory assumed here is valid for particles with a large inertia, such as particles in air. But in the case of particles in liquid, the effect of the inertia is small and the hydrodynamic interaction plays an important role so that the trajectory becomes complicated, as illustrated in Figure 8.8b. The trajectory for this case was solved by Equation 8.16 with © 2006 by Taylor & Francis Group, LLC
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FB ⫽ 0, and the correction factor, αs, for the rapid shear coagulation rate was obtained numerically as shown in Figure 8.8.21 If the contributions of the Brownian and shear coagulations are additive, the change of Nt is given by dN t ⎛ 4a kT ⎞ ⎛ 4a gf ⎞ ⫽⫺ ⎜ B ⎟ N t2 ⫺ ⎜ S ⎟ N t ⎝ p ⎠ ⎝ 3m ⎠ dt
(8.33)
This equation is easily solved with the proper initial condition, and it is found that the prediction agrees with the experimental results quantitatively.22 Taking all the contributions of the hydrodynamic interaction and interaction potentials into account except that of the Brownian motion, the relative trajectory of a pair of particles in the shear flow was calculated by Equation 8.16, and the regions of coagulation and dispersion between equal spherical particles are clarified, as shown in Figure 8.9. 23 This diagram indicates that the stability of suspensions is determined by the balance between the static and dynamic interactions. Coagulation in Turbulent Flow It is not easy to estimate the coagulation rate of particles in turbulent flow because of the complicated flow field. Saffman and Turner 24 derived an expression for the rapid coagulation rate in a turbulent
10
Unretarded 1
s (–)
Smoluchowski Theory
10–1 1 10 100 10–2 10–1
10
1
A
FIGURE 8.8
103
102
104
105
(–)
Regions of coagulation and dispersion.
103
102
Dispersion
Secondary minimum Coagulation
Primary minimum coagulation 101 10–1
100
101
102
103
104
FIGURE 8.9 © 2006 by Taylor & Francis Group, LLC
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flow without taking the hydrodynamic interaction into account. Their expression was modified by introducing the correction factor, as, for the rapid shear coagulation rate as follows25: ⎛ ⌽⎞ J ij ⫽ 1.429aS Rij3 ⎜ ⎟ ni n j ⎝ v⎠
(8.34)
where Φ is the energy dissipation per unit mass of fluid and v is the kinetic viscosity of the medium. Substituting Equation 8.34 into Equation 8.24, the change of the particle size distribution can be predicted. It is confirmed that the prediction by Equation 8.34 coincides with experimental results quantitatively.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
Koch, W. and Friedlander, S. K., J. Colloid Interface Sci., 140, 419–427, 1990. Tsantilis, S. and Pratsinis, S. E., AIChE J., 46, 407–415, 2000. Smoluchowski, M. W., Phys. Z., 17, 557–571, 1916. Fuchs, N. A., The Mechanics of Aerosols, Dover, New York, 1964, pp. 288–302. Sitarski, M. and Seinfeld, J. H., J. Colloid Interface Sci., 61, 261–271, 1977. Okuyama, K., Kousaka, Y., and Hayashi, K., J. Colloid Interface Sci., 101, 98–109, 1984. Kruis, F. E., Kusters, K. A., and Pratsinis, S. E., Aerosol Sci. Technol., 19, 514–526, 1993. Marlow, W. H., J. Chem. Phys., 73, 6284–6287, 1980. Friedlander, S. K., Smoke, Dust, and Haze: Fundamentals of Aerosol Dynamics, 2nd Ed., Oxford University Press, New York, 2000, pp. 210–219. Yoshida, T., Okuyama, K., Kousaka, Y., and Kida, Y., J. Chem. Eng. Jpn., 8, 317–322, 1975. Saffman, P. G. and Turner, J. S., J. Fluid Mech., 1, 16–30, 1956. Okuyama, K., Kousaka, Y., and Yoshida, T., J. Aerosol Sci., 9, 399–410, 1978. Shaw, D. T., Recent Development in Aerosol Science, Wiley, New York, 1978. Hunter, R. J., Foundations of Colloid Science, Vol. 1, Clarendon Press, Oxford, 1987. Israelachvili, N. J., Intermolecular and Surface Force, Academic Press, London, 1985. Kruyt, H. R., Colloid Science, Elsevier, Amsterdam, 1952. Higashitani, K. and Matsuno, Y., J. Chem. Eng. Jpn., 12, 460, 1979. Higashitani, K., Tanaka, T., and Matsuno, Y., J. Colloid Interface Sci., 63, 551, 1978. Troelstra, S. A. and Kruyt, H. R., Kolloid-Beihefte, 54, 225, 1942. Smoluchowski, M., Z. Phys. Chem., 92, 129, 1917. Higashitani, K., Ogawa, R., Hosokawa, G., and Matsuno, Y., J. Chem. Eng. Jpn., 15, 299, 1982. Higashitani, K., Gyoushu Kougaku, Nikkankogyo Shinbunsha, Tokyo, 1982, p. 53. Zeichner, G. R. and Showalter, W. R., AIChE J., 23, 243, 1977. Saffman, P. G. and Turner, J. S., J. Fluid Mech., 1, 16, 1956. Higashitani, K., Yamauchi, K., Matsuno, Y., and Hosokawa, G., J. Chem. Eng. Jpn., 16, 299, 1983.
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2.9
Viscosity of Slurry Hiromoto Usui Kobe University, Nada-ku, Kobe, Japan
2.9.1
INTRODUCTION
The technical term slurry contains quite a wide concept of solid–liquid mixtures. If a solid–liquid mixture has some fluidity, we call it a slurry. However, suspended solid particles vary in size from very fine colloidal particles to sedimentable coarse particles. Also, the surface characteristics change according to the combination of solid material and liquid phase. Therefore, the flow characteristics of slurry change widely. The solid concentration greatly affects the viscosity of slurry. In addition to the effect of solid concentration, the effects of particle size distribution, the shape, and the surface properties of particles on the viscosity of slurry are significant. In this section, viscosity behavior of a wide variety of slurries is described, and a brief description of the measuring technique of slurry viscosity is also given.
2.9.2
BASIC FLOW CHARACTERISTICS
The representative flow models for various kinds of slurries are shown in Figure 9.1, in terms of the plot of shear stress t versus shear rate (du/dr).1 Figure 9.1a represents the Newtonian model, t ⫽ mN
du dr
(9.1)
where mN is a Newtonian viscosity. Figure 9.1b represents the power-law model, ⎛ du ⎞ t⫽K⎜ ⎟ ⎝ dr ⎠
n
(9.2)
where K and n are material constants. If n is less than unity, this model represents a pseudoplastic fluid, and if n is greater than unity, a dilatant fluid is simulated by this model. When n is equal to unity, this model coincides with the Newtonian fluid model. Figure 9.1c indicates the Bingham plastic model, t ⫺ t y ⫽ mB
du dr
(9.3)
where mB and ty are Bingham plastic viscosity and yield stress, respectively. Figure 9.1d indicates the Herschel–Bulkley model, ⎛ du ⎞ t ⫺ ty ⫽ K ⎜ ⎟ ⎝ dr ⎠
n
(9.4)
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FIGURE 9.1 Classification of fluid models: (a) Newtonian model; (b) power-law model; (c) Bingham model; (d) Herschel–Bulkley model.
More complex models have been proposed for non-Newtonian fluids. However, the foregoing models are simple and widely used for engineering problems.
2.9.3 TIME-DEPENDENT FLOW CHARACTERISTICS Highly loaded slurries of very fine particles (e.g., plaster, clay, paint pigment, and ceramic powder) show not only the non-Newtonian behavior described above but also time-dependent flow characteristics. A slurry often shows the time-dependent decrease of apparent viscosity with increase in shear rate, but the viscosity is recovered when the slurry is settled at rest. This reversible flow characteristic change under constant temperature is called thixotropy. The basic idea for a phenomenological thixotropy model was proposed by Cheng and Evans,2 and articles on thixotropy models have been recently reviewed by Galassi et al.3 and Barnes.4 Rheological characteristics of certain kinds of suspension are remarkably changed by applying an external electric field. This phenomenon is called an electrorheology effect. Electrorheological 5–7 fluids have been extensively investigated because they are expected to be applicable as new functional fluids to many engineering purposes.
2.9.4 VISCOSITY EQUATIONS FOR SUSPENSIONS OF SPHERICAL PARTICLES OF NARROW PARTICLE SIZE DISTRIBUTION Superlative reviews on this subject are given by Thomas8 and Metzner.9 Figure 9.2 shows the increase in viscosity with the solid concentration. There appears to be no effect of particle size on fluid viscosity. The volume fraction of solid particle, f, is a unique function for the relative viscosity, mr. The classic Einstein equation,10 mr ⫽ 1 ⫹ 2.5f
(9.5)
may represent accurately the behavior of the suspension within only a vanishingly small range of solid concentration. Guth and Simha11 proposed the following viscosity equation, which is applicable to a higher solid concentration range. They modified the Einstein equation by taking the effect of interaction between solid particles into account. mr ⫽ 1 ⫹ 2.5f ⫹ 14.1f 2
(9.6)
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Relative viscosity ur (-)
102
10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
Volume fraction of solid o (-) FIGURE 9.2 Relative viscosity versus concentration behavior for suspensions of spheres having narrow particle size distributions. Particle diameters range from 0.1 to 400 mm. The solid line depicts Equation 12.7, and keys for the data are given in the article by Thomas. [From Thomas, D. G., J. Colloid Sci., 20, 267, 1965. With permission.]
Thomas8 recommended the following empirical equation: mr ⫽ 1 ⫹ 2.5f ⫹ 10.05f 2 ⫹ A exp( Bf)
(9.7)
with A = 0.00273 and B = 16.6. This equation gives a prediction similar to the oft-quoted Moony equation12: ⎛ 2.5f ⎞ mr ⫽ exp ⎜ ⎝ 1⫺ k f ⎟⎠
(9.8)
where the parameter k, called the crowding factor, is determined empirically and usually in the range of 1.3 ⱕ k ⱕ 1.9. The empirical equation proposed by Kitano et al.13 f⎞ ⎛ mr ⫽ ⎜ 1⫺ ⎟ ⎝ A⎠
⫺2
(9.9)
gives a good prediction for highly loaded slurries. The single empirical constant A has a value of 0.680 for suspensions of smooth spheres. All the equations mentioned above are applicable for © 2006 by Taylor & Francis Group, LLC
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completely dispersed suspensions. However, the agglomeration of suspended particles affects significantly the fluidity of slurries. The effect of agglomeration on the non-Newtonian flow characteristics has recently been discussed by Usui et al.14
2.9.5 EFFECT OF PARTICLE SIZE DISTRIBUTION ON SLURRY VISCOSITY Most systems of interest are, of course, more complex than the monodisperse suspensions considered thus far. If particles of a wide size distribution are employed, the maximum possible packing density of particles (1/k in Equation 9.8 and A in Equation 9.9) will change considerably. This change causes a reduction in slurry viscosity. Chong et al.15 investigated the effect of bimodal particle size distribution on the slurry viscosity. They showed experimentally that the viscosity of a bimodal suspension is reduced to almost 14%, in magnitude, of unimodal suspension viscosity at f = 0.6 and d/D = 0.137, where d and D are the small and large particle diameters, respectively. The volume fraction of maximum packing in bimodal systems was investigated experimentally by McGeary.16 Using the experimental results of McGeary, Lee17 proposed a prediction method for the maximum packing volume fraction of multimodal suspensions. Recently, the estimation method of the packing density of particle mixtures by a linear-mixture packing 18 model was proposed by Yu and Standish Funk 19 modified the theoretical equation proposed by Andreasen and Andersen20 to give an optimum size distribution for particles in a limited size range. Recently, Usui21 has developed a non-Newtonian viscosity prediction model, which is applicable for nonspherical particle suspension system with particle size distribution.
2.9.6 MEASUREMENT OF SLURRY VISCOSITY BY A CAPILLARY VISCOMETER A capillary viscometer is the most simple and the most frequently used viscometer. The pressurized fluid is forced to flow through a capillary of known diameter D, and the volumetric flow rate is measured. The wall shear stress tW is calculated by
tW ⫽
D⌬P 4L
(9.10)
where ΔP and L are the pressure drop and capillary length, respectively. The relationships between the flow rate and the wall shear stress for the fluid models are given as follows: 1. Newtonian model:
Q⫽
pD 3 t W (Hagen–Poiseuille equation) 32 mN
(9.11)
2. Power-law model:
Q⫽
npD 3 8(3n ⫹1)K
1
n
tW
(9.12)
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3. Bingham model at tW > ty: ⎡ 1 ⎛ ty ⎞ 1 ⎛ ty ⎞ 4 ⎤ pD 3 Q⫽ t W ⎢1⫺ ⎜ ⎟ ⫹ ⎜ ⎟ ⎥ (Buckingham–Reiner equation) 32 mB ⎢⎣ 3 ⎝ t W ⎠ 3 ⎝ t W ⎠ ⎥⎦
(9.13)
4. Herschel–Bulkley model: pD 3 4 n ⎛ t W ⎞ Q= 32 3n ⫹1 ⎜⎝ K ⎟⎠
1
n
ty ⎞ ⎛ ⎜⎝ 1⫺ t ⎟⎠ W
⎧⎪ t y ⎞ ⎤ ⎫⎪ 1 ⎛ ty ⎞ ⎡ 2n ⎛ t y ⎞ ⎛ n 1 ⫺ 1 ⫹ ⎥⎬ ⎢ ⎨1⫺ ⎜ ⎟ ⎜ ⎟⎜ t W ⎟⎠ ⎥⎦ ⎪⎭ ⎪⎩ 2 n ⫹1 ⎝ t W ⎠ ⎢⎣ n ⫹1 ⎝ t W ⎠ ⎝
(9.14)
If it is not known what kind of non-Newtonian fluid model is suitable for a given slurry, one cannot use the equations shown above. In this case, a more complex data acquisition technique, called Maron–Krieger’s method,22 should be employed.
2.9.7 MEASUREMENT OF SLURRY VISCOSITY BY A ROTATING VISCOMETER The rotating viscometers are subdivided into three categories: coaxial rotating cylinders, cone and plate, and plate and plate viscometers. The clearance between cone and plate or between plate and plate is generally so narrow that the use of a coaxial rotating cylinder viscometer is preferred for the measurement of slurry viscosity. For the case of an outer cylinder rotating (Couette type) viscometer, the relationship between the rotating speed V and torque T is given as follows: 1. Newtonian model: V⫽
n ⎛ 1 1⎞ ⫺ 4pL mN ⎜⎝ R12 R22 ⎟⎠
(9.15)
where L, R1, and R2 are the length of the inner cylinder, and radii of inner and outer cylinders, respectively. 2. Power-law model:
V⫽
n 2K
1
n
⎛ 1 ⎞ ⎜⎝ 2pR 2 L ⎟⎠ 1
1
n
2
2
R2 n ⫺ R1 n 2
(9.16)
R2 n
3. Bingham model at T ⬎ 2pR22Lty:
V⫽
R n ⎛ 1 1 ⎞ ty ⫺ 2⎟⫺ ln 2 2 ⎜ 4pL m B ⎝ R1 R1 ⎠ m B R1
(9.17)
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The estimation of shear rate, du/dr, on the surface of the inner cylinder is complex if the fluid is non-Newtonian. The reader should be careful in determining the shear rate using the general description of a Couette viscometer.22
REFERENCES 1. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, Wiley, New York, 1960, p. 10. 2. Cheng, D. C.-H. and Evans, F. Br. J. Appl. Phys., 16, 1599, 1965. 3. Galassi, C., Rastelli, E., and Laspin, R. Science, Technology, and Application of Colloidal Suspensions, American Ceramics Society, 1995, p. 3. 4. Barnes, H. A. J. Non-Newtonian Fluid Mech., 70, 1, 1997. 5. Brooks, D. A., in Proceedings of the Eleventh International Congress on Rheology, Vol. 2, Elsevier Science, Amsterdam, 1992, p. 763. 6. Otsubo, Y. and Watanabe, K. Nihon Rheology Gakkaishi (J. Soc. Rheol. Jpn.), 18, 111, 1990. 7. Hasegawa, Y., Isobe, T., Senna, M., and Otsubo, Y., J. Soc. Rheol. Jpn., 27, 131, 1999. 8. Thomas, D. G., J. Colloid Sci., 20, 267, 1965. 9. Metzner, A. B., J. Rheol., 29, 739, 1985. 10. Einstein, A. Ann. Phys., 19, 289, 1906. 11. Guth, E. and Simha, R., Kolloid Z., 74, 266, 1936. 12. Mooney, M., J. Colloid Sci., 6, 162, 1951. 13. Kitano, T., Kataoka, T., and Shirota, T., Rheol. Acta, 20, 207, 1981. 14. Usui, H., Kishimoto, K., and Suzuki, H., Chem. Eng. Sci., 56, 2979, 2001. 15. Chong, J. S., Christensen, E. B., and Baer, A. D., J. Appl. Polym. Sci., 15, 2007, 1971. 16. McGreary, R. K., J. Am. Ceram. Soc., 44, 513, 1961. 17. Lee, D. I., J. Paint Technol., 42, 579, 1970. 18. Yu, A. B. and Standish, N., Ind. Eng. Chem. Res., 30, 1372, 1991. 19. Funk, J. E., U.S. Patent 4,282,006, 1979. 20. Andreasen, A. H. M. and Andersen, J., Kolloid Z., 50, 217, 1930. 21. Usui, H., J. Chem. Eng. Jpn., 35, 815, 2002. 22. Middleman, S., The Flow of High Polymer, Wiley, New York, 1968, p. 13.
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2.10
Particle Impact Breakage Mojtaba Ghadiri University of Leeds, Leeds, United Kingdom
In processing of particulate solids, there are circumstances under which interparticle and particle– wall impacts cause damage to the particles. In some cases, such as comminution, this is desirable, while in other cases it is undesirable, as for example in attrition of weak and friable particles in pneumatic conveying. These processes share the same underlying mechanisms, namely, the formation and propagation of various types of crack. It is therefore of great interest to identify and quantify the conditions under which particulate solids are damaged.
2.10.1
IMPACT FORCE
Impact of particles causes a transient stress whose magnitude depends on a number of factors such as impact velocity, particle size, material properties, and contact geometry.
Elastic If the transient stress does not exceed the plastic yield stress, then the impact is treated as elastic. The maximum impact force, F, is commonly obtained by assuming that the relationship between the impact force and displacement is the same as that of a static elastic contact, given by Hertz analysis.1 For normal impact between two spheres F = 1.283 m 3 5 R 1 5 E ∗2 5 V 6 5
(10.1)
where 1/m = 1/m1 + 1/m2, 1/R = 1/R1 + 1/R2, 1/E* = (1 – ν12)/E1 + (1 – ν22)/E2, V is the impact velocity, m is the mass, R is the radius, E is Young’s modulus, ν is Poisson’s ratio, and subscripts 1 and 2 refer to the two impacting bodies. Particle damage in this case is by brittle failure.
Plastic The maximum contact force in this case is obtained by first calculating the onset of yield. The contact stress remains at its yield value and if the material does not strain-harden, the maximum force can be estimated from the contact area. The calculation of contact area is, in turn, based on the impact energy balancing the work of plastic deformation, from which the maximum contact force can be calculated2: ⎛ 4 rY ⎞ F = p R2 ⎜ ⎝ 3 ⎟⎠
12
V (10.2)
where r is the particle density, Y is the yield stress, and V is the normal impact velocity. Particle breakdown in this case is by plastic rupture. 205 © 2006 by Taylor & Francis Group, LLC
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Elastic–Plastic A mathematical treatment of the process has been proposed by Ning and Thornton,3 where Hertz analysis is used to describe the preyield behavior, and a “coupled” elastic/plastic analysis is used to describe the postyield behavior. The maximum impact force, F, is given by Ning2 as F 6 1 2 ⫽ V VY ) ⫺ ( FY 5 5
(10.3)
where FY and VY are the impact force and velocity at the onset of yield, respectively, given by 2
FY ⫽ p 3 R 2Y 3 /(6E ∗ )
(10.4)
and ⎛ p ⎞ VY ⫽ ⎜ ∗ ⎟ ⎝ 2E ⎠
2
⎛ 2⎞ ⎜⎝ 5r ⎟⎠
12
Y52
(10.5)
Particle breakdown in this case is by fracture, which is preceded by limited plastic deformation.
2.10.2
MODE OF BREAKAGE
The current understanding of particle breakage has recently been collated in a special issue of Powder Technology, edited by Salman,4 covering the classic brittle and semibrittle failure modes. Schönert,5 Salman et al.,6 and Wu et al.7 have provided detailed reviews of failure of spheres and discs under both impact and quasi-static loading. Failure of particles is a process that depends on material properties and the mode of loading. The latter is related to the strain rate and contact geometry. Increasing the strain rate from quasi-static to impact can cause a switch in the failure mode from semibrittle to brittle. The geometric effects are in practice related to particle size and shape, as they affect the size of the contact area. If large particles of a material fail by the brittle mode, it is likely that reducing the size can switch the mode of failure to the semibrittle failure and ultimately to the ductile failure. Therefore, the determination of critical transition conditions is important and will be addressed below.
Brittle Failure Mode This mode of failure is due to the presence of preexisting internal or surface flaws. In their absence, particles are strong and can only fail when shear deformations can generate microcracks. Surface flaws often produce orange-segmented fragments and play a major role in damage when the elastic compliance of the particle or the contacting surface is high, producing large tensile hoop stresses.8 Failure due to internal flaws is dominant when the elastic compliance of the contact is low. 8 It usually produces diametrical cracks, splitting the particles into fragments, as the diametrical plane is under the greatest tensile stress.9 When a very large preexisting flaw is present elsewhere, a crack initiating from the flaw follows the tensile stress trajectory. A deterministic analysis of particle breakdown in the case of brittle failure requires a knowledge of the size and position of the flaws. In the absence of this information, the empirical determination of the crushing strength of the particles is the only way to characterize the breakdown. The interpretation of data is commonly carried out by statistical analysis, for example, by the use of the Weibull distribution.10,11 Highly localized loading in this mode of failure may lead to the formation of Hertzian cone cracks. Oblique impacts cause tilting of the cone angle, as the tensile stress trajectories are modified © 2006 by Taylor & Francis Group, LLC
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by the frictional traction, and this can produce small chips from the particles, which is responsible for the erosive wear of the particles.12
Semibrittle Failure Mode This failure mode is identified by limited plastic flow, which is responsible for crack initiation. Plastic flow occurs because the impact stress exceeds the onset of yield, whose characteristics are defined by the critical elastic–plastic transition size, as defined by Puttick. 13 The plastic zone produces compressive radial stresses and tensile hoop stresses. The latter type of stress propagates radial and median cracks, initiated from the plastic zone. When the load is removed, the residual tensile stresses, formed by the elastic unloading component, generate subsurface lateral cracks. Gorham et al.14 and Chaudhri15 have investigated the impact breakage of spheres failing in this mode, using polymethyl methacrylate (PMMA) spheres. The most important properties here are the hardness, Young’s modulus, and toughness (see Section 1.5). Qualitatively “hard” materials undergo less plastic deformation than “soft” materials but can store greater residual stresses, depending on the extent of elastic deformation in the hinterland beyond the plastic zone. Therefore, their tendency for generation of lateral cracks is greater than that of soft materials. Impact damage analysis of particles in this mode of failure has been carried out by indentation fracture mechanics.16 Particle breakdown in the semibrittle failure mode can be characterized by crack morphology and extension: particle fragmentation occurs by the formation of median and radial cracks, and chipping occurs by the formation of lateral cracks.
Ductile Failure Mode Soft materials, such as some polymers, are usually damaged under this failure mode. Ploughing and cutting are the main two mechanisms of material removal. 17 Slip-line field plasticity analysis has been widely used to calculate the deformation pattern.18 Important factors in the process here are the attack angle, the ratio of Young’s modulus to hardness of the surface, the ratio of the hardness values of the two surfaces, and the shear strength of the interface, in other words, the ratio of the shear stress at the interface, and the shear yield stress of the material. In ductile failure, cracking does not readily occur, but instead the plastic ruptures. The breakdown of particulate solids by this mode of failure has not been widely investigated so far. Analysis of Breakage for the Brittle Failure Mode Weibull analysis 19 is commonly used to fit experimental observations of breakage in this failure mode, where the probability of breakage, S, is related to the applied stress, s, using two fitting parameters, z and ss, representing a characteristic flaw density and strength, respectively. ⎡ ⎛ s ⎞m ⎤ S = 1⫺ exp ⎢⫺z ⎜ ⎟ ⎥ ⎢⎣ ⎝ s s ⎠ ⎥⎦
(10.6)
Vogel and Peukert11 have recently applied the above analysis to the impact breakage of particles by relating the applied stress to the incident kinetic energy Wk using Weichert’s approach20: S ⫽ 1⫺ exp ⎡⎣⫺ f mat x (Wk ⫺ Wk ,min )⎤⎦
(10.7)
where f mat and W k,min are two fitting parameters similar to those of Equation 10.6, and x is particle size. In line with the significance of the parameters of Equation 10.6, Vogel and Peukert 11 suggest f mat and W k,min reflect the material properties and the minimum kinetic energy that cause breakage, © 2006 by Taylor & Francis Group, LLC
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respectively. Similar work has also been done by Salman et al.,21 who related the fraction of unbroken particles to the impact velocity by the use of Weiball analysis.
2.10.3 ANALYSIS OF BREAKAGE FOR THE SEMIBRITTLE FAILURE MODE To address the impact breakage of particulate solids on a fundamental basis, fracture mechanics is used to define the conditions for propagation of various types of cracks, namely, radial and median cracks for the fragmentation, and lateral cracks for chipping.
Chipping Figure 10.1 shows a typical example of chipping. A sequence of images from the impact process has been recorded by a high-speed digital video recorder at 27,000 frames per second, where the detachment of a large chip can be seen.22 The chipping process has been described theoretically by a mechanistic model developed by Ghadiri and Zhang.16 The formulation of the model is based on the indentation fracture mechanics of lateral cracks. A fractional loss per impact, j, is defined as the ratio of the volume of chips removed from a particle to the volume of the original (mother) particle, and it is used as a measure of the breakage propensity break under impact conditions. According to the model, j is given by
j ⫽ ah ⫽ a
r V 2DH K c2
(10.8)
where h is a dimensionless group describing the attrition propensity, r is the particle density, V is the impact velocity, D is a linear dimension of the particle, H is the hardness, Kc is the fracture toughness, and ␣ is a proportionality factor, which depends on particle shape and impact geometry and is determined experimentally.
Fragmentation Figure 10.2 shows a sequence of digital video images of a particle fragmenting on impact. The recording speed is 27,000 frames per second, as for Figure 10.1.22,23 Fragmentation occurs when the radial or median cracks extend to the full length of the particles. The formation of two large fragments by the propagation of a median crack can be seen in Figure 10.2. At higher impact energies, a more extensive cracking takes place, producing a larger number of fragments. However, at present there is no theory that can relate the product size distribution to the impact conditions and material properties in a predictive way. The force for fracture of a sphere of diameter D can be estimated based on indentation fracture. Based on the relationship proposed by Ghadiri and Zhang16 for crack extension, the fragmentation force is given by Ffr ∝ K c4 3 D 4 3 H⫺1 3
(10.9)
Transition Velocities The transition velocities from plastic deformation to chipping and from chipping to fragmentation are important features of particle breakage by impact. The dependence of the transition velocities on particle size and material properties has not been widely quantified, and it can only be construed by theoretical considerations at this stage. © 2006 by Taylor & Francis Group, LLC
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Frame no FIGURE 10.1 High-speed video sequence of impact of porous alumina catalyst carrier beads that broke by chipping when impacted at 16 m s–1, recorded at 27,000 fps. [From Couroyer, C., Ghadiri, M., Laval, P., Brunard, N., and Kolenda, F., Oil and Gas Science and Technology—Revue de l’Institut Français du Pétrole, 55, 67–85, 2000. With permission.]
Frame no FIGURE 10.2 High-speed video sequence of impact of porous alumina catalyst carrier beads that broke by fragmentation when impacted at 20 m s–1, recorded at 27,000 fps. [From Couroyer, C., Ghadiri, M., Laval, P., Brunard, N., and Kolenda, F., Oil and Gas Science and Technology—Revue de l’Institut Français du Pétrole, 55, 67–85, 2000. With permission.]
Plastic Deformation–Chipping Transition Marshall et al.24 have shown that there is a critical load to cause lateral fracture, given by ⎛ K ⎞ Fcl ∝ E ⎜ c ⎟ ⎝ H ⎠
4
(10.10)
Hutchings25 used this criterion to predict the minimum particle size that can cause erosion of surfaces by impact. The same approach can be followed here to define the critical transition velocities for chipping and fragmentation. In the case of a round (or relatively flat) contact between a particle and target, the critical transition velocity for plastic deformation–chipping is given by 4
⎛ K ⎞ E −1 2 −2 Vch ∝ ⎜ c ⎟ r D ⎝ H ⎠ H1 2
(10.11)
The above expression indicates that the critical velocity for the onset of chipping is inversely proportional to the square of the particle size. Chipping–Fragmentation Transition. A similar approach can be followed to specify the threshold conditions for particle fragmentation. Hutchings 25 specified a critical particle size below which no fragmentation occurs, based on the indentation fracture model of Hagan.26 The critical load for indentation fracture proposed by Hagan26 is given by Fcf ∝
K c4 H3
(10.12)
This equation can be combined with the impact dynamics by making the same assumptions as in the approach used for the chipping case: the contact deformation under impact follows a quasi-static © 2006 by Taylor & Francis Group, LLC
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indentation model. This enables the impact force to be described in terms of the hardness times the impression area, to give a threshold velocity for fragmentation: 4
⎛ K ⎞ Vfr ∝ ⎜ c ⎟ H 1 2 r −1 2 D −2 ⎝ H ⎠
(10.13)
It is interesting to note that the dependence of V on the particle size D is the same as that of chipping. Limit of Breakdown. Equation 10.11 has a lower limit of validity for particle size. This limit is given by an ultimate particle size below which the particles can only be deformed plastically and cannot be fractured at all, irrespective of impact velocity. There are several models for this limit, proposed by Kendall,27 Puttick,13 and Hagan.26 These models are all based on energy requirements for crack nucleation and propagation. The model of Hagan26 appears to provide a closer agreement with the experimental evidence and is therefore given below. ⎛K ⎞ Dc ≅ 30 ⎜ c ⎟ ⎝ H⎠
2
(10.14)
In conclusion, impact damage depends on the mode of failure, which in turn depends on material properties and contact geometry. The rate of breakage and particle transition size and velocities are all affected by mechanical characteristics such as hardness, toughness, and stiffness. Therefore, the determination of these parameters is important for a better understanding of particle breakage (see Section 1.5).
2.10.4 ANALYSIS OF BREAKAGE OF AGGLOMERATES The failure mode of agglomerates can cover macroscopically the three modes described in Section 2.10.2 because of their large degree of freedom arising from many factors that can influence agglomerate strength, such as void fraction, primary particle size, interparticle bond characteristics, density, and so forth. Consequently the breakage map of agglomerates is not established yet. The disintegration of weak agglomerates failing macroscopically in a “ductile” mode has been studied by Boerefijn et al. 28 Patterns of failure of large agglomerates of glass ballotini bonded together with a brittle glue have been reported by Subero and Ghadiri29 and more recently for various dry and wet granules by Salman et al.6 For the simple case of auto-adhesive primary particles, where the interparticle adhesion follows the JKR model,30 extensive work has been reported in the literature based on the development of the distinct element analysis of agglomerates by Thornton and his coworkers (see, e.g., Thornton et al.31 and Thornton and Liu32). The effects of interface energy, impact angle, and agglomerate morphology have been investigated by distinct element analysis by Subero et al.,33 Moreno et al.,34 and Golchert et al.,35 respectively. Kafui and Thornton36 and Moreno37 have simulated the impact damage of agglomerates on collision with a wall by the distinct element method. The simulation of impact fragmentation of an agglomerate carried out by Moreno37 is shown in Figure 10.3, where the fragments formed at the end of impact are shown with different (gray) density levels according to the number of particles in each fragment. The agglomerate is made of 10,000 spheres 100 μm in diameter, having the surface energy of 3.5 J m–2, elastic modulus of 31 Gpa, and packing density of 0.55, and it has been impacted at velocity of 2 m s–1. Kafui and Thornton36 suggested that the extent of damage described by the damage ratio, Δ, (i.e., the number of broken interparticle bonds divided by the total number of bonds present in the agglomerate) is related to the Weber number, We: We ⫽
r V 2D ⌫
(10.15)
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FIGURE 10.3 Simulation of impact fragmentation of an agglomerate by the distinct element method. [From Moreno, R., Ph.D. Dissertation, University of Surrey, 2003. With permission.]
where r, D, and ⌫ are the primary particle density, diameter, and surface energy, and V is the impact velocity. The breakage propensity parameter h given by Equation 10.8 incorporates the Weber number, because Kc can be related to the surface energy ⌫ by the use of linear elastic fracture mechanics. For example, for the case of plane strain, K c2 = 2 EG 1⫺n 2
(10.16)
it then follows that h∝
rV2 D H ⫻ G E
(10.17)
The form H/E is attributed to the elastic–plastic deformation characteristics of the agglomerate. When the simulation results are interpreted in the form of fractional loss per impact, Thornton et al.37 report that at low impact velocities, corresponding to the chipping regime, the fractional loss per impact varies linearly with the Weber number, which is in agreement with the model of Ghadiri and Zhang.16 To model agglomerate breakage, Moreno38 explored a simple case where the energy required for breakage was linearly related to the incident kinetic energy. Considering the work spent to break a bond, he found that the damage ratio is given by ⌬∝
r V 2 D ⎛ ED ⎞ ⫻⎜ ⎝ ⌫ ⎟⎠ ⌫
2/3
(10.18)
Clearly the Weber number and other dimensionless groups such as ED/⌫ influence the breakage of agglomerates, as demonstrated by the numerical simulations of Moreno.38 More extensive work is required to describe the breakage characteristics of agglomerates with binders.
REFERENCES 1. Johnson, K. L., Contact Mechanics, Cambridge University Press, Cambridge, U.K., 1985. 2. Ning, Z., Ph.D. Dissertation, Aston University, U.K., 1995. 3. Ning, Z. and Thornton, C., in Powders and Grains 93, Thornton, C., Ed., Balkema, Rotterdam, 1993, pp. 33–35. © 2006 by Taylor & Francis Group, LLC
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4. Salman, A. D., Powder Technol., 1, 143–144, 1, 2004. 5. Schönert, K., Powder Technol., 143–144, 2–18, 2004. 6. Salman, A. D., Reynolds, G. K., Fu, J. S., Cheong, Y. S., Biggs, C. A., Adams, M. J., Gorham, D. A., Lukenics, J., and Hounslow, M. J., Powder Technol., 143–144, 19–30, 2004. 7. Wu, S. Z., Chau, K. T., and Yu, T. X., Powder Technol., 143–144, 41–55, 2004. 8. Shipway, P. H., Hutchings, I. M., Powder Technol., 76, 23–30, 1993. 9. Salman, A. D., Gorham, D. A., Verba, A., Wear, 186–187, 92–98, 1995. 10. Van den Born, I. C., Ph.D. Dissertation, University of Groningen, 1992. 11. Vogel, L. and Peukert, W., Powder Technol., 129, 101–110, 2003. 12. Lawn, B. R., in Fundamentals of Friction: Macroscopic and Microscopic Processes, Singer, I. L. and Pollock, H. M., Eds., Kluwer Academic Publishers, London, 1991, pp. 137–165. 13. Puttick, K. E., J. Phys. D: Appl. Phys., 13, 2249–2262, 1980. 14. Gorham, D. A., Salman, A. D., and Pitt, M. J., Powder Technol., 138, 229–238, 2003. 15. Chaudhri, M. M., Powder Technol., 143–144, 31–40, 2004. 16. Ghadiri, M. and Zhang, Z., Chem. Eng. Sci., 57, 3659–3669, 2002. 17. Hutchings, I. M., Powder Technol., 76, 3–13, 1993. 18. Childs, T. H. C., in Fundamentals of Friction: Macroscopic and Microscopic Processes, Singer, I. L. and Pollock, H. M., Eds., Kluwer Academic Publishers, London, 1991, pp. 209–226. 19. Weiball, W., J. Appl. Mech., 9, 293–297, 1951. 20. Weichert, R., Zement -Kalk-Gips, 45 (Suppl. 1), 1–8, 1992. 21. Salman, A. D., Biggs, C. A., Fu, J., Angyal, L., Szabo, M., and Hounslow, M. J., Powder Technol., 128, 36–46, 2002. 22. Couroyer, C., Ghadiri, M., Laval, P., Brunard, N., and Kolenda, F., Oil and Gas Science and Technology— Revue de l’Institut Français du Pètrole, 55, 67–85, 2000. 23. Couroyer, C., Ph.D. Dissertation, University of Surrey, 2000. 24. Marshall, D. B., Lawn, B. R., and Evans, A. G., J. Am. Ceram. Soc., 65(suppl. 11), 561–566, 1982. 25. Hutchings, I. M., in Erosion of Ceramic Materials, Ritter, J. E., Ed., Trans Tech Publications, 1992, pp. 75–92. 26. Hagan, J. T., J. Mater. Sci., 16, 2909–2911, 1981. 27. Kendall, K., Nature, 272, 710–711, 1978. 28. Boerefijin, R., Ning, Z., and Ghadiri, M., Int. J. Pharm., 172 (Suppl. 1–2), 199–209, 1998. 29. Subero, J. and Ghadiri, M., Powder Technol., 120 (Suppl. 3), 232–243, 2001. 30. Johnson, K. L., Kendall, K., and Roberts, A. D., Proceedings of the Royal Society of London A, 324, 301–313, 1971. 31. Thornton, C., Ciomocos, M. T., and Adams, M. J., Powder Technol., 105, 74–82, 1999. 32. Thornton, C. and Liu, L., Powder Technol., 143–144, 110–116, 2004. 33. Subero, J., Ning, Z., Ghadiri, M., and Thornton, C., Powder Technol., 105(suppl. 1–3), 66–73, 1999. 34. Moreno, R., Ghadiri, M., and Antony, S. J., Powder Technol., 130 (Suppl. 1–3), 132–137, 2003. 35. Golchert, D., Moreno, R., Ghadiri, M., and Litster, J., Powder Technol., 143–144, 84–96, 2004. 36. Kafui, K. D. and Thornton, C., in Powders and Grains 93, Thornton, C., Ed., Balkema, Rotterdam, 1993 , pp. 401–406. 37. Thornton, C., Kafui, D., and Ciomocos, T.,. paper presented at the IFPRI Annual Conference, Urbana, Illinois, 1995. 38. Moreno, R., Ph.D. Dissertation, University of Surrey, 2003.Author Queries
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2.11
Sintering Kikuo Okuyama Hiroshima University, Higashi-Hiroshima, Japan
Sintering or densification is an irreversible thermodynamic phenomenon to convert unstable packed powder having excess free energy to stable sintered agglomerates. The sintering phenomenon involves the fusion of particles, volume reduction, decrease in porosity, and increase in grain size.
2.11.1
MECHANISMS OF SOLID-PHASE SINTERING
Sintering kinetics have been studied by many investigators from both experimental and theoretical points of view. Coble1 divided the sintering process into three successive elementary stages, as shown in Figure 11.1. The initial stage corresponds to neck formation and growth, the intermediate stage to the growth of cylindrical vacancies, and the final stage to diffusion and disappearance of the vacancies.
Formation of Necks Crystalline particles usually contain dislocations at the surface, created while they were produced. These dislocations are moved or recovered at an initial stage of sintering to form a neck at the contact point of particles. The phenomenon usually begins to be observed at the absolute temperature ratio to the melting point ∝ = 0.23.
Neck Growth Neck growth is caused by mass transfer such as evaporation–condensation, diffusion, and plastic flow. The principal mass transfer depends on the composition of the material, sintering conditions, and the sintering step. Evaporation–Condensation The material evaporates at convex particle points or on the surface of necks with concave curvatures. The process makes round grains. Diffusion The equilibrium concentration of atomic or ionic vacancies at the particle surface and necks for crystalline particles varies with the chemical potential at the respective location, which is relatively higher than in the interior of particles. Hence, diffusion occurs from the inside (volumetric diffusion) to the surface (surface diffusion) to the crystalline grain boundary (grain boundary diffusion). Surface diffusion usually occurs at a lower temperature (∝ = 0.33 to 0.45) and there is little access of coalesced particles, whereas at a higher temperature (∝ = 0.42 to 0.8), volumetric diffusion becomes active and the growth rate of necks is increased due to an increase in the mass transfer rate. Plastic Flow (Viscous Flow) Sintering mechanisms of amorphous materials such as glass and resine are usually controlled by plastic flow. Plastic flow also plays a significant role in rapid shrinkage and densification at the initial stage of sintering. 213 © 2006 by Taylor & Francis Group, LLC
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neck formation and growth
shrinkage of pores
diffusion and disappearance of vacancies
FIGURE 11.1 Sintering of multiple primary particles.
FIGURE 11.2 Two-sphere model of initial-stage sintering.
Mechanism of Shrinkage and Bloating of Pores Shrinkage and bloating of pores at the final stage of sintering are phenomena due to sink or source of mass transfer, and therefore volumetric diffusion and grain boundary diffusion play major roles regardless of mass transfer mechanisms such as evaporation– condensation and surface diffusion.
2.11.2
MODELING OF SINTERING OF AGGLOMERATES
Coblentz et al.2 proposed sintering rate equations based on several existing sintering models. Ashby3 presented a sintering diagram to identify the dominant mechanisms of sintering. Models describing the sintering of crystalline agglomerates have been presented by numerous researchers. One of the most representative and popular modes uses the ideal geometry illustrated in Figure 11.2. The center-to-center approach and the neck growth between two equal spheres are described to evaluate the rate of sintering in the initial stage. Because the neck growth rate depends on the sintering mechanism, it is important to know that what mechanism is dominant. The change in neck radius l with time t is given as4 n
⎛ 2l ⎞ Kt ⎜ d ⎟ ⫽ dm ⎝ p0 ⎠ p
(11.1)
where dp and dp0 are the primary particle diameters at t = t and t = 0, respectively. K, m, and n are the constants which depend on the physical properties and dominant mass transport mechanisms for sintering, as shown in Table 11.1. © 2006 by Taylor & Francis Group, LLC
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TABLE 11.1 Values of Constants in Equation 11.1 m 1
n 2
Mechanism Plastic flow or viscous flow
2
3
Evaporation and condensation
3
5
Volume diffusion
4
6
Grain boundary diffusion
4
7
Surface diffusion
When the neck size reaches a certain value, a channel-like vacancy is formed (intermediate stage). In the final stage of sintering, the vacancy grows to be spherical, and sintering rates are expressed in terms of the diffusion of vacancies. There is no kinetic expression that holds over all the stages of sintering. From many scaling factors available to describe the shrinkage due to sintering, the surface area or the density of the sintered body is usually chosen. Over all the stages of sintering, the reduction rate of surface area due to sintering is approximately given as5 das 1 ⫽⫺ (as ⫺ asc ) dt t
(11.2)
where asc is the surface area of the final single sphere after complete fusion. t is the rate constant called the sintering time. Assuming that the sintering of two spheres is complete when the ratio of neck radius l to the primary particle radius dp/2 reaches 0.83, t is given by t⫽
(2lf ⁄dp 0 )n dpm K
⫽ Adpm exp (
E ) RgT
(11.3)
where lf is the neck radius at the equilibrium state, E is the activation energy for self-diffusion, and A is a constant depending on the sintering mechanism. For example, for grain boundary diffusion2 A is given by
A=
kT (2lf ⁄dp 0 )6 12bD0 gV
(11.4)
where k, D0, γ, and ⍀ are respectively the Boltzmann constant, preexponential factor for the diffusion coefficient, surface tension, and atomic volume. Figure 11.3 shows the temperature dependency of t for ultrafine silver and titania particles as a function of primary particle diameter dp. t is a strong function of temperature and changes also with dp. If t is constant (i.e., isothermal sintering without grain growth), Equation 11.2 indicates that the surface area of the agglomerates decays exponentially. However, t does not, in general, remain constant because of the increase in the primary particle diameter during sintering. The primary particle diameter dp is related to the surface area of an agglomerate by
dp ⫽
6V as
(11.5)
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FIGURE 11.3 The change in sintering time, τ, with temperature; dp is the primary particle diameter.
where V is the volume of the agglomerate that remains constant over the entire sintering process. Equation 11.5 states that dp is inversely proportional to the surface area—thus, that the growth rate of primary particles can be evaluated from the observed decrease in as. Ultrafine particle synthesis via gas-phase reaction is reported by many researchers. In the synthesis processes, however, many phenomena affect particle morphology (e.g., coagulation, condensation, crystallization, and sintering). Sintering is especially important because it influences particle size, shape, and crystal structure. The sintering of silver and titania agglomerates consisting of nanometer primary particles in a heated gas flow was investigated experimentally. The densification of agglomerates accompanying primary particle growth was explained quantitatively by solving the equation of sintering, Equation 11.2, under the temperature profile. 6,7 The smaller the primary particles are, the shorter the time required for sintering. The sintering rate, therefore, slows as primary particles grow in size. This size dependency in sintering rate will be a problem in sintering of nanophase materials from this point of view, as sintering at low temperature without grain growth is necessary to obtain a compact of high density.7
2.11.3
SINTERING PROCESS OF PACKED POWDER
Solid-phase sintering of packed powder of a single composition is conducted at a temperature lower than the melting point. Materials having an inexact melting point begin to sinter near the softening temperature, whereas materials having a clear melting point begin to sinter at three fifths of the melting temperature (in kelvin). Bonding between particles takes place at contact points to form necks that grow with time, resulting in shrinkage of the pores and forming isolated spherical voids as shown in Figure 11.4. The densification of packed powder due to sintering depends on the density of the compact before sintering and, hence, is evaluated by the densification parameter D:
D⫽
rs ⫺ rg rt ⫺ rg
(11.6)
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FIGURE 11.4 Structures of sintered grounds of spherical copper particles observed by an optical microscope (sintered in hydrogen after being packed with tapping): (a) 1273 K (1000°C), 5 min; (b) 1273 K, 23 min; (c) 1223 K (950°C), 6 h; (d) 1223 K, 20 h.
Where rs is the density of the sintered body, rg is the density before sintering, and rt is the net density of the particles. When solid-phase sintering progresses smoothly, the empirical equation D = Ktn is widely used as the relationship between sintering time t and densification parameter D. The constant K depends on the temperature, and the exponent n depends on the transfer mechanism of materials and is usually less than unity. The sintering process of mixed materials depends primarily on wettability between particles of different types. Mutual solubility and diffusivity are also significant. A binary mixture of A and B with surface energies gA and gB, is respectively, and with the interfacial free energy gAB does not mutually wet if gAB > gA ⫽ gB. Hence, the sintering progresses selectively between particles of the same material. In this case, if a third material reactive to both components is added in small amounts or if the atmosphere is controlled, sintering becomes feasible. Reaction between particles is often applied to sintering (reaction sintering), but the sintered ground is sometimes bloated or broken if the reaction product is of less density. Mutual diffusion plays a major role in solid-phase sintering. When there is a large difference in the mutual diffusivities, pores remain where particles with larger diffusivity migrate. Grooves and pores are also formed around the neck between particles of different composition.8 When mass transfer is one-sided, segregation or concentration distribution takes place to form heterogeneous sintering.
REFERENCES 1. Coble, R. L., J. Appl. Phys., 32, 787, 1961. 2. Coblentz, W. S., Dynys, J. M., Cannon, R. M., and Brook, R. J., Mater. Sci. Res., 13, 141, 1980. © 2006 by Taylor & Francis Group, LLC
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Ashby, M. F., Acta Metallurgica, 22, 275, 1974. German, R. M. and Munir, Z. A., J. Ann. Ceram. Soc., 71, 225, 1976. Koch, W. and Friedlander, K., J. Colloid Interface Sci., 140, 209, 1990. Shimada, M., Seto, T., and Okuyama, K., J. Chem. Eng. Jpn., 27, 795, 1994. Siegel, R. W., Ramasamy, S., Hahn, H., Zongquan, L., Ting, L., and Gronsky, R., J. Mater. Res., 3, 1367, 1988. 8. Kuczynski, G. C., Hooten, N. A., and Gibbon. C. F., Sintering and Related Phenomena, Gordon and Breach, New York, 1967. 9. Seto, T., Shimada, M., and Okuyama, K., Aerosol Sci. Technol., 23, 183, 1995.
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2.12
Ignition and Combustion Reaction Hisao Makino, Hirofumi Tsuji, and Ryoichi Kurose Central Research Institute of Electric Power Industry, Yokosuka, Kanagawa, Japan
2.12.1
COMBUSTION PROFILE
Fuel Properties There are many kinds of solid fuels, including coal, oil sand, oil shale, refuse fuel, and biomass, and coal is the most abundant among these solid fuels. The most common and useful methods for analyzing these solid fuels are the proximate and ultimate analyses. Proximate analysis is used to quantify the amounts of moisture, volatile matter, fixed carbon, and ash contents. The amounts of moisture and volatile matter contents of a sample of solid fuel are evaluated by measuring the weight losses in air at 107 ⫾ 2⬚C for 1 h and in an inert gas at 900 ⫾ 20⬚C for 7 min., respectively. The amount of fixed carbon content is supposed to correspond to the weight loss after further heating in air at 815 ⫾ 10⬚C for 1 h, and the residual is regarded as ash. Thus, the dried sample consists of combustible matter and ash, and the combustible matter comprises volatile matter, which easily volatilizes to the gas phase, and fixed carbon, which remains in the char even at high temperature. On the other hand, the major elements such as carbon, hydrogen, oxygen, nitrogen, and sulfur in solid fuels are evaluated using ultimate analysis. In general, ultimate analysis is roughly related to proximate analysis: for example, coals with high fixed carbon content contain much carbon, and coals with high volatile matter content contain much hydrogen (see Table 12.1).
Combustion Process of Solid Fuel Figure 12.12 shows a schematic of the typical coal combustion processes on pulverized coal, which is one of the most common methods for burning solid fuels. The combustion processes are as follows: (1) moisture in solid fuels is immediately vaporized when solid fuels enter into a high-temperature region in a furnace, (2) evolution of volatile matter (devolatilization) takes place, (3) the volatilized gas is ignited and volatile combustion occurs, (4) char combustion (combustion of fixed carbon) follows the volatile combustion, (5) the combustion is terminated (a piece of the fixed carbon remains in the ash).
2.12.2
DEVOLATILIZATION AND IGNITION
The combustion of solid fuels begins by the ignition of volatilized gas after the evolution of volatile matter takes place in a high-temperature region. Although the devolatilization is believed to finish in about 100 ms, it is very difficult to accurately understand the devolatilization mechanism because devolatilization is an extremely complicated phenomenon. The devolatilization processes are strongly affected by coal properties, temperature, gas compositions, and so on, and these
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TABLE 12.1
Analysis Results of Solid Fuels
High heating value*
kcal/kg
Coal A 7,270
Low heating value*2
kcal/kg
9,980
2
Moisture
wt%
Coal B 7,030
Refuse fuel*1 —
Biomass (Australian Bagasse) —
6,750
—
—
52.7*2
44.0–53.0*2
Proximate analysis 2.4*2 1.1*2 3
11.6*
3
*2
9.5
4.0–30.0*3
Ash
wt%
12.2*
Volatile matter
wt%
29.1*3
34.1*3
—
75.0–87.0*4
wt%
3
54.3*
3
—
—
1.59
—
—
26.45*2
43.0–52.05*4
Fixed carbon Fuel ratio
—
58.7*
2.02
Ultimate analysis 74.1*3 72.5*3
Carbon
wt%
Hydrogen
wt%
5.29*3
5.57*3
3.93*2
5.2–6.9*4
wt%
3
3
0.65*2
0.2–0.4*4
Nitrogen
1.54*
1.67*
3
3
Oxygen
wt%
6.5*
Total sulfur
wt%
0.34*3
0.32*3
wt%
3
3
Combustible sulfur
0.33*
8.4*
*2
40.8–52.0*4
0.08*2
0.02–0.11*4
16.21
0.31*
—
—
1
An example of Japanese refuse fuel in Japanese urban areas. Source: Makino, H. and Ito, S., J. Soc. Powder Technol. Japan, 34, 247–254, 1997. 2 Equilibrium moisture basis. 3 Dry basis. 4 Dry ash-free basis.
Preheating zone
Flame zone (Volatile combustion region) 1000~ 1700
Particle temperature
Flame
Air and Pulverized coal
Burner
Post-Flame zone (Char combustion region)
Volatile matter
Char combustion Air
Ignition
End point of devolatilization
Start point of Devolatilization Sudden rise in particle temperature (Radiative and convective heat transfer) Particle heating (Flame radiation) Distance from burner
FIGURE 12.1 Combustion profile of a coal particle in pulverized coal combustion. [From Tominaga, H., in Sangyou Nenshou Gijutsu, The Energy Conservation Center, Japan, 2000, pp. 93–100. With permission.] © 2006 by Taylor & Francis Group, LLC
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conditions change moment to moment in actual combustion fields. The devolatilization rate of volatile matter is commonly modeled by a single-step Arrhenius reaction scheme,3 such as dV ⫽ K v (V * ⫺V ) dt
(12.1)
⎛ E ⎞ K v ⫽ Av exp ⎜⫺ v ⎟ ⎝ RTp ⎠
(12.2)
where V* and V indicate the total volatile matter content in a solid fuel particle and volatilized mass released from the particle, Tp the particle temperature, and R the universal gas constant. Kinematic parameters of preexponential factor Av and activation energy Ev in Equation 12.2 are determined experimentally.4 Besides, a two-step Arrhenius reaction scheme5 is also proposed to improve the adaptability. The definition of ignition temperature has not been explicitly settled yet, since the measurements of ignition temperature of solid fuels with complicated properties are very difficult. Typically, the ignition temperature of brown coals, whose volatile matter content is relatively high, is about 250⬚C, and that of anthracite, which is a low-volatile coal, is about 500⬚C. The ignition temperature of bituminous coals, which are generally used in pulverized coal combustion in Japanese coal-fired thermal power plants, is 300–400⬚C. For the solid fuels with nonvolatile matter, the ignition temperature is regarded as the temperature at which the heat generated by the combustion on the surface of a solid fuel is beyond the heat loss due to convection and radiation.
2.12.3
GASEOUS COMBUSTION
The combustion of the volatilized gas due to the evolution of volatile matter with combustion air takes place in the gaseous phase. The simple model of gaseous combustion is as follows. The chemical mechanism has two global reactions: Ca H b Oc ⫹ 0.5O2 → aCO ⫹ bH 2 O CO ⫹ O2 → CO2
(12.3)
where CaHbOc is the volatilized fuel gas, and a, b, c, a, and b are governed by the coal property. The gaseous combustion provided in Equation 12.3 is often calculated using the combination of the kinetics and eddy dissipation models.6 Regarding the kinetics, the rate of reaction for reactants such as CaHbOc is given as an Arrhenius expression: ⎛ Eg ⎞ d e Rg ⫽ Ag exp ⎜⫺ ⎟ [ Reactant ] [O2 ] RT ⎝ g⎠
(12.4)
where [c] means the mol fractions of chemical species c.The values of the preexponential factor Ag, activation energy Eg, and orders d and e are determined experimentally.7,8 Since the gaseous combustion between volatilized gas and combustion air is strongly affected by the mixing of them, it should be discussed with flow behavior. The reaction rate of volatilized gas is fast, but soot particles are formed under the condition that the mixing of volatilized gases and combustion air is not enough. Figure 12.2 9 shows the emission characteristics of fine particles of less than 1 μm against the excess O2 concentration in the exhaust gas. As the excess O2 concentration decreases, the concentration of particles of the order of 0.1 μm, which are thought to be soot, increases. © 2006 by Taylor & Francis Group, LLC
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10 10
O 2 =6% O 2 =4% O 2 =3%
10 9
dn/dlogDp [number/cm3N]
10 8
10 7
10 6
10 5
10 4
10 3
0.001
0.01
0.1
1
10
100
Particle size [ m] FIGURE 12.2 Emission characteristics of fine particles from pulverized coal combustion. [From Makino, H., J. Aerosol Research Japan, 4, 206–210, 1989. With permission.]
2.12.4
SOLID COMBUSTION
The combustion of fixed carbon, which is often referred to as char, is explained. The char burning rate is modeled calculated using Field et al.’s model10: ⎛ K K ⎞ dC ⫽⫺ ⎜ c d ⎟ Pg pDp 2 dt ⎝ Kc ⫹ Kd ⎠ 5.06 ⫻ 10⫺7 ⎛ Tp ⫹ Tg ⎞ Kd ⫽ ⎜⎝ 2 ⎟⎠ Dp ⎛ E ⎞ K c ⫽ Ac exp ⎜⫺ c ⎟ ⎝ RTp ⎠
(12.5)
0.75
(12.6)
(12.7)
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where C is the char mass, K c and Kd are the chemical and diffusion rate coefficients, respectively, and Pg is the partial pressure of oxygen in the bulk gas. This model is obtained under the assumption that the char burning rate is controlled by both the chemical reaction rate and the diffusion rate of oxygen to the surface of the char particle. The values of the kinematic parameters of the preexponential factor Ac and activation energy Ec in Equation 12.7 are determined experimentally.4 It is considered that the char burning rate is dominated by the chemical reaction rate at a temperature less than 1000⬚C, whereas it is dominated by the diffusion rate at higher temperatures. The remaining particles consist of ash and char. If the particle temperature is higher than the melting points of the particles, the particles become spherical due to the surface tension and solidify again as the particle temperature decreases. The remaining combustible char is exhausted as pure char particles or contained in ash particles.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Makino, H. and Ito, S., J. Soc. Powder Technol. Jpn., 34, 247–254, 1997. Tominaga, H., in Sangyou Nenshou Gijutsu, The Energy Conservation Center, Japan, 2000, pp. 93–100. Van Krevelen, D. W., Van Heerden, C., and Huntjens, F. J., Fuel, 30, 253–258, 1951. Kurose, R., Tsuji, H., and Makino, H., Fuel, 80, 1457–1465, 2001. Kobayashi, H., Howard, J. B., and Sarofin, A. F., in Sixteenth Symposium (International) on Combustion, The Combustion Institute, 1976, pp. 411–425. Magnussen, B. F. and Hjertager, B. W., in Sixteenth Symposium (International) on Combustion, The Combustion Institute, 1976, pp. 719–729. Borman, G. L. and Ragland, K. W., in Combustion Engineering, McGraw-Hill, 1998, pp. 120–122. Kurose, R., Makino, H., and Suzuki, A., Fuel, 83, 693–703, 2004. Makino, H., J. Aerosol Res. Jpn., 4, 206–210, 1989. Field, M. A., Gill, D. W., Morgan, B. B., and Hawksley, P. G. W., The Combustion of Pulverised Coal, British Coal Utilisation Research Association, Leatherhead, Surrey, 1967.
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2.13
Solubility and Dissolution Rate Yoshiaki Kawashima Gifu Pharmaceutical University, Mitahora-Higashi, Gifu, Japan
2.13.1
SOLUBILITY OF FINE PARTICLES
Finely divided particles have a greater solubility than large particles. Therefore, the smaller particles will dissolve and their mass will reprecipitate on the larger particles. But solubility cannot be simply related to gross particle size, because it depends somewhat on the particle (crystal) face exposed, surface roughness, and irregularity. Under ideal conditions, the solubility of spherical particles is expressed by log
2s m M 1 1 a2 ⫽ ( ⫺ ) a1 2.303rRT r2 r1
(13.1)
Assuming that the activity of the solution is proportional to the molar concentration, Equation 13.1 can be rewritten as log
2.13.2
2s m M 1 S2 ⫽ S1 2.303rRT r2
(13.2)
FACTORS TO INCREASE SOLUBILITY
Particle (Crystal) Size The size and shape of fine particles (diameter, ≅µm) affect the solubility, which increases with decreasing particle size as predicted by Equation 13.2.
Crystalline Form (Polymorph and Amorphous Form) Polymorphism occurs owing to different molecular arrangements in the solid phase, resulting in two or more crystalline forms. The difference in crystal energy of polymorphs generally leads to different physical properties, such as melting point, solubility, density, and so on, although the crystals are chemically identical. Metastable solid polymorphs having higher thermodynamic activity increase solubility, resulting in the improved bioavailability of a poorly soluble drug. Chloramphenicol palmitate is a representative drug. The solubility of polymorph B of chloramphenicol palmitate is roughly two times that of polymorph A.1 Phenylbutazone has five different polymorphic forms. Among them Form I is thermodynamically most stable, and its equilibrium solubility is the lowest.2 The polymorphism effect of cimetidine with four different crystalline forms (A, B, C, D) on stress ulceration in the rat was reported.3 Amorphous forms have clearly the highest free energy, resulting in the largest solubility ratio. Amorphous forms are sometimes used rather than crystalline forms, to increase the solubility and bioavailability of antibiotics (e.g., novobiocin).4 Solubility ratios of some other amorphous drugs compared to those of crystalline form are listed in Table 13.1.5,6 225 © 2006 by Taylor & Francis Group, LLC
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TABLE 13.1 Solubility Ratio for Amorphaous to Crystalline Form Drug (Temp. in ºC) Caffeine (25)
More Soluble Phase Less Soluble Phase Amorphaous/crystalline
Solubility Ratio 6.5
Ref. (5)
Diacetylmorphine (25)
Amorphaous/crystalline
16
(5)
Theophylline (17)
Amorphaous/crystalline
58
(5)
Theobromine (16)
Amorphaous/crystalline
50
(5)
Morphine (20)
Amorphaous/crystalline
268
(5)
Hydrochlorothiazide (37)
Amorphaous/crystalline
1.1
(6)
Bendrofluazide (37)
Amorphaous/crystalline
2.8
(6)
Cyclothiazide (37)
Amorphaous/crystalline
6.2
(6)
Cyclopenthiazide (37)
Amorphaous/crystalline
8.3
(6)
Polythiazide (37)
Amorphaous/crystalline
9.8
(6)
Cosolvents A solute is frequently more soluble in a mixture of solvents (e.g., water and water-miscible organic solvent) than in one solvent alone (e.g., water). This phenomenon is known as cosolvency, and the mixed solvents are called cosolvents. The solubility of phenobarbital in a water–alcohol–glycerin mixture increases dramatically compared to that in water.7 A linear relationship between the logarithm of the observed solubility (Sm) of nonpolar nonelectrolyte solutes in a cosolvent–water mixture and the volume fraction of the cosolvent (f) was found, as shown in Figure 13.1.8 log Sm ⫽ Sw ⫹ s f
(13.3)
Surfactants Surface active agents in solution form the micelles above the critical micelle concentration at which insoluble or poorly soluble solids are solubilized. This solubilization results in increasing the solubility of particles. Comprehensive reviews of solubilization in surfactant systems were carried out by Swarbrick9 and Elworthy et al.10
Complexation When poorly soluble substances interact with a second substance added in solution to form soluble complexes, the solubilities of poorly soluble substances can be improved. The additive, called a solubilizing agent, whose optimum concentration in solution increases solubility, should be properly chosen. Representative solubilizing agents are listed in Table 13.2.
2.13.3 THEORIES OF DISSOLUTION Diffusion in a Liquid Film When a solid particle is agitated in liquid and allowed to dissolve, Noyes and Whitney,11 Nernst,12 and Brünner and Tolloczko13 assumed a stagnant liquid film, called a diffusion layer, around the particle surface. The concentrations at the solid surface and at the outside of the diffusion layer are assumed to be © 2006 by Taylor & Francis Group, LLC
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FIGURE 13.1 Dependence of solubility of some alkyl p-aminobenzoates upon solvent composition. All measurements were performed at 37ºC. Key: z, ethyl; S, butyl; T, hexyl; , octyl; , dodecyl. [From Yalkowsky, S. H., Flynn, G. L., and Amidon, G. L., J. Pharm. Sci., 61, 983–984, 1972. With permission.]
equal to the particle solubility (Cs) and the bulk concentration (C), respectively, as shown in Figure 13.2. A steady state is assumed and Fick’s first law is employed to derive the dissolution rate equation: dC DS S ⫽ (Cs ⫺ C ) ⫽ k (Cs ⫺ C ) dt Vd V
(13.4)
The thickness of the diffusion layer d at the liquid–solid interface of the rotating disk at a constant angular velocity was derived by Levich14: d ⫽ 1.612 ⫻ D 3 V 6 W ⫺ 2 1
1
1
(13.5)
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TABLE 13.2 Substances
Solubilizing Agents for Poorly Soluble
Poorly soluble substances Iodine
Solubilizing agents Potassium iodide
Caffeine
Sodium benzoate, sodium salicylate
Theobromine
Sodium salicylate
Calcium theobromine
Calcium salicylate
Mephenesin
Salicylic acid
Quinine hydrochloride
Urethane, carbamide
Theophylline
Sodium acetate, ethylenediamine
Riboflavin
Nicotinic amide
FIGURE 13.2 Concentration profile at solid– liquid interface in Noyes–Nernst model. [From Nernst, W., Z. Phys. Chem., 47, 52–55, 1904; Noyes, A. and Whitney, W. I., Z. Phys. Chem., 23, 689–692, 1897. With permission.]
Hixon and Crowell15 derived a dissolution rate equation, called the cube root law for spherical, cubic, or cylindrical monodispersed particles, assuming an isotropic dissolution and sink condition (Cs>>C ):
W0 3 ⫺ W 3 ⫽ ( 1
1
pNP 13 2 DCs t ) 6 dP
(13.6)
Surface Renewal of Solvent Surrounding Particle by Eddy Diffusion Danckwerts16 assumed that the surface of a solid particle dispersed in liquid is continuously renewed with fresh liquid by eddy diffusion. According to this model, there is not a stagnant liquid film and the solute concentration at the surface of the particle is not Cs, but a lower limiting concentration, © 2006 by Taylor & Francis Group, LLC
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Ci. The solute dissolves into renewed fresh liquid during its stay at the interface. The solvent with the dissolved solute is continuously replaced by fresh solvent. The model proposed is described by Equation 13.7: dm ⫽ S gD (Cs ⫺ C ) dt
(13.7)
Reaction in a Diffusion Layer When the dissolved molecules or ions from solid particles react in the diffusion layer, the Nernst– Brunner model exhibited schematically in Figure 13.3 is applicable. Here, the solid particle is an acid, HA, the solute is a base, BOH, and d is the effective diffusion layer in which the dissolved HA and the BOH diffused from bulk react. Dissolutions of benzoic acid, sulfadiazine, and triethanolamine in a strong aqueous basic solution are explained by this model.
Double-Layer (Reaction and Diffusion Layers) Model Kawashima and Takenaka17 proposed a double-layer (i.e., reaction and diffusion layers) model for acid neutralization with antacid (e.g., magnesium carbonate) by considering hydrogen ion transfer from the bulk through the diffusion layer into the neutralizing reaction layer on the solid surface. A steady concentration of hydrogen ions in the reaction layer is established, and the net rate of change in hydrogen ion concentration is set to zero. By assuming the neutralization reaction in the reaction layer is of first order with respect to hydrogen ion concentration, the acidity change (i.e., pH change) is described by Equation 13.8:
pH ⫺ pH o ⫽
k1 k2 A t 2.303V (k1 A ⫹ k2 )
(13.8)
FIGURE 13.3 Concentration profile in Nernst– Brunner Model. [From Brünner, E. and Tolloczko, S. T., Z. Phys. Chem., 47, 56–102, 1904; Nernst, W., Z. Phys. Chem., 47, 52–55, 1904. With permission.] © 2006 by Taylor & Francis Group, LLC
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Chemical Equilibrium in a Diffusion Layer Higuchi et al.18 postulated that chemical equilibrium is established between the dissolved molecules and ions from solid particles in the diffusion layer. The dissolution of sulfadiazine particles in dibasic sodium phosphate is explained by this mechanism.
Dissolution of a Metastable Solid Metastable solid particles such as anhydride dissolve faster than stable solids such as hydride. The concentration of the resultant solution becomes higher than that of the saturated solution of a stable solid (i.e., supersaturated). The concentration of a supersaturated solution gradually decreases to the solubility of a stable solid. Nogami et al.19 found that the dissolution of r-hydroxybenzoic acid and phenobarbital belongs to the foregoing case and proposed a model modified with the crystallization theory.
Dissolution Controlled by Reaction at the Solid–Liquid Interface When the dissolution of a solid is controlled by the reaction at the solid–liquid interface, the active energy for dissolution is higher (e.g., >15 to 16 Kcal/mol) than that for dissolution-controlled diffusion (e.g., 5–6 Kcal/mol), although the rate equation of dissolution is identical with Equation 13.4.
Consecutive Process of Reaction and Diffusion In this mechanism, dissolution is controlled by both the reaction at the solid–liquid interface and the diffusion of solute molecules in the diffusion layer. Therefore, the process is consecutive. Dissolutions of metastable crystals of prednisolone and barbital obey this mechanism.
2.13.4
MEASUREMENT OF DISSOLUTION RATE
The rotating disk method shown in Figure 13.420 and the stationary disk method shown in Figure 13.5 are useful to measure intrinsic dissolution rate, as the rate is controlled only by agitation speed in those methods. A typical constant surface area disk, shown in Figure 13.4, is constructed by mounting a tablet on a holder with paraffin wax such that the single surface of the tablet is exposed to the dissolution medium. Wood et al.21 devised an assembly for compression of the tablet, which can also be used as the tablet holder during dissolution. The apparent dissolution rate can be measured by a simple beaker method.22 The column method in Figure 13.623 can measure the dissolution rate automatically. The flow-through apparatus is operated under either closed mode when the fluid is recirculated or is of fixed volume, or open mode when there is continuous supply of the fluids. Such facilities can resolve some of the problems associated with nonsink conditions. The material under testing is placed in the vertically mounted dissolution cell in Figure 13.6. Yajima et al.24 have devised a minicolumn running in open mode to evaluate the bitterness of clarithromycin dry syrup and found the threshold of the dissolution rate well correlated to bitterness (Figure 13.7).
2.13.5
METHODS TO INCREASE THE DISSOLUTION RATE
Agitation Speed The diffusion model indicates that decreasing the thickness of the diffusion layer increases the dissolution rate. The thickness of the diffusion layer is a function of the stirring rate, as described previously. The dissolution rate is correlated experimentally with the stirring rate, as represented by K d ⫽ aN vb
(13.9)
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FIGURE 13.4 Rotating disk apparatus for dissolution rate measurement. m, induction motor and gearbox; n, stainless steel shaft; o, brass sleeve; p′, disk holder; q, disk; u, flask; v, scrubbing bottle. [From Nogami, H., Nagai, T., and Suzuki, A., Chem. Pharm. Bull., 14, 329–338, 1966. With permission.]
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FIGURE 13.5 Stationary disk apparatus for dissolution rate measurement. B, stirrer; C, die holder; D, die; G, sampling port; H, solvent; I, jacked beaker; J, inlet for water. [From Simonelli, A. P., Mehta, S. C., and Higuchi, W. I., J. Pharm. Sci., 58, 538–549, 1969. With permission.]
With the constant surface area disk method, Hersey and Marty25 relates the intrinsic dissolution rate to the rotation rate of disk by dW ⫽ K i Rn0.5 Sdt
(13.10)
Diffusion Coefficient The diffusion coefficient, D, is defined by Einstein as represented by D⫽
T 6phgm
(13.11)
Increasing the diffusion coefficient increases the dissolution rate, which can be accomplished by decreasing the viscosity of the dissolution medium.
Surface Area The increasing effective surface area, which contacts the dissolution medium directly, effectively increases the dissolution rate. Grinding the material is one of the methods. Poorly soluble pharmaceuticals ground with microcrystalline cellulose, methylcellulose, chitin, or chitosan using a ball mill significantly increase the dissolution rate and solubility. In this process, crystalline drugs become disordered frequently, which results in increasing the solubilities.26 Takahata et al.27 found that copulverizing © 2006 by Taylor & Francis Group, LLC
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FIGURE 13.6 Apparatus for measurement of dissolution rate by column method. A, synchronous motor; B, particle bed; C, cell; F1, F2, screens; H, heat exchangeer; h, height of cell; P1, P2, volumetric pump; R, liquid reservoir; x, circulation factor; Q, volumetric flow rate. [From Langenbucher, F., J. Pharm. Sci., 58, 1265–1272, 1969. With permission.] 233
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Mini-column
CAM dry syrup
Pu mp
Co tton
Te st liquid
FIGURE 13.7 Minicolumn apparatus for evaluating the bitterness of CAM dry syrup.
poorly soluble pharmaceuticals, such as griseofulvin, oxolinic acid, phenytoin, and so forth, with water-soluble additives (e.g., mannitol, sorbitol, glucose, glycine, etc.) was significantly useful to submicronize such drug particles without changing crystallinities. The submicronized particles dramatically increased their dissolution rates in aqueous medium and bioavailabilities in dogs more than those of conventionally pulverized particles (average diameter, 2–3 µm) without copulverizing additives. Müller et al.28 prepared nanosuspension (nanoparticle <1 µm suspension) by high-pressure homogenization of a poorly soluble drug microparticle suspension with surfactant to improve its bioavailability by increasing saturation solubility.
Solubility The dissolution rate can be controlled by the solubility, as indicated by the dissolution model. The solubility varies with the solution pH, salt formation, solubilization by surface-active agents, change in crystal form, complexation, and sufficient reduction in particle size. Polymorphs The use of metastable polymorphic forms of a chemical is effective in increasing the solubility and dissolution rate. Many organic chemicals and drugs, such as prednisolone, aspirin, ampicillin, barbital, benzoic acid, sulfathiazole, chloramphenicol, and novobiocin, exhibit polymorphisms. Amorphous forms are also effective in increasing the dissolution rate. Takeuchi et al.29 created a stable supersaturated system of indomethacin by dissolving its amorphous forms with porous silica prepared by spray drying. Coprecipitate and Inclusion Compound Coprecipitation of a poorly soluble drug, such as sulfathiazole30 and griseofulvin,31 with polyvinylpyrrolidone, polyethylene glycol, or urea can increase the dissolution rate. An inclusion compound with cyclodextrin can also increase the dissolution rate.32 The increase in rate is mainly due to the increase in solubility and/or the decrease in crystallinity of the drug by inclusion complexation. The solubilities of cyclodextrin–drug complexes can be improved by utilizing chemically modified cyclodextrin such as dimethyl-b-cyclodextrin. Dimethyl-b-cyclodextrin facilitates particularly the solubilities of steroid hormones, cardiac glycosides, and fat-soluble vitamins.33 Significant solubility © 2006 by Taylor & Francis Group, LLC
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TABLE 13.3 Solubility Enhancement through the Use of High Concentrations of HP--CD Solute Estriol
%HP--CD 50
Solubility Enhancement 13,666
Estradiol
40
7,000
Progesterone
40
2,266
Spironolactone
40
1,400
Testosterone
40
1,461
Digoxin
50
971
Dexamethasone
50
240
Chlorthalidone
50
87.5
Diphenylhydantoin
50
57
Furosamide
50
24
Nitroglycerin
40
8.3
Acetamidopen
50
6
Apomorphine
50
5.8
Theophylline
50
1.3
enhancement of drug with high concentration of hydroxypropyl-b-cyclodextrin was found by Pitha et al.,34 as shown in Table 13.3. Dispersing on Adsorbent or Disintegrant Poorly soluble drugs can increase their dissolution rate by being adsorbed in fine adsorbents such as fumed silicon dioxide, charcoal, or montomorillonite clay.35,36 Takeuchi et al.37 improved the dissolution rate of a poorly water-soluble drug (e.g., tolbutamide) by depositing the fine drug crystals on a disintegrant (e.g., partly pregelatinized cornstarch by a spray-drying solvent deposition method).
Notation A a a1 a2 b C Cs D f K Kd Ki k1 k2 M m
Surface area of antacid Constant Activity of the solute in a solution produced with larger particle Activity of the solute in a solution produced with smaller particle Constant Concentration of solute in bulk liquid Solubility of particle Diffusion coefficient of solute Volume fraction of cosolvent Constant Dissolution rate constant Intrinsic dissolution rate constant Mass transfer coefficient Reaction rate constant Molecular weight of particle Dissolved mass
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N Nv pHo R Rn r1 r2 rm S Sw S1 S2 T t V W Wo X g d h k n r s sm v
Number of particles Agitation speed Initial pH Gas constant Rotation rate of disk Radius of larger particle Radius of smaller particle Radius of molecule Surface area Solubility of drug in water Solubility of larger particle Solubility of smaller particle Absolute temperature Time Volume of medium Weight of particles at time t Initial weight of particles Distance from solid surface Rate of surface renewal Thickness of diffusion layer Viscosity of medium Boltzmann constant Kinematic viscosity of medium Density of particle Slope of a plot of log Sm against f Mean interfacial tension Angular velocity
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.
Aguiar, A. J., Krc, J., Kinkel, A. W., and Samyn, J. C., J. Pharm. Sci., 56, 847–853, 1967. Tuladhar, M. D., Carless, J. E., and Summers, M. P., J. Pharm. Pharmacol., 35(5), 269–274, 1983. Kokubo, H., Morimoto, K., Ishida, T., Inoue, M., and Morisaka, K., Int. J. Pharm., 35, 181–183, 1987. Mullins, J. D. and Macek, T. J., J. Am. Pharm. Assoc. Sci. Ed., 49, 245–248, 1960. Toffoli, F., Avico, U., Signoretti, C. E., DiFrancesco, R., and Di Palumbo, V. S., Ann. Chem., 63, 1–4, 1973. Corrigan, O. I., Holohan, E. M., Sabra, K. Int. J. Pharm., 18, 195–200, 1984. Krause, G. M. and Cross, J. M., J. Am. Pharm. Assoc. Sci. Ed., 40, 137–139, 1951. Yalkowsky, S. H., Flynn, G. L., and Amidon, G. L., J. Pharm. Sci., 61, 983–984, 1972. Swarbrick, J., J. Pharm. Sci., 54, 1229–1237, 1965. Elworthy, P. H., Florence, A. T., and Macfarlane, C. B., in Solubilization by Surface Active Agents, Chapman & Hall, London, 1968, p. 335. Noyes, A. and Whitney, W. I., Z. Phys. Chem., 23, 689–692, 1897. Nernst, W., Z. Phys. Chem., 47, 52–55, 1904. Brünner, E. and Tolloczko, S. T., Z. Phys. Chem., 47, 56–102, 1904. Levich, V. G., Acta Physicochim URSS, 17, 257–307, 1942. Hixon, A. and Crowell, J., Ind. Eng. Chem., 23, 923–931, 1931. Danckwerts, P. V., Ind. Eng. Chem., 43, 1460–1467, 1951. Kawashima, Y. and Takenaka, H., J. Pharm. Sci., 63, 1546–1551, 1974. Higuchi, W. I., Parrott, E. L., Wurster, D. E., and Higuchi, T. J., Am. Pharm. Assoc. Sci. Ed., 47, 376–383, 1958. Nogami, H., Nagai, T., and Yotsuyanagi, T., Chem. Pharm. Bull., 17, 499–509, 1969.
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237
Nogami, H., Nagai, T., and Suzuki, A., Chem. Pharm. Bull., 14, 329–338, 1966. Wood, J. H., Syarto, J. E., and Letterman, H., J. Pharm. Sci., 54, 1068, 1965. Levy, G. and Hayes, B. A. N., Engl. J. Med., 262, 1053–1058, 1960. Langenbucher, F., J. Pharm. Sci., 58, 1265–1272, 1969. Yajima, T., Fukushima, Y., Itai, S., and Kawashima, Y., Chem. Pharm. Bull., 50(2), 147–152, 2002. Hersey, J. A., Marty, J., and Man, J., Chem. Aerosol News, 46(6), 43*, 1975. Nakai, Y., Yakugaku Zasshi, 105, 801–811, 1985. Takahata, H., Nishioka, Y., and Osawa, T., Funtai to Kogyo (Powder and Industry), 24, 53*, 1982. Müller, R. H., Becker, R., et al., Int. Patent PCT/EP95/04401, 1996. Takeuchi, H., Nagira, S., Yamamoto, H., and Kawashima, Y., Powder Technol., 141, 187–195, 2004. Simonelli, A. P., Mehta, S. C., and Higuchi, W. I., J. Pharm. Sci., 58, 538–549, 1969. Chiou, W. L. and Riegelman, S., J. Pharm. Sci., 58, 1505–1510, 1969. Hamada, Y., Nambu, N., and Nagai, T., Chem. Pharm. Bull., 23, 1205–1211, 1975. Uekama, K. and Otagiri, M., in CRC Critical Reviews in Therapeutic Drug Carrier Systems, Vol. 3, CRC Press, New York, 1987, p. 1. Pitha, J., Milecki, H., Fales, H., Pannell, L., and Uekama, K., Int. J. Pharm., 29, 73–82, 1986. Kreuter, J., Acta Pharm. Fenn., 90, 95–98, 1981. Chiou, W. L. and Riegelman, S., J. Pharm. Sci., 60, 1376–1380, 1971. Takeuchi, H., Handa, T., and Kawashima, Y., J. Pharm. Pharmacol., 39, 769–773, 1987.
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2.14
Mechanochemistry Mamoru Senna Keio University, Yokohama City, Kanagawa, Japan
2.14.1 TERMINOLOGY AND CONCEPT Mechanochemistry, first defined by Ostwald1 at the beginning of the twentieth century, deals with an interplay between mechanical energy and chemical states of matters. Early mechanochemical studies were focused on the chemical transformation induced by gravitational and kinetic energy, as well as energies stored in solids. Mechanochemical phenomena were then broadly divided into two categories: those related with chemical reactions per se, or those dealing with the change in the activity or reactivity of solids. These are called mechanochemical reactions and mechanical activation, respectively. The boundary between these subdivisions is diffuse, however, since both of them involve common physicochemical changes of solids under the influence of mechanical energy. Mechanochemistry was regarded from the beginning as an independent discipline of chemistry, like electro-, photo-, or irradiation chemistry. Biological tissues can also deform as a consequence of chemical changes, as in the origin of every movement of organisms, or in animals through nutrition. However, the latter aspect of the definition is excluded from this chapter, since it is far from powder technology. Tribochemistry, or impact chemistry, came from a different origin but deals with almost the same subjects as mechanochemistry.2 These different but similar expressions are being unified as mechanochemistry. There are a number of monographs or books devoted, at least partly, to this field. They are authored, among others, by Boldyrev and Meyer,3 Kubo,4 Beke,5 Heinicke,2 Butyagin,6 Tkacova,7 Juhasz and Opoczky,8 Gutman,9,10 Balaz,11 and Avvakumov et al.12
2.14.2
PHENOMENOLOGY OF MECHANOCHEMISTRY
Mechanochemistry originated from industrial phenomenology, in other words, from mineral processing and the related practices of comminution and grinding. Downsizing of solid particles by grinding or milling cannot continue infinitely to individual atoms but is limited generally to a single micron regime or, at most, to a fraction of micrometer, as far as the size of separately available single particles is concerned. The lower limit of comminution comes mainly from two factors: microplasticity13 and agglomeration. The first concept is understood as an increasing tendency of plastic deformation when the particle size becomes smaller than several micrometers.14 Below this critical size, it becomes much more difficult to induce fracture via a crack formation and propagation. Increase in the surface free energy prevents powders from limitless size decrease but promotes agglomeration. Fine particles are more than a fragment of solids with a smaller dimension. Excess surface energy, which plays an important role in mechanochemistry, is partly due to a small radius of curvature and surface defects. In addition, the downsizing operation brings about severe defects, which directly combine with surface and bulk chemical properties of solids. Mechanochemistry is, therefore, a very general concept, which all scientists and engineers should understand, as long as they are dealing with fine particulate materials. Mechanochemical reaction during milling has long been known. Phase transformations of crystalline solids are extensively studied, not only from a metastable to a stable phase, but also many unusual transformations, from an otherwise stable phase to a metastable phase. An apparent equilibrium or mechanochemical stationary state is often attained. Likewise, mechanochemical dissociation of hydroxides and carbonates has been widely studied. Addition reactions, in their broadest definition, 239 © 2006 by Taylor & Francis Group, LLC
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can be divided into two types, according to their reaction partner: with gases or liquids existing in a grinding milieu as a continuum, and with solids during milling a solid mixture. In the former category, stress corrosion or tribosorption is also included. A very large number of mechanochemical solid–solid reactions are reported, including mechanical alloying. Concrete examples are given below in their respective sections.
2.14.3 THEORETICAL BACKGROUND The theoretical approach to mechanochemistry is complicated and manifold. Introduction of grinding limits due to the physicochemical changes of solids in the fine grinding was one of the earliest trials. Khodakov 15 took the following factors into account in his energy density equation for fine grinding: (1) plastic and elastic deformation, (2) surface free energy, and (3) excess work consumed by particle interaction and comminution of particles. All these factors are directly related to excess free energy and, hence, mechanical activation. A triboplasma model16 is one of the most famous models, visually explaining triboluminescence and other processes of relaxation. Efforts were mainly paid to verify that mechanochemical processes are fundamentally different from thermal processes, taking place in very limited areas, termed hot spots, during mechanical stressing. This is evidenced from different kinetic orders between thermal and mechanical ways, as well as different decomposition products.17 It was established that the state of excitation under impact stressing is more likely to be that of photochemistry in the sense that electronic energy can be in some of the excited states.17 Electron excitation is not possible by usual thermal processes. Factors associated with the periodical nature of practical machines, notably high-intensity mills, where stressing and relaxation take place simultaneously, were introduced.18,19 Accumulation of energy and related mechanochemical changes, then, are described as a function of frequency of mechanical stressing. There was a trial to explain the apparent stationary state of mechanochemical change as that of higher order, on the basis of thermodynamics of irreversible systems.20 Energy transfer from any equipment to the body of fine particles is decisive for the argument of mechanochemical changes. It is important to notice that most of the energy once absorbed is converted into a joule effect.6 This can be regarded as a useless dissipation of energy, lowering the grinding efficiency, from the viewpoint of comminution.21 However, a certain amount of energy may be retained in the substance to give an excess energy, which serves as the origin of elevated activity and reactivity of solids. Higher reactivity of solids is beneficial in many industrial aspects. Energy storage in the particles as a result of mechanical treatment, often called as excess energy, or excess free energy when entropy term is also taken into account, is after all the source of all the mechanochemical phenomena.6 Mechanochemical processes observed ex situ are the consequences of relaxation of the excess energy. Relaxation per reaction can be divided into physical and chemical ones. Butyagin6 summarized these relations by using a concept of energy yield. The relative importance of the mechanochemical changes is determined by the excess free energy and the relative amount of physical and chemical relaxation times. Computer simulation studies have also been carried out semistatically22 or dynamically.6 Modification of Gibbs thermodynamics was tried by Gutman9 in order to apply it to the mechanochemical processes. He tried to explain dissolution of mechanically activated solids, notably metals, by using a concept of concentration polarization because of surface heterogeneity. This, together with stress corrosion, is still to be researched in the area of electrochemistry.
2.14.4
STRUCTURAL CHANGE OF SOLIDS UNDER MECHANICAL STRESS
Morphological changes observable under a conventional scanning electron microscope inevitably bring about changes of the internal structure of solids simultaneously. Direct observation of microplasticity © 2006 by Taylor & Francis Group, LLC
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was thoroughly made by Hess.14 A number of model experiments were carried out for the purpose of basic understanding of the topochemical nature of mechanochemistry. Some were compared with laser chemistry or tribology.23 Silicon single crystals were observed as one of the most easily available nearly ideal solids. Reactivity studies of Si(111) with well-defined stressing by indentation or scratching were carried out in detail for the purpose of studying the topochemical elementary processes,24 as shown in Figure 14.1. Every plastic deformation is related to a number of dislocations, which can store elastic energy in the core. Increase in the dislocation density further contributes to structural degradation and ends, in an extreme case, with amorphization. A trend of amorphization depends strongly on the nature of chemical bonds in a crystal. More anisotropic crystals tend to cleave with less extensive structural damage, while crystals with homopolar, isotropic bonds with larger free space tend to amorphization on mechanical treatment.25 Thus, formation of lattice imperfection and amorphization of solids during fine grinding are very frequently observed and evaluated.26 Broadening of X-ray diffraction peaks has been interpreted in detail.27 Even glasses, which are noncrystalline from the beginning, are further damaged under mechanical stress by loosing connectivity of their structural units, notably SiO44–, and the density of dangling bonds increases.28 When structural imperfection induces some kind of stacking faults, serving as embryos of other crystalline phases, topotactic polymorphic transformation can take place. These, together with other types of transformations have been observed, for example, for PbO, PbO2, Sb2O3, ZrO2, ZnS, γ-Fe2O3, γ-Al2O3, and CaCO3.29 Phase transformation of organic crystals can be used to improve the rate of dissolution or solubility,30 which can be utilized for pharmaceutical purposes. It is also to be noted that there is an opposite process of amorphization or decomposition during mechanical treatments, namely, crystallization and grain growth. Grinding of iron oxyhydroxide is one such example.31 There is an idea of phase equilibrium under mechanochemical condition.32,33 Particularly interesting is the formation of the metastable, high-pressure form. Some of the phase transformation systems have been studied with reference to their high-pressure chemistry.34 While hydrostatic pressure can describe the stability range of the high-pressure form, they can be formed only when kinetic conditions
FIGURE 14.1 Scanning electron micrograph of a single crystal of Si(111), indented at 1.96 N and etched by a 1 N aqueous solution of HF. [From Katayama, K. and Senna, M., Solid State Ionics, 73, 127, 1994. With permission.] © 2006 by Taylor & Francis Group, LLC
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are fulfilled, and shear stress is inevitable. This is associated with a fundamental profile of mechanochemical structural change processes, where isotropic hydrostatic pressure plays only a partial role. Hydroxides, carbonates, chlorates, nitrates, or similar thermally decomposable compounds can also be decomposed under mechanical stresses.8,35,36 Enhanced reactivity of solids at the moment of structural change, including phase transformation, is known as the Hedvall effect.37 This concept seems also to work in the course of mechanical treatment of solid mixtures. Carbonates or hydroxides, as well as some crystalline phases, easy to transform under mechanical stress, are therefore often superior to corresponding oxides as a starting material for solid-state reactions, although mechanochemical decomposition takes place via routes different from thermal ones. Organic solids can be caused to react with either organic or inorganic materials by grinding a mixture.38,39 These phenomena can also be expected to be used in the field of pharmaceutics.
2.14.5
MECHANOCHEMICAL SOLID-STATE REACTION AND MECHANICAL ALLOYING
There have been numerous attempts to synthesize various complex compounds only by mechanochemical process, for example, by grinding a mixture of Ag2SO4 with ZnS or CdS.40 Formation of zinc ferrite has also been reported.41 For these mechanochemical solid-state reactions, material transport is accomplished via short-range diffusion by repeated close contact of dissimilar particles with the aid of high-density defects. Mechanical alloying (MA) is one of the simplest reactions, but a very important example of in situ reaction. Studies on MA were stimulated for the production of a Ni-based superalloy with finely dispersed oxide particles 42 by high-energy ball milling. Afterward, an enormous number of trials were made to obtain various alloys by using energy-intensive mills. From its birth about a quarter century ago until recently, MA has been developed exclusively at the hands of metallurgists and metal physicists. Nowadays, they are jointly working with materials scientists, including mechanochemists. Metals and oxides have also been ground together to obtain an exchange reaction similar to thermal reactions.43,44 Mechanochemical reactions between solids and gases have been extensively studied. One of the most outstanding studies is that between hydrogen or hydrocarbons and silicon carbide.45 Grinding SiC in a hydrogen atmosphere produced various saturated and unsaturated hydrocarbon compounds. This is strong evidence of elevated reactivity of SiC under the influence of mechanical stress. Reactions with a gaseous reaction partner with more complex systems have been compiled by Juhasz and Opoczky. The formation of metal carbonyls has been extensively studied.35 Starting from Ni + CO or NiCO3 and H2S + CO, Ni(CO)4 was obtained. For the latter, the following mechanisms are proposed: (1) NiCO3→NiO + CO2; (2) NiO + H2S→NiS + H2O; and (3) NiS + 4CO + H2O→Ni(CO)4 + H2S. A direct synthesis of silicon tetrachloride and gaseous chlorine has also been reported.47 A mechanochemical route has an advantage of producing complex or composite materials by the use of apparatus similar to or identical with grinding mills, which are popular in most of industries. At the same time, mechanochemistry has turned out to be a versatile tool, enabling chemical reactions which are quite laborious or not possible via conventional synthetic methods. For solidstate mechanochemical reactions, however, reactions often take hundreds of hours, and hence the products have often suffered from serious contamination from the grinding elements. This is one of the reasons why people often hesitate to apply mechanochemical processes to industrial practices.
2.14.6
SOFT MECHANOCHEMICAL PROCESSES AND THEIR APPLICATION
Mechanochemical synthesis of precursors coupled with subsequent heat treatments can open a new way of synthesizing complex oxide materials,48 since initial complex formation via a mechanical route can be done quickly enough under gentle conditions when a starting mixture contains ingredients with hydroxyl groups or water.49 Examples and mechanistic arguments are given below. © 2006 by Taylor & Francis Group, LLC
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Mg(OH)2 remains crystalline even after intensive vibromilling for two days. When grinding with TiO2, however, amorphization takes place in two hours, accompanied by dehydration.50 One of the predominating mechanisms of this kind of complex formation is a proton transfer from solid acids such as silica or titania to a basic site of the hydroxides, where proton affinity is sufficiently large.51 As a consequence of the proton transfer, two metallic species, for example, magnesium and titanium, are cross-linked, mediated by an oxygen atom. This is nothing but an incipient chemical reaction between Mg(OH)2 and TiO2, leading to a chemical complex. A complex oxide such as MgTiO3 is conventionally prepared by mixing and firing MgO and TiO2 powders above 1700 K. Grinding a mixture a little improves the fabrication process. However, when we replace MgO with Mg(OH)2, the synthesis becomes much easier. As a matter of fact, MgTiO3 was observed as a single phase on heating a mechanochemically treated mixture, while other mixtures resulted in different phases as well, as shown in Figure 14.2.52 There are a number of similar examples of the system Sr–O–Ti leading to SrTiO3.53
FIGURE 14.2 X-ray diffraction patterns of equimolar Mg(OH)2/TiO2 mixtures after calcining at 1173 K for 2 h with (a) and without (b) preliminary grinding for 1 h. A, anatase; R, rutile; M, MgO; squares, MgTi2O5; circles, MgTiO3. [From Baek et al.52 With permission.] © 2006 by Taylor & Francis Group, LLC
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Very recently, some complex ferroelectric materials were synthesized on the basis of abovementioned precursor mechanochemistry.52 In contrast to many complicated procedures, application of mechanical stresses on precursors comprising PbO, MgO, Nb2O5, TiO2, and Mg(OH)2 in a simple stoichiometric proportion brought about pure perovskite already at the stage of calcination. Firing at increasing temperatures brought about crystal growth, enabling the maximum dielectric constants to reach as high as 30,000, with characteristics of a relaxer, as shown in Figure 14.3.
2.14.7
RECENT DEVELOPMENTS AND FUTURE OUTLOOK
Concepts and applications of mechanochemistry are ever expanding. In recent developments in mechanochemistry, it is particularly noteworthy that successful application is being directed to many high-profile technological fields. In the case of materials for microelectronics, it is essential to obtain phase pure and micrograined material. For that purpose, a soft mechanochemical process is particularly useful, since firing at low temperature suppresses grain growth. In the field of fast-developing high-frequency telecommunication, hexaferrites are needed, but they are generally not easy to obtain as a pure phase. Phase pure Y-phase hexaferrites were obtained by firing at temperatures as low as 1000oC after appropriate mechanical activation of the starting mixture.54 Likewise, ferroelectric relaxers such as solid solutions of PMN and PZN, the latter being a similar perovskite where Mg in PMN is substituted by Zn, are available with a PZN fraction up to 0.7 via a soft-mechanochemical route.55 In the field of organic chemistry, the concepts and principles of mechanochemistry are also increasingly utilized. In the field of polymeric species, there is a tradition of mechanochemistry for polymers.2 A similar concept has been extended to degradation of polymers, for environmental purposes.56 Much more innovative, however, is the systematic application of mechanochemistry to organic synthesis. A number of peculiar phenomena in a mortar were reported without recognizing that they definitely involved mechanochemistry. 57 The mechanisms of such mechanochemical organic reactions
FIGURE 14.3 Change in the dielectric properties of PT-PMN with temperature after firing at 1473 K for 2 h with (a) and without (b) preliminary grinding for 1 h. [From Baek et al.52 With permission.] © 2006 by Taylor & Francis Group, LLC
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were also elucidated via a computer simulation study.58 Mechanochromism is also an interesting phenomenon and related topic.59 Science and technology associated with mechanochemistry are developing with increasing acceleration. Reference to one of the latest reviews is recommended.60
REFERENCES 1. Ostwald, W., Handbuch der allgemeinen Chemie, Vol. 1, Akademie-Verlagsanstalt, Leipzig, 1919. 2. Heinicke, G., in Tribochemistry, Akademie-Verlag, Berlin, 1984, pp. 335–341. 3. Boldyrev, V. V. and Meyer, K., Festkoerperchemie, VEB Deutscher Verlag fuer Grundstoffindustrie, Leipzig, 1973. 4. Kubo, T., Introduction to Mechanochemistry, 2nd Ed., Tokyo Kagaku Dojin, Tokyo, 1978. 5. Beke, B., The Process of Fine Grinding, Martinus Nijhoff, The Hague, 1981. 6. Butyagin, P. Yu., Sov. Sci. Rev. B,. 14(1), 1989. 7. Tkacova, K., Mechanical Activation of Minerals, Elsevier, Amsterdam, 1989. 8. Juhasz, A. Z. and Opoczky, L., Mechanical Activation of Minerals by Grinding, Akademiai Kiado, Budapest, 1990. 9. Gutman, E. M., Mechanochemistry of Solid Surfaces, World Scientific, Singapore, 1994. 10. Gutman, E. M., Mechanochemistry of Materials, Cambridge International Science Publishing, 2000. 11. Balaz, P., Extractive Metallurgy of Activated Minerals, Elsevier, 2000. 12. Avvakumov, E., Senna, M., and Kosova, N., Soft Mechanochemical Synthesis. A Basics for New Chemical Technologies, Kluwer Academic Publishers, 2001. 13. Smekal, A. G., Anz. Oesterr. Akad. Wiss., 92, 733, 1955. 14. Hess, W., Dissertation, Univ. Karlsruhe, 1980. 15. Khodakov, G. S., Physics of Grinding, Izd. Nauka, Moscow, 1972. 16. Thiessen, P.A., Meyer, K., Heinicke, G., in Abh. dtsch. Akad. Wiss. Berlin, Kl. Chem. Geol., Biol., Berlin, 1966, p. 11. 17. Boldyrev, V. V. and Heinicke, G., Z. Chem., 19, 353, 1979. 18. Lyakhov, N. Z. and Boldyrev, V. V., Izv. Sib. Otd. Nauk. SSSR. Ser. Thim., 3, 1982. 19. Lyakhov, N. Z., in Proceedings of the Second Japan–Russia Symposium on Mechanochemistry, Jimbo, G., Senna, M., and Kuwahara, Y., Eds., Soc. Powder Technol., Japan, Kyoto, pp. 59 and 291. 20. Thiessen, K. P., Z. Phys. Chem. Leipzig, 260, 403, 1979. 21. Rumpf, H., Aufber.-Tech., 2, 59, 1973. 22. Karagedov, G. R., in Proceedings of the Fourth Japan–Russia Symposium on Mechanochemistry, Jimbo, G., Kuwahara, Y., and Senna, M., Eds., Soc. Powder Technol., Japan, Kyoto, p. 137. 23. Meyer, K. and Meier, W., Krist. Tech., 3, 399, 1968. 24. Katayama, K. and Senna, M., Solid State Ionics, 73, 127, 1994. 25. Steinike, U., Kretzschmer, U., Ebert, I., Henning, H.-P., Reactivity of Solids, 4, 1, 1987. 26. Fricke, R. and Gwinner, E., Z. Phys. Chem., A183, 165, 1938. 27. Hall, W. H., Proc. Phys. Soc., 62A, 741, 1949. 28. Zachariasen, W. H., J. Am. Chem. Soc., 54, 3841, 1932. 29. Senna, M., Cryst. Res. Technol., 20, 209, 1985. 30. Otsuka, M. and Kaneniwa, N., J. Pharm. Sci., 75, 506, 1986. 31. Mendelovici, E., Villalba, R., and Sagaraz, A., Mater. Res. Bull., 17, 241, 1982. 32. Schrader, R. and Hoffmann, B., Z. Anorg. Allgem. Chem., 369, 41, 1969. 33. Iguchi, Y. and Senna, M., Powder Technol., 43, 155, 1985. 34. Dachille, F. and Roy, R., Nature, 186, 34 and 71, 1960. 35. Heinicke, G. and Harenz, H., Z. Anorg. Allgem. Chem., 329, 185, 1963. 36. Nonat, A. and Mutin, J. C., Mater. Chem., 7, 455 and 479, 1982. 37. Hedvall, J., Einfuehrung in die Festkoerperchemie, Friedrich Vieweg u. Sohn Verl., Braunschweig, 1952. 38. Nakai, Y., Yamamoto, K., Terada, K., and Kajiyama, A., Chem. Pharm. Bull., 33, 5110, 1985. 39. Dushkin, A. V., Nagovitsina, E. V., Boldyrev, V. V., and Druganov, A. G., Siberian J. Chem., 5, 75, 1991. 40. Lin, I. J. and Somasundaran, P., Powder Technol., 6, 171, 1972. 41. Lin, I. J. and Nadiv, S., Mater. Sci. Eng., 39, 193, 1979. © 2006 by Taylor & Francis Group, LLC
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246 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60.
Powder Technology Handbook Benjamin, J. S., Met. Trans., 1, 2943, 1970. Takacs, L., Mater. Lett., 13, 119, 1992. Yang, H. and MacCormick, P. G., J. Solid State Chem., 110, 136, 1994. Heinicke, G. and Hennig, H.-P., Silikattech, 14, 86, 1967. Opoczky, L., Powder Technol., 17, 1–7, 1977. See also Ref. 8. Koester, A., Angew. Chem., 69, 563, 1957. Senna, M., Solid State Ionics, 63–65, 3, 1993. Avvakumov, E. G., Chemistry for Sustainable Development, 2, 1, 1994. Liao, J. and Senna, M., Solid State Ionics, 66, 313, 1994. Watanabe, T., Liao, J., and Senna, M., J. Solid State Chem., 115, 390, 1995. Baek, D., Isobe, T., and Senna, M., Solid State Ionics, in press, 1996. Kamei, Isobe, T., and Senna, M., Mater. Sci. Eng., B, in press, 1996. Temuujina, J., Aoyamaa, M., Senna, M., Masukob, T., Andob, C., and Kishib, H., J. Solid State Chem., in press, 2004. Shinohara, S., Baek, J. G., Isobe, Isobe, Senna, M., J. Am. Ceram. Soc., 83, 208, 2000. Mio, H., Saeki, S., Kano, J., Saito, F., Environ. Sci. Technol., 36 (6), 1344, 2002. Murata, Y., Kato, N., Fujiwara, K., and Komatsu, K., J. Org. Chem., 64, 3483, 1999. Fajar Pradipta, M., Watanabe, H., and Senna, M., Solid State Ionics, in press, 2004. Tipikin, D. S., Russ. J. Phys. Chem., 75, 1720, 2001; Zh. Fiz. Khim., 75, 1876–1879, 2001. Zhang, D. L., Progr. Mater. Sci., 49, 537–560, 2004.
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Part III Fundamental Properties of Powder Beds
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3.1
Adsorption Characteristics Masatoshi Chikazawa and Takashi Takei Tokyo Metropolitan University, Hachioji, Tokyo, Japan
3.1.1
INTRODUCTION
Adsorption has been defined as the phenomenon that occurs when concentration of a component in an interfacial layer of vapor–solid, vapor–liquid, liquid–solid, or liquid–liquid becomes higher or lower than that of the bulk phase. The former case is called a positive adsorption and the latter a negative adsorption. The adsorption phenomenon should be distinguished from absorption, where adsorbate molecules penetrate the interior of a solid. In this section the interactions between vapors and solids, or between liquids and solids, are described. Vapor–solid systems are dealt with in detail. Generally, adsorption phenomena are classified into two types: physical adsorption and chemical adsorption. In physical adsorption, attractive forces between individual adsorbate molecules and atoms or ions composing a solid surface originate in a van der Waals force that contains dispersion force and orientation forces of permanent and induced dipoles. If the solid surface consists of ions or polar groups, they will produce an electric field that induces dipoles in the adsorbed molecules. Moreover, if the adsorbate molecules possess permanent dipoles, their dipoles interact with the field, and hence, oriented adsorption occurs. On the other hand, chemical adsorption takes place only in special cases of vapor–solid and liquid–solid systems where adsorbate molecules are adsorbed by chemical bonds. Various adsorption phenomena are classified into these two types through detailed discussion of their mechanism. The adsorption mechanism, such as the existence of a chemical bond, can be studied by ultraviolet (UV), infrared (IR), and electron spin resonance (ESR) spectroscopy methods. Other methods (e.g., measurement of adsorption heats and adsorption isotherms) are necessary for further accurate classification. Generally the heat of physical adsorption is smaller than that of chemical adsorption, and the rate of physical adsorption is faster than that of chemical adsorption. Moreover, physical adsorption and desorption processes are reversible, and a relatively large amount of molecules is adsorbed on a solid surface at temperatures below the boiling point of the adsorptive or under high concentration of adsorptive in solution. Namely, multilayer adsorption occurs. On the other hand, in chemical adsorption, chemical bonds must be formed between a solid surface and adsorbed individual molecule, so the multilayer observed in physical adsorption does not occur. Hence the coverage θ V/Vm is always lower than unity, where V is the amount of adsorbed molecules and Vm is that for monolayer completion. Powder properties, ease of handling, and various problems in powder processes are closely related to surface properties. Hence inspection of the surface properties of powder is very important. In order to estimate the surface properties such as surface area, pore structure, and surface chemical properties, adsorption techniques are widely used.
3.1.2 ADSORPTION MEASUREMENT Measuring Methods Vapor–Solid System Measurements of the amount of adsorbed molecules are carried out by direct and indirect methods. Gravimetric and volumetric methods correspond to the direct method. In volumetric methods the 249 © 2006 by Taylor & Francis Group, LLC
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adsorbed amount is calculated from the vapor pressure of adsorptive and known volume (dead volume) of an apparatus. On the other hand, IR and UV spectroscopy and other instrumental methods are classified as indirect methods. Generally, a direct method is more accurate than an indirect method. The adsorbed amount is expressed by ml STP or μmol per unit surface area or per unit weight of a sample powder. Liquid–Solid System An adsorbed amount of molecules or ions is determined from the difference in concentration of a component in a solution before and after adsorption. An adsorption isotherm in a liquid–solid system is similar to that in a vapor–solid system. For example, adsorption isotherms of alcohols or fatty acids that have a long hydrocarbon chain belong to the Langmuir-type isotherm. On the other hand, acetic acid adsorption on active carbon belongs to the Freundlich-type isotherm. The particular point of adsorption in a liquid–solid system is the existence of competition adsorption of solvent and adsorptive molecules.
Preparation of Samples The pretreatment of samples is very important to obtain an accurate adsorbed amount. A clean surface is required for the adsorption measurement. The powder particles are usually handled or reserved under atmosphere. Therefore, water molecules are adsorbed according to their relative vapor pressure surrounding the powder. To obtain a clean surface of powder particles, elimination of the adsorbed water molecules is necessary. Especially in metal oxides, water molecules are chemically adsorbed, and various types of hydroxyl groups are formed on their surfaces. These hydroxyl groups are characteristic for each substance. Since the bonding strength of the surface hydroxyl groups varies with the nature of the metal oxides, the elimination temperatures of the hydroxyl groups are different from each other. In Figure 1.1 the contents of the surface hydroxyl groups on several oxides are shown as a function of the pretreatment temperature T.1 In all samples, the hydroxyl groups are markedly eliminated at temperatures above 400°C. When the temperature reaches 800°C, almost all hydroxyl groups are removed from the surfaces. The existence of surface hydroxyl groups has serious effects on the surface properties, such as selective adsorbability, water vapor affinity, catalytic activity, chemical activity, and electric field strength. Moreover, the hydroxyl groups play an important role in the adsorption of polar substances and molecules that can form hydrogen bonds. From these points of view, careful attention should be paid to determine the treatment temperature of sample powders. The surface properties may also be changed for other reasons. For example, the release of CO2 from CaCO3 powder is found at temperatures above 300°C.2 In the cases of oxides, oxygen defects are produced by high-temperature treatment, and hence, the samples must be kept under oxygen pressure after pretreatment to remove the defects. A decrease in surface area usually occurs in the hightemperature treatment, which stabilizes the surface condition, causing disappearance of active sites and growing stable crystal planes.
3.1.3 THEORY OF ADSORPTION ISOTHERMS Types of Isotherms The adsorption phenomenon gives valuable information on the physical and chemical properties of a solid surface and is represented by an adsorption isotherm. In the case of a vapor–solid system, the isotherm is usually expressed by the relationship between the adsorbed amount V and the equilibrium vapor pressure P of an adsorptive at a fixed temperature T. V = fT ( P )
(1.1)
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On the other hand, in the case of a liquid–solid system, concentration (C ) of adsorptive in a solution is used instead of vapor pressure. V fT (C )
(1.2)
The other expressions obtained at a fixed pressure or at a constant amount of adsorbed molecules are given as follows: V fP (T )
(1.3)
V fV (T )
(1.4)
Many equations for the adsorption isotherm are introduced by assuming various adsorption mechanisms. The isotherms of physical adsorption are grouped conveniently into eight classes. The typical classification is proposed by IUPAC.3 Besides these isotherms, types of isotherms are shown in Figure 1.2, which is useful for an understanding of adsorption phenomena.
FIGURE 1.1 Change in surface OH groups by heat treatment. {: α-Fe2O3;
: ZnO; z: TiO2. [From Morimoto, T., Nagao, M., and Tokuda, F., Bull. Chem. Soc. Jpn., 41, 1533–1537, 1968. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.2 Types of adsorption isotherms. This figure represents adsorption isotherms of gas–solid system. P0, saturated vapor pressure at measurement temperature. In the case of a liquid–solid system, the abscissa indicates a concentration (C ) of solute.
Equations of Adsorption Isotherms Henry Equation The relationship between an adsorbed amount V and an equilibrium pressure P is expressed by V kH P
(1.5)
where kH is a constant. This equation holds for a small adsorbed amount below 1% of the monolayer capacity. Even in the cases of the other equations, the adsorbed amounts at early stages are represented approximately by the Henry equation. The amount of O2 or N2 adsorbed at 0° C by a silica gel increases linearly with increasing pressure up to 1 atm pressure, and the isotherm is given by the Henry equation.4 Langmuir Equation This equation is introduced from the most basic adsorption model, in which two assumptions are made. The first assumption is that the heat of adsorption is the same for every molecule and that there is no interaction among the adsorbed molecules. The second is that every molecule that collides with a molecule already adsorbed on a site reflects immediately to the vapor phase. Therefore, multilayer adsorption does not occur, and the maximum amount of adsorption is obtained at the © 2006 by Taylor & Francis Group, LLC
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monolayer completion, where the particle surface is just covered with closely packed molecules adsorbed on the specific adsorption sites. From such adsorption mechanisms, the Langmuir equation is derived as follows.5 Adsorption velocity is proportional to the number of molecules that strike the bare surface of powder particles. On the other hand, desorption velocity is closely related to the adsorbed amount or the surface coverage θ. In equilibrium, the velocities of adsorption and desorption are equal to each other, and hence the following equation is derived: kad (1 u)P kd u
(1.6)
where kad and kd represent the rate constants of adsorption and desorption, respectively. The value of u is given by V/Vm. Hence Equation 1.6 is rewritten as follows: V
Vm (kad / kd )P AVm P 1 (kad / kd )P 1 AP
(1.7)
Especially when the equilibrium pressure is very small, Equation 1.7 will accord with the Henry equation (Equation 1.5). Freundlich Equation The Freundlich equation is represented by Equation 1.8 and widely used in the case of liquid–solid systems: V kF P1/ n
(1.8)
where kF is a constant and n is a value in the range 1 to 10. This equation was obtained experimentally. If the isosteric heat of adsorption decreases linearly with increase in ln V, the adsorption isotherm is expressed by the Freundlich equation. It is known that the measurement temperature and pretreatment temperature of a sample affect the n value in Equation 1.8. BET Equation In order to understand the physical adsorption, the BET equation is the most widely used. This equation is derived by expansion of the Langmuir adsorption model to multilayer adsorption under a few assumptions. The isosteric heat of adsorption for molecules other than in the first layer is equal to the liquefaction heat of an adsorptive, interaction among the adsorbed molecules at same layers is not produced, and at saturated vapor pressure the number of adsorbed layers is infinite. From these assumptions, the BET equation is obtained6: V
Vm CX (1 X )(1 X CX )
(1.9)
where X is the relative vapor pressure P/Po, C is the value related to the strength of adsorption force, Po is the saturated vapor pressure at measurement temperature, and Vm is the monolayer capacity. Detailed derivation of the BET equation is abbreviated here. If the adsorbed layer is restricted to the nth layer, the BET equation becomes V=
Vm CX ⎡⎣1 (n 1) X n nX n1 ⎤⎦
(1 X ) ⎡⎣1 (C 1) X CX n1 ⎤⎦
(1.10)
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In the case of n 1, Equation 1.10 accords with the Langmuir equation. The C value, which is often used to evaluate the surface property, is expressed by ⎛ E EL ⎞ C exp ⎜ 1 ⎝ RT ⎟⎠
(1.11)
where E1 and E L are the isosteric heat of the first-layer adsorption and the liquefaction heat of adsorptive, respectively. Therefore, when C is large enough, the molecules are strongly adsorbed and the adsorption isotherm at low pressure becomes steeper. When C is less than 2, the adsorption isotherm is categorized into type III, according to the classification of IUPAC. The BET equation usually holds in the relative vapor pressure range 0.05 X 0.35. The adsorbed amount calculated from the BET equation at low pressure below X 0.05 becomes smaller than the experimental value, but at high pressures above X 0.35 it becomes larger, conversely. However, employment of an appropriate value of n in the modified BET equation (Equation 1.10) can expand its use up to X 0.7. There are many comments about the BET equation. Neglect of the interaction force produced among the adsorbed molecules, as well as the occurrence of the nth layer adsorption before completion of the (n – l)th layer, are pointed out as being illogical. In the latter case, the configurational entropy increases and the adsorbed amounts calculated from the BET equation at high pressures above P/P0 0.35 become larger than experimental values. Many elaborate amendments of the BET equation have been attempted, but a more useful and concise equation has not yet been deduced. Jura–Harkins Equation With respect to the adsorption layer formed on a solid surface, Equation 1.12 is proposed by Jura and Harkins7: p b aa
(1.12)
where p is the two-dimensional pressure, a the surface area occupied by one adsorbed molecule, and a and b are constants. The two-dimensional pressure p is given from the Gibbs adsorption equation. p RT ∫ d (ln P ) P
0
(1.13)
where G is the molar amount per unit surface area. The adsorbed molar amount per unit surface area is given as
1 V V0 S a
(1.14)
where the values V and V0 are the volume of adsorbed amount and one molar volume at standard pressure and temperature, respectively. From Equation 1.12 and Equation 1.13 the following equations are deduced: ⎛ 1⎞ dp ad a aV0 Sd ⎜ ⎟ ⎝V⎠
(1.15)
dp = RT d (ln P )
(1.16)
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Hence, using Equation 1.14 through Equation 1.16, the following equation is obtained: ⎛ P ⎞ ⎛ aS 2V02 ⎞ ⎛ 1 1⎞ 2⎟ ln ⎜ ⎟ ⎜ ’2 ⎜ ⎟ V ⎠ ⎝ P0 ⎠ ⎝ 2 RT ⎠ ⎝ V0
(1.17)
Since the value V′0 is the adsorbed amount at saturated pressure P 0, it becomes infinite, theoretically. Consequently, the following equation is obtained: ⎛ P⎞ A ln ⎜ ⎟ 2 V ⎝ P0 ⎠ where A is constant and related to the surface area S of a powder, as follows: S kJH A
(1.18)
(1.19)
where kJH is a constant to be corrected by using a sample of known surface area. Because the potential energy of one molecule adsorbed on a liquid film with thickness h is proportional to h–3 or V–3, Equation 1.20 is derived by Frenkel8: ⎛ P⎞ C ln ⎜ ⎟ 3 P V ⎝ 0⎠
(1.20)
On the other hand, the general formula (Equation 1.21) is proposed by Halsey,9 assuming that the potential energy of an adsorbed molecule is proportional to 1/hs (i.e., 1/V s): ⎛ P⎞ D ln ⎜ ⎟ S V ⎝ P0 ⎠
(1.21)
where h is the distance between particle surface and an adsorbed molecule, and generally the value of s ranges from 2 to 3. When s is large enough, the adsorption force is specific and short ranged, whereas when the value of s is small, the force is long ranged and originates in the van der Waals force. In the case of N2 adsorption on anatase, the s value is equal to 2.67. Various adsorption equations are listed in Table 1.1, in which expressions convenient for plotting experimental data are also shown.
3.1.4 ADSORPTION VELOCITY Velocity of physical adsorption is different from that of chemical adsorption, comparing the per unit surface area and at fixed temperature and pressure. Usually, the velocity of physical adsorption on a plain solid surface is faster than that of chemical adsorption. However, when a sample is porous and adsorptive pressure becomes high, capillary condensation occurs. In this case, a long time is necessary to attain an equilibrium. Therefore, as the adsorption velocity depends on such geometric properties and on the diffusion rate of adsorption heat, detailed discussion of adsorption velocity is difficult. The equation of adsorption velocity derived by Langmuir is ⎛ dV ⎞ ⎜⎝ ⎟ kad (Vm V ) kdV dt ⎠
(1.22)
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TABLE 1.1
Equations of Adsorption Isotherms
Type of Adsorption Isotherm
Equation
Henry type
V kH P
Langmuir type
V
Freundlich type BET type Jura-Harkins type Frenkel type Halsey type
V kH P
Vm AP 1 AP
V KF P
V
Expression for Plotting Experimental Data
1
P 1 1 P V Vm A Vm 1 In V InK F InP n
n
X 1 C 1 X V (1 X ) VmC VmC
VmCX (1 X )(1 X CX ) D VS (2 ≤ S ≤ 3) InX
InX
D VS
As this equation is derived for a fixed pressure, the constant kad contains a pressure term, kd represents a desorption rate constant and Vm is the monolayer capacity. Denoting the adsorbed amount in equilibrium at a fixed pressure by Ve, the adsorbed amount at time t becomes
}
V = Ve {1 − exp [ − (kad + kd )t ]
(1.23)
3.1.5 ADSORBED STATE OF ADSORBATE Investigation of the state of adsorbed molecules (i.e., gaseous, liquidlike, or solidlike state) gives very important information regarding adsorption phenomena. Moreover, the determinations of adsorption structures and sites are also valuable. Chemical adsorption and physical adsorption are different from each other. This section deals with physical adsorption. The adsorbed states are usually estimated from thermodynamic quantities such as isosteric heat of adsorption and differential molar entropy. Moreover, the adsorbed state can also be discussed based on the changes in dielectric constant and surface conductivity. On the other hand, adsorption mechanisms are studied by IR, UV, and nuclear magnetic resonance (NMR) spectroscopy methods. Especially, IR spectroscopy has become a well-established technique for the determination of adsorption sites and structures. For example, from pyridine adsorption on oxide surfaces, Brönsted and Lewis acid sites were investigated, and the adsorption mechanism was discussed.10 Generally, the surfaces of SiO2, TiO2, α-Fe2O3, and ZnO powders are covered with the various OH groups which are formed by chemical adsorption of H2O under atmosphere or preparation methods (precipitation of hydroxides and sol–gel method). The effects of the quantity and quality of OH groups on adsorption phenomena are also studied by the IR spectroscopy method. Usually, samples for measuring the infrared spectrum are prepared by pressing powder samples into self-supporting disks. The sample disks thus obtained are mounted in the cell, which can be evacuated and heated in order to control the adsorption state. Various cells have been designed by many researchers.11 When it is difficult to prepare a self-supporting disk, diffuse reflectance spectroscopy (DRS) and photoacoustic spectroscopy (PAS) are used. These methods are appropriate because of easy sample preparation without special treatments for grinding and pressing samples. Generally, the sample for DRS is diluted with KBr or KCl powder. Therefore, the measurement responsibility of DRS depends on the degree of dilution and particle sizes of diluents and samples.
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In the case of PAS, a spectrum can be obtained without dilution of the sample, and this spectroscopy can be applied to any samples with different shapes (powder, pellet, and film). Moreover, PAS can obtain the depth profile of samples. When a powder sample is very fine and well dispersed in the liquid, which is transparent for infrared rays, a spectrum of the sample can be obtained in the liquid as it is. For example, surface groups on fine silica powders were clearly observed by dispersing the powder samples in carbon tetrachloride.12 The interaction between the adsorbed molecules and surface functional groups such as hydroxyl groups is investigated from changes in absorption bands deduced by perturbations of adsorbed molecules. The adsorption of molecules on the surface functional groups causes the shift and decrease of the absorption peaks of surface functional groups. The correlation between differential heats (Qa) of adsorption and peak shift (Δn) of the absorption band of hydroxyl groups on a silica surface is shown in Figure 1.3.13 The adsorption characteristics and adsorbed states largely depend on chemical composition and geometric structure of solid surfaces. The geometric structure of solid surfaces contains surface roughness, pore structure, defect sites, and difference in crystal face. Generally, the chemical composition of real solid surfaces is different from that of the bulk phase by the chemisorption of oxygen, carbon dioxide, and water molecules or by surface modification. Therefore, the real surfaces of powders will be covered with surface functional groups (e.g., hydroxyl, carbonyl, carboxyl, and various alkyl groups). These surface functional groups act as the dominant factor of the adsorption characteristics and adsorbed states. The adsorption behavior and adsorbed states closely relate to the combination between the nature of the powder surface and that of the adsorbate molecule. Especially, in the case of polar adsorbate molecules, such effects are remarkably influenced by the surface functional groups. Figure 1.4 shows the effects of surface functional groups of silica powder treated with various silane reagents on water vapor adsorption.14 It is important to estimate the number and types of the surface
FIGURE 1.3 The correlation of stretching bands of silica surface hydroxyl groups and differential heats of adsorption. 1, benzene; 2, toluene; 3, p-xylene; 4, mesitylene. [From Galkin, G. A., Kiselev, A. V., and Lygin, V. I., Trans. Faraday Soc., 60, 431–439, 1964. Reproduced by permission of The Royal Society of Chemistry.]
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FIGURE 1.4 Water adsorption isotherms on the silane-treated silica. [From Zettlemoyer, Z. C. and Hsing, H. H., J. Colloid Interface Sci., 58, 263–274, 1977. With permission.]
FIGURE 1.5 Adsorption isotherms of Kr on alkali halides. {, NaCl (400°C heat treatment ); , RbCl (300°C heat treatment);
, KCl (400°C heat treatment). [Reprinted with permission from Takaishi, T. and Saito, M., J. Phys. Chem., 71, 453–454, 1967, copyright (1967) American Chemical Society.]
functional groups. These groups are investigated by spectroscopic (IR, Raman, X-ray photoelectron spectroscopy [XPS], NMR) and chemical reaction methods. If a powder sample has a pore structure, the specific surface area of this powder will be very large. Moreover, capillary condensation and micropore filling phenomena occur through gas adsorption. In these cases, the physical properties of the adsorbed layers are speculated to be different from those of adsorbed layers formed on flat surfaces. Capillary condensation is observed in mesoporous materials © 2006 by Taylor & Francis Group, LLC
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Cp/(cal/mol.deg)
30
22
14
6 60
70
80 T/(K)
FIGURE 1.6 Change in heat capacity of Ar layer formed on rutile surface, 2.9 Vm, 4.0 Vm;, 4.6Vm.
and shows the increase of adsorption capacity at a lower pressure than the saturated vapor pressure. On the other hand, the micropore-filling phenomenon is found in microporous materials (e.g., activated carbon and zeolite). The microporous materials have extremely high specific surface areas (1000 m 2 /g) and show strong adsorption force due to the effect of the pore wall. The adsorbate molecules are filled up in the micropore at very low relative pressure, and the adsorption isotherm exhibits the Langmuirtype adsorption isotherm. Adsorption isotherms of Kr on alkali halides are depicted in Figure 1.5.15 The sharp increases are noted in the adsorption isotherms, and they become progressively sharper with increasing pretreatment temperature. These annealing processes make the surface more homogeneous. Therefore, the sharp increase due to two-dimensional condensation is considered to be strongly affected by surface conditions. The molar heat capacities of Ar adsorbed on rutile are illustrated in Figure 1.6.16 The peaks in the heat capacity curves are considered to be ascribed to the phase transition. The peak becomes larger and appears at higher temperature with increased amount adsorbed. The lowering phase-transition temperature of an adsorbed layer can be explained as follows. Because the liquid structure and physical properties of the adsorbed layer are influenced by the solid surface, solidification of the adsorbed layer becomes more difficult. The same results were observed in N2 adsorption on TiO217 and H2O adsorption on SiO218. The Clausius–Clapeyron plot and differential scanning calorimetry (DSC) measurement are other methods for determination of the phase transition temperature.
3.1.6 ESTIMATION OF SURFACE PROPERTIES BY ADSORPTION METHOD Physical and Chemical Properties of Powder Surfaces Various phenomena concerning powder materials depend largely on physical and chemical properties of their surfaces. In powder processes such as classification, mixing, and comminution, adhesive force and surface energy play important roles. On the other hand, the wettability of powder surfaces is significant in filtration, sedimentation, and floatation. Moreover, reactivity, solubility, and activity © 2006 by Taylor & Francis Group, LLC
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of the catalyst are closely related to chemical properties of powder surfaces. Information about the chemical properties of powder surfaces is obtainable from adsorption characteristics of various adsorptives such as the isosteric heat of adsorption and adsorbed amount. Generally, the adsorption heat determined at very low coverage is large, and the adsorption is considered to occur at active sites. On the other hand, adsorption heat measured at high coverage contains lateral interaction produced among adsorbed molecules. Therefore, the adsorption heat calculated at a coverage of u0.5 is used to estimate the surface property. The changes in surface property of modified powder particles are shown in Figure 1.7 and Figure 1.8.19 Their changes are expressed by the BET C value
FIGURE 1.7 Change in water vapor affinity of CaF2 coated with oleate by an amount V. SKr, SHO; surface area determined by Kr and H2O adsorption, respectively; {, the surface area ratio; z, C value for H2O adsorption. Vm 0.26 mmol/g. [From Hall, P. G., Lovell, V. M., and Frinkelstein, N. P., Trans. Faraday Soc., 66, 1520–1529, 1970. With permission.]
FIGURE 1.8 Isosteric heat of H2O adsorption on CaF2 modified with oleate. Vm 0.26 mmol/g. Adsorbed amounts (mmol/g): {, 0; , 0.125; , 0.243;
, 0.272; ×, 0.406; S, 0.592; z, 2.043. [From Hall, P. G., Lovell, V. M., and Frinkelstein, N. P., Trans. Faraday Soc., 66, 1520–1529, 1970. With permission.] © 2006 by Taylor & Francis Group, LLC
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and isosteric heat of adsorption. The affinity of water vapor decreases markedly when the amount of potassium oleate increases and it covers the particle surfaces with a complete monolayer. The molecular diameter of a potassium oleate is calculated to be about 2.1 nm from the surface area and the adsorbed amount of oleate. Heat treatment usually makes particle surfaces more homogeneous, and such results obtained for carbon black are shown in Figure 1.9.20 The surface property of untreated carbon black is inhomogeneous, but it becomes more homogeneous with an increase in heat treatment temperature. These results are obtained from the comparison of adsorption heat. The adsorption heat of Ar molecules at low coverage is large and decreases gradually with the adsorbed amount. The high value of the adsorption heat at the early stage disappears with an increase in treatment temperature. At coverage at about u l, lateral interactions will arise among adsorbed molecules, and an apparent maximum appears in the heat curves. The maximum due to the lateral interaction becomes increasingly prominent with progress of the graphitization by the heat treatment. From these results the lateral interaction is presumed to be effectively generated when the surface homogeneity is attained. After that, multilayer adsorption occurs, and the adsorption heat rapidly decreases approaching the liquefaction heat at adsorbed amount V above Vm. The surface properties are also estimated from the heat of immersion. This immersion heat of particle surfaces for a certain liquid is related to the adsorption heat of the corresponding vapor molecules. The measurement of the immersion heat is relatively easy, compared with that of the adsorption heat. When water is used as the immersion liquid, the heat of immersion represents the degree of hydrophilic or hydrophobic properties of the surface. The change in the immersion heat of a silica powder modified with trimethylsilyl groups for water is shown in Figure 1.10.21 The hydrophilic property of the surface decreases with an increase in the concentration of trimethylsilyl groups.
FIGURE 1.9 Differential heat of Ar adsorption on carbon black heat treated at high temperatures. [From Beebe, R. A. and Young, D. M., J. Phys. Chem., 58, 93–96, 1954. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.10 Effect of the amount of the combined trimethylsilyl group on the heat of adsorption. [From Tsutsumi, K. and Takahashi, H., Colloid Polym. Sci., 263, 506–511, 1985. With permission.]
Surface Geometrical Properties It is difficult to describe geometric irregularities of real particle surfaces. Avnir estimated the geometric structure of the surface using the fractal dimension D.22,23 The fractal dimension of solid surfaces is investigated by measurements of physical adsorption of the gases, which have various molecular sizes, and is calculated as log n = ( −
1 D) log s + C 2
(1.24)
where n is the monolayer capacity of adsorbate molecules on the surface, s is the effective crosssectional area of one adsorbate molecule, and C is a constant value. The value of D is calculated from the slope of a linear line, which is obtained by plotting log n versus log s. Generally, this value for the real surfaces ranges from 2 to 3. For typical solid surfaces the values are obtained as follows. In the cases of graphite and monmorillonite, which have a smooth surface, their D values are nearly 2. However, porous samples that have high specific surface areas and porous structures, such as silica gel and charcoal, are nearly 3. The study of the relationship between the fractal dimension and various adsorption phenomena will become more important in the future. Powder with a large specific surface area often has a porous structure. The pore size, shape, and distribution are different in each sample. Generally, the diameter for a cylindrical pore and the distance between two side walls for a slit-shaped pore are used as representative pore sizes. The pores are classified into three types according to the pore size: micropores for below 2 nm, transitional or intermediate pores for 2–50 nm, and macropores for above 50 nm. In the determination of pore sizes by the adsorption method, the Kelvin equation is applied, and pore radii in the range of 0.5 to 30 nm are measured accurately. In the case of N2 adsorption, if the pore structure is presumed © 2006 by Taylor & Francis Group, LLC
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to be cylindrical, the Kelvin equation holds for the radius above 2 nm but fails below 2 nm.24 This result is considered as follows. The liquid properties of capillary condensed liquid, such as molar volume and surface tension, are different from those of bulk liquid. The Kelvin equation expresses the relationship between pore radius r and vapor pressure P at which capillary condensation occurs. Denoting the molar volume by V, the adsorbate surface tension by s, and the contact angle of condensed liquid by u, it becomes ⎛ P⎞ 2V s cos u ln ⎜ ⎟ = − rRT ⎝ P0 ⎠
(1.25)
The pore size below 2 nm should be calculated by the t-method,25 the D-R method,26 and the H-K method.27
REFERENCES 1. Morimoto, T., Nagao, M., and Tokuda, F., Bull. Chem. Soc. Jpn., 41, 1533–1537, 1968. 2. Morimoto, T., Kishi, J., Okada, O., and Kadota, T., Bull. Chem. Soc. Jpn., 53, 1918–1921, 1980. 3. Sing, K. S. W., Everett, D. H., Haul, R. A. W., Moscou, L., Pierotti, R. A., Rouqe rol, J., and Siemieniewska, T., Pure Appl. Chem., 57, 603–619, 1985. 4. Lambert, B., and Peel, D. H. P., Proc. Roy. Soc. London, A144, 205, 1934. 5. Langmuir, I., J. Am. Chem. Soc., 40, 1361–1403, 1918. 6. Brunauer, S., Emmett, P. H., and Teller, E., J. Am. Chem. Soc., 60, 309–319, 1938. 7. Jura, G. and Harkins, W. D., J. Chem. Phys., 11, 430–432, 1943. 8. Frenkel, J., Kinetic Theory of Liquids, Clarendon Press, Oxford, 1946, p. 332. 9. Halsey, G. D., J. Chem. Phys., 16, 931–937, 1948. 10. Parry, E. P., J. Catal., 2, 371–379, 1963. 11. Bell, A. T., in Vibrational Spectroscopy of Molecules on Surfaces, Yates, J. T. and Madey, T. E., Eds., Plenum Press, 1987, p. 116. 12. Tripp, C. P. and Hair, M. L., Langmuir, 8, 1961–1967, 1992. 13. Galkin, G. A., Kiselev, A. V., and Lygin, V. I., Trans. Faraday Soc., 60, 431–439, 1964. 14. Zettlemoyer, Z. C. and Hsing, H. H., J. Colloid Interface Sci., 58, 263–274, 1977. 15. Takaishi, T. and Saito, M., J. Phys. Chem., 71, 453–454, 1967. 16. Morrison, J. A. and Drain, L. E., J, Chem. Phys., 19, 1063, 1951. 17. Morrison, J. A., Drain, L. E., and Dugdale, J. S., Can. J. Chem., 30, 890, 1952. 18. Plooster, M. N. and Gitlim, S. N., J. Phys. Chem., 75, 3322–3326, 1971. 19. Hall, P. G., Lovell, V. M., and Frinkelstein, N. P., Trans. Faraday Soc., 66, 1520–1529, 1970. 20. Beebe, R. A. and Young, D. M., J. Phys. Chem., 58, 93–96, 1954. 21. Tsutsumi, K. and Takahashi, H., Colloid Polym. Sci., 263, 506–511, 1985. 22. Pfeifer, P. and Avnir, D., J. Chem. Phys., 79, 3558–3565, 1983. 23. Avnir, D., Farin, D., and Pfeifer, P., J. Chem. Phys., 79, 3566–3571, 1983. 24. Harris, M. R., Chem. Ind. (London), 268–269, 1965. 25. Lippens, B. C. and Boer, J. H., J. Catal., 4, 319–323, 1965. 26. Dubinin, M. M. and Stoeckli, H. F., J. Colloid Interface Sci., 75, 34–42, 1980. 27. Horvath, G. and Kawazoe, K., J. Chem. Eng. Jpn., 16, 470–475, 1996.
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3.2
Moisture Content Satoru Watano Osaka Prefecture University, Sakai, Osaka, Japan
3.2.1
BOUND WATER AND FREE WATER
Water is achromatic and transparent, and it has three phases (gas, liquid, and solid), depending on the temperature and pressure. Due to its unique characteristics, various existing forms are observed.1 In a particulate system, existing forms of water are classified as water in a single particle or water in a particulate (bulk) system, as illustrated in Figure 2.1. Generally, water is categorized into two classes: bound water and free water. Bound water, which exists inside crystals and in aqueous solutions, gels, organisms, soil, and so forth, is stuck to materials by hydrogen bonding. As shown in Figure 2.1, bound water includes combined water, occluded water, adsorption water, and hygroscopic water. Combined water is also called crystal water, which has the following types, depending on the structure and the bonding conditions: 1. 2. 3. 4.
Constitution water: Water exists in compounds as a hydroxyl group. Coordinated water: Water forms a chelate ion in a chelate complex. Anion water: Water is stuck to anions by hydrogen bonding. Lattice water: Water exists inside a crystal lattice without having direct bonding with either cations or anions. 5. Zeolite water: The vacancy of a crystal is occupied with water molecules, and the crystal structure does not change after dehydration. Occluded water is captured inside a solid (particle), and it exists as a solid solution or hydrate. Adsorption water is called either surface adsorption water or hygroscopic water; the former is bound by van der Waals force, while the latter is captured among particles. Contrary to bound water, free water is captured by materials with relatively weak bonding strength and is called adhesive water or swelling water. In a particulate system, damping conditions by free water are categorized into funicular, capillary, and moving states. Especially in a soil particulate system, consistency of soil and the phase dramatically change depending on the water content. In general, the transition point between the different phases is called the Atterberg limit (shrinkage, plastic, and liquid limits).2 The condition of water distribution and bonding strength of water in a particulate system is described as a function of suction potential S (m),which is defined as
S⫽⫺
RT ⎛ P ⎞ ln Mg ⎜⎝ P0 ⎟⎠
(2.1)
where P is the water vapor pressure inside the particulate system, Po is the vapor pressure on the free surface of pure water, R is the gas constant ( ⫽ 8.315 J/mol K), M is the molecular weight of water ( ⫽ 18.02 g/mol), and g is the gravitational acceleration ( ⫽ 9.81 ⫻ 10–3 J/gm). Suction potential is also related to the pF value as pF ⫽ log S
(2.2) 265
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1)
2)
A.P. (MPa) Combined wa te r
shrinkage limit
1) wate r heigh t 2) atmospheric pressure
FIGURE 2.1
Existing forms of water.
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TABLE 2.1 Existing Water Forms in Solid, and Water Removal Temperature Bonding Strength Weak
Name and Bonding State
Water Removal Temperature (⬚C)
Water bound by adhesion or water retained through holes
Dried air at room temperature
Occluded water Absorbed water
250⬃300
Outer surface
100⬃130
Submicron channels
Strong
⬃600
Crystalline water
Dependent on substance
Constituent water
Dependent on substance
Zeolitic water
600⬃1000
The relationship between suction potential and the existing water forms is also explained in Figure 2.1. Another way of measuring bonding strength of water in a particulate system is to use the water removal temperature in heating. Table 2.1 explains the existing water forms in a solid and their removal temperature.
3.2.2 METHODS FOR DETERMINING MOISTURE CONTENT IN A PARTICULATE SYSTEM The appropriate method for measuring the moisture content is strongly dependent on the water form in a sample. The simplest way of measuring moisture content in a sample is to measure weight loss after drying. Other methods, such as chemical, electric, and optical methods, as well as the nuclear magnetic resonance (NMR) spectrometer method and method of applying radioactive rays, are also available. These details are shown in Section 6.6 On-Line Measurement of Moisture Content. Determination methods for water content have been established in various national and international standards such as JIS and ISO.
REFERENCES 1. Gotoh, K., Masuda, H., and Higashitani, K., Eds., Powder Technology Handbook, 2nd Ed. Marcel Dekker, New York, 1997, pp. 265–276. 2. Smith, K. A. and Mullins, C. E., Eds., Soil and Environmental Analysis: Physical Methods, Marcel Dekker, New York, 2000. 3. Kawamura, M. and Toyama, S., J. Soc. Powder Technol. Jpn., 15, 292–297, 1978.
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3.3
Electrical Properties Hiroaki Masuda Kyoto University, Katsura, Kyoto, Japan
Ken-ichiro Tanoue Yamaguchi University, Ube, Yamaguchi, Japan
Yasufumi Otsubo Chiba University, Image-ku, Chiba, Japan
3.3.1
IN GASEOUS STATE
The specific resistance, dielectric constant, specific charge, and contact potential difference of particles are discussed in this section. These electrical properties are necessary for studying the behavior of electrically charged particles, electrostatic precipitation, electrostatic powder coating, electrostatic powder imaging, and so on.
Specific Resistance The electric resistance of a powder bed is one of the most important properties, especially in relation to electrostatic precipitation. If the specific resistance rd of particles is lower than 102 m, the particles will immediately be neutralized on the dust collecting electrode. Further, they are oppositely charged by the electrode and reentrained by gas flow in the presence of electrostatic repulsion between the particles and the collecting electrode. On the other hand, if the specific resistance is higher than 10 9 m, back discharge from the collecting electrode will occur, resulting in a decrease in the precipitation efficiency.1 The specific resistance of highly resistive powder is strongly dependent on the ambient temperature and moisture. It takes a maximum at some temperature (100 to 200°C) and decreases with increasing absolute humidity.2 It also depends on the chemical composition of particles. The specific resistance can be measured by (a) parallel-plate electrodes, (b) cylindrical electrodes, or (c) needle-plate electrodes, as shown schematically in Figure 3.1. After particles are packed or deposited between the electrodes, the applied voltage V and electric current I through the powder bed are measured. Then the specific resistance rd can be calculated from the following equations3: 1. Parallel-plate electrodes:
AV d I
(3.1)
2p l V ln ( b a ) I
(3.2)
rd 2. Cylindrical electrodes:
rd
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FIGURE 3.1 Electrodes for measuring specific resistance.
3. Needle-plate electrodes:
rd where V I A d l a b V L
A V L I
(3.3)
applied voltage (V) electric current (A) area of the main electrode (m2) gap between the electrodes (m) length of the cylindrical electrode (m) outside diameter of the inner cylindrical electrode (m) inside diameter of the outer cylindrical electrode (m) electrical potential of the exploring needle (V) distance between the exploring needle and the plate electrode (m)
The specific resistance depends on the packing density of the powder bed. Therefore, measurements should be carried out after precise setting of the packing density. Needle-plate electrodes are suitable for measurement under the same conditions as in an industrial electrostatic precipitator. © 2006 by Taylor & Francis Group, LLC
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f ( p ) ( p 1)
TABLE 3.1 –p f(–p )
( 2) p
1
2
5
10
20
50
100
0
0.250
0.571
0.750
0.864
0.942
0.971
1
TABLE 3.2
g( p ) 3p
( 2) p
p
1
2
5
10
20
50
100
g(p)
1
1.50
2.14
2.50
2.73
2.88
2.94
3
The specific resistance is usually higher for a powder bed of smaller packing density. The applied voltage should be kept as low as possible to avoid a void discharge. It may take a long time to reach a constant current, especially for high-resistivity powder, because the current is partly absorbed in the powder bed as charge accumulates between particles. The ambient temperature and the humidity should be controlled. A special wind tunnel called a race track may be utilized in the needle-plate electrodes method.3
Dielectric Constants The dielectric constant of particles is necessary to estimate the dielectrophoretic force or the maximum charge attainable by field charging. The dielectrophoretic force is a function of ( p – )/(p 2), where p is the dielectric constant of the particle and is that of the medium. If the medium is air, is close to 0 ( 8.85 × 10–12 F/m) of vacuum. The ratio p p/0 is called the specific dielectric constant. Numerical values of f( p) ( p–1)/( p2) are listed in Table 3.1, showing its insensitivity to the specific dielectric constant. The maximum charge acquired by field charging depends on the function g( p) 3 p /( p2), whose numerical values are listed in Table 3.2. The dielectric constant of the powder bed is called the apparent dielectric constant a, which is a function of the packing fraction f of particles as follows: 1. Maxwell model4–5
a
( (
) )
(
)
a 3 2 (1 f ) p 1 0 3 (1 f ) p 1
(3.4a)
2. Rayleigh model6 ⎛ 2 ⎞ p 1 a p 1 3f ⎜ f 1.65 f10 13 ⎟ ⎜⎝ p 1 ⎟⎠ 0 p ( 4 3)
1
(3.4b)
3. Böettcher model5,7,8
(
)
a 1 f p 1 3a p 2
(3.4c)
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4. Lichtnecker model a p f
(3.4d)
a k f p k (1 f )
(3.4e)
5. Lichtneker–Rother model
The apparent dielectric constant a can be measured by the parallel-plate or cylindrical electrodes shown in Figure 3.1. a 0
C1 C0
(3.5)
where C0 is the electric capacitance of the measuring system without powder and C1 is the capacitance with powder between the electrodes. The capacitance C0 is given by the following equations: 1. Parallel-plate electrodes: A C0 0 d
(3.6)
2. Cylindrical electrodes:
C0
2pl 0 ln ( b a )
(3.7)
The apparent dielectric constant depends on the packing fraction, temperature, and humidity, as in the case of the specific resistance. The humidity strongly affects the measurement because the specific dielectric constant of water is about 10 times larger (81 at 18ºC) than that of particles. The dielectric constant of a particle p can be measured by use of the standard liquid listed in Table 3.3. If the capacitance of a cell filled with the reference liquid does not change by adding particles into the cell, the dielectric constant of the particles is equal to that of the liquid. The dielectric constant of the reference liquid is adjustable by blending it with different liquids. The following relation is useful in an estimation of the dielectric constant of a multicomponent particle: 1
* av (1 )av
(3.8)
where n ⎛ 1⎞ av ∑ ni i and ⎜ ⎟ ∑ i ⎝ ⎠ av i
(3.9a,b)
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TABLE 3.3 Dielectric Constants of Standard Liquids Specific Dielectric Constant 2.023
Liquid Cyclohexane Benzene
2.284
Chlorobenzene
5.708
Acetone
21.3
Nitrobenzene
35.7
Specific Charge The particle charge q divided by the particle mass mp is called the specific charge. The specific charge is sometimes approximated by the total charge of powder divided by the mass of the powder. Since the charge on a particle is approximately proportional to its surface area in various electrification processes, the specific charge is almost inversely proportional to the particle size. Therefore, electrostatic phenomena may become predominant for smaller particles. If a fine particle is electrically charged, its motion is usually determined only by the ratio of the electrostatic force and the fluid viscous force. The total charge of powder is measured by an electrically shielded metal vessel, known as the Faraday cage. Powder is put into the Faraday cage and the potential V is measured by a voltmeter. The charge Q is, then, given by the following equation: Q CV
(3.10)
where C is the capacitance of the Faraday cage. The capacitance C is calibrated with a steel ball charged in an electric field supplying sufficient unipolar ions, according to the following equation for the charge of the sphere: q 3p 0 DP2 E
(3.11)
where E (V/m) is the electric field strength. Hence the specific charge Q/w is obtainable for known sample mass w. The specific charge of suspended particles will be obtained through isokinetic sampling. Figure 3.2 shows the Faraday cage available for this purpose. The suspended particles are collected on a filter set in the Faraday cage. If the sampling of particles is impossible, a spherical cage made of a steel mesh may be used. The potential at the center of the spherical cage is given by the equation V
Q 8p 0 R
(3.12)
where Q is the total charge of particles in the cage. The specific charge of a particle is obtainable from the particle trajectory in an electric field. Figure 3.3 depicts parallel-plate electrodes where charged particles are introduced. The charge q can be obtained from the following equation: q
3pmDp ud CC Ex
(3.13)
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FIGURE 3.2 Faraday cage for measuring specific charge of suspended particles.
FIGURE 3.3 particles.
Parallel-plate electrodes for measuring a specific charge of individual
where u is the mean air velocity, d is the distance between upper and lower electrodes, CC is Cunningham’s slip correction factor, and x is the position of deposited particle. The electric field strength E is calculated as follows: E
V d ( t r )
(3.14)
where V is the applied voltage, t is the thickness of the slide glass, and r is the specific dielectric constant of the glass. The particle will suspend in the space when the electric force qE balances the gravitational force mp g. Hence the specific charge can be obtained from the relation q/mp g/E. If the particle sediments in a horizontal electric field, it will deposit at a point (x, y) according to the relation x/y qE/mp g. Hence the distribution of the specific charge will be obtained. 9 The particle motion in a DC electric field superimposed upon an acoustic field or AC electric field can also be utilized in the simultaneous measurement of particle size and charge distributions.10 © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.4 difference.
Measuring system for the contact potential
Contact Potential Difference The contact potential difference of particles is essential for the estimation of the charge polarities and the amount of the charge acquired in various powder handling processes. It relates to the electrostatic powder coating, electrostatic powder imaging, and so on. For pure materials, including metals, there are some reference values available. However, it should be measured for particles, because small amounts of impurities and imperfections and the surface states may change the electrostatic characteristics of the particles. Figure 3.4 shows schematically the measurement apparatus, which is based on the Kelvin method.11–12 It consists of an upper electrode (1), lower electrode with packed powder layer (2), motor (3), digital-power voltage supply, and a digital electrometer for electric current measurement. The upper electrode moves up and down sinusoidally with a constant frequency. The timing to supply voltage and to detect electric current is controlled by the GP-IB (General Purpose Interface Bus) of a computer. The main device was set in an autodesiccator (5) with an electric shield (4) for noise reduction. The upper electrode is gold plated while the lower electrode is nickel plated so as to have enough strength against forcible contact or friction of the packed powder. Electric current generated by the cyclic movement of the upper electrode becomes zero, when the applied voltage V to the upper electrode is equal to the apparent contact potential difference. And the contact potential difference is obtained if the charge of the powder layer is fully diminished. The effective work function of the powder is estimated by the equation f P f Au eVPAu
(3.15)
where, fP is the effective work function of the powder, fAu the work function of Au, and VP/Au is the contact potential difference of powder against Au.
3.3.2
IN NONAQUEOUS SOLUTION
On the application of electric fields to a suspension of polarizable particles dispersed in an insulating liquid, each particle acquires a dipole. In high electric fields, the dipole–dipole interactions affect © 2006 by Taylor & Francis Group, LLC
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the particle arrangement which, in turn, leads to changes in physical properties such as rheology, electric resistance, and light transmittance. The nonlinear effects of electric fields on the physical properties of suspensions will be described in relation to the field-induced structure.
Column Structures in Electric Fields When the dielectric constants of the particles and liquid are substantially different, an electric field generates a dipolar field around the particle. The particles attract one another if they are aligned along the field direction, whereas the particles in the plane perpendicular to the field direction repel one another. Because many polarized particles dispersed in a liquid interact in this way, chains of particles are formed to span the electrode gap. As a simplified approach, most existing models assume that all of the particles align into chains of a single particle width and equal spacing. Although the ideal single-chain model is helpful to understand the basic process of particle arrangement, the chainlike structures are unlikely to be constructed by the separated chains. The induced structure in electric fields must have the configuration which minimizes the dipolar interaction energy. The force between two particles is attractive if the two particles are aligned within 55° of the electric field.13 When placed near a chain, a particle will be repelled by the nearest particles in the chain. The particle will be attracted, however, by the chained particles far above and below it. When many particles are placed near the chain, they will be attracted to it and can form a second chain. Consequently, because of long-range interactions, the polarization forces between particles cause the chains to form columns. The structural changes from noninteracting particles to thick columns spanning the electrode gap are schematically shown in Figure 3.5. The column may be constructed by three types of chains: fully developed chains connecting the electrodes, chains attached at only one end to an electrode, and drifting chains. According to the theory on the column structures by Tao and Sun,14 the dipolar interaction energy per particle strongly depends on the lattice structure. For example, in a simple cubic lattice, a particle in one chain has four particles in other chains as its nearest neighbors, and the centers of particles lie in a plane perpendicular to the field vector. Because the forces between neighboring particles are strongly repulsive, the particles rarely construct a simple cubic lattice. In a quiescent state, the most stable structure is a body-centered tetragonal lattice. Therefore, the suspension in electric fields comprises the columns with the body-centered tetragonal lattice. The fully developed columns spanning the electrodes’ gap can be achieved even at a particle concentration of 1 vol%.
Electrorheology In response to electric fields, the suspensions often exhibit a rapid and reversible transition between viscous liquids and rigid solids. This phenomenon was first reported by Winslow15 and is referred to as the Winslow effect or electrorheological (ER) effect. ER fluids are very attractive as effective vehicles in new devices for controlling the motion of liquids through a narrow channel. Possible devices
E
FIGURE 3.5
Column formation of particles on the application of electric fields.
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include stop valves, brakes, and dampers. Figure 3.6 shows a schematic diagram of an ER clutch. The torque transmitted between two elements can be controlled by the field strength. The columns fully developed between the electrodes can accumulate elastic energy, unless the strain exceeds some critical value. The suspensions in electric fields behave as solids under low stress or strain. When the suspensions are subjected to steady shear at low rates, the rupture and reformation of column structures are constantly repeated to produce a constant stress. In general, the constant stress in the limit of zero shear rate is considered to correspond to the yield stress. The yield stress is defined as a critical stress below which no flow can be observed under the conditions of experimentation and above which the substance is a liquid. The rheological changes of suspensions on the application or removal of electric fields can be characterized as a reversible sol–gel transition. In very high shear fields, the hydrodynamic forces dominate the electric forces. Therefore, the stress linearly increases with shear rate. Figure 3.7 shows the flow curves in electric fields of 0 and 2.0 kV/mm for an ER fluid at a particle concentration of 30 vol%. In the absence of electric fields, the stress is proportional to shear rate, and the slope gives the Newtonian viscosity. In electric fields, significant flow occurs only after the yield stress of about 100 Pa has been exceeded. The plastic viscosity, which is determined as the slope of the flow curve for electrified suspension, is comparable to the Newtonian viscosity without an electric field. It is generally accepted16,17 that the suspensions are converted from Newtonian liquids to Bingham bodies on the application of electric fields. The time scale of transition is on the order of 1 ms. The deformation and rupture process of columns is very complicated. On the assumption that the particles all align into chains of single-particle width and equal spacing, Marshall et al.18 derived a Bingham constitutive equation of ER fluids. In the analysis, all the chains spanning the electrode gap rupture in the center and immediately swing back to re-form with the nearest chain on the opposite electrode. Based on the bulk polarization theory, the yield stress s0 is given by s 0 ∝ fε 0 ε c b2 E 2
FIGURE 3.6
(3.16)
ER clutch.
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( Pa )
200
E= 2 kV mm-1
Stress
100
E=0 0 0
50 Shear
rate
100 ( s-1 )
FIGURE 3.7 Transition from Newtonian flow to Bingham flow for an ER suspension.
b
p c p 2 c
(3.17)
where f is the volume concentration of particles, 0 is the permittivity of free space, c and p are the dielectric constants of medium and particles, respectively, and E is the field strength. This theory is successful in correlating the yield stress of suspensions and material functions of components. However, according to the bulk polarization theory that ignores the surface effects, the only method for formulating excellent ER fluids is a dielectric mismatch. In actual suspensions, the interparticle forces, due to surface polarization, play an essential role in the ER effects. For example, the ER behavior of silica suspensions is governed by the amount of surface silanol groups. The surface modification of particles may have great potential to improve the ER performance of suspensions.19
Electric Resistance Provided that the medium is completely insulating, an electric current is attributed to the particle– medium interface or particle itself. Although the current arises only from the electrical contact of two particles, the chains spanning the electrode gap are primarily responsible for susceptible current as an overall response of static suspensions. Because the chain formation is promoted at high electric fields, the current density rapidly increases with field strength. In addition, ER fluids typically operate at electric fields of 3–5 kV/mm. The local field strength near the contact area can reach values as high as 30–100 kV/mm, where partial breakdown of oil is possible. Hence, this mechanism also leads to nonohmic behavior in which the resistance decreases with increasing field strength. Figure 3.8 shows the current density plotted against the field strength for a 40-vol% silica suspension in a quiescent state under a steady shear rate of 200 s–1.20 The plots for each shear condition lie on a straight line. In shear fields, the rupture and reformation of columns are constantly repeated. Although the current density is decreased by the application of shear fields, the effect of shear rate is not very strong. Although the ionic and electronic migration along the columns spanning the electrodes is the main origin of current in the static suspensions, a different mechanism may be couched in the charge transfer of suspensions under shear flow. © 2006 by Taylor & Francis Group, LLC
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102
101
100 100
Electric field E (k V mm-1) FIGURE 3.8 Current density plotted against the field strength for static and flowing suspensions.
If the suspension is completely dispersed to primary particles in electric fields, a charged particle approaches the electrode of the opposite charge. When the particle touches the electrode surface, a transfer of charge to the particle occurs until the electrode and particle reach the same potential. Then the particle is repelled from the electrode and moves toward the other electrode. Because the process is continuously repeated, the oscillatory motion of charged particles leads to the current. Presumably, the oscillatory motion of drifting chains between electrodes may contribute to conductivity. Because of combined effects of various mechanisms, the relation between the current density and field strength has not been quantitatively established. However, several authors have reported that the current density A can be expressed by A En
(3.18)
From the slopes of straight lines in Figure 3.8, the n values were determined to be 2.5 for the static suspension and 2.7 for the flowing suspension. The n value has been found to be 2.5–3.0, independent of shear rate.
Light Transmittance In suspensions without electric fields, the particles are randomly and uniformly dispersed by Brownian motion. However, the column structures are developed and the optical anisotropy is induced in electric fields. Figure 3.9 shows the light transmittance plotted against the thickness in electric fields of 0 and 2 kV/mm for an ER suspension at a particle concentration of 7 vol%. The light emitted from a lamp is introduced parallel to the electric field. The transmittance in a zero electric field rapidly decreases with thickness. The plots are approximated by a straight line, the slope of which gives the absorption coefficient of suspension. The exponential decay of transmittance with thickness shows that in the absence of electric fields, the particles are randomly dispersed in the medium. In electric fields, the transmittance is almost constant irrespective of thickness. The most © 2006 by Taylor & Francis Group, LLC
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(–)
100
Transmittance
T
E = 2 kV mm-1
10-1 E=0
10-2 0
0.5
Thickness
1.0
H
(mm)
FIGURE 3.9 Thickness dependence of light transmittance for an ER suspension in electric fields of 0 and 2 kV/mm.
significant feature is that the transmittance of suspension is drastically increased in electric fields. The clearing effect is markedly enhanced as the cell thickness is increased. If the suspension does not absorb the irradiation energy, the transmittance must be equal to unity and independent of thickness. However, the results indicate that the energy is absorbed in the suspension, although the light intensity is considered to be relatively homogeneous in the direction parallel to the field vector. The light may be absorbed in a very thin layer near the surface exposed to the light source. In electric fields, the dilute suspension consists of a collection of discrete columns; hence, the induced structures are strongly anisotropic. The plane perpendicular to the optical path is occupied by many discrete columns and a continuous medium. In ordinary suspensions, the particles are highly absorbing. Because of high optical density, the columns do not transmit light, and the vacant area without particles is regarded as transparent. Considering that the irradiation energy is absorbed mainly by the particles and the fraction absorbed by the medium is negligible, the cross section of optical path comprises dark (column) and transparent (medium) regions. Because the fraction of vacant area increases with field strength, the transmittance is markedly increased and is not strongly affected by the cell thickness. The light absorption in electric fields is a purely geometrical effect, which depends only on the cross section of columns.21 The electrooptical effect of suspensions, due to the column formation, can provide the basis for passive display devices.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
White, H. J., in Industrial Electrostatic Precipitation, Addison-Wesley, Reading, Mass., 1963, p. 319. Masuda, S. and Mizuno, A., Proc. Inst. Electrostat. Jpn., 2, 59, 1978. Masuda, S., Denki Gakkaishi, 80, 1790, 1960. Maxwell, J. C., Electricity and Magnetism, Vol. 1, Clarendon, Oxford, 1892. Louge, M. and Opie, M., Powder Technol., 62, 85, 1990. Rayleigh, W. R., Philos. Mag. Ser. 5, 34, 481, 1892. Böettcher, C. J. F., Rec. Trav. Chim., 64, 47, 1945.
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281
Yadav, A. S. and Parshad, R., J. Phys. D Appl. Phys., 5, 1469, 1972. Masuda, H., Gotoh, K., and Orita, N., J. Aerosol Res. Jpn., 8, 325–332, 1993. Mazumder, M. K., KONA, 11, 105–118, 1993. Thomson, W. (later Lord Kelvin), Phil. Mag., 46, 1898. Masuda, H., Itakura, T., Gotoh, K., Takahashi, T., and Teshima, T., Adv. Powder Technol., 6, 295–303, 1995. Halsey, T. C., Science, 258, 761, 1992. Tao, R. and Sun, J. M., Phys. Rev. Lett., 67, 398, 1991. Winslow, W. M., J. Appl. Phys., 20, 1137, 1949. Block, H. and Kelly, J. P., J. Phys. D Appl. Phys., 21, 1661, 1988. Jorda, T. C. and Shaw, M. T., IEEE Trans. Electr. Insul., E1–24, 849, 1989. Marshal, L., Zukoski, C. F., and Goodwin, J. W., J. Chem. Soc. Faraday Trans. 1, 85, 2785, 1989. Otsubo, Y. and Edamura, K., J. Colloid Interface Sci., 168, 230, 1994. Otsubo, Y., Sekine, M., and Katayama, S., J. Rheol., 36, 479, 1992. Otsubo, Y., Edamura, K., and Akashi, K., J. Colloid Interface Sci., 177, 250, 1996.
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3.4
Magnetic Properties Toyohisa Fujita University of Tokyo, Tokyo, Japan
3.4.1
MAGNETIC FORCE ON A PARTICLE1
The magnetic dipole force Fm acting on a paramagnetic particle in vacuum is given by Fm ⫽ ⵜ∫ v (B · H) dV
(4.1)
where B is the magnetic polarization of a particle, V is the volume of a particle, and H is the magnetic field strength. ⵜ is the operator of the gradient and acting on a scalar w that can be written as ⵜ w ⫽ id w /dx ⫹ jd w /dy ⫹ kd w /dz
(4.2)
where i, j, k are the unit vectors in the directions x, y, z, respectively. If the particle is sufficiently small it can be reduced to a point dipole moment m ⫽ BV. The force on such a point dipole is F m ⫽ (m · ⵜ)H
(4.3)
Permeability of a spherical paramagnetic or diamagnetic particle is given as ⫽ (1 ⫹ )0
(4.4)
where is the volume magnetic susceptibility and 0 is the permeability of free space numerically equal to 4p · 10–7 H/m. The magnetic polarization of the particle is given by B ⫽ 0 H/(1 ⫹ /3)
(4.5)
Combining Equation 4.5 and Equation 4.3, the force on a small spherical weakly magnetic particle placed in the external magnetic field can be written as Fm ⫽ [m0kV/(1 ⫹ k/3)](Hⵜ)H
(4.6)
or in a simplified form (assuming that k << 1), Fm ⫽ (1/2)m0kV ⵜ(H 2)
(4.7)
If a paramagnetic particle (volume magnetic susceptibility: k p) is immersed in the fluid (volume magnetic susceptibility:kf), the magnetic force per unit volume acting on a particle is given by Equation 4.7 (using k ⫽ kp – kf). For practical calculations it is sometimes advantageous to replace the magnetic field strength by the magnetic induction B. Then, Equation 4.7 reads as follows: Fm ⫽ (k/m0)VBⵜB
(4.8) 283
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Here B is considered as the external magnetic induction, and ⵜB is the gradient of the magnetic induction. Thus, in the direction of x, the magnetic force Fmx can be written by the following equation (where B ⫽ moH and magnetization kHx ⫽ Ix), Fmx ⫽ m0Vk(Hx · dHx/dx) ⫽ m0VIx · dHx/dx
(4.9)
Magnetic force is proportional to the product of the external magnetic field and the field gradient and has the direction of the gradient. In a homogeneous magnetic field, in which ⵜB ⫽ 0 or d Hx/dx ⫽ 0, the force to change the position of a particle is zero. Along the axis of symmetry, and for values of the external magnetic field Hx0 lower than the bulk saturation value Hs of the ferromagnetic rod filament, the magnetic field strength is given by the expression Hx ⫽ Hxo(1 ⫹ a2/r2)
(4.10)
where a is the rod radius and r is the radius from the center of the rod, as shown in Figure 4.1. For Hxo > Hs, Hx is given by Hx ⫽ Hxo ⫹ Hs(a2/r2)
(4.11)
The magnetic field gradient for Hxo < Hs along the x axis of the particle, when the magnetization of the particles is small, is given as dHx/dx ⫽ –2Hxo(a2/r3)
(4.12)
If the Equation 4.10, Equation 4.12, and Equation 4.6 are combined, the approximate expression for the magnetic force on a pointlike spherical weakly magnetic particle is given as Fm ⫽ –(8/3)pm0kb3 Hxo(1 ⫹ a2/r2)Hxo(a2/r3)
(4.13)
where b is the particle radius. Equation 4.13 can be rewritten for a matched system a ⫽ 3b as Fm ⫽ –(75/128)pm0kb2Hxo2
(4.14)
Ferromagnetic wire matrix Particle
a r
b
Appliect magnctic field Ho X
Hxo
FIGURE 4.1 Cross section of spherical particle of radius b, attached to ferromagnetic wire rod of radius a (magnetized uniform magnetic field Hxo). © 2006 by Taylor & Francis Group, LLC
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TABLE 4.1
Characteristics of Temporary Magnetic Alloys
Alloy Silicon iron
Constituentsa 4Si
Initial Permeability 500
Maximum Permeability 7,000
Coercive Force (Oe) 0.5
Saturated Magnetism (G) 19,700
Curie Point (ⴗC) 690
Density (g/cm) 7.65
45Permalloy (V)b
45Ni
3,000–5,000
50,000–70,000
0.06–0.12
16,000
—
8.17
78Permalloy (V)
78.5Ni
30,000–70,000
80,000–200,000
0.01–0.02
10,800
200
8.60
4Mo, 79Ni
20,000
100,000
0.05
8,700
460
8.72
4–79Permalloy Supermalloy Mumetal
5Mo, 79Ni
70,000–150,000
400,000–700,000
0.01–0.05
7,500–8,100
400
8.77
5Cu,2Cr,77Ni
20,000
100,000
0.05
6,500
—
8.58
50Co
800
5,000
2.0
24,500
980
8.3
1,8V,49Co
800
4,500
0.8
24,000
980
8.2
Permender VanadiumPermender (V) Alperm
16A1
3,000
55,000
0.04
8,000
400
6.5
Sendust
5A1,10Si
30,000
120,000
0.05
10,000
500
7.0
a b
Residual constituent is Fe. (V) ⫽ vacuum dissolution.
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Several competing forces act on the particles placed in a magnetic separator. These are the force of gravity, the inertial force, the hydrodynamic drag, and surface and interparticle forces. The force of gravity Fg can be written as Fg ⫽ rVg
(4.15)
where r is the density of the particle, and g is the acceleration due to gravity. The hydrodynamic drag Fd on a particle of small radius is given by Stoke’s law, Fd ⫽ 6phbvp
(4.16)
where h is the dynamic viscosity of the fluid, b is the particle radius, and vp is the relative velocity of the particle with respect to the fluid. Magnetic particles will be separated from “ nonmagnetic” (i.e., more magnetic particles from less magnetic particles), if the following conditions are met: mag Fmag and F mnonmag ⬍ ∑F cnonmag m ⬎∑ F c
(4.17)
where Fc is a competing force, while Fmag and F nonmag are forces acting on magnetic and nonmagnetic particles respectively. Although these conditions are clearly defined, a complication arises because the relative significance of the forces is determined mainly by the particle size. It can be seen from Equation 4.8, Equation 4.15, and Equation 4.16 that, since Fm⬀ b3 or b2, the competing forces have the following dependence on particles size: Fd ⬀ b1 and Fg ⬀ b3
(4.18)
In dry magnetic separation, where Fd is usually negligible, the particle size does not affect the efficiency of separation significantly because of the same particle size dependence of the magnetic force and of the force of gravity. On the other hand, when the hydrodynamic drag is important, selectivity of the separation will be influenced by particle size distribution. With decreasing particle size the relative importance of the hydrodynamic drag increases in comparison to the magnetic force. The nonselective nature of the magnetic force is illustrated in Table 4.1. It can be seen that the magnetic force exerted on a coarse weakly magnetic particle is similar to the one exerted on a smaller and more strongly magnetic particle. Both particles will appear in the same product of separation unless the competing forces acting on particles of different sizes is different.
3.4.2 FERROMAGNETIC PROPERTIES OF A SMALL PARTICLE2 Ferromagnetic material is an ensemble of small magnetic domains within which the magnetic moment is unidirectional. The boundary between these domains is called the domain wall. When a diameter d of spherical particle contains width d of several domains, the domain wall energy Uw is given by the following formula: Uw ⫽ s(pr2)2r/d
(4.19)
where s is a wall energy. On the other hand, as a magnetic static energy, Um is two times larger than the single domain’s one, Um is given by Um ⫽ (Is2/6mο)(4pr3/3)(d/2r)
(4.20)
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FIGURE 4.2 Variation of magnetic properties with particle size [From Mitsui, S., in Jisei Buturi no Shinpo, S. Chikazumi, Ed., Agne, Tokyo, 1964, p. 242. With permission.]
FIGURE 4.3 Dependence of coercive force of magnetite on particle size. [From Gottschalk. V H., Physics 6, 127, 1935. With permission.]
where Is is saturation magnetization of ferromagnetic material. Uw in Equation 4.19 is proportional to area, while Um in Equation 4.20 is proportional to volume. When the size of a ferromagnetic material decreases and reaches the critical small size, the wall energy becomes larger than the magnetic static energy. At this critical size the magnetic domain can not be divided. When the size of ferromagnetic particles is reduced sufficiently and reaches the critical small size, the domain wall disappears (i.e., the particles have a single domain structure), and the susceptibility and coercive force of particles vary as shown schematically in Figure 4.4. There is a particle size under which particles are superparamagnetic, and the coercive force of particles is negligibly small by kT where k is Boltzmann constant and T is absolute temperature. © 2006 by Taylor & Francis Group, LLC
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Yu [o]
3
2
r Fo2O2
Fe2O2
1
0.2
0.4
0.5
0.8
1.0
X-5
FIGURE 4.4 Critical particle size for single domain structure; a and b are the minor and major axes of the particle, respectively. [From Morrish, A. H. and Yu, S. P., J. Appl. Phys., 26, 1049, 1955. With permission.]
A typical dependence of the coercive force of magnetite particles on the particle size is shown in Figure 4.3. Figure 4.4 shows the variation of critical particle size under which Fe3O4 and ␥-Fe2O3 particles have a single domain structure with the aspect ratio. Magnetic fluid is one example of an apparent superparamagnetic behavior described by the same Langevin’s law,6 where the Brownian or Neel mechanisms lead.
3.4.3
MAGNETISM OF VARIOUS MATERIALS
The development of permanent magnetic materials and the improvement on their magnetic properties have been particularly studied during the last 30 years. Figure 4.5 illustrates the history of improvement in the energy product of permanent magnets.7 Probably the most significant innovation step was made in the late 1970s when rare-earth magnets became available. Ferromagnetic materials are classified into soft and hard magnetic materials. Hard magnetic materials are used as permanent magnets, which have large coercive force. Typical properties of hard magnetic materials of artificial alloys, ferrites, and bonded in polymer are listed in Table 4.2.2 On the other hand, properties of soft magnetic materials are listed in Tables 4.3–4.5.2 Soft iron as well as DC relays, plungers, pole pieces, solenoids, and brakes for intermittent use are used, since the magnetic requirements are low and the cost must also be low. The high permeability, low coercive force, and high saturation magnetization of soft magnetic materials are used in certain applications. In the past, industry employed ferrites at low power levels; however, nowadays higher power levels of ferrites have been developed. Relative susceptibility of minerals is listed in Table 4.6.2 Using the magnetization kH ⫽ I, permeability m is given in the following formula: B ⫽ (k ⫹ mο)H ⫽ m⌯
(4.21)
Here the relative susceptibility means k/mο (SI unit) that is 4pr times of susceptibility in CGS units (r, g/cm3). © 2006 by Taylor & Francis Group, LLC
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FIGURE 4.5 Progress in development of permanent magnetic materials. [Svoboda, J. and Fujita, T., in Proceedings of the Twenty-second International Mineral Processing Congress (IMPC), Cape Town, 2003, pp. 261–269. With permission.]
TABLE 4.2
Characteristics of Permanent Magnetic Alloys Coercive Force(Oe) 200–250
Residual Magnetic Flux Density (G) 9,000–10,000
(BH)max ⴛ 10ⴚ6 (G·Oe) 1.0
Density (g/cm3) 8.2
Alloy 36Co steel
Constitutentsa 0.7C,36Co
Alnico 2
12.5Co
470–590
7,000–8,000
1.4–1.6
7.1
Alnico 5
24Co,14Ni
660–620
12,800–13,400
4.5–5.3
7.3
Alnico 5(DG)
24Co,14Ni
640
13,100
5.5
7.3
New KS steel
27Co,18Ni
950–1,050
5,500–6,300
1.6–2.1
7.4
Vicalloy 2
52Co,14V
510
10,000
3.5
—
Alnico 2(S)
2.5Co,17Ni,10Al,6Cu
540
6,900
1.4
7
Alnico 5(S)
24Co,15Ni,8Al,3Cu
575
10,000
3.5
6.6
a
Residual constituent is Fe.
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TABLE 4.3
Ferrite Mn-Zn
Ni-Zn
Characteristics of Temporal Magnetic Ferrite
F
Initial Permeability 1000
Saturated Magnetism (G) 3200
Residual Magnetic Flux Density (G) 1200
F4
1300
4000
1000
0.3
F5
2000
3900
900
0.3
H
2100
4000
1250
0.2
J
800
4200
2000
0.4
B2
1200
4000
2500
0.3
B3
1400
4200
1700
0.3
B5
1100
4500
1500
0.3
N3b
60
4000
2300
4
N3d
100
3000
1500
6
TABLE 4.4
Characteristics of Barium Ferrite Magnet
Barium Ferrite Magnet
Residual Magnetic Flux Density
Coercive Force (Oe)
(BH)max (G·Oe) ⴛ 10ⴚ6
Q-1
N.O.
2300–2500
1800–2000
1.0–1.2
Q-6
O
4100–4400
1600–1900
More than 3.6
Q-7
O
3300–3700
2700–3000
More than 2.6
Q-8
O
3800–4100
2100–2500
More than 3.4
Q-10
O
3400
1900
TABLE 4.5
Coercive Force (Oe) 0.3
2.4
Characteristics of Rare Cobalt Magnet Magnetic Induction (kG) 12.5
Anisotropic Magnetic Field (kOe) 107
Curie Point (ⴗC) 920
(BHmax) (MG·Oe) 39
Sm2(Co3.8Fe0.2)17
14.0
83
890
49
Ce(CoFe)17
11.5
15
780
—
Pr(CoFe)17
14.0
20
880
—
Y(CoFe)17
12.7
15–18
950
—
SmCo5
11.6
290
720
31
Magnet Sm2Co17
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TABLE 4.6
Susceptibility of Minerals
Chemical Composition (Fe,Zn,Mn)O(FeMn)2O3
Specific Susceptibility (ⴛ10ⴚ6) 455–640
Siderite
FeCO2
84.2, 142.6
Zircon
ZrSiO4
⫺0.17, ⫹0.732
Minerals Franklinite
Corundum
Al2O3
⫺0.34
Quartz
SiO2
⫺0.461, ⫺0.466
Rutile
TiO2
1.96, 2.09
Pyrite
FeS2
0.98
Sphalerite
ZnS
⫺0.264
Dolomite
CaMgC2O8
0.787, 1.20
Apatite
Ca5Cl(PO4)3
⫺2.64
CuFeS2
0.85
MgAl2O4
0.62
Galena
PbS
⫺0.350
Rock salt
NaCl
⫺0.50
Chalcopyrite Spinel
Tourmaline Graphite
H9Al2(BOH)2Si4O19
1.12, 0.748
C
⫺2.2, ⫺14.2
CaF2
⫺0.285
Aragonite
CaCO3
⫺0.392, ⫺0.444
Calcite
CaCO3
⫺0.363, ⫺0.405
Fluorite
Ruby
Al2O3
0.47
Topaz
(AlF)2SiO4
⫺0.42
Be3Al2(SiO3)6
0.826, 0.386
Epidote
Beryl
Hca2(Al, Fe)3Si3O13
23.8, 23.9
Hornblende
(Ca, Mg, Fe)8Si9O26
24.0, 18.0
CaMgSiO6
26.6, 22.7
Augite
CaMg(SiO3)7
8.8
Columbite
—
33.6–43.2
Monazite
—
18.1
Pitchblende
—
13.0
Gadolinite
—
5.83
Manganite
Diopside
Mn2O3·H2O
490
Cobaltite
CoAsS
5.8
Feldspar
—
1.6
Limestone
—
6
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REFERENCES 1. Svoboda, J., Magnetic Methods for the Treatment of Minerals, Elsevier, New York, 1987, pp. 3–9. 2. Yashima, S. and Fujita, T., J. Soc. Powder Technol. Jpn., 28, 257–266, 1991; Ushiki, K., Powder Technology Handbook, 2nd Ed. Macel Dekkar, New York, 1977. 3. Gottschalk. V H., Physics 6, 127, 1935. 4. Mitsui, S., in Jisei Buturi no Shinpo, S. Chikazumi, Ed., Agne, Tokyo, 1964, p. 242. 5. Morrish, A. H. and Yu, S. P., J. Appl. Phys., 26, 1049, 1955. 6. Rosensweig, R. E., Ferohydrodynamics, Cambridge University Press, London, 1985, pp. 61–62. 7. Svoboda, J. and Fujita, T., in Proceedings of the Twenty-second International Mineral Processing Congress (IMPC), Cape Town, 2003, pp. 261–269.
Acknowledgment The author gratefully appreciates Dr. Svoboda’s useful discussions.
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3.5
Packing Properties Michitaka Suzuki University of Hyogo, Himeji, Hyogo, Japan
It is well known that the packing of particulates is one of the most fundamental and important powder-handling processes, and various properties of the particulate assembly are basically determined by the geometrical arrangement of individual particles. Void fraction or voidage is a most popular and simple expression of packing characteristics of a powder bed. In an actual powder system, void fraction is defined by 1
rb M 1 rp r pV
(5.1)
where rb, rp, M, and V are apparent density, particle density, and mass and volume of powder bed, respectively. Particle volume fraction or packing fraction defined as f 1 is frequently used also to express the packing structure in place of the void fraction. In actual powder, the void fraction is changing with the particle size.1 As shown in Figure 5.1, the void fraction shows the constant value c over certain critical particle size Dpc, and below this size, the void fraction increases with decrease in the particle size. Roller2 proposed the following equation as a relationship between the void fraction and particle size. c
(Dp Dpc )
1 ⎛1 ⎞ ⎛ Dp ⎞ 1 ⎜ 1⎟ ⎜ ⎟ ⎝ c ⎠ ⎝ Dpc ⎠
n
(Dp Dpc )
(5.2a)
(5.2b)
Index n and the critical particle size Dpc change with environmental conditions, packing method, and types of powder. Generally n takes the value between 0 and 1.
3.5.1 PACKING OF EQUAL SPHERES Many researchers have investigated the assembly of equal spheres because of its simplicity and its convenience in theoretical work. Regular packing is the easiest to use to describe internal structure as a set of unit cells. There are six typical unit cells, as shown in Figure 5.2. Graton and Fraser3 characterized the internal structure of the unit cells, consisting of a few primary spheres in which geometrical features such as the angle and distance between neighboring sphere centers, the void fraction, and the coordination number (or the number of contact points on a particle surface) are obtained. Closest or rhombohedral packing is used frequently as a reference system. In actual packing of particles the void fraction and coordination number vary widely and continuously, but in regular packings there exist only discrete values: void fraction 0.2594 (rhombohedral), 0.3119 (tetragonal–sphenoidal), 0.3954 (orthorhombic), and 0.4764 (cubic) and coordination number Nc 12, 10, 8, and 6, respectively. Heesh and Laves4 proposed a regular packing of equal spheres supported by wire frames as a model structure of particle assembly 293 © 2006 by Taylor & Francis Group, LLC
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FIGURE 5.1 Effect of particle size on void fraction of monosized powder bed. [From Suzuki, M., Sato, H., Hasegawa, M., and Hirota, M., Powder Technol., 118, 53–57, 2001. With permission.]
FIGURE 5.2
Regular packing structures of equal spheres.
with high void fraction. But this type of spherical structure scarcely exists in actual particulate assemblies and has fixed values of void fraction and coordination number. Smith et al.5 proposed a hybrid model obtained by mixing the simple cubic and rhombohedral unit cells, and the model equation (7) in Table 5.1 shows the continuous relation between the void fraction and the coordination number Nc. Shinohara et al.6 and Shinohara and Tanaka7 proposed another hybrid model, which is a random mixture of the simple cubic and rhombohedral unit cells and © 2006 by Taylor & Francis Group, LLC
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TABLE 5.1 Expressions of the Relationship between Void Fraction and Average Coordination Number Expression
Equation No. in Fig. 5.3
N c 3.1
(1) Rumpf 1958
N c 2e2.4(1)
(2) Meissner et al. 1964
1.072 0.1193 N c 0.0043 N c
2
N c 22.47 33.39 ( 0.5)
(
)
(3) Ridgeway et al. 1967 (4) Haughey and Beveridge 1969
2 N c ⎡⎣8p 0.7273 3 ⎤⎦ (1 )
(5) Nagao 1978
N c 1.611.48 ( 0.82)
(6) Nakagaki and Sunada 1968
N c 4.28 103 17.3 2.00 (0.82 ) N c 26.49 10.73 (1 ) ( 0.595)
(7) Smith et al. 1929
N c 20.7 (1 ) p 4.35 (0.3 0.53)
(8) Gotoh 1978
N c 36 (1 ) (0.53 ) 2.812 (1 )
1 / 3
Nc
( b x )
2
{1 (bx) } 2
(9) Suzuki et al. 1980
b / x 7.318 102 2.193 3.357 2 3.1943 N c (32 13)( 7 8 )
(10) Ouchiyama et al. 1980
can explain a wide and continuous change in the void fraction and the coordination number. These are useful models for close-packing structures with local order. Gotoh8,9 and Iwata and Homma10 made similar studies of random packing in terms of the combination of regular packings. Recently, Bargiel and Tory11 investigated the transition from dense random packing and measured the disorder of ultradense irregular packing of equal spheres. Many investigations have been made of the coordination number in a random assembly of equal spheres; these include studies by Smith et al.,5 Rumpf,12 Bernal and Mason,13 Meissner et al.,14 Wade,15 Ridgway and Tarbuck,16 Arakawa,17 Haughey and Beveridge,18 Bernal et al.,19 and Nagao,20 and a number of empirical relations have been proposed. Nakagaki and a Sunada,21,22 Bennett,23 and Tory et al.24 made computer simulations of three-dimensional random packings from which they obtained the relationship between the average coordination number Nc and the void fraction. Figure 5.3 shows a comparison between the results calculated from the equations in Table 5.1 and practical experiments and computer experiments. Equation 5.1 through Equation 5.5 (the dashed curves) express the empirical relations, whereas Equation 5.6 and Equation 5.7 (the dashed curves) are obtained from the random mixture model of regular packings. Gotoh and Finney,25 Suzuki et al.,26 and Ouchiyama and Tanaka27 reported the theoretical relations in Equation 5.8 through Equation 5.10 (the solid curves). Suzuki and Oshima28 compared the calculated results obtained by Equation 5.1 through Equation 5.10 and the four kinds of computer simulated results, showing that Nakagaki et al.’s empirical formula agreed well with three simulation results over a wide range © 2006 by Taylor & Francis Group, LLC
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FIGURE 5.3 Relationship between void fraction and average coordination number for monosized spheres.
of the void fraction. Some of the above simulation results corresponded well to Suzuki et al.’s and Gotoh’s model formula and Nagao’s empirical formula in a certain void fraction. Suzuki et al.29 investigated the effect of size distribution on the relation using the model calculation and computer simulation results. Figure 5.4 shows the relation between the void fraction and the average coordination number of the random-packed bed with log-uniform size distribution. Based on the results, the coordination number is not so affected with the size distribution of particles, but the void fraction becomes smaller for wider distribution of particle size. It means that the value of the coordination number, estimated by a conventional equation for equal spheres from void fraction data, is overestimated for randomly packed beds of multicomponent spheres with wider particle size distributions. Knowledge of the spatial structure of particles in a random assembly such as the radial distribution function, the first-layer neighbors, and the average nearest-neighbor spacing is important in dealing with the deformation of a powder bed, the electrostatic and liquid bridge forces between particles, radiative and convective heat transfer in particle systems, and so on. The radial distribution function g(R) is the local number density of particles, normalized by its bulk-mean value, at a distance R from a central reference sphere. The analytical result in the Percus–Yevick approximation is depicted in Figure 5.5, in which the damped oscillation implies the short-range order in the particle arrangement. The number density exhibits a shell-like distribution in the polar coordinate about the central sphere, and the spheres in the innermost shell are regarded as first-layer neighbors. Gotoh 30 © 2006 by Taylor & Francis Group, LLC
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FIGURE 5.4 Effect of size distribution on the relation between void fraction and average coordination number for monosized spheres.
FIGURE 5.5 Radial distribution function; F(r), bulk-mean particle volume fraction. [From Gotoh, K., J. Soc. Powder Technol. Jpn., 15, 726–733, 1980. With permission.] © 2006 by Taylor & Francis Group, LLC
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and Suzuki et al.31 discussed the structure of the first-layer neighbors. Chandrasekhar,32 Bansal and Ardel,33 and Gotoh et al.34 discussed the average center-to-center distance between a central sphere and its nearest neighbor. Figure 5.6 depicts the relationship between the bulk-mean particle volume fraction f (1 – ) and the distance of the first-layer neighbors from a central reference sphere in units of the sphere diameter. R1, Rm, and Rn in Figure 5.6 express the outer edge and average positions of the first layer and the average nearest-neighbor distance, respectively. The number of first-layer neighbors is calculated from the radial distribution function as follows30: Z
(∫
R1
1
)
4pR 2 dR N v g( R)
(5.3)
in which Nv 6f/p 6(1–)/p, Nv is the number density of the particles, and is the bulk-mean particle volume fraction. The unit cell of the particle assembly can be expressed by a Voronoi polyhedron as illustrated in Figure 5.7. A plane bisects each line joining the central sphere to its neighbors; the innermost volume enclosed by these planes is the Voronoi polyhedron. Tanemura35 discussed random tessellation using the Voronoi polyhedron. In random close packing the Voronoi polyhedron has 14.25 faces on average.
2
R (–)
R1
1.5 Rm
Rn
1 0
0.5
FIGURE 5.6 Relationship between packing fractions; F and distance of the first-layer neighbors from a central sphere in units of the sphere diameter. R1, outer edge; Rm, average position; Rn, nearest neighbor. [From Gotoh, K., J. Soc. Powder Technol. Jpn., 16, 709–713, 1979. With permission.] © 2006 by Taylor & Francis Group, LLC
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Fractal dimensions of particle beds are also used to describe the particulate assemblies. Witten and Samder36 and Tohno and Takahashi37 expressed the particle arrangement in an aerosol agglomerate by the fractal dimension. Kamiya et al.38 used it to characterize the packing structure of an aggregate in a particle bed. Suzuki and Oshima39 obtained a fractal dimension of a packed particle system by the scaling method. The fractal dimension enables us quantitatively to distinguish the difference in packing structures that cannot be expressed by the relation between coordination number and void fraction.
3.5.2 PACKING OF MULTISIZED PARTICLES Actual powders have wide distributions in size; hence, a packed bed can be regarded as a mixture of different-sized particles. Horsfield, 40 White and Walton, 41 and Hudson42 have discussed the regular packings of multisized spheres, in which the interstices of an array of spheres are filled with smaller spheres. The void fractions of the Horsfield packings are listed in Table 5.2. Although the
TABLE 5.2 Calculated Results of the Horsfield Model for Multisized Regular Packing of Spheres Fraction
Dp1/Dp2
Relative No. of Spheres
Void
1
1
0.2595
Second spheres
0.414
1
0.207
Third spheres
0.225
2
0.190
Fourth spheres
0.177
8
0.158
Fifth spheres
0.116
8
0.149
Filler
Fine
Infinite number
0.039
First sphere
FIGURE 5.7 Two-dimensional illustration of Voronoi polyhedron. © 2006 by Taylor & Francis Group, LLC
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packing densities are expected to be very high theoretically, in practice it is difficult to construct these packings. Generally speaking, powder has a random structure, and regular packing hardly exists in practice. Experimental data of the void fraction and coordination number in a multisized particle bed have been reported by Furnas,43 Sohn and Moreland,44 Arakawa and Nishino,45 and Suzuki and Oshima.46 Powell,47 Dinger and Funk,48 Suzuki and Oshima,49 and Ito et al.50 devised a computer simulation for random packing of multisized spheres and obtained the simulation data. Many different models have been conceived for the random packing of multisized particles, by Furnas,51 Westman,52 Tokumitsu,53 Kawamura et al.,54 Abe et al.,55 Ouchiyama and Tanaka,27,56,57 Suzuki and Oshima,46 and Suzuki et al.,58,59 and the void or packing fraction is estimated. For example, Furnas estimated the void fraction of a binary powder mixture as follows: 1
1 1 Sv1
(5.4a)
1
1 2 1 Sv1 2
(5.4b)
where 1 is the void fraction of a coarse monosized powder bed, 2 is the void fraction of the fines, respectively, and Sv1 is the fractional volume of coarse powder 1 in the binary mixture. Equation 5.3a means that the interstices between coarse particles are filled with fine particles, and Equation 5.3b means that the coarse particles are dispersed in the fine powder bed. If the two calculated voidages are not equal to each other, the higher voidage should be chosen. The typical results calculated from these equations are shown in Figure 5.8 for 1 2 0.35, 0.4, 0.45, and 0.5. The voidage of the binary mixture becomes much smaller than that of monosized powder bed and exhibits the minimum value, because the finer powder fills the interstices of the coarse powder. Most investigators, except Ouchiyama and Suzuki et al., treated the packing as a continuous medium and did not consider the relation between the void or packing fraction and the coordination number. Furthermore, Ouchiyama and Tanaka’s model assumes an equal void fraction for the packing of each size of sphere, but this is not the case in the actual powders. Cross et al. 60 modified the model for expressing the matrix structure of composite materials. Here Suzuki’s model for estimating the void fraction of a packed bed of m different-sized spheres is explained. There are m m basic types of contact between a reference sphere and others (see Figure 5.9). The void fractions about the central spheres are denoted by (1,1),(1,2),…….,(m,m), which are defined by Equation 5.7 below. The bulk-mean void fraction of the bed becomes m
∑ Sv j b j j , 1
m
j ∑ Sa k ( j , k )
(5.5)
1
where Svj is the fractional volume of sphere j, 2 is the partial void fraction about sphere j, Sak is the fractional area of sphere k about sphere j, and bj is a constant that is calculated from the measured void fraction of the bed composed of sphere j by putting Sak 1 and Svj 1 in Equation 5.4: bj
j ( j , j )
(5.6)
The next step is to derive the partial void fraction (j,k). N(j,k) of sphere k of diameter Dpk are in direct contract with a reference sphere j of diameter Dpj, as shown in Figure 5.10. A spherical region of radius — OD (shown by the dashed curve) is considered, where O is the center of the reference sphere j and D is
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FIGURE 5.8
Calculated results from Furnas’s equations.
FIGURE 5.9 Fundamental combinations of contact in multicomponent random mixture of spheres. [From Suzuki, M., Yagi, A., Watanabe, T., and Oshima, T., Int. Chem. Eng., 26, 491–498, 1986. With permission.]
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Contact particle k
V1 C E
Dpk
D
B
V2
A
Dpj
O Reference particle j
Hypothetical sphere FIGURE 5.10 Model of a reference particle j in direct contact with particle k. [From Suzuki, M. and Oshima, T., Powder Technol., 43, 147–153, 1985. With permission.]
the contact point of the two adjacent k spheres. The void fraction (j,k) of the spherical region can be determined as follows:
( j , k ) 1
(p 6) Dp 3j N ( j , k ) (V1 V2 ) Vs
(5.7)
where V1 is the partial volume of the spherical region, V2 is that of the contact particle (see the — cross-hatched area in Figure 5.9), and Vs the volume of the spherical region of radius OD. The coordination number N(j,k) can be calculated from Suzuki and Oshima’s model61 for random packing of different-sized spheres. A packed bed of two sizes of glass beads is formed for the particle size ratio Dp1 /Dp2 of 2, 2,83, 3.99, and 8.02. Plotted in Figure 5.11 are the experiments, and the curves depict the results calculated by Equation 5.5. Figure 5.12 shows the relationship between the standard deviation(lnsg) of log-normal size distribution and the void fraction . The void fraction becomes smaller for wider particle size distributions. Figure 5.13 shows the relationship between the Fuller constant q of the Gaudin–Schuhmann (Andreasen) size distribution and the void fraction . The minimum void fraction is obtainable for © 2006 by Taylor & Francis Group, LLC
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FIGURE 5.11 Relationship between fractional volume of fine spheres and void fraction of two-component mixture. [From Suzuki, M., Yagi, A., Watanabe, T., and Oshima, T., Int. Chem. Eng., 26, 491–498, 1986. With permission.]
the Fuller constant of 0.5–0.8 in this case. This is because the interstices among particles decrease for q 0.5 to 0.8. It was also reported by Kawamura et al.54 experimentally that the void fraction could become minimum at q 0.6, and Dinger and Funk62 showed that optimum packing occurs for the distributions with q 0.37. Suzuki and Oshima49 reported that the q value at minimum void fraction changes with the adhesive property of fine powder and the width of size distribution. Because the calculated results are in good agreement with the experimental and simulation data, the present model may be used in the practical applications.
Notation b Dp1, Dp2
constant in Equation (h) in Table 5.1 (m) diameter of coarse particle 1 or fine particle 2 in a binary mixture (m)
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FIGURE 5.12 Relationship between standard deviation of lognormal size distribution and void. [From Suzuki, M., Oshima, T., Ichiba, H., and Hasegawa, I., KONA Powder Sci. Technol. Jpn., 4, 4–12, 1986. With permission.]
Dpc Dpj, Dpk g(R) M m n– N(j, k) Nc Nc Nv q R Rl Rm Rn Sak Sv1 Svk V
critical particle size of Roller’s equation (m) diameter of particle j or k in a multicomponent mixture (m) radial distribution function mass of powder in a packed bed (kg) number of components in a multicomponent mixture index of Roller’s equation coordination number for particle j in direct contact with particle k coordination number in a regular packed bed of equal-sized spheres average coordination number in a bed of equal-sized spheres number density of spheres Fuller constant of Andreasen (Gaudin–Schumann) distribution distance from a central reference spheres in units of sphere diameter outer edge of the first layer in units of sphere diameter average position of the first-layer neighbors in units of sphere diameter average nearest-neighbor distance in units of sphere diameter fractional area of particle k in a multicomponent mixture fractional volume of coarse powder in a binary mixture fractional volume of particle k in a multicomponent mixture volume of powder (m3)
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FIGURE 5.13 Relationship between Fuller constant of Gaudin–Schumann size distribution and void fraction. [From Suzuki, M., Oshima, T., Ichiba, H., and Hasegawa, I., KONA Powder Sci. Technol. Jpn., 4, 4–12, 1986. With permission.]
Vs V1, V2 Z bj _1, 2 c (j, k) j j rb, r sg w w (R)
volume of a hypothetical sphere in Figure 5.7 (m3) volume of a spherical segment in Figure 5.7 (m3) number of the first-layer neighbors proportionality constant in Equation 2.2 void fraction void fraction of coarse monosized powder bed or void fraction of fine one void fraction of coarse monosized powder over critical particle size partial void fraction around particle j in direct contact with particle k partial void fraction around particle j void fraction in a bed of uniformly sized particles apparent and particle density of powder (kg/m3) geometric standard deviation of log-normal distribution bulk-mean particle volume fraction or packing fraction ( 1 – ) radial distribution function
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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.
Suzuki, M., Sato, H., Hasegawa, M., and Hirota, M., Powder Technol., 118, 53–57, 2001. Roller, P. S., Ind. Eng. Chem., 22, 1206–1208, 1930. Graton, L. C. and Fraser, H. J., J. Geology, 43, 785–909, 1935. Heesh, H. and Laves, F., Z. Kristallor, 85, 433–453, 1933. Smith, W. O., Foote, P. D., and Busang, P. F., Phys. Rev., 34, 1271–1274, 1929. Shinohara, K., Kobayashi, H., Gotoh, K., and Tanaka, T., J. Soc. Powder Technol. Jpn., 2, 352–356, 1965. Shinohara, K. and Tanaka, T., Kagaku Kogaku, 32, 88–93, 1968. Gotoh, K., Nature Phys. Sci., 231, 108–110, 1971. Gotoh, K., J. Soc. Powder Technol. Jpn., 15, 220–226, 1978. Iwata, H. and Homma, T., Powder Technol., 10, 79–83, 1974. Bargiel, M. and Tory, E. M., Adv. Powder Technol., 12, 533–557, 2001. Rumpf, H., Chem. Eng. Technol., 30, 144–158, 1958. Bernal, D. J. and Mason, J., Nature, 188, 910–911, 1960. Meissner, H. P., Michael, A. S., and Kaiser, R., Ind. Eng. Chem. Process Des. Dev., 3, 202–205, 1964. Wade, W. E., J. Phys. Chem., 69, 322–326, 1965. Ridgway, K. and Tarbuck, K. J., Br. Chem. Eng., 12, 384–388, 1967. Arakawa, M., J. Soc. Mater. Sci. Jpn., 16, 319–321, 1967. Haughey, D. P. and Beveridge, G. S. G., Can. J. Chem. Eng., 47, 130–140, 1969. Bernal, D. J., Cherry, I. A., Finney, J. L., and Knight, K. R., J. Phys., E3, 388–390, 1970. Nagao, T., Trans. J.S.M.E., 44, 1912, 1978. Nakagaki, M. and Sunada, H., Yakugaku Zasshi, 83, 73–78, 1963. Nakagaki, M. and Sunada, H., Yakugaku Zasshi, 88, 705–709, 1968. Bennett, H., J. Appl. Phys., 43, 2727–2734, 1972. Tory, R. M., Church, B. H., Tam, M. K., and Ratner, M., Can. J. Chem. Eng., 51, 484–493, 1973. Gotoh, K. and Finney, J. L., Nature, 252, 202–205, 1974. Suzuki, M. Makino, K., Yamada, M., and Iinoya, K., Int. Chem. Eng., 21, 482–488, 1981. Ouchiyama, N. and Tanaka, T., Ind. Eng. Chem. Fundam., 19, 555–560, 1980. Suzuki, M. and Oshima, T., KONA Powder Sci. Technol. Jpn., 7, 22–28, 1989. Suzuki, M., Kada, H., and Hirota, M., Adv. Powder Technol., 10, 353–365, 1999. Gotoh, K., J. Soc. Powder Technol. Jpn., 16, 709–713, 1979. Suzuki, M., Makino, K., Tamamura, T., and Iinoya, K., Int. Chem. Eng., 21, 284–293, 1981. Chandrasekhar, S., Rev. Mod. Phys., 15, 1–89, 1943. Bansal, P. P. and Ardel, A. J., Metallography, 5, 97–111, 1972. Gotoh, K., Jodrey, W. S., and Tory, E. M., Powder Technol., 21, 285–287, 1978. Tanemura, M., J. Microscopy, 151, 247–255, 1988. Witten, T.A. and Samder, L. M., Phys. Rev. Lett., 47, 1400–1403, 1981. Tohno, T. and Takahashi, K., Aerosol Res., 2, 117–127, 1987. Kamiya, H., Yagi, E., Jimbo, G., J. Soc. Powder Technol. Jpn., 30, 148–154, 1993. Suzuki, M. and Oshima, T., J. Soc. Powder Technol. Jpn., 26, 250–254, 1989. Horsfield, H. T., J. Soc. Chem. Ind., 53, 107–115, 1934. White, H. E. and Walton, S. F., J. Am. Ceram. Soc., 20, 155–166, 1937. Hudson, D. R., J. Appl. Phys., 20, 154–162, 1949. Furnas, C. C., U.S. Bur. Mines Rep. Invest., No. 2894, 1928. Sohn, H. Y. and Moreland, C., Can. J. Chem. Eng., 46, 162–167, 1968. Arakawa, M. and Nishino, M., J. Soc. Mater. Sci. Jpn., 22, 658–662, 1973. Suzuki, M. and Oshima, T., Powder Technol., 43, 147–153, 1985. Powell, M. J., Powder Technol., 25, 45–52, 1980. Dinger, D. R. and Funk, J. E., Interceram., 42, 150–152, 1993. Suzuki, M. and Oshima, T., Powder Technol., 44, 213–218, 1985. Ito, T., Wanibe, Y., and Sakao, H., J. Jpn. Inst. Met., 50, 740, 1986. Furnas, C. C., Ind. Eng. Chem., 23, 1052–1058, 1931. Westman, A. E. R., J. Am. Ceram. Soc., 19, 127–129, 1936. Tokumitsu, Z., J. Soc. Mater. Sci. Jpn., 13, 752–758, 1964. Kawamura, J., Aoki, E., and Okuzawa, K., Kagaku Kogaku, 35, 777–783, 1971.
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Abe, E., Hirosue, H., and Yokota, A., J. Soc. Powder Technol. Jpn., 15, 458–462, 1978. Ouchiyama, N. and Tanaka, T., Ind. Eng. Chem. Fundam., 22, 66–71, 1981. Ouchiyama, N. and Tanaka, T., Ind. Eng. Chem. Fundam., 25, 125–129, 1986. Suzuki, M., Yagi, A., Watanabe, T., and Oshima, T., Int. Chem. Eng., 26, 491–498, 1986. Suzuki, M., Oshima, T., Ichiba, H., and Hasegawa, I., KONA Powder. Sci. Technol. Jpn., 4, 4–12, 1986. Cross, M., Douglas, W. H., and Fields, R. P., Powder Technol., 43, 27–36, 1985. Suzuki, M. and Oshima, T., Powder Technol., 35, 159–166, 1983. Dinger, D. R. and Funk, J. E., Interceram., 43, 87–89, 1994.
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3.6
Capillarity of Porous Media Hironobu Imakoma Kobe University, Nada-ku, Kobe, Japan
Minoru Miyahara Kyoto University, Katsura, Kyoto, Japan A macroscopic body composed of fine particles naturally possesses microscopic void spaces with a size several to ten times smaller than the particle diameter. Sometimes particles themselves may have some porosity. Such pore spaces can bring functional characteristics such as adsorptive capacity, selective permeation, and dielectric properties, while in some cases they stand as a deficit. With increasing demand for functionality, recent powder-based manufacturing has tended to use finer particles of submicron size down to the nanometer range. As the result the characterization of capillarity in the nanometer range is getting more and more important. In this section, two typical pore-size characterization methods, nitrogen adsorption and mercury porosimetry, are described with their theoretical basis and with attention paid especially to the nanometer range. Some other methods are explained briefly.
3.6.1 COMMON PHENOMENON: YOUNG–LAPLACE EFFECT An important phenomenon that commonly applies to nitrogen adsorption and mercury porosimetry is the so-called Young–Laplace effect. As shown in Figure 6.1a, the surface of liquid that wets the wall will form a hemispherical meniscus in a sufficiently thin tube to satisfy that the contact angle u is zero. The Young–Laplace equation describes the pressure difference ΔP brought by the pulling force of surface tension g, which is given by Equation 6.1, including the case of partial wetting:
⌬P ⬅ P ⫺ P ’⫽
2g cos u r
(6.1)
More generally, menisci with arbitrary shape can be described with two principal curvature radii, r1 and r2. 1 1 ⌬P ⫽ g ( ⫹ ) cos u r1 r2
(6.2)
For a hemi-cylindrical meniscus formed in slit space of width W, for example, Equation 6.2 will be ΔP ⫽ 2gcosu /W because the two principal radii are 2/W and 0. This equation resembles Equation 6.1, but note that r is the radius of the space while W denotes the span between walls. The above has macroscopic and classical understandings. On the nanoscale, especially the so-called single-nano length, it fails to express the phenomena because of the hindering characteristics of a liquid surface and the effects of potential energy exerted from pore walls, for instance. With attention paid to the single-nano scale, each method is described in the following. 309 © 2006 by Taylor & Francis Group, LLC
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r
r P
P
P’
P’
(a) Perfect wetting
(b) Partial wetting
FIGURE 6.1 effect.
3.6.2
'
Young–Laplace
NITROGEN ADSORPTION METHOD
Pore size distributions, as well as the specific surface area, can be estimated from nitrogen adsorption isotherms at liquid-nitrogen temperature, or the amount adsorbed against relative pressure [ ⫽ (equilibrium pressure p)/(saturated vapor pressure ps)]. Measurement is done either by the volumetric method, which detects pressure variation of nitrogen gas introduced in an adsorption system with constant volume, or the gravimetric method, which measures weight variation of a sample contacting with gas of given relative pressure. In the following, description relating to measurement refers to the former method unless otherwise specified, since the great majority of commercially available automated apparatuses are based on it.
Principle: Capillary Condensation and the Kelvin Equation Suppose that liquid nitrogen exists in a cylindrical pore as shown in Figure 6.1a. The pressure in liquid phase is lowered by ΔP ⫽ 2g /r more than that in gas phase, which results in a lower free energy than a normal liquid with flat surface by vΔP, (v being the molar volume of liquid). Then the equilibrium vapor pressure of pore liquid pg must be smaller than the saturated vapor pressure ps. Conversely, the vapor with a pressure smaller than ps can condensate if it goes into a pore, which is termed the capillary condensation phenomenon. The equilibrium pressure pg must satisfy Equation 6.3 since the difference of free energy from ps equals vΔP. ln
pg ps
⫽⫺
2v g rRT
(6.3)
This is the so-called Kelvin equation, which was derived originally by Lord Kelvin for vapor pressure of a small droplet without the negative sign: the difference apparently comes from the concave geometry. The equation gives the pore radius r from the relative pressure pg/ps at which the condensation occurs. The above stands as the principle for estimating pore size distribution from a nitrogen isotherm. Further to be considered for an adsorption system, however, is the adsorbed film with thickness t on the walls contacting with nitrogen vapor (see Figure 6.2). The condensation phase is formed within the core space excluding the films, with a contact angle of zero. The radius for a cylindrical pore is then given by rp ⫽ r ⫹ t ⫽
2 vg ⫹t RT ln( ps Ⲑpg )
(6.4)
and the width of a slit pore will be © 2006 by Taylor & Francis Group, LLC
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t
rp r pg
FIGURE 6.2 Surface adsorption film and condensation phase.
W ⫽ 2r ⫹ 2t ⫽
2 vg ⫹ 2t RT ln( ps Ⲑpg )
(6.5)
Many detailed calculation schemes were proposed by, for example, Cranston and Inkley,1 Dollimore and Heal,2 and Barell et al.3 Major differences between these methods are the data or the equation for adsorption thickness t, and the schemes of calculation themselves do not produce significant differences if common data for t are used. Therefore one need not worry which method is installed in a commercial apparatus; one should pay more attention to the t-data in the software and consider how precise and how abundant for various materials they are.
Measurable Range of Pore Size Many automated adsorption apparatuses declare the upper limit of the pore size to be approximately 100 nm, which corresponds to a relative pressure of approximately 0.98. In general, however, the accuracy of the adsorption measurement decreases as the equilibrium pressure approaches the saturated vapor pressure, because the liquid nitrogen temperature will fluctuate following variation in atmospheric pressure. For safety one should understand the upper limit of reliability to be approximately 0.95 in relative pressure, corresponding to a pore size of about 40 nm. Measurement for larger pores should be done with mercury porosimetry. As for the lower limit, pore size analysis based on the capillary condensation will lose its basis if the size goes down below about 2 nm, because the condensation phenomenon itself (or the firstorder phase transition) does not occur in such small space. Much of the research conducted in the 1970s and 1980s discussed pore sizes in the range of 1–2 nm for, for example, activated carbons, but nowadays such analysis and discussion cannot be accepted. Instead, the present understanding of adsorption phenomena in such small pores is the so-called micropore filling, meaning that the gas molecules are gradually filled into the pore space by strongly attractive potential energy exerted by pore walls. Some pore analysis methods based on this mechanism are available in the literature,4,5 among which the so-called t-plot method6 and related ones7,8 are suitable for simple and reliable analysis of micropore size distribution. Refer to the literature for details of these methods. Further rigorous analysis of micropore size distribution is presently under development by many researchers. Most of the approaches are based on the statistical thermodynamics method, such as molecular simulation and the density functional theory, which produce so-called local adsorption isotherms for a series of various sizes of micropore. An experimental nitrogen isotherm is expressed as a convolution integral of the local adsorption isotherms and the pore size distribution function, the latter thus can be determined through minimization of the error between measured and predicted isotherms. Note that, however, this kind of technique needs further development. Sometimes calculated pore size distribution suffers from an artifact of nonexistent pores, which may result from surface heterogeneity. One has to pay much attention to this kind of artifact, especially if the resultant distribution has bimodal or multimodal distribution. © 2006 by Taylor & Francis Group, LLC
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Kelvin Model’s Deficit: Underestimation in Single-Nano Range The estimation based on the Kelvin model as given by Equation 6.4 and Equation 6.5 works well for pores larger than 10 nm. It has was pointed out in the late 1980s, however, that the model underestimates the so-called single-nano range of pores.9,10 The research in that area was rather scientific and showed this deficit by complicated statistical thermodynamics techniques or molecular simulations with great computational costs. No method based on simplicity and convenience was available even in the 1990s, which forced people to use the Kelvin model, though knowing its inaccuracy. A condensation model with a simple concept and easy calculation has been recently proposed11–14 and is explained briefly below. The point is that the attractive potential energy from pore walls and the stronger surface tension of a curved interface will enhance the condensation in nanoscale pores. The basic equation is RT ln
pg ps
⫽⫺
2 vg ( r ) ⫹ ⌬f(r ) r(r )
(6.6)
in which the free energy for condensation (right-hand side) is compensated not only by the Young– Laplace effect with local curvature dependent surface tension g (r), but also by the attractive energy of pore walls relative to the condensing liquid Δf(r). The latter effect can be determined13 from adsorption thickness data that are usually included in the automated adsorption apparatuses. An iterative calculation is needed to determine the relation between pore size and relative pressure because the curvature of the liquid surface depends on location in the pore, but it remains an easy calculation at a pocket-computer level. For further details, see the original papers. The degree of underestimation by the Kelvin model stays almost constant regardless of the pore size but varies depending on the pore-wall potential energy. Some examples of difference between Kelvin-based prediction and the true pore size are about 1 nm or slightly greater for carbon materials, about 1 nm or less for silica materials, and 0.5–0.7 nm for ordered mesoporous silicates (MCM-41). Thus one should understand that, if the BJH method gives peak pore size as 3 nm for a silica gel, the true size is about 4 nm or slightly less than that. The model based on Equation 6.6 can give the true pore size from the nitrogen isotherm without any additional measurement.
Hysteresis If an adsorption isotherm goes with capillary condensation, the hysteresis between adsorption process and desorption results in most cases . The classification of the hysteresis into four types is given by IUPAC, as shown in Figure 6.3. There have been long discussions on which branch should be used for pore-size determination, but it is still quite difficult to obtain a general conclusion. Neimark and coworkers,15,16 for example, reported that the adsorption branch for MCM-41, which possesses almost ideally cylindrical pores, is of spinodal process with higher condensation pressure than thermodynamic equilibrium because the surface adsorption layer has to grow up locally into the central portion of the pore space. The desorption process, on the other hand, does not suffer from such an energy barrier, resulting in an equilibrium desorption. This mechanism seems to be accepted by many researchers recently, but some groups have raised questions about it. By limiting the topic into the structure made up by aggregated particles or a sintered porous body, however, the following understanding can be achieved. The characteristics of this kind of structure would be that there must exist a particles contact point at the end of the pore space. Then the contact points and their vicinity are able to provide
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H1
H2
H3
H4
FIGURE 6.3 Classification of hysteresis defined by IUPAC.
nuclei for the condensation, and an energy barrier, as is the case for MCM-41, would not stand upon the condensation process. Therefore the condensation process for this kind of materials follows an equilibrium path. Another feature is that pores are connected through narrower spaces between particles, or the pore network is formed with connecting “necks,” In this case, the desorption process itself is in equilibrium, but the so-called ink-bottle effect gives hysteresis in which the evaporation of condensate in a pore space is not possible until the pressure goes down to a value corresponding to the size of the neck. The evaporated volume at this hindered process, then, does not provide the pore volume of the pore size corresponding to this pressure. This process exhibits H2-type hysteresis, and the adsorption branch should be used to calculate the size distribution of the pore space. Examples of materials include silica gels and porous polymer gels. Many providers of such materials use desorption branches for showing porous characteristics to users because it gives sharp distribution. One should notice, however, that the peak in the distribution simply gives the neck size, and the real distribution would be broader in most cases. As clarified by percolation theory, the extrusion process for wetting fluid is the same as the intrusion process for nonwetting fluid. Therefore the intrusion curve of mercury porosimetry, which is more often used than the extrusion, similarly gives the neck size. Another important topic related to hysteresis is the end-closure point of the desorption branch. As seen in clay materials and activated carbons, a hysteresis loop of types H3 and H4 often closes at the relative pressure of 0.40–0.45. As described in detail in the literature,17 this closure of hysteresis will result from the spinodal evaporation of condensed liquid when exceeding its tensile limit, which is determined not by the pore size but merely by the nature of liquid. This phenomenon should be considered when one uses the desorption branch for characterization.
3.6.3 MERCURY INTRUSION METHOD (MERCURY POROSIMETRY) An evacuated sample is immersed in mercury, and pressurization of the system cause mercury’s intrusion into the pore space. The detection and analysis of the intruded volume against applied pressure gives the pore size distribution. Many automated apparatuses are commercially available.
Principle: Nonwetting Fluid and Washburn Equation Mercury does not wet almost all of solid surface, which corresponds to the upper portion in Figure 6.1b. Also, P′ ⫽ 0 stands because the pore space is evacuated before measurement. Equation 6.1 then will be
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P⫽
2g cos u 4g cos u ′ , or D ⫽ 2r ⫽⫺ r P
(6.7)
which gives the relation between applied pressure P and the pore diameter D. u′ is the contact angle for mercury, which is set to be 140° in most cases. The above equation is essentially the same as the Young–Laplace equation, but the right-hand side is called the Washburn equation. Since the intruded volume directly means the volume of pores larger than a size corresponding to the pressure, the differentiation of the volume with respect to the pore size gives the pore-size distribution.
Measurable Range of Pore Size Many automated porosimetry apparatuses declare the lower limit of the pore size to be 3–4 nm, which corresponds to the applied pressure of about 4000 atm. At this high pressure, the porous framework might be deformed or some other influence may occur. It is also important to point out that the surface of mercury in a single-nanometer range may be different from the surface of bulk liquid, which may result in a hindered contact angle. Much attention, then, should be paid to the reliability of the data in this range, for which the gas adsorption method has far superior accuracy and reliability. The upper limit of mercury porosimetry would be around several hundred microns. The detection of cracks or supermacro pores, which is difficult to measure by gas adsorption, is precisely done by the intrusion method.
Hysteresis The extrusion process with decreasing pressure generally gives a different path, or hysteresis. Further, it would be almost always the case that a certain amount of mercury remains in the pores even after complete release of the pressure. The extrusion process, therefore, would not be suitable for analysis, and the intrusion branch is used in general. This corresponds to the analysis of the desorption branch in gas adsorption, and one should notice that the analysis gives the neck size for aggregated or sintered bodies.
3.6.4
OTHER TECHNIQUES OF INTEREST
Bubble-Point Method This technique detects perforating pores while the gas adsorption method and mercury porosimetry cannot distinguish those from dead-end pores. Because of this feature the method is often applied to filters, membranes, cloths, or those porous materials whose permeation properties are of importance. Depending on the wettability, the porous material is immersed in freon or water. Pressurized air or nitrogen is then introduced to one side of the material. At a pressure corresponding to maximum size of the perforating pore, the gas starts to permeate. Other than the detection of the maximum pore size, the size distribution can be estimated by applying higher pressures, because smaller pores start to open with increased pressure. Detectable pore size is usually above several tens of nanometers with freon, or a few hundred nanometers with water.
Thermoporometry: Detection of Freezing Point Depression Based upon the Gibbs–Thomson equation, which assumes that freezing point depression in a pore from bulk temperature is inversely proportional to the pore radius, the size distribution is estimated from calorimetric measurement. It may sometimes be the case that the pore structure when wetted varies after drying because of capillary suction pressure or de-swelling of the base material. The gas adsorption or mercury porosimetry cannot characterize such porous materials because both methods
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need evacuation before measurement. This method may be used to overcome the above problem: the measurement goes as wetted. However, not much has been clarified for the freezing behavior in confined space. Recent study, for example, has clarified that the freezing point may even be higher than the bulk freezing point, depending on the physicochemical nature of the pore walls.18 The method is thus especially controversial if the single-nano range is concerned, and it is better not to rely on it for the smaller range of pore sizes.
Small Angle X-ray Scattering X-ray scattering with angles smaller than 10° can probe porous characteristics in the singlenanometer range. Since X-rays can detect not only open pores but also closed (or isolated) pores, measurements for low-permittivity materials are often seen as an application. One has to be careful, however, because the resulting space distribution or correlation length does not necessarily relate to the scale of the pores but has resulted from the electron density distribution. Further, an ordered material such as MCM-41 would give clear signals showing its lattice size or the periodicity of the regular pores, but not much sensitivity can be expected for materials with a disordered or random nature.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Cranston, R. W. and Inkley, F. A., Adv. Catal., 9, 143, 1957. Dollimore, D. and Heal, G. R., J. Appl. Chem., 14, 109, 1964. Barell, E. P., Joyner, L. G., and Halenda, P. P., J. Am. Chem. Soc., 73, 373, 1951. Horvath, G. and Kawazoe, K., J. Chem. Eng. Jpn., 16, 470, 1983. Saito, A. and Foley, H. C., AIChE J., 37, 429, 1991. Lippens, B. C. and de Boer, J. H., J. Catal., 4, 319, 1965. Sing, K. S. W., in Surface Area Determination, Everett, D. H. and Ottewill, R. H., Eds., Butterworths, London, 1970, p. 25. Mikhail, R. S., Brunauer, S., and Bodor, E. E., J. Colloid Interface Sci., 26, 45, 1968. Evans, R., Marconi, U. M. B., and Tarazona, P., J. Chem. Phys., 84, 2376, 1986. Miyahara, M., Yoshioka, T., and Okazaki, M., J. Chem. Phys., 106, 8124, 1997. Miyahara, M., Yoshioka, T., and Okazaki, M., J. Chem. Eng. Jpn., 30, 274, 1997. Miyahara, M., Kanda, H., Yoshioka, T., and Okazaki, M., Langmuir, 16, 4293, 2000. Miyahara, M., Yoshioka, T., Nakamura, J., and Okazaki, M., J. Chem. Eng. Jpn., 33m 103, 2000. Kanda, H., Miyahara, M., Yoshioka, T., and Okazaki, M., Langmuir, 16, 6622, 2000. Vishnyakov, A. and Neimark, A. V., J. Phys. Chem. B, 105, 7009, 2001. Neimark, A. V., Ravikovitch, P., and Vishnyakov, A., Phys. Rev. E, 65, 2002. Ravikovitch, P. and Neimark, A. V., Langmuir, 18, 1550, 2002. Miyahara, M. and Gubbins, K. E., J. Chem. Phys., 106, 2865, 1997.
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3.7
Permeation (Flow through Porous Medium) Chikao Kanaoka Ishikawa National College of Technology Ishikawa, Japan
Permeation is fluid flow through interstices among many discrete particles. Flows of groundwater, crude oil, and natural gas are typical permeations taking place in nature. This type of flow is widely utilized in industrial processes in which both particles and fluid flow play important roles simultaneously, such as a catalytic reaction, filtration, and so on. Thanks to the extensive number of studies on porous media since D’arcy, macroscopic flow characteristics in a porous medium can be evaluated fairly well, although the microscopic flow pattern inside the medium is complex and not amenable to rigorous solution by the Navier–Stokes equation. In this section, general theories developed to explain resistance to flow through a porous medium are described, and the resistances of granular and fibrous packed beds are outlined as typical examples of high and low packing densities, respectively.
3.7.1 RESISTANCE TO FLOW THROUGH A POROUS MEDIUM A porous medium is composed of many discrete particles with different shapes and sizes, as shown in Figure 7.1a. The packing density of the medium is high and its structure is sophisticated. Porous media can be simplified as an assembly of many tiny channels or uniformly spaced spherical particles of equal size, as shown in Figure 7.1b and 7.1c. For the case of the former model, flow resistance can be predicted by analogy with that of a straight circular tube, and for the case of the latter, it can be calculated from the fluid drag acting on individual particles. These two models are usually called the “channel” and “drag” theories, respectively. Of course, pressure drop can be predicted by either model, but, in general, the former is believed to be suited for a flow-through porous medium with high packing density and the latter for one with low packing density.
Channel Theory Because the porous system is assumed to be a bundle of tiny channels with noncircular cross sections, the pressure drop of the system is identical with that for one of those channels. When fluid with density r and viscosity µ flows through a straight tube with diameter D and length L, the pressure drop P is expressible by the so-called Hagen–Poiseuille equation for laminar flow and by Fanning’s equation in general: Hagen–Poiseuille equation:
Fanning’s equation:
P
32 L mu D2
P 4 f
L ru 2 D 2
(7.1)
(7.2)
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FIGURE 7.1
Models of a porous medium.
Here, Fanning’s friction factor f is expressed by Equation 7.3 in a laminar flow region, which becomes dependent not only on the Reynolds number Re but also on the surface roughness in the turbulent flow region: f 16
m 16 Dur Re
(7.3)
Equation 7.1 through Equation 7.3 are applicable to noncircular channels by introducing the hydraulic radius m defined as m
Cross-sectional area of a channel Wetted periphery
(7.4)
The value of m calculated from Equation 7.4 equals one quarter of the tube diameter for circular tubes and about one quarter of the equivalent diameter for other cross-sectional shapes. The definition above is not applicable, as it is to the porous medium; hence, it is modified as follows.
Cross-sectional area of a channel tube length Wetted peripphery tube length Tube volume Wetted area of a tube Total pore volume Total surface area of particles in a porous meddium Porosity Specific surface area based on the volume of porous medium
m
SB Sv (1 )
(7.5)
where SB and Sv are the specific surface areas based on packed layer volume and particle volume, respectively. © 2006 by Taylor & Francis Group, LLC
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According to Carman’s tortuousity model,1 the time required for fluid to travel an equivalent channel is equal to the actual penetration time to pass through the bed. Hence, the following relation is obtained: Le L ue u
(7.6)
Substituting Equation 7.5 and Equation 7.6 into Equation 7.1 gives P
32 Le mue
( 4m )2
2 2 Le mSB2 Le L (u ) ⎛ Le ⎞ SB L mu 2 ⎜⎝ L ⎟⎠ 2 3 2
2 ⎛ L ⎞ S (1) L mu 2⎜ e ⎟ v ⎝ L⎠ 3 2
2
(7.7)
As mentioned earlier, the hydraulic radius calculated from Equation 7.5 is one fourth of the tube diameter for circular tubes. Hence, four times the hydraulic radius is used in Equation 7.7. Le/L is the ratio of the equivalent channel length to the bed thickness, and it is called the tortuosity. For noncircular channels, the numerical constant in the last two expressions in Equation 7.7 differs from 2. Therefore, we write the following expression for the general case of a channel model by replacing the numerical constant 2 by ko: 2 2 ⎛ L ⎞ S L mu ⎛ L ⎞ S (1 − ) L mu P k0 ⎜ e ⎟ B 3 k0 ⎜ e ⎟ v ⎝ L⎠ ⎝ L⎠ 3 2
2
2
(7.8)
Equation 7.8 is the expression based on the channel model, and the pressure drop can be evaluated by knowing the values ko and Le/L; k = ko(Le/L)2 is usually called the Kozeny constant and is dependent on particle shape, packing structure, and so on. is about 5 for most cases. Carman obtained the following well-known expression called the Kozeny–Carman equation: Sv2 (1) L mu SB2 L mu 5 3 3 2
P 5
(7.9)
Because Equation 7.9 is derived from the Hagen–Poiseuille equation, Equation 7.1, its applicable range is limited to the laminar flow region. In the turbulent flow region, Fanning’s equation, Equation 7.2, has to be used by modifying the Reynolds number and friction factor as follows2: Re
2f
m(u / )r ur ur ur m SB m mSB mSv (1)
DPm L r (u )
2
DPSB
L r (u )
2
DP 3 DP 3 L ru 2 SB L ru 2 Sv (1)
(7.10)
(7.11)
Equation 7.11 can also be applicable to the laminar flow region. Substituting Equation 7.10 and Equation 7.11 into Equation 7.9, one obtains 2f
mS (1) DP3 5 5 v L ru Sv (1) ur Re 2
(7.12)
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In Equation 7.12 the friction factor is inversely proportional to Reynolds number. This agrees well with the experiments for Re < 2, as shown in Figure 7.2. The friction factor for Re > 2 has to be determined experimentally.3,4
Drag Theory The pressure drop of a porous medium has to be equal to the fluid drag experienced by particles inside the medium: N
PA ∑ Rmi
(7.13)
i1
where A is the cross-sectional area of a porous medium, Rm i is the fluid drag experienced by the ith particle, and N is the total number of particles in the medium. Let us consider a uniformly packed bed of spherical particles with diameter Dp, depth L, and porosity , as shown in Figure 7.1c. It is considered that fluid flow around each particle in the bed is identical, and hence the fluid drag acting on it is also the same, yielding Rmi= Rm. Rm is expressed by the following equation: Rm Rf ( x )
(7.14)
R expresses the fluid drag acting on a single spherical particle in infinite field and f() is the correction term due to the change in packing density and usually referred to as the porosity function. R and N are given by the following equations: pDp2 ru 2 4 2
(7.15)
6 AL (1) pDp3
(7.16)
R CD N
where C D is the drag coefficient of a single sphere.
FIGURE 7.2 Correlation of experimental friction factor and Reynolds number. © 2006 by Taylor & Francis Group, LLC
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Substituting Equation 7.14 through Equation 7.16 into Equation 7.13, it becomes 3 L ru 2 P CD (1) f () 2 Dp 2
(7.17)
By adopting Steinour’s6 expression for f(), which was obtained for hindered settling, Equation 7.17 becomes f () P
25 (1) 33
25 (1)2 L ru 2 CD 2 3 Dp 2
(7.18)
(7.19)
For turbulent flow, the following Burke and Plummer expression7 obtained from the dimensional analysis is popular: P
3.5 1 L ru 2 4 3 Dp 2
(7.20)
One of the most effective expressions for both laminar and turbulent flow regions is the following Ergun’s equation 7, which is a sum of fluid drags proportional to the first and second powers of the fluid velocity: P (1 )2 mu 1 ru 2 150 1.75 3 3 2 L Dp Dp
(7.21)
In the form of a modified friction factor, it becomes 2f
150(1 ) 1.75 Rep
(7.22)
where Rep
Dpur m
(7.23)
Figure 7.3 is the correlation between 2f, and calculated values from the Kozeny–Carman and the Burke–Plummer equations are also shown. As one can see from the figure, Ergun’s expression agrees well with the Kozeny–Carman equation in the laminar flow region1,8 and with the Burke–Plummer equation in the turbulent flow region.
3.7.2
PRESSURE DROP ACROSS A FIBROUS MAT
A fibrous mat can be considered as a special kind of porous medium composed of particles with an extremely large aspect ratio, to which both channel and drag theories are applicable. The porosity of a fibrous mat is usually higher than 85%; thus, the average interfiber distance becomes larger than several times the fiber diameter. This means that interference effects among neighboring fibers are © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.3 others.
Comparison of Ergun’s correlation with
substantially smaller compared with the ordinary granular packed bed. Hence, most of the equations proposed have been derived based on the drag theory. According to the drag theory, the pressure drop P across a fibrous mat with cross-sectional area A and thickness L is related to the fluid drag acting on fibers in the mat as follows: PA Flf AL
(7.24)
where F and lf denote the fluid drag acting on a fiber with unit length and the total fiber length in a unit mat volume, respectively, and they are defined by the following equations: F CD Df
lf
ru 2 2
4a pDf2
(7.25)
(7.26)
Substitution of the preceding two equations into Equation 7.24 yields P 4CD
aL ru 2 Df 2
(7.27)
where u denotes the average fluid velocity in the mat and CD is usually given as a function of the Reynolds number based on the approaching velocity ue. Furthermore, there exist some fibers parallel to the fluid flow. Hence, it would be better in practice to define P by P 4CDe
aL ru 2 Df 2
(7.28)
Table 7.1 summarizes the proposed effective drag coefficients CDe. In the table, the expression of Langmuir is derived from the channel theory. © 2006 by Taylor & Francis Group, LLC
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TABLE 7.1
323
Drag Coefficient of Fiber
Researchers Kozeny and Carman
Drag Coefficient (811Kf /Ref)α (1−α)2
Remarks Channel theory
Langmuir (1942)
8pB 1 Ref ln a 2a a2 / 2 3 / 2
Channel theory, B = 1.4
Davies (1952)
32p 0.5 a (1 56a2 ) Ref
Dimensional analysis
Lamb (1932)
8p 1 Ref 2 ln Ref
Perpendicular to the flow, isolated fiber
Iberall (1950)
8p Ref
Parallel to the flow, isolated fiber
Iberall (1950)
Semi-empirical equation 4.8p ⎛ 2.4 ln Ref ⎞ Ref ⎜⎝ 2 ln Ref ⎟⎠
Chen (1955)
k2 2 Ref ln k3a0.5
k2 = 6.1, k3 = 0.64
Happel (1959)
8p 1 Ref ln a 2a a2 / 2 3 / 2
Tube bank parallel to the flow
Happel (1959)
16p 1 Ref ln a (1 − a2 ) (1 + a2 )
Tube bank perpendicular to the flow
Kuwabara (1959)
16p 1 Ref ln a 2a a2 / 2 3 / 2
Tube bank perpendicular to the flow
Kimura and Iinoya
0.6 4.7 / Ref 11 / Ref 1 a
Empirical 10–3 < Ref < 10–2, 3 < Df < 270 m
Notation A CD D Df Dp F f f() k ko L Le
Cross-sectional area of porous medium (m2) Drag coefficient Diameter (m) Fiber diameter (m) Particle diameter (m) Fluid drag acting on a fiber with unit length (N/m) Fanning’s friction factor Porosity function Kozeny constant Numerical constant used in Equation 7.8 Thickness (m) Equivalent channel length (m)
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lf m N P R Rm Re Ref Rep SB Sv u ue a µ
Total fiber length per unit filter volume (m/m3) Hydraulic radius (m) Total number of spheres in a porous medium (m–3) Pressure drop (Pa) Fluid resistance acting on an isolated particle (N) Fluid resistance acting on a particle in a porous medium Reynolds number (Dur/µ) Reynolds number based on fiber diameter (Dfur/µ) Reynolds number based on particle diameter (Dpur/µ) Specific surface of particle based on the bed volume (m2/m3) Specific surface of particle based on the particle volume (m2/m3) Fluid velocity (m/s) Effective fluid velocity in a porous medium (m/s) Fiber packing density Porosity Viscosity (Pa·S)
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Carman, P. C., Trans. Inst. Chem. Eng., 15, 150–156, 1937. Blake, F. E., Trans. Am. Inst. Chem. Eng., 14, 415–421, 1922. Chilton, T. H. and Colburn, A. D., Ind. Chem. Eng., 23, 913–918, 1931. Brownell, L. E. and Katz, D. L., Chem. Eng. Prog., 43, 537–548, 1947. Leva, M., Fluidization, McGraw-Hill, New York, 1959. Steinour, H., Ind. Eng. Chem., 36, 618–624, 1944. Burke, S. P. and Plummer, W. B., Ind. Eng. Chem., 20, 1196, 1928. Ergun, S., Chem. Eng. Prog., 48, 89–94, 1952. Kozeny, J., Sitzungsber. Akad. Wissensch. Wien, 136, 271–306, 1927. Langmuir, I., OSRD Rep. No. 865, 1942. Davies, C. N., Proc. Inst. Mech. Eng. (London), 131, 185–213, 1952. Iberall, A. S., J, Res. Natl. Bur. Stand., 45, 85–108, 1950. Chen, C. Y., Chem. Rev., 55; 595, 1955. Happel, J., AIChE J., 5, 174–177, 1959. Kuwabara, S., J. Phys. Soc. Jpn., 14, 527–532, 1959. Kimura, N. and Iinoya, K., Kagaku Kogaku, 33, 1008–1013, 1969.
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3.8
Specific Surface Area Masatoshi Chikazawa and Takashi Takei Tokyo Metropolitan University, Hachioji, Tokyo, Japan
3.8.1
DEFINITION OF SPECIFIC SURFACE AREA
Specific surface area Sw (m2/kg) of a powder is one of the basic properties of the powder and is generally represented by the surface area of total particles contained in a unit mass of powder. This value implies the internal and external surfaces which can be measured using various probes, such as gases and liquids. In the case of adsorption methods, adsorptive gas as a probe molecule must be accessible to all of the surfaces in cavities, cracks, and micropores. Nowadays, powders used as raw materials and intermediate manufactured goods have become more important, and industrial needs concerning particle size, particle shape, purity and uniformity, of powders have become more stringent. For example, for fine powders, purity and uniformity are necessary for manufacturing of precision materials such as electronics and fine ceramics. When the particle size decreases markedly, powdery phenomena depend largely on surface properties. Therefore, characterization of the powder surface becomes increasingly significant with a decrease in the diameter of powder particles. The surface area of spherical particle type having a diameter of Di is obtained from (8.1)
S ⫽ pDi2 The mass of the particle is
W⫽
pDi3 r 6
(8.2)
If distributions of particle size and shape in a powder sample are known, the surface area of the powder can be calculated. For example, when the particle is spherical and the number of particles having diameter of Di is ni, the specific surface area of the powder is given by
Sw ⫽
∑ n pD ∑ n pD r / 6 2 i
i
i
3 i
(8.3)
If the particle size is assumed to be uniform as for the spherical or cubic type, Equation 8.3 is reduced to Sw ⫽
6 Dm r
(8.4)
where Dm represents the specific surface area diameter. Generally, the specific surface area of a powder is determined by gas adsorption, permeability, or heat-of-immersion methods. 325 © 2006 by Taylor & Francis Group, LLC
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3.8.2 ADSORPTION METHOD The surface area of a powder is generally determined by the adsorption method. In this case, it is necessary to obtain the monolayer capacity and cross-sectional area of an adsorbate molecule. To estimate the monolayer capacity Vm, the Langmuir or BET (Brunauer-Emmett-Teller) equation has been applied to experimental adsorption data. The Langmuir equation can be rewritten as P 1 P ⫽ ⫹ P V bVm Vm
(8.5)
where V and Vm are the adsorbed amount at vapor pressure P and the monolayer completion, respectively. The b value is a constant. Therefore, the plot of P/V against P should be a straight line with slope 1/Vm. The Vm value is obtained from the reciprocal of the slope. On the other hand, the BET equation can be represented by P 1 C ⫺1 P ⫽ ⫹ V ( P0 ⫺ P ) Vm C Vm C P0
(8.6)
The plots of P/V(P0 – P) against P/P0 should therefore exhibit a straight line with slope (C – 1)/(VmC) and intercept 1/(VmC). These plots are called BET plots. The BET plots usually hold within the relative pressure range 0.05–0.35. The slope (C – 1)/(VmC) and intercept values 1/VmC, are determined graphically or by linear regression. Therefore, the monolayer capacity Vm can be derived from the reciprocal of the summation between the slope and intercept values, as shown in
Vm ⫽
1 ⎛ C ⫺1 1 ⎞ ⎜⎝ V C ⫹ V C ⎟⎠ m
⫺1
(8.7)
m
To calculate the surface areas of powders, another important value is the cross-sectional area A of the adsorbate molecules. The surface area S (m2) of the powder can be obtained from the product of the monolayer capacity Vm and cross-sectional area A of an adsorbate molecule, using S⫽
Vm AN A (m 2 ) 22.4 ⫻10⫺3
(8.8)
where NA is Avogadro’s number. A reasonable estimation of the cross-sectional area of an adsorbate molecule is generally obtained from the liquid density of the adsorptive at measurement temperature using Equation 8.9 with several assumptions: (a) the shape of an adsorptive molecule is spherical, (b) the liquid structure of the adsorptive is the closest packing structure with 12 nearest neighbors, and (c) the adsorptive molecules are adsorbed on the particle surface with the 6 nearest neighbors in the close-packed hexagonal arrangement: ⎛ M ⎞ A ⫽ 1.091 ⎜ ⎝ rN A ⎟⎠
2Ⲑ3
(8.9)
The factor 1.091 in Equation 8.9 is a packing factor deduced from the assumptions described above. © 2006 by Taylor & Francis Group, LLC
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The recommended molecular cross-sectional areas of adsorbate gases are summarized in Table 8.1. The nitrogen molecule is generally used as an adsorptive molecule. If the surface area is relatively small, the argon or krypton molecule is used in order to precisely measure adsorbed amounts.
Single-Point Method The BET plots usually show a linear relationship in the pressure range of 0.05 ⬍ P/P0 ⬍ 0.35. C in the BET equation is a constant that expresses a magnitude of the interaction between solid surface and adsorptive molecules. When the value of C is reasonably high, the intercept value 1/(VmC) becomes zero. Then the BET plots are recognized to be a linear line passing close to the origin. In this case, Equation 8.6 is rewritten as P 1 P ⫽ V ( P0 ⫺ P ) Vm P0
(8.10)
⎛ P⎞ Vm ⫽ V ⎜ 1⫺ ⎟ P0 ⎠ ⎝
(8.11)
Equation 8.10 is reduced to
Consequently, the surface area can be determined by measuring one adsorbed amount at vapor pressure P. This method is called the “single-point method.” Usually, inorganic materials have large C values for N2 adsorption. However, because the single-point method is deduced by simplifying the BET equation, there is some intrinsic error. The error in the single-point method can be evaluated using the BET equation. Table 8.2 summarizes the difference in the specific surface areas determined by the single-point method and the multipoint method. These results suggest that the adsorption measurement should be done at high relative pressure in which the BET plots exhibit a straight line.
Volumetric Method This method is the most standard method. A simplified volumetric apparatus is shown in Figure 8.1. A dosing volume in an adsorption apparatus must be determined so that adsorbed amounts can be calculated. The volume is obtained by expanding noble gases such as He and N2 from known volume
TABLE 8.1
Gas N2
Cross⫺Sectional Areas of Adsorbate Gases
Measurement Temperature (°C) ⫺196
Bulk State* L
s2 (nm2) 0.162
Gas CO2
⫺183
G
0.170
O2
⫺183
L
0.141
Ar
⫺183
L
0.143
⫺196
S
0.138
Ethane
Kr
⫺196
S
0.202
n⫺Butane
Xe
⫺183
S
CO
⫺183
L
NH3
Measurement Temperature (°C) ⫺56.5
Bulk State* G
s2 (nm2) 0.170
⫺78
S
0.208
⫺32.5
L
0.154
⫺78
S
0.141
⫺183
S
0.220
0
L
0.444
0.232
20
G
0.479
0.166
25
G
0.510
* Phase of bulk state at measurement temperature: G, gas; L, liquid; S, solid. © 2006 by Taylor & Francis Group, LLC
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TABLE 8.2 Difference, Represented by Ss /S m Values, in Specific Surface Areas Determined by Single-Point and Multipoint Methods Using the BET Equation Ss/Sm C Value 10
P/P0 ⫽ 0.35 0.843
⫽ 0.3 0.811
⫽ 0.25 0.769
⫽ 0.2 0.714
⫽ 0.1 0.526
20
0.915
0.896
0.870
0.833
0.690
30
0.942
0.928
0.909
0.882
0.769
50
0.964
0.955
0.943
0.926
0.847
70
0.974
0.968
0.959
0.946
0.886
100
0.982
0.977
0.971
0.962
0.917
200
0.991
0.988
0.985
0.980
0.957
500
0.996
0.995
0.994
0.992
0.982
Note: The Ss and Sm values are surface areas determined by the single-point method and multipoint method, respectively. The values P/P0 are the relative pressure at which single-point measurings are performed.
FIGURE 8.1
Diagram of a volumetric apparatus.
to a volume of an objective part. It is necessary to measure the pressures before and after expanding the noble gas. An equilibrium pressure is determined using a pressure transducer, for example, a capacitance manometer (baratron). To obtain an adsorption isotherm, a known amount of adsorptive gas is introduced into a sample tube. The sample adsorbs the adsorptive molecules, and the vapor pressure decreases gradually until equilibrium is attained. The introducing pressure and equilibrium pressure must be measured precisely. The adsorbed amount is determined from the difference between the introduced amount and the residual amount. The residual amount is evaluated from the same procedure using He gas. In this case, adsorption of He molecules does not occur. Therefore, the adsorbed amount is determined from the difference between the amount of introduced adsorptive molecules and the amount of He gas introduced into © 2006 by Taylor & Francis Group, LLC
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the sample tube, comparing at the same equilibrium pressure. By repeating this procedure, adsorbed amounts at various pressures can be obtained. Generally, estimation of residual amounts of adsorptive molecules is performed using He gas, then a dead volume is determined. The residual amounts of adsorptive molecules at the other vapor pressures are calculated using this dead volume. The dead volume must be measured immediately before or after the adsorption measurement. During the measurement, the surface level of liquid nitrogen in a cooling bath should be maintained constant within 1 mm at least 5 cm above the powder sample.
Carrier Gas Method A mixture gas of known concentration of adsorptive in a nonadsorbable gas such as He is passed through a sample cell which is cooled in a liquid nitrogen bath. As a result of adsorption, the concentration of adsorptive decreases. This concentration change was first detected by Nelsen and Eggertsen,1 using a thermal conductivity detector. The nitrogen molecule is generally used as an adsorptive. When the sample cell is immersed in the liquid nitrogen, only nitrogen gas is adsorbed and its concentration changes. This concentration change generates an adsorption peak as a function of time. After removing the coolant, the adsorbed molecules are degassed, and a desorption peak is recorded in the opposite direction. The adsorption and desorption processes are recorded as shown in Figure 8.2.
FIGURE 8.2 Measurement of adsorption volume by a dynamic method. (a) Nelsen method, (b) elution method. © 2006 by Taylor & Francis Group, LLC
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To calculate the surface area from the adsorption or desorption peak, it is necessary to calibrate the gas detector in an adsorption apparatus. The calibration is performed by injecting a known amount of adsorptive gas and by comparing this amount with a peak area generated. Adsorbed amounts are determined at various partial pressures of mixed gas, and using the BET equation, the monolayer capacity Vm is calculated. However, the carrier gas method is generally used for the single-point method. The apparatus developed by Suito et al.2 is shown in Figure 8.3.
Gravimetric Method In this method, the adsorption amount is determined by measuring the increase in a sample weight. For example, when a monolayer adsorption is accomplished on a powder surface by nitrogen molecules, the increasing weight is calculated to be 0.286 mg/m 2 . If the specific surface area of a powder is 100 m2/g, the 1-g powder sample adsorbs 28.6 mg nitrogen molecules. This weight change can be measured sufficiently well by an ordinary balance. However, when the surface area is small, a correction due to buoyancy produced by volumes of sample powder and sample supporter is necessary. In the case of nitrogen adsorption, the buoyancy of a 1-ml volume at a relative pressure of 0.3 and at temperature 77 K is calculated to be 0.0594 mg.
3.8.3
HEAT OF IMMERSION
When a solid surface is immersed in a liquid that does not dissolve the solid, heat is generated as a result of the interaction between the solid surface and the liquid molecules. This heat is called the heat of immersion. A diagram of the enthalpy changes deduced by immersion and heat of adsorption is shown in Figure 8.4. The value hi represents the heat of immersion of the solid and is equal to the enthalpy change between the solid surface and solid–liquid interface. The Q and Q’ values are heat of adsorption and real heat of adsorption, respectively. The symbol HL is the liquefaction heat of adsorptive gas. When powder particles are placed in an adsorptive vapor, adsorption of the vapor molecules occurs. If the resulting particles are immersed in the liquid which consists of the adsorptive molecules, the adsorption
FIGURE 8.3 Schematic diagram of a dynamic method. (An evaporator is necessary in order to use an organic adsorptive with a high boiling point.) © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.4 Schematic diagram of heat change. S, solid; L,: liquid; V, vapor; hi, immersion heat; HL, liquefaction heat of vapor; hL, enthalpy of liquid surface; Q, adsorption heat; Q′, real adsorption heat.
layer disappears, and only a solid–liquid interface appears instead. In such a case, the heat of immersion per unit surface area is shown by applying the Gibbs–Helmholtz equation as follows: ⎛ ⭸(gs ⫺ gsl ) ⎞ hi ⫽ gs ⫺ gsl ⫺ T ⎜ ⎟⎠ ⎝ ⭸T p
(8.12)
where gs and gsl are the free energies of solid surface having a adsorbed layer and solid–liquid interface, respectively. On the other hand, Equation 8.13 is well known as the Young equation: gs ⫽ gsl ⫹ g l cos u
(8.13)
where u is the contact angle of a liquid droplet formed on a solid surface, and γl represents the free energy of liquid surface. From Equation 8.12 and Equation 8.13, the following equation is obtained: ⎛ ⭸g cos u ⎞ hi ⫽ g l cos u ⫺ T ⎜ l ⎝ ⭸T ⎟⎠ p
(8.14)
when u is zero, Equation 8.14 is rewritten ⎛ ⭸g ⎞ hi ⫽ g l ⫺ T ⎜ l ⎟ ⎝ ⭸T ⎠ p
(8.15)
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liquid. If the entire surface of the powder particles is covered with adsorbate molecules thick enough, the external surface of the liquid layer newly formed by adsorption would be identical in nature with the surface of bulk liquid adsorptive. The thickness of the liquid layer necessary for the application of this method is five to seven layers. However, Partyka et al.3 reported that two molecular layers was adequate in the case of water layers. When the particle surfaces thus treated are immersed in the liquid adsorptive, the enthalpy change per unit surface area is considered to be equal to the enthalpy hL per unit surface area of the liquid adsorptive. Using the principle of this immersion heat, Harkins and Jura4 developed an “absolute method” to determine the surface areas of powders. The surface area of the powder is regarded to be in accord with the external surface of the liquid adsorbed layer, if the particle diameters are large enough comparing the thickness of the liquid layer. This method is independent of adsorption isotherms and is only based on calorimetry. The magnitude of the immersion heat depends only on the surface area and is independent of the surface properties of powder particles. Therefore the surface area is obtained by S⫽
hi hL
(8.16)
A practical measuring procedure is as follows: Adsorptive molecules are adsorbed on powder particles under appropriate vapor pressure. After adsorption equilibrium, a sample tube is sealed and then the sample tube is placed in a calorimeter. After attainment of temperature equilibrium, the sample tube is broken by pressing a breaking rod. The heat evolved by immersing the powder particles into the adsorptive liquid is measured.
3.8.4
PERMEAMETRY
The surface area of powder can be determined by measuring the pressure drop in the course of fluid flow through a packed powder bed. This method has been widely used on account of the simple apparatus, facility of operation, and rapidness of measuring, compared with other methods. The measuring method is based on the Koseny–Carman equation,5 1 ⎡⎛ ⌬PtA ⎞ ⎛ 3 ⎞ ⎤ Sw ⫽ ⎢ ⎜ ⎜ ⎟⎥ r ⎢⎝ kkhLV ⎟⎠ ⎝ (1⫺ )2 ⎠ ⎥ ⎦ ⎣
1Ⲑ 2
(8.17)
where Sw is the specific surface area of the sample powder, ΔP is the pressure drop, t is time, A is crosssectional area of the powder bed, L is length of the powder bed, is the porosity of the powder bed, r is the true density of the sample powder, kk is the Kozeny constant which is usually taken as 5, h is the viscosity coefficient of the fluid, and V is the volume of fluid passed through the powder bed during time t. Therefore, the specific surface area Sw of the powder bed whose porosity is known can be determined by measuring the flow rate V/t and pressure drop ΔP of fluid. However, in order to use this equation, it is necessary that the stream of permeating fluid is a viscous flow and the Hagen– Poseuilles law can be applied. The surface area thus obtained expresses the external surface area and does not contain internal surfaces in the micro pores and cracks. When air is used as a fluid material, the mean free path of gas molecules is calculated to be about l ⫽ 1.1 ⫻ 10–5cm at 15 °C under 1 atm. In this system, it is necessary for the viscous flow that the diameter D of the flow path is about 10 times larger than l, that is, larger than about 1m. The gas flow mechanism changes with decrease in the particle size of powders from viscous flow to © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.5
Schematic apparatus for permeability measurement.
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Knudsen flow which is governed by the rate of molecular diffusion. In the range of 1 ⬍ D/l ⬍ 10, both mechanisms coexist, and the following equation holds: V ′L P′ 3 4Z v 2 ⫽ ⫹ ⌬PA 5hSv2 RT (1⫺ )2 3 Sv (1⫺ )
(8.18)
rSw ⫽ Sv
(8.19)
where V’ is the velocity (mol/s) of fluid gas and P’ is the average pressure in the powder bed. Z is a constant and is related to the shape of the path. v is the rate of molecular diffusion. The first term of the right-hand side is the contribution of viscous flow, and the second term is that of Knudsen flow. The plots of V’ L/Δ PA versus P’ should show a linear line. Then, the surface area estimated from a slope of the linear line originates in the viscous flow, and it is assumed to be related with an outer surface area of agglomerate particles. This value depends on the porosity of a powder bed. On the other hand, the surface area calculated from the intercept value is attributed to the Knudsen flow and is considered to represent the total surface area of primary particles. The specific surface area determined by this method agrees well with that calculated from the particle size distribution obtained by electron microscopy.6 The determination of Z is difficult, so this value is obtained experimentally. Derjaguin7 determined the Z to be 0.69, and Arakawa and Suito6 obtained 0.47. Apparatuses and a practical measuring procedure are as follows. Gas and liquid are used as fluid materials. Apparatuses commercially available are based on air permeametry, and many of them are applicable for viscous flow using the Koseny–Carman equation. Typical apparatuses are shown in Figure 8.5. The surface areas measured by various apparatuses do not agree each other, and the differences attain about 20–30% in some cases. However, reproducibility in each apparatus is very high. In this method, the specific surface area Sw varies with the porosity of the sample bed. In general, Sw increases with a decrease in the porosity and reaches a constant value below a certain porosity. The Kozeny constant kk in Equation 8.14 is taken as 5. However, this value is affected by tortuosity of the flow path, which depends largely on the packing degree of the sample bed.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Nelsen, G. and Eggertsen, F. T., Anal. Chem., 30, 1387–1390, 1958. Suito, E., Arakawa, M., Sakata, M., and Natsuhara, Y., Zairyo, 15, 178–183, 1966. Partyka, S., Rouquerol, F., and Rouquerol, J., J. Colloid Interface Sci., 68, 21–31, 1979. Harkins, W. D. and Jura, G., J. Am. Chem. Soc., 66, 1362–1366, 1944. Carmen, P. C., Flow of Gases through Porous Media, Butterworth, London, 1956. Arakawa, M. and Suito, E., Kogyo Kagaku Zasshi, 63, 556, 1960. B Derjaguin, C. R. Acad. Sci. URSS, 53, 623, 1946.
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3.9
Mechanical Properties of a Powder Bed Michitaka Suzuki University of Hyogo, Himeji, Hyogo, Japan
The mechanical properties of powder such as adhesion, cohesion, and flowability may affect powder handling. These properties of powder can be measured by a tensile, shearing, and compression test of the powder bed. In this section, we discuss how to measure the mechanical properties of a powder bed and how to describe the mechanical strength or flowability of a powder bed based on the powder mechanics. Various types of shear testers have been devised to measure the powder yield locus (PYL), consolidation yield locus (CYL), and critical state line (CSL), which are projections of the flow surface, consolidation surface, and CSL, respectively, onto the s- plane in the Roscoe diagram (see Figure 9.9). Several kinds of tensile testers have been also developed to measure the tensile property, which is represented by a curve t1t2 in the Roscoe diagram.
3.9.1
SHEARING STRENGTH OF A POWDER BED
Triaxial Compression Test The schematic diagram of a triaxial compression tester is shown in Figure 9.1. A cylindrical specimen is placed into the space surrounded by two end plates and a rubber side membrane, to which vertical and lateral pressures are applied independently. From the pressures at the break point, the Mohr circle with the maximum principal stress s1 and the minimum principal stress s3 can be obtained, as shown later in Figure 9.13. The yield locus is obtainable as an envelope of a family of the Mohr circles at the critical stress condition. Although the triaxial compression tester is used widely in soil mechanics,1,2 it is complex in structure and has the disadvantage that the failure surface of test samples becomes irregular. Hence, it is not widely used in the field of powder technology except for a few cases. 3,4 Aoki et al. 5 modified the tensile tester of the split-cell type, called the Cohe tester, and made a simplified triaxial tester to measure the PYL of loosely packed powders in a low-stress region. Arthur et al. 6 designed a new biaxial tester with flexible membranes stretched to the retaining guides. The tester imposes known uniform principal stresses on a precisely compacted cubical sample powder and measures the resulting strains in a simple and immediate manner.
Direct Shear Test A direct shear tester has a simple structure and the testing procedure is also simple. Hence, it is widely used in powder technology, especially in the field of bulk materials handling. The direct shear tester usually consists of two cells: one cell is placed on the other cell, which is fixed. They are filled with the particulate material under examination, the test sample is consolidated to attain a prescribed packing density, and then under a compression load, a horizontal shear force is applied to the upper cell so as to measure the yield point. The relation between the compression load and shear force at the yield point gives the yield locus for the prescribed packing density. Various types of testers have been proposed for this purpose. 335 © 2006 by Taylor & Francis Group, LLC
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FIGURE 9.1
Triaxial compression tester.
Jenike’s Shear Tester The shear tester developed by Jenike et al.7 is the most popular and widely used of those designed with a hopper, bin, or silo.8–13 Recently, Jenike’s shear tester and test procedure became an international standard.14 A cross-sectional view of Jenike’s shear cell is shown in Figure 9.2. A test sample is filled in the cell and consolidated by twisting under normal stress sp. Then the ring is sheared under the same normal stress sp until 95% of the yield force measured beforehand is attained, and then the normal stress is replaced by a smaller stress, s (<sp). The cell is sheared again until the specimen reaches yield. In this way, a relationship between s and is obtained. Hirota et al.15 reported coincidence of the results obtained by the Jenike shear tester and his own tester, which is a parallel-plate tester, using the same procedure. On the Jenike shear tester, Haaker16 investigated the effect of the ratio of externally applied load during consolidation and preshearing. Richmond and Gardner11 devised shear blades in the cell to avoid the slip of powder on the wall while in shearing. Tsunakawa and Aoki 17,18 modified the Jenike cell to keep a constant volume and obtained the yield locus from a single shear test by recording the normal and shear stresses simultaneously on an X-Y recorder. Kirby19 subsequently presented a query for their measuring procedure. Terashita et al.20 measured the normal stress distribution in the cell and noted that the normal stress does not apply accurately on the shear surface because of the wall friction. After correcting the normal stress on the shear surface, they obtained the CSL. Matsumoto et al. 21 modified the direct shear tester to measure the actual normal stress on the shear surface. These types of shear tester have limitations in effective shear surface, and Hirota et al.22 pointed out that the size of the shear cell affects the measuring results substantially. Hidaka et al.23 investigated the particle assemblies in the shearing © 2006 by Taylor & Francis Group, LLC
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FIGURE 9.2 Jenike’s shear tester. [From Jenike, A. W., Elsey, P. J., and Wolley, R. H., Am. Soc. Test. Mater. Proc., 60, 1168–1190, 1960. With permission.]
process by computer simulation based on the distinct element method, and discussed the acoustic emission from the shear flow of granular materials.
Annular Ring Shear Tester Annular ring or torsion shear testers are used for cylindrical powder beds. The major advantages of this type are that the area of the shear surface does not change during the test, there is no limitation in the amount of angular displacement, and the shear characteristics after the yield can also be measured. Walker24 made the annular shear tester shown in Figure 9.3. Suzuki et al.,25 Gotoh et al.,26 and Yamada et al.27 used similar shear testers. Scarlett and Todd28 designed a split annular ring shear cell to achieve a simple flow profile of powder, but its structure was complicated. Simple Shear Tester Roscoe et al.29 made a simple shear tester in which the stress can be determined accurately. As shown in Figure 9.4, the guided plunger stresses the test powder vertically, and the shear force is applied to the bottom unit of the tester. This shear cell is the only one of the translational type that can determine the void fraction of the powder bed in the shear zone. Enstad30 developed an annular tester to achieve an unlimited amount of displacement and to make the shear zone wider, as with the simple shear tester, but its structure and measuring procedure are complicated. Parallel-Plate Shear Tester Hiestand and Wilcox31 and Budny32 devised a parallel-plate tester, and Hirota et al.33 installed electric rod heaters in the plates, as shown in Figure 9.5, in order to measure the shear properties at high temperatures up to 800 K. The test powder is placed between two solid plates, the bottom plate is fixed, and the normal and shear forces are applied to the upper movable plate. No sidewall exists, so that it is unnecessary to consider the wall friction. Small expansion and contraction on the order of 1 μm are detected during the shear process.34 Hirota et al.35,36 also devised tensile and compression testers similar to the parallel plates shear tester. © 2006 by Taylor & Francis Group, LLC
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FIGURE 9.3 Annular ring shear tester. [From Walker, D. M., Powder Technol., 1, 228–236, 1967. With permission.]
3.9.2 ADHESION OF A POWDER BED The tensile strength of powder bed st can be measured by several kinds of tensile testers, which are classified to two types: one is based on a vertical tensile method and the other a horizontal tensile method.
Horizontal Tensile Method The tensile force in this method is applied perpendicular to the compression load for consolidation of powder. Ashton et al.37 developed a split-cell-type tensile tester, which is shown in Figure 9.6. © 2006 by Taylor & Francis Group, LLC
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FIGURE 9.4 Simple shear tester. [From Schwedes, J., Powder Technol., 11, 59–67, 1975. With permission.]
Test powder
Normal force
Movable plate Shear force
Fixed plate Electric rod heaters FIGURE 9.5 Parallel-plate shear tester for high temperature shear tests. [From Hirota, M., Oshima, T., and Naitoh, M., J. Soc. Powder Technol. Jpn., 19, 337–342, 1982. With permission.]
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Test powder is compacted into the split cell, and half of the cell is fixed on a plate. The other half of the cell is put on ball bearings for movement. The tensile strength of the test powder is measured by a load cell connected to the movable cell. The tensile strength of a loosely packed powder bed can also be measured using this type of tester because the friction of the movable cell is reduced by the ball bearing, but the cleaning of the balls after each test is time-consuming. Schotton and Harb38 measured the tensile strength of powder using the same kinds of tester on a slanted plate by changing the angle of the plate. Jimbo and Yamazaki 39 developed a modified split-cell bearing-type tester, which consists of two movable cells. The friction of the tester can be reduced, but the structure of the tester is complicated. Yokoyama et al.40 devised a frictionless split cell by hanging one half of the cell, like a swing, as shown in Figure 9.7, the Cohe tester. This is a tester to measure the tensile strength of a loosely packed powder bed in a high void fraction region.
Vertical Tensile Method The tensile load can be applied vertically to the compacted powder; it means that the direction of the tensile force is the same as that of the compression force. Shinohara and Tanaka,41 Arakawa,42 and Suzuki et al.25 developed the tensile testers based on this method. Arakawa42 developed the verticaltype tensile tester from an electric balance, as shown in Figure 9.8. He also tried to explain the effect of a particle packing structure on tensile strength of a powder bed by a two-dimensional particle model experiment. Hirota et al.36 developed a parallel-plate-type shear and tensile tester to obtain the yield loci under the normal stress ranging from positive to negative (including the adhesion force) and measured the PYL in the tensile region, but the operation of the tester was not so easy.
load cell
amplifier
recorder
FIGURE 9.6 Horizontal tensile tester. [From Ashton, M. D., Farley, R., and Valentin, F. H. H., J. Sci. Instrum., 41, 763–765, 1964. With permission.] © 2006 by Taylor & Francis Group, LLC
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The tensile strength measured by the vertical tensile method is usually larger than that measured by the horizontal tensile method, because the direction of compression force at preconsolidation is the same as the direction of tensile force in the vertical method and is perpendicular to the tensile force in the horizontal method. The relation between the tensile strengths by these methods is not clear yet.
3.9.3 YIELDING CHARACTERISTICS OF A POWDER BED Figure 9.9 shows the concept of the Roscoe diagram or Cambridge model proposed by Roscoe et al.,43,44 which expresses the yielding characteristics of a powder bed in relation to shear, tensile, and compression strengths (see section 3.9.1). The coordinates of the diagram are the shear stress , the normal stress s, and the void fraction of the powder bed. The curve t1t2 on the plane = 0, called the tensile property, depicts the relationship between tensile strength and void fraction. The curve f1f2 on the plane = 0, called the compressive property or consolidation line, is the relation between compression strength and void fraction. Only the region between these two curves is possible. In the three-dimensional diagram, two failure surfaces, called the flow surface and the consolidation surface, exist between these two curves. The intersection of them is the curve e1e2 in Figure 9.9, which is called the CSL. The flow surface between curves t1-t2 and e1-e2 corresponds to the plastic flow with expansion, and the projection of the flow surface onto the s- plane is called the powder yield locus, or simply the yield locus. The consolidation surface between curves f1f2 and e1e2 corresponds to plastic deformation with contraction, whose projection onto a s- plane is called
FIGURE 9.7 Horizontal tensile tester called the Cohe tester. [From Yokoyama, T., Fujii, K., and Yokoyama, T., Powder Technol., 32, 55–62, 1982. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 9.8 Vertical tensile tester. [From Arakawa, M., J. Soc. Mat. Sci. Jpn., 29, 881–886, 1980. With permission.]
the CYL. The elastic condition corresponds to the region below the surface in Figure 9.9, while the region above the surfaces is actually impossible. Consider the shear process of a powder bed with the void fraction b in the three-dimensional diagram. At the normal stress sb, the shear stress increases from point C to point E in Figure 9.9, where the steady flow occurs. In this case the stress versus strain curve passes over the line OPES in Figure 9.10. A set of τ and s on the CSL can be obtained from τb and sb of point E in Figure 9.10. When s = sa (sa < sb), τ increases from point H to point P in Figure 9.9, where yielding starts with expansion, and then it decreases gradually along PZ with increasing void fraction from b to a until the powder bed begins to flow steadily at point Z. A set of and s on the PYL can be obtained from d and sa of point P in Figure 9.10. When s = se (se > sb), τ increases from point F to Q in Figure 9.9, where the yielding starts with contraction, and then it increases gradually along QK with decreasing void fraction from b to e until the powder bed starts to flow steadily at point K. A set of and s on the CYL can be obtained from e and se of point Q in Figure 9.10. As a result, the yield locus, which is the relation between and s at the yield point of the powder bed, can be depicted by the curve I⬘A⬘P⬘SQ⬘, which is the projection © 2006 by Taylor & Francis Group, LLC
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FIGURE 9.9 Roscoe’s condition diagram of powder. [From Roscoe, K. H., Schofield, A. N., and Wroth, C. P., Geotechnique, 8, 22–53, 1958. With permission.]
of the curve IAPEQ in Figure 9.9 onto the s- plane. Point Y expresses the preconsolidation state at the starting point of yielding. The yield locus at the state of steady flow leads the curve OZ’SK’B, which is the projection of the curve e1ZEKe2 in Figure 9.9 onto the s- plane and equals the CSL proposed by Schwedes.45 The void fraction decreases with increasing s, and and s are specified for a given value of . If the Roscoe diagram is obtained for a powder bed, the static mechanics of the bed can be discussed quantitatively. The examples of the experimental data are shown in Figure 9.11 and Figure 9.12. These diagrams were obtained from the compression, shear, and tensile testers of parallel-plate type by use of the three-dimensional computer graphic technique.46 Figure 9.11 is for flyash and Figure 9.12 for precipitated calcium carbonate powder. Because the yield locus becomes a straight line on the s- plane, the flyash might be a Coulombic powder, while the upper surface of the yield locus of the calcium carbonate powder looks like a mountain, and it is found to be a cohesive bulk material. If a powder is a rigid plastic or Coulombic solid, its yield locus can be expressed by t ⫽ mi d ⫹ t c ⫽ s tan fi⫹ t c
(9.1)
where and s are the shear and normal stresses, mi the internal friction coefficient, fi the internal friction angle, and c the cohesion (kPa). Using Equation 9.1, the mechanical properties of powder, including the CSL, can be expressed by two parameters, mi or fi and c. The Coulomb equation is applicable in general to cohesionless coarse powders. Because the yield locus is the envelope of a © 2006 by Taylor & Francis Group, LLC
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FIGURE 9.10 Conceptual diagram of shearing process. [From Hirota, M., Oshima, T., and Naitoh, M., J. Soc. Powder Technol. Jpn., 19, 337–342, 1982. With permission.]
family of yield stress circles (Mohr circles), it should cross the s axis perpendicularly at point I′, which corresponds to the tensile strength of the powder bed. As one can see from Figure 9.13, however, the Coulomb equation does not cross the s axis at a right angle, a contradiction explained by Jenike et al.,7 Umeya et al.,47 and Makino et al.48 On the assumption that the particle arrangement is a face-centered cubic and the attractive force between particles is a function of the separation of nearest neighbors with a maximum limit of separation, Ashton et al.49 derived the following relation: n
⎛t⎞ s ⎜⎝ t ⎟⎠ ⫽ s ⫹1 c t
(9.2)
where st is the tensile strength, c is the cohesion of the powder bed, and n is the shear index. In the case of n = 1, Equation 9.2 reduces to Equation 9.1; n is found to be independent of the void fraction and it ranges from about 1 for free-flowing powders up to 2 for very cohesive ones. Umeya et al.47 derived the same result experimentally. The values of st and c in Equation 9.1 and Equation 9.2 vary with the void fraction. Williams and Birks50 suggested that if a family of yield loci are plotted on the /sL versus s/sL diagram, where sL is the normal stress at the end point of a yield locus, they come to lie on a single PYL curve, which is a straight line on logarithmic scales. Hirota et al. 36 pointed out that it is difficult to measure sL; also, Williams’s linearization method cannot apply to the CYL. Therefore, they recommended to use a /(sc – st) versus (s – st)/(sc – st) diagram in order to express various PYL and CYL curves with a single curve. © 2006 by Taylor & Francis Group, LLC
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FIGURE 9.11 Roscoe diagram based on experimental data of flyash powder. [From Suzuki, M., Hirota, M., and Oshima, T., J. Soc. Powder Technol. Jpn, 24, 311–314, 1987. With permission.]
FIGURE 9.12 Roscoe diagram based on experimental data of calcium carbonate powder. [From Suzuki, M., Hirota, M., and Oshima, T., J. Soc. Powder Technol. Jpn, 24, 311–314, 1987. With permission.]
Based on a theoretical model, Rumpf51 derived the following expression for the tensile strength of randomly packed bed of equal spheres: st ⫽
H 1⫺ Nc p Dp 2
(9.3)
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FIGURE 9.13 Yield locus of a Coulomb powder.
where is the void fraction, Nc is the average coordination number. H is the cohesion force at a contact point, and Dp the particle diameter. Molerus52 supposed that Equation 9.3 holds for the relation between the stress and the applied force under the isotropic or hydrostatic pressure, and he analyzed the shear mechanism of a powder bed. Makino et al.48 proposed a model for the yield of a powder bed, taking into account a particle arrangement and interparticle force of a Lennard–Jones type. They discussed a yield condition under constant strain, and the PYL and CYL were made obtainable for any void fraction from only four parameters determined by tensile and compression tests. Yamada et al.27 subsequently reduced the number of parameters required to estimate the PYL and CYL of a powder bed. But, it is difficult to express the interparticle friction by the Lennard–Jones type of potential. Nagao53 proposed a more precise theory of powder mechanics, including equation of stress and moment equilibrium, equation of continuity, a geometric relation, stress–strain relations, and boundary conditions. Based on the fundamental equations, he calculated the stress distribution in compressed powder beds by use of the finite-element method. 54 A principle of similarity in the mechanics of granular materials was also investigated.55. Kanatani 56 established a theory of continuum mechanics for the flow of granular materials, which is an application of the theory of polar fluids. He derived theoretically various characteristics of granular materials, such as the critical stress condition, angle of repose, and the effects of coupling stress in the Couette flow between two parallel plates. Tsubaki and Jimbo57 reported that Rumpf’s equation for tensile strength of a powder bed coincides with Nagao’s theory for the stress–strain relation, Molerus’s research on the yield of cohesive powders 52 and Kanatani’s theory for flow of granular materials. Hence it can be said that Rumpf’s equation is applicable not only to the tensile strength but also to the shear and compression strengths of the powder bed. © 2006 by Taylor & Francis Group, LLC
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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45.
Newland, P. L. and Allely, B. H., Geotechnique, 7, 17–34, 1957. Rowe, P. W., Proc. R. Soc. London A, 269, 500–527, 1962. Masuda, Y. and Sakai, N., J. Jpn. Soc. Powder and Powder Metall., 19, 1–16, 1972. Takagi, F. and Sugita, M., J. Soc. Powder Technol. Jpn., 16, 277–282, 1979. Aoki, R., Suganuma, A., and Motone, M., J. Soc. Powder Technol. Jpn., 9, 538–542, 1982. Arthur, J. R. F., Dunstan, T., and Enstad, G. G., Int. J. Bulk Solids Storage Silos, 1, 7–10, 1985. Jenike, A. W., Elsey, P. J., and Wolley, R. H., Am. Soc. Test. Mater. Proc., 60, 1168–1190, 1960. Jenike, A. W., Trans. Inst. Chem. Eng., 40, 264–271, 1962. Jenike, A. W., Trans. Soc. Min. Eng., 235, 267–275, 1966. Jenike, A. W., Powder Technol., 11, 89–91, 1975. Richmond, O. and Gardner, G. C., Chem. Eng. Sci., 17, 1071–1078, 1962. Williams, J. C., Chem. Process Eng., 46, 173–179, 1965. Eisenhart-Rothe, M. and Peschl, I. A. S. Z., Powder testing techniques for bulk materials equipment, in Solid Handling, McGraw-Hill, New York, 1981. ISO 11697, 1994. Hirota, M., Kobayashi, T., Tajiri, H., Murata, H., Wakabayashi, M., and Oshima, T., J. Soc. Powder Technol. Jpn., 22, 144–149, 1985. Haaker, G., Powder Technol., 51, 231–236, 1987. Tsunakawa, H. and Aoki, R., J. Soc. Powder Technol. Jpn., 11, 263–268, 1974. Tsunakawa, H. and Aoki, R., Powder Technol., 33, 249–256, 1982. Kirby, J. M., Powder Technol., 39, 291–292, 1984. Terashita, K., Miyanami, K., Yamamoto, T., and Yano, T., J. Soc. Powder Technol. Jpn., 15, 526–534, 1978. Matsumoto, K., Yoshida, M., Suganuma, A., Aoki, R., and Murata, H., J. Soc. Powder Technol., 19, 653–660, 1982. Hirota, M., Oshima, T., and Hashimoto, S. J. Soc. Powder Technol. Jpn., 16, 198–206, 1979. Hidaka, J., Kinboshi, T., and Miwa, S., J. Soc. Powder Technol. Jpn., 26, 77–84, 1989. Walker, D. M., Powder Technol., 1, 228–236, 1967. Suzuki, M., Makino, K., Iinoya, K., and Watanabe, K., J. Soc. Powder Technol. Jpn., 17, 559–564, 1980. Gotoh, A., Kawamura, M., Matsushima, H., and Tsunakawa, H., J. Soc. Powder Technol. Jpn., 21, 131–136, 1984. Yamada, M., Kuramitsu, K., and Makino, K., Kagaku Kogaku Ronbunshu, 12, 408–413, 1986. Scarlett, B. and Todd, A. C., Trans. ASME Ser. B, 91, 478–488, 1969. Roscoe, K. H., Arthur, J. R. F., and James, R. G., Civil Eng. Public Works Rev., 58, 873–876, 1963. Enstad, G. G., Proc. Eur. Symp. Particle. Technol., B, 997, 1980. Hiestand, E. N. and .Wilcox, C. J., J. Pharm. Sci., 58, 1403–1410, 1969. Budny, T. J., Powder Technol., 23, 197–201, 1979. Hirota, M., Oshima, T., and Naitoh, M., J. Soc. Powder Technol. Jpn., 19, 337–342, 1982. Hirota, M., Kobayashi, T., Sano, O., and Oshima, T., J. Soc. Powder Technol. Jpn., 21, 137–142, 1984. Hirota, M., Oshima, T., Ishihara, T., and Kanazawa, A., J. Soc. Mater. Sci. Jpn., 33, 1125–1129, 1984. Hirota, M., Kobayashi, T., and Oshima, T., J. Soc. Powder Technol. Jpn., 22, 271–277, 1985. Ashton, M. D., Farley, R., and Valentin, F. H. H., J. Sci. Instrum., 41, 763–765, 1964. Schotton, E. and Harb, N., J. Pharm. Pharmacol., 18, 175–178, 1966. Jimbo, G. and Yamazaki, R., European Symposium Particle Technology Preprints, Vol. B, 1064–1074, 1980. Yokoyama, T., Fujii, K., and Yokoyama, T., Powder Technol., 32, 55–62, 1982. Shinohara, K. and Tanaka, T., J. Chem. Eng. Jpn., 8, 46–50, 1975. Arakawa, M., J. Soc. Mat. Sci. Jpn., 29, 881–886, 1980. Roscoe, K. H., Schofield, A. N., and Wroth, C. P., Geotechnique, 8, 22–53, 1958. Roscoe, K. H., Schofield, A. N., and Thurairajah, A., Geotechnique, 13, 211–240, 1963. Schwedes, J., Powder Technol., 11, 59–67, 1975.
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46. Suzuki, M., Hirota, M., and Oshima, T., J. Soc. Powder Technol. Jpn, 24, 311–314, 1987. 47. Umeya, K., Kitamori, N., Araki, Y., and Mima, H., J. Soc. Mater. Sci. Jpn., 15, 166–171, 1966. 48. Makino, K., Saiwai, K., Suzuki, M., Tamamura, T., and Iinoya, K., Int. Chem. Eng., 21, 229–235, 1981. 49. Ashton, M. D., Cheng, D. C. H., Farley, R., and Valentine, F. H. H., Rheol. Acta, 4, 206–218, 1965. 50. Williams, J. C. and Birks, A. H., Powder Technol., 1, 199–206, 1967. 51. Rumpf, H., Chem. Eng. Technol., 42, 538–540, 1970. 52. Molerus, O., Powder Technol., 12, 259–275, 1975. 53. Nagao, T., Trans. Jpn. Soc. Mech. Eng., 33, 229–241, 1967. 54. Nagao, T. and Katayama, S., Jpn. Soc. Mech. Eng., 46, 355–366, 1980. 55. Nagao, T., J. Soc. Powder Technol. Jpn., 21, 398–405, 1984. 56. Kanatani, K., J. Soc. Powder Technol. Jpn., 16, 445–452, 1979. 57. Tsubaki, J. and Jimbo, G., Powder Technol., 37, 219–227, 1984.
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3.10
Fluidity of Powder Toyokazu Yokoyama Hosokawa Powder Technology Research Institute, Hirakata, Osaka, Japan
3.10.1
DEFINITION OF FLUIDITY
The fluidity of powder is defined intuitively as the ease of flow and relates to the change of mutual position of individual particles forming the powder bed. The fluidity of powder is strongly related to physical properties such as frictional force and cohesive force of the particles. The dynamic behavior of powder seems to be determined basically by interparticle forces and packing structure. Powder flow in various industrial processes takes place in different ways that can hardly be described in a universal form. Table 10.1 classifies the type of powder flow from the practical viewpoint.1 Based on the source of energy exerted on the particles, powder flow is classified as (1) gravitational flow, (2) mechanically forced flow, (3) vibration flow, (4) compression flow, and (5) fluidized flow, which appear simultaneously in actual processes in most cases.
3.10.2
MEASUREMENT OF FLUIDITY
As the phenomena of powder flow are complicated and involve the combined flow patterns listed in Table 10.1, a suitable method of measurement is required for individual cases. Although there are a variety of methods to determine fluidity and related properties, it is difficult to find a general result among them. Therefore, it is essential for measurement to clarify the measuring conditions and powder properties. Furthermore, attention should be paid to the range applicable to the powderhandling processes.
Gravitational Flow The flow from hoppers or through an orifice is often considered to express the fluidity of powder. In this case, the magnitude and uniformity of flow rate, as well as the tendency of choking, are regarded as the criteria of the fluidity. The flow pattern and segregation are also related to the fluidity of powder. In the field of metallurgy, the time required for a powder sample of 50 g to be discharged by gravitational force from the funnel shown in Figure 10.1 is often used as a measure to evaluate fluidity. For cohesive powders, Irani et al.2 proposed a double-funnel method with larger openings. Even if the flow rate is high, a flow with a large fluctuation cannot be regarded as a good measure of fluidity. Several reports have been presented on the uniformity of the discharge rate, especially in the pharmaceutical field. Gold et al.,3 for example, studied the discharge rate and its fluctuation experimentally. Cohesive powders tend to form a bridge near the outlet of the container. The minimum opening size free of choking corresponds to the critical condition for powder flow and can be a measure of the fluidity. For the Coulomb powders, the critical choking aperture can be obtained from Mohr’s stress circle. The mass-discharge rate is proportional to the product of the height of the surface level and the square root of the orifice diameter in the case of liquid; it is independent of the height 349 © 2006 by Taylor & Francis Group, LLC
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TABLE 10.1 Type of Flow and Fluidity Type of Flow Gravitational flow
Process/Phenomena Discharge from bin and hopper, chute, sand-glass, tumbling mixer, moving bed, packing
Expression of Fluidity Discharge rate, angle of wall friction, angle of repose, critical discharge opening
Mechanically forced flow
Powder mixing, chain conveyor, screw conveyor, table feeder, ribbon mixer, rotary feeder, extruder
Angle of internal friction, angle of wall friction, mixing resistance
Compression flow
Briquetting, tableting
Compressibility, angle of wall friction, angle of internal friction
Vibration flow
Vibrating feeder, vibrating conveyor, vibrating screen, packing, discharge
Angle of repose, discharge rate, compressibility, bulk density
Fluidized flow
Fluidized bed, pneumatic conveyor, air slide, aerated vibration dryer
Angle of repose, minimum fluidization velocity, aeration resistance, apparent viscosity
Source: Hayakawa, S., Funtai Bussei Sokuteiho, Asakura, Tokyo. 1973, p. 80.
of the powder bed and proportional to the orifice diameter to the power of 2.5 to 3 in the case of powders. Numerous experimental equations 4 have been proposed for the flow rate of powder discharged by gravitational force. Matsusaka et al. investigated the flow of fine powder using a microfeeder with a vibrating capillary tube having a diameter of less than 1.6 mm.5 Figure 10.2 shows the effect of the inner diameter of a capillary tube d on maximum powder flow rate W max. It was confirmed that fine particles of about 10 μm diameter are successfully discharged using a vibrating capillary tube, and that the maximum flow rate is proportional to the 2–2.5th power of the capillary diameter, which is a little low compared with that for the conventional orifice. This flow rate is also regarded as a measure of fluidity of fine powder. The angle of repose is defined as the angle from the horizontal plane to the free surface of a pile of powder under the critical stress in the gravitational field. There are three principal ways to determine the angle of repose, as shown in Figure 10.3. The angle of repose tends to increase in the following order: injection method, discharge method, and tilting method. The major factors influencing the angle of repose are size distribution,6 surface roughness of the particles, void fraction of the powder bed,7 and moisture content. In the case of the injection method, the injection rate, falling distance,8 and size of the conical pile of powder9 influence the angle of repose.
Forced Flow Mechanically Forced Flow The powder flow actuated by mechanical force has been studied in relation to torque using various types of rotary viscometers: those of Stomer, Couette, Green, and MacMichael. Iiyama and Aoki10 obtained an experimental equation to find the torque for the vortex flow of standard silica sands, in which the apparatus shown in Figure 10.4 was used with various-shaped agitating blades. © 2006 by Taylor & Francis Group, LLC
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FIGURE 10.1 Device for the measurement of fluidity. [From Japanese Industrial Standard Z 2502 (1979).]
Compression Flow In a large vessel such as a huge silo, the value of k in Janssen’s equation is used as a measure of the fluidity. 11 A compression test also gives a measure for the fluidity of powder. A number of equations have been proposed for the relationship between the external pressure acting on the powder bed and the change of volume. One typical experimental equation12 related the apparent volume reduction ratio C to the applied pressure P (Pa): P 1 P ⫽ ⫹ C ab a
(10.1)
where a and b are constants, and a is considered to increase with decreasing fluidity of the powder. © 2006 by Taylor & Francis Group, LLC
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FIGURE 10.2 Effect of the inner diameter of a capillary tube d on maximum powder flow rate Wmax.
Vibration Flow Vibration changes the packing structure of powder bed and, hence, leads to the change of the fluidity. Miwa 13 proposed the following experimental relation between the mass flow rate through sieves and sieve openings: Ws ⫽ kDsn Dp⫺2 ra
(10.2)
where Ws (tons/h) is the mass flow rate, ra (tons/m 3) is the particle specific gravity, D s (mm) is the sieve opening, Dp (mm) is the particle diameter, and n is a constant (about 2.7). The factor k, called the sieve fluidity index, expresses the ease with which particles pass through sieves and is considered to be a measure of fluidity under dynamic conditions. Arakawa14 studied vibration fluidity in relation to the packing structure, paying attention to the vibration in the electric resistance of the electroconductive particles in vibrating motion. Under the © 2006 by Taylor & Francis Group, LLC
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FIGURE 10.3 Methods to measure the angle of repose.
condition of gravitational acceleration, the relation between the electric resistance ratio and the fluidity f is shown in Figure 10.5 and described by the equation ⎛ R ⎞ w ⫽ a exp ⎜⫺b 0 ⎟ R ⎝ x⎠
(10.3)
where a and b are constants, Rx and R 0 are electric resistances of the powder beds in flow and in the most densely packed state, respectively. © 2006 by Taylor & Francis Group, LLC
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FIGURE 10.4 Equipment to determine the mixing resistance.
Sato et al. 15 evaluated the flow characteristics of powders by a vibrating powder tester. A detecting sphere attached to the force transducer is displaced horizontally within the powder bed. The fluidity was discussed in terms of an apparent viscosity obtained from the relationship between the force and displacement velocity.
Shear Properties and Fluidity In powder mechanics, the ultimate balanced condition is estimated from the yield locus obtained by shearing tests. The fluidity defined on the basis of the yield locus indicates the condition where the force balance is about to be lost and is related strongly to the chocking in the pneumatic conveyor or the container outlet. Jenike 16 defined the yield stress f(kPa) and the maximum compaction stress s1 from two Mohr’s stress circles coming in contact with the yield locus, and he called the ratio s1 /f the flow function (FF), which is regarded as a measure of the fluidity. The fluidity increases with the flow function. The following Warren–Spring equation, proposed by Ashton et al., is widely applied to evaluate the shear stress: n
s ⫹T ⎛t⎞ ⎜⎝ ⎟⎠ ⫽ C T
(10.4)
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10
5
1
0.5 0
0.2
0.4
0.6
0.8
1.0
R0/ Rx FIGURE 10.5 Relationship between R o /R x and the fluidity of a vibrating powder bed.
where C (kPa) is the cohesive stress, T (kPa) is the tensile strength, s (kPa) is the normal stress, and t (kPa) is the shear stress. Farley and Valentin 17 took the shear index n as a measure of fluidity and proposed an experimental equation. Stainforth and Berry18 found a linear relationship between s1 /f and se /T and defined the gradient as the fluidity, where se (kPa) is the normal stress at the terminal of the yield locus. One-dimensional yield stress f (kPa) and the bulk density ra (kg/m3) increase with increasing maximum compaction stress s1 (kPa). The ratio f/r a is considered to be another measure of the fluidity: FI ⫽
f ra g
(10.5)
where g is the acceleration due to gravity. The flow index FI becomes larger as the fluidity decreases. Tsunakawa19 studied the relationship between FI and the maximum compaction stress s1 and obtained the following experimental equation from Figure 10.6: FI ⫽ as 1b
(10.6)
where the coefficient a and index b, which are specific to each powder, increase as the fluidity decreases. © 2006 by Taylor & Francis Group, LLC
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100 8 6
Flow index FI [ m ]
4
Lactose Calcium carbonate P-30 White Alundum 2500 # Calcium carbonate P-70
2
10–1 8 6 100
2
4
6
8 10
2
4
6
8 102
FIGURE 10.6 Relationship between the flow index and the maximum compaction stress.
FIGURE 10.7 Relationship between the critical discharge opening and the flowability index. © 2006 by Taylor & Francis Group, LLC
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Fluidized Flow When a powder bed is aerated and agitated, the particles have less opportunity to come in contact with each other, and the powder bed exhibits fluidlike properties. The apparent viscosity of the aerated powder bed measured with a Stomer viscometer20 decreases with increasing aeration rate. Diekman and Forsythe21 regarded the time for the apparent viscosity to reach a certain value as a measure of the fluidity.
Total Evaluation In practical powder-handling processes, the various types of powder flow shown in Table 10.1 occur simultaneously in most cases. Therefore, it is necessary to evaluate the fluidity from the practical viewpoint for the design of suitable processes. Carr 22 proposed a method for the evaluation of fluidity by the addition of several indexes from his abundant experience with a variety of powders. This method was not developed on a theoretical basis but is a numerical synthetic evaluation of fluidity from several different factors. A powder tester has been developed to conduct these measurements within one unit.23 The flowability is obtained with the powder tester as a sum of the indexes converted from the four properties according to a chart: (1) angle of repose, (2) compressibility, (3) angle of spatula, and (4) cohesiveness (or uniformity). It gives an indication of how to select a means of preventing the formation of powder bridge. The floodability index is related to the flushing tendency of powder. It is given as a sum of four indexes converted from the flowability index and the following three properties, using another chart: (1) angle of collapse, (2) the difference between angles of repose and collapse, and (3) dispersibility. In general, the compressibility is more reproducible than the angles and correlates well with the flowability index. Suzuki and Maruko24 found the relationship between the critical outlet width for the powder discharge and Carr’s flowability index experimentally, as shown in Figure 10.7. The flowability index has also been investigated in relation to the sieve fluidity index 25 and the coefficient of mixing rate of the horizontal cylindrical mixer.26
3.10.3
FACTORS AFFECTING FLUIDITY
The fluidity of powder is influenced by various properties of the particles forming the powder, such as particle size and its distribution, shape and surface roughness of the particles, and other interparticle forces. As a general tendency, finer powders show less flowability. Tsunakawa 27,28 obtained an experimental equation to estimate the flow index of fine powders with the particle true density r s (kg/m 3) and voidage from the specific surface diameter Dsp (μm) and major consolidation stress 1 (kPa) accounting for the mean contact number κ: FI ⫽ k ( grs )
⫺1
Dsp
⫺ 2b
⎛ t s 1b ⎞ k ⎜ ⎟ ⎝ k (1⫺ ) ⎠
10.7
where κ and b are constants. Additionally, the influence of voidage has been further discussed.
3.10.4
IMPROVEMENT OF FLUIDITY
The fluidity of a powder can be improved by changing its physical properties, such as moisture content and particle size and shape, by means of drying, grinding,29 classification, and granulation. The fluidity is also improved by altering the dynamic contact of particles, making use of pulsed air pressure. Experimental work on a vibrating two-dimensional hopper30 and the discharge rate through an orifice by vibration 31 and compressed air have been reported. © 2006 by Taylor & Francis Group, LLC
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FIGURE 10.8 The change in angle of repose of composite powder produced by a mechanical treatment method compared with that produced by a simple mixing.
A small amount of additive can improve the fluidity of a powder. The mechanism is not yet elucidated clearly but is suggested to be attributable to the additives’ properties, alteration of the particle arrangement, and the discharge of electrostatic charge, as confirmed experimentally by Nash et al.32 and Jimbo.33 Although the optimum additive fraction varies with the sort of additives, it is effective generally at a low mixing fraction in the range of 0.1–2%. It is well known that an additive has a different magnitude of influence on the fluidity, depending on the type of powder. 34 Furthermore, it has been pointed out that the mechanical treatment for making composite particles changes the fluidity to a great extent. Figure 10.835 shows the changes in the angle of repose of polymethylmethacrylate particles (average particle size 5 μm) treated with TiO2 particles (0.015 μm). It is hardly changed by the simple mixing but decreases drastically by intensive mechanical treatment forming composite particles. It has been also made clear that fluidity shows a maximum value at a certain condition, and excessive treatment deteriorates the fluidity when the surface particles are buried into the core particles.36 It should also be noted that moisture affects the fluidity of powder considerably.
REFERENCES 1. Hayakawa, S., in Funtai Bussei Sokuteiho, Asakura, Tokyo. 1973, p. 80. 2. Irani, R. R., Callis, C. F., and Liu, J., Ind. Eng. Chem., 51, 1285–1288, 1959. © 2006 by Taylor & Francis Group, LLC
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Fluidity of Powder 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.
359
Gold, G., Duvall, R. N., and Palermo, B. T., J. Pharm. Sci., 55, 1133–1136, 1966. Miwa, S., in Funryutai Kogaku, Asakura, Tokyo, 1972, p. 211. Matsusaka, S., Yamamoto, K., and Masuda, H., Adv. Powder Technol., 7, 141–151, 1996. Pilpel, N., Br. Chem. Eng., 11, 699–702, 1966. Otsubo, K., J. Res. Assoc. Powder Technol. Jpn., 2, 179–188, 1965. Aoki, R., J. Res. Assoc. Powder Technol. Jpn., 6, 3–8, 1969. Train, D., J. Pharm. Pharmacol., 10 (Suppl.), 127T–135T, 1958. Iiyama, E. and Aoki, R., J. Chem. Eng. Jpn., 24, 205–212, 1960. Stepanoff, A. J., in Gravity Flow of Bulk Solids and Transport of Solids in Suspension, WileyInterscience, New York, 1969, p. 26. Kawakita, K., Funtai to Kogyo, 9(2), 59–64, 1977. Miwa, S., Funsai (The Micromeritics), 15, 8–11, 1970. Arakawa, M., J. Soc. Mater. Sci. Jpn., 20, 776–780, 1971. Sato, M., Shigemura, T., Hamano, F., Fujimoto, T., and Miyanami, K., J. Soc. Powder Technol. Jpn., 27, 308–314, 1990. Jenike, A. W., Utah Unlv. Eng. Exp. Station, 108, 1961. Farley, R. and Valentin, F. H. H., Powder Technol., 1, 344–354, 1967. Stainforth, P. T. and Berry, R. E. E., Powder Technol., 8, 243–251, 1973. Tsunakawa, H., J. Soc. Powder Technol. Jpn., 19, 516–521, 1982. Matheson, G. L., Herbst, W. A., and Holt, P. H., Ind. Eng. Chem., 41, 1099–1104, 1949. Diekman, R. and Forsythe, W. L., Jr., Ind. Eng. Chem., 45, 1174–1177, l953. Carr, R. L., Chem. Eng., 72, 163–168, 1965. Yokoyama, T. and Urayama, K., J. Res. Assoc. Powder Technol. Jpn., 6, 264–272, 1969. Suzuki, A. and Maruko, O., Funsai (The Micromeritics), 18, 80–88, 1973. Miwa, S. and Shimizu, S., Funsai (The Micromerilics), 16, 103–116, 1971. Yano, T., Terashita, K., and Yamazaki, T., Funsai (The Micromeritics), 16, 96–102, 1971. Tsunakawa, H., J. Soc. Powder Technol. Jpn., 27, 4–10, 1990. Tsunakawa, H., J. Soc. Powder Technol. Jpn., 30, 318–323, 1993. Oshima, R., Zhang, Y., Hirota, M., Suzuki, M., and Nakagawa, T., J. Soc. Powder Technol. Jpn., 30, 496–501, 1993. Suzuki, A. and Tanaka, T., Powder Technol., 6, 301–308, 1972. Yoshida, T. and Kousaka, Y., J. Res. Assoc. Powder Technol. Jpn., 6, 194–201, 1969. Nash, J. H., Leiter, G. G., and Johnson, A. P., Ind. Eng. Chem. Prod. Res. Dev., 4(2), 140–145, 1965. Jimbo, G., in Jitsuyo Funryutai Purosesu to Gijutsu, Kagaku Kogyo, Tokyo, 1977, p. 186. Hayashi, S., J. Res. Assoc. Powder Technol. Jpn., 6, 286–291, 1969. Yokoyama, T., Urayama, K., Naito, M., Kato, M., and Yokoyama, T., KONA, 5, 59–68, 1987. Terashita, K., Umeda, K., and Miyanami, K., J. Soc. Powder Technol., 27, 457–462, 1990.
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3.11
Blockage of Storage Vessels Kunio Shinohara Hokkaido University, Sapporo, Japan
Hiroshi Takahashi Muroran Institute of Technology, Muroran, Japan Among various kinds of bridging phenomena, blockage of storage vessels is dealt with here. It causes stoppage of solids flow and thus leads to unstable operation of powder handling processes, shortages of production capacity, and degradation of product quality. It is one of the serious problems in the operation of storage and supply, including breakage of vessels, unpredictable discharge rate of solids, nonuniform residence time and distribution of particles, and segregation of solids mixtures.
3.11.1
PHENOMENA AND FACTORS
Based on particle properties, blockage phenomena are classified into three kinds. One is a geometrical interlocking of coarse particles at an outlet of the container. Statistically, it occurs when the opening size is smaller than several times the particle diameter. It is based on friction between particles and/or a vessel wall. The particle shape also affects the flow criterion. The second kind of blockage is a mechanical stable arch of cohesive fine powders formed inside the bed. It is directly governed by solids pressure and strength of powder mass and, thus, is closely related to the dimension and shape of the container, storage weights, filling methods, and storage periods. The third blockage is caused by caking with solid bonds due to chemical reaction or physical change, or by an adhered layer to the wall due to electrostatic or physicochemical attractive force. It stems mainly from powder properties, irrespective of the vessel shape. Thus, in addition to the chemical and physical constitution, it is affected by the water content of the powder, humidity and temperature of the operating atmosphere, period of resting time, and solids pressure exerted. From the operational viewpoint, the blockage is sometimes caused by tapping, vibration, and aeration, in addition to gravity. Hammering and vibration are often useful to promote solids flow due to impact force but are dangerous in that they can compact and strengthen the powder bed at the same time. Proper conditions of intensity and frequency of impact are necessary. Aeration from below the bed can support the arch against discharge, though it is effective to reduce wall friction and caking owing to bed expansion. The air pressure and the air flow rate or the aeration period are relevant to this type of blockage.
3.11.2
MECHANISMS AND FLOW CRITERIA
Interlocking of Coarse Particles Even cohesionless particles are able to form a stable arch at a converging section in the container. It happens when the particles are so arranged as to support the weight of solids above the arch by 361 © 2006 by Taylor & Francis Group, LLC
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the frictional force on the wall surface and still not collapse the arch by the reaction force. Thus, the critical opening size ratio to the particle diameter depends hardly at all on particle density but considerably on particle shape. The following are known experimental equations. For general granules,1 Ds 1.8 0.038ws1.8 ds
(11.1)
Dc 2.3 0.071ws1.8 ds
(11.2)
where Ds and Dc are the minimum dimensions of a slit and a circular orifice, respectively, ds is a specific surface diameter of the particle, and ws is the shape coefficient based on the specific surface of the particle of circle equivalent diameter, dc; that is, as compared with Equation 11.1, experimental results with a two-dimensional hopper indicated more variation in ws 10, and that the critical dimension was smaller for spheres and cylinders due to easy orientation for discharge around the outlet, but larger for prisms due to plane contact with each other. For coarse sands and pebbles,2
Do
(1 c ) j 2c
(d 0.081) tan w p
i
(11.3)
where Do is the diameter of a circular orifice or the shortest side length of a rectangular outlet, and c is the length ratio of the long side to the short side. j is a constant of 2.6 for uniform size or 2.4 for general granules, dp is the arithmetic mean diameter of the largest and smallest particles, and wi is the angle of internal friction or the repose angle. In any case, this type of blockage is a rather particular case in storage vessels and seldom occurs in practice except on a screen of aperture size near the particle’s.
Arching of Cohesive Powders Fine powders tend to behave as a continuum due to particle cohesion, and the flow criterion is estimated on the basis of powder mechanics and flow models. In case of gravity flow of cohesive powders, there are two types of flow obstruction, stable arching and piping. Let us consider the criterion to the formation of a cohesive arch in mass-flow bins at first. Assume the following: The solids material consists of a stack of self-supporting free arches, one upon another, and the element arch has uniform vertical thickness T, as shown in Figure 11.1. The arch is formed along the direction of maximum principal stress in the passive state of stress. Perpendicular to the direction of the arch, the minimum principal stress equals zero, because a stress-free surface is assumed. In practice, the material would collapse in the arch due to stress from above, and thus the self-supporting arch would define an upper bound of the critical span for the blockage. Consider a conical hopper with a smooth wall. For the material of the element arch about to fail within itself at the wall, Mohr’s circle MR is shown in Figure 11.2, passing through the origin of T-s coordinates and being tangential to the powder yield locus (PYL). The wall yield locus (WYL) is assumed to be linear. Both can be determined by shear cell test.3 ww is the angle of material–wall friction. fc is the unconfined yield strength of the material as a function of the consolidating — stress, which can be determined by shear cell test. The shear stress acting on the wall surface is ST. The vertical plane makes a as a half of the hopper apex angle anticlockwise from the wall plane, equivalent © 2006 by Taylor & Francis Group, LLC
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FIGURE 11.1 Arch of uniform thickness.
FIGURE 11.2 Mohr’s circles along free arch.
to 2a from point S to point U along the circle MR. Therefore, the shear stress on the vertical plane at — — the foot of the arch is UV, which is expressed as UV ( fc / 2) sin 2(a ww). Since sum of the vertical — forces is given as pDT(UV ) (pD2/4)Trbg, the limiting diameter Dc for a free arch is obtained as4 fc 1 r b gDc 2 sin 2 ( a ww )
(11.4)
where rb and g are the bulk density of the material and acceleration due to gravity, respectively. Thus, the minimum diameter for discharge Dc depends upon the values of powder properties like fc and ww and hopper geometry like a. In the case of a plane-walled wedge-shaped hopper, the limiting span for the slit opening S can be derived as a half of Dc, given by Equation 11.4. In order to evaluate the critical diameter free from blockage, it is necessary that the strength–stress characteristics of powder, fc vs. ss, and the major consolidating stress, sf, set up in the arched field are known. In the case where both PYL and WYL are given by Coulomb-type equations, the limiting diameter is derived by the same consideration as mentioned above.5
Dc
b
4Cs (1 sin wi ) sin 2 ( a b) r b g cos wi ww sin1 (sin ww / sin wi ) 2
(11.5)
(11.6)
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where Cs is the cohesive shear strength of powder and w i is the angle of internal friction, defined by PYL: T s tan wi Cs. As for WYL, it is assumed that the cohesive tensile strength of powder and that between powder and wall plate are equal (WYL: T s tan ww Cw, Cw Cs cot wi tan ww). The properties Cs, wi, and rb are a function of the void fraction, and these increase with consolidating stress. Therefore, the limiting diameter usually increases with the hopper apex angle as well as the bed height.5 The horizontal normal stress sx is assumed to be constant across the arch. Describing Mohr’s stress circle at the horizontal distance x from the apex of the arch by Mx in Figure 11.2, the vertical plane at x is — — characterized by the intersection Z of the line UV and Mx due to constant sx OV across the arch. The angle l between the major principal plane and the vertical plane in Figure 11.1 corresponds to the angle ∠ ZCxE 2l on Mohr’s circle Mx. Accordingly, the following relation holds. Txy dy ZV tan l dx OV OV
(11.7)
The shear stress Txy over the vertical plane at x is given by taking the vertical force balance as Txy —
rbgx/2, and the relation OV ( fc /2) (1 cos2(aww) is derived from Mohr’s circle. Hence, the profile
of the free arch is parabolic, using Equation 11.4. y
x 2 sin 2(a ww ) Dc 1 cos 2 ( a ww )
(11.8)
In the case of a hopper with a rough wall, the value of 45° for a ww in Equation 11.4 and Equation 11.8 is chosen so as to maximize the vertical component of the supporting forces at the — foot of the arch, that is, UV in Figure 11.2. The following is Jenike’s method3,6,7 to determine the critical span of the outlet for flow–no flow criterion. Figure 11.3 illustrates the profiles of the stresses along the wall that are required for
FIGURE 11.3 Determination of critical size of outlet. [From Jenike, A. W., Powder Technol., 1, 237, 1967/1968. With permission.] © 2006 by Taylor & Francis Group, LLC
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the design calculation, where sf is the major principal stress set up in a passive stress field during flow, and fc is the unconfined yield strength representing the strength of the self-supporting arch. As the powder fills the bin, it becomes consolidated under the major consolidating stress (i.e., the major principal stress in the active stress fields ss ). s s after filling is also shown in Figure 11.3. For each value of the major consolidating stress ss, there is a corresponding unconfined yield stress fc. Wherever sf exceeds fc, the powder does not have sufficient strength to support an arch. Therefore, the minimum size of the outlet for mass flow to continue free from arching is determined by the condition s f fc
(11.9)
The condition that ensures steady gravity flow is sf fc, and blockage with a stable arch is sf fc. Jenike defined flow function (FF) and flow factor (ff) as FF ff
ss fc
ss sf
(11.10)
(11.11)
FF depends on the powder property and is determined by shear cell tests as a function of major consolidating stress ss. ff represents the stress condition set up in a hopper indicating the flowability from the hopper. The critical diameter Dc is evaluated by sf 1 r b gDc H ( a)
(11.12)
H(a) is given in graphical form in Figure 11.4, which is derived from the refined arch analysis by considering the variation in thickness of the arch.3,6,8 In the case of a slit aperture of length L longer than three times the width B, Dc represents B. ff is also given in graphical form as a function
FIGURE 11.4 H(a) versus a for conical and pyramidal hoppers. [From Jenike, A. W., Bull. Utah Eng. Exp. Station, 108, 1961; Jenike, A. W., Bull. Utah Eng. Exp. Station, 123, 1964. With permission.] © 2006 by Taylor & Francis Group, LLC
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of effective angle of internal friction d, ww, and a, as shown in Figure 11.5. Reading the value of ff from Figure 11.5 with a given set of d, ww, and a, we can draw a straight line, sf (1/ff)ss, on sf ss rectangular coordinates. The critical value of sf defined by Equation 11.9 is obtained from the intersection of the above-mentioned ff straight line and the FF curve by plotting fc against ss, which is obtained by shear tests. The critical span Dc is then calculated from Equation 11.12. The assumption of constant ff in the hopper means that the profiles of s f and ss are similar, as shown in Figure 11.3. However, if the profile of s s is given as a broken line, the value of ff would not be constant. In such a situation the critical span evaluated would considerably differ from that obtained by the above method.9 For a steep hopper with a smooth wall, Walker’s method is also useful to predict the critical span. The major principal stress exerted during flow, sf, is generally proportional to the span D.4 ⎛ D⎞ s f K ⎜ ⎟ rb g ⎝ 2⎠
(11.13)
The value of K is determined theoretically. Equation 11.4 means that the material can bridge if the unconfined yield strength develops to the critical value fc required for arching due to consolidation during flow under the major principal stress sf. Based on Equation 11.4 and Equation 11.13, Walker defined the critical flow factor of the hopper FFc for the arch to be formed by FFc
sf K sin 2 ( a ww ) fc
(11.14)
The critical unconfined strength fc is obtained from the intersection of the FFc line and the FF curve. Substituting A into fc in Equation 11.4, the critical diameter Dc can be predicted by A 1 ; a ww 45° rb gDc 2 sin 2(a ww )
(11.15)
FIGURE 11.5 ff as ww versus a for conical hoppers. [From Jenike, A. W., Bull. Utah Eng. Exp. Station, 108, 1961; Jenike, A. W., Bull. Utah Eng. Exp. Station, 123, 1964. With permission.]
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For the steep and smooth-walled hopper, such a procedure as is described above is easier than Jenike’s method, because the unknown factor for the calculation is only A. In funnel flow the solids flow down through a channel formed within the material. According to Jenike’s method,6,7 a lower bound of the critical diameter, Df, for no-piping is estimated by
Df
G(wt ) s rb g f
(11.16)
where wt is a static angle of internal friction. The function G (w t) and a flow factor ff p (s s /s f ) for no-piping are reproduced in Figure 11.6 and Figure 11.7. The value of sf in Equation 11.16 is obtained by drawing the ff p line on the FF graph and finding the intersection, which enables Df to be predicted. Nevertheless, piping flow would appear in practice, when the powder head descends lower than a critical level. Based on a block-flow model,10,11 the flow criteria are generally derived under gravity, tapping, and aeration. The model assumes that the portion of the powder directly above the outlet initiates discharge as blocks of aperture size. The minimum opening size of a conical hopper, Do, is obtained under gravity and tapping as
Do
‘
⎡ 1 Yo a1 ⎞ ⎤ km b ⎛ 4 ⎢Cs i ⎜ 1 ⎥ rb g (1 I ) ⎢⎣ a ⎝ ( a 1) (1 Yo ) ⎟⎠ ⎥⎦
(11.17)
1 Yo a1 ⎞ 4 k mi H ⎛ 1Yo a 1 ⎜⎝ 2 ( a 1)(1Yo ) ⎟⎠
FIGURE 11.6 Function G(wt). [From Arnold, P. C., McLean, A. G., and Roberts, A. W., in Bulk Solids, TUNRA Bulk Solids Handling Research Associates, Australia, 1980, p. 3E11, p. 3E30. With permission.]
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FIGURE 11.7 Flow factor for no piping, ffp. [From Arnold, P. C., McLean, A. G., and Roberts, A. W., in Bulk Solids, TUNRA Bulk Solids Handling Research Associates, Australia, 1980, p. 3E11, p. 3E30. With permission.]
where
(
a 2mw cot a sin 2 a k cos2 a b 2Cw cot a k
1 sin wi 1 sin wi
)
(11.18) (11.19) (11.20)
and Yo (height of outlet from hopper apex)/(height of solid surface from hopper apex, H), I is intensity of tapping impact, µi tan wi, µw tan ww, and Cw is the cohesive shear strength between the powder and the wall. The equation indicates that a higher bed requires a larger outlet and higher intensity of impact.
3.11.3
METHODS OF PREVENTING BLOCKAGE
The methods of preventing blockage can be devised on the basis of the above-mentioned mechanisms, and some of them have been adopted industrially. But, sometimes, in reality they may require inverse treatments. Careful and perspective attention to driving discharge against resistance should, therefore, be paid in practice, so that the most appropriate way can be chosen for individual purposes. In designing storage vessels, a bigger outlet is the most effective, and other contrivances are also useful, such as asymmetric shapes with a vertical wall, a steeper wall for the hopper portion, smooth surfaces with smaller angles of wall friction, avoidance of acute corners, and proper inserts. Adjustments of particle properties are preferable, such as uniform and large size, less moisture content, and increased flowability with additives and particle surface treatments, besides smaller amounts and shorter times of storage, if possible. © 2006 by Taylor & Francis Group, LLC
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Mechanical operations and devices are also helpful, such as arch breakers with fluidization and air jets, eccentric vibrating mortors, hammering, agitating breakers with screws, chain conveyors and so forth.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Langmaid, R. N. and Rose, H. E., J. Inst. Fuel, 30, 166, 1957. Pikon, J., Sasiadek, B., and Drozdz, M., Process Technol. Int., 17, 888, 1972. Jenike, A. W., Bull. Utah Eng. Exp. Station, No. 108, 1961. Walker, D. M., Chem. Eng. Sci., 20, 975, 1966. Aoki, R., in Kagaku Kogaku Binran, Maruzen, Ed., Society of Chemical Engineers, Tokyo, p. 1002. Jenike, A. W., Bull. Utah Eng. Exp. Station, No. 123, 1964. Jenike, A. W., Powder Technol., 1, 237, 1967/1968. Arnold, P. C., McLean, A. G., and Roberts, A. W., in Bulk Solids, TUNRA Bulk Solids Handling Research Associates, Australia, 1980, p. 3E11, p. 3E30. 9. Tsunakawa, H., J. Soc. Powder Technol. Jpn., 18, 405, 1981. 10. Shinohara, K. and Tanaka, T., Chem. Eng. Sci., 30, 369, 1975. 11. Shinohara, K. and Tanaka, T., Ind. Eng. Chem. Process Des. Dev., 14, 1, 1975.
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3.12
Segregation of Particles Kunio Shinohara Hokkaido University, Sapporo, Japan
3.12.1
DEFINITION AND IMPORTANCE
In almost all industrial processes in which particulate materials are handled, mixtures that differ in particle properties are often subjected to relative movements. When each component (which may have the same physical properties but different chemical properties) is randomly dispersed throughout the particle bed in a microscopic sense, the result is referred to as “well mixed.” But if one of the key components congregates or is condensed locally in the macroscopic sense, it is called “segregation,” as a sort of separation phenomenon in the dense phase. The mechanisms by which mixing and segregation occur are identical in principle. In mixing, the scale of scrutiny and the partial composition are important as an index of the quality of mixing, but in segregation the location of segregated species is of main concern. Segregation is usually undesirable when product homogeneity is required, but it may be helpful for separation. Especially in the chemical, pharmaceutical, agricultural, and smelting industries, segregation causes serious problems such as uneven quality of fertilizers and tablets, fluctuating packet weight, low mechanical strength of compacts and abrasives, poor refractory materials, and low rates of contact and reaction. In sieving or classification, segregation during stratification or dispersion of particles facilitates subsequent separation.
3.12.2
RELATED OPERATIONS
Segregation is encountered in various operations accompanied by deformation or flow of solids mixtures.
Gravitational Operations In piling up solids on a plane for storage, the component of lower flowability, being smaller, denser, more angular, more frictional, more cohesive, or less resilient particles of relatively smaller fraction (here called a segregating component), collects around the pouring point and forms an inverted conical portion inside the conical heap.1 While, in the initial feeding of solids mixture at a greater rate from a higher position into the narrower cylinder, the larger component of continuous size distribution becomes rich at the center of the bottom plate due to stronger impact.2 In moving beds and storage vessels such as hoppers and bins, the segregating component is deposited in a core region of shape similar to the container after central filling. The mixture flows down with a nearly flat top surface in a mass-flow hopper of apex angle smaller than about 30° and exhibits little segregation during discharge. But in a funnel-flow hopper of larger apex angle, the central region above the outlet first discharges with an excess of the segregating component collected due to segregation in filling, and then the other component in the peripheral portions follows. This causes serious separation with a lapse of discharge time. In the operation of a blast furnace the combined segregation takes place in part of a moving bed and a kind of segregation in filling occurs in the bottom dead zone. The size segregation is usually remarkable.3 Flowing out from a chute linked perpendicularly to a lower conveyor belt, the segregating component tends to settle down near the bottom and the remainder to the outer side of the belt.4 371 © 2006 by Taylor & Francis Group, LLC
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Within a shear testing cell, finer particles move through a deforming bed under large shear strain and congregate near the bottom. Particle size ratio and shape affect the percolation velocity markedly, but normal stress does not.5
Rotational Operations In a rotating horizontal cylinder at a low rate of revolution, the segregating component congregates in the core region of its cross section, and segregation bands extend in the axial direction with stripes of the excess component and the mixture with critical composition. In the case of a rotating horizontal cone, the fine component collects near the wide end and the coarse one near the opposite end, but at a high speed of revolution the order becomes reversed, as is often the case with a conical ball mill.6 Inside a V-shaped mixer the denser component is apt to settle down in any stagnant region.7
Vibratory Operations In a container under tapping8 or an inclined trough under vibration9 smaller particles tend to sink to the bottom, while larger ones float up through the bed. An intermediate layer is present in the mixture. Inverse density segregation of a mixture as lower glass and upper lead particles of the same size will take place in a vertically vibrated cylinder at a higher critical velocity amplitude of 1.0 m/s below the frequency of 90 Hz.10 Differential decrease in mass median diameter of a particle mixture is proportional to the root of the falling height due to tapping impact. Particles circulate upward at the center of the deeper bed. In contrast to larger particles moving to the center, smaller ones descend faster in the vicinity of the cylindrical wall.11
Aeration Operations In a fluidized bed at low air velocity the larger or denser particles become the segregating component as in a liquid. The segregation is caused by a difference in density rather than in size of particles, and it is considerably reduced at larger superficial velocities12 In a spouted bed, larger or denser particles congregate near the center of a circulating annular region above an inlet nozzle. At higher gas velocity the situation in the radial direction becomes inversed, but the segregating component is still abundant around the bottom.13 In pneumatic conveying pipes similar segregation occurs according to particle density and shape.
3.12.3
FUNDAMENTAL MECHANISMS
Since segregation is the inverse of mixing, the mechanism seems to involve the three basic processes of mixing: diffusion, shear, and convection. It is governed by the physical properties of particles, the dimensions of the vessels, and the operational conditions; but these have not yet been elucidated. As a basis for the general case, some simple segregation mechanisms under gravity alone are therefore considered here.
Trajectory Effect When a particle of size d and density r is projected into a fluid of viscosity µ with initial horizontal velocity vh, larger or denser ones can fly farther horizontally by a distance L14: L
vh rd 2 18m
(12.1)
Thus it is possible to cause size or density segregation due to air drag during flying or feeding. But, the denser particle will collect near the feed point in practice. © 2006 by Taylor & Francis Group, LLC
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Rolling Effect After a single particle of radius r moves down a plate of height H and inclination I, it travels a distance L on a horizontal plane. L is written for the rolling and sliding motions, respectively, as follows: L rH (cos2 I )
1 fr /(r tan I ) fr
(12.2)
1 fs tan I fs
(12.3)
L H (cos2 I )
where fr and fs are friction coefficients of rolling and sliding motions, respectively. These suggest that the larger or less frictional particles roll down farther and that only friction segregation takes place during sliding motion, irrespective of particle density.15 According to the motion analysis of an ellipsoidal particle over an inclined plate, it is possible for the smaller or more spherical particle to fly farther from the bottom end of the plate16 But smaller particles will collect near the feed point in practice.
Stumbling Effect It is easier for larger particles to roll over obstacles of height k and to travel farther due to the larger angular velocity w:
(
2 ⎛ ⎡ 2 g r (sin I ) r 2 (r k ) ( cos I )(r k )⎤ r 2 j 2 ⎜ ⎣⎢ ⎦⎥ w⎜ r (r k ) j 2 ⎜ ⎝
)
⎞ ⎟ ⎟ ⎟ ⎠
1/ 2
(12.4)
where j is the radius of gyration. Hence size segregation may occur on the heap surface.17
Push-Away Effect A sphere of diameter dt and density rt placed over two particles on a two-dimensional plane can push away the bottom ones of db and rb in inverse directions against the sliding friction. The criterion is represented by18
(d d ) (r r ) tan u ( d d ) ( r r ) 2 3
b
t
t
b
3
t
b
t
a
(12.5)
b
where ua is a half of the top angle of the isosceles triangle formed by the centers of three spheres. This suggests the possibilities of density and size segregation in terms of size and density ratios. But, larger particles usually tend to go farther.
Percolation Whenever voidage and flowability increase with deformation of the particle bed, the smaller component may easily percolate through the interstices between larger particles. Such size segregation happens under shear strain or while in flow, even for a size ratio as small as 1.53. For the radial dispersion of © 2006 by Taylor & Francis Group, LLC
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small particles through the packed bed, the number of small particles, N, having centers within radius rd at time t is expressed by19 rd No ln No N 4 Er t
(12.6)
where Er is the radial dispersion coefficient and N0 is the total number of percolated particles. A discharge equation for hypothetical hoppers formed within a flowing bed gives the relative falling velocity of small particles.20
Combined Effects With respect to surface segregation of a binary solids mixture in a horizontally rotating cylinder, the combined effect of pushing away and percolation is analyzed to give the relationship among size ratio, density ratio, and volume fraction of coarse particles.21 The ability for a coarse particle to sink in a solids bed based on the potential energy to the work to open a void must be combined with the probability of not finding a void of size large enough for percolation of fines to give a segregation parameter, S,
S
rc ⎛ 1Vc ( drc 1) ⎞ ⎛ ⎧ 1 ⎜ 1 − exp ⎨ ⎟ ⎜ drc ⎝ 1Vc (rc 1) ⎠ ⎝ ⎩ 2 1 ⎤⎫ ⎡ ⎡⎛ ⎞ 1 ⎪ ⎛ 1 ⎞ ⎤ ⎞ ⎥ ⎢⎜ 1 ⎟ 1 ⎬ ⎢1 exp ⎜⎝3 ⎟⎠ ⎥ ⎟ ⎥⎪ ⎣ ⎢⎝ 1Vc ( drc 1) ⎠ ⎦ ⎠ ⎦⎭ ⎣
(12.7)
where rrc and d rc are the density and size ratios of a coarse particle to a fine one, respectively, V c is the volume fraction of coarses, and is the voidage of coarses. S will correlate with the mixing index, M. M 1
Vci Vco
Vc
(12.8)
where subscripts i and o denote inner and outer regions of the cascading layer in the cylinder, and
Vc is the maximum difference of Vc in both regions. Thus, M = 1 means perfect mixing, 0 M 1 segregation with excess coarse particles or percolation of fines, and 1 M 2 segregation due to sinking of coarse particles.
Screening Model When multicomponent mixtures of different sizes flow down an inclined heap surface, the smaller component is screened with the larger one to settle down on the stationary underlayer. Then the oversize fraction fox of the surface layer is represented by applying Equation 12.9 for continuous sieving as
log
fox , n fox , n1
log
fi , n fi, n1
( cn cn1 ) x
(12.9)
where fi is the oversize fraction in the feed, c is a constant, and the subscripts x and n denote, respectively, the distance from the feed point along the surface and the screen number from the top © 2006 by Taylor & Francis Group, LLC
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of the standard screen piled up. The finer component becomes dilute close to the periphery, and size segregation becomes conspicuous with a large c value that increases with smaller feed rate, larger flowability, and wider size range. 1
Screening Layer Model When a binary mixture of solids differing in particle size, density, or shape flows down a heap surface of inclination I, the segregating component of initial mixing ratio Mi is depleted in the mixture at the penetration rate Q by a screening process and descends over the stationary underlayer until it reaches the vessel wall. Then the smaller particles are packed into the interstices in the underlayer at the packing rate P. The segregation process is illustrated in Figure 12.1 and described by material balance in each sublayer of thickness h, velocity v, and voidage 22 as
( h v ) Q cos I 1 M i
hr r r Mi
t
x 1r
(hrs vr )
hrs Q cos I
t
x M i (1 rs ) ( hs vs ) Q P hs cos I 1s t x
hsp
t
P cos I 1s
(P = 0 except size segregation)
(12.10)
As a result, general segregation patterns during the filling of vessels are obtained as the mixing ratio versus the distance from the feed point, x.
FIGURE 12.1 Segregation process during flow on a solids heap. [From Shinohara, K. and Miyata, S. Ind. Eng. Chem. Process Des. Dev., 23, 423–428, 1984. With permission.] © 2006 by Taylor & Francis Group, LLC
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PATTERNS AND DEGREES
The mechanisms leading to the prediction of patterns or degrees of segregation are useful in engineering applications. For storage and supply, there are three cases or typical patterns of segregation: over heap surface, inside layer, and their combination. Surface segregation occurs when feeding the mixture onto the heap. The segregation patterns in the heap are described by variations in the mixing fraction of the segregating component with the traveling distance from the feed point along the heap surface. Figure 12.2 depicts a pattern of density segregation by the screening layer model. Then the degree of segregation is represented approximately by the distance ratio of the segregating core region to the heap periphery. The region increases with higher mixing fraction and feed rate. A bigger difference in particle properties yields the smaller region. In the case of a dead man at the bottom of a blast furnace the multipoint feeding generates the peak of the mixing fraction at the middle of the inclined heap surface, as shown in Figure 12.3.23 A typical segregation pattern of a five-component mixture is illustrated in Figure 12.4 for the case where the feed composition consisted of equal proportions of all five components. The smallest particles (GB1) were found to collect close to the central feed line of a vessel, and the largest particles (GB5) to collect close to the vessel wall.24 Thus, the mixture could be considered to consist of a key component of smallest particles (GB1) and another “pseudo” component containing the remaining larger ones. In the case of multicomponent density segregation, the segregation index Is will experimentally be correlated with operational conditions and particle properties.25 I s 1.25 M
− 0.653
F − 0.150C − 0.219 D 0.631 S − 0.125
(12.11)
where F is the normalized feed rate, C is the number of components, D is the density ratio of densest to pseudolight components, and S is the size ratio of smallest to pseudolarge components.
FIGURE 12.2 Density segregation pattern in a two-dimensional bin by central feed. [From Shinohara, K. and Miyata, S. Ind. Eng. Chem. Process Des. Dev., 23, 423–428, 1984. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.3 Size segregation pattern over a static dead man in a blast furnace by uniform feed. [From Shinohara, K. and Saitoh, J. ISIJ Int., 33, 672–680, 1993. With permission.]
1.0
GB1 GB2 GB3 GB4 GB5
Concentration 1[-]
0.8
0.6
approximation
0.4
0.2
0.0 0.0
0.2
0.4
0.6
0.8
1.0
Scaled distance from center along heap line, [-] FIGURE 12.4 Size segregation pattern for five-component mixture of equal proportions at Ft = 12 cm3/s. [From Shinohara, K., Golman, B., and Nakata, T., Adv. Powder Technol., 12, 33–43, 2001. With permission.] © 2006 by Taylor & Francis Group, LLC
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A kind of segregation index of a multisized particle mixture during the filling of a two-dimensional hopper could, for example, be defined by the concept of partial separation efficiency. 26 The sharpness of separation, which is defined as a slope at the complete or initial mixing fraction of each component, will change along the heap surface. Then the slope of the sharpness of separation at the intersect xis on the scaled distance line will give the intensity of segregation as well as Ssb, as illustrated in Figure 12.5. xis corresponds to the segregation zone. Inside layer segregation refers to separation of the mixture during emptying vessels. The segregation patterns are expressed as the change in mixing fraction of the discharge with time. Figure 12.6 shows one of the patterns where small particles are initially placed on large ones separately. Size segregation of a binary mixture during discharge is affected by the size ratio of particles and the initial and dynamic mixing ratio as well.27 This could be explained through the microstructure by defining the limiting fines fractions on a number basis, where large particles are completely covered by the fines. The maximum packing density during flow is attained in the range 40–60 wt% of fines, and the peak becomes low and sifts to the larger fraction with a smaller size ratio of coarses to fines. As a result, the discharge rate of the binary mixture, Wmix, could be expressed by modification of monosized granules with respect to the so-called empty annulus.28 Wmix 0.58mix (Vf , dr ) g1 / 2 ( Do kmix dmix )5 / 2
(12.12)
where kmix varies between 1 and 2 depending on the kind of continuous phase and on the weight mean diameter of the mixture, dmix. In the case of a ternary mixture of different-sized particles, the continuous phase boundaries can be drawn on a triangle diagram as a function of different size ratios.29 The discharge rate is estimated by substituting the mean diameter of the continuous phase into dmix in Equation 12.12. Thus, the discharge rate under size segregation will change with the particle size ratios and the mixture
Sharpness of separation, Ss
Segregation indices, [-] x is is
Ssb x
xis is
0 Ssb xb
Scaled distance from center along heap line, l [-] FIGURE 12.5 Definition of segregation indices by sharpness of separation with distance along heap line. [From Shinohara, K. and Golman, B. Adv. Powder Technol., 13, 93–107, 2002. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.6 Size segregation pattern during discharge from a two-dimensional hopper. [From Shinohara, K., Shoji, K., and Tanaka, T. Ind. Eng. Chem. Process Des. Dev., 9, 174–180, 1970. With permission.]
composition through microstructural phase transformation in the flow field. But, the rate or extent of segregation during discharge is not predicted here. A combination of these two mechanisms gives the general patterns of segregation. They depend largely on the flow pattern inside the bin and include mixing as well as segregation. In these cases the standard deviation of the segregating component or various other definitions of the degree of mixing are usually adopted as a measure. In the case of a moving bed, where the continuous heaping from above and withdrawal from below the bed occur simultaneously, the segregation pattern generated by the filling of the hopper will give rise to the initial condition of a segregation pattern during withdrawal on the basis of the velocity profile assumed by Equation 12.13 and the penetration effect described by Equation 12.14 and Equation 12.15. ⎛ sin ww ⎞ ⎫⎪ ⎛ u ⎞ ⎤ ⎤ A ⎡ ⎡ ⎧⎪ Vr ⎢cos ⎢ ⎨ww arcsin ⎜ ⎥⎥ ⎬ r ⎢⎣ ⎢⎣ ⎩⎪ ⎝ sin wi ⎟⎠ ⎭⎪ ⎜⎝ uw ⎟⎠ ⎥⎦ ⎥⎦
((
) )
uw / cw
(12.13)
r M rti Vr (rQm cos u) (Qm sin u) 0
r
r
u
(12.14)
Qm qm ( M rti )n
(12.15)
where Vr is the radial velocity, Qm is the penetration flux, rti true density of the ith component, and qm and n are constants. As a result, the segregation profiles in the circumferential direction u and the radial one r are drawn in Figure 12.7.30 In the case of a batch operation, where the particles are withdrawn after the filling of the hopper without feeding, for example, the segregation pattern of a binary mixture of different densities during emptying of a two-dimensional hopper, the segregation time zone expands and the peak decreases with increasing initial mixing ratio of the denser component and the feed rate and with decreasing density ratio, as shown in Figure 12.8.31 © 2006 by Taylor & Francis Group, LLC
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380
80
40
0
-40 Initial Mixing Ratio, M, 0.091 0.188 0.338 experimental calculated
-80 r = 59 cm
0.05
0.00
0.10
(a)
0.15
0.20
0.25
Inclination angle, [rad]
Normalized mixing ratio, (M-M )/M [%] I I
100 Initial Mixing Ratio, M, 0.091
0.188
0.338
80
experimental calculated = 0 rad
60
40
20
0
-20
10
20
30
40
50
60
(b)
FIGURE 12.7 Size segregation profile inside moving bed (a) in the u direction and (b) in the radial direction. [From Shinohara, K. and Golman, B. Chem. Eng. Sci., 57, 277–285, 2002. With permission.]
3.12.5
MINIMIZING METHODS
As long as solids mixtures are handled, segregation is inevitable. Hence some ways to minimize segregation are proposed on the basis of the segregation mechanisms. The particle properties are usually adjusted beforehand, and if possible, it is effective to make them uniform by agglomeration with mixing, surface treatment, humidification, and so on. It may also be useful to increase the initial mixing fraction of the segregating component. Regarding design, tall bins or hoppers of small apex angle are preferable to shorten the flow length during filling and to promote mass flow during discharge. But for already installed bins, vertical partitions or proper insertions will be of some help.32 Multiple outlets, slit openings, and multiple dicharge pipes will improve the flow pattern. Multiple fixed feeding spouts, fixed deflectors, or horizontally moving feeders also prevent heap formation. The following will be useful to reduce segregation such as higher feed rate: less handling, avoidance of vibration, and shorter falling distance. Multiple passes of the mixture through funnel-flow hoppers also results in mixing or reducing segregation to some extent.33 © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.8 Effect of initial mixing ratio on outlet time segregation during discharge. [From Shinohara, K. and Golman, B. Adv. Powder Technol., 14, 333–347, 2003. With permission.]
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.
Miwa, S., Res. Assoc. Powder Technol. Jpn., 26, 1–14, 1960. Gotoh, K., Maki, T., and Masuda, H., J. Soc. Powder Technol. Jpn., 31, 842–849, 1994. Van Denburg, J. F. and Bauer, W. C., Chem. Eng., (Sept. 28), 135–142, 1964. Johanson, J. R., Chem. Eng., (May 8), 183–188, 1978. Bridgwater, J., Cook, H. H., and Drahun, J. A., Ind. Chem. Eng. Symp. Ser., 69, 171–191, 1983 Sugimoto, M. and Yamamoto, K., Kagaku Kogaku Ronbunshu, 5, 335–340, 1979. Yamaguchi, K., J. Res. Assoc. Powder Technol. Jpn., 14, 520–529, 1977. Hayashi, T., Sasano, M., Tsutsumi, Y., Kawakita, K., and Ikeda, C., J. Soc. Mater. Sci., 19, 84–92, 1970. Williams, J. C. and Shields, G., Powder Technol., 1, 134–142, 1967. Ohyama, Y. and Uchidate, I., J. Soc. Powder Technol. Jpn., 35,218–221, 1998. Yubuta, K., Gotoh, K., and Masuda, H., J. Soc. Powder Technol. Jpn., 32, 89–96, 1995. Rowe, P. N. and Nienow, A. W., Powder Technol., 15, 141–147, 1976. Uemaki, O., Yamada, R., and Kugo, M., Can. J. Chem. Eng., 61, 303–307, 1983. Williams, J. C., Powder Technol., 15, 245–251, 1976. Matthee, H., Powder Technol., 1, 265–271, 1967/1968. Shinohara, K., Powder Technol., 48, 151–159, 1986. Miwa, S., in Funryutai Kogaku, Asakura, Tokyo, 1972, p. 222–230. Tanaka, T., Ind. Eng. Chem. Process Des. Dev., 10, 332–340, 1971. Bridgwater, J., Sharpe, N. W., and Stocker, D. C., Trans. Inst. Chem. Eng., 47, T114–119, 1969. Shinohara, K., Shoji, K., and Tanaka, T. Ind. Eng. Chem. Process Des. Dev., 9, 174–180, 1970. Alonso, M., Satoh, M., and Miyanami, K., Powder Technol., 68, 145–152, 1991. Shinohara, K. and Miyata, S. Ind. Eng. Chem. Process Des. Dev., 23, 423–428, 1984.
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382 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33.
Powder Technology Handbook Shinohara, K. and Saitoh, J. ISIJ Int., 33, 672–680, 1993. Shinohara, K., Golman, B., and Nakata, T., Adv. Powder Technol., 12, 33–43, 2001. Shinohara, K., Golman, B., and Mitsui, T. Powder Hand. Proc., 14, 91–95, 2002. Shinohara, K. and Golman, B. Adv. Powder Technol., 13, 93–107, 2002. Arteaga, P. and Tuzun, U., Chem. Eng. Sci., 45, 205–223, 1990. Beverloo, W. A., Leniger, H. A., and Van de Velde, J., Chem. Eng. Sci., 15, 260–269, 1961. Tuzun, U. and Arteaga, P., Chem. Eng. Sci., 47, 1619–1634, 1992. Shinohara, K. and Golman, B. Chem. Eng. Sci., 57, 277–285, 2002. Shinohara, K. and Golman, B. Adv. Powder Technol., 14, 333–347, 2003. Peacock, H. M., J. Inst. Fuel, 11, 230–239, 1938. Kawai, S., Bull. Fac. Eng. Kanazawa Univ., 2, 187–194, 1959.
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3.13
Vibrational and Acoustic Characteristics Jusuke Hidaka Doshisha University, Kyotanabe, Kyoto, Japan
In many treatments of powder in industrial processes, mechanical vibration and sonic energy have been used to increase the efficiency of the operation, and to enhance the transport rates in various industrial operations such as emulsification, drying, and agglomeration. Furthermore, acoustic noise from powder industrial processes has been used for instrumentation of the state variables in the powder industrial processes. Sound, generally, is radiated by the vibration of a surface, the rapid dilation of fluid medium, and the formation of fluctuating eddies in a fluid. In the instrumentation of a particulate system, impact and frictional sound between particles are important to measure the state variables of the processes.
3.13.1
BEHAVIOR OF A PARTICLE ON A VIBRATING PLATE
A plate with an incline of b to the horizontal plane is vibrating with amplitude A, and angular frequency v in direction AB as shown in Figure 13.1. Then, the x and y components of displacement of the plate are given, respectively, by x = A sin v t cos(a b), y A sin v t sin (a b)
(13.1)
and the velocity and acceleration of the plate are described as follows: x Av cos v t cos(a b) y Av cos v t sin(a b) x Av2sin v t cos(a b) y Av2sin v t sin(a b)
(13.2)
The acceleration b of a plate along the direction AB and the component c of the acceleration in gravitational direction are described as follows: x 2 y2 b av2sin v t sin ( a b) cb cos b
(13.3)
The maximum value of acceleration of the plate is rv2 when vt p/2, and then c r v2
sin ( a b) cos b
(13.4)
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FIGURE 13.1 Vibrating plate.
The ratio of the component c in the gravitational direction of acceleration of the plate to gravitation acceleration g is given by
Kv K
sin ( a b)
(13.5)
cos b
where k is centrifugal effect(= K = rv2/g). The relation between displacement and acceleration of the vibrating plate is illustrated in Figure 13.2. For Kv > 1, a particle on the vibrating plate leaves the plate with the velocity of the plate at that time. When Kv equals one, the phase anglefL is given by r v2 sin f L sin w L
(
)
sin ( a b) cos b
g
1
r v 2 . sin ( a b) ⁄ cos b g
1 Kv
(13.6)
The particle contacts with the vibrating plate at 0 w wL and jumps out of the plate at wL w. The coordinate (xL, y L) of the colliding point between the flying particle and the plate is given by xL r sin wL cos ( a b) yL r sin wL sin ( a + b)
(13.7)
The coordinate (x, y) of the flying particle that jumped out of the vibrating plate at velocity L (r coswL) is described as follows x ′ xL vL ( cos a / cos b)
y′ = yL vL sin ( a b) u′ (1/2 ) g u′ 2 cos b
(13.8)
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FIGURE 13.2 Acceleration, displacement of vibrating plate, and motion of particle on the plate.
where u⬘ is the time when the particle jumps out of the plate. We obtained the time and phase angle at the maximum height of the flying particle as follows:
u′ m
vL sin ( a b) g cos b
wm w L +
vL sin ( a b) g cos b
v wL
cos wL sin wL
(13.9)
Then, the flying duration of particle is given by (w0wL)/v, and during the flying, the particle takes the distance along the vibrating plate as follows: ⎛ cos a ⎞ (w0 − wL ) vL ⎜ ⎝ cos b ⎟⎠ v
(13.10)
where w0 is the phase angle at colliding point of the flying particle with vibrating plate. If the phase angle at the colliding time equals wL (w0), the particle jumps again out of the vibrating plate as soon as the flying particle collides with the plate. The displacement of vibrating particle is shown in Figure 13.2.
3.13.2
BEHAVIOR OF A VIBRATING PARTICLE BED
The behavior of a vibrating bed depends on the particle size distribution, particle density, frequency, and amplitude of the applied vibration. 1 The flow behavior of a vibrating bed can be classified into four patterns, as shown in Figure 13.3. The type in Figure 13.2a denotes uniform consolidation, the type in Figure 13.3b denotes the surface flow near the side wall, the type in Figure 13.3c shows inward circulation, and the type in Figure 13.3d shows outward circulation. © 2006 by Taylor & Francis Group, LLC
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FIGURE 13.3 Flow pattern of a vibrating powder bed.
3.13.3
GENERATING MECHANISM OF IMPACT SOUND BETWEEN TWO PARTICLES
Impact sound from spherical particles stems from the particle surfaces suddenly accelerated by the colliding particles.2–4 An acceleration waveform acting on two particles colliding elastically can be obtained using Hertz’s elastic contact theory described below. When spherical particles 1 and 2, having diameters of a1 and a2, respectively, as shown in Figure 13.4, collide with each other at relative speed v0, the following equation holds for the elastic deformation j(t) and deformation acceleration j(t): j(t ) q1q2 j(t )3 2
(13.11)
where
q1
m1 m2 4 ⎛ 1 ⎞ , q2 3p ⎜⎝ d1 d2 ⎟⎠ m1 m2
d1,2
1 v12,2
a1 a2 a1 a2
(13.12)
pE1,2
and m1 and m2 are the masses of particle 1 and 2, n1 and n2 are their Poisson’s ratios, and E1 and E2 are Young’s moduli. Elastic deformation j(t) can be expressed approximately as
j(t ) jm sin
1.068n0 t jm
(13.13)
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FIGURE 13.4 Collision between two elastic particles.
where jm is the maximum deformation of the colliding particles and can be expressed as 2
1
⎡15pn02 ( d1 d2 ) m1 m2 ⎤ 5 ⎛ a1 a2 ⎞ 5 jm ⎢ ⎥ ⎜ ⎟ 16 ( m1 m2 ) ⎥⎦ ⎝ a1 a2 ⎠ ⎢⎣
(13.14)
Assuming that particle deformation acceleration equals gravity center acceleration, the acceleration waveform [a( t ) 1 / 2 ⋅ j( t ) ] of the particle surfaces due to collision can be obtained from Equation 13.11 and Equation 13.13. Let us determine the sound pressure waveform (impulse response) radiated by a particle when unit impulse acceleration acts on it. The velocity potential f(r,u,t) of sound generating when a spherical particle Figure 13.5b oscillates reciprocatingly along the z axis at a speed of U0 exp( jvt) satisfies the following equation:
2 w 2 w 1
⎛
w ⎞ 2 ⎜ sin u ⎟⎠ + k w 0 r r r 2 sin u u ⎝
u
r 2
(13.15)
where k is the wave number. When one has solved Equation 13.15 using the boundary condition that indicates that the speed of the particle surface equals that of the medium, the following velocity potential is obtained: ( a 3 U (1 jkr ) cos u ⋅ e ⎣ w 2 0 r ⎡⎣2 (1 jka ) k 2 a 2 ⎤⎦
j ⎡. vt − k r − a )⎤⎦
(13.16)
Based on Equation 13.16, we can obtain the sound pressure radiated by the particle subjected to unit impulse acceleration as follows, using the method employed by Koss et al.:
Pimp
r0Ca cos u ⎡ ⎛ C ⎞ ⎛ a⎞ ⎛C ⎞⎤ ⎛C ⎞ ⎢cos ⎜⎝ a t⬘⎟⎠ ⎜⎝ 1 r ⎟⎠ sin ⎜⎝ a t⬘⎟⎠ ⎥ e ⎜⎝ a t⬘⎟⎠ r ⎣ ⎦
(13.17)
where t = t – (r – a)/C, r0 is medium density, while C is sound speed. © 2006 by Taylor & Francis Group, LLC
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The pressure waveform of sound radiated by a particle when subjected to a given acceleration j(t) can generally be expressed as follows: 1
P (r , u, t ) ∫ Pimp ( t t ) j ( t ) d t *
(13.18)
0
If we approximate the acceleration resulting from Equation 13.11 and Equation 13.13 by j(t) 1/2·jm sin bt for easier calculation, we obtain the following equation from Equation 13.17 and Equation 13.18 where b p/T, and T is the contact time between two colliding particles: When 0 < t < T, P (ri , ui , ti )
rο m a i3 cos ui 1 2 8 b 4 4li4 ri
(
)
(
⎧⎪⎛ 2ri ⎞ 1⎟ ⎡⎣ 8lli3 b 4li b3 cos b ti 8li2 b 2 sin b ti ⎤⎦ ⎨⎜ a ⎠ ⎩⎪⎝ i
(
)
)
4b 4 sin b t 8li3 b 4li b3 cos b ti i ⎛ 2l ⎞ ⎜ i 1⎟ ⎡⎣ 4b3li 8bli3 cos li ti − 8bli3 4b3li sin li ti ⎤⎦ exp (li ti ) ⎝ ai ⎠
(
)
⎡ ⎢ 4b3li 8bli3 cos li ⎣
(
)
(
)
⎛ π⎞ 3 3 ⎜⎝ ti − 2l ⎟⎠ 8bli 4b li sin li i
(
)
⎛ π ⎞⎤ ⎜⎝ ti 2l ⎟⎠ ⎥ exp (li ti ) i ⎦
rο am ai3 cos ui sin b ti 2ri2
(13.19)
When T < t, P (ri , ui , ti ) =
r0 m ai3 cos ui 1 2 8 b 4 + 4li4 ri
(
(
)
)
{(
⎞ ⎪⎧⎛ 2ri − 1⎟ ⎡⎣ 4b3li − 8bli3 cos li ( ti − T ) ⎨⎜ ⎠ ⎪⎩⎝ ai
)
8bli3 4b3li sin li ( ti T )⎤⎦ exp ⎡⎣li ( ti T )⎤⎦
(
)
(
)
}
⎡⎣ 4b3li 8bli3 cos li ti 4b3li 8bli3 sin li ti ⎤⎦ exp (li ti ) ⎛ p⎞ ⎡⎣ 8bli3 4b3li cos li ⎜ ti T ⎟ 2 li ⎠ ⎝
(
)
⎛ p ⎞⎤ 8bli3 4b3li sin li ⎜ ti T ⎟ ⎥ exp ⎡⎣li ( ti T )⎤⎦ 2li ⎠ ⎥⎦ ⎝
(
)
⎡ ⎛ p⎞ ⎢ 8bli3 4b3li cos li ⎜ ti ⎟ 8bli3 4b3li 2 li ⎠ ⎝ ⎢⎣
(
)
⎫⎪ ⎛ p ⎞⎤ sin li ⎜ ti ⎟ ⎥ exp (li ti )⎬ 2li ⎠ ⎦⎥ ⎝ ⎭⎪
(
) (13.20)
where the subscript i is an index of the spherical particle, li C/ai. Sound pressure P(r,u,t) from two colliding particles at the observation point M(r,u), shown in Figure 13.5, must allow for the sum of sound pressure radiated from particles 1 and 2. © 2006 by Taylor & Francis Group, LLC
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FIGURE 13.5 Impact sound emitted from two colliding particles.
P (r , u, t ) P ( r1 , u1 , t1 ) P ( r2 , u2 , t2 )
(13.21)
Figure 13.6 shows the comparison of a measured pressure waveform and a calculated one. The relationship between the peak pressure of sound Pm and the impact velocity v0 is shown by the following equation:
Pm n0
6 5
(13.22)
The frequency of maximum peak fm in the frequency spectrum of the impact sound is shown experimentally by the following equation as shown in Figure 13.7. fm Dp 1
(13.23)
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Based on these relations, we can measure the state variables of powder industrial processes with the acoustic noise emitted from the processes.
3.13.4
FRICTIONAL SOUND FROM A GRANULAR BED
The sound pressure radiated from the friction between two particles is ordinarily too small to be detected with a microphone, but a granular bed of particles in contact with each other can radiate the frictional sound of which the parameters relate to the frictional properties of the bed.5,6 In the flow and deformation of a granular bed, successive intermittent slip lines are formed in the bed, and the periodicity of the intermittent shear yields relates to the frictional properties of the bed. The frictional sound results from the dilation of particles in the vicinity of the slip line. As results of the dilation, the surface of a granular bed vibrates as a piston, and the sound is radiated into space. Consider, for the sake of simplicity, a plate penetrating at constant velocity into a dense, infinitely wide bed. In the bed, the successive slip lines as shown in an X-ray radiograph (Figure 13.8) are formed, and the intervals, h, between the slip lines are shown by the following equation: h
B ⎧ ⎛ p ws ⎞ ⎛ p we ⎞ ⎫ ⎨tan ⎜ ⎟ tan ⎜ ⎟ ⎬ ⎝ 4 2 ⎠⎭ 2⎩ ⎝4 2⎠
(13.24)
where B is the diameter of the penetrating plate, fs is the static angle of internal friction, and fe is the dynamic angle of internal friction. The sound pressure from the surface of the bed can be © 2006 by Taylor & Francis Group, LLC
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FIGURE 13.6 Pressure waveforms and frequency spectra of impact sound. 391
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FIGURE 13.7 Relationship among peak pressure and frequency of impact sound, impact velocity, and particle diameter.
estimated by the model of sound emission from a piston in an infinite wall. Let the circular piston of radius, a, as shown in Figure 13.9, oscillate with a small amplitude and with a velocity given by U U0 exp(–jvt). The sound pressure p along the center axis is given by the following equation: P j2 r CU sin
kr 2
( 1 a 1) exp ⎧⎨⎩vt j kr2 ( 1a 1)⎫⎬⎭ 2
2
(13.25)
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FIGURE 13.8 X-ray radiograph of slip lines in a powder bed.
FIGURE 13.9 Sound emission by a flat, circular piston in a finite wall.
where k is the wave number, a = a/r. Thus, the relationship between the peak sound pressure Pm and the radius a is shown as follows: Pm 2r CU 0
sin
kr 2
( 1a 1) 2
(13.26)
The increase a in the radius of the oscillating surface of the bed at each shear yield is given as follows: a
h ⎛p w ⎞ sin ⎜ s ⎟ ⎝4 2⎠
(13.27)
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The amplitude of a vibrating surface is shown by the equation A
h ⎛ 1 ⎞ b ⎜⎝ l2 l1 ⎟⎠
(13.28)
where b is the coefficient for describing the average distance between particles in the bed, l1 is the linear concentration of particles before rupture, and l2 is the linear concentration of particles after rupture. Figure 13.10 is a comparison the measured sound pressure waveform with the calculated one. The frequency of the frictional sound can be estimated by the following equation: fa
Vp v 2p h
(13.29)
where Vp is the penetrating speed of the plate into the powder bed. The parameters of this sound closely relate to the frictional properties and the formation of rupture layers in the granular bed.
3.13.5 VIBRATION OF A SMALL PARTICLE IN A SOUND WAVE The vibration of a small particle in a sound wave is of some importance in a variety of areas, such as in acoustic coagulation of aerosols.7–8 This motion has been studied by Brandt and Hiedemann.9 The amplitude Xm of a fluid medium can be written in terms of angular frequency v, wavelength l, maximum amplitude Am, and the distance from the node of the wave to the specified particle l. ⎛ 2p l ⎞ X m Am sin v t sin ⎜ ⎝ l ⎟⎠
(13.30)
The force acting on the oscillating sphere can be presented by the well-known Stokes law for drag as follows: dxp ⎞ ⎛ dx R 3pm Dp 3 pm Dp ⎜ m dt ⎟⎠ ⎝ dt
(13.31)
FIGURE 13.10 Comparison of the measured sound pressure waveform with the calculated one. © 2006 by Taylor & Francis Group, LLC
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where µ is the coefficient of viscosity of a medium, v is the relative velocity between the particle and the fluid, Xp is the amplitude of the particle. The motion equation of the particle in a sound wave is given by
rp
pDp3 d 2 X p 6 dt
2
⎡ ⎛ 2pl ⎞ dX p ⎤ 3pm Dp ⎢ v A cos v t sin ⎜ ⎥ ⎝ l ⎟⎠ dt ⎦ ⎣
(13.32)
The solution yields
Xp
Am sin ( 2pl / l) sin ( v t b) ⎡ ⎛ pr D 2 f ⎞ 2 ⎤ p ⎢1⎜ ⎟ ⎥ 9 m ⎢ ⎝ ⎠ ⎥⎦ ⎣
0.5
(13.33)
where f is the frequency of the sound wave, and b is the phase difference of the particle and medium in oscillation. The ratio Xp/Xm yields Xp Xm
1 ⎡ ⎛ pr Dp2 f ⎞ ⎤ 2 ⎢1⎜ ⎟ ⎥ 9 m ⎠ ⎥⎦ ⎢⎣ ⎝
0.5
(13.34)
where Xm is the amplitude of the medium. The phase difference b is given by ⎛ prp Dp f ⎞ b tan1 ⎜ ⎟ ⎝ 9m ⎠
(13.35)
Figure 13.11 shows the calculated values of Xp/Xm.
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3.13.6 ATTENUATION OF SOUND IN A SUSPENSION OF PARTICLES When a sound wave propagates in a dilute suspension, it is attenuated by the particle-wave interaction, such as the oscillation of the particle and the scattering of the incident wave.10,11 If a plane sound wave propagates in the suspension, then the decay of sound pressure is given by P P0 exp (gx )
(13.36)
where P0 is the sound pressure of incident wave at x 0, g is the damping coefficient, and x is the distance along the path. The decay of energy can also be expected to be exponential: E ( t ) E0 exp (2ge t )
(13.37)
where E0 is the value of E at t 0. If k is the energy removal rate due to one sphere, the energy dissipation rate per unit volume E is given by En⋅k
(13.38)
where n is the number of particles per unit volume. For very small particles, the energy loss rate due to scattering is very small. Therefore the power that must be spent instantaneously to maintain the motion is represented by hp(np – n) in terms of the velocity of particle vp and that of fluid v. Thus, the average energy dissipation rate per unit volume is
(
E n ⋅ hp n p v
)
(13.39)
For a plane sound wave, the energy per unit volume E0 equals (1/2) rn20 where v0 is the velocity amplitude of the medium. Thus the coefficient of amplitude attenuation is obtained by Equation 13.37
g
(
R np v rC n
2 0
) n ⋅ Re ⎡⎣ R% (v%
)
v% ⎤ ⎦ 2 rC n02 p
v a 2 /2l 1, rp r 1
(13.40)
(
(13.41)
For
)
R% 6pm n% p n% , n% p n% n0
j vt 1 j vt
From Equation 13.40 and Equation 13.41, we obtain g
3h mk na v2 t 2 C 1 v2 t 2
(13.42)
where µk is the coefficient of kinetic viscosity of the medium. Rewriting Equation 13.42 in terms of ~
and gC / v and c (4 / 3)a 3 n( rp /r), Equation 13.42 gives
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2 2 %1c v t a 2 2 2 1 v t
(13.43)
The above equation shows that the attenuation per wavelength has a maximum at vt = 1. Thus, the sound waves having the frequency of the order of t−1 are damped the most by the attenuation. The frequency for maximum attenuation per wavelength is given by the following equation: f
1 9mk rp 4p a 2 r
(13.44)
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
Tamura, S. and Aizawa, T., Int. J. Mod. Phys., 7, 1829, 1993. Hidaka, J., Shimosaka, A., and Miwa, S., KONA, 7, 4, 1989. Hidaka, J., Shimosaka, A., Ito, H., and Miwa, S., KONA, 10, 175, 1992. Hidaka, J. and Shimosaka, A., Int. J. Mod. Phys. 7, 1965, 1993. Hidaka, J., Kirimoto, Y., Miwa, S., and Makino, K., Int. Chem. Eng., 27, 514, 1987. Hidaka, J., Miwa, S., and Makino, K., Int. Chem. Eng., 28, 99, 1988. Hoffman, T. L. and Koopman, G. H., J. Acoust. Soc. Am., 101, 3421, 1997. Song L., Koopman, G. H., and Hoffman, T. L., 116, 208, 1994. Brandt, O. and Hiedemann, Trans. Faraday Soc., 32, 1101, 1936. Temkin, S., Elements of Acoustics, John Wiley & Sons, New York, 1981, p. 455. Richard, S. D., Leighton, T. G., and Brown, N. R., J. Acoust. Soc. Am., 114, 1841, 2003.
© 2006 by Taylor & Francis Group, LLC
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Part IV Particle Generation and Fundamentals
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4.1
Aerosol Particle Generation Richard C. Flagan California Institute of Technology, Pasadena, California, USA
An aerosol is a suspension of solid or liquid particles in a gas. Aerosols are generated by a wide range of methods involving either mechanical dispersion of condensed phase materials or condensation from the vapor phase. In this section, we focus on the formation of particles by physical processes, spray technologies, and condensation. The formation of aerosol particles by chemical reactions is discussed in the next section. Table 1.1 classifies the different physical methods for generation of aerosol particles. Representative methods are described below.
4.1.1
CONDENSATION METHODS
Generation of Mists Mists are suspensions of liquid particles produced by vapor condensation onto foreign nuclei. Mists may be generated inadvertently, for example, by releasing hot, vapor-laden gases into a cooler atmosphere, leading to visible plumes. Mists are also generated in the laboratory as a source of relatively uniform organic particles. Table 1.2 lists some of the organic liquids that are commonly used for laboratory mists, along with their physical properties. The classical apparatus for production of laboratory mists is the Sinclair–LaMer aerosol generator1 that is illustrated in Figure 1.1. The apparatus consists of a vapor source, a source of foreign nuclei, a reheater, and a condenser. The vaporizer is maintained at a constant temperature, typically 100°C to 200°C, to provide a uniform concentration of vapor in the carrier gas. Solid nuclei are generated by a spark source, as illustrated, or by vaporization of mineral salts such as NaCl or AgCl, or metals, commonly silver. The seeded vapor then passes through the reheater to homogenize the mixture. Subsequent cooling in the condenser causes the vapor to condense on the seed particles. The size distribution of the resulting aerosol droplets depends on the temperatures of the vaporizer, reheater, and cooling tube, the number concentration of the seed nuclei, and the gas flow rate. The breadth of the size distribution is influenced primarily by the temperature of the reheater. This generator can produce relatively monodisperse aerosols with geometric standard deviations, sG, on particle diameter, Dp, less than 1.2 can be produced with this generator. Typical particle sizes range from 0.03 µm to 2 µm. Number concentrations may be as high as 107 to 108 particles/cm3. One important variant of the Sinclair–LaMer generator is the Rapaport–Weinstock generator,2 which uses atomized droplets as the liquid to be evaporated. Impurities or seed materials in the atomized solution cause each droplet to leave a nonvolatile residue after evaporation. Each such residue particle then acts as a condensation nucleus for subsequent vapor condensation.
Fume Generation Fumes are aerosols of ultrafine solid particles produced by evaporation of metals or salts and subsequent condensation of their vapors, usually in high-temperature systems. Electrical resistance heaters, high-frequency induction furnaces, and infrared ovens are used to heat materials to the temperature at
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TABLE 1.1 Aerosol Generation Methods Methods
Particles
Material
Seeded liquid Evaporation/condensation
Droplets
DOP, DOS, DBP, glycerol
Solid evaporation Condensation
Solid particles
Metals, mineral salts, stearic acid
Gas evaporation method (low pressure)
Nanoparticles
Metals
Droplets
DOP, mineral oil
Solid particles
Powders, beads, fibers
Droplets
H2SO4, photochemical smog
Solid particles
NH4CI, GaAs, Si, SiC, TiO2
Solid particles
Carbon black, TiO2, SiO2, smoke soot
Condensation aerosols
Liquid atomization Pressure atomizers Nebulizers Ultrasonic nebulization Electrospray Spray pyrolysis Mechanical power resuspension Chemical reaction Gas-phase chemical reaction
Combustion
TABLE 1.2
Organic Liquids Used in the Laboratory Generation of Mists Specific gravity
Melting point (°C)
Boiling point (°C)
Compound
Formula
Molecular Weight
Dibutylphthalate (DBP)
C16H22O4
278.4
1.05
-35
340
Dioctylphthalate (DOP)
C24H38O4
390.6
0.99
-55
386
Dioctylsebacate (DOS)
C26H50O4
426.7
0.92
-60
377
Linoleic acid (LA)
C18H30O2
278.5
0.91
—
194
Oleic acid (OA)
C18H34O2
282.5
0.89
14
360
Stearic acid (SA)
C18H36O2
284.5
0.85
71
370
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dilution gas
aerosol out
condensor tube
reheater
electric spark nuclei source filtered carrier gas
aerosol
FIGURE 1.1 Evaporation condensation generator developed by Sinclair and LaMer.
which they will evaporate. Thermal plasmas and pulsed laser ablation are also used, particularly for compound materials for which composition control becomes a problem when less vigorously heated sources are employed. Homogeneous nucleation and condensation of the vapors occur when the hot, vapor-laden gases are cooled. Particle sizes are generally below 1 µm. Rapid cooling, often by dilution with a cold gas, favors the formation of ultrafine particles (Dp < 0.1 µm). A major mechanism of particle growth is Brownian coagulation. In addition to promoting condensation, dilution serves to slow particle coagulation. Nonetheless, agglomerate particles are commonly produced by this method. Fumes produced by direct evaporation of metals have recently received considerable attention in the production of so-called nanostructured materials. Metals or ceramics produced by consolidation of particles finer than 20 nm diameter have a large fraction of their atoms in grain boundary regions, leading to physical properties that differ dramatically from those of materials with coarser grain structures. These properties have been demonstrated using particulate materials generated by the so-called gas evaporation method in which a metal is evaporated directly into a low-density gas, as illustrated in Figure 1.2. Vapors diffuse rapidly from the hot source into the cold surrounding gas © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.2 Gas evaporation method apparatus for synthesizing nanoparticles by evaporation of a metal into a low-density, inert gas. Particles are deposited by thermophoretic diffusion on a cold substrate. Following collection of the nanoparticles, reactant gases such as oxygen may be introduced to produce oxides or other materials. Particles are scraped from the deposition substrate for collection and consolidation.
where they homogeneously nucleate to form nanoparticles. The particles grow by coagulation, forming agglomerates that are collected by thermophoretic deposition on a cold substrate. Although the gas evaporation method is a valuable laboratory tool, particle production rates are small. Numerous efforts are focusing on continuous flow systems that operate at higher pressures with short residence times to increase rates of production of nanoparticles.
4.1.2
LIQUID ATOMIZATION
Liquid atomization is the production of droplets in the act of dispersing a bulk liquid into a gas. The production of droplets from a bulk liquid requires work to produce the additional surface area. Fluid energy is supplied either by the fluid being sprayed or by a second fluid. The ratio of © 2006 by Taylor & Francis Group, LLC
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the kinetic energy of the surrounding fluid to the energy required to form a droplet is the Weber number
We ⫽
rV 2 Dp s
where V is the relative velocity between the liquid and the surrounding gas, and r is the gas density. Droplets form when the Weber number exceeds a critical value. The droplet diameter scales inversely with the square of the velocity and is proportional to the surface tension. Atomization of a liquid can produce liquid particles directly, or solid materials by evaporation of a liquid solvent to leave a residue of a nonvolatile, solid solute or by chemical reactions. Evaporation may also play a critical role in the production of liquid droplets since solvents may be added to reduce the viscosity or to lower the concentration of the solute residue. The final size of the dried particles is directly related to that of the solution being sprayed and the densities of the two phases, that is, dp f = dd (Cvs + Iv)1/3 where Cvs and Iv are the volume fractions of solute and impurities in the solution, respectively. Impurities in the solution limit the size reduction that can be achieved by spraying dilute solutions.
Pressure Atomizers Release of a pressurized liquid through an appropriately designed nozzle disperses the liquid as droplets. The range of sizes produced depends on the nozzle design, the fluid properties, and the flow rate of the liquid. Such atomizers are routinely used to deliver liquid fuels to combustion systems ranging from gas turbines to diesel engines to large boilers and furnaces. Pressure atomizers generally produce relatively large droplets, dp > 10 µm, and often much larger, due to the difficulty in producing large velocities in spraying a relatively viscous liquid. Pressure atomizers can, however, efficiently process large volumes of liquid.
Two-Fluid Atomizers and Nebulizers One way to increase the energy imparted to the liquid and, thereby, to reduce the droplet size is to use a high-velocity gas flow. Air assist and two-fluid atomizers use this technology for large-scale dispersal of a liquid phase into fine droplets. Nebulizers are the laboratory implementation of this technology. The operating principles of the two classes of devices are the same: a high velocity, often sonic or near-sonic, gas flow is blown past the liquid to produce the capillary instabilities that break the liquid into fine droplets. A common laboratory nebulizer is illustrated in Figure 1.3. In this device the Bernouli effect of a high-velocity gas jet blowing past a small tube that supplies the liquid reduces the pressure at the capillary outlet, allowing the liquid to be siphoned from a reservoir. Alternatively, the liquid may be fed by a pump to ensure a constant flow rate. The gas jet breaks the liquid into fine droplets and then impinges on a target. Large particles impact on the target for collection or recycle, so only droplets below a critical size are conveyed from the nebulizer in the carrier gas flow. While properly designed air assist atomizers can disperse the entire liquid feed into a gas flow, nebulizers disperse only a fraction of the liquid. Because of their simplicity of design and operation, however, nebulizers find numerous applications in the laboratory and in medicine. Nebulizers typically produce particles in the 1 to 10 µm diameter size range at number concentrations between 106 and 107 cm−3. © 2006 by Taylor & Francis Group, LLC
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AEROSOL OUT
COMPRESSED AIR IN
LIQUIDIN
EXCESS LIQUIDTO CLOSED RESERVOIR FIGURE 1.3 Constant rate nebulizer of Liu and Lee4 for dispersal of liquid as fine droplets.
Ultrasonic Nebulizers Ultrasonic nebulizers such as the one illustrated in Figure 1.4 use high-frequency acoustic energy to disperse a liquid as fine droplets. The size of droplets depends on the applied frequency f(Hz). Mercer3 suggests an empirical relation for the droplet diameter, ⎛ 8ps ⎞ dd ⫽ 0.34 ⎜ 2 ⎟ ⎝ rf ⎠
13
where s and rl are the surface tension and the density of the liquid, respectively. For frequencies of 0.1 to 10 MHz, droplet diameters are typically 5 to 10 µm. Ultrasonic nebulizers and atomizers have been applied to aqueous solutions, organic liquids, and even molten metals.
Electrospray The energy needed to produce the droplet surface area can also be supplied electrostatically. When a liquid is sprayed into an electric field, charge separation occurs. The extreme limit of this charge separation is the electrospray, which has found extensive use as a method for the introduction of fragile, high molecular weight species into mass spectrometers, but which has also been proposed © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.4
407
Ultrasonic nebulizer.
for the production of particulate materials,5 spraying of liquid fuels,6 and the production of calibration particles. When an electric field is applied to a droplet at the end of a capillary, the droplet will distort into a so-called Taylor cone. This sharp cone is established in the hydrostatic limit when there is no flow through the capillary and it has a half-angle of 49.3. If the electric field is increased above a critical value, flow is induced. Ultimately, this leads to the ejection of liquid from the apex of the cone. That liquid breaks into small droplets that are highly charged. Evaporation of the liquid increases the surface charge density on the droplets. The repulsive forces on a particle carrying q charges balance the surface tension forces at the so-called Rayleigh limit, that is,
(
q ⫽ 8p ⑀0 R 3p
)
1 2
(1.1)
where 0 is the permittivity of free space. As the droplet shrinks below this critical size, it becomes unstable, forms a cusp, and then emits a significant fraction of its charge and a small fraction of its mass as even smaller droplets. 7 A cascade of these Coulombic explosion events may produce finer and finer droplets. The electrostatically driven flow through the electrospray is small. This has limited the use of the electrospray to applications in instruments of analytic chemistry, although arrays of electrosprays can be used to enhance the throughput.5
4.1.3
POWDER DISPERSION
Powders can be dispersed into an air flow by a variety of methods. Most systems that are employed for such dispersion consist of a constant rate feeder and an entrainment apparatus. One such system is © 2006 by Taylor & Francis Group, LLC
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the Wright dust feeder that is illustrated in Figure 1.5. In this device, powder that has been compacted into a rotating cup is scraped from the compact and entrained into a carrier gas flow. Large agglomerates are removed from the gas flow by an impactor. Another approach to powder dispersion is a fluidized bed in which glass or metal beads of about 100 µm in diameter are fluidized to facilitate the dispersal of a finer, dry powder that is continuously fed into the bed. 8 Fine particles are entrained into the carrier gas and carried out as an aerosol, while coarser particles fall back into the bed. The aerosol generation rate can be controlled through the feed rate of the powder and the carrier gas flow rate.
4.1.4
GENERATION OF MONODISPERSE PARTICLES
Uniformly sized particles are needed to calibrate aerosol measurement instruments, to evaluate filter efficiencies, and for inhalation studies and a wide range of research applications. A variety of sources are used to produce so-called monodisperse particles. Each produces a range of particle sizes. For many applications such as filter testing, a polydisperse aerosol with a geometric standard deviation of 1.2 or 1.3 is sufficiently monodisperse. Such aerosols in the 0.1 to 1 µm size range can readily be generated by vapor condensation (Section 4.1.1). Some nebulizers can produce particles with this degree of uniformity for sizes ranging from 0.5 µm to several microns in diameter. Production of aerosols of greater uniformity requires special methods.
Polystyrene Lattices Instrument calibration frequently requires highly monodisperse aerosols. Light-scattering instruments are frequently calibrated in the 0.08 to 10 µm size range using aerosols produced by spraying dilute suspensions of commercially available uniformly sized polymeric particles, most commonly
FIGURE 1.5
Schematic of the Wright dust feeder.
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polystyrene latex (PSL). To prevent the formation of agglomerate particles, the PSL suspension must be sufficiently dilute to ensure that the probability is low that a droplet produced during atomization will contain multiple PSL spheres. Furthermore, very high purity solvents (typically water) should be used to dilute the concentrated suspension since nonvolatile solutes may alter the sizes of the dried particles. Even when high purity solvents are used, surfactants that are added to the suspension to prevent agglomeration during storage and other solutes produce small residue particles, so the resulting aerosol generally has a peak at the size of the PSL particles and a broad residue distribution.
Spinning Disk Generators Spinning disk aerosol generators disperse a liquid that is supplied to the top surface of a rapidly rotating disk.9 The spinning disk may be driven by a motor or pneumatically. The size of the droplets, Dd, produced is related to the angular frequency, v and diameter, d, of the disk, that is., Dd v
rld ⫽K s
where K is an empirically determined constant, ranging from 3.8 to 4.12. Disks ranging from 2 to 8 cm in diameter are typically operated at angular velocities of 102 to 105 rad/s. Droplets are typically produce at a rate of 10 6 to 10 8 particles/min and dispersed into a gas flow smaller than 1 m 3 min −1 . The standard deviation of the droplet size is often quite small, although much smaller “satellite” particles are sometimes generated as the droplets break from the spinning disk. Some versions of the device eliminate satellite droplets from the product aerosol by surrounding the disk with a small sheath flow. Only those particle that are large enough to penetrate through the sheath flow are carried from the generator in the main flow.
Vibrating Orifice Aerosol Generator A capillary jet is unstable and will disintegrate into droplets given sufficient time. Although a range of particle sizes is generally produced, a very uniform aerosol can be generated as the capillary jet is excited at an appropriate frequency. The resulting droplet size is determined by the volumetric flow rate of the liquid, Q (cm3 s−1), and the excitation frequency, f(Hz), namely, ⎛ 6Q ⎞ Dd ⫽ ⎜ ⎟ ⎝ pf ⎠
13
Not all frequencies will produce monodisperse particles. Some frequencies will produce one or more satellite particles. A small jet of air impinging on the aerosol jet from the side is commonly used to determine whether a single droplet size is produced (indicated by a single stream in the deflected jet) or multiple sizes (indicated by multiple streams in the deflected jet). A variety of systems has been constructed to produce monodisperse particles by this method. The most common version, which is shown in Figure 1.6, is the vibrating orifice aerosol generator (VOAG) introduced by Berglund and Liu.10 In the VOAG, a piezoelectric crystal directly vibrates the orifice, thereby exciting the capillary jet. Numerous other modes of excitation have been used in other implementations of this technology. Once a suitable frequency is found, an air flow is introduced along with the aerosol through a small orifice, dispersing the aerosol throughout the gas and minimizing coagulation of the uniform particles produced by the VOAG.
Polydisperse Aerosol Classification Another way to produce an aerosol comprised of uniformly sized particles is to classify a polydisperse aerosol. Below 0.08 µm, the primary method for generating monodisperse aerosols is © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.6 Vibrating orifice aerosol generator developed by Berglund and Liu. [From Berglund, R. N. and Liu, B. Y. H., Environ. Sci. Technol., 7, 147–153, 1973. With permission.]
differential mobility classification.11,12 In this method, charged particles are caused to migrate across a particle-free sheath flow by an imposed electric field. Particles are classified in terms of their electrical mobilities, Z⫽
qD kBT
where q is the charge on the particle, D is the particle diffusivity, kB is the Boltzmann constant, and T is the temperature. A small flow that is extracted through an opening in the electrode opposite from the aerosol entrance downstream of the aerosol entrance carries particles that migrate across the gap between electrodes during the flow time. Ideally, the particles carry only one elementary charge (q = ±e), so the particle mobility provides a direct measure of the diffusivity and, indirectly, of the particle size through the Stokes–Einstein relation D=
kTCc 3pmd p
where µ is the viscosity, and ⎛ ⎡ g ⎤⎞ Cc ⫽ 1 ⫹ Kn ⎜ a ⫹ b exp ⎢⫺ ⎥⎟ ⎝ ⎣ Kn ⎦⎠
(1.2)
is a slip correction factor that accounts for noncontinuum effects for particles with sizes comparable to or smaller than the mean-free-path of the gas molecules, l. The empirically determined coefficients are © 2006 by Taylor & Francis Group, LLC
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a = 1.207, b = 0.596, and g = 0.999.13 The differential mobility analyzer can produce aerosols ranging from 1 µm to as small as 0.001 µm with geometric standard deviations of 1.1 or smaller, although the distributions broaden for very small particle sizes.14,16 For particle sizes above about 0.1 µm diameter, multiple charging leads to multiple peaks in the size distribution of the transmitted aerosol, although methods have been developed to minimize this effect. Classification of particles at larger sizes can be accomplished by inertial separation using a combination of inertial impaction ontos a substrate and a virtual impactor. The size distribution produced by this method is generally broader than that produced by the differential mobility analyzer, although there is no fundamental reason why sharper inertial classification cannot be achieved.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Sinclair, D. and LaMer, V. K., Chem. Rev., 44, 245–267, 1949. Rapaport, E. and Weinstock, S. E., Experimentia, 11, 364–363, 1955. Mercer, T. T., Arch. Intern. Med., 13, 39–50, 1973. Liu, B. Y. H. and Lee, K. W., Am. Ind. Hygiene Assoc., 36, 861–865, 1975. Rulison, A. J. and Flagan, R. C., Rev. Sci. Instrum., 64, 683–686, 1993. Gomez, A. and Chen, G., Combust. Sci. Technol., 96, 47–59, 1994. Smith, J. N., Flagan, R. C., and Beauchamp, J. L. J., Chem. Phys. A, 106, 9957–9967, 2002. Guichard, J. C., in Fine Particles, Liu, B. Y. H., Ed., Academic Press, New York, 1976, p. 173. Mitchell, J. P. J., Aerosol Sci., 15, 35–45, 1984. Berglund, R. N. and Liu, B. Y. H., Environ. Sci. Technol., 7, 147–153, 1973. Knutson, E. O. and Whitby, K. T. J. , Aerosol Sci., 6, 443–451, 1975. Knutson, E.O. and Whitby, K. T. J., Aerosol Sci., 6, 453–460, 1975. Allen, M. D. and Raabe, O. G., Aerosol Sci. Technol., 4, 269–286, 1985. Kousaka, Y., Okuyama, K., and Adachi, M., Aerosol Sci. Technol., 4, 209–225, 1985. Kousaka, Y., Okuyama, K., Adachi, M., and Mimura, T. J., Chem. Eng. Jpn., 19, 401–407, 1986. Stolzenburg, M. R., Ph.D. Thesis, University of Minnesota, 1986.
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4.2
Generation of Particles by Reaction Richard C. Flagan California Institute of Technology, Pasadena, California, USA
Yasushige Mori Doshisha University, Kyotanabe, Kyoto, Japan
4.2.1
GAS-PHASE TECHNIQUES
Particle synthesis by reactions in the gas phase begins with seed generation, generally by a burst of homogeneous nucleation when reactions produce condensible products or when volatile products are quenched. This may be physical nucleation of a single condensible species, but often involves complex chemical reactions. Once seed particles are present, they may grow by condensation or physical vapor deposition, chemical vapor deposition, or coagulation. The latter mechanism is the collisional aggregation of small particles to form larger ones. The processes involved in particle synthesis from the vapor phase are summarized in Figure 2.1. Because it is a second-order process, coagulation dominates when the number concentrations of particles in the nucleation burst is large, which is normally the case in industrial powder synthesis reactors. Since liquid particles coalesce upon coagulation, spherical particles result unless the viscosity is extremely high. Coagulation of solid particles or of very high viscosity liquids, notably silica, often results in the formation of agglomerate particles comprised of a large number of smaller primary particles. Such agglomerate formation frequently begins after a period of coalescent coagulation in high-temperature regions of the reactor, leading to a relatively uniform primary particle size that is sometimes misinterpreted as evidence that coagulation has ceased. Instead, coagulation accelerates once agglomerates begin to form due to the increased Collision cross section that result from agglomerate formation. Strong bonds may form between the primary particles by solid-phase sintering, leading to hard agglomerates that are often difficult to disperse. Such agglomerates are undesirable for many applications but impart desirable properties in others, notably when used to reinforce soft polymers.1
Coagulation and Aggregation Kinetics Particle growth by coagulation is described by the coagulation equation, also known as the Smoluchowski equation, ∞ dn 1 v ⫽ ∫ b ( v%, v ⫺ v% ) n ( v% ) n ( v ⫺ v% ) dv ⫺ ∫ b ( v%, v ) n ( v% ) n ( v ) dv% 0 0 dt 2
(2.1)
where n(v) is the number concentration of particles with volumes between v and v + dv, b (v, ~ v ) is the collision frequency function for particles of volume v with those of volume ~ v . The structure of the particles, and their size relative to the mean-free-path of the gas molecules, l, determines the form for b. Most aerosol synthesis reactors produce small particles at elevated temperatures, so that Rp
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Growth by coalescent coagulation
Precursor vapor
or
Growth by aggregation
Growth by vapor and cluster deposition Precursor or vapor FIGURE 2.1
Mechanisms of particle formation and growth from vapor-phase precursors.
Coagulating liquid particles are generally spherical, but solid aggregates form more complex structures. In many cases the structure of particulate aggregates approaches a self-similar form wherein the particle mass scales with the radius of gyration of the agglomerate, Rg, as ⎛ Rg ⎞ mp 艐 m0 ⎜ ⎟ ⎝ R0 ⎠
Df
(2.2)
where Df is the mass fractal dimension, which is generally smaller than the Euclidean dimension of 3, often about 1.8 for highly agglomerated particles, and m0 ⫽ rpv0 is the mass of the primary particle. © 2006 by Taylor & Francis Group, LLC
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The particle size distribution that results when particles grow by coagulation approaches an asymptotic shape known as the self-preserving particle size distribution. For particles that grow as dense spheres, the size distribution is frequently approximated as log normal (even though it is notably asymmetric), with a geometric standard deviation of approximately 1.45 if the particles are much smaller than the mean free path of the gas molecules (free-molecular regime), and 1.35 if they are larger than the mean free path (continuum regime). Agglomerate particles also approach a self-preserving particle size distribution, albeit with a different geometric standard deviation that depends on the agglomerate structure. The self-preserving particle size distribution model allows one to predict the decay in the total particle number concentration as2 2
v
v
1
5
v
⫺ ⫺ ⫺ dN ⫽⫺21⫺v akrv03 Df f Df 2 N 2 Df dt
(2.3)
where a is a dimensionless constant whose value depends on the fractal dimension and the shape of the self-preserving size distribution (a ⫽ 6.67 for dense spheres, Df ⫽ 3), 1
v
1
⎛ 6 k bT ⎞ 2 ⎛ 3 ⎞ Df ⫺2 2⫺ D6 Rg f k⫽ ⎜ ⎟ ⎟ ⎜ ⎝ rp ⎠ ⎝ 4p ⎠
(2.4)
v ⫽ Min(2, Df ), kb is the Boltzmann constant, and T is the temperature, and f is the volume fraction of particulate material. The mean particle volume is –v ⫽ f/N. Aerosols with size distributions that are narrower than the self-preserving distribution can be generated if growth by vapor deposition or condensation dominates over coagulation. This generally requires relatively low particle concentrations, which leads to low production rates, or very short growth times. Charging the particles can inhibit coagulation to facilitate growth by vapor deposition to produce a narrow size distribution.
Agglomeration Control Highly desirable for applications such as polymer reinforcement, rigid agglomerate particles are deleterious for many others. Because agglomerate particles have a much larger collision cross section than do dense spherical particles of the same volume, once agglomerates begin to form, they quickly scavenge the remaining small dense particles and become the dominant form of particle. In many particle synthesis technologies, particles are initially formed at high temperature where coagulating particles quickly sinter into dense spheres. If the time scale for that coalescence is small compared to the time between collisions, dense particles will be grown. If the coalescence time is very long compared to the time between collisions, aggregates will form, but the necks between the primary particles will remain small, and the agglomerates can be broken apart in later processing. If, however, the two times are comparable, significant necking will occur, and hard agglomerates will be formed. Following high temperatures during the growth phase by rapid quenching to a temperature so low that sintering is insignificant enables formation of powders that can readily be dispersed as primary particles.
Types of Particles Synthesized Table 2.1 lists some of the many approaches to powder synthesis and the powders produced by those methods. Major commercial aerosol–synthsized particulate materials include carbon black, titanium dioxide, and fume silica, although a much wider range of compositions can be generated. Carbon black is used as structural reinforcement in polymer composites employed in tires and a wide range of other © 2006 by Taylor & Francis Group, LLC
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TABLE 2.1
Methods for Production of Fine and Ultrafine Powders
Method Fine Powder
Description
Powder Charactereristics
Ref.
Evaporation and condensation
Vapor of a low-boiling-point metal is cooled.
Zn
16
Melt atomization
Molten metal is sprayed.
Metals: DP >10 mm
Spray pyrolysis
Solution containing salt or other precursor compound is sprayed. Solvent is evaporated and precursor is reacted at elevated temperature.
Metal oxide
Gas evaporation method
Metal is vaporized into cool, low-pressure gas.
Metals: DP > 20mm
7, 10
Thermal decomposition aerosol reactor
Thermal decomposition of precursor vapor in heated reactor.
Fe, Ni: from carbonyls; Si from SiH4; Al2O3, SiO2, TiO2 from alkoxides 5 nm < DP < 1mm
19, 20
Aerosol flow reactor
Reaction of vapor-phase precursors in heated flow reactor.
S, SiC, Si3N4 AIN, TiN, BC, etc. 5nm < DP <1mm
2, 3
Flame synthesis
Combustion of precursor vapor in hydrocarbon flame.
Carbon black, TiO2, SiO2 10 nm < DP < 1mm
25
Ultrafine Powder
Combustion of precursor vapor in Na flame Thermal plasma reactor
Decomposition of vapor or liquid precursor or direct evaporation of material in thermal plasma leading to particle formation in postplasma region.
Metals, carbides, nitrides, borides, oxides
13, 14, 27, 28
Laser synthesis
Laser-induced reactions of vapor-phase precursors.
Si, SiC, Si3N4, WC
4, 5
applications. Titanium dioxide particles of the rutile phase in the near-submicron size range (typically 0.2 to 0.3 m) have a high refractive index that scatters visible light efficiently. Such particles are used to increase the opacity of paints and pigments. The structures of the particles can vary widely: structural carbon blacks are generally ramified agglomerate particles, while the dense nonagglomerated particles are generally preferred for pigment applications. With the exception of TiO2 pigment particles, most particles produced from the vapor phase are in the ultrafine size range (Dp < 0.1 m). With potential applications in magnetic storage media, catalysis, ceramics, and electronic devices among many others, ultrafine particles are the subject of considerable interest. Even smaller particles in the nanometer size range (Dp < 0.05 m) have attracted considerable interest because they can impart unique physical, chemical, electronic, and optical properties to structures fabricated from them. Ceramics and metals produced from such nanoparticles acquire unusual physical properties as a result of the large numbers of atoms that reside in grain boundary regions when the particles are consolidated to form a macroscopic structure. The high surface energies of such materials lower melting points and allow consolidation at much lower temperatures than would be required © 2006 by Taylor & Francis Group, LLC
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for composites of larger particles. Quantum confinement of charge carriers in nanoparticles increase the band gaps of semiconductor materials, altering optical, electronic, and chemical properties. These special properties have greatly increased interest in nanoparticle synthesis. Flame Synthesis The predominant technology used to produce particles from the vapor phase is flame synthesis. Carbon black, titanium dioxide, and fume silica particles are produced by this method in which a precursor vapor-phase compound is burned to produce condensible liquid or solid-phase products. For carbon black synthesis, combustion of a hydrocarbon fuel (either fuel oil or natural gas) heats the reactants to the high temperature required to drive the chemical reactions and provides the material from which the particles are formed. Soot particles are rapidly produced in the fuel-rich flame and then coagulate to generate agglomerate particles that are quenched and then separated from the gas flow. Titanium dioxide particles are produced by combustion of titanium tetrachloride (TiCl4). The oxidation of TiCl4 is not sufficiently exothermic to raise the reactants to a high enough temperature to achieve rapid reactions, so another energy source is generally required. Most commonly, that energy is supplied by combustion of a hydrocarbon fuel, although thermal plasmas have been used to supply the energy while avoiding the introduction of hydrogen that leads to production of large quantities of HCl vapor that severely aggravate corrosion problems in TiO2 synthesis reactors. The slow reaction kinetics of the TiCl4 precursor allow chemical vapor deposition to play a more important role in the growth of TiO2 particles than it does in carbon black synthesis. Combined with the low melting point of TiO2, that aids in the production of dense particles. Flame synthesis methods can be extended to a wide range of oxide materials. Metallic, carbide, boride, and nitride particles can be produced using more exotic flames. For example, combustion of TiCl4 by sodium vapor has been used to produce TiBr2 particles.3 Thermal Reactions of Volatile Precursors A wide variety of particulate materials can be synthesized by gas-phase reactions of volatile precursors. Combustion reactions were discussed above. Thermal decomposition of metal carbonyls has been used to generate ultrafine Ni and Fe powders. Silicon particles ranging in size from 1 nm to 5 m have been generated by thermal decomposition of silane gas (SiH4). Silicon carbide and silicon carbide, which have promising applications for ceramics, have been synthesized by vapor-phase reactions. SiC has been produced by gas-phase reactions of CH3SiCl3 or of SiH4 with C2H4. Si3N4 is produced by reaction of SiH4 with NH3. Both amorphous and highly crystalline particles with sizes ranging from 1 nm to 0.2 m have been produced by these routes. Moreover, a wide range of compound particles have been synthesized, including particles with a core of one material and a shell of another. As one example, Si nanoparticles encapsulated within SiO2 have been produced by first growing silicon nanoparticles by silane pyrolysis and then forming an oxide coating on that core. Two coating methods were demonstrated: (1) chemical vapor deposition of tetraethyl orthosilicate and (2) partial oxidation of the outer portion of the silicon particle. Laser Synthesis of Ultrafine Particles Laser-induced reactions of precursor gases have been used to produce a variety of particulate materials including Si, SiC, and Si3N4.4,5 Laser irradiation is used to provide the energy needed to drive the chemical reactions. Ultrafine particles have been generated by this route. Nonagglomerated SiC and Si3 N4 powders were produced when the SiH 4 precursor was first decomposed by laser irradiation, producing molten silicon particles that were later reacted with C2H4 or NH3 that was injected downstream of the primary reaction zone. Agglomerate particles were produced when the reactants were injected simultaneously. © 2006 by Taylor & Francis Group, LLC
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Gas Evaporation Method Evaporation of a metal into a low-density gas has long been used to produce ultrafine metal particles. 6–10 The material is evaporated into an inert gas where ultrafine particles are produced by vapor condensation as the vapors diffuse into the cool gas surrounding the vapor source. The method has been applied to a wide range of materials. Pure metals have been produced by direct evaporation, as has SiC,11 producing particles ranging in size from 2 to 200 nm.7 The size of particles produced depends on the gas and the pressure, with finer particles being produced at lower pressures, consistent with particle growth primarily by coagulation. Plasma Synthesis of Ultrafine Particles Thermal plasmas produce temperatures in the range of 5000 to 10,000 K. At such high temperatures, precursor materials can be vaporized, or precursor compounds can be decomposed.12–13 The resulting vapors condense when the vapors cool, generating large numbers of ultrafine particles by homogeneous nucleation. Gas-phase reactions can be carried out to produce compound materials such as SiC, Si3N4, or TiO2. Particles can be produced in large quantities by this method, with particle sizes ranging from 10 nm to nearly 1 m. Lower-intensity plasmas provide a highly reactive environment that has been employed in synthesis of particles with a wide range of compositions from gas-phase precursors. Inductively coupled plasmas have also been used to synthesize highly uniform silicon nanoparticles at slightly reduced pressures.15 Microwave plasmas have been applied to a wide range of metal and ceramic nanoparticle syntheses.
4.2.2
LIQUID-PHASE TECHNIQUES
When a gradual increase in the concentration of the required materials for particles in solution is produced by the chemical reaction, or the physical techniques, precipitation usually does not occur as soon as the concentration reaches the saturation value.29 As the concentration reaches a certain degree of supersaturation value, nucleation is induced and then the nuclei ultimately grow into the particles. Methods of preparing powders by the supersaturation of the concentration of the required materials could be classified into the techniques listed in Table 2.2 where inorganic or organic materials are separated. The chemical technique is the precipitation from homogeneous solution by chemical reaction or chemical equilibrium. On the other hand, the physical technique means that the solution is first divided into small isolated droplets, and then the powder is made by the evaporation of the solvent or by the chemical reaction in the individual droplets. The solvent evaporation methods in inorganic powder and the mechanical dispersion methods in organic powder in Table 2.2 may be of the physical technique.
Inorganic Powder If the generation rate of inorganic powder is fast, most of the particles will be in the amorphous state or the aggregate state consisting of many small crystals, so that the shape of the powder is nearly spherical. The particles will grow to nearly a single crystal by a slow generation speed.30 The generation method of the oxide powder generally consists of two steps: formation of precursor salt or hydrated oxide, and thermal decomposition.
Solvent Evaporation Methods These methods are of physical techniques; that is, the source solution is divided into small droplets of liquid, and the powder is formed by the evaporation of the solvent in the liquid droplets. These methods are applicable for any multicomponent solution and are superior to the precipitation methods, because © 2006 by Taylor & Francis Group, LLC
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Methods of Preparing Powders from Solution
Inorganic Powder Solvent evaporation methods
Freeze drying Spray drying Spray pyrolysis
Precipitation methods
Addition of precipitating agent Hydrolysis Redox reaction Decomposition of compound
Special reaction field
Hydrothermal Supercritical fluid Microemulsion
Organic Powder Mechanical dispersion methods Mixing methods interfacial reaction Capillary methods Phase-separation methods
Other method
Suspension polymerization Emulsion polymerization Mini-emulsion polymerization Vibration nozzle Porous membrane Soap-free polymerization Micro-emulsion polymerization Inverse micelle polymerization (Nonaqueous) dispersion polymerization Coacervation Seeding polymerization
they keep homogeneity in the source solution if the solute cannot evaporate.31 The powders produced are usually spherical and porous, and their size can be controlled by the solute concentration in the feed solution. Spray methods are the usual way to disperse to small droplets. These are called freeze drying, spray drying, and spray pyrolysis, depending on the operating temperature. In the freeze-drying technique, the solutions are atomized into the refrigerant, and then the sublimation of the solvent produces the dry powders. The specific surface area of the powder decreases as the temperature increases in the sublimation process. The spray-drying technique uses the chemical reaction between the droplets and the environmental hot gas. In spray pyrolysis, a solution is atomized in an atmosphere of temperature high enough to decompose the metal salt into the final oxide. Roy 32 reported a laboratory-scale process for the production of powders using this method. Sometimes the high temperature is provided by burning combustible solvents such as alcohol. The metal or alloy powder can be produced directly from a solution by the reduced reaction in hydrogen or nitrogen gases.
Precipitation Methods In the precipitation methods, homogeneous nucleation and growth can occur in the solution, and then the LaMer model33 is taken as the principle to produce monodispersed particles, in which the nucleation stage should be separated from the growth stage. Sugimoto34 reviewed the particle formation process to control size and shape based on this model, with many experimental examples, including the process for organic powders. Recently, some studies for particle formation used new analytical techniques, and the results that were found could be explained by the aggregation model rather than the LaMer model.35 Precipitation methods are classified into the addition of precipitating agents, hydrolysis, redox reaction, and decomposition techniques by the reaction type. Techniques of the addition of precipitating agents are further classified as to the formation of metal hydroxides, coprecipitation, and homogeneous precipitation. The metal hydroxide precipitates from © 2006 by Taylor & Francis Group, LLC
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an aqueous solution of metal salt by adjusting certain pH values. When the simultaneous precipitation occurs at a certain pH value from the solution mixed with various metal salts, all metal hydroxides are precipitated, and the polycomponent oxide powder is prepared from the metal hydroxides mixture by thermal decomposition. This technique is called coprecipitation. However, homogeneity of the metal hydroxides mixture is difficult to achieve because of the different conditions for precipitation of each hydroxide. It is necessary to make a multicomponent powder where the precipitated conditions are nearly the same among the metal components, and each precipitation rate is fast. Moreover, it may be necessary to have operating conditions such as the addition of a large amount of precipitating agent and vigorous agitation in the solution. In the special case of compound precipitation, there is a useful precursor compound with a desired composition such as BaTiO(C2O4)2 or CaZrO(C2O4)2. The homogeneous precipitation technique is a different operation from the above techniques of the addition of precipitating agents, where precipitating agents are produced by a chemical reaction instead of addition in the solution. This technique is superior to precipitated metal hydroxide, because the concentration of the precipitating agent becomes more homogeneous in the solution. For example, urea, thioacetamide, or dimethyl oxalate can be used in hydrolysis as a precipitating agent, instead of ammonia, hydrogen sulfide, or oxalic acid. Sacks et al. 36 reported the formation of aluminum oxide by homogeneous precipitation and thermal decomposition techniques from the mixture of aluminum sulfate hydrate and urea solutions. In the category of hydrolysis, there is formation from the corresponding metal alkoxides in alcohol solution as well as from metal salt solutions. The technique to produce particles by hydrolysis of metal alkoxides is called the alkoxide method. The particles prepared by this technique are normally amorphous metal oxide and hydrated. This technique has several advantages, such as simple and rapid reaction around room temperature, the easy achievement of high purity of the product because it is free from inorganic ions of all kinds, and the strong possibility of the formation of a complex component product from the mixture of metal alkoxides. Uniform and spherical silica particles are often prepared by this technique, as reported by Stöber et al.37 The metal hydroxide solution is easily generated by hydrolysis from the metal salt solution, especially with a small ionic radius metal such as Si4+ or A13+, or with a high ionic charged metal such as Fe3– or Ti4+. Metal oxide particles are produced by thermal decomposition. When metal salts are highly purified, the particles with high purity will be produced very easily, and the average size and size distribution of particles can be controlled by an aging process. Matijevic developed many procedures for uniform metal (hydrous) oxide particles, and some are reviewed, for example, the formation of iron oxide.38 In the redox reaction technique, 39 fine metal particles, such as noble metals and sulfur, are produced by reducing or oxidizing the metal salt or the metal chelate complex solutions. Many kinds of complex or protective colloid agents are frequently employed to moderate the reaction speed and to stabilize the generated particles, so that highly monodispersed particles are obtained. Matijevic also reported that preparation of fine powders of some noble metals and copper, using this technique, was combined with the following technique, called decomposition of the compound.40 Some organic compounds such as EDTA, triethanol amine, thioacetamide, and urea can be used in the technique of decomposition of compound. Haruta et al. obtained molybdenum and cobalt sulfide particles by hydrolysis of thioacetamide promoted with hydrazine in ammonium orthomolybdate and cobalt acetate, respectively.41
Special Reaction Field The hydrothermal method is one method for producing metal (hydrous) oxide crystalline particles in aqueous solution under high temperature and high pressure. This method can be classified as hydrothermal oxidation, hydrothermal precipitation, hydrothermal synthesis, hydrothermal decomposition, hydrothermal crystallization, or hydrothermal reduction, according to the reaction mechanism. When an autoclave is used as a reaction vessel, it takes several hours or days to produce fine metal oxide particles, because of the slow dehydration rate. Recently, Adschiri et al. proposed a new process using supercritical water and showed the synthesis time of AlOOH particles was a few minutes.42 © 2006 by Taylor & Francis Group, LLC
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The methods of particle formation using supercritical fluids are the rapid expansion of the supercritical solution method43 and the gas antisolvent recrystallization method reported by Gallagher et al. 44 As the operating temperature is relatively low when carbon dioxide supercritical fluid is used, these methods are now being applied to make the products for the medical and food industries. Ultrafine particles, which sometimes have quantum size effects, can be produced by the reaction in pools of water in (water-in-oil) microemulsions. The redox reactions can be used by mixing more than two kinds of microemulsion with the reactant in the water pool, or by adding gas or an aqueous solution into the microemulsion. Nagy et al. prepared uniform and very fine nickel boride and cobalt boride particles for catalysis, and also reviewed the formation of many metal particles.45 Fendler’s review is focused on novel particle formation by the use of a surfactant assembly, such as reversed micelles, microemulsions, vesicles, and bilayer liquid membranes.46
Organic Powder Organic polymer particles are produced by chemical reactions among the monomers, which are dissolved or dispersed in a solution. Dispersion methods to make monomer droplets are roughly classified into mechanical dispersion methods and phase-separation methods. The former methods need mechanical energy from agitation or pressure difference, and a dispersing agent, such as a surfactant or polymer, is often used for easy dispersion. On the other hand, the droplets of the monomer can be made spontaneously and quite stable in the latter methods. Seeding polymerization cannot be classified within the two above methods, because the polymer particles already exist in the system. These polymer particles are used as seeds and grow by polymerization between the surface of the particles and the monomer supplied in the system.47
Mechanical Dispersion Methods Suspension polymerization, emulsion polymerization, and mini-emulsion polymerization are included in the mechanical dispersion methods, depending on the droplet size of the monomer. The interfacial reaction technique, which acts at the surface of the dispersed droplets, is also in this category. In this technique, microcapsules can be easily produced, because the chemical reaction rate to make solids decreases rapidly for the high diffusion resistance of the solid film already produced at the interface, and the inner materials can not be completely converted to solid. Sometimes this technique is applied to inorganic particles, as Nakahara et al. reported of the microcapsule formation of spherical calcium carbonate particles.48 Suspension polymerization is one of the methods of polymer formation that uses vigorous agitation of the aqueous solution contained in the dispersing agent and hydrophobic monomer contained in the oil-soluble initiator for polymerization.49–50 The polymerization occurs in the dispersed small droplets of the monomer. This technique operates easily, but the products have average diameters of 2–10 m and wide size distributions. Sometimes fine particles or water-soluble polymers can be used as the dispersing agent. In the case of emulsion polymerization, a water-soluble initiator is used with a surfactant, of which concentration is more than the critical micelle concentration. The polymerization is initiated at the surface of the monomer solubilized in micelle, and a fresh monomer comes from the large-sized organic droplets stabilized by the surfactant.50 The products are typically 0.5-m-diameter particles with narrow distribution. In the mini-emulsion polymerization technique, fine oil droplets are made with a small amount of the surfactant, adding to the slightly water-soluble materials like hexadecane or long-chain alcohol.51 The initiator of both water-soluble types and oil-soluble types can be used. The polymer produced by this technique is 0.1 1m in diameter and relatively polydispersed. Mixing methods essentially lead to particle size distribution. On the other hand, capillary methods are one idea to control particle size. The droplets of uniform size are produced by the vibration from the breakup of the liquid jet through a nozzle or a capillary. This vibration is usually created by mechanical systems such as piezoelectric, ultrasonic, or pressure difference units. Sometimes © 2006 by Taylor & Francis Group, LLC
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an electrostatic spray can be used. Most droplets, however, are dispersed into the gas to make a uniform-size polymer.52 The porous membrane or tube can be also used to produce the uniform droplets. Omi et al. reported how to make polymer particles of about 8 m with uniform size by using a porous tube called Shirasu porous glass.53
Phase-Separation Methods The dispersion phase for the reaction can be created spontaneously or by adding a small amount of mechanical energy. Soap-free polymerization is one of the phase-separation methods. In this technique, a very small amount of monomer dissolved in the aqueous solvent becomes the nuclei for the polymer particles. The residue of the initiator for polymerization, which has a charge in the aqueous solution, remains on the surface of the products, and so the stability of the dispersion state of the products is kept due to the repulsion for the surface charge. The particle size of the products is around 1 m and monodispersed. The greatest advantage for this technique is that the surface of the particles is free of the surfactant.54 On the other hand, the micro-emulsion polymerization technique uses a large amount of surfactant with the monomer and the solvent (usually water). The polymerization occurs in a micelle system of high concentration. The particle size of this product is in the range from 5 to 50 nm. Inverse micelle polymerization is the same technique as micro-emulsion polymerization, but a water-soluble monomer is used and produces polymer particles 5–100 nm in diameter in the reversed micelle.55 In the dispersion polymerization technique including nonaqueous types, the amphiphilic polymer is used as the dispersing agent with the organic solvent, for example, an alcohol, which is a good solvent for the monomer but is a poor solvent for the polymer. The particle size of the products is from a few micrometers to a few tens of micrometers. They are relatively uniform and can be controlled by the degree of the compatibility between the polymer and the solvent.56 The coacervation technique also uses a poor solvent for the deposition of the polymer phase, and it can be controlled by the operated temperature or by the progress of the polymerization. Monodispersed but porous particles, of from a few micrometers to a few tens of micrometers, are produced.57 The polymer produced by the coacervation has a high surface activity, so that when the other materials are dispersed in the system, the coacervation occurs on the surface of the other materials and polymer membrane covering it. This technique can be applied to make microcapsules.58
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18. Zachariah, M. R. and Huzarewicz, S., J. Mater. Res., 6, 264–269, 1991. 19. Alam, M. K. and Flagan, R. C., Aerosol Sci. Technol., 5, 237–248, 1986. 20. Okuyama, K., Kousaka, Y., Tonge, N., Yamamoto, S., Wu, J. J., Flagan, R. C., and Seinfeld, J. H., AIChEJ, 32, 2010–2019, 1992. 21. Flagan, R. C. and Lunden, M. M., Mater. Sci. Eng. A Struct., 204, 113–124, 1995. 22. Wu, J. J., Flagan, R. C., and Gregory, O. J., Appl. Phys. Lett., 49, 82–84, 1986. 23. Griffin, G. L., J. Am. Ceram. Soc., 75, 3209–3214, 1992. 24. Akhtar, M. K., Pratsinis, S. E., and Mastrangelo, S. V. R., J. Am. Ceram. Soc., 75, 3408–3416, 1992. 25. Ulrich, G. D., Combust. Sci. Technol., 4, 47, 1971. 26. Pratsinis, S. E., Prog. Energy Combust. Sci., 24, 197–219, 1998. 27. Holmgren, J. D., Gibson, J. O., and Sheer, C., J. Electrochem. Soc., 111, 362–369, 1964. 28. Yoshida, T., Kawasaki, A., Nakagawa, K. J., and Akashi, K., J. Mater. Sci., 14, 1624–1630, 1979. 29. Sugimoto, T., Ed., Fine Particles: Synthesis, Characterization, and Mechanisums of Growth, Marcel Dekker, New York, 2000. 30. Sugimoto, T., Monodispersed Particles, Elsevier Science, Amsterdam, 2001. 31. Neilson, M. L., Hamilton, R. J., and Walsh, R. J., in Kuhn, B., Lamprey, W. H., and Sheer, C., Eds., Ultrafine Particles, John Wiley & Sons, New York, 1963, p. 181. 32. Roy, D. M., Neurgaonkar, R. R., O’Holleran, T. P., and Roy, R., Am. Ceram. Soc. Bull., 56, 1023–1024, 1977. 33. LaMer, V. K. and Dinegar, R. H., J. Am. Chem. Soc., 72, 4847, 1950. 34. Sugimoto, T., Adv. Colloid Interface Sci., 28, 65, 1987. 35. Calvert, P., Nature, 367, 119, 1994. 36. Sacks, M. D., Tseng, T. Y., and Lee, S. Y., Am. Ceram. Soc. Bull., 63, 301–310, 1984. 37. Stöber, W., Fink, A., and Bohn, B., J. Colloid Interface Sci., 26, 62, 1968. 38. Blesa, M. A. and Matijevic, B., Adv. Colloid Interface Sci., 29, 173–221, 1989. 39. House, H. O., Modern Synthetic Reactions, 2nd Ed., W. A. Benjamin, Redwood, CA, 1972. 40. Matijevic, B., Faraday Discuss., 92, 229–239, 1991. 41. Haruta, M., Lamaitre, J., Delannay, F., and Delmon, B., J. Colloid Interface Sci., 101, 59, 1984. 42. Adschiri, T. K., Kanazawa, K., and Arai, K., J. Am. Ceram. Soc., 75, 1019–1022, 1992. 43. Mohamed, R. S., Halverson, D. S., Debenedetti, D. G., and Prud’homme, R. K., in Johnston, K. P. and Penninger, J. M. L., Eds., Supercritical Fluids: Science and Technology, ACS Symposium Series, No. 406, American Chemical Society, Washington, DC, 1989, p. 355. 44. Gallagher, P. M., Coffey, M. P., Krukomis, V. I., and Klasutis, N., in Johnston, K. P. and Penninger, J. M. L., Eds., Supercritical Fluids: Science and Technology, ACS Symposium Series, No. 406, American Chemical Society, Washington, DC, 1989, p. 334. 45. Nagy, J. B., Derouane, B. G., Gourgue, A., Lufimpadio, N., Ravet, I., and Verfaillie, J. P., In Mittal, K. L., Ed., Surfactants in Solution, Vol. 10, Plenum Press, New York, 1989, pp. 1–43. 46. Fendler, J. H., Chem. Rev., 87, 877, 1987. 47. Ugeistad, J., Mork, P. C., Mufutakhamba, H. R., Soleimy, B., Nordhuns, I., Schmid, R., Berge, A., Ellingson, T., and Aune, O., Science and Technology of Polymer Colloids, Vol. 1, NATO ASI Series B, No. 67, Kluwer Academic, Dordrecht, 1983. 48. Nakahara, Y., Mizuguchi, M., and Miyata, K., J. Colloid Interface Sci., 68, 401–407, 1979. 49. Bishop. R. B., Practical Polymerization of Polystyrene, Cahners Publ., Des Plaines, IA, 1971. 50. Blackley, D. C., Emulsion Polymerization: Theory and Practice, Applied Science Publ., London, 1970. 51. Barnette, D. T. and Schork, F. J., Chem. Eng. Prog., 83(6), 25–30, 1987. 52. Panagiotou, T. and Levendis, Y. A., J. Appl. Polymer Sci., 43, 1549–1558, 1991. 53. Omi, S., Katami, K., and Iso, M., J. Appl. Polymer Sci., 51, 1, 1994. 54. Ceska, G. W., J. Appl. Polymer Sci., 18, 2493–2499, 1974. 55. Candau, F. and Orrewill, R. H., Eds., Scientific Methods for the Study of Polymeric Colloids and Their Applications, NATO ASI Series C, No. 303, Kluwer Academic Publ., Dordrecht, 1990. 56. Barrett, K. B. J., Ed., Dispersion Polymerization in Organic Media, John Wiley & Sons, New York, 1970. 57. Hou, W. H. and Lloyd, T. B., J. Appl. Polymer Sci., 45, 1783–1788, 1992. 58. Nixon, J. R., Ed., Microencapsulation, Marcel Dekker, New York, 1976.
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4.3
Crystallization Masakuni Matsuoka Tokyo University of Agriculture and Technology, Koganei, Tokyo, Japan
Industrial crystallization to produce particulate products with desired mean size and size distribution from a liquid phase is reviewed briefly here. By controlling operating conditions and using properly designed apparatuses, desired products are, in principle, obtainable. In this section, fundamentals of crystallization kinetics and phenomena are described, together with their applications to practical operations.
4.3.1 CRYSTALLIZATION PHENOMENA AND KINETICS In order for crystallization to proceed, the establishment of supersaturation is essential. This is done either by the cooling or evaporating of solvents, or a combination of these two operations. Chemical reactions to form insoluble materials are also applied for this purpose. Crystallization phenomena usually include nucleation and growth of crystals.
Nucleation The formation of crystal nuclei can be classified into primary and the secondary nucleation categories, as shown in Table 3.1.1 Primary nucleation refers to the spontaneous formation of crystals from clear supersaturated solutions and is further classified into homogeneous and the heterogeneous nucleation. The latter occurs when foreign solid surfaces, such as vessel walls, stirrer blades, and/ or dust particles, exist in the solution and act as nucleation sites. Theoretical investigations have mainly been developed for homogeneous primary nucleation; however, quantitative correlations without empirical constants are not yet possible. The classical theory predicts the dependence of the nucleation rate B [#/(m3 s)] on the solution saturation ratio S ( ⫽ x/x*), in which x is the solution concentration in terms of mole or mass fraction, and x* is the saturation concentration at the operating temperature as B ⫽ Al exp[ ⫺ A2T ⫺3 (ln S )⫺2 ] ⬇ Al exp( ⫺ A2T ⫺3 s⫺2 )
(3.1)
where Al and A2 are constants, T [K] is the solution temperature, and s ( ⫽ S – 1 ⫽ (x – x*)/x*) is the supersaturation ratio or simply the supersaturation of the solution. The last equality is held only if the supersaturation is so small that ln S ⫽ 1n(1 + s) ⫽ s. The secondary nucleation predominates in practical crystallizers where a large number of crystal particles are present in the solution. Although a number of mechanisms of the secondary nucleation have been proposed, only empirical correlations are available to correlate the rate of nucleation and operating conditions such as solution supersaturation s, flow conditions in terms of stirring rates N [rps], and slurry density MT [kg/m3]. A typical correlation is as follows: B ⫽ kb N a M Ti s b ⫽ kb ⬘N a M Tj G i
(3.2)
where k b and kb⬘ are the coefficients for secondary nucleation and the exponents a, j, b, and i are experimentally determined for each system and crystallizer. G [m/s] stands for the growth rate of 425 © 2006 by Taylor & Francis Group, LLC
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TABLE 3.1 Classification of Nucleation and Growth Phenomena System Phenomena
Clear
Suspension
Nucleation
Primary nucleation Homogeneous nucleation Heterogeneous nucleation
Secondary nucleation Contact nucleation Fluid shear nucleation Breaking/attrition Agglomeration (negative)
Growth
Primary growth
Secondary growth Agglomeration Growth enhancement by microcrystals Breakage/attrition (negative)
Source: Matsuoka, M., J. Chem. Eng. Jpn., 35, 1025–1037, 2002.
crystalline particles as described in the following section. Typical values of these exponents are summarized by van Rosmalen and van der Heijden2 as a ⫽ 1.5 – 4, j ⫽ 1 or 2 (depending on the collision mechanisms involved), b ⫽ l – 3, and i ( ⫽ b/g) ⫽ 0.5 – 3. The second equality is held because the growth rate is a function of the solution supersaturation and will be mentioned in the next section.
Crystal Growth Crystal particles grow with the driving force of solution supersaturation. The rate processes involved in crystal growth kinetics are the mass transfer of crystallizing component(s) from the bulk to the surface in the solution, the surface integration in which the crystallizing components are incorporated into the crystal lattice, and, finally, the heat transfer of the latent heat of crystallization. The first two processes occur in series, whereas the last parallels them. For mass transfer rates, the conventional treatment is applicable, and the following expression can be used:
1⫺ wi dW ⫽ kd AC ln dt 1⫺ w
(3.3)
where W [kg] and A [m2] are the mass and the surface area of a crystal, C is molar or mass density of the solution, and xi denotes the solution concentration at the surface and can be equal to the bulk concentration (x) when the solution is completely agitated. Theoretical treatments of the surface integration rate have been developed for single crystals. These treatments include the Burton–Cabrera–Frank (BCF) theory and the birth and spread (B&S) model, which correlate the linear growth rate R hkl [m/s] of a specific crystallographic surface of (hkl) with the surface supersaturation ss ( ⫽ (xi – x*)/x*). The dependence of Rhkl on ss is different for each model, as shown in Figure 3.1, so that from an engineering viewpoint, it is enough to express the relation by the power law with g ⫽ 1–2: Rhkl ⫽ kg ssg
(3.4)
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Linear growth rate Rhkl [m/s]
10–6
10–7
Kossel model BCF theory B&S model uniform growth
10–8
0.01
0.1
1
FIGURE 3.1 Linear grown rate versus supersaturation for several models of −7 crystal growth. Coefficients and constants are taken to satisfy G ⫽ 10 m/s when s ⫽ 0.1.
As the mass and the surface area of a crystal particle can be expressed by the use of its characteristic dimension (diameter) L [m], and the shape factors fv and fs as W ⫽ rsfvL3 and A ⫽ fsL2, the mass rate of crystal growth can be converted into the linear growth rate G ( ⫽ dL/dt): r fv dL r fv 1 dW ⫽3 S ⫽3 S G A dt fs dt fs
(3.5)
The two linear growth rates are different, as Rhkl is the advancement of the crystallographic surface (hkl), corresponding to the increasing rate of a radius, whereas G is considered to be that of a diameter. Hence, G ⬇ 2Rhkl. In addition, heat transfer plays an important role in particular systems with high solution concentration, large heat of crystallization, and large temperature-dependent solubility.3 Therefore, the © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.2 Solution conditions at the growing crystal surface at different flow rates. [From Matsuoka, M. and Garside, J., J. Cryst. Growth, 129, 385–393, 1993. With permission.]
solution conditions at the crystal surface can be a measure of the relative importance of the three rate processes. For example, they approach the bulk conditions as the relative velocities between the crystal and the solution increase as shown in Figure 3.2.4
4.3.2
OPERATION AND DESIGN OF CRYSTALLIZERS
Analysis with Population Balance Equation Besides respective measurements of the nucleation and growth rates, the rates can be determined experimentally if the crystal size distribution (CSD) of the product is known. For a simple wellstirred tank crystallizer, operated continuously at steady state, the number balance of particles of size L leads to the following mathematical relation between the growth and the nucleation rates5: ⎛ L ⎞ B ⎛ L ⎞ n( L ) ⫽ n(0) exp ⎜⫺ ⎟ ⫽ exp ⎜⫺ ⎟ ⎝ Gt ⎠ G ⎝ Gt ⎠
(3.6)
where n(L) [#/(m3m)] denotes the population density of particles of size L, and t [s] is the mean resident time (i.e., the crystallizer volume divided by the volume flow rate). This type of crystallizer is known as a mixed-suspension mixed-product removal (MSMPR) crystallizer, in which complete mixing is assumed for both the slurry in the vessel and the product. Equation 3.6 was derived from the population balance with the assumptions of steady state, size-independent growth rate, zero size nuclei, no particles in the feed stream, and no agglomeration or breakage of crystal particles in the crystallizer. By plotting the population density of products against size on a semilogarithmic paper, and knowing the residence time, one can determine the growth and the nucleation rates simultaneously from the slope and the intersect of the straight line, respectively, as shown as a bold line in Figure 3.3. The growth rate thus determined is often different from the one discussed in the preceding section, in that effects of neglected phenomena of agglomeration or breakage are all included. Product © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.3 Ideal semilogarithmic population density plot (bold line) with some deviations from ideality. [Modified from Garside, J., Mersmann, A., and Nyvlt, J., Eds., Measurement of Crystal Growth Rates, FECE Working Party on Industrial Crystallization, München, 1990, p. 160. With permission.]
crystals from industrial crystallizers are usually agglomerated, and their corners are rounded due to collisions between the particles or impeller blades. However, the growth and the nucleation rates thus determined are the effective parameters that control the product particles having the CSD measured. From Equation 3.6, physically meaningful particle characteristics can be obtained: The mean size LM is given as Gt, that is, (growth rate) ⫻ (residence time); and the total number of particles per unit volume NT is equal to Bt, that is, (nucleation rate) ⫻ (residence time). For mass-based mean size, 3Gt can be derived as the modal or dominant size where the distribution is maximum.
Crystallization Phenomena in Crystallizers In actual crystallizers, crystallization phenomena can be very different from those occurring during the growth of single crystalline particles. Crystals of the same size can grow at different rates, or the rates can be different among the particles of different sizes under the same supersaturation. These are known as growth rate dispersion (GRD) or size-dependent growth rates (SDG), respectively.2 Recently, experimental evidence has been reported for the effects of microcrystals on larger crystal particles. During the growth of a crystal particle from a clear supersaturated solution, induced nucleation causes a sudden increase in the growth rate. This phenomenon is called “growth rate enhancement (GRE) by microcrystals”6 and is considered to be common in actual crystallizers where nucleation is continuously occurring and large numbers of particles coexist in the mother liquor. The agglomeration of particles is a similar but different phenomenon and results in polycrystals, whereas GRE can produce perfect, single crystal-like particles. These phenomena result in curved population density plots. The curves in small size regions are often concave upward, suggesting either slower growth rate, from Equation 3.6, or secondary © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.4 Factors affecting the CSD and their interrelations. [From Randolph, A. D. and Larson, M. A., in Theory of Particulate Processes, 2nd Ed., Academic Press, San Diego, 1988, p. 201. With permission.]
Crystallization
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FIGURE 3.5 permission.]
Control of CSD utilizing ripening phenomena. [From Garside, J., personal communication (special lecture at TUAT), 1989. With 431
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nucleation by attrition, although the same effects can be obtained by poor achievement of MSMPR conditions. Some typical effects of factors causing deviations from the ideal density plot are shown in Figure 3.3. The diagram is modified from that of Garside et al. 7 Accordingly the growth of crystals in suspension systems may be different from the one in clear solution, and therefore regarded as secondary growth in contrast to the primary growth where single crystalline particles are growing from clear solutions. This classification is already given in Table 3.1 (see work by Matsuoka8).
Design of Crystallizers With an MSMPR crystallizer, the interrelations among the factors influencing the product CSD are illustrated in Figure 3.4.9 Many factors such as feed concentration, residence time, and operating temperature can change the product CSD through the solution supersaturation with feedback loops from the CSD, which defines the suspension density and total surface areas. Still, it is not easy to tell what happens on the CSD when one of these factors is changed. For example, the answer to a question such as, “Does the mean size increase when the residence time increases?” depends strictly on the dependencies of both the nucleation and growth rates on the supersaturation. In most cases, where b ⬎ g (i.e., i ⬎ 1), the answer is “Yes,” provided the solution supersaturation is kept constant. As long as MSMPR crystallizers are used, the product possesses a broad CSD, and in order to obtain uniformly sized particles, classification, in principle, is essential. Garside, 10 however, illustrated a process to produce very fine particles with a narrow CSD by the use of a series of MSMPR crystallizers. This can be achieved by utilizing size-dependent solubilities of crystallizing component(s) and controlled supersaturations in the crystallizers (Figure 3.5). In the first small tank, with higher supersaturation, rapid nucleation occurs with very small nuclei, whereas in the second tank, operated at lower supersaturation with longer residence time, smaller nuclei tend to dissolve, and relatively large particles can survive, so that sharply distributed particles are produced. This occurs because of the Ostwald ripening phenomena; the size-dependent solubilities given by the Gibbs–Thomson equation can cause small particles to dissolve even in the supersaturated solutions. The Gibbs–Thomson equation is written as
ln
x( L ) 4⌬Ga M ⫽ x (∞ ) RT rS L
(3.7)
where x(L) is the solubility in terms of mole fraction of particles of diameter L, ⌬Ga [J/m2] is the surface tension, M is the molecular weight, and rs is the density of crystals. For moderately soluble materials, particles of which size is a few microns are large enough to assume that their solubility is size independent. For sparingly soluble materials of that size, however, they usually show sizedependent solubilities, hence the ripening can effectively proceed.
REFERENCES 1. Mullin, J. W., in Crystallization, 3rd Ed., Butterworth-Heinemann, Oxford, 1993, p. 172. 2. Van Rosmalen, G. M. and van der Heijden, A. E., in Science and Technology of Crystal Growth, van der Eerden, J. P. and Bruinsma, O. S. L., Eds., Kluwer Academic Publishers, Dordrecht, 1995, p. 259. 3. Matsuoka, M. and Garside, J., Chem. Eng. Sci., 46, 183–192, 1991. 4. Matsuoka, M. and Garside, J., J. Cryst. Growth, 129, 385–393, 1993. 5. Randolph, A. D. and Larson. M. A., in Theory of Particulate Processes, 2nd Ed., Academic Press, San Diego, 1988, p. 80. © 2006 by Taylor & Francis Group, LLC
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6. Matsuoka, M. and Eguchi, N., J. Phys. D Appl. Phys., 26, B162–B167, 1993. 7. Garside, J., Mersmann, A., and Nyvlt, J., Eds., Measurement of Crystal Growth Rates, FECE Working Party on Industrial Crystallization, München, 1990. 8. Matsuoka, M., J. Chem. Eng. Jpn., 35, 1025–1037, 2002. 9. Randolph, A. D. and Larson, M. A., in Theory of Particulate Processes, 2nd Ed., Academic Press, San Diego, 1988, p. 201. 10. Garside, J., personal communication (special lecture at TUAT), 1989.
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4.4
Design and Formation of Composite Particles Hideki Ichikawa and Yoshinobu Fukumori Kobe Gakuin University, Kobe, Japan
Particles have been processed by various methods for composite particle formation depending on required size and function (Figure 4.1). Particles larger than 200 µm can be successfully processed by a fluid bed. Many kinds of composite particles prepared by a fluid bed are on the market; for example, most recent pharmaceuticals are composites for controlled drug release, including sustained release, prolonged release, delayed release, and taste masking. For smaller particles, agglomeration has been successfully carried out to get free-flowing coarse particles, but their coating for forming multilayered structures without core-particle agglomeration is difficult, because of the adhesive and cohesive properties of such fine particles.1 So, liquid-phase processes, including emulsifying processes and phase separation, have been applied to make fine composite par2 ticles. In this section, the fluid bed process and the emulsifying process are described, mostly with examples limited to the pharmaceutical field, since the methods and applications are diverse among industrial fields.
FIGURE 4.1
Pharmaceutical composite particles and their preparation techniques.
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4.4.1 COMPOSITE PARTICLE FORMATION IN THE FLUID BED PROCESS Apparatus in the Fluid Bed Process Fluid bed processors have been used for drying, agglomeration, and coating. In addition to a simple fluid bed, the tumbling, agitating, centrifugal, and spiral flow fluid bed and the spouted bed with or without draft tube have been developed for improving process performance. Among many types of surface modification or composite formation processes, fluid bed processes are characterized by their simple, easy formation of multilayers on particles, leading to commercial production of many types of functional particulate systems. Typical fluid bed processors are illustrated in Figure 4.2. Figure 4.2A shows a simple fluid bed, usually with a conical shape at the bottom of the chamber, inducing spouted particle flow that more or less depends on the angle of the conical chamber. The spray is supplied from the top toward the fluid bed in agglomeration, but into the bed in a tangential direction in coating. Figure 4.2B shows an example of a tumbling fluid bed processor. The fluidization air is supplied through the slit between the turntable and the chamber. The tumbling forces particles to be centrifuged toward the chamber wall; then, the particles are blown up by the slit air. These make a circulating fluid bed, in which the spray is supplied from the top or in tangential mode. The rolling of particles on the turntable makes the wet particles roundish and compact. Figure 4.2C shows a typical assembly of the Wurster process, a kind of spouted bed process assisted with a draft tube. The particles fluidized in the annular part between the draft tube and the chamber are introduced into the draft tube due to accelerated air flow from the bottom and, then, spouted from the draft tube. The particle velocity is reduced in the upper expansion chamber, leading to return of particles to the fluid bed in the annular part. During this circulating flow, the particles are sprayed in the draft tube; at the same time, they have a chance to be exposed to the spray-air jet, which can exert a strong separation force on the particles. Usually, the particles are easily agglomerated or coagulated during spraying, but the above strong air-jet can disintegrate the agglomerates. These characteristics of the Wurster process make it possible to coat finer particles 3 or to produce finer agglomerates, 4 compared with the other types of fluid bed processes. The types of fluidization, as proposed by Geldart, depend on particle size and density. 5 For example, since the particle density of pharmaceutical powders is mostly around 1.5 g/cm3, the particles are categorized into larger than 900 µm (D type), 900–150 µm (B type), 150–20 µm (A type), and smaller than 20 µm (C type) particles from Geldart’s fluidization map. The A and C types are
FIGURE 4.2 Typical fluid bed equipment for film-coating of pharmaceutical particulates. © 2006 by Taylor & Francis Group, LLC
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cohesive and adhesive; therefore, they are agglomerated into D and B types in order to achieve free-flowing properties. This is the main purpose of agglomeration. The D type cannot be fluidized due to too large particle size; therefore, the coating is simply processed using a rotating pan. The B type always exhibits bubbling fluidization, which induces inhomogeneous particle flow; in practical terms, this is disadvantageous because such a flow, especially in the spray zone, leads to poor coating performance in yield and homogeneity of the product. In order to achieve homogeneous particle flow of B-type particles, the particle flow patterns that are different from simple fluidization, such as those in the tumbling, centrifugal, and spiral fluid bed and the spouted bed, have been required. This is the reason that many different types of fluid bed processor have been developed so far. The A-type particles can be fluidized homogeneously without air bubbles, but the separation force from fluidization is not sufficient to avoid agglomeration during spraying the liquid solution. This prevents the fluid bed coating process from being industrially applied to these small particles in spite of much research attempting fine particle coating technologies. The C-type particles cannot be fluidized due to their small size. When particles are fluidized or spouted under spraying, separation force is exerted to the particles and they are more or less agglomerated by the spray solution. Balance between the separation force and the binding strength of the binder or coating material determines the degree of agglomeration, that is, agglomerate size. 6,7 Each fluid bed processor can generate its inherent separation force, surely depending on the operating conditions such as air flow pattern and inlet air flow rate. The excessive separation force can even disintegrate core particles. Thus, depending on the requisite of final agglomerate size or the core particle size to be discretely coated, the optimal processor should be selected (Figure 4.1).
Material in the Fluid Bed Process Table 4.1 lists typical binders and coating materials for agglomeration and coating. The watersoluble polymers are used as binders in agglomeration, and some of them are also used as coating materials for water-proofing or taste masking. As each type of water-soluble polymer becomes higher in molecular weight, it contributes more to the increase in the viscosity of its solution and the interparticle binding strength, leading to more enhanced particle growth in agglomeration and coating.6,7 The commercially available polymeric dispersions, which have been most widely used as coating material, are classified into three types based on the preparation methods 8: (1) latexes synthesized by emulsion polymerization; (2) pseudolatexes prepared by emulsifying processes such as emulsion-solvent evaporation, phase inversion, and solvent change; and (3) dispersions of micronized polymeric powders. Eudragit L30D and NE30D are acrylic copolymer latexes synthesized by emulsion polymerization.9 The particle sizes of these latexes are in a submicron order. L30D is a copolymer of ethyl acrylate (EA) as an ester component with methacrylic acid (MA) (MA/EA 1:1). It is used for enteric coating because of the presence of carboxyl groups in the copolymer. NE30D is a copolymer of ester components only, EA and MMA (2:1). The films formed from NE30D have a very low softening temperature and hence are flexible and expandable even under indoor conditions. Cellulose derivatives cannot be synthesized directly in latexes; therefore, they are prepared as pseudolatexes (Aquacoat, Aquateric,10 Surelease11), or micronized powders (Aqoat [HPMCAS],12 EC N-l0F). While the pseudolatexes can be prepared as submicron particles mostly by emulsifying processes, the micronized powders have mean particle sizes of a few micrometers. Poly(VAP) is also supplied as a micronized powder (Coateric).11 Eudragit RS and RL are terpolymers of EA and MMA as ester components with trimethylammonioethyl methacrylate chloride (TAMCl) as hydrophilic quaternary ammonium groups; RS and RL are 1:2:0.1 and 1:2:0.2 terpoly (EA/MMA/TAMCl), respectively. Because Eudragit RS and RL contain MMA-rich ester components (EA/MMA 1:2), they have softening temperatures higher than those of NE30D (EA/MMA 2:1) and form hard films © 2006 by Taylor & Francis Group, LLC
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TABLE 4.1 Type Solutions
Trade name Kollidon VA64
Supplier
Solubilization
Application form
BASF
Water-soluble
Aqueous solution
Components 6:4 Poly(Vinylpyrrolidone/Vinylacetate)
Kollidon 20, 30
BASF
Water-soluble
Aqueous solution
Polyvinylpyrrolidone (PVP)
Opadry
Coloracon
Water-soluble
Aqueous solution
Mix of water- soluble polymer, plasticizer and pigment
TC-5
Shin-Etsu
Water-soluble
Aqueous solution
Hydroxypropylmethylcellulose (HPMC)
HPC-SL, SSL, SSM
Nippon Soda
Water-soluble
Aqueous-solution
Hydroxypropylcellulose (HPC)
CMEC
Freund
Enteric
Aqueous ethanolic soln
Carboxymethylethylcellulose (CMEC)
HPMCP
Shin-Etsu
Enteric
Aqueous ethanolic soln
Hydroxypropylmethylcellulose phthalate (HPMC)
Eudragit E 100
Rhom Pharm
Acid-soluble
Organic solvent soln
Poly(BMA/MMA/DAEMA)
Eudragit L/S
Rhom Pharm
Enteric
Organic solvent soln
Poly(MMA/MAA)
Eudragit RS 100/RL100
Rhom Pharm
Water-soluble
Organic solvent soln
1:2:0.1/1:2:0.2 poly(EA/MMA/TAMCl)
EC N-10F
Shin-Etsu
Water-insoluble
Aqueous dispersion
Ethylcellulose (EC)
Aquacoat
Asahi-Kasei
Water-insoluble
Pseudo-latex
Ethylcellulose (EC)
Surerease
Coloracon
Water-insoluble
Pseudo-latex
Ethylcellulose (EC)
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Dispersions
Examples of Commercially Available Coating Materials
Cores
Rhom Pharm
Water-insoluble
Pseudo-latex
1:2:01 poly(EA/MMA/TAMACl)
Eudragit RL30D
Rhom Pharm
Water-insoluble
Pseudo-latex
1:2:02 poly(EA/MMA/TAMACl)
Eudragit NE30D
Rhom Pharm
Water-insoluble
Latex
2:1 Poly(EA/MMA)
Aqoat
Shin-Etsu
Enteric
Aqueous dispersion
Hydroxypropylmethycellulose acetate succinate
Aquateric
FMC
Enteric
Aqueous dispersion
Cellulose acetate phthalate (CAP)
Kollicoat MAE30D/DP
BASF
Enteric
Aqueous dispersion
Poly(EA/MMA)
Eudragit L30D
Rhom Pharm
Enteric
Latex
1: 1 Poly(MMA/EA)
Lubri Wax 101/103
Freund
Water-insoluble
Powder
Hydrogenated caster oil/hydrogenated rapessed oil
Polishing wax
Freund
Water-insoluble
Powder
Carnauba wax
PEP 101
Freund
Water-insoluble
Powder
Poly (Ethylene oxide/propyloene oxide)
Nonpareil101/103/105
Freund
Granule
Granules of Sucrose-Starch/Sucrose/Lactose Microcrystalline cellulose
Granule
Granules of microcrystalline cellulose
Celphere 102/203/305/507 Asahi-Kasei
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Eudragit RS30D
MAA, methacrylic acid; MMA, methyl methacrylate; EA, ethyl acrylate; TAMCl, trimethylammonioethylmethascrylate chloride; DAEMA, dimethylaminoethylmethacrylate; BMA, butyl methacrylate.
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under indoor conditions. Eudragit RS and RL powders are easily transformed into pseudolatexes by emulsifying their powders in hot water without additives.9 It is costly to ship aqueous dispersions around the world; therefore, the Aquateric pseudolatex is supplied as a spray-dried powder; it is redispersed just before use.10 The film formation process from aqueous dispersion is shown schematically in Figure 4.3. The mechanisms of film formation from aqueous polymeric dispersions have been discussed for a long time, and many theories have been proposed. 8–10,13 Fusion and film formation of polymeric particles during the coating process can be explained by the wet sintering theory for particles suspended in water, the capillary pressure theory for particle layers containing water in various degrees of saturation, and the dry sintering theory for dry particle layers.
Particle Design in the Fluid Bed Process The typical particle structures produced from the agglomeration process are shown in Figure 4.4. Simple agglomerates (Figure 4.4A) are prepared by a fine powder mixed with fine additive powders in a fluid bed, and then, binder solution is sprayed from the top of the simple fluid bed (Figure 4.2A), followed by drying. In this case, since the particles are agglomerated in the fluidization air flow, they are usually very porous. The tumbling fluid bed or the spouted bed process assisted with a draft tube (Figure 4.2B and 4.2C) can be applied if more compact agglomerates are required. When a fine powder is agglomerated with coarse carrier particles (cores in Table 4.1), fine-powder-layered agglomerates shown in Figure 4.4B can be prepared. 14 In the latter process, the tumbling fluid bed is often used, because tumbling on the turntable contributes to efficient layering of the fine powder. The fine-powder-layered agglomerates are often used as core particles in the coating process. Among many kinds of surface modification process, the fluid bed processes can most easily produce multilayered particle structures with each layer being monolithic, a random multiphase structure, an ordered multiphase structure, and so on (Figure 4.5). Combination of different components
FIGURE 4.3 Film formation from latex. © 2006 by Taylor & Francis Group, LLC
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FIGURE 4.4
FIGURE 4.5
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Structure of agglomerates.
Possible structure and characteristics of microencapsulated particles.
and layers can produce almost infinite types of functional particles. As an example, designs and preparations of thermosensitive controlled-release particles are described below. Controlled-release technology based on externally temperature-activated release can find applications in diverse industrial fields.1,15–17 In the pharmaceutical area, for example, deviation of the body temperature from the normal temperature (37˚C) in the physiological presence of pathogens or pyrogens can be utilized as a useful stimulus that induces the release of therapeutic agents from a thermosensitive controlled-release system. Physically controlled temperature using a heat source such as microwaves from outside the body can also be used for temperature-activated antitumor drug release, combined with local hyperthermic treatment of cancer. © 2006 by Taylor & Francis Group, LLC
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The membranes of the thermosensitive controlled-release microcapsules were constructed by a random mixing of Aquacoat (Table 4.1) with latex particles having a poly(EA/MMA/2-hydroxyethyl methacrylate) core and a poly(N-isopropylacrylamide (NIPAAm)) shell (Figure 4.6). This is an example where the membrane has the random two-phase structure in Figure 4.5. The microcapsules exhibited a thermosensitive release of water-soluble drug.18 The mechanism is explained in Figure 4.6. When the temperature was changed in a stepwise manner from 30˚C to 50˚C, the microcapsules showed an “on-off ” pulsatile release. This on-off response was reversible. Alternatively, this type of thermosensitive microcapsule can be prepared even with already established pharmaceutical ingredients. As is well known, HPC, the commonly used binder and coating substrate (Table 4.1), has a lower critical solution temperature (LCST), around 44˚C, and its water solubility drastically changes across the LCST. Negatively thermosensitive drugrelease microcapsules with an HPC layer were thus designed by utilizing the thermally reversible dissolution/precipitation process resulting from the LCST phenomena19 (Figure 4.7). In this case,
FIGURE 4.6 release.
FIGURE 4.7 release.
Composite particle exhibiting positively thermosensitive drug
Composite particle exhibiting negatively thermosensitive drug
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particles are constructed with multilayers of monolithic structure (Figure 4.5). The release rate from the microcapsules with a sandwiched HPC layer was found to be drastically decreased when the temperature came near the LCST: the release rate at 50˚C was approximately 10 times lower than that at 30˚C. Unlike many types of millisized devices that have been developed so far, the microparticles mentioned here were around 100 µm in diameter. Such small dimensions provided a sharp release rate change in response to temperature change. This may be one of advantages of fine particle coating technology. The above demonstrated flexibility in designing membrane structures also offers considerable advantages over conventional microencapsulation methods.
4.4.2 PARTICLE DESIGN IN THE EMULSIFYING PROCESS Particulate systems, including microcapsule, liposome, microemulsion, polymeric nanoparticles, and micelles, have been developed for pharmaceutical use. Their application as drug carriers for cancer therapies has attracted much attention recently. The particle structures to be constructed are essentially the same as those of microparticles produced by the fluid bed process. The present authors have been developing particulate drug delivery systems for gadolinium neutron-capture therapy (Gd-NCT) for cancer. Gd-NCT is a cancer therapy that utilizes g-rays, X-rays, and electrons emitted in vivo as a result of the nuclear neutron-capture reaction with administered gadolinium-157.20–24 A key factor for success in Gd-NCT at the present stage is a device whereby Gd is delivered efficiently and is retained in tumor for a sufficient period during thermal-neutron irradiation. Here, as examples of composite formation in liquid phase, we describe preparation of gadolinium-containing lipid nanoparticles and chitosan nanoparticles, which have been designed and prepared in order to accumulate gadolinium in a tumor.
Lipid Nanoparticles Gadolinium-diethylenetriaminepentaacetic acid (Gd-DTPA) is traditionally used clinically as an MRI contrast agent. It is reported that Gd-DTPA is rapidly eliminated from systemic circulation and hardly accumulates in tumors because of its high hydrophilicity. Meanwhile, the Gd concentration in tumors required to obtain the minimum tumor inactivation effect was estimated to be as high as about 100 µg Gd-157/ml.22 One of the major obstacles to using lipid particles as drug carriers is their rapid removal from systemic circulation by the reticuloendothelial system (RES). To avoid RES uptake, attempts have been made to coat lipid particles with a hydrophilic polymer such as polyethylene glycol (PEG). In addition, it is reported that drug carriers smaller than 100 nm are expected to more easily avoid uptake by RES. From these perspectives, a hydrophobic stearylamide (SA) derivative of Gd-DTPA was synthesized. Soybean oil and lecithin were employed to be the standard components of the parenterally administerable nanoparticulate systems because of their biocompatible properties. Gd-containing nanoparticles were surface modified with a cosurfactant having polyoxyethylene (POE) units. They were designed and prepared to fulfill the following requirements: particle size smaller than 100 nm; high Gd content so as to achieve a Gd concentration of 100 µg/g in the tumor in vivo; and surface properties giving rise to prolonged circulation in the blood (Figure 4.8).25,26 Lipid nanoparticles containing Gd-DTPA-SA (Gd-nano-LPs) were prepared by a thin-film hydration method combined with a sonication (bath-type sonication) method. Formulations and the particle size of lipid emulsions are listed in Table 4.2. As shown in Table 4.2, use of a cosurfactant led to a marked reduction in the size of the resultant nanoparticles: the particle size was less than 100 nm. Additionally, the use of HCO-60 was found to be effective to reduce the particle size even in the high-Gd formulation. At 48 h after intraperitoneal injection of the lipid nanoparticles containing HCO-60 in tumor-bearing hamsters, the Gd © 2006 by Taylor & Francis Group, LLC
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FIGURE 4.8
TABLE 4.2
Structure of Gd-nano-LP.
Formulation of Gd-DTPA Derivative-Containing Nano-LPs Standard-Gd Formulation
High-Gd Formulation
Plain Gd-nano-LP
Cosurfactant Gd-nano-LP
HEPC a
500 mg
500 mg
250mg
Gd-DTPA-SA
250 mg
250 mg
500 mg
Soybean oil
2 ml
2 ml
2 ml
Cosurfactant
—
750 mgb
750 mgc
23 ml
23 ml
23 ml
250 nm
92,76,78 nm
84 nm
Water Particle size a
L– Phosphatidylcholine hydrogenated (from egg yolk).
b
Myrj 53, Brij 700, or HCO-60.
c
HCO-60.
concentration in the tumors reached 107 µg/g tumor (wet basis). This result suggested that the potent antitumor effect might be obtained using the HCO-60 Gd-nano-LPs in the high-Gd formulation as a Gd carrier in Gd-NCT.
Chitosan Nanoparticles As an alternative approach, biodegradable and highly gadopentetate-loaded chitosan nanoparticles (Gd-nano-CP) were prepared by a novel emulsion-droplet coalescence technique.27–29 Chitosan, a polysaccharide, has been widely studied as a material for drug delivery systems due to its bioerodible, biocompatible, bioadhesive, and bioactive characteristics. These distinctive features led us to use chitosan as a promising material for design and preparation of Gd-loaded nanoparticulate devices. Further, its amino pendants dangling from sugar backbones may be useful for enrichment of Gd-DTPA into the nanoparticles by a possible ionic interaction between two freecarboxylic groups of Gd-DTPA and the amino groups of chitosan. The preparation process of Gd-nano-CPs is shown in Figure 4.9. Chitosan was dissolved in an aqueous solution of Gd-DTPA. This solution was added to the paraffin liquid containing Arlacel C © 2006 by Taylor & Francis Group, LLC
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FIGURE 4.9 technique.
445
Preparation of Gd-nano-CP by emulsion-droplet coalescence
and was stirred using a high-speed homogenizer to form a water-in-oil (w/o) emulsion A. Similarly, a w/o emulsion B consisting of sodium hydroxide solution and paraffin liquid containing Arlacel C was prepared. The emulsions A and B were mixed and stirred to solidify chitosan as a result of collision and coalescence between droplets of each emulsion. With 100% deacetylated chitosan, mean particle size and Gd content were 426 nm and 9.3% (corresponding to 32.4% Gd-DTPA), respectively. The mechanism of Gd-nano-CP generation is unclear in detail, but it is presumed that the incorporation of Gd resulted from ionic interaction between two free carboxylic groups in Gd-DTPA molecules and polyamino groups in the chitosan molecules. Release of Gd-DTPA from Gd-nano-CPs in an isotonic phosphate buffer saline solution (PBS) at 37˚C was hardly recognized during 7 days (1.8%) in spite of high water solubility of Gd-DTPA. In contrast, in human plasma, Gd-nano-CPs released 55% and 91% of Gd-DTPA during 3 and 24 h at 37˚C, respectively. Neutron irradiation testing was carried out in vivo to confirm its therapeutic potential in Gd-NCT, using B16F10 melanoma-bearing male C57BL/6 mice. Gd-nano-CP suspension or Gd-DTPA solution as a control was intratumorally injected twice at a dose of Gd 1200 µg 24 and 8 hours before thermal-neutron irradiation. The thermal-neutron irradiation was performed only once for each tumor site with a flux of 2 10 9 n/cm 2 /s for 60 min at Kyoto University Research Reactor (Japan). When the change of tumor volume in mice after the thermal-neutron irradiation was examined, the tumor growth was significantly suppressed in the Gd-nano-CP-administered group despite the radio-resistive melanoma model, compared with the cases of no Gd–administered or Gd-DTPA solution–administered group. As is generally recognized, Gd-DTPA in solution used in ordinary Gd-NCT trials can be eliminated very rapidly from tumor tissue. The enhanced Gd-NCT effects by Gd-nano-CP might result from the high gadolinium quantity retained in tumor tissue.
4.4.3
SUMMARY
Using fluid bed processes and processors along with the appropriate materials and their welldesigned formulation and particulate structure make it possible to prepare highly functional par© 2006 by Taylor & Francis Group, LLC
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ticles, as demonstrated here. However, this method has an unavoidable limit in the size of particles that it can efficiently process because of easy agglomeration of particles smaller than 200 µm during spraying. In order to expand applications of this process, some new or improved fluidization technologies will be required. Very fine, especially nanosized, composite particles can be produced through liquid-phase processes. When applied to cancer therapy, there would be many factors affecting their biodistribution. Their size and surface properties have to be regulated depending on treatment strategy, but this cannot yet be done well. Since these particles have many attractive applications, as demonstrated here, development of novel technologies of composite formation is expected.
REFERENCES 1. Jono, K., Ichikawa, H., Miyamoto, M., and Fukumori, Y., Powder Technol., 113, 269–277, 2000. 2. Fukumori, Y. and Ichikawa, H., in Encyclopedia of Pharmaceutical Technology, DOI:10.1081/E-EPT 120018218, Marcel Dekker, New York, 2003, pp. 1–7. 3. Ichikawa, H., Tokumitsu, H., Jono, K., Osako, Y., and Fukumori, Y., Chem. Pharm. Bull., 42 (6), 1308–1314, 1994. 4. Ichikawa, H. and Fukumori, Y., Int. J. Pharm., 180, 195–210, 1999. 5. Geldart, D., Powder Technol., 7, 285–292, 1973. 6. Fukumori, Y., Ichikawa, H., Jono, K., Takeuchi, Y., and Fukuda, Y., Chem. Pharm. Bull., 40 (8), 2159– 2163, 1992. 7. Fukumori, Y., Ichikawa, H., Jono, K., Fukuda, T., and Osako, Y., Chem. Pharm. Bull., 41 (4), 725–730, 1993. 8. Fukumori, Y., in Multiparticulate Oral Drug Delivery, Marcel Dekker, New York, 1994, pp. 79–111. 9. Lehmann, K. O. R., in Aqueous Polymeric Coatings for Pharmaceutical Dosage Forms, Marcel Dekker, New York, 1989, pp. 153–246. 10. Steuernagel, C. R., in Aqueous Polymeric Coatings for Pharmaceutical Dosage Forms, Marcel Dekker, New York, 1989, pp. 1–62. 11. Moore. K. L., in Aqueous Polymeric Coatings for Pharmaceutical Dosage Forms, Marcel Dekker, New York, 1989, pp. 303–316. 12. Nagai, T., Sekikawa, F., and Hoshi, N., in Aqueous Polymeric Coatings for Pharmaceutical Dosage Forms, Marcel Dekker, New York, 1989, pp. 81–152. 13. Muroi, S., High Polymer Latex Adhesives, Kobunnshi-Kankokai, Kyoto, Japan, 1984. 14. Fukumori, Y., Yamaoka, Y., Ichikawa, H., Fukuda, T., Takeuchi, Y., and Osako., Y., Chem. Pharm. Bull., 36 (4), 1491–1501, 1988. 15. Kaneko, Y., Sakai, K., and Okano, T., Biorelated Polymers and Gels, Academic Press, Boston, 1998, pp. 29–70. 16. Ichikawa, H. and Fukumori, Y., STP Pharm., 7 (6), 529–545, 1997. 17. Peppas, N. A., Bures, P., Leobandung, W., and Ichikawa, H., Eur. J. Pharm. Biopharm., 50 (1), 27–46, 2000. 18. Ichikawa, H. and Fukumori, Y., J. Controlled Release, 63, 107–119, 2000. 19. Ichikawa, H. and Fukumori, Y., Chem. Pharm. Bull., 47 (8), 1102–1107, 1999. 20. Allen, B. J., Mcgregor, B. J., and Martin, R. F., Strahlenther. Onkol., 165, 156–158, 1989. 21. Akine, Y., Tokita, N., Matsumoto, T., Oyama, H., Egawa, S., and Aizawa, O., Strahlenther. Onkol., 166, 831–833, 1990. 22. Akine, Y., Tokita, N., Tokuuye, K., Satoh, M., Fukumori, Y., Tokumitsu, H., Kanamori, R., Kobayashi, T., and Kanda, K., J. Cancer Res. Clin. Oncol., 119, 71–73, 1992. 23. Fukumori, Y., Ichikawa, H., Tokumitsu, H., Miyamoto, M., Jono, K., Kanamori, R., Akine, Y., and Tokita, N., Chem. Pharm. Bull., 41 (6), 1144–1148, 1993. 24. Miyamoto, M., Ichikawa, H., Fukumori, Y., Akine, Y., and Tokuuye, K., Chem. Pharm. Bull., 45 (12), 2043–2050, 1997. 25. Miyamoto, M., Hirano, K., Ichikawa, H., Fukumori, Y., Akine, Y., and Tokuuye, K., Chem. Pharm. Bull., 47 (2), 203–208, 1999.
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26. Miyamoto, M., Hirano, K., Ichikawa, H., Fukumori, Y., Akine, Y., and Tokuuye, K., Biol. Pharm. Bull., 22 (12), 1331–1340, 1999. 27. Tokumitsu, H., Ichikawa, H., and Fukumori, Y., Pharm. Res., 16 (12), 1830–1835, 1999. 28. Tokumitsu, H., Hiratsuka, J., Sakurai, J., Kobayashi, T., Ichikawa, H., and Fukumori, Y., Cancer Lett., 150, 177–182, 2000. 29. Tokumitsu, H. et al., STP Pharm. Sci., 10 (1), 39–49, 2000.
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4.5
Dispersion of Particles Kuniaki Gotoh Okayama University, Okayama, Japan
Hiroaki Masuda and Ko Higashitani Kyoto University, Katsura, Kyoto, Japan
4.5.1
PARTICLE DISPERSION IN GASEOUS STATE
In order to disperse the agglomeration of particles in a gaseous state, external force larger than the adhesive force between primary particles in the agglomeration should be applied to the agglomeration. The dispersio methods can be classified by the methods of applying dispersion forces. Here, we outline methods for dispersion of agglomerated particles in a gaseous state, with particular attention to the external forces that can be utilized for the dispersion of agglomerates.
Dispersion Force Dispersion Force Induced by Acceleration or Deceleration of Airflow One of the external forces that can be utilized for the dispersion of agglomerated particles is a fluid resistance induced by the acceleration or deceleration of a fluid. Here, it is assumed that agglomerated particle can be modeled by a doublet consisting of two primary particles having different sizes, as shown in Figure 5.1. When the agglomerated particle is in a uniform flow field, the motion equation of each primary particle can be written as follows1,2:
mA mB
dup dt dup dt
RfA Fd (5.1) RfB Fd
where RfA and RfB are the fluid resistance forces acting on each particle, up is particle velocity, and Fd is the force acting on each particle. For the agglomeration, the force Fd acts as a dispersion force. mA and mB are the mass of particle A and particle B respectively. These masses can be calculated by the particle diameters DpA, DpB and the particle mass density rpA, rpB. p 3 rpA DpA 6 p 3 mB rpB DpB 6 mA
(5.2)
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FIGURE 5.1 Agglomerated particle in uniform flow.
If the particle Reynolds number is in the range of the Stokes law, the resistance force Rf can be expressed by 3 RfA 3pmur DpA
(5.3)
3 RfB 3pmur DpB
where u r is the relative velocity between particle and fluid. Substituting Equation 5.2 and Equation 5.3 into Equation 5.1 and rearranging under the assumption that the mass density of the agglomerated particle is the same as that of the primary particles (rp rpArpB) leads to the following equation expressing the dispersion force Fd.
Fd 3pmur DpA DpB
(D
2 pA
(D
pB
DpA
)
2 DpA DpB DpB
)
(5.4)
When we consider the resistance force beyond the Stokes regime, the resistance force Rf is expressed by the following equation2:
Rf
p 2 2⎛ 4.8 ⎞ Dv rf ur ⎜ 0.55 ⎟ ⎝ 8 Re ⎠
(5.5)
Dv ⬅ DpA DpB Re Dv rf ur m rf : fluid density © 2006 by Taylor & Francis Group, LLC
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In this case, the dispersion force Fd can be expressed by 2 Fd {0.119 rf ur2 DpB Dv2 ( kA Dv kB )
(
3/ 2 3/ 2 2.07 mrf ur3 DpB Dv kA Dv3 / 2 kB
(
)
)
(5.6)
9.05mur DpB Dv kA Dv2 kB }/( Dv3 1) The above two expressions for the dispersion forces are assumed for single agglomerated particles. The effect of neighboring particles on the dispersion force has also been reported.3 Figure 5.2 shows the calculated result of dispersion force expressed by Equation 5.4.4 In the calculation, the diameter of particle A was kept constant at 1 µm, and the diameter of particle B was varied. The difference of the diameters is expressed by the ratio of diameters DpB/DpA for the case of DpB < DpA and DpA/DpB for the case of DpA < DpB. When the diameter ratio is unity (i.e., the diameter of particle B is the same as that of particle A), the dispersion force induced by the acceleration or deceleration is zero. In addition, when the ratio of DpA/DpB and DpB/DpA is zero (i.e.. diameter of particle B is quite a lot bigger or smaller than that of A), the dispersion force induced by the acceleration or deceleration is also zero. In other words, the dispersion induced by the acceleration or deceleration of fluid is effective for the agglomeration that consists of primary particles having an adequate size ratio. Dispersion Force Induced by Shear Flow Field When agglomerated particles are dipped into a shear flow field having a velocity gradient g, the dispersion force is induced by the shear flow. Bagster and Tomi 5 studied the force induced by the flow theoretically and reported that the shear stress t acting on the agglomerated particles has its maximum in the central plane of the agglomerated particle< and the maximum value is expressed by following equation: t 8.5mg
(5.7)
FIGURE 5.2 Estimation of dispersible region. (a) Comparison between dispersion force and adhesive force. (b) Dispersible region. © 2006 by Taylor & Francis Group, LLC
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On the other hand, when an agglomerated particle consisting of two primary particles, shown in Figure 5.3, is dipped into the shear flow field having velocity gradient g, it can be considered that the bending moment sg is also induced. In the case where the diameter of particle B is quite a bit larger than that of particle A, the bending moment sg is expressed by the following equation1: s ⬇
93mg pc 3
(5.8)
where c is a ratio of the diameter of the circular contact area to the diameter of the particle. Although the coefficient c depends on the contact configuration, the value is, in general, quite a bit smaller than unity. Thus, the value of the bending moment sg calculated by Equation 5.8 is larger than the
FIGURE 5.3 Agglomerated particle in shear flow field. © 2006 by Taylor & Francis Group, LLC
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value of the shear stress t calculated by Equation 5.7. This fact means that the bending moment dominates the dispersion force in the shear flow field. Impaction Force When an agglomerated particle having mass m collides with an obstacle with velocity vi, the impaction force Fi expressed by the following equation arises1: Fi mvi t
(5.9)
where t is the duration time of the impaction. On the other hand, compression stress si is generated in the central plane of the agglomeration. If the agglomeration is modeled by a single spherical particle having a diameter of Dpag, the compression stress si can be expressed by1 2 ⎛v ⎞ s i rp Dpag ⎜ i ⎟ ⎝ t ⎠ 3
(5.10)
When we consider the dispersion by the impaction against an obstacle, the impaction probability h should be taken into consideration. The probability h is a function of the shape of the obstacle and of the inertia parameter C or Stokes number S k . The inertia parameter C is defined by C
rpuDpag 18mD
Sk 2
(5.11)
where u and D are representative velocity and representative length, respectively. The dependence of the impaction probability h on the obstacle shape can be found in the literature describing the prediction of collection efficiency of particles on the impactor or filters. In general, the impaction probability h increases with the inertia parameter C. It can be found by Equation 5.9 through Equation 5.11 that the impaction force Fi, the compression stress, and the inertia parameter increase with the diameter of the agglomerated particles. It means that dispersion by impaction is effective for the larger agglomeration, independently of the structure of the agglomeration. The impaction force described above is the force induced by the collision of the agglomerated particle. The force induced by the collision of other obstacle with the agglomerated particle is also utilized for the dispersion. Two of the dispersers that utilize the collision force are a ball mill type and a fluidized bed type disperser. In these dispersers, not only the impaction force but also the friction or attrition force are induced and act as the dispersion force. Other Dispersion Forces The dispersion forces described above assume a simple flow field and simple particle behavior. However, the flow in a real disperser is complex, and in many case, the flow is turbulent. The turbulent flow induces dispersion force. Although the mechanism of the dispersion in the turbulent flow is not well known, it was reported that the mass median diameter Dp50 of an aerosol passing through the disperser such as the orifice type6 and the tube type7 is proportional to the energy dissipation rate −0.2, which is calculated by the pressure drop P in the disperser. If the diameter of the agglomerated particles is smaller than the microscale of the vortex in the turbulence, the shear stress t and the bending stress sg can be expressed by the flowing equations1: t ⬵ 3.1 rf m sg ⬵
10.8 rf m c3
(5.12)
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Disperser in Gaseous State Orifice An orifice is a plate having a smaller hole than the diameter of the pipe, as shown in Figure 5.4. It makes a converging and expanding flow, inducing a rapid acceleration and a rapid deceleration. The mass median diameter D p50 of aerosol particles dispersed by the orifice correlates with the energy dissipation rate calculated by the pressure drop P at the orifice 6: Dp 50 Dp 50 s 31.3 −0.2 0.4Pu d0
(5.13)
D _ p50s: mass median diameter of primary particles u : average air velocity d 0 : diameter of orifice Capillary Tube When the flow in a capillary tube is turbulent, the tube can be used as a disperser. Yamamoto et al. 7 reported that the mass median diameter Dp50 of aerosol particles dispersed by the capillary tube correlates with the energy dissipation rate calculated by the pressure drop P. Dp50 15 − 0.2 Pu 2.5d
(5.14)
d: diameter of capillary tube Ejector1,2 A schematic cross-sectional view of an ejector is shown in Figure 5.5. Pressurized primary air is introduced into the ejector. The air jet generated by the nozzle induces a pressure lower than the atmospheric pressure. The low pressure causes a secondary airflow. When a powder containing agglomerated particles is fed with the secondary airflow, the agglomerated particles are dispersed at the merging point of the primary and the secondary flow by the forces induced by the acceleration and shear flow field.
FIGURE 5.4 Orifice. © 2006 by Taylor & Francis Group, LLC
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Venturi1,2 A venturi has a throat that causes converging and expanding flows, as shown in Figure 5.6. In the venturi, low pressure is induced at the throat, resulting in a secondary airflow through the inlet tube. If agglomerated particles are fed with the secondary airflow, the agglomerated particles suffer from the forces induced by the acceleration and shear flow field and are dispersed. Fluidized Bed8 In a fluidized bed, a particle suffers mechanical forces such as impaction and attrition by the neighboring fluidizing particles. When the airflow velocity above the fluidized bed is higher than the terminal settling velocity of the primary particle fluidizing, the particles dispersed by the mechanical forces are entrained to the airflow. Thus, the fluidized bed can be utilized as a disperser. When particles do not fluidize because of their adhesive force, mixing in large particles such as glass or metal beads as the fluidizing medium is effective for dispersion (Figure 5.7). The large particles are fluidized easily and generate impaction and attrition forces that act as dispersion forces on the adhesive particles. Mixer-Type Disperser4,8,9 The mixer-type disperser shown in Figure 5.8 is one of the mechanical dispersers with a rotating obstacle. The disperser consists of the impeller rotating at high speed (from 4000 to 20,000, in general), an inlet tube attached to the top of the vessel, and outlet tube attached tangentially to the
FIGURE 5.5
Ejector.
FIGURE 5.6
Venturi.
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FIGURE 5.7
Fluidized bed.
vessel. By the rotation of the impeller, low pressure is generated above the impeller, and it causes suction flow through the inlet tube. The powder containing agglomerated particles is fed with the suction flow. The agglomerated particles are dispersed by several forces; these are the forces induced by acceleration and shear flow and/or impaction force to the impeller and the vessel wall.
4.5.2
PARTICLE DISPERSION IN LIQUID STATE
Particles in suspensions are thermodynamically unstable and tend to coagulate each other. Because well-dispersed suspensions are required in various particulate processes, it is important to establish the dispersion technology of particles in solutions. The term “dispersion” has been used in two different ways: the dispersion that will be discussed in this section represents the breakup of coagulated particles (flocs) by external forces, but it is often used to imply that a suspension is stable, such that particles do not coagulate even if they collide with each other. The dispersion of this meaning is discussed in Section 2.8.2. Flow fields, collisions, high-pressure differences, and ultrasonic fields have been employed as the fundamental principle to deflocculate flocs in various commercial dispersers. In the deflocculation by flow fields, particles will be deflocculated by the hydrodynamic drag force exerted on coagulated particles. In the collision deflocculation flocs are broken by their collision to the solid surface or the other flocs, but the contribution of this deflocculation will be small because the inertia of microscopic particles in small in liquids. Particles are deflocculated when the suspension experiences either the high-pressure drop or the ultrasonic exposure. The cavitation, vibration, or turbu© 2006 by Taylor & Francis Group, LLC
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Mixer-type disperser.
lence has been considered as the possible mechanism for these deflocculations, but enough data to clarify the mechanism have not been reported. Flow and ultrasonic fields have been popularly employed in many commercial devices to deflocculate flocs in suspensions. Here, the fundamentals of the deflocculation by the flow and ultrasonic fields are examined and their characteristics are compared.
Theoretical Background of Deflocculation by Flow Fields The flow fields employed in various dispersers are usually very complicated, but they will be decomposed into a few fundamental flows: the shear, elongational, and turbulent flows. Because the turbulent flow is too complicated to know the details of the flow field, the mechanism of turbulent deflocculation is hard to clarify, but the features of the turbulent deflocculation can be estimated using the characteristics of the deflocculation by shear flow, because flocs will be broken by the shearing flow in turbulent eddies. The hydrodynamic theory for the relative trajectory between a pair of particles of equal size in the shear and elongational flows was developed.10 Applying this trajectory analysis to a pair of coagulated particles located in a shearing plane, the conditions under which particles remain coagulated or dispersed are calculated, as illustrated in Figure 5.9, where NR is the dimensionless measure of the electrostatic repulsive force against the attractive force acting between particles 6mpa3g/A, and NF is the measure of the strength of flow field against the attractive force between particles, aco2 /A, where m and are the viscosity and permitivity of the medium, respectively, a and co are the radius and surface potential of particles, respectively, g is the shear rate, and A is the Hamaker constant.11, 12 The boundary condition between coagulated and dispersed states can be expressed analytically also by the following equation when a pair of particles is in a shearing plane12: NP =
( a / d)2 pkaN 12
R
(5.15)
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dispersed state
ka = 10
105 ka = 100
NR [-]
104 103 102 coagulated
101 state 100 2 10
103
104
105
106
107
NR [-] FIGURE 5.9 Boundary between coagulated and dispersed states for a pair of particles in a shearing plane.
where d is the minimum gap between the surfaces of coagulated particles, k = (2n0z2e2/kT)0.5, n0 and z are the number concentration and valency of ions, respectively, e is the elementary charge, k is the Boltzmann constant, and T is the temperature. This equation enables us to estimate on what conditions the shear-induced deflocculation depends. Figure 5.9 shows that d/a and ka, especially d/a, are the important parameters to control the deflocculation. The value of d is often assumed to be 4Å for bare surfaces, which is attributable to the Born repulsion between the surfaces. When molecules are absorbed on the surface, the value of d is closely correlated with the thickness of the adsorbed layer of molecules on the surface. The adsorbed layers of surfactants and polymers are often used to stabilize particles in solutions. It is also possible to evaluate numerically the deflocculation process of a pair of particles tilted from a shearing plane and a pair of unequal deflocculated than the flocs described above. Applying the trajectory analysis for a pair of flocs in an elongational flow, the coagulationdispersion map similar to Figure 5.9 will be obtained. When the map for the elongational deflocculation is compared with Figure 5.9, it is found that flocs are deflocculated at the smaller dissipation energy of flow in the elongational deflocculation. This prediction confirms are the experimental finding that the elongational flow is more effective to deflocculate particles than the shear flow.13
Deflocculation by Flow Fields It is difficult to experimentally realize the process where flocs are deflocculated solely by the simple shear or the elongational flow. Here, the characteristics of the deflocculation by the orifice contractile flow and rotational flow generated by a rotary disk are examined experimentally. These are the fundamental flows employed in the commercial devices such as dispersers with rotational blades; thus, it is important to understand the deflocculation by these flow fields. Deflocculation by Orifice Contractile Flow14 The orifice contractile flow can be realized by placing a plate with a small hole (an orifice) in the Poisuille flow. The fluid is contracted and abruptly accelerated just before the orifice. The flow along © 2006 by Taylor & Francis Group, LLC
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the centerline may be regarded as an elongational flow and the flow near the wall inside the orifice is regarded as a high shear flow. Figure 5.10 shows the dependence of the number-averaged diameter, Dav, of deflocculated particles on the energy dissipation of the orifice contractile flow, , for three orifice diameters d, where flocs of monodispersed latex particles of diameter Do = 0.71, 0.91, and 1.27 µm coagulated in 1 M KCI solution are employed. It is important to note that Dav depends on but not on d, as for as the value of Do is the same, and that the relation Dav versus is expressed by Dav ~0.035, irrespective of Do. The maximum number of constituent particles in a floc, imax, also shows the similar dependence on , Do, and d, and the relation imax versus is expressed by imax ~0.11. These equations are well supported by the data given elsewhere.15,16 The results indicate that the energy dissipation of the flow and the size of constituent particles in the floc are the most important parameters to control the degree of deflocculation. Deflocculation by Flow Generated by Rotary Disk17 Because dispersers with rotary blades are widely used, it is fundamentally important to know the characteristics of the deflocculation by the rotational flow generated by a rotary disk, as schematically shown in Figure 5.11. Because the fluid flows along the tube with the rotational motion generated by the rotor, the flow is a combined flow of the rotational, contractile, and shear flows. Figure 5.12a shows the dependence of the average diameter of broken flocs, Dav, on the rotational speed of the disk, n, for various gaps between the rotor and the inside wall of the cylinder, d. The value of Dav decreases with n, as expected . It is interesting to note that Dav increases with d when d 0 2.0 mm, but it is independent of d at d 0 2.0 mm, whereas when the same data are plotted against the shear rate g as in Figure 5.12b, the data of d 0 2.0 mm fall on a single line. These results indicate the flocs are broken by the rotational and/or contractile flow at the upstream before the rotary disk but not by the shear flow generated within the gap when d 0 2.0 mm. On the other hand, flocs are broken mainly by the shearing flow in the gap when the gap is widely open. A high-shear flow has been considered to be effective to break flocs and a very small gap between the rotor and cylinder wall has been designed to gain a high-shear flow in many commercial devices. However, according to the above results that deflocculation is not attributable to the shear flow 2.0 1.8 1.6 key
1.4
d[cm] 0.1 0.05 0.01
1.2
1.0 108
109
1010
1011
FIGURE 5.10 Dependence of the number-averaged diameter Dav of particles deflocculated by the orifice contractile flow on the energy dissipation of the flow, d is the orifice diameter, and D0 is the diameter of the constituent particle in the floc. © 2006 by Taylor & Francis Group, LLC
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within the gap when the gap is sufficiently small, a sufficient deflocculation will be achieved by employing a thin disk, thus reducing the power required to rotate the disk considerably.
Deflocculation by Ultrasonic Field18 Ultrasonic dispersers are widely used, but there have been very few systematic investigations on the deflocculation by ultrasonication. Here, the characteristics of the ultrasonic deflocculation that the author has clarified are explained. Figures 5.13a and 5.13b show the dependence of Dav and imax, respectively, on the frequency and power Ws of sonic generator, the radiation time tr, and the volume of suspensions Vf , where flocs of latex particles coagulated in 1 M KCI solution are employed. It is important to know that all the data are expressed by single curves irrespective of various experimental conditions. These results indicate that the degree of ultrasonic deflocculation is solely determined by the total sonic energy radiated to the floc solution of unit volume, Et( = Wstr /Vf), so long as the size of constituent particles is the same. This information is very important because the degree of deflocculation is easily controlled by adjusting the combination of the values of Ws, tr, and Vf. It is also found that the ultrasonication is strong enough to deflocculate flocs into particles with the initial size distribution when a sufficient amount of energy (E 0 4 107 J/m3 in this case) is radiated. It is known that the sonic vibration contributes not only to the deflocculation but also to the coagulation of suspending particles. Hence, it is necessary to make deflocculated particles free from coagulating again. This can be done by either diluting the suspension to minimize the collision frequency or dosing surfactants to form the adsorbed layer that stabilize the particles, just after flocs are deflocculated. Comparison among Deflocculations by Flow and Ultrasonic Fields18 Here the features of the deflocculations examined above are compared. Figure 5.14 shows the size distributions of the flocs with nearly the same average size that are deflocculated by the orifice contractile flow, rotational flow, and ultrasonication. It is found that the size distributions of the flocs deflocculated by fluid flows are similar to each other, but they differ from that by ultrasonication. Fractions of single constituent particles and large coagulated particles are much higher in the sonic deflocculation than in the deflocculation by the fluid flows. This result implies that flocs are split into smaller agglomerates in the case of flow-induced deflocculation, whereas single particles are ripped off one by one from the floc surface in the case of sonic deflocculation. The former mechanism is called “splitting” and the latter, “erosion.” This difference in deflocculation mechanism suggests that the flow-induced deflocculation with the subsequent sonic deflocculation will be the effective process to deflocculate particles.
rotor
contractile and shearing flows
shearing flow
FIGURE 5.11 Dependence of Dav of flocs deflocculated by the rotational disk flow on various experimental conditions. (a) Dependence of Dav on g; (b) dependence of Dav on n. © 2006 by Taylor & Francis Group, LLC
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1.4 8 (mm) 0.5 1.0 2.0 3.0 5.0
1.3
1.2
1.1
1.0
0.9 103
105
104
(a) 1.4 8 (mm) 0.5 1.0 2.0 3.0 5.0
1.3
1.2
1.1
1.0
0.9
(b)
6000
10,000 14,000 n (rpm)
18,000
FIGURE 5.12 Dependence of Dav and imax on Et ( Wstr/Vf) under various experimental conditions. (a) Dav versus Et under various conditions; (b) imax versus Et under various conditions.
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1.4
28kHz V1[CM3] 8 40 200
1.3
4 8 W 24 (W) 37
1.2
60kHz 20W
1.1
1.0
0.9
105
Et
(a) 50
108 109
106 107 [J/m3]
28kHz V1[CM3] 8 40 200 4 8 w 24 (W) 37
imax [–]
40
60kHz 20W
30
20
10
0
(b)
105
106 107
108 109
Et [J/m3]
FIGURE 5.13 Comparison between size distributions of flocs dispersed by (a) orifice contractile flow, (b) rotational disk flow, and (c) ultrasonication.
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40
(a) Dav=1.09um
Number fraction (-)
20
0 40
(b) Dav=1.10um
20
0 40
(c) Dav=1.10um
20
0 0
10
20
30
Number of particles in a floc i (-) FIGURE 5.14 Schematics of two mechanisms of deflocculation: (a) splitting, (b) erosion.
(a)
(b) FIGURE 5.15
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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Kousaka, Y., Okuyama, K., Shimizu, A., and Yoshida, T., J. Chem. Eng. Jpn., 12, 152–159, 1979. Kousaka, Y., Endo, N., Horiuchi, T., and Niida, R., Kagaku Kogaku Ronbunshu, 18, 233–239, 1992. Yuu, S. and Oda, T., AIChE J., 29, 191–198, 1983. Gotoh, K., Takahashi, M., and Masuda, H., J. Soc. Powder Technol. Jpn., 29, 11–17, 1992. Bagster, D. F. and Tomi, D., Chem. Eng. Sci., 29, 1773–1783, 1974. Yamamoto, H. and Suganuma, A., Kagaku Kogaku Ronbunshu, 9, 183–188, 1983. Yamamoto, H., Suganuma, A., and Kunii, T., Kagaku Kogaku Ronbunshu, 3, 12–18, 1977. Masuda, H., Fushiro, S., and Iinoya, K., J. Res. Assoc. Powder Technol. Jpn., 14, 3–10, 1977. Gotoh, K., Asaoka, H., and Masuda, H., Adv. Powder Technol., 5, 323–337, 1994. Batchelor, G. k. and Green, J.T., J. Fluid Mech., 56, 375, 1972. Higashitani, K., Tsutsumi, T., and Iimura, K., Proc. of China Japan Symposium on Particology, Beijing, 1996, p. 1. Higashitani, K., Iimura, K., and Vakarelski, I.U. KONA, 18, 6, 2000. van de Ven, T. G. M., Colloidal Hydrodynamics, Academic Press, London, 1989, p. 531 Higashitani, K., Tanise, N.,Yoshiba, A., Kondo, A., and Murata, H., J. Chem. Eng. Japan, 25, 502, 1992. Sonntag, R. and Russel, W., J. Colloid Interf. Sci., 115, 390, 1987. Higashitani, K., Inada, N., and Ochi, T., Colloid Surfaces, 56, 13, 1991. Higashitani, K. and Biryushi, K., Asakura Shoten, Tokyo, 1994, p. 43. Higashitani, K., Yoshida, K., Tanise, N., and Murata, H., Colloids Surfaces A, 81, 167, 1993.
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4.6
Electrical Charge Control Motoaki Adachi Osaka Prefecture University, Sakai, Osaka, Japan
Kikuo Okuyama Hiroshima University, Higashi-Hiroshima, Hiroshima, Japan
Hiroaki Masuda, Shuji Matsusaka, and Ko Higashitani Kyoto University, Katsura, Kyoto, Japan
4.6.1 IN GASEOUS STATE The charging process of particles by gaseous ions depends on two physical mechanisms: field and diffusion charging. When ions drift along electric field lines and impinge on the particle, the charging process is referred to as field charging. Diffusion charging is due to thermal collisions between particles and ions. Photoelectron charging, in which electrons are emitted from particles by UV irradiation, is also used to control the particle charge.
Gaseous Ions Production Methods Gaseous ions are produced by three methods of irradiations: UV or other radiations, discharges in gas, and fissions of droplets, as shown in Table 6.1. In -ray (≅5 MeV) and -ray (≅0.7 MeV) irradiations, high-energy particles collide with air molecules and produce primary positive ions and electrons. The primary ions grow metastable positive ions by ion–molecule reactions and ion clustering. The free electrons attach molecules with high electron affinity, such as O2 or H2O, and form negative ions. In soft X-ray (≅10 keV) and vacuum UV-ray (≅3~10 eV) irradiations, gas atoms and molecules are excited by multiphoton adsorption and emit electrons. These primary positive ions and electrons change to metastable positive and negative ions at atmospheric and low pressures but do not change in vacuum. When UV-rays (≅5 eV) irradiate a metal film, photoelectrons are emitted from the film. Photoelectrons become metastable negative ions by electron attachment and ion clustering at atmospheric and low pressures. Corona discharge is produced in an inequality electric field between a needle electrode and a plate electrode or between a line electrode and a cylinder electrode. When energy higher than the ionization potential of gas is given to casual electrons, the accelerated electrons collide with gas molecules and produce pairs of a new electron and a primary positive ion near the needle or the line electrode (discharge electrode). Either the electrons or primary positive ions are drawn toward the plate or cylinder electrode (collection electrode) according to the polarity of the discharge voltage. During the travel from the discharge electrode to the collection electrode, the electrons and primary positive ions change to metastable negative and positive ions, respectively. When water droplets are broken by a collision with an obstacle, like a waterfall, the space is filled by negative ions. This is called the Lenard effect. In a droplet suspended in air, anions generally form a layer near a surface of the droplet. By the collision of a droplet, many droplets with very small sizes are produced from a surface layer, and a few droplets with relatively large size are produced from the 465 © 2006 by Taylor & Francis Group, LLC
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Methods for Generation of Ions
Method
Principle
Products
Maximum ion Concentration
Other Applications
Irradiation -ray
Ionization of gas
Bipolar ions
107 cm-3
Static charge elimination
-ray
Ionization of gas
Bipolar ions
107 cm-3
Static charge elimination
8
-3
Soft X-ray
Photoionization of gas
Bipolar ions
10 cm
Static charge elimination
UV-ray
Photoionization in vacuum
Positive ions and electrons
105 cm-3
Static charge elimination
UV-ray
Photoelectron emission from metals
Photoelectrons or negative ions
105 cm-3
Contamination control
DC corona
Ionization of gas
Unipolar ions and ozone
107 cm-3
Electrostatic precipitator Indoor air cleaner
AC corona
Ionization of gas
Bipolar ions and ozone
107 cm-3
Static charge elimination
Collision
Lenard effect
Negative ions and charged droplets
104 cm-3
Physical therapy Sterilization
Electrospray
Reyleigh fission
Unipolar ions and charged droplets
106 cm-3
Spray pyrolysis Mass spectrometer
Discharge
Fission of droplets
body of an original droplet. As a result, the very small droplets have a negative charge and the large droplets have a positive one. The very small droplets with negative charge evaporate and become negative cluster ions consisting of 10~30 water molecules. In electrospray, DC high voltage is applied to the nozzle and droplets have a high unipolar charge. When the repulsive force between charges accumulated on the droplet is stronger than the surface tension, that is Reyligh limit, and the droplets are broken into very small droplets. Two processes for ion formation are considered. One is that liquid molecules evaporate from the very small droplets and the sizes of the droplets reduce to ions. Another is that ions evaporate directly from the surfaces of droplets. Formation Process and Reactivity The primary ions and electrons produced by the high-energy irradiation and the discharge change with time by the following ion–molecule reactions. e– + A → A– A+ + B → A + B+ A– + B → A + B– A+Bn + B + M → A+Bn+1 + M A−Bn + B + M → A−Bn+1 + M
Electron attachment Charge transfer Charge transfer Clustering Clustering
Ions that have high chemical reactivity, such as O–, O2–, O3–, H+, and N2O+, are generated generally in a first stage of growth processes. Furthermore, the ions and electrons often produce radicals and excited molecules by the following recombination reactions: AB+ + e−→ A· + B· AB+ + e− → AB* + hn
Dissociation recombination Radiative recombination
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AB+ + e− + M → A· + B· + M AB+ + C− → AC+ + B·
Three body recombination Ion–ion recombination
In air, ·O, ·OH, and O3 are produced by the above process. These radicals and excited molecules induce further chain reactions with other chemical reactants. Effects of the radicals and excited molecules on the particle materials should be considered when the ions are used to control the electrical charge. Physical Properties The physical properties of ions depend on the ion–molecule reaction and ion-clustering processes. Table 6.2 shows ion properties in air. Electrical mobility was measured by Bricard,9 and other properties were calculated from them. The positive ions are H+(H2O)n (n 58) and negative ions are O–(H2O)n, O2–(H2O)n, OH–(H2O)n, NO–(H2O)n, CO3–(H2O)n, and so forth.10 The number concentrations N of unipolar and bipolar ions are obtained by Ns
I B eAE S ei
for unipolar ion
⎛ I ⎞ N N ⎜ s ⎟ ⎝ ai eV ⎠
(6.1)
1/ 2
for bipolar ion
(6.2)
where I is the ion current in the electric field of intensity E, Is is the saturation ion current, A is the area of the electrode, V is the volume of the ion-generation chamber, Bei is the electrical mobility of ions, e is the elementary charge ( 1.602 10–19 C) and ai is the recombination coefficient of a bipolar ion. The superscripts are the polarity of the ion.
Field Charging Field charging tends to dominate the charging for particles lager than 2 µm in diameter under a strong electric field. The particle charge q can be evaluated from the equation introduced by White 11 : q
q∞ (eBeis N s t 4 0) 1 (eBeis N s t 4 0 )
(6.3)
2 ⎛ 2( 1) ⎞ p EDp q∞ ⎜ 1 1 1 2 ⎟⎠ e ⎝
(6.4)
where q is the number of elementary charges captured by a particle, q is the saturation charge, Dp is the particle diameter, t is the charging time, 0 is the dielectric constant of gas ( 8.855 10–12 F/m), and 1 is the specific dielectric constant of particles. The average charges q av calculated
TABLE 6.2
Polarity Positive ion Negative ion
Physical Properties of Air Ions Electrical mobility,
Diffusion coefficient,
Property Mean thermal velocity,
Bei (m2 s-1 V-1)
Di (m2 s-1)
ci (m s-1)
1.4x10
-4 -4
1.9x10
-6
3.53x10 4.80x10
-6
Mean free path,
Recombination coefficient,
li (m)
i (m2 s-1)
2
1.46x10
2
1.94x10-8
2.18x10 2.48x10
-8
1.6x10-12
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by Equation 6.3 and Equation 6.4 at N+t 1013 s/m3 are shown in Figure 6.1. The particle charge q decreases steeply as the particle size decreases. For the charging of high-resistivity particles, Equation 6.3 and Equation 6.4 cannot be applied, and numerical computation is necessary.12
Diffusion Charging Diffusion charging is caused by the kinetic energy of an ion and the electrostatic energy between a particle and an ion in the absence of an external electric field. Charging is classified into bipolar charging, where particles are mixed with both negative and positive ions, and unipolar charging, where they are exposed to either ion. This mechanism dominates the charging in the size range D p 0.2 µm even in strong electric fields because field charging does not work. Combination Probability of an Ion with a Particle In the theoretical estimation for the diffusion charging, a combination probability b of an ion and a particle is necessary because diffusion charging is the coagulation process of ions and particles. The following equation, derived by Fuchs,13 can be applied in the whole range of particle size: ⎛ f( l) ⎞ bqs pcis jl2 exp ⎜ ⎝ kT ⎟⎠
2 s ⎡ ⎛ f( l) ⎞ ci jl 1 exp ⎢ ⎜⎝ ⎟ kT ⎠ 2 Di s ⎣
exp[f( l) kT ] ⎤ dr ⎥ r2 l ⎦
∞
∫
(6.5)
where l is the radius of the image sphere ( ≅ lis + Dp/2), k is the Boltzmann constant ( 1.380 10–23 J/K), T is the absolute temperature, Di is the diffusion coefficient of ion, c i is the mean thermal velocity, li is the mean free path of ions, and j is the correction coefficient of free molecule collision by the electrostatic force between a particle and an ion. f(l) is the electrostatic potential between a particle and an ion given by ∞
f( l) ∫ Fe (r )dr r
FIGURE 6.1
Dp 3 − 1 e2 qe2 1 4p 0 r 1 + 1 8p 0 r 2r 2 (4r 2 − Dp 2 )
(6.6)
Average particle charge by unipolar ions.
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Figure 6.2 shows the calculated results of Equation 6.5 using the ion properties listed in Table 6.2. The combination probability for a negative ion is larger than that of a positive ion because a negative ion has larger ion properties than a positive ion. Diffusion Charging by Unipolar Ions Diffusion charging can be evaluated theoretically by two approaches: deterministic and stochastic, using the combination probability b. In the deterministic approach, the time change in particle charges q is expressed by dq bqs N s dt
(6.7)
In the stochastic approach, the time-dependent change in charge distribution is expressed by the following birth–death equations: dn0 b0s n0 N s dt dnq dt
FIGURE 6.2
(
)
bq1s nq1 bqs nq N s , q 0 and s = +
(6.8) (6.9)
Combination probability of an ion and a particle.
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dnq or
dt
(
)
bqs+1 nq +1 bq s nq N s , q 0 and s = −
(6.10)
where n is the particle number concentration and the subscripts are the numbers of elementary charges and their polarity. The above equations were solved analytically by Natanson14 under the initial condition that particles were uncharged:
(
n0 = exp b0s N s t nT
)
(6.11)
s s ⎛ q1 ⎞ q exp(bj N t ) = ⎜ ∏ bk s N s ⎟ ∑ q , q 0 and s nT ⎝ k0 ⎠ i0 s s s ∏ bi bj N
nq
(
i=0 j≠ i
(6.12)
)
(
)
s s ⎛ q1 s s ⎞ q exp bj N t ∏b N ⎟ ∑ q , q < 0 and s nT ⎜⎝ k0 k ⎠ i0 s s s b b N ∏ i j
nq
i=0 j≠ i
or
(
(6.13)
)
The average charges qav calculated by Equation 6.11 and Equation 6.12 at N+t 1013 s/m3 are shown in Figure 6.1. A necessary condition to bring uncharged particles to the desired charge distribution is obtained by15 N st
(
ln 1 npT nT
)
(6.14)
s 0
b
( exp (33.49 1.389 ln D 0.08163 ln
) D )
b0 exp 33.69 1.479 ln Dp 0.04214 ln 2 Dp 0.01001 ln 3 Dp
(6.15)
b0
(6.16)
p
2
Dp 0.01448 ln 3
p
where npT/nT is the desired number ratio of total charged particles to total particles. Equation 6.15 and Equation 6.16 are approximate equations of the theoretical curves shown in Figure 6.2. Dp is measured in nanometers in these equations. A numerical computation is necessary to solve Equation 6.5, Equation 6.11, and Equation 6.12. Although the following equation was derived under incorrect assumptions by White,11 it is useful for making a rough estimate of the diffusion charging:
q=
2p 0 Dp kT e2
⎛ Dp cis e2 N s t ⎞ ln ⎜ 1 + ⎟ 8 0 kT ⎠ ⎝
(6.17)
However, it should be noted that results given by Equation 6.17 are about two times higher than strict solutions. In the range 0.2 µm Dp 2 µm, particles are charged by both field and diffusion charging. Liu and Kapadia16 solved this problem numerically. If errors of 50–100% are allowed, the sum of Equation 6.3 and Equation 6.17 is available as a simple estimation. © 2006 by Taylor & Francis Group, LLC
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Diffusion Charging by Bipolar Ions (Neutralization) Bipolar diffusion charging is expressed generally by birth–death equations because the deterministic approach cannot reflect the difference of ion properties:
dnq dt
dno b−1 n1 N b0 n0 N b1 n1 N b0 n0 N dt
(6.18)
bq1 nq1 N bq nq N − bq1 nq −1 N + − bq+ nq N + q > 0
(6.19)
When the number concentration of bipolar ions is sufficiently high under sufficient charging time, particle charge reaches the equilibrium state. The equilibrium distribution at steady state is obtained from the above equations.17 n0 1 nT S nq nT
q
=
p 1
=
nT
S
+ p1
bp−
)
q
nq where
P (b
P (b
p1
p 1
bp
(6.20)
q 1
S,
)
q 1
S,
⎡ ⎛ b ⎞⎤ ⎡ ⎛ b ⎞⎤ S ⎢P⎜ S ⎢P⎜ ⎟⎥ ⎟⎠ ⎥ 1 b ⎠ b ⎝ ⎝ ⎢⎣ ⎢⎣ ⎥⎦ ⎥⎦
∞
∞
P 1
P 1
P 1
(6.21)
(6.22)
∞
∞
P 1
P 1
P1
(6.23)
P
P
Figure 6.3 shows the equilibrium charge distribution in the size range of 0.001 µm Dp 10 µm. Fractions of negatively charged particles are higher than those of positively charged particles due to differences in ion properties. In the size range of Dp 0.5 µm, strict solutions agree with the Boltzmann charge distribution18 within 30% error: nq nT
(
exp q 2 e2 4p 0 Dp kT
S exp (p e
∞
2 2
P ∞
)
4p 0 Dp kT
)
(6.24)
The necessary condition for which q0 charged particles attain the Boltzmann charge distribution is obtained by N b 0 kT
(
ln 0.1 q 0
)
Di e2
(6.25)
where Nb is the number concentration of ion pair ( N+ N-).
Photoelectron Charging When a solid is irradiated by UV light, electrons are emitted from the surface. This phenomenon is called a photoelectron emission and is used as a principle of surface analysis by photoelectron © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.3
Equilibrium charge distribution by bipolar ions.
spectroscopy. The photoelectron emission is generally explained by the three-step model shown in Figure 6.4. Electrons in the solid are first optically excited into states of high energy; then they move to the surface of the solid and escape into vacuum or the gas phase. Therefore, electrons can be emitted from the surface when the photon energy hn is higher than the work function fw of the solid. Photoelectron emission is also caused by UV irradiation to particles suspended in a gas stream (aerosol). In this case, aerosol particles are charged positively because of the escape of electrons. The potential barrier fq which must be overcome by an electron to escape from the q charged particle is expressed by19 fq fw
e2 ( q 1) 2p 0 Dp
5 e2 8 2p 0 Dp
(6.26)
When fq is higher than hn, the particle charges reach to a maximum value qmax obtained from Equation 6.26. qmax
hv fw 5 2p 0 Dp 2 8 e
(6.27)
The probability zq which an electron is emitted from the q charged particle is obtained by
(
zq C hv fq
)
m
I pDp 2 4hv
(6.28)
where C and m are material-dependent empirical constants, and I is the irradiation intensity. © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.4 solid.
473
Illustration of three-step model for photoelectron emission from a
4.6.2 CHARGE CONTROL BY CONTACT ELECTRIFICATION IN GASEOUS STATE When two different materials come into contact, charge is transferred according to the contact potential difference, which depends on the work functions of the two materials; thus particle charging can be controlled by changing the wall material. If the work function of the wall is close to that of the particles, the particle charging will be repressed. Furthermore, it is possible to give the opposite polarity to the particles using a different material. Since the charge transfer occurs in the contact area, the surface property is more significant than the bulk property. If the wall is covered with the oxidation layer or the layer is removed by particle collision, the surface property will be changed, and thus the particle charging will fluctuate. To avoid such instabilities, the wall surface is treated chemically or mechanically. Surface treatment such as surface coating,20 attaching a charge-control agent, 21 and chemical modification is effective to change the work function of particles. Figure 6.5 shows the contact potential difference of alumina particles coated with stearic acid.20 The value of the contact potential difference varies according to the thickness of the coating layer but is almost constant where the thickness is larger. Therefore, particle charging can be controlled by the thickness of the coating layer. The amount of charge transferred by elastic collision is proportional to the maximum contact area. When a spherical particle collides with a wall, the maximum contact area S can be approximated by22 S 1.36 ke2 / 5 rp 2 / 5 Dp 2 ni 4 / 5
(6.29)
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FIGURE 6.5 Contact potential difference of coated particles against the reference (Au). [From Yoshida, H., Fukuzono, T., Ami, H., Iguchi, Y., and Masuda, H., J. Soc. Powder Technol. Jpn., 29, 504–510, 1992. With permission.]
where ke is the elasticity parameter, rp is the density of the sphere, Dp is the particle diameter, and ni is the impact (incident) velocity. If the particle is not spherical, the particle shape should be taken into account.23 The elasticity parameter ke is given by ke
1 n12 1 n22 E1 E2
(6.30)
where n is Poisson’s ratio, E is Young’s modulus, and subscripts 1 and 2 represent the particle and the wall, respectively. To reduce the charge transferred by impact, it is desirable to choose materials having a larger Young’s modulus, that is, to use harder materials. The net contact area depends on the surface state: the net contact area decreases with increasing roughness .24 If there is no further electrification, the charge on the surface decreases in the course of time, which is called electric leakage. For conductive particles, the charge tends to leak out for a short time. It is more difficult for particles having high electric resistance to release their charge, but it is easier if they are mixed with conductive fine particles or fillers, because of the higher conductivity. Hydrophilic surface modification with a surfactant is also effective to release the charge. Charge reduction is prominent at higher humidity25; then the maximum charge on the particles is lower even though they are charged repeatedly. If the wall is made of nonconductive material, the charge on the wall is also taken into consideration. The charge forms an electric field so as to reduce the effective contact potential difference, and also causes electric discharge or dust explosion; therefore, the charge on the wall should be released. One of the effective methods for this is to induce dielectric breakdown using a thin dielectric wall reinforced with a conductive material such as metal.26
4.6.3
IN LIQUID STATE
As described in Section 2.5.2, particles in solutions bear a surface charge as a result of the adsorption of ions or the dissociation of functional groups on the particle surface. Then, because of the © 2006 by Taylor & Francis Group, LLC
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electroneutrality principle, counterions are attracted toward the particle surface, as illustrated in Figure 6.6. Some of the ions are adsorbed directly on the surface to form the so-called Stern layer. When the counterions excessively adsorb on the surface with specific affinity, sign reversal of the surface charge may occur. The rest of the ions in the bulk are in thermal motion, balancing the electrical attractive force to form a diffuse double layer. The potential at the particle surface is called the surface potential, c0, and the potential at the Stern layer is called the Stern potential, cd. Because the structure within the Stern layer is not well known, the Stern potential is often regarded as the surface potential of particles for many engineering purposes. Hence the Stern potential is expressed by c 0 in the succeeding subsection unless the specific effect of the Stern layer is discussed.
Potential and Charge in the Diffuse Layer27 The Poisson equation holds between the volume density of charge r and the potential c in the diffuse double layer: 2 c
r
(6.31)
where is the permitivity of the medium. The charge density is given by the sum of all the ionic contributions in the unit volume: r ∑ ni zi e
(6.32)
where ni and zi are the number concentration and valency of ions of type i, respectively, and e is the elementary charge. The Boltzmann equation holds at the equilibrium where the electrostatic and chemical potentials of ions balance: ⎛ z ec ⎞ ni n0i exp ⎜ i ⎟ ⎝ kT ⎠
(6.33)
FIGURE 6.6 Model of Stern layer and diffuse layer: (a)a small amount of ions is adsorbed on the surface; (b) an excessive amount of ions is adsorbed because of their high affinity to the surface. © 2006 by Taylor & Francis Group, LLC
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where n0i is the bulk concentration of ions of type i, k is the Boltzmann constant, and T is the temperature. Substitution of Equation 6.32 and Equation 6.33 into Equation 6.31 results in the Poisson– Boltzmann equation: ⎛ n z e⎞ ⎛ z ec ⎞ ∇2 c ⎜ 0 i i ⎟ exp ⎜ i ⎟ ⎝ ⎠ ⎝ kT ⎠
(6.34)
This equation cannot be solved explicitly in general. However, when the Debye–Huckel approximation that c is small everywhere [i.e., zec < kT (c < 25.7/z mV at 25°C)] holds, and the electrolyte is symmetric with valency z z z and concentration n 0 n 0 n 0, the equation is simplified and the analytical solution is derived. When the surface is a flat plate, the potential distribution is given by c+ c0 e xp(kx ) ⎛ 2n z 2 e2 ⎞ k⎜ 0 ⎝ kT ⎟⎠
(6.35)
0.5
(6.36)
where x is the distance from the plate surface, k is a measure of the ionic concentration of the bulk solution, and 1/k is called the thickness of the diffuse layer. The charge density of the diffuse layer, sd, is given by ∞
s d ∫ rdx 0
(6.37)
Then, Equation 6.37 is rewritten as follows, using Equation 6.32 and Equation 6.34: ⎛ 4 n ze ⎞ ⎛ zec0 ⎞ ⎛ dc ⎞ s d ⎜ ⎟ ⎜ 0 ⎟ sinh ⎜ ⎝ dx ⎠ x 0 ⎝ k ⎠ ⎝ 2 kT ⎟⎠
(6.38)
When the effect of the Stern layer is regarded as negligible, the surface charge, sd, is given by ⎛ 4 n ze ⎞ ⎛ zec0 ⎞ s 0 = s d = ⎜ 0 ⎟ sinh ⎜ ⎝ k ⎠ ⎝ 2 kT ⎟⎠
(6.39)
This shows the correlation between the surface potential and the charge density of the diffuse layer. A similar argument can be developed for the potential around a spherical particle of radius a, and the potential in the diffuse layer is given as ⎛ c a⎞ c ⎜ 0 ⎟ exp[ k(r a )] ⎝ r ⎠
(6.40)
The total charge of the particle Q is then written as ∞
Q = ∫ 4pr 2 rdr 4p a(1 ka )c0 a
(6.41)
Charge Control by Electrolytes In order to control the charge of the particle surface, it is necessary to know not only the charging mechanism but also how the surface charge varies with the surrounding conditions. This is illustrated using a simple model for surface charging.28 © 2006 by Taylor & Francis Group, LLC
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When the particle surface is charged by the complete dissociation of “strong” functional groups, such as OSO3H → OSO3 H, for polystyrene latex particles in an aqueous solution, the surface charge s0 depends only on the density of dissociation sites, N ( [OSO3]), as shown by Equation 6.42, where [ ] denotes the site density or solution concentration: s0 FN
(6.42)
where F is the Faraday constant. When the particle surface is charged by the dissociation of “weak” functional groups, such as COOH, the degree of dissociation is influenced greatly by the property of the aqueous solution. Suppose there is a weak-acid functional group described by SH. Then it dissociates as K ⎯⎯ ⎯ → S Hs SH ← ⎯
(6.43)
where K [S][Hs]/[SH] and [Hs] is the interfacial concentration of protons, which is related to the concentration in the bulk [H] by [HS] [H]exp(ec0/kT). Because the density of the surface site, N, is given by N [SH] [S], the surface charge is written as 0 F[S] =
{ 1 [H
FN ]exp(ec0 / kT ) / K
}
(6.44)
When K , Equation 6.44 coincides with Equation 6.42. Then the values of 0 and c0 will be obtained by solving Equation 6.39 and Equation 6.44 simultaneously. Hence, it is known that 0 and c0 are the functions of N, K, [H], and k, respectively. Because the values of N and K are usually fixed in a given solution, the surface charge may be controlled by [H] and k (i.e., the pH and ionic concentration of the solution). The description of the surface charge of amphoteric materials, such as metal oxides and proteins, is more complicated than that of the monofunctional surface explained above. The dissociation of a metal oxide surface can be written as K1 K2 ⎯⎯ ⎯⎯ → − SH H 2 O ← ⎯⎯ ⎯⎯ → S + H H 2 O SH2 OH ←
(6.45)
It is clear that the surface charge varies from positive to negative as the value of pH increases. An argument similar to that for the monofunctional surface using a generalized 1pK model enables us to evaluate the surface charge of amphoteric materials.12 The predicted surface charge and potential are drawn in Figure 6.7. The estimated values might not be correct quantitatively, but the qualitative features are well predicted: (1) the positive charge decreases with pH at low pH, the charge increases negatively with pH at high pH, and there is the point of zero charge (p.z.c.) in between; and (2) the absolute value of the surface potential decreases, but that of the surface charge increases, as the ionic concentration in the bulk increases. It is clear that the pH and ionic concentration are the important parameters to control the charge and potential of particle surfaces.
Surface Charge and Specific Adsorption There are ions that have a certain affinity for the surface that cannot be explained by the simple principles described above. They are called specifically adsorbing ions. When these ions exist, the surface charge or potential in any particular system is not easy to control, although it seems there exists © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.7 Surface charge and potential predicted using the generalized 11-pK model. Ce is the electrolyte concentration.29 [From Koopal, L.L., in Coagulation and Flocculation, Bobias, B., Ed., Marcel Dekker, New York, 1993, p.101.]
some rule to govern the specific adsorption. It is known experimentally that the adsorption depends on various conditions, such as the pH, the electrolyte concentration, and the type of electrolyte in the solution. Some typical data are presented in Figure 6.8, showing the electrophoretic mobility of rutile particles as a function of pH in Ca(NO3)2 solutions and for various nitrates at the constant concentration.13 When no specifically adsorbing ion exists, the mobility must decrease monotonically with pH, as predicted in Figure 5.10. The increase of the mobility at high pH in Figure 6.8 indicates the existence of specific adsorption. The data in Figure 6.8 a shows that cations adsorb specifically on the negatively charged surface, and the degree of specific adsorption increases with the magnitude of surface charge and the ionic concentration. Figure 6.8 shows that the adsorption depends also on the type of electrolyte, and the adsorbing ability of this system can be written as Ba Sr Ca Mg. It is known that the ability of specific adsorption follows the Hofmeister or lyotropic series given below, if the valency of ions is the same. These series coincide with the order of the radii of ions that are given in parentheses (angstroms). Because the degree of hydration of ions decreases with the increasing radius of ions, the explanation of the higher adsorbing ability of larger ions is that less energy is needed for the ions to be dehydrated in their adsorption on the surface: Cs(1.81) Rb(1.66) K(1.52) Na(1.13) Li(0.73) Ba(1.49) Sr(1.32) Ca(1.14) Mg(0.71) I(2.06) Br(1.82) Cl(1.67) F(1.19) © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.8 Dependence of electrophoretic mobility on pH for rutile (a) in solutions of various concentrations of Ca(NO3)2 and (b) in 0.33103 M Ca(NO3)2 solutions with various types of cations. [From Furstenau, D.W., Manmotian, D., and Raghavan, in Adsorption from Aqueous Solutions, Tewari, P.H., Ed., Plenum Press, New York, 1981, p. 81.] 479
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Charge Control by Surfactants The surface charge of particles in aqueous solutions is controllable using surfactants. When the alkyl group of a surfactant has a high affinity to the surface, the aliphatic tail is adsorbed on the surface and the polar head is directed toward the solution, as shown in Figure 6.9. Then particles are dispersed and stable because of the interparticle repulsive force caused by the charge of the adsorbed surfactants. When the bare surface is charged as shown in Figure 6.9 and surfactants of the opposite charge are dosed in the solution, the head groups are adsorbed on the surface because of the electrostatic attractive force between them. When the first adsorbed layer is formed completely, the surface becomes hydrophobic, and particles will be unstable because of the hydrophobic attractive force. When the surfactant is dosed further, the secondary adsorbed layer will be formed such that the polar heads are directed toward the solution. Then the surface charge is reversed and the suspension becomes stable again. An example is given in Figure 6.10, showing the change of the zeta potential of AgI sol with the concentration of anionic surfactants.29 It is important to note that the longer the alkyl group is, the less molar concentration of surfactant is needed to vary the charge of the surface. Because the longer the aliphatic tail is, the more segregated from water surfactants are, surfactants with the long tail are adsorbed on the surface effectively.
FIGURE 6.9 Schematic of the surfactant adsorption, the charge, and the stability of particles. © 2006 by Taylor & Francis Group, LLC
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It is possible to control the surface charge in such a way that the secondary surfactants of different kinds are dosed after the first layer is formed. In this case, the final surface charge will be given by the charge of the secondary surfactants.
Charge Control by Polymers Water-soluble polymers are very effective agents to control the surface charge of particles in aqueous solutions. Various polymers, which differ in charge density, hydrophobicity, molecular weight, and so forth, have been synthesized, and even polymers which have several different properties simultaneously are able to be synthesized these days. Hence, it is possible to prepare a polymer which has a high affinity to the particular surface and the property desired as the adsorbed layer at the same time. High molecular weight polymers which are opposite to the particle surface in charge are popularly employed to control the charge of particles in solutions. When highly charged polymers are dosed, they adsorb on the particle surface with strong electrostatic attractive force, and the surface charge is altered even by a small dosage. Typical data are presented in Figure 6.11,
FIGURE 6.10 Dependance of the zeta potential of AgI sol on the surfactant concentration: (1) CH3(CH2)9SO4Na; (2) CH3(CH2)11SO4Na; (3) CH3(CH2)13SO4Na.
FIGURE 6.11 Zeta potential of latex particles versus the concentration of dosed cationic polymers. Cp and Mw are the concentration and molecular weight of dosed polymers, respectively. © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.12 Schematic of the surface charge by ions in micelles in nonaqueous medium.
showing the change of the zeta potential of negatively charged polystyrene lattices with the concentration of dosed cationic polymers.30 The strong coulombic force acts between the polymers and surface, and almost all polymers are adsorbed on the surface unless the amount of dosage is too large. It is interesting to note that the surface charge depends only on the total weight of dosed polymers but does not depend on the molecular weight of the polymer. This is because the charge density of the adsorbed layer is determined by total charge (i.e., the total weight) of adsorbed polymers, as far as the structure of polymer is the same and all the polymers dosed are adsorbed on the surface.
Charge in Nonaqueous Medium The charge of particles in nonaqueous solutions varies sensitively with the property of the medium, water, and surfactants dissolved. There is no general rule to explain the charging mechanism in nonaqueous solutions at present. Nonaqueous solutions are classified into polar and nonpolar solutions. Because the characteristics of particles in polar solutions are more or less similar to those in aqueous solutions, the characteristics of the charge of particles in nonpolar solutions are considered here. It is almost impossible to eliminate a small amount of water stuck on the particle surface even if the suspension is carefully prepared. The water dissociates as H2O → ← H OH . This proton combines with either the particle surface (S) or the medium molecules (M), depending on the relative acidic and basic balance between the surface and medium. When the basic strength of the surface is greater than that of the medium, the proton will combine with the particle surface and the surface will be charged positively, as shown below. But if it is the reverse, the surface will be negatively charged: −SH MOH
S M H OH
SOH MH
(6.46)
This mechanism was successfully confirmed by the experimental data in Table 5.2. 31 If surfactants exist in nonaqueous solvents, ions are able to be solubilized into reversed micelles of surfactants, as shown in Figure 6.12. If these charged micelles are adsorbed on the surface, the surface will be charged. The charge in this case depends on the combination of the properties or the surfactant, the medium, and the surface, as illustrated in Table 6.3.32 It is known that the existence of water in these systems affects sensitively the charge of the particle surface.33 © 2006 by Taylor & Francis Group, LLC
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TABLE 6.3 of Watera
483
Particle Charge in Organic Solvents Containing a Small Amount
Particle
Solvent 2,2,4-Trimethylpentane Benzene (40) (270)
Methyl ethyl ketone (650)
Ethylacetate (700)
Carbon black
—
—
—
Toluidine red
—
—
—
Titanium dioxide
—
a Values in parentheses are water content (ppm). Source: Tamaribuchi, T. and Smith, M. L., J. Colloid Interface Sci., 22, 404, 1966.
REFERENCES 1. Kousaka, Y., Adachi, M., Okuyama, K., Kitada, N., and Motouchi, T., Aerosol Sci. Technol., 2, 421–427, 1983. 2. Lui, B. Y. H. and Pui, D. Y. H., Aerosol Sci. Technol., 5, 465–472, 1974. 3. Han, B., Shimada, M., Choi, M., and Okuyama, K., Aerosol Sci. Technol., 37, 330–341, 2003. 4. Inaba, H., Ohmi, T., Yoshida, T., and Okada, T., J. Electrostatics, 33, 15–42, 1994. 5. Shimada, M., Cho, S. C., Tamura, T., Adachi, M., and Fuhii, T., J. Aerosol Sci., 4, 649–661, 1997. 6. Akasaki, M., in Handbook of Electrostatics, The Institute of Electrostatics, Japan, Ed., Ohm-Sha, Tokyo, 1981, pp. 217–218 (in Japanese). 7. Moore, A. D., in Electrostatics and Its Applications, John Wiley, New York, 1973, Chap. 4. 8. Kebarle, P. and Tang, L., Anal. Chem., 65, A972–A986, 1993. 9. Bricard, J., in Problems of Atmospheric and Space Electricity, Coroniti, S. C., Ed., Elsevier, Amsterdam, 1965, pp. 82–117. 10. Mohnen, V. A., in Electrical Processes in Atmospheres, Dolezalk, H. and Reiter, R., Steinkopff Verlag, Darmstadt, 1977, pp. 1–16. 11. White, H. J., AIEE Trans., 70, 1186–1191, 1951. 12. Masuda, S. and Washizu, M., J Electrostat., 6, 57–67, 1979. 13. Fuchs, N. A., Geofis. Pura Appl., 56, 185–193, 1963. 14. Natanson, G. L., Sov. Phys., 5, 538–551, 1960. 15. Adachi, M., Okuyama, K., and Kousaka, Y., in Proceedings of the Second International Conference on Electrostatic Precipitation, Kyoto, 1984, pp. 698–701. 16. Liu, B. Y. H. and Kapadia, A., J. Aerosol Sci., 9, 277–242, 1978. 17. Adachi, M., Kousaka, Y., and Okuyama, K., J. Aerosol Sci., 16, 109–123, 1985. 18. Keefe, D., Nolan, P. J., and Rich, T. A., Proc. R. Ir. Acad., 60-A, 27–45, 1959. 19. Maisels, A., Jordan, F., and Fissan, H., J. Appl. Phys., 91, 3377–3383, 2002. 20. Yoshida, H., Fukuzono, T., Ami, H., Iguchi, Y., and Masuda, H., J. Soc. Powder Technol., Jpn., 29, 504– 510, 1992. 21. Itakura, T., Masuda, H., Ohtsuka, C., and Matsusaka, S., J. Electrostat., 38, 213–226, 1996. 22. Timoshenko, S. P. and Goodier, J. N., Theory of Elasticity, 3rd Ed., McGraw-Hill, New York, 1970, pp. 409–422. 23. Masuda, H. and Iinoya, K., AIChE J., 24, 950–956, 1978. 24. Ema, A., Tanoue, K., Maruyama, H., and Masuda, H., J. Powder Technol. Jpn., 38, 695–701, 2001. 25. Nomura, T., Taniguchi, N., and Masuda, H., J. Soc. Powder Technol. Jpn., 36, 168–173, 1999. 26. Matsusaka, S., Nishida, T., Gotoh, Y., and Masuda, H., Adv. Powder Technol., 14, 127–138, 2003. 27. Hunter, R. J., Foundations of Colloid Science, Vol. 1., Clarendon Press, Oxford, 1987. 28. James, R. O., in Polymer Colloids, Buscall, R., Corner, T., and Stageman, J. R., Eds., Elsevier, London, 1985, p. 69. © 2006 by Taylor & Francis Group, LLC
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29. Watanabe, A., Bull. Inst. Chem. Res. Kyoto Univ., 38,179, 1960. 30. Higashitani, K., Kage, A., and Arao, E., Dispersion and Aggregation, Moudgil, B. M. and Somasundaran, P., Eds., Engineering Foundation, New York, 1994, p. 191. 31. Tamaribuchi, T. and Smith, M. L., J. Colloid Interface Sci., 22, 404, 1966. 32. Kitahara, A., Amano, M., Kawasaki, S., and Kon-no, K., Colloid Polymer Sci., 255, 1118, 1977. 33. Cooper, W. D., J. Chem. Soc. Faraday Trans., 1, 70, 864, 1974.
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4.7
Surface Modification Mamoru Senna Keio University, Yokohama, Kanagawa, Japan
4.7.1
PURPOSE OF SURFACE MODIFICATION
The surface of powder particles is often modified to adapt them to their final uses. The purposes of surface modification are very diverse. They range from simple organophilication of inorganic substances by a coupling agent to very complicated complex formation with the guest species under well-controlled distribution and interactions with a substrate. Introduction of electronically or biologically functional groups belongs to an important part of a fast expanding palette of techniques. Surface modification and formation of composite or complex particles are almost indistinguishable, since the change of the outermost surface almost always influences the physicochemical properties of the near-surface region. It is therefore more appropriate to discuss the method and consequences of surface modification together with those of complex or composite particles. Modern surface modification techniques aim at finer particles down to nanosized ones under precisely controlled homogeneity. A number of elegant methods including in situ surface reactions are being developed. After a brief description of modern techniques, some case studies and examples are given. The significance of characterization and related techniques is also referred to.
4.7.2
METHODS OF SURFACE MODIFICATION
The technology of surface modification of fine particles has been traditionally developed in the pigment and paint industries. They have been dealing exclusively with fine particles. Similar techniques have been applied for rather traditional fillers, for example, calcium carbonate, kaolin, or carbon black. Soaking of powders in an appropriate solution containing inorganic salts, dispersants or surfactants, as well as coupling agents, is quite easy and popular. However, this is always followed by a drying process, which brings about a nuisance of agglomeration due to capillary pressure at the final stage of solvent evaporation. In huge contrast to those traditional fields, many of the modern surface coating and modification methods are being applied to fine particles. Specific techniques are becoming more precision oriented under conditions as mild as, and as energy-saving as possible. One of the most important aspects for any kind of surface modification is the chemical affinity between the substrate and newly settled surface species. The weakest interaction is physical adsorption, which is by van der Waals forces, followed by various chemical adsorptions. Surface nucleation and graft polymerization root the active centers on the host surface, so that they result in stronger attachment to the substrate, as compared to any kind of adsorption. Frequently, those active centers must be artificially introduced by etching, adsorption of various activators, irradiation of electromagnetic beams, or mechanical stressing. Methods of surface modification are conventionally divided into physical and chemical ones. Physical methods are subdivided into mechanical treatment, irradiation, sputtering, or similar techniques, combined with vacuum technology. Chemical methods begin with the above-mentioned soaking and resulting adsorption, and surface deposition with or without in situ surface reaction. In contrast to a posttreatment, that is, to put some new materials on the surface of already matured solid surfaces, there are also many chemical processes, where host particles and surface layers are synthesized simultaneously. The latter is generally called an in situ reaction. Organic polymers are very often subject to such an in situ synthesis to give desired properties by choosing an appropriate initiator1 or adding 485 © 2006 by Taylor & Francis Group, LLC
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different monomers at an interval.2 A chemical process of surface modification is often regarded as immobilization of chemical species on the substrate surface. This subject is discussed at the end of this chapter (4.6.8). To make uniform layers or films on the surface of the host particles is very common in the field of surface modification. They are generally categorized as microencapsulation.3 Surface polymerization, coacervation, or even mechanical deposition with and without a subsequent heat or chemical treatment belong to this category. Electroless deposition is also a well-developed chemical method, leading to a similar result. The guest species are mostly metallic ones,4 but not exclusively. While mechanical routes are fairly free from choice of combination between the host and guest species, 5 most of the surface chemical methods are conditional with respect to interfacial affinity. On the other hand, the former has serious drawbacks of structural degradation and contamination after prolonged handling in a machine similar to mixers or grinding mills. The thickness of activated layers is distributed from a few atomic layers to several micrometers.6 In the latter case, the activated near-surface region has its specific “bulk” properties as well. Chemical affinity at the host–guest interface within the scope of conventional colloid and interface science is one of the fundamental prerequisites for a chemical route. Active centers are often provided on the host surface for coating with a film. The density and uniformity of such active centers are decisive for uniform nucleation and growth of the surface species, which covers the surface of the host particles continuously and uniformly. For the purpose of the controlled release of drugs or fertilizers, control of the properties of the surface film is very important. 7,8
4.7.3
CONVENTIONAL TREATMENTS WITH SURFACTANTS, COUPLING AGENTS, AND SIMPLE HEATING
Every surface-active or coupling agent can be made suitable for surface modification, provided the agent firmly adsorbs on the surface of the substrate particles. This has long been done by soaking immersion or impregnation.9 Most of the supported catalysts are prepared by soaking the carrier in an aqueous solution of the active species, followed by an appropriate heat treatment.10 Since supported catalyses are quite sensitive whether and to what extent the chemical interaction takes place between the host surface and the guest species, it is important to learn from the entire preparation process of supported catalyses, not only the method of preparation but also characterization for better surface modification. For the control of chemical interaction at the host–guest interface, subsequent heat treatment is also of vital importance. Some classical examples are given below for the surface modification of pigments.11 Titania, as one of the most frequently used pigments, is normally surface treated by inorganic salts such as aluminum sulfate or sodium silicate.12 By choosing subsequent heat treatment conditions, the microstructure, notably pores and fissures, is controllable. Organophilication is very usual and accomplished by using various organic reagents with polar groups, for example, n-buthanol, decyl amine, and tetramethyrol cyclohexanol.13 Calcium carbonate and kaolin are frequently made organophilic by using, for example, poly(acrylic acid) or acetates of alkyl amine.13 Various coupling agents are also used, for example, titanates, chromates, or silane coupling agents. The surface of carbon black can be made hydrophilic by simple heating to give oxidation products. Carbon black can also be surface treated by chemicals such as dodecyl benzene sulfate.13 Thermal treatments to introduce a controlled surface oxide layer are not restricted to carbon black but used on ultrafine metal particles to avoid self-ignition. 14
4.7.4 MICROENCAPSULATION AND NANOCOATING Microencapsulation is a kind of surface coating, developed mainly in the pharmaceutical industry for the specific purpose of a better drug delivery system and toxicity protection. 15 In this context, one © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.1 Scanning, as well as transmission, electron micrographs of titania encapsulated by poly(methylmethacrylate). Without (a) and with (b) pretreatment by SDS. Kindly supplied by Professor Masahiro Hasagawa.
of the most popular ways of drug coating is coating the drug with gelatin sponges through coacervation. For successful coacervation, the core particles, being mostly drug particles, should not be too soluble in the coacervation solution. As long as an aqueous solution of gelatinlike substance is used, an initiator of phase separation should also be hydrophilic. Pretreatment of the core particles by polymeric ions often facilitates coacervation by attracting a larger amount of gelatin to form surface layers of high quality. Encapsulation of inorganic particles is also becoming popular. Ultrafine silicon particles, for example, can be coated by carbon with a relatively simple method: exposing the core silicon particles in a vapor of benzene diluted by argon and fire at temperatures as high as 1000°C. 16 The obtained carbon layer is considerably graphitized. Graphite basal planes are found parallel to the surface of the silicon particles. © 2006 by Taylor & Francis Group, LLC
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Surface treatment via an aqueous polymerization is particularly useful. Like most chemical processes, one of the most important factors for the purpose of uniform coating is a pretreatment. In the case of coating titania with poly(methacrylic acid), preadsorption of sodium dodecyl sulfate is indispensable. For the purpose of thickness control of the surface layer, successive addition of a monomer is recommended rather than applying the desired total amount at the beginning.17) When poly(methylmetacrylate) (PMMA) is to be coated on the surface of titania, it is necessary to use surfactants such as sodium dodecyl sulfate (SDS) prior to in situ polymerization. When surface pretreatment is insufficient, localized, clusterlike polymerization takes place, in contrast to the uniform polymeric surface layer achieved by the use of SDS pretreatments. A similar technique is applicable to coat ultrafine magnetic particles with PMMA. Successful and unsuccessful surface layers are easily distinguished on the transmission electron micrographs shown in Figure 7.1. Those coated magnetic fine particles are not only used for a normal magnetic fluid, but also made bioactive after subsequent biochemical treatments. Ultrafine metallic iron with very thin protective oxide layers was successfully modified by adding acrolein for the purpose of further, higher-order surface modification, for example, by enzymes. 18 Polymeric particles or microspheres can be modified with various dissimilar polymeric species as well. They have been developed to the level of surface design for biochemical or medical uses.19 To prepare a desired surface for subsequent treatment, it is important to choose an appropriate initiator. A comonomer may be most appropriate. After polymerization of styrene (acrylamide copolymer microspheres), the surface can be carboxylated under controlled hydrolysis. A Hofmann reaction brings about, in contrast, a series of amphiphilic surfaces with a regulated isoelectric point by controlling the temperature and time of the surface reaction.20
4.7.5 POLYMERIZATION AND PRECIPITATION IN SITU An in situ synthesis of coated materials can be subdivided into several categories. By using a subtle difference in the polarity, a mixture of monomers often brings about various core-shell polymeric particles in one step. Core-shell polymerization is an example of in situ surface modification and results in a number of useful properties on the surface of microspherical particles.21 Seeded polymerization is a similar technique but with wider variation and at the cost of an extra step.22 When inorganic precipitation is carried out in the presence of polymerizable species, fine inorganic particles are often coated in situ by polymer films. Magnetite fine particles are obtained, for example, in PVA solution, resulting in the precipitates of a well-dispersed nanometer regime with considerably large saturation magnetization.23 Mn-doped ZnS is prepared in the presence of methacrylic acid in a methanolic medium to give nanoparticles with brilliant electroluminescent properties.24 They show quantum dot effects with a blue shift of the absorption edge.
4.7.6
MECHANICAL ROUTES AND APPARATUS
Via mechanical routes, solid particles are covered relatively easily by various solids in the form of gels, fine particles, or films. Apart from traditional coating machines, there are a number of machines available on the market today for the purpose of mechanical surface modification or coating. There are numerous reports of using such commercial apparatuses to achieve, for example, acquired superplasticity after surface modification. 25,26 When the surface of the host materials is modified with another solid species, chemical interaction or solid-state reactions occur at the interface .27,28 Such a mixture can be used for many practical purposes including graded functional materials.29
4.7.7
CHARACTERIZATION OF COATED PARTICLES
Morphological observation is by no means sufficient to analyze the results of surface modification. If the purpose is simple enough, for example, to prevent agglomeration due to a hygroscopic nature, © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.2 (a) Transmission electron micrograph of sliced polystyrene with electroplated Ni. (b) Scheme explaining the method of evaluating fluctuation of the thickness of the surface layer in the radial direction, to be evaluated from the variance of the local thickness. Subdivision up to 36 sections per particle suffices, in general, for a reliable result.
conventional measurements of various mechanical tests of the compacts could suffice. Rheological measurements are quite often used for the evaluation of surface modification.30 Conversely, surfacemodified particles can be used as dispersing particles for electrorheological fluids, which is gathering increasing interest. © 2006 by Taylor & Francis Group, LLC
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Chemical interaction between the core and surface species, as well as in the surface layers, can conveniently be examined by various surface analyses including X-ray photoelectron spectroscopy.31 If particles are embedded and cut into a very thin slice of a nanometer regime, transmission electron microscopy could reveal the microstructure at the interface, as well as the extent of mechanochemical short-range diffusion, provided the microscope is equipped with a tool for local elemental analysis. At the same time, microhomogeneity within a modified single particle can be quantitatively determined by using statistical variance obtained from repeated local analyses of the composition.32 Topochemical distribution of the deliberately introduced surface species, for example, by electroless deposition, can be made, combined with the above-mentioned ultrathin slice technique, by sectioning the particle into several radial elements and evaluating the statistical fluctuation of the thickness.33 As shown in Figure 7.2, radial fluctuation and the average thickness of the surface layer can be determined from the repeated measurements of the thickness on the cross-sectional view of surface-coated particles.
4.7.8
REMARKS AND RECENT DEVELOPMENTS
To create a functional surface on powder particles is one of the basic issues for surface modification. Silica is one of the best examples in this context, for inorganic materials, as well as for inorganic–organic composites.34 To immobilize the appropriate chemical species, their selectivity, irreversibility, and stability are of particular importance.35 Biomimetic methods are also used to modify nonbiological materials. Coating of polymeric microspheres by uniformly grown inorganic particles is only one example of such a tendency.36 For tissue engineering including controlling cellular responses, various techniques of surface modification are applied.37 There have been many attempts to modify natural organic products, such as leather powder, by copolymerization to gain stronger affinity with a matrix phase to obtain artificial leather.38 An entirely different aspect of modern surface modification methods is to carry out in situ methods, with or without passing through a dry state, when a process is wet and the end state of use is also wet. Drying after surface modification is not only energetically unfavorable but also brings about hazardous, undesired agglomeration.39 An all-dry process can avoid undesirable agglomeration. One such technique is plasma-aided surface polymerization. It enables surface modification of fine particles with cross-linked, tough polymer layers from otherwise unpolymerizable species. Thus, the technology of surface modification of powder particles is in the process of rapid expansion. A vast number of new technologies are introduced every year at symposia, meetings, and expositions, dealing with functional fine particles. The number of patents in this direction is also rapidly increasing. More systematic studies are required to impart those attractive technologies a sound scientific and theoretical basis for further development.
REFERENCES 1. 2. 3. 4. 5. 6. 7.
Sugiyama, K., Ohga, K., and Kukukawa, K., Macromol. Chem. Phys., 195, 1341, 1994. Inomata, Y., Wada, T., and Sugi, Y., J. Biomater. Sci. Polym. Ed., 5, 293, 1994. Shiba, M., Tomioka, S., and Kondo, T., Bull. Cem. Soc. Jpn., 46, 2584, 1973. Jackson, R. L., J. Electrochem. Soc., 137, 95, 1990. Mizota, K., Fujiwara, S., and Senna, M., Mater. Sci. Eng. B, 10, 139, 1991. Bernhardt, C. and Heegn, H., Freiberger Forsch.-H., A602, 49, 1978. Fukumori, Y., Ichikawa, H., Yamaoka, Y., Akino, E., Takeuchi, Y., Fukuda, T., Kanamori, R., and Osako, Y., Chem. Pharm. Bull., 39, 164, 1991. 8. Nakahara, K., Shikizai, 59, 543, 1986.
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Surface Modification 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.
491
Perfitt, G. D., Dispersion of Powders in Liquids, Applied Science Publ., London, 1981. Delmon, B., in Reactivity of Solids, Barret, P. and Dufour, L., Eds., Elsevier, Amsterdam, 1984, p. 81. Seino, M., Shikizai, 40, 163, 1967. Perfitt, G. D. and Sing, K. S. W., Characterization of Powder Surfaces, Academic, London, 1976. Moriyama, Y., in Surface Modification, Chem. Rev., No. 44, Nishiyama, Y., Ed., Gakkai Shuppan Center, Tokyo, 1984, p. 118. Kashu, S., in Ultrafine Particles, Science and Application, Chem. Rev., Vol. 48, Chemical Society of Japan, Ed., Gakkai Shuppan Center, Tokyo, 1985, p. 135. Nixon, J. R., Saleh, A. H., Khalil, J. R., and Carless, J. E., J. Pharm. Pharmacol., 20, 528, 1968. Iijima , S., J. Surface Sci. Soc. Jpn., 8, 325, 1987. Hasegawa, M., Arai, K., and Saito, S., J. Polym. Sci. A Polym. Chem., 25, 3231, 1987. Miyamoto, H., J. Surface Sci. Soc. Jpn., 8, 345, 1987. Kato, T., Fujimoto, K., and Kawaguchi, H., Polym. Gels Networks, 4, 237, 1993. Kawaguchi, H., Hoshino, H., Amagasa, H., and Ohtsuka, Y., J. Colloid Interface Sci., 97, 465, 1984. Okubo, K. and Hattori, H., Colloid Polym. Sci., 271, 1157, 1993. Kobayashi, K. and Senna, M., J. Appl. Polym. Sci., 46, 27, 1992. Lee, J. W. and Senna, M., Colloid Polym. Sci., 273, 76, 1995. Yu, I., Isobe, T., and Senna, M., Mater. Eng. Sci. B, in press, 1995. Yokoyama, T., Urayama, K., Naito, M., and Yokoyama, T., Kona, 5, 59, 1987. Alonso, M., Satoh, M., and Miyanami, K., Powder Technol., 59, 45, 1987. Mizota, K., Fujiwara, S., and Senna, M., Mater. Sci. Eng. B,10, 139, 1991. Saito, I. and Senna, M., Kona, in press, 1995. Tanno, K., Yokoyama, T., and Urayama, K., J. Soc. Powd. Technol. Jpn., 27,153, 1990. Otsubo, Y., Colloid Surf., 58, 73, 1991. Saito, I. and Senna, M., Kona, in press, 1995 Komatsubara, S., Isobe, T., and Senna, M., J. Am. Ceram. Soc., 77, 278, 1994. Nakajima, S., Koga, T., Isobe, T., and Senna, M., Mater. Sci. Eng. B, in press, 1995. Jal, P. K., Patel, S., and Mishra, B. K., Talanta, 62, 1005–1028, 2004. Deorkar, N. V. and Tavlarides, L. L., Ind. Eng. Chem. Res., 36, 39, 1997. Tarasevich, B. J. and Rieke, P. C., Mater. Res. Soc. Symp. Proc., 174, 51, 1991. Shin, H., Jo, S., and Mikos, A. G., Biomaterials, 24, 4353–4364, 2003. Nakajima, Y., Isobe, T., and Senna, M., Mater. Sci. Eng. B, 10,139, 1996. Nakajima, S., Koga, T., Isobe, T., and Senna, M., Mater. Sci. Eng. B, in press, 1995.
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4.8
Standard Powders and Particles Hideto Yoshida Hiroshima University, Higashi-Hiroshima, Japan
Hiroaki Masuda Kyoto University, Katsura, Kyoto, Japan There are several standard particles currently in scientific use. For example, uniformly sized spherical polystyrene latex particles are used as a calibration standard in electron microscope studies. Natural pollens and spores are also monosized and nearly spherical, but they are sometimes swelled by moisture and have irregular surfaces. For example, the diameter of lycopodium particles measured by means of a microscope is about 30 µm, whereas it is about 24 µm when a liquid sedimentation method is used, because of surface irregularities. A specially made precipitated calcium carbonate is also monosized and is cubic in shape. Standard powders for industrial use generally have fairly narrow size distributions. In the United States, AC fine and coarse dusts are defined by the Society of Automotive Engineers and are utilized for performance tests of automobile air cleaners. They have the same size distributions as Japanese Standard Powders (JIS Z 8901) No. 2 and No. 3 or No. 7 and No. 8, respectively. Japan has also defined many types of standard powders and particles for industrial tests according to JIS Z 8901. They are sold by the Association of Powder Process Industry and Engineering (APPIE), Japan. Figure 8.1 shows some information on their size distributions, and Table 8.1 lists
FIGURE 8.1 8901).
Size distributions of standard test powders (JIS Z
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the materials they are prepared from and their applications. No. 8 Kanto loam powder is used mainly for the performance tests of automobile engine air cleaners in Japan, because it is easily dispersed in air by a mechanical or an air-jet-type powder disperser, shown in Figure 8.2 and Figure 8.3, respectively. Nearly monosized glass beads and white fused alumina are also defined according to JIS Z 8901, and their sizes are shown in Table 8.2 and Table 8.3.
TABLE 8.1 Materials and Application Examples of Standard Particles (JIS Z8901) Type No. 1
Material Silica sand (coarse)
Test Application Examples Abrasion or life test of machine; performance test of chemical plant
No. 2
Silica sand (medium)
Abrasion or life test of machine
No. 3
Silica sand (fine)
Abrasion or life test of machine
No. 4
Talc (fine)
Dust collector performance test
No. 5
Flyash (fine)
Dust collector performance test
No. 6
Portland cement
Airtightness test of car lamps
No. 7
Kanto loam (medium)
Dust collector performance test; abrasion or life test of machine
No. 8
Kanto loam (fine)
Dust collector performance test; abrasion or life test of machine
No. 9
Talc (ultrafine)
Performance test of high-efficiency dust collector air filter
No. 10
Flyash (ultrafine)
Performance test of high-efficiency dust collector air filter
No. 11
Kanto loam (ultrafine)
Performance test of high-efficiency dust collector air filter
No. 12
Carbon black (ultrafine)
Performance test of high-efficiency dust collector
No. 15
Mixed dust
Prefilter performance test
No. 16
Calcium (fine)
Carbonate air Classifier
No. 17
Calcium (ultrafine)
Carbonate air Classifier
FIGURE 8.2
Mixer-type disperser.
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FIGURE 8.3 disperser.
Nozzle-jet-type
TABLE 8.2 Size Distribution of Glass Beads Standard Powders in Japan (JIS Z 8901) Codea
No.
Oversize in mass basis 90% Size (µm) ⬎26
50% Median size (µm) 30 ⫾ 1.0
10% Size (µm) ⬍34
1
GBL30
2
GBL40
⬎37
41 ⫾ 1.0
⬍45
3
GBL60
⬎55
59 ⫾ 1.0
⬍63
4
GBL100
⬎95
0 ⫾ 1.0
⬍105
5
GBM20
⬎18
22 ⫾ 1.0
⬍26
6
GBM30
⬎26
30 ⫾ 1.0
⬍34
7
GBM40
⬎37
41 ⫾ 1.0
⬍45
GBL, sodalime silicate glass (rp ⫽ 2.1−2.5 g/cm3); GBM, barium titanate glass (rp ⫽ 4.0−4.2 g/cm3).
a
Table 8.4 presents several reference powders for certified size distributions, defined in Europe, and Table 8.5 presents several reference powders for the same purpose in the United States. Recently, Japan also has defined two types of standard powders, MBP1-10 and MBP10-100. The particle size ranges are from 1 to 10 µm and 10 to 100 µm, respectively. They consist of barium titanate glass beads, and their sizes are controlled to nearly log-normal distributions within the specified size range. Table 8.6 shows some of their physical properties and Figure 8.4 shows photographs of the test particles. Figure 8.5 and Figure 8.6 show particle size distributions that were carefully measured by a scanning electron microscope with a particle count of more than about 10,000 particles. The APPIE committee, Japan, recognizes these particles as a standard reference particle in Japan.1,2 © 2006 by Taylor & Francis Group, LLC
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TABLE 8.3 Size Distributions of White Fused Alumina Standard Powder in Japan (JIS Z 8901) Oversize in Mass Basis No.
1
92% Size (µm) ⬎0.8
50% Median size (µm) 2 ⫾ 0.4
3% Size (µm) ⬍5
2
⬎2.0
4 ⫾ 0.5
⬍11
3
⬎4.5
8 ⫾ 0.6
⬍20
4
⬎9.0
14 ⫾ 1
⬍31
5
⬎20
30 ⫾ 2
⬍58
6
⬎40
57 ⫾ 3
⬍103
Note: p ⫽ 3.9 4.0 g/cm3
TABLE 8.4
IRMM—SMT in Europe 0BCR Before 1995
CRM No.
Size Range Description Stokes diam.
Unit Size
(µm) 0.35–3.50
Price (ECU)
066
Quartz powder
10 g
125
067
Quartz powder
Stokes diam.
068
Quartz sand
Vol. diam.
069
Quartz powder
070
Quartz powder
130
Quartz powder
131
Quartz powder
Vol. diam.
480–1800
200 g
125
132
Quartz gravel
Vol. diam.
1400–5000
700 g
125
2.4–32
10 g
125
160–630
100 g
125
Stokes diam.
14–90
10 g
125
Stokes diam.
1.2–20
10g
125
Vol. diam.
50–220
50 g
125
165
Latex
Sphere.
2.223 ⫾ 0.013
1 vial
100
166
Latex
Sphere.
4.821 ⫾ 0.019
1 vial
100
167
Latex
Sphere.
9.475 ⫾ 0.018
1 vial
100
169
Alpha
alumina
0.104 ⫾ 0.012
60
100
170
Alpha
alumina
1.05 ⫾ 0.05
50
100
171
Alumina
2.95 ⫾ 0.13
50
100
172
Quartz
2.56 ⫾ 0.10
10
100
8.23 ⫾ 0.21
46
100
173
Rutile
titania
Note: Joint Research Centre Institute for Reference Materials and Measurements (IRMM), Retieseweb, B-2440 Geel, Belgium. Attention: Management of Reference Materials (MRM) Unit. Tel: 32 14 57 17 19, Fax: 32 14 59 04 06. Standards, Measurements, and Testing (SMT) Program.
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TABLE 8.5 SRM No.
497
NIST-SRM (301) in the United States (1996) Description
114p
Portland cement
659
Silicon nitride (particle
Particle Size (µm)
Unit issued Price (US $) 114.00
Certif. Date
45 µm oversize 8.24%
Set (20)
May 94
0.2-10
Set (5)
207.00
Mar. 92
size) 1003b
Glass (particle size)
10-60
25 g
159.00
Sept. 93
1004a
Glass (particle size)
40-170
70 g
154.00
Dec. 93
1017b
Glass (particle size)
100-310
70 g
252.00
Aug. 95
1018b
Glass (particle size)
225-780
74 g
In prep.
-
1019a
Glass (particle size)
760-2160
200 g
179.00
Oct. 84
1690
Polystylene (particle size)
0.895
5 ml
391.00
Dec. 82
1691
Polystylene (particle size)
0.269
5 ml
39.00
May 84
1692
Polystylene (particle size)
2.982
5 ml
384.00
May 91
1960
Polystylene (particle size)
9.89
5 ml
805.00
Apr. 85
1961
Polystylene (particle size)
29.64
5 ml
806.00
Jan. 87
1963
Polystylene (particle size)
0.1007
5 ml
504.00
Nov. 93
1965
Polystylene (on slide) (particle size)
9.94
1 slide
147.00
Jan. 87
1978
Zirconium oxide (particle size)
0.33-2.19
5g
203.00
Oct. 93
8570
LGCGM calcined kaolin (surf. area)
10.3-10.9 m2/g
10 g (25 g)
90.00
Sept. 94
8571
LGCGM alumina (surf.area)
153.2-158.7 m2/g
10 g (25 g)
90.00
Sept. 94
8572
LGCGM silica (surf.area)
277.6-291.2 m2/g
25 g
90.00
Sept. 94
Note: Purchase orders (in English) for all NIST SRMs/RMs should be directed to: National Institute of Standards and Technology (NIST), Standard Reference Materials Program (SRM), Room 204, Building 202, Gaithersburg, MD 20899-000l, USA; Tel.: (301) 975-6766, Fax: (301) 948-3730, e-mail:
[email protected]
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FIGURE 8.4
Photographs of test particles.
TABLE 8.6
Physical Properties of Test Particles MBP1-10 n⫽14806
MBP10-100 n⫽10515
Dp50 (µm) (mass base)
4.76
36.50
⫽lng (-)
0.312
0.395
g (-)
1.37
1.48
p (g/cm3)
4.19
4.10
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FIGURE 8.5 Particle size distribution for MBP1-10 particles. Measured by Electrical Sensing Zone (ESZ) and Scanning Electron Microscope (SEM) methods with the certified ranges as horizontal bars.
FIGURE 8.6 Particle size distribution for MBP10-100 particles. Measured by Electrical Sensing Zone (ESZ) and Scanning Electron Microscope (SEM) methods with the certified ranges as horizontal bars.
REFERENCES 1. Yoshida, H., Masuda, H., et al., Adv. Powder Technol., 12, 79–94, 2001. 2. Yoshida, H., Masuda, H., et al., Adv. Powder Technol. 14, 17–31, 2003. © 2006 by Taylor & Francis Group, LLC
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Part V Powder Handling and Operations
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5.1
Crushing and Grinding Tatsuo Tanaka Hokkaido University West, Sapporo, Japan
Yoshiteru Kanda Yamagata University, Yonezawa, Yamagata, Japan
5.1.1
INTRODUCTION
Comminution is the oldest mechanical unit operation for size reduction of solid materials and an important step in many processes where raw materials are converted into intermediate or final products. The purposes of comminution are to reduce the size, to increase the surface area, and to free the useful materials from their matrices, and it is involved recently in modification of the surface of solids, preparation of the composite materials, and recycling of useful components from industrial wastes. Comminution has a long history, but it is still difficult to control particle size and its distribution. Hence, fundamental analysis and optimum operation have been investigated. A demand for fine or ultrafine particles is increasing in many kinds of industries. The energy efficiency of comminution is very low, and the energy required for comminution increases with a decrease in feed or produced particle size. Research and development to find energy-saving comminution processes have been performed.
5.1.2
COMMINUTION ENERGY
In design, operation, and control of comminution processes, it is necessary to correctly evaluate the comminution energy of solids. In general, the comminution energy (i.e., the size reduction energy) is expressed by a function of a particle size.1
Laws of Comminution Energy Rittinger’s Law Rittinger assumes that the energy consumed is proportional to the produced fresh surface. Because the specific surface area is inversely proportional to the particle size, the specific comminution energy E/M is given by Equation 1.1:
(
)
(
E CR Sp Sf CR′ xp1 xf1 M
)
(1.1)
where Sp and Sf are the specific surface areas of product and feed, respectively, x p and x f are the corresponding particle sizes, and CR and C ′R are constants which depend on the characteristics of materials.
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Kick’s Law Kick’s law assumes that the energy required for comminution is related only to the ratio of the size of the feed particle to the product particle: ⎛x ⎞ ⎛ Sp ⎞ E Ck ln ⎜ f ⎟ Ck′ ln ⎜ ⎟ M ⎝ Sf ⎠ ⎝ xp ⎠
(1.2)
where CK and C′K are constants. Equation 1.2 can be derived by assuming that the strength is independent of the particle size, the energy for size reduction is proportional to the volume of particle, and the ratio of size reduction is constant at each stage of size reduction. Bond’s Law2 Bond suggests that any comminution process can be considered to be an intermediate stage in the breakdown of a particle of infinite size to an infinite number of particles of zero size. Bond’s theory states that the total work useful in breakage is inversely proportional to the square root of the size of the product particles, directly proportional to the length of the crack tips, and directly proportional to the square root of the formed surface: 10 ⎞ ⎛ 10 1/ 2 1/ 2 W Wi ⎜ ⎟ CB′ Sp Sf ⎝ P F⎠
(
)
(1.3)
where W(kWh/t)is the work input and F and P are the particle size in microns at which 80% of the corresponding feed and product passes through the sieve. Wi(kWh/t) is generally called Bond’s work index. The work index is an important factor in designing a comminution processes and has been widely used. Holmes’s Law3 Holmes proposes a modification to Bond’s law, substituting an exponent r, in place of 0.5 in Equation 1.3 as follows: ⎛ 10 10 ⎞ W Wi ⎜ r r ⎟ ⎝P F ⎠
(1.4)
Values of r which Holmes determined for materials are tabulated in Table 1.1.4
5.1.3
CRUSHING OF SINGLE PARTICLES
In principle, the mechanism of size reduction of solids is based on the fracture of a single particle and its accumulation during comminuting operations.
Fracture Properties of Solids In a system composed of an elastic sphere gripped by a pair of rigid parallel platens, the load-deformation curve can be predicted by the theories of Hertz as summarized by Timoshenko and Goodier.5 The © 2006 by Taylor & Francis Group, LLC
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TABLE 1.1 Values of r Determined by Holmes Holmes Exponent Materials
(r)
Amygdaloid
0.25
Malartic
0.40
Springs
0.53
Sandstone
0.66
Morenci
0.73
East Malartic
0.42
Chino Nevada Consolidated
0.65
Real del Monte
0.57
La Luz
0.34
Kelowna Exploratory
0.39
Utah Copper
0.50
elastic strain energy, E (J), input to a sphere up to the instant of fracture is given by the integral of the load acting through the deformation: ⎛ 1 v 2 ⎞ E 0.832 ⎜ ⎝ Y ⎟⎠
2/3
x −1 / 3 p 5 / 3
(1.5)
where Y (Pa) is Yougth’s modulus, v (-), Poisson’s ratio, x (m) the diameter of the sphere (particle size), and P (N) is the fracture load. In this system, the compression strength of the sphere, S, is given by Hiramatsu et al., 6 and the specific fracture energy E/M (J/kg) is given by ⎛ 1 v 2 ⎞ E C1 r −1p 2 / 3 ⎜ M ⎝ Y ⎟⎠
2/3
(1.6)
S5/3
where p (kg/m3) is density. The relationship between the specific fracture energy and the strength for quartz and marble are shown in Figure 1.1.7 On the other hand, when two spherical particles, 1 and 2, collide with each other, the maximum stress, Smax, generated inside the particles is expressed by a function of particle size, x, relative velocity, v (m/s), and mechanical properties8: ⎛ mm ⎞ Smax = C2 ⎜ 1 2 ⎟ ⎝ m1 m2 ⎠
1/ 5
⎛2 2⎞ 2 / 5 ⎜ ⎟ ⎝ x1 x2 ⎠
3/5
⎛ 1 v12 1 v22 ⎞ ⎜⎝ Y Y ⎟⎠ 1 2
4 / 5
(1.7)
where m1 (kg) and m2 are the mass of the particles, and C2 is a constant. The subscripts 1 and 2 denote two particles. © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.1 Relationship between strength S and specific fracture energy E/M.
Variation of Fracture Energy with Particle Size Strength is a structure-sensitive property and changes with specimen volume. From a statistical consideration of the distribution of the presence of minute flaws, 9 Weibull10 and Epstein11 showed that the mean strength of the specimen, S, is proportional to the (–1/m) power of the specimen volume, V (m3):
(
)
S S0V01 / m V 1 / m
(1.8)
where S0 (Pa) is the strength of unit volume V0 (m3), and m is Weibull’s coefficient of uniformity. Experimental data lines determined by the least squares method for bolosilicate glass and quartz are shown in Figure 1.212 From Equation 1.6 and Equation 1.8, the relationship between specific fracture energy or fracture energy of a single particle, E, and particle size x is obtained as fallows:
2 E 5 / 3 m 1 ( 2 m5) / 3 m ⎛ 1 v ⎞ C3 ( 6 ) r p ⎜ M ⎝ Y ⎟⎠
E C4 (6 )
5 / 3m
⎛ 1 v 2 ⎞ p (5 m5) / 3 m ⎜ ⎝ Y ⎟⎠
2/3
2/3
(S V )
1/ m 5 / 3 o o
(S V )
1/ m 5 / 3 o o
x5 / m
(1.9)
x (3 m5) / m
(1.10)
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FIGURE 1.2 specimen.
Variation of strength S with volume V of
The calculated result for quartz is shown in Figure 1.3. 12 It is important to note that the specific fracture energy increases rapidly for smaller particle size (less than approximately 500 μm), namely, the requirement of large amounts of energy in fine or ultrafine grinding can be presumed. The strength and the specific fracture energy increase also with an increase in loading rate. 13
Crushing Resistance and Grindability The importance of crushing resistance or grindability of solid materials and energy efficiency of comminuting equipment have been recognized in determining comminution processes in a variety of industries. Grindability is obtained from a strictly defined experiment. The two typical methods are the following. Hardgrove Grindability Index (JIS M 8861,1993) The machine to measure the grindability consists of a top-rotating ring with eight balls 1 in. in diameter. A load of 64 0.5 lb is applied on the top-rotating ring. Fifty grams of material sieved between 1.19 and 0.59 mm is ground for the period of 60 revolutions. The Hardgrove grindability index, H.G.I., is defined as H.G.I. 13 6.93w
(1.11)
where w (g) is the mass of ground product finer than 75 μm. Bond’s Work Index (JIS M 4002,1976) Bond’s work index WI, defined in Equation 1.3, 14 is given by. Wi
1.1 44.5 P
0.23 1
0.82 b.p
G
(10
P ′ 10 / F
)
(1.12)
where P1 is the sieve opening in micron for test grindability, Gbp (g/rev) is the ball mill grindability, P′ is the product size in microns (80% of product finer than size P1 passes), and F is the feed size in microns (80% of feed passes). A standard ball mill is 12 in. (305 mm) in internal diameter and 12 in. in internal length, charged with 285 balls, as tabulated in Table 1.2. The lowest limit of the total mass of balls is 19.5 kg. The amount of feed material is 700 cm3 bulk volume, composed of © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.3 Relationship between size x and specific fracture energy E/M, or fracture energy E.
TABLE 1.2 Composition of Steel Balls for Measurement by Bond’s Work Index Diameter (mm)
No.of Balls
36.5
43
30.2
67
25.4
10
19.1
71
15.9
94
Sum
285
particles finer than 3360 μm. The mill is rotated a number of times so as to yield a circulating load of 250% at 70 rev/min, where the circulating load is defined as the component ratio of the oversize to the undersize. The process is continued until the net mass of undersize produced per revolution becomes constant Gb,p in Equation 1.12. Table 1.3 shows the work index measured by wet process.15 In fine grinding, when P in Equation 1.3 is smaller than 70 μm, the work index, Wi, is multiplied by a factor f to account for the increased work input. The factor f is found from the following empirical equation16: f
P 10.3 ( P 70 m ) 1.145P
(1.13)
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TABLE 1.3 Average Work Indexes
Material All materials tested
Number tested 1211
Average Specific Work gravity index 14.42
Andesite
6
2.84
18.25
Barite
7
4.50
4.73
Basalt
3
2.91
17.10
Bauxite
4
2.20
8.78
Cement clinker
14
3.15
13.56
Cement raw material
19
2.67
10.51
Coke Copper ore
7
1.31
15.18
204
3.02
12.73
Diorite
4
2.82
20.90
Dolomite
5
2.74
11.27
Emery
4
3.48
56.70
Feldspar
8
2.59
10.80
Ferro-chrome
9
6.66
7.64
Ferro-manganese Ferro-silicon
5
6.32
8.30
13
4.41
10.01
Flint
5
2.65
26.16
Fluorspar
5
3.01
8.91
Gabbro
4
2.83
18.45
Glass
4
2.58
12.31
Gneiss
3
2.71
20.13
Gold ore
197
2.81
14.93
Granite
36
2.66
15.05
Graphite Gravel Gypsumrock
6
1.75
43.56
15
2.66
16.06
4
2.69
6.73
56
3.55
12.93
Iron ore Hematite Hematite-specular
3
3.28
13.84
Oolitic
6
3.52
11.33
Magnetite
58
3.88
9.97
Taconite
55
3.54
14.60
Lead ore
8
3.45
11.73
Lead-zinc ore
12
3.54
10.57
Limestone
72
2.65
12.54
Manganese ore
12
3.53
12.20
Magnesite
9
3.06
11.13
Molybdenum ore
6
2.70
12.80 (Continued)
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TABLE 1.3 (Continued) Average Work Indexes Average Specific Work gravity index 3.28 13.65
Number tested 8
Material Nickel ore Oilshale
9
1.84
15.84
17
2.74
9.92
Potash ore
8
2.40
8.05
Pyrite ore
6
4.06
8.93
Pyrrhotite ore
3
4.04
9.57
Phosphate rock
Quartzite
8
2.68
9.58
13
2.65
13.57
Rutile ore
4
2.80
12.68
Shale
9
2.63
15.87
Silica sand
5
2.67
14.10
Quartz
Silicon carbide
3
2.75
25.87
Slag
12
2.83
9.39
Slate
2
2.57
14.30
Sodium silicate
3
2.10
13.50
Spodumene ore
3
2.79
10.37
Syenite
3
2.73
13.13
Tin ore
8
3.95
10.90
Titanium ore
14
4.01
12.33
Trap rock
17
2.87
19.32
Zinc ore
12
3.64
11.56
Bond16 proposed a relationship between the work index, Wi, and the Hardgrove grindability index (H.G.I.): Wi
435
(H.G.I.)0.91
(1.14)
Grindability in Fine Grinding When the particle sizes of the products are submicron or micronized particles, it will be difficult to estimate the comminution energy by Equation 1.3, Equation 1.12, and Equation 1.13. Bond2 had proposed Equation 1.15 for measurement of Wi before Equation 1.12. ⎛ P ⎞ Wi 1.116 ⎜ 1 ⎟ ⎝ 100 ⎠
0.5
⋅ Gb.p0.82
(1.15)
Equation 1.15 is simpler than Equation 1.12. There was not a great difference17 between Wi calculated by Equation 1.12 and Wi calculated by Equation 1.15. Figure 1.4 shows the relationship © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.4 Relationship between grinding time t and mass fraction finer than size xc, Qxc.
between the mass fraction Qx (−) finer than the particle size (μm) and grinding time (min) in a ball c mill grinding of silica glass.18 In the early stage of grinding, a zero-order increasing rate is applicable, as shown in following equations. Qxc kxc t
(1.16)
Wxc Qxc ⋅ Ws kxc ⋅ Ws ⋅ t
(1.17)
where Wx·c is the mass of product finer than a size x c, and Ws is the mass of the feed. From Equation 1.15 through Equation 1.17, the following equations can be obtained:
(
Wi P10.5 ⋅ Gbp 0.82 xc 0.5 ⋅ k xc ⋅ Ws
(
Wic xc 0.5 ⋅ k xc ⋅ Ws
)
0.82
)
0.82
(1.18) (1.19)
where Wi is proportional to Wi, which was proposed by Bond. Wi could be estimated by the examic c nation of the zero-order increasing rate constant of the mass fraction less than a sieving size using an arbitrary ball mill. Figure 1.5 shows the relationship between sieving size, xc, and Wi for silica glass.18 It was prec sumed that the work index could be approximately constant to a sieving size of 20 μm and increased in the range of a size less than 20 μm. It was also found that large amounts of energy are necessary to produce fine or ultrafine particles. © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.5 Relationship between sieving size xc and corresponding work index WI, . c
5.1.4
KINETICS OF COMMINUTION
A pulverizing machine is to be designed and operated by pursuing the comminution process of particle assemblage with the elapsed time. The size distribution after fracture of a single particle is derived by applying a stochastic theory by Gilvarry,19 Gaudin and Meloy,20 or Broadbent and Callcot.21 These theories are based on the idea that some microcracks preexisting in the solid body are activated with stress and absorb elastic strain energy, so that the cracks develop rapidly and collide with each other to yield fragments of distributed sizes. The undersize cumulative fraction B(g,x) produced from a single particle of size g is called a breakage function and is written as follows: Gilvarry: ⎡ ⎛ x ⎞ ⎛ x ⎞2 ⎛ x ⎞3⎤ B (g, x ) 1 exp ⎢ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎥ ⎝ c3 ⎠ ⎥⎦ ⎢⎣ ⎝ c1 ⎠ ⎝ c2 ⎠
(1.20)
Gaudin and Meloy: ⎡ ⎛ x⎞⎤ B (g, x ) 1 ⎢1 ⎜ ⎟ ⎥ ⎝ g⎠⎦ ⎣
j
(1.21)
Broadbent and Callcot: B (g , x )
1 exp (x ⁄g ) 1 exp (1)
(1.22)
where x is the particle size, c1, c2, and c3 are constants, and j is the number of fragments or 10. They may be approximated as ⎛ x⎞ B (g , x ) ⎜ ⎟ ⎝ g⎠
m
(1.23)
where m is a constant.22 The derivative B/ x is called a distribution function appearing later. © 2006 by Taylor & Francis Group, LLC
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With respect to the mass–size balance in batch grinding, the mass increment of a component of size x during a differential time interval dt is expressed by removal of a portion of the component due to selective grinding and by production of the same component due to selectively grinding a portion of all the coarser particles followed by the distribution to the noted size range, as illustrated in Figure 1.6: 2 D ( x, t ) D ( x , t ) S ( x, t ) t x x xm D (g , t ) B ( g , x ) ∫ S (g , t ) dy x g x
(1.24)
where S(g,t) is the probability density for particles of size g to be selected for grinding and it is called a selection function or rate function, t is the grinding time, and Xm is the maximum size present. Thus, the rate of comminution of particle assemblage is determined by the size distribution after crushing a single particle and by the probability that each particle is selected for crushing within a certain time. Assuming Equation 1.23 for B(g,x) and the empirical relationship S(x,t) = Kxn
(1.25)
for S(x,t) (Bowdish, 1960), the oversize cumulative fraction, R(x,t) = 1 – D(x,t), is obtained by integration of Equation 1.24 using a rate constant, K: m = n: R(x,t) = R(x,0) expKxnt
(1.26)
m n: R(x,t)R(x,0) exp[(mKxnt)v]
(1.27)
where R(x,0) is the initial size distribution and can be regarded as unity if the grinding time t is long enough. m and v are determined only by the value of m/n, as indicated in Figure 1.7. The size distribution is found to vary with grinding time in the Rosin–Rammler type, which was first confirmed experimentally by Chujo23 using a ball mill. The selection function or the rate constant has been experimentally determined by Shoji and Austin24 for ball milling, as shown in Figure 1.6, in which S(x) is depicted in relation to fc, the fractional holdup of particles in a mill, to J, the ball filling degree,
FIGURE 1.6 Explanation of mass balance in comminution process. [From Austin, L.G., and Klimpel, R.R., Ind. Eng. Chem., 56, no. 11, 18–29, 1964.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.7 function.
Approximation of Equation 1.25 by exponential
FIGURE 1.8 Variation of relative absolute rate of breakage with powder and ball filling for dry grinding.
and U, the particle filling in the interstice of balls. The grinding rate is reported by Tanaka25 for ultrafine grinding based on the current experimental works using very small beads in ball, vibration, planetary, and stirred milling, as ⎛ d ⎞ ⎛ x⎞ S ( x ) r ⎜ ⎟ exp ⎜ m ⎟ ⎝ d⎠ ⎝ d⎠
(1.28)
where d is the beads’ diameter, dm the optimum beads’ diameter as a function of the beads density ρ, colliding speed, size, and strength of material crushed. © 2006 by Taylor & Francis Group, LLC
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515
GRINDING OPERATIONS
To reduce the strength or the toughness of the material crushed, wet process grinding, the addition of grinding aids, and cryogenic grinding are proposed. The additives are likely to reduce the surface energy, leading to facilitating grinding of the particles. In addition to this, newly formed surfaces are active enough for stronger chemical bonding, so that surface modification can be expected by the grinding operation. Furthermore, to prepare the size distribution required for product quality, a multipass of ground material through a mill and a classifier are necessary, as noted below.
Internal Classification System The classification mechanism is assembled in a grinding mill. For example, an air-swept mill adopts internal classification by flowing fluids such as air through the mill, where particles finer than a critical size xc are removed from the machine immediately after grinding. Then the following mass balance holds26: ⎛ dRp ( x ) ⎞ xm ⎛ dR (g ) ⎞ ⎛ ∂ B (g , x ) ⎞ F⎜ = H∫ ⎜ S (g ) ⎜ dg ⎟ ⎟ xc ⎝ dg ⎠ ⎝ ∂x ⎟⎠ ⎝ dx ⎠
(1.29)
where F is the feed to a continuous grinding machine, H is the holdup of particles in the machine, and the subscript p denotes product. Using Equation 1.23 and Equation 1.25 along with the conditions
∫
xm xc
⎛ dR ( ) ⎞ ⎜⎝ d ⎟⎠ d 1;
∫
xc
0
⎛ dRp ( x ) ⎞ ⎜ dx ⎟ dx = 1 ⎝ ⎠
the ideal size distribution of the ground material is given in the case where Equation 1.30 is applicable and sorting by a clean-cut classification is assumed: F xcn KH
(1.30)
⎛ x⎞ Rp ( x ) 1 ⎜ ⎟ ⎝x ⎠
n
(1.31)
c
Closed-Circuit Grinding System In contrast to the preceding system, a closed-circuit system is characterized by an external classifier involved in the system. The ground material, D, is continuously sent to a classifier, where only the fine component is removed as the finished product, P. The coarse material is recirculated to the mill, as shown in Figure 1.9. Increasing the circulating load T, it is possible to avoid overcrushing so as to increase the grinding capacity. Controlling the size distribution of the finished product is also possible to some extent. In a clean-cut classifier27, the cumulative oversize fraction or the ground particles, RD, is obtained from Equation 1.26 for the average residence time t0 ( = H/F) in the mill as ⎛ Kt x n ⎞ RD ( x ) exp ⎜ 0 ⎟ ⎝ 1 CL ⎠
(1.32)
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FIGURE 1.9 system.
Typical connection of closed-circuit grinding
where the average residence time is given by the overall residence time t0 ( = H/F) in this circuit and the circulating ratio CL as iav
t0 H H E F (1 CL ) 1 CL
(1.33)
RD ( xc ) T T P F 1 RD ( xc )
(1.34)
CL
hence, the cutoff size xc is given by combining Equation 1.32 and Equation 1.34 as ⎡ ⎛ 1 ⎞⎤ Kt0 xcn ln ⎢1 ⎜ ⎟⎥ ⎣ ⎝ CL ⎠ ⎦
1 CL
(1.35)
The characteristic classification size xc* of x c corresponding to infinite CL is also obtained from Equation 1.34 and Equation 1.35, as ⎛ F ⎞ xc∗n ⎜ ⎝ KH ⎟⎠
(1.36)
As the clean-cut size distribution of product, Rp(x), is written by Rp ( x )
RD ( x ) RD ( xc ) 1 RD ( xc )
(1.37)
the following are obtained using the equations above:
(
)
n ⎛ x ⁄xc∗ ⎞ ⎛ x⎞ ⎟ CL Rp ⎜ ∗ ⎟ (1 CL ) exp ⎜ ⎝ xc ⎠ ⎜⎝ 1 + CL ⎟⎠
(1.38)
"
⎛ xc ⎞ ⎡ ⎛ 1 ⎞⎤ ⎜⎝ x ∗ ⎟⎠ (1 CL ) ln ⎢1 ⎜⎝ CL ⎟⎠ ⎥ ⎣ ⎦ c
(1.39)
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The interrelationship between the above two equations is graphed in Figure 1.10. The chart is used such that a desired size ratio (x30/x70) of product at Rp 30% and 70%, for example, is fitted to the horizontal distance between the two corresponding curves. Hence, xc* is calculated from the value on the abscissa using either x30 or x70; then, xc and CL are read from the ordinate on both sides. The flow system of the closed circuit can be designed by use of CL and F for the transportation equipments and xc for the classifier. The mill design should be made on basis of H in Equation 1.36 for a known grinding rate constant K as well as F and xc*. In the case of a non-clean-cut classifier,28 classification performance can be expressed by two parameters, A and S, in a mathematical model of the partial classification efficiency h (x) as follows: 1
A ⎧ ⎡ 4S ⎡ ⎛ x ⎞ ⎤ ⎫⎪ ⎪ ⎛ ⎞ h ( x ) ⎨1 exp ⎢⎜ ⎟ ⎢1 ⎜ ⎟ ⎥ ⎬ ⎢⎝ A ⎠ ⎢⎣ ⎝ xc ⎠ ⎥⎦ ⎪ ⎪⎩ ⎣ ⎭
(1.40)
where A 1.5 and S 1.0 for air separators, and A 1.0 and S 0.5 for hydrocyclones. Choosing a reference curve in Figure 1.11 to be fitted to the desired cumulative undersize distribution of the finished product, the size corresponding to 1.0 on the abscissa is equal to xc* (for v 1 in Equation 1.27), and the cumulative undersize Dp (1) on the ordinate is connected with CL for specific values of S and A in Figure 1.12. Then, the value of CL indicates xc/xc in Figure 1.13; thus, the cutoff size, xc, is obtained from xc*. Figure 1.14 illustrates some partial classification efficiency curves modeled by use of the two parameters S and A.28 Alteration of the size distribution of the finished product is possible by some combinations of mills and classifiers, as well as feed positions. The analysis is given by Tanaka,29 taking advantage of Figure 1.10. Grinding capacity increases in general with increasing CL due to the reduction of overcrushed fine particles. In this sense, it is worthwhile noting that when CL in Equation 1.38 tends to infinity, Rp (x) becomes the same as Equation 1.31, the internal classification mechanism.
5.1.6
CRUSHING AND GRINDING EQUIPMENT
A major objective of comminution is to liberate minerals for concentration processes. Another objective is to produce particles of a required sizes. Comminution processes generally consist
FIGURE 1.10 Collinear chart for ideal classification. [From Furuya, M., Nakajima, Y., and Tanaka, T., Ind. Eng. Chem. Process. Des. Dev., 10, 449–456, 1971.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.11 Reference curve for size distribution of product. (30) [From Furuya, M., Nakajima, Y., and Tanaka, T., Ind. Eng. Chem. Process. Des. Dev., 12, 18–23, 1973.]
FIGURE 1.12 Relationship among classification parameters, circulating ratio, and undersize fraction of product at x = xc*. [From Furuya, M., Nakajima, Y., and Tanaka, T., Ind. Eng. Chem. Process. Des. Dev., 12, 18–23, 1973.]
of several stages in series. Various types of crushing and grinding equipments have been used industrially as a mechanical way of producing particulate solids. The working phenomena in these types if equipment are complex, and different principles are adopted in the loading, such as compression, shear, cutting, impact, and friction; in the mechanism of force transmission or the mode of motion of grinding media, such as rotation, reciprocation, vibration, agitation, rolling, and acceleration due to fluids; and in the operational method, such as dry or wet system, batch © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.13 Relationship among classification parameters, circulating ratio, and (xc/xc*)n.
FIGURE 1.14 Fractional recovery curves calculated from the mathematical model (Equation 1.35 and Equation 1.36).
or continuous operation, association of internal classification or drying and so on. But, in practice, it is most common to classify comminution processes into four stages by the particle size produced. Although the sizes are not clear cut, they are called the primary (first), intermediate (second), fine (third), and ultrafine (fourth) stages, according to the size of ground product. On the basis of the above classification, typical equipment types and their structure and characteristics are mentioned briefly below.
Crushers Crushers are widely used as the primary stage to produce particles finer than about 10 cm in size. They are classified as jaw, gyratory, and cone crushers based on compression, cutter mill and shredder based on shear, and hammer crusher based on impact.
Jaw Crusher The jaw crusher shown in Figure 1.15 consists essentially of two crushing surfaces, inclined to each other. Material is crushed between a fixed and a movable plate by reciprocating pressure until © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.15 Jaw crusher.
the crushed products become small enough to fall through the narrowest gap between the crushing surfaces.
Gyratory Crusher The essential features of a gyratory crusher are a solid cone on a revolving shaft, placed within a hollow shell which may have vertical or conical sloping sides, as shown in Figure 1.16. Material is crushed when the crushing surfaces approach each other, and the crushed products fall through the discharging chute.
Hammer Crusher, Swing-Hammer Crusher, and Impactor These are used either as a one-step primary crusher or as a secondary crusher for products from a primary crusher. Pivoted hammers are mounted on a horizontal shaft, and crushing takes place by the impact between the hammers and breaker plates. A cylindrical grating or screen can be placed beneath the rotor. Materials are reduced to a size small enough pass through the bars of the grating or screen. Hammers are symmetrically designed. The size of product can be regulated by changing the spacing of the grate bars or the opening of the screen, and also by lengthening or shortening the hammer arms.
Intermediate Crushers Intermediate crushers produce particles finer than about 1 cm. The roller mill, crushing roll, disintegrator, screw mill, edge runner, stamp mill, pin mill, and so on belong to this category. Roll crushers are of many types. They consist of at least one cylinder rotating on its principal axis, which nips material with two surfaces to compress and break the material into pieces. Figure 1.17 shows a typical crushing roll. It consists of two cylinders mounted on horizontal shafts, which are driven in opposite directions. The distance between the cylinders (rollers) is usually made adjustable. The size of feed materials is determined by the diameter of the cylinders, the required size of products, and the angle of nip. Recently, high-pressure roller mills (100–200 MPa) are available to comminute finely brittle materials.30 Figure 1.18 shows a typical roller mill. Roller mills have been actively used for preparation of fine particles. Material is fed to the center of the horizontally rotating table, conveyed to its circumference by centrifugal force, ground by several units of rollers on the concave table, and moved toward the circumference. © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.16 Gyratory crusher.
FIGURE 1.17 Crushing roll.
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FIGURE 1.18 Roller mill.
Fine-Grinding Equipment Fine-grinding equipment produces particles finer than about 10 μm. There are many kinds of machines in this category. They are roughly classified into three types: ball-medium type, medium agitating type, and fluid-energy type. In a ball-medium type, the grinding energy is transferred to materials through media such as balls, rods, and pebbles by moving the mill body. Based on the mode of motion of the mill body, ball-medium mills are classified as tumbling ball mills, vibration mills, and planetary mills. 1. A tumbling mill or a ball mill is most widely used in both wet and dry systems, in batch and continuous operations, and on small and large scales. The optimum rotational speed is usually set at 65 to 80% of critical speed, Nc (rpm), when the balls are attached to the wall due to centrifugation: Nc
42.3 Dm
(1.41)
where Dm is the mill diameter in meters. It is desirable to reduce the ball size in correspondence with the smaller size of the feed materials, as in a compartment mill, shown in Figure 1.19, and a conical ball mill. © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.19 Compartment mill.
2. A vibration mill is driven by eccentric motors to apply a small but frequent impact or shear to the grinding media. Loose bodies or media contained in a shell caused it to vibrate. In contrast to tumbling mills, the media in vibration mills move only a few millimeters through a complex path, shearing as well as impacting the material between them. The apparent amount of media is 75 to 85% by volume, which is about two times as much as filling mills. One of the advantages of vibration mills compared with tumbling mills is the higher grinding rate in the range of fine particles. But it is unsuiTable for heat-sensitive materials. 3. A planetary mill consists of a revolving base disk and rotating mill pots, as shown in Figure 1.20. Materials are ground in a large centrifugal field by the force generated during revolution and rotation. The intensity of acceleration can be increased up to 150g on the scale of gravitational acceleration. It is predicted that the grinding mechanism consists of compressive, abrasive, and shear stresses of the balls. Planetary mills are also used in the study of mechanical alloying and composite particles.31 A fluid-energy mill is widely noted as a jet mill. In a jet mill, the materials are ground by the collision of a particle with a particle, a wall, or a plate of the grinding vessel (chamber). The collision energy is generated by a high-speed jet flow. Fluid-energy mills may be classified in terms of the mill action. In one type of mill, the energy is generated by high-velocity streams at a section or whole periphery of a grinding and classification vessel. There are the Micronizer and Jet-O-Mizer shown in Figure 1.21 and others of this type. In Majac and other mills, two streams convey particles at high velocity into a vessel where they impact on each other. A fluid-energy mill has no movable mechanical parts, and it has advantages such as dry and continuous operation without a temperature rise, and controlling the particle size by the feed rate of materials and the velocity of jet stream. But the energy efficiency is low, and the power cost is high.
Ultrafine Grinding Equipment Ultrafine grinding equipments produce particles finer than about 1 μm. There are the medium agitating mills (stirred media mills) in this category. A medium agitating mill is regarded as one of the most efficient devices for micronizing materials and has been actively used for preparation of ultrafine particles. In this technique, a large number of small grinding media are agitated by impellers, screws, or disks in a vessel. Breakage occurs mainly by collision of the media. It is classified into three types by agitating mode. Medium agitating mills could produce submicron particles. 1. An agitating-tank type utilizes vertical agitation of balls and pebbles in a tank. A typical stirred mill is shown in Figure 1.22. 2. In a flow-tank type, disks, pins, or paddles are attached to a shaft to agitate balls or beads and grind particles in a flowing suspension. © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.20 Planetary mill.
FIGURE 1.21 Jet mill.
3. A tower type is a large standing mill that has a central shaft with a screw to convey balls upward, and the centrifugal force causes them to collide with the wall. Recently, the demand for ultrafine (submicron) particles has been increasing in many kinds of industries. The research and development to find ultrafine grinding processes has been performed for years for wet grinding, grinding aids, and closed-circuit grinding. The number of particles to be ground varies inversely with the cube of the particle size of the feed material. Then, in ultrafine grinding equipment, it is necessary to increase the collision probability of the particle and grinding medium. Figure 1.22 shows a stirred mill. With the mill, media is evenly distributed at the outside of the rotor, creating even shear forces. In this mill, the minimum diameter of media is 0.1 mm. © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.22 Stirred mill.
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REFERENCES 1. Walker, W.H., Lewis, W.K., McAdams, W.H., and Gilliland, E.R., Principles of Chemical Engineering, McGraw-Hill, New York, pp.254, 1937. 2. Bond, E.C., Trans. AIME, 193, 484–494, 1952. 3. Holmes, J.A., Trans. Inst. Chem. Eng., 35, 125–140, 1957. 4. Lowrison, G.C., Crushing and Grinding, Butterworth, London, pp. 54, 1974. 5. Timoshenko, S. and Goodier, J.N., Theory of Elasticity, Mc Graw-Hill, New York, pp. 372, 1951. 6. Hiramatsu, Y., oka, T., and Kiyama, H., Mining Inst. Japan, 81, 1024–1030, 1965. 7. Kanda, Y., Samo, S., and Yashima, S., Powder Technol., 48, 263–267, 1986. 8. Rumpt, H., Chem. Ing. Technol, 31, 323–337, 1959. 9. Griffith A.A., Proceedings of 1st Inter Congr.\Appl.\Mech, pp. 55–63, 1924. 10. Weibull, W., Ing. Vetenshaps Akad Handle, 151, 1939. 11. Epstein, B., J. Appl. Phys, 19, 140–147, 1948. 12. Yashima, S., Kanda, Y., and Sano, S., Powder Technol., 51, 277–282, 1987. 13. Yashima, S., Saito, F., and Hashimoto, H., J. Chem. Eng. Japan, 20, 257–264, 1987. 14. Bond, F.C., Trans. AIME, 217, 139–153, 1960. 15. Bond, F.C., Brit. Chem. Eng., 6, 543–548, 1961. 16. Bond, F.C., Brit. Chem. Eng., 6, 378–385, 1961. 17. Ishihara, T., J. Min. Metal. Inst. Japan, 80, 924–928, 1964. 18. Kotake, N., Soji, H., Hasegawa, M., and Kanda, Y., J. Soc. Powder Technol., Japan, 31,626–630, 1994. 19. Gilvarry, J.J., J. Appl. Phys, 32, 391–399, 1961. 20. Gaudin, A.M., and Meloy, T.P., Trans. AIME, 223,43–51 1962. 21. Broadbent, S.R., Callcot, T.G., J. Inst. Fuel., 29, 524–528, 1956. 22. Nakajima, Y. and Tanaka, T., Ind. Eng. Chem. Process. Des. Dev. 12, 23–25, 1973. 23. Chujo, K., Kagaku Kougaku to Kagakukikai, 7, 1–83, 1949. 24. Shoji, K., Austin, L.G., Smaila, F., Brame, K., and Luckie, P.T., Powder Technol. 31, 121–126, 1982. 25. Tanaka, T., J. Soc. Powder Technol. Jpn. 31, 25–31, 1994 26. Ouchiyama N., Tanaka T., and Nakajima, Y., Ind. Eng. Chem. Process. Des. Dev., 15, 471–473, 1976. 27. Furuya, M., Nakajima, Y., and Tanaka, T., Ind. Eng. Chem. Process. Des. Dev., 10, 449–456, 1971. 28. Furuya, M., Nakajima, Y., and Tanaka, T., Ind. Eng. Chem. Process. Des. Dev., 12, 18–23, 1973. 29. Tanaka, T., J. Soc. Powder Technol. Jpn. 31, 333–337, 1994. 30. Schonert, K., and Lubjuhn, U., 7th European Symposium Comminution Preprints Part 2, 747–763, 1990. 31. Mizutani, U., Takeuchi, T., Fukunaka, T., Murasaki, S., Kaneko, K., and Mater, J., Sci. Lett., 12, 629–632, 1993.
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5.2
Classification Kuniaki Gotoh
Okayama University, Okayama, Japan
Hiroaki Masuda
Kyoto University, Katsura, Kyoto, Japan
Hideto Yoshida
Hiroshima University, Higashi-Hiroshima , Japan
Jusuke Hidaka
Doshisha University, Kyoto, Japan
5.2.1
BASIS OF CLASSIFICATION
The unit operation for the separation of particulate material depending on its characteristics, such as particle diameter, density, shape, and so forth, is called “classification.” However, this term is commonly used to express the size classification that is the separation of particles depending on their diameter. In this section, the basis of size classification is described. Size classification is meant to separate particles of a certain diameter that is called the “cut size” or the “classification point,” and to collect them as fine powder and coarse powder, although separation and collection of dust is aimed to collect all particles suspended in a fluid.
Classification Efficiency When F [kg] of a powder (⫽ raw material) is classified into A [kg] coarse powder and B [kg] fine powder, the total mass balance of the powder can be written as F⫽A⫹B
(2.1)
If the cumulative oversize fractions of the raw material—coarse powder and fine powder—at the cut size Dpc are Rf, Ra, and Rb, respectively, the mass balance of the powder, which is coarser than the cut size Dpc is as follows: FRf ⫽ ARa ⫹ BRb
(2.2)
By the above equations, recovery, which is one of the representative values of classification efficiency, can be expressed. Recovery of coarse powder ra
ra ⫽
ARa Ra ( Rf ⫺ Rb ) ⫽ FRf Rf ( Ra ⫺ Rb )
(2.3)
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Recovery of fine powder rb rb ⫽
B (1⫺ Rb )
( Ra ⫺ Rf )(1⫺ Rb ) F (1⫺ Rf ) ( Ra ⫺ Rb ) (1⫺ Rf ) ⫽
(2.4)
These recoveries are the mass ratio of the mass collected as coarse or fine powder to that fed to the classifier. In the powder collected as coarse powder, finer powder than cut size Dpc is may be included, while coarser powder than cut size Dpc also may be included in the powder collected as fine powder. These contaminations mean the deterioration of classification performance. Therefore, any unnecessary component in the collected powder should be addressed by concern for separation efficiency. Newton’s efficiency hn defined by the following equation is one aspect of separation efficiency in which the unnecessary component is taken into consideration hn ⫽ ra ⫹ rb ⫺1 ⫽
( Rf ⫺ Rb )( Ra ⫺ Rf ) Rf (1⫺ Rf ) ( Ra ⫺ Rb )
(2.5)
Newton’s efficiency is defined as the difference between the recovery of product (useful component) and the residuum of the unnecessary component. As in Equation 2.5, Newton’s efficiency is the same whether the product is the coarse powder or the fine powder. The recoveries and Newton’s efficiency described above represent the classification efficiency by means of one numerical value. These efficiencies are called “total classification efficiency” and “total separation efficiency.” The total classification efficiency is a function of the cut size, because the efficiency is defined by the cumulative oversize fractions at the cut size.
Partial Separation Efficiency In general, the collection probability Er of each particle into the coarse side of the classifier depends on the size of particle. When dF [kg] of the particles having a diameter in the range from Dp to Dp ⫹ dDp is included in the powder fed into a classifier and when dA [kg] of particles among them is classified as coarse, the recovery of the particles is expressed as follows:
( )
Er Dp ⫽ dA dF
(2.6)
The recovery is called “partial separation efficiency” or “partial classification efficiency.” It is often called simply “classification efficiency.” When the frequency of the raw material, coarse powder, and fine powder designated as fF(Dp), fA(Dp), and fB(Dp) respectively, the mass balance of the particles having a diameter in the range from Dp to Dp ⫹ dDp is expressed as follows:
( ) dA ⫽ Af ( D ) dD
dF ⫽ Fff Dp dDp
(2.7)
A
(2.8)
p
p
By substituting Equation 2.7 and Equation 2.8 into Equation 2.6, the partial separation efficiency can be rewritten as the following equation:
( )
( )
Er Dp ⫽ AfA Dp
( )
FfF Dp
(2.9)
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Typical partial separation efficiency shows a curved line. The curved line is called the partial classification curve, partial separation curve, or Tromp curve. The particle diameters corresponding to Er ⫽ 0.25, 0.5, and 0.75 are defined as 25% of the cut size Dp25, 50% of the cut size Dp50, and 75% of the cut size Dp75, respectively. The 50% cut size Dp50 is one of the most important representative values of the characteristics of classification.
Sharpness of Classification The cut size described above is one of the representative values of classification performance. On the other hand, if the separation efficiency can be expressed by a unit function that jumps up from 0 to 1 at the cut size, the classification is an ideal classification. Thus, whether the separation efficiency increases sharply is another characteristic of performance. The gradient of the partial separation efficiency is one of the expressions of the sharpness of the classification. However, because the separation efficiency usually shows a curved line, the gradient depends on the diameter. Therefore, the following indexes are used as representative values of the sharpness of the classification. Index of separation accuracy: k ⫽ Dp 75 Dp 25
(2.10)
Terra index” Ep ⫽
Dp 75 ⫺ Dp 75
(2.11)
2
Incomplete index: I⫽
Dp 75 ⫺ Dp 75 Dp 75 ⫹ Dp 75
(2.12)
As found by the above definitions, index κ for an ideal separation is infinity, while the Terra index Ep and incomplete index I, which are the nondimensional form of the Terra index, are zero for an ideal separation. The reciprocal numbers of these indexes are also used as representative values of the sharpness. Although these indexes are defined by means of the 25% cut size Dp25 and Dp75, describing the 90% cut size or the 100% cut size is preferable for a precise expression of the classification performance, because the gradient of a separation efficiency curve decreases rapidly at the diameter below the 25% cut size and over the 75% cut size.
Outlines of Classifiers There are two types of classifier, depending on the medium of the particle suspension. One is a “wet classifier” using a liquid as the medium of suspension, and the other is a “dry classifier” using a gas as the medium of suspension. In both classifiers, the classification is achieved by applying a certain force to each particle. The applied force causes different particle trajectories reflecting the particle size. By means of the difference of the trajectories, particles are classified into coarse or fine, depending on their size. The force applied to the particles is gravity, centrifugal force, and so on. As for centrifugal force, it is generated by a semifree vortex or forced vortex. In the case of dry classification, inertia is also utilized for making the difference in the trajectories. There are advantaged and disadvantages for wet and dry classifications: 1. Production rate per unit area of a wet classifier is less than that of a dry classifier. In general, the rate of a wet classifier is 1/50 of that in a dry classifier. © 2006 by Taylor & Francis Group, LLC
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2. Dispersion of the particles, which is a pretreatment for classification, into a liquid is easier than that into a gas. Because of that, the sharpness of the wet classification is better than that of the dry classification. 3. Velocity of particles in a liquid is less than that in a gas, when the same magnitude of separation force is applied. It leads to a narrow size range classification. 4. The collection and transportation of the classified particles is easier in a liquid than in a gas. It leads to a low probability of processing trouble. On the other hand, in case of wet classification, a drying process for classified powder is required after the classification. In other words, wet classification requires additional energy and cost for posttreatment. As listed above, wet and dry classifications both have advantages and disadvantages. The selection of wet or dry must be made by taking into account the operations before and after the classification.
5.2.2
DRY CLASSIFICATION
Dry classification is widely used in many industrial processes. Compared to wet classification, dry classification does not need drying and slurry treatment. As a result, dry classification is more widely used compared to wet classification. However, in order to classify particles with a cut size of less than about 3 μm, a suitable particle disperser is necessary before using the dry classification apparatus. Theoretical study about the collection efficiency of cyclones was conducted by Leith and Licht. 1 Numerical calculations of fluid flow and particle motions in a cyclone were conducted by Ayers and Boysan,2 Zhou and Soo,3 and Yamamoto et al.4,5 In order to control the cut size, rotationalblade-type classifiers are generally used. However, it is difficult to shift the cut size in the submicron range by use of the forced-vortex-type classifier. Recent experimental results indicate that it is possible to shift the cut size in the submicron range by use of a flow controlling method applied to the free-vortex-type classifier. Recent industrial interest in dry classification is mainly directed to accurate cut size control in the fine size region. The following are typical air classifiers used in actual industrial processes.
Gravitational Classifiers Particles are classified by the difference in either their settling velocity or falling position. Horizontal flow, vertical flow, and inclined flow classifiers are all used (see also 5.2.3 for the theory).
Inertial Classifiers Particles are classified by the difference in their trajectories. Rectilinear ( elbow-type), curvilinear (impactor), and inclined (louver) classifiers are commercially available. Figure 2.1 shows various inertial classifiers and Figure 2.2 shows a louver separator with plane or special blades. 6
Centrifugal Classifiers Centrifugal force is created either by airflow in cyclone-type classifiers, or by mechanical revolution in air separators. A variety of centrifugal classifiers is available: Micron separator (Figure 2.3), Turbo classifier (Figure 2.4), and Turboplex (Figure 2.5). The characteristics of centrifugal separators are listed in Table 2.1. Figure 2.6 shows an example of the partial separation efficiency of a special type of cyclone with and without blow-down for the submicron range.7 © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.1 Principle of classification by traverse flow.
FIGURE 2.2 Structure of louver classifier. © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.3
Micron separator. (Courtesy of Hosokawa Micron.)
Cut Size Control by Use of Free-Vortex-Type Classifier In order to control the cut size with a gas-cyclone, the use of a movable slide plate, shown in Figure 2.7, is effective. Figure 2.8 shows the classification performance using a movable slide plate and the blow-down method proposed by Yoshida.8, 9 Both methods are effective in decreasing the 50% cut size. The effect of an apex cone set at the upper part of the dust box is to decrease the cut size and to increase the collection efficiency. Figure 2.9 shows the simulated fluid vectors with and without the apex cone. Using the apex cone, it is possible to decrease the magnitude of the fluid velocity component in the dust box. As a result, it is possible to reduce re-entrainment of particles from the dust box to the vortex finder. The cut size of cyclones with an apex cone is smaller than that without an apex cone, and Figure 2.10 shows these results. By use of the apex cone, it is possible to shift the cut size to the fine size region. 9
5.2.3 WET CLASSIFICATION The principle of wet classification is the same as dry classification. Recently, wet classification has been used for producing ceramic particles and in recycling processes. The following facts should be considered regarding wet classification characteristics. 1. Particle dispersion control is easy compared to dry classification. 2. In order to separate particles from liquid, a drying or dewatering process is necessary. © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.4 Turbo classifier. (Courtesy of Nisshin Engineering.)
FIGURE 2.5 Turboplex classifier. (Courtesy of Alpine.) © 2006 by Taylor & Francis Group, LLC
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TABLE 2.1
Characteristics of Centrifugal Air Classifiers
Type of classifier Micron separator Turbo classifier Turboplex
Power (kw) 5–150
Cut size (m) 150–12,000
Capacity (kg/h) 200–2,300
Rotor speed (rpm) 0.75–37
0.5–150
150–24,000
700–12,000
1.5–70
2–180
40–35,000
120–22,000
1–45
FIGURE 2.6 Particle separation efficiency of 70 cyclone classifiers.
3. The viscosity of the slurry decreases with increasing temperature, and the particle sedimentation velocity increases as the slurry temperature increases.
Classification Theory Figure 2.11 shows the two types of classifiers, a vertical-flow type and a horizontal-flow type. For the vertical-flow type, the collection efficiency is calculated by Equation 2.13 and Equation 2.14: Vg ⬍ u0 or VgS⬍ Q : E ⫽ 0
(2.13)
Vg ⬎ u0 or VgS⬎ Q : E ⫽ 1
(2.14)
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FIGURE 2.7
Cyclone with moving slide plate.
FIGURE 2.8 Effect of slide plate and blow down on partial separation efficiency.
An ideal classification is realized for this type. However, the performance deteriorates when fluid turbulence is generated in the classifier. The collection efficiency for the horizontal type without fluid mixing is as follows: Vg S Q
ⱕ1
Vg S Q
⬎1
E⫽
Vg S Q
E ⫽1
(2.15)
(2.16)
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FIGURE 2.9 Calculated fluid vectors with and without apex cone.
For the horizontal type with fluid mixing in the vertical direction, the collection efficiency is calculated by Equation 2.17: E ⫽ 1⫺ exp(⫺
Vg S Q
)
(2.17)
Figure 2.12 shows the relation between the collection efficiency E and the parameter (VgS/Q) for three cases. Ideal classification is realized for a vertical-flow type. However, classification sharpness decreases for the horizontal-flow type with fluid mixing. © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.10 Partial separation efficiency with and without apex cone.
Sedimentation Velocity The particle sedimentation velocity in a fluid is important in estimating the separation efficiency. Assuming a steady state, the Stokes sedimentation velocity is as follows: Vs ⫽
( rp − rf ) Dp 2 g
Rep ⱕ 2
18 m
(2.18)
When the particle Reynolds number is greater than about 2, the sedimentation velocity calculated by the following equations are more accurate: Rep ⫽
Allen region Vs ⫽ (
DpVs r m
4 ( rp ⫺ rf )2 g 2
Newton region Vs ⫽ (
225m rf
)1 / 3 Dp
3g( rp ⫺ rf ) Dp rf
2 ⱕ Rep ⱕ 500
)1 / 2
500 ⱕ Rep
(2.19)
(2.20)
Particle sedimentation velocities calculated by Equation 2.18 through Equation 2.20 are shown in Figure 2.13 for various particle densities. When the particle concentration increases, the particle sedimentation velocity calculated by Equation 2.18 through Equation 2.20 should be corrected using the following equation: Vsc ⫽
Vs F ()
F () ⱖ 1
(2.21)
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Particle feeder Fine particles u
o
v
g
Coarse particles Flow Vertical flow type q
c Q(c+dc)
Qc
H h
u( vg z)
zx
vgcBdx x
L
dx L
FIGURE 2.11 Vertical- and horizontal-flow-type classifier.
1.0 Eq. (2.2)
Collection efficiency E (–)
0.8
Eq. (2.3)
0.6 0.4 Eq. (2.1) 0.2
Vertocal flow Eq (2.1) Horizontal flow E=Vgs/Q Eq. (2.2) Horizontal flow with mixing E=1– exp (–VgS/Q) Eq. (2.3)
0 0
0.5
1.0
1.5
2.0
2.5
3.0
VgS/Q (–) FIGURE 2.12 Relation between collection efficiency and parameter (VgS/Q). © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.13 Sedimentation velocity of spherical particle in water (25°C, 1 atm).
The correction function F() is experimentally determined. F() ⫽ −4.65
(2.22)
where is the volume fraction of the fluid, and the value of F() increases with decreasing . The sedimentation velocity calculated by Equation 2.21 is referred to as a hindered settling velocity.
Various Types of Classifier The typical apparatus used in wet classification includes the following: 1. 2. 3. 4.
Horizontal-flow type Vertical-flow type Hydrocyclone Centrifugal type with wall or blade rotation
Particle separators of the horizontal-flow type are shown in Figure 2.14. The coarse particles are collected on the bottom part and fine particles are collected to the fluid outlet part. Figure 2.15 shows a spiral classifier. This type is widely used because of its simple design and good performance. The coarse particles in the feed are discharged by the spiral flow, while the fine particles are collected in the overflow side.
Hydrocyclone Hydrocyclones are widely used in many industrial processes. Experimental studies of the separation efficiency and pressure drop of a hydrocyclone have been reported.10,11,12 Several flow controlling methods at the outlet pipe of a hydrocyclone have been proposed.13,14 A prediction of the flow within © 2006 by Taylor & Francis Group, LLC
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Feed
Feed i
flow Fine particles tifi FIGURE 2.14 Particle separators of horizontal-flow type.
Fine
icles
e part Coards ) (san spiral tank
fine particles (slime) FIGURE 2.15 Spiral classifier.
a hydrocyclone was made using several turbulence models. 15 Performance of the axial-flow-type hydrocyclone was studied by Sineath and DellaValle.16 Performance between conical and cylindrical hydrocyclones was examined by Chine and Ferrara.17 The characteristics of hydrocyclones are as follows: 1. The size of a hydrocyclone is small compared to other types of wet classifiers. 2. The cut size decreases with an increase in inlet velocity or with a decrease in the cyclone diameter. 3. The pressure drop is about 0.5 to 5 Kg/cm2, depending on the operating conditions. Figure 2.16 shows the general flow pattern in a hydrocyclone. The coarse and fine particles are collected in the underflow and overflow sides, respectively. The ratio of inlet volume flow rate to underflow volume flow rate is referred to as the underflow ratio. The normal operating conditions for a hydrocyclone are as follows: 1 2 3 4 5
Cyclone inlet velocity Cyclone diameter Pressure drop 50% cut size Underflow ratio
u0 ⫽ 2 ~ 10 m/s Dc ⫽ 1 ~ 30 cm ΔP ⫽ 0.5 ~ 5 Kg/cm2 Dpc ⫽ 5 ~ 50 μm Rd ⫽ 5 ~ 20%
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Di=Dc/7
Dc
De=Dc/5 Dc/2
Dc/2
Dc/4 Inlet feed
Dc
Vortex finder Fine particles Air core Coarse particles Dc/10~Dc/6
Dense slarry
FIGURE 2.16 Standard hydrocyclone and flow pattern.
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Figure 2.16 also shows the standard cyclone, and each of the dimensions is indicated using the cyclone diameter Dc.12 The bottom diameter of the conical section is relatively small compared to a gas-cyclone. As the underflow rate increases, the air core near the axis is observed. In order to improve the sharpness of classification, the air core region should be eliminated. The typical separation efficiency curve obtained using the standard hydrocyclone is shown in Figure 2.17. The partial separation efficiency approaches the underflow ratio Rd as the particle diameter decreases. In order to normalize experimental data for different underflow ratio conditions, the corrected partial separation efficiency Δηc calculated by Equation 2.23 is indicated in the figure. ⌬hc ⫽
⌬h ⫺ Rd 1⫺ Rd
(2.23)
where Dp50* denotes a 50% cut size under conditions where Rd is zero. The corrected collection efficiency indicated in Figure 2.17 can approximately be represented by the following equation: ⌬hc ⫽ 1⫺ exp (⫺ (
Dp Dp 50*
⫺ 0.115)3 ) 0.02 ⱕ ⌬hc ⱕ 0.98
(2.24)
Control of Cut Size In order to improve the cut size control easily, the modified hydrocyclone shown in Figure 2.18 has been developed.18,19 The inlet of the hydrocyclone is attached to a movable guide plate. The apex cone at the inlet of the underflow side is also attached. Both underflow and upward flow methods as in gas-cyclone are used to increase classification sharpness in fine size. In order to decrease the 50% cut size, a movable guide plate shown in Figure 2.19 was used. Under constant feed rate conditions, the inlet velocity u increases with a decrease in inlet width b. The inlet width ratio G, defined by the following equation, is used as a variable parameter.
G⫽
b b*
(2.25)
100
100
80
Corrected separation efficiency
Partial separation efficiency
The value of G is equal to one for the standard case. Because of increased centrifugal force and the small radial sedimentation distance, small particles are easily collected for the case of a small inlet
80
60
60
40
40
=20%
R R=10% R=0%
20 0 0
10
20
D*50 20
30
40
50
0
.2
.4
.6
.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
Dp/Dp*50 (–)
Reduced particle diameter FIGURE 2.17 Particle separation efficiency and corrected partial separation efficiency. © 2006 by Taylor & Francis Group, LLC
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width. Typical experimental results using the guide plate are shown in Figure 2.20. The 50% cut size decreases from 10 μm to 7 μm as the inlet width ratio decreases under a constant inlet flow rate. The partial separation efficiency approaches 0.1 for a particle diameter less than 5 μm. In order to increase classification sharpness in the small particle diameter region, upward flow and underflow methods were used simultaneously. Some experimental results for partial separation efficiency are shown in Figure 2.21. In this case, the upward flow, underflow, and movable guide plate were used.
FIGURE 2.18 Experimental apparatus of a hydrocyclone.
FIGURE 2.19 Cross section of a cyclone with a guide plate. © 2006 by Taylor & Francis Group, LLC
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The upward flow ratio was 30%, the underflow ratio was 10%, and the inlet width ratio was changed from 1 to 0.1. The 50% cut size changes from 35 μm to 10 μm as the inlet width ratio decreases from 1 to 0.1. The partial separation efficiency in this case approaches zero as the particle diameter decreases.
FIGURE 2.20 Classification performance using a guide plate and the underflow method.
FIGURE 2.21 Classification performance using a guide plate, underflow, and upward flow methods. © 2006 by Taylor & Francis Group, LLC
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Classification Theory of the Centrifugal-Type Separator Figure 2.22 shows the centrifugal-type separator with a wall rotation.20 The feed slurry is supplied to the bottom part, and classified coarse and fine particles are collected at the upper part of the rotating cylinder. The limiting particle trajectory satisfies the following conditions: z ⫽ 0 : r ⫽ r1, z ⫽ L : r ⫽ r2
(2.26)
Integrating the equation of particle motion, the partial separation efficiency can be represented as follows: 2
⌬h ⫽
2 V Vg v 1 ( 1⫺ exp (⫺ )) r1 2 gQ 1⫺ ( ) r2
(2.27)
The notation V is the volume of the apparatus, and Q is the maximum liquid flow rate defined by the following equation: V ⫽ p (r2 2 ⫺ r12 )L
(2.28)
V L v2 Vg Q⫽ r g ln 2 r1
(2.29)
where v is the rotational speed and Vg the terminal velocity of the particle. In order to separate fine particles, it is necessary to increase the length L and the rotational speed v. The 50% cut size in this type is small compared to a standard hydrocyclone. However special care and maintenance are required at the rotor shaft and collection area of coarse particles in the cylindrical wall.
FIGURE 2.22 Centrifugal-type separator with rotation. © 2006 by Taylor & Francis Group, LLC
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SCREENING
Screening is the separation of solids particles having various sizes into different size portions or classes using a screen surface. The screening surface acts as a multiple go/no-go gauge, and the final products consist of more uniform size than those of the original material. The purposes of screening are (1) to remove fines from a raw material before using grinding equipment, (2) to scalp oversize material or impurities, and (3) to produce or process a commercialgrade product to meet specifications.
Screen Surface Selection of a proper screen surface is very important; opening size, wire or thread diameter, and open area should all be carefully considered. The screening surface may consist of a perforated or punched plate, grizzly bars, wedge-wire sections, woven-wire cloth, and nylon, polyester, or other bolting cloth. Silk bolting cloth, which was widely used in the past, has largely been replaced by nylon cloth. An electroformed sieve, which has uniform apertures, has been also used for the accurate fractionation of fine powder ranging from 3 to 100 μm. 21,22
Equipment Screening machines may be classified into five main categories: grizzlies, revolving screens, vibrating screens, sifters, and air-assisted screening machines. Grizzlies are used primarily for scalping material of 50 mm and coarser size, while revolving screens or trommel screens are generally used for separations of material more than 1 mm in size. The screening machine consists of a cylindrical frame equipped with a wire cloth or perforated plate. The material to be screened is delivered at the upper end, and the oversize is discharged from the lower end. The screens revolve at a relatively low speed of 15 to 20 rpm. Such screens have largely been replaced by vibrating screens, but they are still used for special purposes, for example, in municipal solid waste separation processes. Vibrating screens, whose screening surface vibrates perpendicular to the screening surface with a frequency greater than 600 rpm, are available for fine powders as well as coarse powders. There are a large variety of vibrating screens on the market, but basically they can be divided into two main types: mechanically vibrated screens and electrically vibrated screens. The most important factors for the selection of vibrating screens are amplitude and frequency. The centrifugal effect K is defined by the formula23,24 K⫽
r v2 g
(2.30)
where r ⫽ (amplitude in m)/2, ω ⫽ 2πn, n ⫽ (frequency in rpm)/60, and g is the acceleration of gravity. If the centrifugal effect is too small, the near-size particles will wedge into the screen openings. This reduces the open area available for the passage of other fines and creates probably the single greatest limitation on the capacity of the screen. The proper vibrating action not only keeps the feed materials moving on the screen, but it serves primarily to reduce “blinding” to a minimum. Figure 2.23 shows the correlation between the amplitude and frequency for commercially used values of K. Sifters are characterized by low-speed (300 to 400 rpm), large-amplitude (smaller than about 50 mm) oscillation in a plane essentially parallel to the screening surface. Sifters are usually used for material of 30 μm to 400 μm or more in size and have many applications in chemical processes.25 Most gyratory sifters have an auxiliary vibration caused by balls bouncing against the lower surface of the screen cloth. Figure 2.24 shows the schematics of a typical screening surface and driving mechanisms of vibrating screens and sifters. The air-assisted screening machine, shown in Figure 2.25, has been used for fine powders. The stream of air accelerates the passage of fine particles through the screen and removes the blinding particles in the apertures of screen. © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.23 Centrifugal effect K of screening machines.
Relation between 50% Separation Size and Screening Length According to Gaudin’s probability theory, the passage probability P of particles of size Dp through an opening of size a in the cloth is given by
P⫽
( a ⫺ DP )2 ⫽ ⎛ 1⫺ DP ⎞ 2 a2
⎜⎝
(2.31)
a ⎟⎠
Introducing the concept of the number of passage trials i, the oversize fraction of particles after i trails, ηi, which is the partial separation efficiency, is expressed by ⎡ ⎛ a ⫺ DP ⎞ 2 ⎤ hi ⫽ (1⫺ P ) ⫽ ⎢1⫺ ⎜ ⎟ ⎥ ⎢⎣ ⎝ a ⎠ ⎥⎦
i
i
(2.32)
If i is sufficiently large, Equation 2.32 is written approximately as ⎛ a ⫺ DP ⎞ ln hi ⫽⫺i ⎜ ⎝ a ⎟⎠
2
(2.33)
Hence the particle size at 50% separation efficiency, Dp50, is derived, substituting ηi ⫽ 0.5 into Equation 2.33: DP50 ⫽ a ⫺
0.832 a i
(2.34)
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FIGURE 2.24 (a) Motion of screen surface of vibrating screens and sifters; (b) driving mechanisms of vibrating screens and sifters.
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FIGURE 2.25 Air-assisted screening machine.
If i ⫽ ξl where ξl is the trial of particle passage per unit length of screen, the following equation holds approximately: DP 50 ⫽ a ⫺
0.832 a jl
(2.35)
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
Leith, D. and Licht, W., AIChE Symp. Ser., 126, 196–206, 1972. Ayers, W. H. and Boysan, F., Filtrat. Separat, 22, 39–43, 1985. Zhou, L. X. and Soo, S. L., Powder Technol., 3, 45–53, 1990. Yamamoto, M., Kitamura, O., and Arakawa, C., Trans. Jpn. Soc. Mech. Eng., 59, 1959–1964, 1993. Yamamoto, M. and Kitamura, O., Trans. Jpn. Soc. Mech. Eng., 60, 4002–4009, 1994. Yoshida, H. et al., J. Soc. Powder Technol. Jpn., 36, 454–461, 1999. Iinoya, K. et al., KONA Powder Particle, 11, 223–227, 1993. Yoshida, H. et al., KONA Powder Particle, 12, 178–185, 1994. Yoshida, H. et al., Kagaku Kogaku Ronbunshu, 27, 574–580, 2001. Bradley, D., Ind. Chemist, Sept., 473–485, 1958. Yoshioka, N. and Hotta, Y., Chem. Eng., 19, 632–642, 1955. Yoshioka, N., Ekitai-Cyclone, Nikkan Kogyo, 1962, pp. 38–41. Chu, L. Y., Chen, W. M., and Lee, X. Z., Chem. Eng. Sci., 57, 207–212, 2002.
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14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
Yamamoto, K. and Jiao, X., Trans. Jpn. Soc. Mech. Eng. Ser. B, 63, 133–138, 1997. Petty, C. A. and Parks, S. M., Filtrat. Separat., 28, 28–34, 2001. Sineath, H. H. and DellaValle, J. M., Chem. Eng. Prog., 55, 59–69, 1959. Chine, B. and Ferrara, G., KONA Powder Particle, 15, 170–179, 1997. Yoshida, H. et al., J. Soc. Powder Technol. Jpn., 34, 690–696, 1997. Yoshida, H. et al., J. Soc. Powder Technol. Jpn., 38, 626–632, 2001. Makino, K. et al., Kagaku Kougaku Gairon, Kaitei 11 Han, Sangyo Tosyo, Japan, 1989, pp. 270–272. Hidaka, J. and Miwa, S., Powder Technol., 24, 159–166, 1979. Pierre, B., Jean, C., and Albert, L., Particle Particle Syst., 10, 222–225, 1993. Giunta, J. and Colijn, H., Powder Handling Process., 5, 45–52, 1993. Modrzewski, R. and Wodzinski, P., Powder Handling Process., 10, 167–171, 1998. Sato, Y., Uehara, K., Yasui, A., and Sakata, Y., Trans. Jpn. Soc. Mech. Eng. Part C, 59, 2688–2693, 1993. 26. Ishikawa, S., Shimosaka, A., Shirakawa, Y., and Hidaka, J., J. Chem. Eng. Jpn., 36, 623–629, 2003.
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5.3
Storage (Silo) Minoru Sugita Ohsaki Research Institute, Chiyoda-ku, Tokyo, Japan
5.3.1
GENERAL CHARACTERISTICS OF SILOS
General characteristics of silos used for storing powder and granular materials are as follows: 1. Granular materials can be collected, distributed, and stored in bulk efficiently. 2. Transportation costs, which influence the costs of raw materials and products, can be reduced. 3. Compared with storage on a flat surface such as a floor, a silo’s storing capacity is several times greater in the same space. 4. Equipment cost per unit of storage capacity is small. 5. Automatic loading, unloading, and control of storage volume are possible. 6. Operations such as pressurization, heat insulation, moisture proofing, and fumigation are easily accomplished. 7. Quality change, decomposition, breakage, and damage of stored materials by insects and rats can be prevented. 8. A silo can be incorporated easily as a part of an industrial production system and has laborsaving advantages.
5.3.2
CLASSIFICATION OF SILOS
Shallow Bins and Deep Bins When studying static powder pressure acting on silo walls, silos are classified into shallow bins and deep bins. The classification is based on the following formulas 1,2 : Deep bins: h > 1.5d(h > 1.5a) Shallow bins: h ≤ 1.5d(h ≤ 1.5a) where h is the height of the silo (meters), d is the inside diameter of a circular silo (meters), and a is the length of a short side of a rectangular silo (meters).
Single Bins and Group Bins For a single bin, a circular cross section is frequently used because of some advantages in design and construction. In recent year, coal silos 40–50 m in diameter and as high as about 40 m have been constructed, many of which are independent shallow bins. In addition, large single bins such as cement silos and crinker silos have been constructed. Many steel silos are of the single-bin type. An example of a group bin is the silo for storing grains. Several to several tens of connected bins in a variety of shapes (e.g., circular, rectangular, and hexagonal in cross section) are used to store various types of powder and granular materials in bulk.
Closed and Open Types Most bulk silos are of the closed type, equipped with a roof onto which loading equipment is installed. Some silos are airtight to permit the fumigation of imported grains, for example, using poisonous gas to exterminate vermin. Vacuum silos are also used to prevent powder clogging. 551 © 2006 by Taylor & Francis Group, LLC
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Open silos are used simply for storage and supply of granular material, the quality of which does not change upon exposure to rain or dust. Examples of these are ore silos (bunkers) and crushed limestone silos.
5.3.3
PLANNING SILOS
Calculation of Silo Capacity In designing a silo, its capacity should be determined from the total storing weight of the materials, the types of the materials to be stored, and the conditions of use. Silo capacity has two components: total capacity (geometric capacity) and the capacity of loaded stored materials (effective capacity). Geometric capacity, also called water capacity, is used as a standard value for calculating the fumigation gas to be employed in treating imported grains. Effective capacity is the base for calculating the storing weight and location for taking in materials whose angle of repose should be taken into consideration. If geometric capacity is represented by VW (m3) and effective capacity by VE (m3), the loss volume VL (m) becomes VL = VW – VE (m3) (3.1) The loss volume of a cylindrical silo, illustrated in Figure 3.1, can be determined from the formula 3:
{
p/2 4 VL ⫽ R 3 3F ∫ cos2 x (1⫺ F ) ⫹ F cos2 x dx 0 3 3/2 p/2 ⎫ ⫹ ∫ ⎡⎣(1⫺ F ) ⫹ F cos2 x ⎤⎦ dx ⎬ tan fr ⫽ cfL R 3 tan fr 0 ⎭
(3.2)
where a⎞ ⎛ F ⫽ ⎜ 1⫺ ⎟ ⎝ R⎠
2
(3.3)
For a cylindrical container, Figure 3.1 symbols are used to calculate the effective capacity of silos: 1 VE = πR2H ⫹ –3 πR3 tan a (3.4)
FIGURE 3.1
Symbols used to calculate the effective capacity of silos.
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cf L is the loss coefficient for the cylindrical silo. The effective capacity can be determined from Equation 3.1 through Equation 3.3. Similarly, the calculation of the capacity of a rectangular silo can be made as follows. The loss capacity of a rectangular silo is
VL ⫽
⎛ 1⫹ 1⫹ b2 ⎞ l3 ⎡ 2 ⎢2 b 1 + b2 ⫹ b2 ln ⎜ ⎟ ⫹ ln b ⫹ 1⫹ b tan fr 6⎢ b ⎝ ⎠ ⎣
(
⎤
)⎥⎥⎦ (3.5)
⫽S fL3 tan fr 1 Vw ⫽ LBH 2 ⫹ LBH 3
TABLE 3.1 a/R or b/l
(3.6)
Loss Coefficients f
c L
f
s L
a/R or b/l
f
c L
f
s L
0.00
3.5555
0.50
2.4808
0.2966
0.02
3.5025
0.0100
0.52
2.4509
0.3115
0.04
3.4500
0.0200
0.54
2.4222
0.3267
0.06
3.3982
0.0301
0.56
2.3947
0.3423
0.08
3.3471
0.0403
0.58
2.3683
0.3581
0.10
3.2967
0.0506
0.60
2.3431
0.3742
0.12
3.2471
0.0610
0.62
2.3191
0.3906
0.14
3.1983
0.0715
0.64
2.2962
0.4074
0.16
3.1503
0.0822
0.66
2.2746
0.4245
0.18
3.1031
0.0931
0.68
2.2541
0.4418
0.20
3.0569
0.1041
0.70
2.2349
0.4595
0.22
3.0115
0.1154
0.72
2.2169
0.4776
0.24
2.9670
0.1268
0.74
2.2001
0.4959
0.26
2.9235
0.1384
0.76
2.1845
0.5146
0.28
2.8809
0.1502
0.78
2.1701
0.5336
0.30
2.8394
0.1623
0.80
2.1570
0.5530
0.32
2.7988
0.1746
0.82
2.1451
0.5728
0.34
2.7592
0.1871
0.84
2.1345
0.5927
0.36
2.7206
0.1998
0.86
2.1251
0.6130
0.38
2.6831
0.2129
0.88
2.1169
0.6337
0.40
2.6467
0.2261
0.90
2.1100
0.6547
0.42
2.6113
0.2397
0.92
2.1044
0.6761
0.44
2.5770
0.2535
0.94
2.1000
0.6979
0.46
2.5438
0.2676
0.96
2.0968
0.7199
0.48
2.5117
0.2819
0.98
2.0950
0.7424
—
—
—
1.00
2.0943
0.7651
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where b = b/l and S f L is the loss coefficient for a rectangular silo. Values determined as a function of b/l are listed in Table 3.1. Effective capacity can be determined from Equation 3.1, Equation 3.5, and Equation 3.6.
Unloading Devices Generally, a gravity discharge system is employed for unloading, but various special devices are used to discharge powder materials of high cohesiveness or materials that are liable to segregate, depending on granular size and composition. Although slight differences exist, many silos used to store cement, aluminum, and flour are equipped with an air slide system to discharge powder by blowing air from the bottom. Vibration is also used to discharge cohesive powders. Even when the gravity discharge system is employed, the unloading device itself (e.g., rotary valve, screw feeder, or chain feeder) will differ depending on the materials being stored, and the hopper shape differs depending on the discharge system employed. Care should be taken in construction and design because the discharge system generates a large difference in the powder pressure acting on the silo walls when discharge is taking place. Some reports indicate the differences in dynamic pressure while discharging materials depend on the location of the discharge openings.
5.3.4
DESIGN LOAD
Design Recommendation for Storage Tanks and Their Supports, issued by the Architectural Institute of Japan in June 1983, and revised in March 1990,1 is available for the structural design of silos. The design methods described include, for the first time, Japan’s new earthquake-proofing requirements. According to the Design Recommendation,1 the following loads should be considered: (1) dead load, (2) live load, (3) snow load, (4) wind load, (5) earthquake load, and (6) loads appropriate to the containers, such as impact and absorption due to the movement of bulk materials inside the containers.
5.3.5
LOAD DUE TO BULK MATERIALS
In designing bulk silos, a proper understanding of the behavior and pressure of bulk materials inside silos is clearly necessary. However, many unsolved and unanticipated problems remain. Bulk pressure changes in complexity, depending on various properties of the materials stored and the operating conditions of silos, thus relegating silo design to specialists. According to the ISO 11697, Bases for Design of Structures: Loads Due to Bulk Materials,4 the bulk pressures inside deep bins are discussed for two specified loading conditions. The filling pressures of bulk materials depend mainly on the material properties and the silo geometry. Discharge pressures are also influenced by the flow patterns that arise during the process of emptying. Therefore, an assessment of material flow behavior shall be made for each silo design. In the assessment of bulk-material flow, it is necessary to distinguish among three main flow patterns: 1. Mass flow (Figure 3.2a): A flow profile in which all the stored particles are mobilized during discharge. 2. Funnel flow (Figure 3.2.b–3.2f): A flow profile in which a channel of flowing material develops within a confined zone above the outlet and the material adjacent to the wall near the outlet remains stationary. The flow channel can intersect the wall of the parallel section or extend to the top surface. In the latter case, the pattern is called “internal flow” (Figure 3.2c–3.2e). 3. Expanded flow (Figure 3.2f): A flow profile in which mass flow develops within a steepbottom hopper, combining with a stationary in an upper, less steep hopper at the bottom of the parallel section. The mass flow zone then extends up the wall of the parallel section. © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.3 Limit between mass flow and funnel flow for circular hoppers.
Different pressure distributions are associated with each of the above flow patterns. The conditions necessary for mass flow depend on the inclination of the hopper wall and the wall friction coefficient. They can be estimated using Figure 3.3 for conical and axisymmetrical hoppers, and Figure 3.4 for configurations producing plane flow. The transition regions shown in Figure 3.3 and Figure 3.4 represent conditions in which the flow pattern can change abruptly between mass and funnel flow, thereby producing unsteady flow with pressure oscillations. If such a condition cannot be avoided, the silo shall be designed for both mass flow and funnel flow.
5.3.6
CALCULATION OF STATIC POWDER PRESSURE
In calculating bulk loads, the static powder pressure is the basis to be determined. The Janssen and Reimbert formulas are employed as silo design standards in various countries. The calculation of the static powder pressure in ISO 11967,4 is based on Janssen’s theory. It is derived from a force balance of the powder stored statically in deep bins, taking into consideration the frictional force generated between the powder and the silo walls. If a silo of cross-sectional area A (m 2) and circumference L (m) is filled uniformly with a powder of bulk density g (tons/m3) (Figure 3.5), the vertical pressure Pv (tons/m2) inside the silo on the horizontal plane at x (m) can be expressed by Pv
gA ⎡ ⎛ mKL ⎞ ⎤ x 1⫺exp ⎜ ⎢ ⎝ A ⎟⎠ ⎥⎦ mKL ⎣
(3.7)
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FIGURE 3.4 Limit between mass flow and funnel flow for wedge-shaped hoppers.
and replacing A/L = R (hydraulic radius) by Pv ⫽
gR ⎡ ⎛ mK ⎞ ⎤ 1⫺ exp ⎜⫺ x ⎝ R ⎟⎠ ⎥⎦ mK ⎢⎣
(3.8)
where R = D/4 for cylindrical silos of diameter D. Equation 3.8 is called the Janssen formula. According to the assumption of Janssen’s theory, the horizontal pressure Ph (tons/m2) is proportional to the vertical pressure Pv and can be expressed as follows, where the proportional constant is denoted by K: Ph ⫽K Pv Ph ⫽ KPV ⫽
gR ⎡ ⎛ mK 1⫺ exp ⎜⫺ ⎢ ⎝ R m ⎣
(3.9)
⎞⎤ x⎟ ⎥ ⎠⎦
(3.10)
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Assuming that x → ∝ in Equation 3.8 and Equation 3.10, the maximum static pressure can be obtained as follows:
Pvmax ⫽
gR , K
Phmax =
gR
(3.11)
In calculating the K value, which is called the Janssen coefficient, the Rankine formula, used in soil mechanics, gives the relation to the angle of internal friction i:
K⫽
1⫺ sin fi 1⫹ sin fi
(3.12)
To use the foregoing formulas in practice, it is necessary to know certain physical properties of the powder. The bulk density g (tons/m3) is generally measured in a laboratory. Some powders have increased bulk density due to accumulated consolidation. The angle of internal friction of powder i is measured by the shear cell method or triaxial compression test. The most difficult measurement is that of the friction angle (coefficient) of the powder against the wall. At present, it is difficult to obtain a correct understanding of the influence of the smoothness of silo walls on powder friction.
FIGURE 3.5 filled.
Powder is uniformly
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The static pressure of powder in shallow bins is almost negligible with regard to frictional forces between the silo walls and the powder. According the Design Recommendation,1 the following formulas are specified for shallow bins: Pv = gx
(3.13)
Ph ⫽ K gx
(3.14)
K, called the Rankine constant, can be determined from Equation 3.12 for both shallow and deep bins.
5.3.7
DESIGN PRESSURES
To calculate the design pressure, it is necessary to take into consideration the fact that dynamic pressure can be generated during discharge, and impact pressure can be generated during loading. Figure 3.6 shows distributions of powder pressure measured in actual silos. It is clear that powder pressures measured during discharge are larger than those obtained from the Janssen formula, especially the maximum pressure, which is four to five times higher than the value from Janssen’s theory. In the design standards of various countries, the correction factors for dynamic overpressure during discharge and impact pressure during loading are introduced based on practical data.
FIGURE 3.6
Examples of pressure measurement.
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The correction factor for dynamic pressure is usually taken as 2.0 and that for impact pressure is 1.0–2.0. By multiplying the static pressure in the preceding section by the correction factor, the minimum design powder load is determined. ISO 11697 4 shows the design wall pressure as follows. In silos where the flow zone intersects the wall (i.e., all flow patterns expect internal flow), the design discharge pressures shall be obtained by multiplying the filling loads by the overpressure coefficient C. The value of C shall be related to the silo aspect ratio h/d: for h/d ⬍ 1.0, C ⫽ 1.0 for 1.0 ⬍ h/d ⬍ 1.5, C ⫽ 1.0 ⫹ 0.7(h/d ⫺ 1.0) for h/d ⬎ 1.5, C ⫽ 1.35 These values apply only to materials listed in tables that are indicated in this international standard.
REFERENCES 1. Architectural Institute of Japan, Design Recommendation for Storage Tanks and Their Supports, Architectural Institute of Japan, 1990. 2. Soviet Code, Ch-302-63, 1965. 3. Vaillant, A., Chem. Eng., 69, 148, 1962. 4. ISO 11697, Bases for Design of Structures: Loads Due to Bulk Materials, International Standards Organization, 1995.
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5.4
Feeding Hiroaki Masuda and Shuji Matsusaka Kyoto University, Katsura, Kyoto, Japan
5.4.1
INTRODUCTION
Powder feeders play an important role in powder-handling processes such as feeding raw materials or discharging product materials from a storage vessel. The powder load or mass to be treated in the processes depends on the accuracy of the feeders. Product quality might depend on the powder load, and low-quality materials will be produced if the load is too high for the process. Feeders should be selected carefully, because some feeders might be unsuited to the process, resulting in a failure of feeding itself. Further, dynamic characteristics of the feeders could also affect the accuracy of feeding. If the feeder cannot respond to a fast change in the manipulating signal, a fluctuation of flow caused by a disturbance cannot be suppressed and could cause a dynamic error in the feeding process. The static and dynamic characteristics of feeders will depend on powder properties such as particle size, shape, internal friction coefficient, and powder flowabilities. They will also depend on the operating conditions, including temperature, pressure, and moisture content of powder in the process. Further, feeders are always associated with feed hoppers, and their design might affect the function of feeders. If the feed hopper is not well designed, the flow mode in the hopper will be an unfavorable one called funnel flow, which can cause particle bridging, rat holing, and flushing. Failure of feeding will occur when the selection or maintenance of feeders is inadequate or hopper design is improper. The power supply should be sufficient so that the feeder will operate even under conditions of a light overload caused by a disturbance. Additional parts, such as a shear pin, are usually incorporated in the feeder design to prevent corapus. However, if the shear pin is too weak, the feeder will often malfunction. Variations in powder properties or operating conditions are also important factors in feeding failure. Unstable flow is caused by poor feeder selection or inadequate hopper design. Unstable flow with bridging or flushing will also be caused by a variation in powder properties or operating conditions. The wall surface can be coated with ultrahigh-molecular-weight materials so as to prevent these troubles. Suitable vibrators can also be utilized for this purpose. The addition of a small amount (0.2–5 wt%) of fine particles, such as kaolin, diatomaceous earth, cornstarch, silica gel, or magnesium stearate, will decrease both the wall and internal frictions of powders. Granulation, drying, or encapsulation will also decrease the friction so that the powder flow can be changed from the funnel flow mode to the mass flow mode. The following items are necessary for feeders: 1. They must be suitable for the properties of the powder. 2. The operating range must be sufficiently broad. 3. The static characteristics, such as the relationship between powder feed rate and rotational speed of a drive motor, must be stable, and the repeatability of feed rate should be high. 4. They must have excellent dynamic characteristics. As mentioned earlier, powder properties affect powder flow. It is necessary to know properties such as particle size, particle shape, cohesiveness, frictional property, and abrasive property so as to 561 © 2006 by Taylor & Francis Group, LLC
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TABLE 4.1
Empirical Rule for the Selection of Feedersa Particle Size b
Flowability c
Feeder Gate valve
a b c d e D O O O D
f g h i X O D D
j k l m n o D X D O D D
Rotary
D O O X X
O O X X
D X X D X X
Table
D O O O D
D O D D
O D X O X D
Belt
D O O O O
D O O D
O O O O D O
Screw
O O O D X
O O D D
D X X D D D
Vibrating
O O O O X
D O O D
O X O O X O
a b
c d
Abrasiveness, Etc. d
O, Applicable; D, difficult; X, no use. a, less than 100 μm; b, 100 μm–1 mm; c, 1 mm–1 cm; d, 1 cm–10 cm; e, larger than 10 cm. f, excellent; g, moderate; h, low; i, cohesive. j, abrasive; k, fragile; l, low bulk density; m, high temperature; n, slurry; o, flaky, fibrous.
TABLE 4.2
Characteristics of Feeders as Final Control Means
Feeder Electromagnetic
Operating Method Voltage
Screw
rpma
VS motor
Moderate
Linear
Belt
Gate
Servomotor
Dead time
Nonlinear
Belt speed
VS motor
Moderate
Linear
VS motor
Fast
Linearb
Servomotor
Fast
Nonlinear
Table
rpm
a
Scraper
Dynamics Fast
Statics Nonlinear
Rotary
rpm
VS motor
Moderate
Linearb
Gate valve
Gate
Servomotor
Fast
Nonlinear
a b
a
Control Element SCR
rpm, nominal speed. Restricted to lower rotational speed.
select appropriate feeders. If the feeder is inadequate, it becomes impossible to control the powder feed rate. Table 4.1 shows an empirical rule for the selection of various feeders. The rotary feeder, for example, can be applied to particles between 100 μm and 1 cm in diameter, but it is difficult to utilize for particles below 100 μm. As shown in Table 4.1, no feeder is suitable for use with adhesive powders, and careful attention should be paid to feeder selection.1,2 Each of the feeders is described in detail in the next section. The operating range of a feeder should be wide enough to cover the feed rate range required in the process, with a margin of 10%, so that the feeder will work well under conditions of unexpected disturbance. Also, the dynamic response should be as fast as possible. The dynamic characteristics of typical feeders are outlined in Table 4.2.
5.4.2 VARIOUS FEEDERS Rotary Feeders A rotary feeder consists of a rotor, a rotor case, and a motor drive. Powder in a hopper flows into the rotor space due to gravity and is discharged through the exit after a half revolution. Figure 4.1 shows © 2006 by Taylor & Francis Group, LLC
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FIGURE 4.1
Rotary feeder.
FIGURE 4.2 feeder.
Powder feed rate of a rotary
a rotary feeder applied in a pneumatic transport line. There are various types of rotary feeders, and they are sometimes called rotary valves, rotary dischargers, or vane feeders. The powder feed rate depends on the rotational speed of the rotor, as depicted in Figure 4.2. Corresponding volumetric efficiency decreases with increasing rotational speed. The static characteristics depend on powder properties, depth of rotor space, and inlet area.3–5 The instantaneous powder feed rate fluctuates periodically because of the rotor configuration. Some modifications are incorporated in the rotor (helical rotor) or inlet configuration in order to suppress fluctuations. Although a rotary feeder can be utilized to feed particles against a pressure difference below 2 atm, the feed rate decreases with increasing pressure difference.3 The dynamic characteristic of a rotary feeder is modeled as a first-order time delay.
Screw Feeders A screw feeder consists of a screw, a U-shaped trough or cylindrical casing, and a motor drive. As the screw rotates, particles are forced to move from the hopper to the outlet of the feeder. Figure 4.3 shows a screw feeder with a U-shaped trough. A modified screw (e.g., tapered screw) is utilized so as to obtain a uniform flow pattern in the feed hopper. 6,7 Feeding against a pressure difference is also possible by modifying the screw configuration. For wet powder feeding, a screw feeder consisting of twin screws is suitable. A coil called an auger may also be utilized instead of a normal screw. The powder feed rate is proportional to the rotational speed of the screw as long as the powder compressibility is negligible. The instantaneous powder feed rate fluctuates periodically as the screw rotates. The fluctuation can be suppressed by using a smaller screw and higher rotational speed, or © 2006 by Taylor & Francis Group, LLC
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FIGURE 4.3
Screw feeder.
by using a coil screw (an auger feeder). The dynamic characteristic is modeled as a second- or thirdorder time delay.8
Table Feeders A table feeder consists of a turntable, a scraper, and a motor drive, as shown schematically in Figure 4.4. The powder feed rate can be adjusted by changing either the scraper position or the rotational speed of the turntable. The skirt clearance S in Figure 4.4 is also changed when an extremely wide operational range is required. The powder feed rate is proportional to rotational speed in the practical range of operation, but it is a nonlinear function of the scraper position. The dynamic characteristics are modeled as either a proportional element or a derivative element.9,10 Further, the instantaneous feed rate can be fairly smooth compared with that of the rotary or screw feeder. The flow pattern in the feed hopper is affected by the hopper inclination and the scraper position. Flow distortion can be controlled by using more than two scrapers.11 For fine powders, a special scraper is used to extend the powder uniformly on the turntable before feeding by normal operation.12 Also, in some types of table feeders, a rotating shell is utilized instead of the turntable. A stationary scraper strips off particles from a gap between the rotating shell and a plate. Table feeders of this type are Auto-feeder, Omega feeder, Bailey feeder, Bin-discharger, and Com-Bin feeder. In these feeders, powder is confined in a vessel.
Belt Feeders An endless belt pulls out the powder from a feed hopper as shown schematically in Figure 4.5. Belt feeders are easily combined with load cells, and they work as constant feed weighers or belt scales. Powder feed rate is adjusted by changing the belt speed or a gate opening. The relationship between the feed rate and the belt speed is linear, but it is a nonlinear function of the gate opening. 8 Therefore, flow-rate adjustment by changing the belt speed is more preferable than changing the gate opening. Further, the dead time associated with the gate operation causes deterioration of the dynamic response in this feeding system. For fine powders, a deaeration chamber should be attached. The chamber is also effective to prevent flushing of powder.
Vibrating Feeders Vibrating feeders utilize either electromagnetic or electromechanical drives. Figure 4.6 shows an electromagnetically vibrating feeder. The vibrating trough transports the particles smoothly. The vibration is selected near the resonance frequency. 13 The flow pattern in the feed hopper is affected by the feeder. For fine powders, an appropriate system is necessary to prevent flushing. The dynamic characteristic is modeled as a first-order time delay. 14 © 2006 by Taylor & Francis Group, LLC
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FIGURE 4.4
Table feeder.
FIGURE 4.5
FIGURE 4.6
Belt feeder.
Vibrating feeder (electromagnetic type).
Also, there is a method using a vibrating capillary tube.15,16 Even for micron-sized particles, microfeeding is possible at a constant rate as small as milligrams per second. For adhesive fine powders, ultrasonic vibration is effective in the feeding.
Valves and Dampers Valves and dampers are utilized in controlling the flow rate of free-flowing particles. Valves in common use in powder-handling industries are cut gates, slide valves, flap valves, vibrating dampers, inclined chutes, lock hoppers, and sleeve valves. These valves are specially modified to prevent particle clogging. There is a kind of valve where the powder flow rate is controlled by blowing off a heap of particles through air injection.17 A special chute called an air slide is made of porous material through which air is supplied. The powder feed rate through the air slide is controlled by the use of a gate and the airflow rate supplied. 18 © 2006 by Taylor & Francis Group, LLC
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REFERENCES 1. Aoki, R., Ed., Funryutai no Choso to Kyokyu Sochi (Feeders and Hoppers in Powder Handling Processes), Nikkan Kogyo Shimbun, Tokyo, 1963, Chap. 5. 2. McNaughton, K., Ed., Solids Handling, McGraw-Hill, New York, 1981, Sec. 2. 3. Jotaki, T. and Tomita, Y., J. Res. Assoc. Powder Technol. Jpn., 7, 534, 1970. 4. Masuda, H., Kameda, T., and Iinoya, K., Kagaku Kogaku, 35, 917–924, 1971. 5. Finkbeiner, T., VDI Forschungsh, 563, 1974. 6. Johnason, J. R., Chem. Eng., 76 (Oct. 13), 75, 1969. 7. Bates, L., Trans. ASME, B91, 295, 1969. 8. Masuda, H., Masuda, T., and Iinoya, K., J. Res. Assoc. Powder Technol. Jpn., 7, 479–484, 1970. 9. Masuda, H., Masuda, T., and Iinoya, K., Kagaku Kogaku, 35, 559–565, 1971. 10. Masuda, H., Miura, K., and Iinoya, K., J. Soc. Mater. Sci. Jpn., 21, 577–581, 1972. 11. Masuda, H., Han, Z., Kadowaki, T., and Kawamura, Y., KONA Powder Sci. Technol. Jpn., 2, 16–23, 1984. 12. Masuda, H., Kurahashi, H., Hirota, M., and Iinoya, K., Kagaku Kogaku Ronbunshu, 2, 286–290, 1976. 13. Arima, T., Kagaku Kojo, 9(2), 34, 1965. 14. Iinoya, K. and Gotoh, K., Seigyo Kogaku, 7, 646, 1963. 15. Matsusaka, S., Yamamoto, K., and Masuda, H., Adv. Powder Technol., 7, 141–151, 1996. 16. Matsusaka, S., Urakawa, M., and Masuda, H., Adv. Powder Technol., 6, 283–293, 1995. 17. Bendixen, C. L. and Lohse, G. E., in Symposium on Solid Handling, 1969. 18. Mori, Y., Aoki, R., Oya, K., and Ishikawa, H., Kagaku Kogaku, 19, 16, 1955.
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5.5
Transportation Yuji Tomita Kyushu Institute of Technology, Kitakyushu, Fukuoka, Japan
Hiromoto Usui Kobe University, Nada-ku, Kobe, Japan
5.5.1 TRANSPORTATION IN THE GASEOUS STATE Introduction Transportation of powder in the gaseous state is known as pneumatic conveying, of which familiar application is a vacuum cleaner, and uses air forces acting on particles exposed in an air stream in a pipe. The transport distance is 2 km at most, and to select flexible routes is easy due to pipeline transportation. Determination of an optimum pipeline and air source is required in design for a given transport distance and mass flow rate of given powder. Conveying systems consist of an air source, powder feeder, conveying pipe, gas particle separator, and air filter. In the positive systems (Figure 5.1), powder is fed into an air stream through a feeder at above ambient pressure and is continuously discharged into the outside at destinations. Foreign substances are not mixed in transported materials. When the air pressure at the feeding point is high, a continuous feeding is difficult and a batch system using a blow tank is employed (Figure 5.2). The positive systems are favored to deliver the powder to several destinations. In the negative systems, the powder can be continuously fed into an air stream at ambient pressure, but it is difficult to discharge it continuously to the outside at the destinations. While the available pressure for transport is limited, the powder does not leak out of the pipeline. These systems are used to collect powder from several points and deliver it to one destination. Besides, there are positive–negative systems that take advantage of both systems. While most systems are open, using air as a conveying medium, closed systems are used for specific cases such as toxic, explosive, and hygroscopic powders in a controlled environment. Circulating systems are employed for fluidized beds.
Model of Gas Particle Flow in a Pipeline A working model of gas solid flow in a pipeline is a steady two-phase flow model,
rs cv
r (1 c ) u
dv d dz 4t ( pc ) rs cg ws F ds ds ds d
du d dz 4t { p (1 c )} r (1 c ) g wa F ds ds ds d
(5.1)
(5.2)
where r and rs are the gas and solid material densities, v and u solid and gas velocities, tws and twa the solid and gas wall shear stresses, p the pressure, c the solid concentration, z the height, g the gravitational acceleration, d the pipe diameter, s the coordinates along the pipe axis, and F the 567 © 2006 by Taylor & Francis Group, LLC
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secondary air
blow tank
compressed air porous plate FIGURE 5.1
Positive pressure system.
packed bed Gs
Gs
settled layer
empty tube U
FIGURE 5.2
Blow tank solid conveyor.
interaction force between gas and solid. Based on this model, the pressure drop along the pipeline is given by z2 dv 4G u2 du g ∫ {rs c r (1 c )} dz ∫ 2 2 v1 d z1 p u1 d s2 ⎛ t t wa ⎞ 4∫ ⎜ ws ⎟⎠ ds ∑ psi pai s1 ⎝ d i
( p2 p1 )
4Gs p
∫
v2
(5.3)
where G and Gs are the gas and solid mass flow rates, and Δpsi and Δpai are the local pressure losses in pipeline due to solid and gas.
Flow Patterns of Gas Solid Flow in Pipes Since rs/r ~103 , the influence of gravity on the flow is strong and particle flow patterns depend on pipeline configurations. Figure 5.3 schematically shows a phase diagram for horizontal flow,1 where Δp/L is the pressure drop per unit length and U is the superficial air velocity defined by u(1 − c). © 2006 by Taylor & Francis Group, LLC
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g particle slug
particle slug
settled layer of particles FIGURE 5.3
settled layer of particles
particle slug
settled layer of particles
Phase diagram for horizontal flow.
Transportation of particles is possible in a region enclosed by lines of empty tube, packed bed, and settled layer. Air velocity at which Δp/L becomes minimum for a given Gs is called the saltation velocity Us,2 which is important in the design of conveying system and above which the particles are conveyed suspended in air flow. Matsumoto et al. 3 gives the following equation for Us based on their measurements for particles of 0.29 mm ≤ D ≤ 2.6 mm and pipe of d 26 and 49mm ⎛r⎞ m 0.448 ⎜ s ⎟ ⎝ r⎠
0.50
⎛ ug ⎞ ⎜ ⎟ ⎝ 10 gD ⎠
1.75
⎛ Us ⎞ ⎜ ⎟ ⎝ 10 gd ⎠
3.0
(5.4)
where m Gs /G is the solid loading ratio, D the mean size of particle, and ug the terminal settling velocity of the single particle. Below Us, the flow becomes heterogeneous and takes various patterns depending on the pipe size and particle properties. Low-velocity transport is advantageous to avoid pipeline erosion and degradation of particles. However, there is danger of pipeline blockage if the air velocity is too low. For coarse particles, the extrusion flow is observed near the packed bed line in a short pipeline, where particles are fully filled throughout the pipeline. When decreasing U below Us, the flow becomes intermittent and unstable. Further reduction in U, a stable slug flow (Figure 5.4), appears where there remain particle settled layers along the pipeline. Particles in a slug are always replaced with those in the settled layer that are transported by a definite distance every slug passing. A plug flow appears when the settled layer disappears. When particle flow is heterogeneous and is not continuous, it is difficult to locate such a flow pattern in the phase diagram, and the relation between Δp/L and U depends on conditions such as feeding method and length of pipeline. For fine powder, the flow pattern changes smoothly to fluidized dense phase flow from suspension flow. Figure 5.5 shows a phase diagram for vertical flow, where A is the point of incipient fluidization and B is the terminal settling velocity of a single particle. Vertically upward transportation is possible in a region enclosed by lines of empty tube, packed bed, and fluidization. Fluidization is a counterpart of the settled layer in horizontal flow in a sense that there is no net particle transport while there is particle circulation suspended by the air stream. Corresponding to Us, there is a choking velocity Uc ucec in the vertical flow for which Yang4 gives the following empirical relation:
(
)
pd 2 rs (1 c ) uc ug , Gs 4
(
) 6.8110
2 gd c 4.7 1
(u u )
2
c
g
5
⎛ r⎞ ⎝⎜ r ⎟⎠
22
(5.5)
s
Above Uc the particle flow is suspension. Below Uc various flow patterns are observed and some of them are similar to those in horizontal flow. Figure 5.6 shows a slug flow in a vertical pipe where the particle slugs propagate upward in substantial particle holdups. © 2006 by Taylor & Francis Group, LLC
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packed bed Gs A Gs fluidization
B
empty tube U
FIGURE 5.4 Slug flow in a horizontal pipe.
particle slug
suspended particles
particle slug g suspended particles
particle slug suspended particles FIGURE 5.5 Phase diagram for vertical flow.
Pressure Loss Calculation for Suspension Flow We can use Equation 5.3 to calculate a total pressure loss in pipeline for suspension flow. We need v for calculating the first term on the right hand side of Equation 5.3, and c for the third term. When it is difficult to find them, we can use the following approximation: v ≅ U − ug rs c r (1 c ) ≅ r ( m 1)
(5.6)
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D'
velocity type
A' B' C blockage
A B
D
valve control
FIGURE 5.6 Slug flow in a vertical pipe.
There are many types of correlation for the friction term and the following correlation is recommended:
( l m l ) rU8
2
z
t wa t ws
(5.7)
where l is the pipe friction coefficient for gas and lz is the additional pressure loss coefficient due to powder. Although there are a numerous number of references for various materials, it is best to use the measurement for a given powder, since this term is important. Weber5 gives the following empirical equation for horizontal flow of several granules:
(
lz 4.56 m 0.6 U ⁄ gd
)
1.970
(5.8)
It is a practice for vertical flow to regard the gravitational term as a loss and include it in the additional pressure loss together with the friction term as follows: m lz
4t 1 rU 2 g rs c ws d 2 d
(5.9)
We can approximately assume lz of this definition for the vertical flow as twice that for the horizontal flow. A serious local loss is due to 90° bends, Δpab, Δpsb, which depends on the bend configurations, and for the rough estimate we can use 6 jb ( m 1)
rU 2 pab psb 2
(5.10)
where, jb 1.5, 0.75 and 0.5 for Rb/d 2, 4 and more than 6, respectively, Rb being the radius of the bend. It becomes accurate to consider the change of gas density along the pipeline when calculating the pressure loss. For this purpose we divide the pipeline into many sections, and we assume that © 2006 by Taylor & Francis Group, LLC
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the gas density is constant in each section. Furthermore, we assume that the gas flow is isothermal, and we use r p/RT, where R is the gas constant and T the absolute temperature of the gas. At first, we guess the total pressure loss in a section and calculate the mean pressure in the section. Then we estimate the mean gas density and mean gas velocity U and revise the total pressure loss based on Equation 5.3. If this is close to the first guess, we stop calculation. Otherwise, we repeat the process until a given convergence is obtained. In the positive system, the pressure at a pipe exit is known, and the calculation is from the pipe exit toward the pipe inlet, while in the negative system it is vice versa. The pressure loss is in proportion to U2 and U 4G/rpd2, then it is effective to keep U constant by increasing the pipe diameter in stepwise fashion toward the downstream in order to reduce power consumption, which is approximately estimated by poQo ln {1 + (Δp/po)} where Δp is the total pressure loss along the pipeline, po the ambient pressure, and Qo the volumetric flow rate of free air. Furthermore, it is important to keep U always larger than the design velocity, in particular, at the pipe inlet where U is minimum.
Matching Air Source and Pipeline Characteristics Operating points of air source are crossings between the air source and pipeline characteristics, as shown in Figure 5.7, where Q is the volumetric flow rate of air and p is the delivery pressure of air source or the total pressure loss for a given pipeline. The stable operating points are those at higher flow rate, that is, points A, B, C, and D. The points, A, B, and C are unstable operating points. When the air source is the velocity type and we want to choose point B as an operating point, we can change the operating points by the valve or speed control. In the figure, the exit valve changes the characteristics of the velocity type of the air source. In this case, there is a limit point C that is called blockage velocity, below which the pipeline is blocked. The blockage velocity depends on the characteristics of the conveying system. A positive-displacement type of air source is used for low-velocity transport.
102 Bingham model:
10
1 CWM with Buller cool
10-1 10-2
FIGURE 5.7
10-1
1
10
102 2-102
Operating points of conveying system.
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5.5.2 TRANSPORTATION IN THE LIQUID STATE Hydrotransport of solid particles is now an acknowledged alternative to conventional forms of transport, such as pneumatic or mechanical conveying. In the coal industry, for example, the use of highly loaded coal–water mixtures with reduced viscosity and the direct combustion of fines with a high water content have been proposed. In the transport of minerals and the disposal of waste materials, hydrotransport techniques have been employed for the reduction of manpower and cost. Conveying distances up to 400 km have been utilized. The capacity of a slurry pipeline is usually less than 12 million tons per year, and a pipe diameter is selected in the range 60–500 mm. The following classification of slurries is useful to discuss the hydrotransport technique: 1. Homogeneous or nonsettling 2. Nonhomogeneous or settling
Homogeneous Slurries The design of a slurry-handling system usually involves two steps: selection of pipe diameter and determination of frictional head loss. A homogeneous slurry can be tested in a Couette viscometer without significant settling. Some examples of the shear stress–shear rate relationship are shown in Figure 5.8. Three purely viscous non-Newtonian fluid models 7 (i.e., Bingham model, Hershel– Bulkley model, and Casson model) are compared in this diagram. Also, the power-law model, which shows a straight line in a log-log diagram such as shown in Figure 5.8, can be used to fit the narrow range of rheological data. A suitable rheological model should be selected by taking into account the shear rate range of the practical problem. Laminar flow in cylindrical tubes has been analytically solved, 8 and the relationships between flow rate Q and pressure drop ΔP for some non-Newtonian fluid models are summarized as follows: Power-law model: τ = mg n 1 ⎛ ⎞ pR 3 ⎟ ⎛ t R ⎞ n PR ⎜ Q= , where t R ⎜ 1 + 3 ⎟ ⎜⎝ m ⎟⎠ 2L ⎝ n ⎠
( )
where
(5.11)
Bingham model: t hg·ty
where
4 ⎛ pR 3 t R ⎞ ⎡ ⎛ 3 ⎞ ⎛ t y ⎞ ⎛ 1 ⎞ ⎛ t y ⎞ ⎤ PR ⎢ 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎥ , where t R Q⎜ ⎟ ⎝ ⎠ ⎝ ⎠ 4 4 3 2L n t t ⎝ ⎠ ⎢⎣ ⎝ R ⎠ ⎥⎦ ⎝ R⎠
(5.12)
In the case of turbulent flows, prediction methods are proposed by Kemblowski and Kolodziejski 9 for the power-law model and by Wilson and Thomas10 for the Bingham model.
Nonhomogeneous Slurries Nonhomogeneous slurries cause the settling of solid particles. Thus, the flow situations are significantly different between horizontal and vertical positions. However, the long-distance transportation of slurries is mainly concerned with the horizontal pipeline. Horizontal transportation is discussed © 2006 by Taylor & Francis Group, LLC
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HOMOGENEOUS FLOW HETEROGENEOUS FLOW SALATATION FLOW CLEAR WATER FLOW FLOW WITH A STATIONARY BED
log V
HOMOGENEOUS FLOW
HETEROGENEOUS FLOW
SALATATION FLOW
FLOW WITH A STATIONARY BED FIGURE 5.8 Shear stress–shear rate relationship obtained for a coal–water mixture. Coal concentration is 68.2 wt%. Slurry temperature and pH are 298 K and 8.6, respectively.
in this section. Flow regimes of slurries can be classified into four regions. The friction factor defined by f = (P/L)(D/2rv2) is given for each flow regime as follows11: Flow with a stationary bed: f fw 0.4036C
0.7389
f
0.7717 w
C
0.4054 D
⎛ ⎞ v2 ⎜ Dg s 1 ⎟ ( )⎠ ⎝
1.096
(5.13)
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Saltation flow: ⎛ ⎞ v2 f fw 0.9857C 1.018 fw1.046 CD0.4218 ⎜ ⎟ ⎝ Dg ( s 1) ⎠
1.354
(5.14)
Heterogeneous flow: ⎛ ⎞ v2 f fw 0.5518C 0.8687 fw1.200CD0.1677 ⎜ ⎟ ⎝ Dg ( s 1) ⎠
0.6938
(5.15)
The deposition velocity, or limit-deposit velocity, is the usual lower limit of the slurry transportation velocity. Below this limit, a stationary deposit of particles forms on the bottom of the pipe. This makes stable pipeline operation very difficult. The correlation of deposition velocity was reported by Durand,12 Oroskar and Turian,13 and Gillies and Shook.14 Gillies et al.15 have proposed a method for predicting the pressure drop in the horizontal slurry pipeline flow based on the two-layer model. The hydraulic capsule pipeline is another candidate to transport solid materials. A capsule pipeline system for limestone transportation has been in commercial use since 1983, with a capacity of 2 million tons per year, a transportation distance of 3.2 km, and a pipe diameter of 1.0 m.16 In this case, a pneumatic capsule transport technique is used. A new capsule transportation technique, the coal log pipeline, has been proposed by Liu and Marrero17 and Liu18 and is utilized by the hydraulic driving force. The coal, extruded and compressed into a log shape, is used as the capsule. Consequently, the coal log pipeline is simpler to handle than the container type (hydraulic capsule).
Notation C CD D d f fw g L m n ΔP Q R s n n∞ g· h r rs
Solid volume fraction [ = (4/3)gd(s–1)/ν∞] drag coefficient for a free-falling sphere Pipe diameter Diameter of solid particle Friction factor Friction factor for a liquid flow Gravity acceleration Pipe length Consistency index of the power-law model Power-law index Pressure drop Flow rate Pipe radius ( = rs/r) relative density Cross-sectional averaged velocity Terminal velocity of particle settling in an unbound fluid Shear rate Viscosity Density of liquid Density of solid
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t tR ty
Shear stress Wall shear stress Yield stress
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.
Welschof, G., VDI-Forsch, 492, 1962. Zenz, F. A., Ind. Eng. Chem. Fundam., 3, 65–75, 1964. Matsumoto, S., Hara, M., Saito, S., and Maeda, S., J. Chem. Eng. Jpn., 7, 425–430, 1974. Yang, W. C., Powder Technol., 35, 143–150, 1983. Weber, M., Bulk Solids Handling, 11, 99–102, 1991. Engineering Equipment Users Association, Pneumatic Handling of Powdered Materials, Constable, London, 1963. pp. 52–64. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., Transport Phenomena, Wiley, New York, 1960, p. 10. Skelland, A. H. P., Non-Newtonian Flow and Heat Transfer, Wiley, New York, 1967, pp. 82 and 110. Kemblowski, Z. and Kolodziejski, J., Int. Chem. Eng., 13, 1973, 265. Wilson, K. C. and Thomas, A. D. (1985). Can. J. Chem. Eng., 63:593. Turian, R. M. and Yuan, T.-F., AIChE J., 23, 232, 1977. Durand, R., In Proceedings of the Minnesota International Conference on Hydraulic Conveying, 1953, p. 89. Oroskar, A. R. and Turian, R. M., AIChE J., 26, 550, 1980. Gillies, R. G. and Shook, C. A., Can. J. Chem. Eng., 69, 1225, 1991. Gillies, R. G., Shook, C. A., and Wilson, K. C., Can. J. Chem. Eng., 69, 173, 1991. Kosuge, S., in Proceedings of the Seventh International Symposium on Freight Pipelines, Vol. 1, 1992, p. 13. Liu, H. and Marrero, T. R., U.S. Patent No. 4,946, 317, 1990. Liu, H., Freight Pipelines, Elsevier, New York, 1993, p. 215.
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5.6
Mixing Kei Miyanami Osaka Prefecture University, Sakai, Osaka, Japan
5.6.1
INTRODUCTION
Powder mixing is an operation to make two or more powder ingredients homogeneous with, if necessary, some amount of liquid. In some industrial fields, it is called blending. The size of the powders to be mixed ranges widely, and the states of moisture range from dry to pendular. Because solid particles are subjected to various interactive forces and are not self-diffusive, they cannot be set in motion without any external force such as mechanical agitation. Different external forces can be applied to the powders to be mixed, and many types of mixers have been developed for a variety of applications.
5.6.2
POWDER MIXERS
Table 6.1 shows a classification of various powder mixers, based on the manner by which the powders are set in motion. This table also lists rough ranges of powder properties appropriate to each type of mixer. Typical structures of powder mixers are depicted in Figure 6.1. Although mixer performance should be evaluated on the basis of the powder properties being handled, operating conditions, and the application purpose, the general features of each mixer are as described below.
Rotary Vessel Type The rate of mixing is rather low in a rotary vessel, but a good final degree of mixedness can be expected. The powders to be mixed are charged up to 30–50% of the vessel volume. The rotational speed is set at 50–80% of the critical rotational speed, Ncr, given as N cr ⫽
0.498 Rmax
(s ) ⫺1
(6.1)
where Rmax (m) is the maximum radius of rotation of the mixer. For mixing powders with poor flowability and large differences in particle densities and diameters, various types of guide plates or internals are installed in the rotating vessel, as shown in Figure 6.1a and 6.1b. The addition of mixing aids or an operation at a rotational speed close to Ncr may be useful in some cases.
Stationary Vessel Type With Mechanical Agitation In stationary vessels using mechanical agitation, large amounts of powders can be handled in a small space. Specialized atmospheres as well as normal temperatures and pressures are accessible for multipurpose operations. Some types can be used in both batch and continuous modes.
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Horisontal axis of rotation
Symbol in Fig. 9.1
Water Content Dry
Wet
Abrasive
Properties Large
Cohesive
Over 45, L
35–45, M
Under 35, H
Under 0.01
0.1–0.01
Over 1.0
1.0–0.1
Continuous
Horizontal cylinder Inclined cylinder V-type Double cones Cubic S-type Continuous V-type
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Rotary vessel
Flowability Angle of
Range of Particle Diameter (mm)
Powder-Handling Operation
Mixer
Classification
Batchwise
Operation
Typical Powder Properties
Differences in Powder
Classification of Powder Mixers and Range of Their Servicesa
Small
TABLE 6.1
Mixing
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Horizontal axis of rotaion
Ribbon Screw
(d)
Rod or pin
Mixing
Stationary vessel
Double-axle paddles Vertical axis of rotation
Ribbon Screw Screw in cone High speed Rotating disk Muller
Vibration
(e) (f) (g)
Vibratory mill Sieve
Complex
Gas flow
Moving or fluidized bed
Gravity
Motionless
Internals in rotating vessel
Horizontal cylinder V-type Double cones
(h) (i) (a) (b)
Gas flow and mechanical agitation Vibration and mechanical agitation
(j) 2a
, suitable; , usable.
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FIGURE 6.1
Typical examples of powder mixers.
With Gas-Flow Agitation Stationary vessels using gas-flow agitation are used primarily for batch mode mixing. The powders to be mixed can be charged to more than 70% of the vessel volume. The vessel also serves as a storage container. Additional equipment, including blowers, dust collectors, and pressure regulators, are necessary so the system as a whole usually becomes rather large. Free Falling Due to Gravity (Motionless Type) In motionless stationary vessels, the gravitational flow of powders is repeatedly divided and united to promote mixing, and the degree of mixedness can be adjusted by the number of repetitions.
Complex Type and Others Impellers (sometimes called intensifiers) or other internal parts can be installed inside a rotary vessel. Mechanical agitation can be carried out in addition to agitation by gas flow. Vibratory motion can also be added to stationary vessel mixers. All these efforts are aimed at making the powder motion in mixers as free as possible from the force of gravity, enhancing the rate of mixing and the degree © 2006 by Taylor & Francis Group, LLC
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of mixedness, and extending the applicability of a mixer to powders having fairly large differences in physical properties. Detailed descriptions of powder mixers are available elsewhere (Mixing Technology for Particulate Materials,1 p. 57).
5.6.3
MIXING MECHANISMS
Degree of Mixedness and Its Final Value In conventional powder-mixing operations, a perfectly homogeneous mixture is defined such that the powder component under investigation becomes uniform throughout the mixture. Statistics are used widely to define the degree of mixedness (the degree of homogeneity) for a powder mixture. Let xi (i ⫽ 1, 2, . . . , N ) be the composition of the key component in the ith sample of N spot samples taken randomly from a binary powder mixture. The sample mean –xs is given by N
xs ⫽ ∑ i⫽1
xi N
(6.2)
If the charged composition –x c is known, the sampling procedure can be examined by comparing –x s to –x c. The variance of the samples s 2 is defined in the following ways: ( xi ⫺ x c )2 N i⫽1
(6.3)
( xi ⫺ x s )2 N ⫺1 i⫽1
(6.4)
N
s p2 ⫽ ∑ or N
ss2 ⫽ ∑
ss2 is an unbiased estimation of the population variance with the degree of freedom, v ⫽ N, and the following relation holds with the reliability of 90%:
xs ⫺
1.64ss N
ⱕ xc ⱕ xs ⫹
1.64ss N
( 6.5)
Therefore, the degree of homogeneity for the mixture can be estimated by evaluating the magnitude of the sample variance s2s . In other words, s2s can be a measure of the degree of mixedness and is useful in practical applications. However, s2s is influenced by various measuring conditions and cannot be the universal measure. Table 6.2 lists a variety of dimensionless or normalized expressions proposed so far for the degree of mixedness.2 In this table, sr2 and s02 are, respectively, the sample variances in a perfectly random mixture and in a completely segregated mixture, namely,
s r2 ⫽
(
x c 1⫺ x c n
)
(6.6)
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TABLE 6.2 Typical Expressions for Degree of Mixedness Classification
Expression for Mixedness
Expression for Degree of Mixedness, M I
( 20 ⫺ 2)/( 20 ⫺ 2r )
1
Perfectly Mixed, Mr 1
1 ⫺ /0
0
1
II
( 20 ⫺ 2r )/( 2 ⫺ 2r )
1
⬁
III
r /
r /0
1
1
/0
1
r /0
2
(2 ⫺ 2r )/( 20 ⫺ 2r )
2
Expression for unmixedness
Completely Segregated M0 0
IV
2 ⫺ 2r
V VI
1
0
20 ⫺ 2r
0
1
2
20
2r
2
0
r
and
(
s 02 ⫽ x c 1 ⫺ x c
)
(6.7)
where n is the size of sample or the number of powder particles contained in a spot sample. Although some of the degree of mixedness shown in Table 6.2 is less dependent on measuring conditions as well as the operating conditions of mixers, there is no rationale for the expressions; they are simply conventional and intuitive. The expressions in Table 6.2 are simply transformations from the interval [s02, sr2] to the fixed interval [0, 1], a region of which is magnified in each definition. For a multicomponent mixture, the degree of mixedness can be evaluated by a covariance matrix in the same way as in the binary mixture, but its measurement and calculation procedure are complicated. Therefore, the multicomponent system is regarded as a mixture of the single most important component (the key component) and the others, and then is treated as a binary mixture. In cases of batch mode mixing, only spatial variation in the composition of the key component is a matter of concern. In cases of continuous mixing, however, the time change in the composition at the outlet becomes important in addition to the spatial variation. The degree of mixedness in continuous mixing can be expressed by the magnitude of the time change in the key composition coming out from the outlet of the mixer in a steady operation. Let be xi the key composition in the ith spot sample of N spot samples taken randomly or periodically at a constant time interval from the mixture flow at the outlet of the mixer. The variance s c2 of the time change in xi is obtained in a manner similar to that for binary mixing:
(
)
2
⎡ xi ⫺ x / x0 ⎤ N xi ⎦ , x⫽ s 2c⫽ ∑ ⎣ ∑ N i⫽1 i⫽1 N N
(6.8)
where x0 is the inlet composition of the key component (feed or charge ratio). At a steady state (i. e., –x ⫽ x0), the degree of mixedness s c2 defined by Equation 6.8 is independent of the sampling interval. Also, it has been confirmed that sc2 is practically unaffected by the measuring and operating conditions.3 Detailed practical methods of evaluating the degree of mixedness, including sampling methods, have been described elsewhere (Mixing Technology for Particulate Materials,1 p. 19). © 2006 by Taylor & Francis Group, LLC
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Mechanism of Powder Mixing Roughly speaking, mixing of powders progresses with the following three types of particle motion. Convective Mixing A circulating flow of powders is usually caused by the rotational motion of a mixer vessel, an agitating impeller such as a ribbon or paddle, or gas flow. This circulating flow gives rise to convective mixing and contributes mainly to a macroscopic mixing of bulk powder mixtures. Although the rate of mixing by this mechanism is rather high, its contribution to the microscopic mixing is unexpected. Convective mixing is beneficial for batch mode operations but gives unfavorable effects to continuous mode mixing. Shear Mixing Shear mixing is induced by the momentum exchange between the powder particles having different velocities (velocity distribution). The velocity distribution develops around the agitating impeller and the vessel walls due to compression and extension of bulk powders. It is also developed in the powder layer in rotary vessel mixers and at blowing ports in gas-flow mixers. Shear mixing can enhance semimicroscopic mixing and be beneficial in both batch and continuous operations. Diffusive Mixing Diffusive mixing is caused by the random motion of powder particles—the so-called random walk phenomenon—and is essential for microscopic homogenization. The rate of mixing by this mechanism, however, is low, when compared with convective and shear mixing.
Characteristic Curve of Mixing Powder mixing proceeds in a mixer where the three mechanisms described above take place simultaneously. The characteristic curve of mixing is the plot of the degree of mixedness M (on a logarithmic scale) against the mixing time t (on a linear scale). The mixing time is the time measured from the start of mixing in a batch mode operation, whereas it corresponds to the mean residence time (the powder volume in a mixer divided by its volumetric flow rate) in a continuous mode operation. The characteristic curve of mixing is useful for the performance evaluation of mixers. Figure 6.2 shows a schematic example of the curve, where the standard deviation is plotted on a logarithmic scale. Generally speaking, convective mixing is dominant in the initial stage (I) and the mixing proceeds steadily by both convective and shear mechanisms in the intermediate stage (II). In the final stage (III), the effect of diffusive mixing appears, and a dynamic equilibrium between mixing and segregation is reached. The degree of mixedness M ⬁ at this stage is called the final degree of mixedness, M⬁. Various powder mixers exhibit a variety of patterns in the characteristic curve of mixing ( Mixing Technology for Particulate Materials,1 p. 33). The value of M⬁ is also influenced appreciably by operating conditions and powder properties.
Rate of Mixing The logarithm of the standard deviation changes linearly with time at the initial period I in almost all powder mixers, as shown in Figure 6.2. This can be expressed by the following rate equation of the first order: ds ⫽⫺ K1s, dt
s ⫽ s0
at t ⫽ 0
(6.9)
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FIGURE 6.2
Characteristic curve of mixing process (schematic).
and solving Equation 6.9, one obtains s ⫽ s 0 exp (⫺k1t )
(6.10)
If M is expressed by the classification IV-2 in Table 6.2, the rate equation becomes
(
)
ds 2 ⫽ k2 s 2 ⫺ s r2 , dt
s 2 ⫽ s r2
at t ⫽ 0
(6.11)
and hence, M ⫽ exp(⫺k2 t )
(6.12)
These equations are nothing more than phenomenological expressions of the rate of mixing. The coefficients k1 and k2 in Equation 6.9 or Equation 6.11, which denote the slope of the straight line at the stage I in Figure 6.2, are called the rate constant of mixing process (the dimension is s⫺1). The rate constant of the mixing process is affected appreciably by operating conditions and powder properties.
5.6.4
POWER REQUIREMENT FOR MIXING
The following factors should be taken into account in estimating the power requirement for a powder mixer in steady-state operation: (1) the net energy to keep powders in the mixer in steady motion, (2) the net energy to keep the mixer itself or the agitating impellers in steady motion, and (3) the compensatory energy for friction losses in the driving system of the mixer. The net energy is influenced by the mechanical structure of the mixer, the powder properties, and the operating atmosphere and conditions. Let T (Nm) be the axial torque for rotating the mixer vessel or impeller at a rotational speed Ns (rps). The power P is given by P ⫽ 2pN sT
(6.13)
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Horizontal Cylinder Mixer The torque acting on the rotational axis of a vessel consists of (1) the torque against the gravity force acting on the center of gravity of the powder bed in the vessel, (2) the torque from the force required to drive the powder into steady rotational motion, and (3) the torque from the frictional force between the vessel wall and the powder pressed onto the wall by centrifugal force. After lengthy consideration of these factors, Equation 6.14 has been developed4,5: N2R T ⫽ A+ B s g R L rb g 3
(6.14)
The term on the left-hand side of Equation 6.14 is called the Newton number, and the second term on the right-hand side is called the Froude number. Coefficients A and B , both dimensionless, can be obtained from Figure 6.3 as functions of the charge ratio f, the angle of repose of powder ø, and the friction coefficient of powder at the vessel wall mw.
V-Type Mixer In this case, the axial torque changes periodically at a period of π, exhibiting a maximum torque Tmax and a minimum torque Tmin, because the rotational motion displaces powders from one end to the other in the vessel. Let Rmax be the maximum radius of rotation, then Tmax and Tmin are given by Tj 3 max
R
rb g
⫽ Aj ⫹ B j
N s2 Rmax , g
j ⫽ max or min
(6.15)
The coefficients Aj and Bj can be evaluated from Figure 6.4.
FIGURE 6.3 Coefficients A and B in Equation 6.14 for horizontal cylinder mixer. © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.4 Coefficients A and B in Equation 6.15 for V-type mixer. Angle of apex ⫽ 90 ° ; ratio of cylinder radius to maximum radius of rotation ⫽ 0.5.
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TABLE 6.3
Exponents in Equation 6.16
Type of Mixers Vertical ribbon
␣1
␣2
Exponent in Eq. (6.16) ␣3 ␣4 ␣5 ␣6
0
1.0
0.7
1.5
3.0
Horizontal ribbon
0
1.0
1.2
1.0
Paddle
0
1.0
1.0
1.0
␣7
␣8
1.2
⫺0.8
—
3.3
⫺0.3
0.7
1.2
2.0
—
1.0
—
Stationary Vessel Mixer An empirical formula for estimating the axial torque for a ribbon or paddle impeller in powders having a high flowability has been proposed 6–8 : ⎛ S⎞ T ⫽ KDp␣1 r b␣ 2 ms␣ 3 Z ␣ 4 d ␣ 5 ⎜ ⎟ ⎝ d⎠
␣6
b ␣ 7 f ␣8 ( Nm )
(6.16)
where Dp ⫽ particle diameter (m) rb ⫽ bulk density (kg/m3) ms ⫽ internal friction factor S ⫽ pitch of ribbon impeller (m) b ⫽ width of impeller (m) d ⫽ diameter of impeller (m) f ⫽ charge ratio (–) Z ⫽ height of powder bed (m) Table 6.3 gives the exponents a1 ~ a8 in Equation 6.16, which have been determined experimentally. A marginal torque for setting impellers in rotational motion in a quiescent powder bed should also be considered in the mechanical design.
5.6.5
SELECTION OF MIXERS
How to select a mixer depends on the degree of homogeneity required for the product (final degree of mixedness), the rate of mixing, and the power requirement. The type of operations (batch or continuous modes) and the strength of the powder particles against mechanical agitation should also be examined. When the mixing data with a bench-scale mixer are available to design a large system, the following factors should be kept in mind: 1. Whether the desired final degree of mixedness is attainable 2. How to determine the operating conditions 3. Amount of power and cost necessary For rotary vessel mixers in which a circulating flow of powders exists, the ratio of centrifugal force to gravity force (i.e., the Froude number, Ns2 R / g) can be kept constant in scaling up. Although the Froude number can be evaluated for stationary vessel mixers by taking one half of the outer impeller diameter as the representative radius, R, the flow patterns of the powders in a prototype mixer might © 2006 by Taylor & Francis Group, LLC
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not become geometrically similar to that in a small-scale mixer. For example, if the clearances between the impeller tips and the vessel walls are geometrically scaled up, a dead zone might be formed in the powder flow in the large-scale mixer. If the scale-up is based on the tip speed of the impeller, a dead zone where agitation is insufficient for mixing may be formed around the impeller axis, necessitating an additional design for the mixer structure in some cases. To select the pertinent mixer, not only must the performance and characteristics of various mixers themselves be well understood, but also the mixing operation in the entire process. The purposes of the mixing operation, the powder properties, the capacity of production, the desired final degree of mixedness, maintenance problems, unit and running costs, and related matters must also be well defined. The priority of each of these factors depends on individual cases. Unfortunately, no general organized argument is available for the methodology of selecting the target mixer.
Notation A, B Aj ,Bj b Dp d f g K ki L M M⬁ N Ns Ncr n P R Rmax S T Tj t x0 –x c xi –x s Z a1 ~ a8 ms mw v rb s2
Coefficients in Equation 6.14 Coefficients in Equation 6.15, j⫽max or min Width of impeller Particle diameter of powder Diameter of impeller Charge ratio; volume of powder charged divided by volume of mixer vessel Acceleration due to gravity (m/s2) Coefficient in Equation 6.16 Rate constant of mixing process, defined by Equation 6.9 for i=1, by Equation 6.11 for i ⫽ 2, (s⫺1) Length of horizontal mixer vessel (m) Degree of mixedness Final degree of mixedness Number of spot samples Rotational speed of mixer vessel or impeller (s⫺1) Critical speed of rotation, defined by Equation 6.1 (s -1) Sample size, number of powder particles in a spot sample Power for driving mixer, defined by Equation 6.13 (W) Radius of horizontal mixer vessel (m) Maximum radius of rotation of mixer (V-type) (m) Pitch of ribbon impeller (m) Axial torque for driving mixer (Nm) Maximum or minimum axial torque for driving V-type mixer (Nm) Mixing time (batch operation) or mean residence time (continuous operation) (s) Inlet composition of key component for continuous mixing Charged composition of key component for batch mixing Composition of key component in the ith spot sample Sample mean of, defined by Equation 6.2 Height of powder bed in mixer vessel (m) Exponents in Equation 6.16 Internal friction factor Wall friction factor Degree of freedom in estimation Bulk density of powder (kg/m3) Variance of compositions of key component in samples
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s 02 s c2 s P2 s r2 s 2s f
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Variance of compositions of key component in completely segregated mixture, defined by Equation 6.7 Variance of compositions of key component at outlet of continuous mixer, defined by Equation 6.8 Variance of compositions of key component, defined by Equation 6.3 Variance of compositions of key component in perfectly random mixture, defined by Equation 6.6 Variance of compositions of key component, defined by Equation 6.4 Angle of repose (rad)
REFERENCES 1. Association of Powder Process Industry and Engineering (APPIE), Ed., Mixing Technology for Particulate Materials, Nikkan Kogyo Press, Tokyo, 2001. 2. Yano, T. and Sano, Y., J. Soc. Chem. Eng. Jpn., 29, 214, 1965. 3. Yano, T., Sato, M., and Mineshita, Y., J. Soc. Powder Technol. Jpn., 9, 244, 1972. 4. Sato, M., Yoshikawa, K., Okuyama, N., and Yano, T., J. Soc. Powder Technol. Jpn., 14, 699, 1977. 5. Sato, M., Miyanami, K., and Yano, T., J. Soc. Powder Technol. Jpn., 16, 3, 1979. 6. Novosad, J., Collect. Czech. Chem. Communi., 29, 2681, 1964. 7. Makarov, Yu. I., Apparatus for Mixing of Particulate Materials, Mashinostroenie, Moscow, 1973, p. 118. 8. Sato, M., Abe, Y., Ishii, K., and Yano, T., J. Soc. Powder Technol. Jpn., 14, 411, 1977.
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5.7
Slurry Conditioning JunIchiro Tsubaki Nagoya University, Nagoya, Japan
Makio Naito Osaka University, Ibaraki, Osaka, Japan
5.7.1
SLURRY CHARACTERIZATION
Slurry is usually not a final product but an intermediate one. Slurry as intermediate product is shaped and then dried before final use. Since in the shaping process slurry must have fluidity, measurement of viscosity is very important to characterize the slurry. After shaping, fluidity is not important because the slurry does not need to deform or flow; on the contrary, characterization of the thickening behavior of the slurry becomes important. A settling test is widely done to characterize thickening behavior, and it is reported that the morphology of spray-dried granules can be estimated from this settling test. Slurry composed of dense sediment makes hollow granules, whereas slurry composed of less dense sediment makes solid granules.1,2 The disadvantage of the settling test is that it takes a long time; for example, in the case of submicron particles it takes weeks or months under gravity, and even in a centrifuge it takes days. Consequently the settling test is hard to use for on-site optimization or control of production process in factories. To shorten the measurement time, the following characterizing methods have been proposed. Particles that make a loose sediment coagulate each other and easily network from the bottom in the early settling stage. The bottom of a container supports the mass of the networked particles. Particles that make dense sediment repulse each other and settle freely. In this case, viscous drag supports the particle mass. If we measure the hydrostatic pressure at the bottom of a container during a batch settling test, the pressure changes from Pmax(⫽rsgh) at t ⫽ 0 to Pmin(rgh) up to infinitely. Where r s is the density of the slurry, r is the density of the dispersion medium, and h is the slurry depth. Pmax — Pmin, Pmax—P and P—Pmin stand for the whole particles, the deposited particles on the bottom, and the freely settling particles, respectively. Figure 7.13 shows settling test results of the aqueous slurry of alumina abrasive particles (mean diameter is 3μm) adjusted by pH value. As shown in Figure 7.2,3 the hydrostatic pressure at the bottom can distinguish the coagulation situation in slurry at an early stage. The mass settling flux from the interface between supernatant and slurry zone is calculated by Kynch’s theory, and the depositing mass flux to the bottom can be calculated from the pressure decrease rate dp/dt . Taking the flux ratio as shown in Figure 7.3, 4 the settling behavior can be analyzed more informatively than in the traditional settling test. Constant pressure filtration is also utilized for characterization of thickening behavior. 5 As the pressure changes, the mechanical properties of the cake are analyzed. If the packing fraction of the cake Φ is constant during filtration, the pressure drop Δ P is described by the following Kozeny– Carman’s equation. ⎧⎪ ⌽2 ⌬P ⫽ m ⎨ Rm ⫹ 5Sv2 (1⫺ ⌽)3 ⎩⎪
⎫⎪ dv L⎬ dt ⎭⎪
(7.1)
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FIGURE 7.1 The settling curves of the 35 vol.% aqueous slurry of alumina abrasive particles. The mean diameter is 3 μm and the slurry is adjusted by pH value.
Where L is the cake thickness, v is the filtrate volume per unit area, t is the filtration time, Sv is the specific surface area of the particles, μ is the viscosity of the filtrate, and Rm is the resistance of the filter medium. Equation 7.1 is rewritten as follows:
(
⌬P dt ⌽ ⫽ ⌽L ⫹ Rm’ 2 5mSv dv (1 ⫺ ⌽ )3
)
(7.2)
The packing fraction of the cake can be read from the plots of the left side vs. ΦL, standing for the cake height as shown in Figure 7.4. Since the cake height is defined as the cake volume per unit area, ΦL can be calculated from the following mass balance equation: FL ⫽ f ( v ⫹ L )
(7.3)
where ø is the slurry volume concentration. If the plots are on a strait line, the cake has no packing fraction distribution. On the contrary, if the plots are on the curve (a) shown in Figure 7.5, it is suggested that the packing fraction decreases with the cake height. And the curve (b) suggests that the packing fraction increases with the cake height. Moreover, the curve (c) suggests that the primary particles coagulate each other and the settling velocity of the agglomerates is not negligibly small. © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.2 The hydrostatic pressure at the container bottom. The depth of the sample slurry is 90 mm.
FIGURE 7.3 The ratio of depositing to settling mass flux. © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.4 The constant pressure filtration results of aqueous alumina slurry expressed by Equation 7.2. The particle size is 0.48 μm, and polyacrylic ammonium is added to be the same absorbed.
FIGURE 7.5 Filtration pattern of constant pressure filtration.
5.7.2
SLURRY PREPARATION
Slurries are widely used in industry as an intermediate material during the fabrication process of final products, as well as being products themselves. The purposes of slurry usage cover a variety of areas. As intermediate materials, slurry is used for preparing spray-dried granules or for slip-casting green © 2006 by Taylor & Francis Group, LLC
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bodies to make ceramics parts. As a product, slurry is used as a material for chemical–mechanical polishing and is a key tool for high-tech polishing techniques. Therefore, the preparation of a slurry is very important, and its preparation method should be decided by considering various factors related to the dispersion process. They mainly concern the wetting behavior between the suspension medium and the particle surface, the dispersion mechanism of the particles in the liquid, and the stability of dispersion state. In this section, these factors, being considered important for achieving a better slurry preparation, are explained. Practical examples of slurry preparation and its influences in the manufacturing process on the improvement of ceramic products are given to provide understanding of the importance of slurry preparation technique.
Wetting between Suspension Medium and Particle Surface It is important to evaluate the wetting behavior between the suspension medium and the particle surface to understand the dispersion behavior of particles in the liquid. The direct measurement of the wetting ability between the surface of particle and the liquid is very difficult. Therefore, various kinds of indirect methods have been proposed to determine the wetting ability of a powder. One of the well-known methods is based on the evaluation of the penetration phenomena of a liquid into the powder bed. By measuring the penetration rate of liquid into the powder bed, the contact angle is calculated by using the Washburn equation.7 To improve the wetting ability, a combination of liquid and particles should be considered, and a low contact angle corresponds to a better dispersion of the particles. Modification of the particle surface for changing the wetting ability is also effective to improve the dispersion of particles.
Dispersion of Particles into a Liquid After the combination of liquid and solid particles is decided, a good wetting of liquid onto the surface of suspended particles should be achieved. Dispersed particles and the dispersion should be stable in the liquid for a long time. In this regard, the dispersion mechanism between particles and suspension liquid should be taken into account. In an aqueous system, the basic mechanism is based on the DLVO theory. The zeta potential is an important factor to evaluate the stability of particles in liquid. It can be changed by controlling several factors. The interaction between particles can be also controlled by adsorbing dispersants as well as polymers on the surface of particles. By using these methods, optimum dispersion conditions should be selected. Mechanical dispersion techniques are also crucial to achieve better dispersion of particles. When the powder has powder aggregates, they must be disintegrated into primary particles. This can be achieved by the application of ultrasound or by milling. For example, agitated ball milling is very effective to reduce agglomerates and disperse them into primary particles. There are different kinds of mechanical dispersion methods. Depending on the strength of aggregates, a suitable method should be selected. Mechanical dispersion methods also depend on the ranges of shear rate of the slurry.8
Stability of Suspension After the particles are dispersed in the liquid, the dispersion should be stable until it is actually used. To prevent the sedimentation of powder particles, a continuous stirring of the slurry may also be effective. For example, a ferrite powder has magnetic properties, and the true density is relatively high, therefore, continuous stirring is crucial to maintain the stability of the suspension.
Slurry Conditioning for Better Ceramics Ceramic manufacturing processes are based on various powder processing techniques. They are classified according to the shaping process used. These include the granule compaction process, slip-casting, and extrusion and injection molding. The slurry properties actually influence the © 2006 by Taylor & Francis Group, LLC
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properties of the spray-dried granules and green bodies, thus leading to a change of the quality of the resultant ceramics. The granule compaction process of alumina is introduced to explain the effect of slurry preparation conditions on the properties of the resultant ceramics. Table 7.1 shows the data of slurry preparation conditions and its apparent viscosity.9,10 The density, fracture toughness, average fracture strength, and Weibull modulus are also listed in the same table.9,10 The table clearly shows that the properties of sintered ceramics change with slurry preparation conditions. Although a low soda alumina (AL160-SG4, Showadenko, Japan) was used, the strength of the sintered ceramics changes from 363 to 486 Mpa, depending on the slurry preparation conditions. The correlation between slurry preparation conditions and the properties of sintered ceramics is explained by the property change of spray-dried granules. Table 7.2 shows the structure and strength of spray-dried granules according to each set of slurry preparation conditions.9,10 The internal structure of the granules was observed by the liquid immersion method.11 The compressive strength was measured by a compressive strength tester for single granules,12 and the average strength and Weibull modulus were determined subsequently. As shown in Table 7.2, granules prepared from the slurry with higher apparent viscosity (#4) have a close-packed structure. They have a low average compressive strength compared to that obtained under conditions #1–3, but the Weibull modulus is relatively low. Therefore, the granules made by the conditions of #4 lead to a uniform green body by pressing, which leads to a higher average strength of the corresponding sintered body. However, the low Weibull modulus of the granules allows the green body to contain a low amount of extremely high compressive strength granules, working as a fracture origin in the sintered ceramics. As a result, this leads to higher average strength and a lower Weibull modulus of ceramics. On the other hand, the low-viscosity slurry leads to a dimple structure of the granules during spray-drying,10 and a high Weibull modulus of granules compared to that obtained under condition #4. The average compressive strength of the granules increases with the amount of dispersant (ammonium polyacrylate). As higher strength granules are harder to fracture during the compaction process,
TABLE 7.1 Ceramics No.
Slurry Preparation Conditions and Properties of the Sintered
pH
Dispersant Amount
Viscosity
Density
[mass%]
[mPa.s]
[kg/m3] 3
Fracture Toughness
Average Strength
[MPam1/2]
[MPa]
Weibull modulus
#1
10
0.2
43
3.91×10
3.7
486
20
#2
9.1
0.5
22
3.94×103
3.8
430
16
54
3.89×10
3
3.8
363
13
3.94×10
3
3.8
480
8
#3 #4
8.1 9.0
2.0 0.2
TABLE 7.2 Granules
125
Structure and Strength of Spray-Dried
No.
Internal structure
#1
Dimple structure
Average compressive strength [MPa] 0.31
Weibull modulus 10
#2
Dimple structure
0.91
6
#3
Dimple structure
5.4
10
#4
Closed packed structure
0.32
3
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it leads to lower fracture strength of the sintered ceramics. With a low-viscosity slurry, a higher Weibull modulus of ceramics is obtained because all granules have a dimple structure, leading to uniformly distributed pores in the sintered ceramics. As a result, the ceramics made under condition #1 reached the highest fracture strength and Weibull modulus of the four slurry conditions. When granules are made by spray-drying, the dispersant will work as solid bridges to increase the compressive strength of granules, as shown in Table 7.2.9 Therefore, slurry conditioning is a key issue to achieve a higher quality of products, and slurry should be processed according to its purposes.
REFERENCES 1. Tsubaki, J., Yamakawa, H., Mmori, T., and Mori, H., J. Ceram. Soc. Jpn., 110, 894–898, 2002. 2. Mahdjoub, H., Roy, P., Filiatre, C., Betrand, G., and Coddet, C., J. Euro. Ceram. Soc., 23, 1637–1648, 2003. 3. Tsubaki, J., Kuno, K., Inamine, I., and Miyazawa, M., J. Soc. Powder Technol.., 40, 432–437. 4. Kuno, K., Analysis of the Settling and Depositing Process in Dense Slurry, Master’s thesis, Nagoya University, Nagoya, Japan, 2002. 5. Tsubaki, J., Kim, H., Mori, T., Sugimoto, T., Mori, H., and Sasaki, N., J. Soc. Powder Technol., 40, 438–4432. 6. Ato, K., Analysis of Cake Forming Process During Constant Pressure Filtration, Bachelor thesis, Nagoya University, Nagoya, Japan, 2004. 7. Washburn, E. W., Phys. Rev., 17, 273–278, 1921. 8. Reed, J. S., Ed.; Principles of Ceramics Processing, John Wiley, New York, 1995, pp. 282–288. 9. Abe, H., Hotta, T., Kuroyama, T., Yasutomi, Y., Naito, M., Kamiya, H., and Uematsu, K., Ceram. Trans., 112, 809–814, 2001. 10. Abe, H., Naito, M., Hotta, T., Kamiya, H., and Uematsu, K., Powder Technol., 134, 58–64, 2003. 11. Uematsu, K., Powder Technol., 88, 291–298, 1996. 12. Naito, M., Nakahira, K., Hotta, T., Ito, A., Yokoyama, T., and Kamiya, H., Powder Technol., 95, 214– 219, 1998.
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5.8
Granulation Isao Sekiguchi Chuo University, Bunkyo-ku, Tokyo, Japan
5.8.1 GRANULATION MECHANISMS Outline of Granulation Techniques The heading “granulation” in this section is used as a main title to describe all forms of granules, small compacts, and small grains or prills. The granulation techniques in a variety of industrial fields are adopted to gain the advantages of solids handling, as listed in Table 8.1. There are two principal modes of granulation: granule growth modes and machine-made product modes. In the former, granulation is normally achieved by tumbling, agitating, or fluidizing raw materials in the presence of liquid binders, and the final products (i.e., agglomerates) have a wide distribution in granule sizes. The latter includes granulation processes in which materials are forced to flow in a plastic or sticky condition through dies or screens or in molds or other devices. Forming droplets from a fused material or suspension such as spray prilling or drying also belongs in the machine-made product mode.
Main Mechanisms of Bonding The important mechanisms of bonding between solid particles can be classified into four principal groups. Deformation and Breakage of Solid Particles In the compaction of powders, adjacent particles are pressed together so that if the particles have a plasticity, the compacting action leads to permanent bonding at the contact points.1 If the failure of brittle solid particles or dry agglomerates within a bed occurs during compaction,2,3,4 the powder compact is deformable and easy to handle. Heating of Solid Particles5 Hot powders heated to an appropriate high temperature just before the compaction bring about various active bonding forces among particles owing to the improved contact surface of particles due to alterable surface structure, desorption, chemical activation, gentle sintering, and so on. At further elevated temperature, immediately below the melting point of the constituent components, sintering or nodulizing causes the formation of hard agglomerated materials in a rotary kiln or hot fluidized bed. Addition of Binding Liquids The lower the activation energy to form droplet nuclei on the surface of fine particles, the gentler agglomeration formed by moistening the powder with a water vapor or other solvent occurs. Some amount of liquid binder is necessary to achieve some degree of plasticity of powders. The choice of liquid binder affects the properties of powders to be agglomerated. Especially the interparticle force among the particles created by the presence of nonviscous liquids shows a dependence on the value of scos u, where s is the surface tension of the liquid and u is the wetting contact angle on the particle surface.6–8 The interparticle force caused by highly viscous binding materials such as liquid 599 © 2006 by Taylor & Francis Group, LLC
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TABLE 8.1 Beneficial Effects of Granulation or Compaction To prepare particulate materials for more convenient solids handling or processing Nonsegregating blends: to achieve more uniform composition of mixture of solids and to prevent segregation Compaction: to ensure uniform filling and pressing in a die or mold Densification: to densify particulates for packing or feeding with-out dust losses or hazard Flowability: to reduce the tendency of particulates, generally hygroscopic, to form lumps, and also to improve the flow properties To make a product suitable for chemical reaction, heat, and/or mass transfer Solubility: to prevent the formation of undissolved lumps of fine powders in fluid phases Permeability: to achieve a more stable flow of fluids through fixed beds of particulates Release: to obtain controlled release of chemicals in granular materials Immobilization: to create a so-called carrier in which the activity of catalyst or biocatalyst may remain
paraffin, pitch, or asphalt 9–12 enhances the viscous and plastic deformation of the binding material that precedes the fracture. Addition of Foreign Powders A very small amount of solid lubricants such as talc, stearate, magnesium oxide, or paraffin wax is added to the compacting material to improve the strength of compact. The addition of special substances such as clay, bentonite, or other plasticizers to damp powders makes it possible to produce stronger agglomerates in a tumbling or agitating granulator.
Testing Methods for Evaluating the Agglomerative Granulation of Damp Powders Flocculent masses of damp powders prepared by mixing dry powder with water or other liquid binders have a marked tendency for agglomerative granulation under the tumbling, vibrating, or agitating action in the various devices. A new testing device to examine how the generation and growth of well-defined agglomerates are made from the damp powders has been developed. 13 Figure 8.1 shows a schematic diagram of a tapping-type gyratory agglomerator. This testing device consists basically of a shaking pan with a rubber bottom (disk diameter = 200 mm) tapped at the central part through a conic swing of the pendulum. The tapping action is induced by an eccentric circular motion of the rubber disk and by the subsequent circular swing of the pendulum rod when the lower front of the pendulum strikes the cylindrical obstacles located at the cyclic positions, as shown in Figure 8.2. Damp powders on the rubber disk accompanied by both the gyratory motion and the subsequent tapping action, as mentioned above, change from a flocculent mass to harder and denser agglomerates in a visible state. The agglomeration behavior of damp powder is subjected to the method of preparing the feed and is often brought into a powdery or pasty state. The quality of the agglomerate at various liquid contents Cd can be evaluated by an agglomeration index defined as lg = (Liquid binder content for preparing a damp powder)/(Liquid binder content at a plastic limit)
Cd [0 /(1 0 )]( r l / rs )
(8.1)
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FIGURE 8.1
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Schematic diagram of a tapping-type gyratory agglomerator.
where o is the void fraction in a close-packed bed of dry powder made by a tapping action. In addition, the plastic limit of the damp powder in Equation 8.1 can be determined by an agitating torque method in which the torque exhibits a distinct maximum of a liquid binder content corresponding to the plastic limit.14 An acceptable value of Ig for establishing stable agglomerates favorable to a tumbling or agitating granulation generally lies in the range of about 0.50 to 0.75. As an example, Figure 8.3 shows a relationship between agglomerate diameter Xv∞ and vibration intensity G for calcium carbonate (volume mean particle diameter = 10.2 μm, o = 0.476) methanol system. The agglomerate diameter in the figure is the final equilibrium value in the agglomerate growth. The growth of equal-sized agglomerates becomes excellent in both ranges 0.55 Ig 0.65 and 0 < G 38.4. It is apparent that the agglomerates in such a region become progressively larger in lower values of Ig with increasing vibration intensity. In this region, an important characteristic of agglomerate growth is such that the agglomerates suddenly grow to larger ones in the course of the agglomeration process under sufficient vibration intensity, as shown in Figure 8.4. This tendency is remarkable in lower values of Ig because of the combined action of breakage and coalescence. However, smaller agglomerates formed near the critical value of Ig 0.65 experience simple random coalescence in the nuclei region, yielding extremely different agglomerate sizes.
Granulation Due to Granule Growth Modes A drum granulator in batch operation is important in characterizing the agglomeration behavior of damp powders. The growth mechanism of granules in this operation is divided into the following three stages: (1) nucleation, nuclear growth, and coalescence of nuclei or small granules (nuclear growth region), (2) rapid coalescence of small granules (transition region), and (3) layering of fragments onto large granules, and compaction by tumbling (balling growth region). © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.2 A circular swing of the pendulum rod striking the cylindrical obstacles, which are located at the cyclic positions.
The growth rate of granules in the nuclear growth region is expressed in most cases as follows15: dXa kn Xa 3 dt
(8.2)
where Xa is the arithmetic mean diameter of granules on a mass basis and kn is the time-invariant rate constant of granule growth by coalescence. If a nuclear or smaller agglomerate assembly in the floclike mass of damp powders has a relatively narrow size distribution, a rate equation of granule growth in the first stage is derived from a random disappearance–coalescence model, to which Equation 8.2 is applicable. The size distribution of granule originating from a floclike mass of damp powders, as mentioned above, is formulated with the aid of statistical mechanics to become16 Rm ( X ) exp(BX 3 )
(8.3)
where Rm(X) is the cumulative mass fraction of granules coarser than diameter X, and B is a constant. In the case of a damp powder having a relatively narrow-sized floclike mass prepared by screening or kneading, the granule size data obey Equation 8.2 and Equation 8.3. The careful preparation of feed material is a prerequisite for a successful granulation in the early stage of tumbling granulation. © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.3 methanol.
603
Agglomeration criteria for calcium carbonate powder agglomerated with
The granule growth period in the transition region is very short, and the rate of granule growth obtained from the plot of dXa/dt versus t has a momentary maximum.17 Formation of large granules in the balling growth region is probable because the layering mechanism of fragments onto the growing granules is extremely dependent on the breakage of granules due to impaction, compaction, or abrasion. The rates of granule growth in the above three regions are given by the following18: 1. Nucleii growth region: dXa/dt ∝ Xa 2. Transitional growth region: dXa/dt ∝ 1/Xa 3. Ball growth region: dXa/dt ∝ 1/Xa 3 For an unstable granulation period, some generalized macroscopic-population balance models have been proposed. A majority of these models describe the self-preserving size spectra of granulated materials in batch tumbling granulation systems.17–24 The point of granulation in a hot fluidized bed with spray is that a liquid binder, suspension, or melt is atomized into the fluidized bed of the particles to be granulated. 25–30 Typical growth of granules formed by a batch fluidized-bed-spraying system is shown in Figure 8.5. The formation of granules at a time interval between A and B is very stable, but the subsequent granulation step forms larger granules or crumbs, so that the fluidized bed becomes extremely bubbly until defluidization occurs. Normal operation of the bed is carried out for the period A to B, in which it is a matter of course that the heat and mass balance must be satisfied. Further treatments of granulation in the granule growth mode, including pan-type granulators, are summarized in Table 8.2. Spouted beds © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.4
Effect of vibration intensities on the growth rate of agglomerates.
developed as the improved type of fluidized bed are often employed to produce granules or coating particles. Spray liquid is introduced as a spray into the conical base of the bed together with the hot spouting gas. This granulation method is usually suitable for the coating or layering particle process because of the rapid evaporation of spray liquid in the spouting gas stream at high temperature.
Machine-Made Product Modes The compaction of solid particles or small granules in a confined space can be carried out by punch, piston, roll, or other means of applying pressure. The degree of consolidation achieved during compacting press depends mainly on the stress–strain relation2 and on the physical and chemical properties of solid particles. The compaction mechanism of solid particles is divided into four stages as follows: (1) increase in the resistance due to the mutual friction of movable particles, (2) breakage of the interlocking among adjacent particles, (3) deformation and fracture of constituent particles, and (4) work-hardening of the particle bed. Such behavior seems to occur in the compaction of narrow solid particles. For a wide variety of solid particles, many empirical equations are proposed for the relationship between compacting pressure and void fraction.31 As many different machines are used for making the desired compacts, several important subjects should be described briefly here. The forming mechanism of a thick disk-shaped compact in a cylindrical die32,33 is expressed by FU ⎛ 4m kL ⎞ exp ⎜ w ⎟ ⎝ D ⎠ FL
(8.4)
where FU and FL are respectively the applied and transmitted forces, k is the ratio of radial to axial stress, m w is the coefficient of wall friction, and D and L are the diameter and thickness of a compact. © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.5 Rate of granule growth in a hot fluidized-bed spray granulator. Seed particles, sodium chloride(Xa = 0.000606 m); binder, gelatine solution(20.0 wt%). W = 6 kg; Fp = 0.00063 kg/s ; u = 0.703 m/s; air temperature = 353 K.
Further analysis reveals that Equation 8.4 has some serious drawbacks. Nevertheless, the expression is straightforward and simple to use. In roll compactions, except briquetting machines, particulate material with or without binders is fed into the compacting zone between two smooth rolls, as shown in Figure 8.6, and the compacted sheet can be cut to the desired granule size. The relationship between the void fraction of compacted sheet and the angle of nip un is given by cosun 1
DS ⎛ 1 p ⎞ 1 DB ⎜⎝ 1 n ⎟⎠
(8.5)
where p is the void fraction of compacted sheet and n is the void fraction of the material between two rolls at the nip angle un. The horizontal force FH exerted on the roll-bearing blocks is given by FH
pDR B un p cos du 360 ∫0
(8.6)
where B is the length of the active roll surface and p is the pressure exerted by the material undergoing compaction on a certain portion of the roll surface. To integrate Equation 8.6, it is necessary to measure the compaction pressure distribution and the void fraction along the circumference of the roll.34 FH in Equation 8.6 is derived from the theory of solids conveyance between two smooth rolls.35 © 2006 by Taylor & Francis Group, LLC
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TABLE 8.2 Typical Operating Data for Pan or Drum Granulators and Fluidized-Bed Spray Granulators Granulator type
Feed Conditions
Operating Conditions
Pan
1. Powder and spray droplets Nopt = (0.40–0.75)Nc 2. Continuous operation Nc=2538(sin/D)1/2 3. Ig=0.50–0.75 H = (0.10–0.25)D
Drum
1. Damp powders, or powder Nopt = (0.45–0.85)Nc and spray droplets h = 0–8 2. Batch or continuous operation 3. Ig = 0.50–0.75
Residence-time Distribution
1. Backmixing at a ring dam
2. Ideal pistol flow
Fluidized-bed 1. Powder and spray droplets umf < u0 < 0.5uf spray 2. Batch or continuous operation
Ideal perfect mixing flow
Pan
W/Fp = K(Fp/PpNdD3) x dV/dt= K1(Ve – V) Larger granules due to size(Ha1/2r/DeNd) (Ohabayashi, (Tohata et al., 1966. Rate classifying properties of 1972) of granule growth due to inclined pan can be separated in an elliptical orbit model on part at the discharge rim the bottom of pan (Macavi, 1965)
Drum
W/FP = (La1/2r/DhNd) x (FP/ρPNd) a(H/D)b (Sekiguchi et al., 1970)
Xa/Xa,0 = exp(knZd/3)* Xa= Xa,0 + ksZd +
Equation (8.2) Ref. 16 Granule size distribution taken both sizes of upper and lower limits into consideration (Capes and Danckwerts, 1965; Ref. 17)
Granule population balance, (Ref. 18, 20) Fluidized-bed W/FP = AfLf(1 - ef)FP spray Nv = f(u0/uf)
Xa/Xa,0 = exp(Ct)‡ Xa/Xa,0 = 1/(1 – Ct)§ (Ref, 40)
Mass balance in a batch‡ or continuous operation§
*
Granule growth by coalescence. Granule growth by layering (Umeya, K. and Sekiguchi, I., J. Soc. Mater. Sci. Jpn., 24, 664–668, 1975). ‡ Batch operation (Harada, K., Fujita, J., and Yoshimura, M., Kagaku Kogaku, 33, 793–799, 1969). § Continuous operation (Harada, K., Fujita, J., and Yoshimura, M., Kagaku Kogaku, 33, 793–799, 1969). †
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FIGURE 8.6 Schematic diagram of a roll compaction. [Kurtz, B. E. and Barduhn, A. J., Chem. Eng. Prog., 56, 67–72, 1960. With permission.]
In the application of screw-type extruders, the delivery rate of a screw rotating within the barrel should be exactly equal to the flow rate of material through the die. The delivery mechanism of the screw has some difficulties, but there is a theory of the screw conveyance of solids in clay extrusion. 36 On the other hand, in connection with extruding flow characteristics of carbon paste 37 or other solid–liquid mixtures,38 the pressure P required to extrude the material through a nozzle or multihole die is expressed as (V V0 ) K ( P P0 )
(8.7)
where V is the velocity of piston movement, Po and Vo are the ultimate limit of both P and V, and K is a constant related to the fluidity of the material through the dies and related to the structure of a nozzle or multihole die. Further useful granulation methods are outlined in the following section.
5.8.2 GRANULATORS Figure 8.7 shows the basic designs of various granulation equipments.
Tumbling Granulation Methods Typical inclined pan granulators consist mainly of an inclined disk with a rim and a granule size separating ability. Raw materials are fed continuously from above onto the nearly central part of the © 2006 by Taylor & Francis Group, LLC
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FIGURE 8.7
Typical granulation methods and equipment.
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Granulation
FIGURE 8.7
609
(continued)
disk, as shown in Figure 8.8, and larger granules formed are discharged beyond the rim. Binding liquid is sprayed onto incoming fine particles and smaller granules at a spraying zone on the disk. To form narrow granules, the Ig value defined by Equation 8.1 is adjusted in the range of about 0.5 to 0.75. Other operating conditions are given Table 8.2. A batch drum granulator is often available in laboratories experimenting with damp powders. In industrial practice, this type is used for the continuous granulation of damp powders within a slightly inclined rotating cylinder. If the equipment is made suitable for an ideal piston flow, experimental data for the batch drum granulator are applicable as is. Of course, it is necessary to have a good tumbling action of growing granules in the drum. Therefore, the inside wall of the drum is covered with a rugged rubber sheet or wire net and equipped with an adjustable or reciprocating scraper. When the granule product discharged has a wide size distribution, more than two rotating drums are used in a closed-circuit system. A rotating conical drum granulator has a granule size separating ability, and its best example is found in the study of simultaneous granulation and separation in a horizontal rotating vessel.39
Fluidized-Bed Spray Methods The basic method for making granules in a fluidized bed (i.e., the fluidized bed spray granulator) can be obtained by the combination of a hot gaseous fluidized bed and a two-fluid spray nozzle. A binding liquid or solution is sprayed onto a bed surface of fluidized particles or is sprayed directly into the bed. It tends to show that compared to seed particles, larger spray droplets generally result in the formation of agglomerates, which is influenced by a wide variation of operating conditions. In relation to granule growth of simultaneous coating and agglomeration in a hot fluidized bed, the size © 2006 by Taylor & Francis Group, LLC
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distribution and growth rate of granules under continuous operation can be derived from the material balance of the bed particles.40–43 The general principal in the spouted bed spray granulator is to spray solution,44–47 melt,48 or slurry49,50 onto seed particles that circulate in a bed spouted by hot gas. This type of granulator is used to granulate several materials of industrial interest.
Agitation Methods The mixing or blending of solid–liquid mixtures not only results in the dispersion of liquid in powders but is also suitable for the formation of small granules or agglomerates. A pugmill is generally used in the pretreatment for further granulating operation, but its application is limited to special fields such as fertilizer formulation or clay preparation. Other mixers, operated by means of pins, pegs, or blades, are utilized in a similar manner as the above. Granular products formed by those mixers tend to show a wide size distribution. The high-speed flow-type mixer is of growing importance in the formation of very small granules and is often used to coat coarse particles with a he-particle layer rather than coalesce with each other. The horizontal pan mixer consists mainly of a rotating mixing pan, a counterrotating blade, and a stationary scraper for cleaning the interior surface of the pan. Dry powders fed into the pan are wetted by adding a liquid binder so that granules are formed depending, on the moisture content and the rotational speed of the pan or the blades.
FIGURE 8.8
Flow of material in an inclined pan granulator.
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Forced Screening Methods A comminuting granulator consists of rotating vertical knives and a chamber, the lower part of which holds interchangeable screens. This machine is advantageous for the disintegration of dry clumps or cakes, or larger agglomerates. In contrast to this, the tornado mill, a type of screen granulator, is well suited for the disintegration of sticky agglomerates, in which moist material passes through a perforated screen by means of the high-speed rotor with a row of cutting knives. The oscillating granulator is used for processing large granulated materials, and the disintegration mechanism depends on a rubbing action created by a combination of a rotor bar and a wire screen with a suitable aperture.
Compaction Methods The roll compaction machine consists of two opposing counterrotating rolls and a feeding arrangement such as gravity feeder or tapered screw feeder. The product, compacted in sheet form, is granulated by a simple disintegrator or mill. The principal feature of a roll briquetting machine is similar to roll compaction, and the feed material, with or without a binding agent, is compacted to egg- or pillow-shaped briquettes, according to the pockets on the two rolls. The formation of a tablet on a single-punch machine is usually as follows. Granular material or powder is fed from a hopper in a die when the lower punch is at the descent position. The excess filling from hopper is scraped off by rotating the die table. Subsequently, the upper punch is lowered to press the powder, and then the compacted tablet is ejected when both punches are raised. The production capacity of this machine is about 100 tablets per minute. High-speed rotary tabulating machines can produce up to about 8000 tablets per minute.
Extrusion Methods The screw-type extruder used in the formation of small rodlike pellets consists of two main parts: a screw and a cylindrical or flat multihole die placed at the forward position of the barrel. Moist powder containing a binding liquid is introduced to the barrel through a hopper and kneaded sufficiently to form a plastic mass that is extruded through the multihole die. The die roll extrusion machine known as a pellet mill is also used extensively. In a typical pellet mill, the material fed into the interior of a large ring-shaped die roll is pressed by means of one or two smaller press rollers arranged eccentrically inside the die roller. At that time, adjustable knives on the exterior of the die roller cut the rodlike extrudates into pellets. Another machine of extruding granulation extrudes plastic or sticky material through a vertical perforated cylinder attached to the top hopper. The extruding force introduced by a multiarmed spreader decreases with the increased resistance of material flow through the holes of a perforated cylinder, so that the preparation of feed material becomes important.
Fusion Methods The prilling method yields a granular product that is essentially spherical and uniform in size. The principle is that of solidifying droplets of melt in air, water, or oil. In air prilling, for example, molten material is dispersed by spraying it from the top of the prilling tower, and the falling droplets solidify in the cooling air stream. 51,52 Other fusion methods include a spouted bed and a steel belt or drum flaker. It is necessary to have more detailed knowledge of granulation techniques in practice because so many different machines have been used to obtain the desired granules in a variety of industries. For more detailed information, readers are referred to Handbook of Granulation, published in 1991.53
Notation FL Fp Fu
Force transmitted through powder mass in cylindrical die (Pa) Mass flow rate (kg/s) Force applied to powder in cylindrical die (Pa)
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H K k kn ks L Lc Lf Nc Nd Nopt Nv P p Po Rm t uo ut umf V V Vo Ve W X Xa Xp Zd r f n p n
w p
Depth of pan, height of ring dam in drum (m) Constant concerning the fluidity of solid–liquid mixture through a nozzle (m 3 /N s) Ratio of radial to axial stress Rate constant of granule growth by coalescence (s–1) Rate constant of granule growth by layering (kg/m2 s) Thickness of powder mass in cylindrical die (m) Height of fixed bed (m) Height of fluidized bed (m) Critical rotating speed of drum or pan (s–1) Rotating speed of drum or pan (s–1) Optimum rotating speed of drum or pan (s–1) Expansion ratio ( = L/Lc) Pressure exerted by the material undergoing compaction certain portion of roll surface (Pa) Compacting or extruding pressure (Pa) Ultimate limit of extruding pressure (Pa) Cumulative mass fraction of granules coarser than a stated size Time (s) _ Superficial gas velocity in a fluidized bed (m/s) Terminal velocity of a particle in upward direction (m/s) Minimum fluidization velocity (m/s) Volume of a granule (m3) Velocity of piston movement (m/s) Extruding piston velocity at the ultimate pressure (m/s) Ultimate volume of a granule (m3) Holdup (kg) Diameter of granule (m) Arithmetic mean diameter of granules given by mass fraction in a size distribution (m) Diameter of solid particle (m) Cumulative drum revolution Angle of repose (deg) Void fraction Void fraction of fluidized bed Void fraction of the material between the rolls at the angle of nip Void fraction of the product when there is no plastic deformation during the compaction process Angle of inclination to the horizontal (deg) Angle of inclined pan (deg) Angle of nip (deg) Coefficient of wall friction Density of particle (kg/m3)
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Dahneke, B. J., Colloid lnterface Sci., 40, 1–13, 1972. Umeya, K. and Sekiguchi, I., J. Soc. Mater. Sci. Jpn., 24, 608–612, 1975. Takahashi, M., Kobayashi, T., and Suzuki, S., J. Soc. Mater. Sci. Jpn., 32, 953–957, 1983. Mort, P. R., Sabia, R., Niesz, D. E., and Riman, R. E., Powder Technol., 79, 111–119, 1994. Sekiguchi, I., Kagaku Kogaku, 32, 745–747, 1968. Newitt, D. M. and Conway-Jones, J. M., Trans. Inst. Chem. Eng., 36, 422–441, 1958. Rumpf, H., in Agglomeration, Knepper, W. A., Ed., John Wiley, New York, 1962, pp. 379–418. Pietsch, W. E., Hoffman, E., and Rumpf, H., Ind. Eng. Chem. Prod. Res., 8, 58–62, 1969.
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Granulation 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53.
613
Umeya, K. and Sekiguchi, I., Kagaku Kogaku, 37, 704–712, 1973. Sekiguchi, I. and Umeya, K., Kagaku Kogaku, 37, 744–747, 1973. Mazzone, D. N., Tardos, G. I., and Pfeffer, R., J. Colloid Interface Sci., 113, 544–556, 1986. Mazzone, D. N., Tardos, G. I., and Pfeffer, R., Powder Technol., 51, 71–83, 1987. Sekiguchi, I. and Ohta, Y., J. Powder Technol. Jpn., 28, 232–238, 1991. Michaels, A. S. and Puzinaskas, V., Chem. Eng. Prog., 59, 604–614, 1954. Umeya, K. and Sekiguchi, I., J. Soc. Mater. Sci. Jpn., 24, 664–668, 1975. Sekiguchi, I. and Tohata, H., Kagaku Kogaku, 32, 1012–1020, 1968. Kapur, P. C. and Fuerstenau, D. M., Ind. Eng. Chem. Process. Des. Dev., 8, 56–62, 1966. Ouchiyama, N. and Tanaka, T., Ind. Eng. Chem. Process. Des. Dev., 13, 383–389, 1974. Kapur, P. C. and Fuerstenau, D. M., Trans. AIME, 229, 348–355, 1964. Kapur, P. C., Chem. Eng. Sci., 27, 1863–1869, 1972. Ouchiyama, N. and Tanaka, T., Ind. Eng. Chem. Process. Des. Dev., 14, 286–289, 1975. Sastry, K. V. S. and Fuerstenau, D. M., Trans. AIME, 250, 64–67, 1971. Sastry, K. V. S. and Fuerstenau, D. M., Trans. AIME, 252, 254–258, 1972. Sastry, K. V. S., Int. J. Mining Process., 2, 187–203, 1975. Dencs, B. and Ormos, Z., Powder Technol., 31, 85–91, 1982. Dencs, B. and Ormos, Z. Powder Technol., 31, 93–99, 1982. Smith, P. G. and Nienow, A. W., Chem. Eng. Sci., 38, 1223–1231, 1983. Smith, P. G. and Nienow, A. W., Chem. Eng. Sci., 38, 1233–1240, 1983. Huang, C. C. and Kono, H. O., Powder Technol., 55, 35–49, 1988. Ennis, B. J., Tardos, G., and Pfeffer, R., Powder Technol., 65, 257–272, 1991. Kawakita, K. and Ludde, K H., Powder Technol., 4, 57–60, 1971. Spencer, R. S. and Gilmore, G. D., J. Appl. Phys., 21, 527–531, 1950. Train, D. and Twis, D., J. Trams. Inst. Chem, Eng., 35, 258–266, 1957. Kurtz, B. E. and Barduhn, A. J., Chem. Eng. Prog., 56, 67–72, 1960. Murakami, K., Kobunshi Kagaku, 17, 571–576, 1960. Parks, J. R. and Hill, M. J., J. Am. Ceram. Soc., 42, 1–6, 1959. Jimbo, G., Iwamoto, F., Hosoya, M., Sugiyama, Y., and Moro, Y., Kagaku Kogaku, 36, 654–660, 1972. Jimbo, G. and Kambe, T., J. Soc. Powder Technol. Jpn., 9, 451–457, 1972. Sugimoto, M. and Kawakami, T., Kagaku Kogaku Ronbunshu, 8, 530–532, 1982. Harada, K. and Fujita, J., Kagaku Kogaku, 31, 790–794, 1967. Harada, K., Fujita, J., and Yoshimura, M., Kagaku Kogaku, 33, 793–799, 1969. Harada, K., Fujita, J., and Yoshimura, M., Kagaku Kogaku, 34, 102–104, 1970. Harada, K., Kagaku Kogaku, 36, 1237–1243, 1972. Uemaki, O. and Mathur, K. B., Ind. Eng. Chem. Process. Des. Dev., 15, 504–508, 1976. Singiser, R. E., Heiser, A. L., and Prillig, E. B., Chem. Eng. Prog., 62, 107–111, 1966. Robinson, T. and Waldie, B., Trans. Inst. Chem. Eng., 57, 121–127, 1979. Kurcharski, J. and Kmiec, A., Can. J. Chem. Eng., 61, 435–439, 1983. Weiss, P. J. and Meisen, A., Can. J. Chem. Eng., 61, 440–447, 1983. Pavarini, P. J. and Coury, J. R., Powder Technol., 53, 97–103, 1987. Liu, L. X. and Litster, J. D., Powder Technol., 74, 215–230, 1993. Tohata, H. and Sekiguchi, I., Kagaku Kogaku, 26, 818–825, 1962. Tohata, H., Sekiguchi, I., and Suzuki, H., Kagaku Kogaku, 32, 5–55, 1968. Association of Powder Process Industry and Engineering, Zouryuu Handbook, Ohmsha, Tokyo, 1991. [In Japanese.]
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5.9
Kneading and Plastic Forming Minoru Takahashi Nagoya Institute of Technology, Tajimi, Aichi, Japan
The term “kneading” is sometimes distinguished from “mixing” to emphasize compounding of powder with liquid having a high viscosity, or coating of a powder surface with liquid. In industrial applications, kneading is usually combined with the plastic forming processes. In this chapter, the fundamentals of kneading and plastic forming processes are described.
5.9.1
KNEADING
Powder–Gas–Liquid Dispersion System The packing conditions of powder–gas–liquid mixture systems are ideally classified in Table 9.1. 1 The systems are distinguished by the differences in the existence and continuity of each phase. Kneading can be defined as the mixing operation in the mud system. Kneading of either film type or matrix type is employed.2 Film-type kneading is the thin coating of the surface of each particle with liquid in the funicular state, whereas matrix-type kneading is the coating of the surface of each particle with a large amount of liquid or the dispersing of particles into the liquid in the capillary state.
Critical Powder Volume Concentration Compounds for plastic forming in a narrow sense should be prepared to a capillary state where the voids among particles are saturated with liquid. The powder concentration corresponding to a capillary state is called a critical powder volume concentration (CPVC). CPVC is very dependent on the size, size distribution, and shape of particles and can be estimated experimentally from the mixing torque versus liquid content curve using a torque rheometer.3 With the addition of liquid, torque gradually increases, reaches a maximum, and then decreases sharply, as shown in Figure 9.1.1,4 CPVC is determined by the powder concentration at the maximum torque. The figure also shows that the condition of the powder–gas–liquid system shifts as follows: powder → (powder ⫹ granules) → granules → (granules ⫹ paste) → paste → slurry.
Mechanisms and Machines Kneading is attained by the mechanisms of convection, shear, and diffusion, as is dry or slurry mixing. However, the convection and shear mechanisms are relatively important in the kneading process because the mixture systems, having a quite high viscosity, are dealt. Therefore, kneading machines capable of producing high shear force must be used to coat the particles with liquid. Also, kneading machines with high shear speed will be required to shorten kneading time. Representative kneaders and their applicabilities are listed in Table 9.2.2 Dispersion of particles into the matrix becomes more critical with decreasing particle size and increasing powder concentration. Special kneading with the assistance of high temperature or vacuum will be useful for spreading the liquid on the particle surface or removal of trapped air. Another route for attaining sufficient dispersion is surface modification of the particles, which improves the wetting between the particles and the organic medium and promotes penetration of the viscous fluid into the particle agglomerates. 615 © 2006 by Taylor & Francis Group, LLC
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TABLE 9.1
Packing Characteristics of a Powder–Air–Water System Water content
Particle (Solid phase)
Continuous
Cont
Cont
Discont
Discont
Discont
Discont
Air (Gaseous phase)
Continuous
Cont
Discont
—
—
—
—
Discontinuous
Cont
Cont
Cont Discont
Cont Cont
DisCont Cont
Discont Cont
Pendular S, P
Funicular S. F-1 F-2
Capillar S. C-1 C-2
Slurry S. S-1 S-2
c-2
c-4
c-6
Water full (Liquid phase I) Free water (Liquid phase II) Stage Critical point.
c-1
c-3
P.L. (Plastic Limit) Rheological System Mix
Power System Dry mix
c-5
L. L. (Liquids Limit) Mud System
Semi-Dry (or Semi-Wet) mix
Wet mix
Slurry System Slurry mix
Source: Umeya, K., J. Soc. Rheol. Jpn., 13, 145–166, 1985.
FIGURE 9.1 Pelletizing effect in powder–air–water system. [From Umeya, K., J. Soc. Rheol. Jpn., 13, 145–166, 1985. With permission.]
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TABLE 9.2 Types and Factors of Kneaders Material Factor Type
Wheel
Name
Sketch
Particle size a
Kneader Factor
b
Binder Viscosity content
c
Power (kw)
Kneading Kneading Force d Speed d
Simpson
M–C
FI–C
H
0.75–150
4
2
Wetpan
M–C
FI–FII
H
5.8–15
4
2
Counter Flow
M–C
FI–FII
M–H
7.5–11
3
2
M–C
FI–FII
M–H
15–75
4
4
F–C
FI–S
L–H
3.7–37
5
1–2
F
S
L–M
0.2–37
5
1–2
M–C
P–FII
M–C
2.2–37
1
3
F–M
FI–C
H
0.25–1500
5
2
F
FII–C
H
0.75–19
5
2
M–F
P–FI
L
3.7–95
2
5
M–F
P–FI
L
1.8–200
2
5
F–C
FI–C
M–H
0.2–15
2
2
F
C–S
H
0.4–200
5
2
Taper Roll
F
C–S
H
2.2–200
5
2
Banbury
F
FII–C
H
5.6–1900
5
2
Continuous
F
FII–C
M–H
0.75–300
5
2
F–C
P–C
L–H
0.1–2.3
3
2
Wheel
Speed muller Ball mill
Ball
Ball
Sand Grinder Paddie
Beads Blade Blade
Kneader Auger Blade
vac
Blade
Turbulizer Henschal
Propeller
Vertical
Blade
Roll Mill Roller
Roll
Roller
Other
Pestle
Screw Pestle
F, Fine <50 μm; M, middle 50–500 μm; C, course >500 μm. Refer to Table 8.1. c L, Low <10 mPa.s; M, middle 102–103 mPa.s; H, high >104 mPa.s. d Estimated value, 5–1; highest kneading force and speed, f. Source: Hashimoto, K., in Powder Technology Handbook, Iinoya, K., Gotoh, K., and Higashitani, K., Eds., Marcel Dekker, New York, 1990, p. 653. a
b
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PLASTIC FORMING
Plastic forming methods among a variety of forming methods include wet pressing, jiggering, extrusion, and injection molding. Plastic bodies prepared by kneading processes are treated in these forming methods.
Wet Pressing Wet pressing is similar to dry pressing and is often called semidry pressing, or dust pressing. The forming cycle consists of three stages: (a) filling raw powders into the die cavity, (b) compaction, and (c) ejection of the pressed body, as shown in Figure 9.2. Usually, feed material containing 5–10% water is pressed in dies made of steel, cast iron, or tungsten carbide. The packing state of the feed mixture is in the range of funicular I to funicular II in Table 9.1. The feed is occasionally prepared by crushing the filtered cake of slurry without the use of a kneader. Compared with the dry feed, the wet feed deforms plastically during compression. Therefore, it is possible to fabricate parts with complex contours by this method. However, dimensional tolerances of wet pressing are only held to ⫾2%, lower than those of dry pressing.5 The pressed body after ejection from the die must be treated carefully because it will deform easily from the force of gravity.
Jiggering Jiggering is one of the oldest plastic forming techniques, where a skilled hand-worker can make any hollowware from plastic clay supported on a rotating wheel. Presently, highly automated jiggering systems are used for the industrial fabrication of hollowware.6 A schematic of a jiggering machine, or roller machine. The jiggering process consists of three stages: (1) putting a slice of plastic feed onto a spinning plaster mold, (2) pressing the slice by a hot performance die, and (3) jiggering. A slice of de-aired extrudate is commonly used for feed material. Sticking of the plastic feed to the die is a serious problem. To prevent this, lubrication is produced by heating metal roller tools in order to generate steam and facilitate release, or a lubricant is sprayed onto a tool to minimize sticking.5
Extrusion A compound adjusted to a capillary state will show a plastic flow necessary for extrusion. If the powder concentration is too low, a squeezing of liquid from the mixture will occur during extrusion at high pressure. On the other hand, too high a concentration will bring about the breakage of the
FIGURE 9.2 component.
Sequence of wet pressing for a ceramic
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extruding machine because of insufficient plasticity. Granulated feed will be fully consolidated in a compression zone of the extruding machine. In the nonclay system, organic additives are added to promote plasticity. Extrusion is usually carried out by an auger having mixing equipment. In the mixing before extrusion, feed material is shredded and de-aired under vacuum. Major stages of extrusion are (1) feeding, (2) compression in a barrel, (3) flow through a die, and (4) ejection. The exit, with a uniform section, is supported on a suitable surface to prevent distortion and is cut to proper lengths. The very complex parts such as honeycomb structures are fabricated by the use of a specially designed die.7 Sometimes, the columnar exit can be used as blanks for further machining. The plastic mixture after compression can be characterized as a Bingham body, which shows two flow modes in a tube: one is laminar flow and the other is plug flow. The laminar flow occurs in the outside portion of the tube, where the shear stress is greater than the yield stress of the body; the laminar flow occurs in the inside portion. The laminar flow causes orientation of particles along its axis if platelike particles such as clay particles are involved in the body. However, the inside portion has the same random orientation as the original feed.8 Inevitably, this nonuniform packing structure of the exit results in a differential shrinkage during drying and sintering. The cracks arising from the differential shrinkage often become serious problems in extrusion forming.
Injection Molding Raw powders must be mixed with an organic vehicle consisting of a binder, lubricant, plasticizer, and dispersing or wetting agent. Thermoplastic or thermosetting resins are commonly used as the binders. The thermoplastic resins are preferable for recycling of a large amount of scrap left in the sprue, runner, and gate of a molding die set. High powder concentration in the compound is generally desired in the case of ceramic injection moldings because the organic vehicle should be removed before sintering. However, too high a concentration over the CPVC results in inadequate fluidity of the compound during mixing and molding. A twin screw extruder and a kneader with pressurizing equipment have been verified to be effective for mixing heavily loaded fine fillers and viscous organic systems.9 It should be noted that thermal or mechanical degradation of thermoplastic resins will sometimes occur during mixing.10 The molding processes consist of four stages: (1) preheating of feed materials within an injection machine, (2) injection of the plasticized feed into the cavity, (3) solidification of the cast body by heating and/or cooling, and (4) ejection of the body. Parts accompanied by thermoplastic or thermoset resins can be made by a screw injection machine. The injected body should be free of defects such as weld lines, pores, and cracks.
REFERENCES 1. Umeya, K., J. Soc. Rheol. Jpn., 13, 145–166, 1985. 2. Hashimoto, K., in Powder Technology Handbook, Iinoya, K., Gotoh, K., and Higashitani, K., Eds., Marcel Dekker, New York, 1990, pp. 653. 3. Markhoff, C. J., Mutsuddy, B. C., and Lennon, J. W., in Forming of Ceramics, Mangels, J. A. and Messing, G. L., Eds., American Ceramic Society, Ohio, 1984, pp. 246–250. 4. Michales, A. S. and Puzinauskas, V., Chem. Eng. Prog., 50, 604–614, 1954. 5. Reed, R. S., Introduction to the Principles of Ceramic Processing. John Wiley, New York, 1988, pp. 355–379. 6. Gould, R. I. and Lux, J., in Ceramic Fabrication Processes, Kingery, W. D., Ed., MIT Press, Cambridge, 1958, pp. 98–107. 7. Richerson, D. W., in Modern Ceramic Engineering, Marcel Dekker, New York, 1982, pp. 178–216. 8. Norton, F. H., in Fine Ceramics, McGraw-Hill, New York, 1970, pp. 130–156. 9. Edirisinghe, M. J. and Evans, J. R. G., Mater. Sci. Eng., A109, 17–26, 1989. 10. Takahashi, M., Suzuki, S., Nitanda, H., and Arai, E., J. Am. Ceram. Soc., 71, 1093–1099, 1988. © 2006 by Taylor & Francis Group, LLC
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5.10
Drying Hironobu Imakoma Kobe University, Nada-ku, Kobe, Japan
Morio Okazaki* Kyoto University, Kyoto, Japan
5.10.1
DRYING CHARACTERISTICS OF WET PARTICULATE AND POWDERED MATERIALS
Water Content and Drying Characteristic Curve Drying is the removal of relatively small amounts of water (including all states of water molecules, e.g., liquid, adsorbed, bounded, and gaseous) from solids by a hot gas stream or airstream, or other heating sources. To express the amount of water contained, one uses two types of (mean) water content: dry-basis water content w (kg water per kg dry material) and wet-basis water content w′ (kg water per kg wet material). The former is much more convenient for calculation than the latter. w⫽
w⬘ 1 − w⬘
(10.1)
w 1⫹ w
(10.2)
w⬘⫽
Suppose that a wet nonhygroscopic porous material is being set in a hot airstream of constant temperature, humidity, and velocity. The water content w and the material temperature tm change with time during drying, as shown in Figure 10.1. The drying process consists of the following three periods: (I) preheating period, (II) constant drying-rate period, and (III) decreasing dryingrate period. Figure 10.2 shows the drying rate R in relation to the water content w. This curve is called the drying characteristic curve. During period II, free water exists on the surface of the material, and the material temperature tm remains constant and is equal to the wet-bulb temperature of the hot air, tw. All of the heat supplied to the material is consumed in water evaporation. Hence, the drying rate becomes constant. This period continues until the water content becomes the critical value wc, which decreases with reduction of the material size. During period III, there is no free water on the surface. Because the evaporation surface proceeds to the inside of the material and diffusion of water vapor in the material takes place, the drying rate decreases with time until finally drying ends at the equilibrium water content we. Hot-air heat is consumed for heating up the material temperature as well as for water evaporation, and the temperature rises from tw until it finally reaches the hot-air temperature. In the case of other materials, except for a nonhygroscopic porous one, one can also observe more or less similar drying characteristic curves. However, shapes of curves in period III are quite different from one another and also from the shape of the nonhygroscopic case. *Retired.
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FIGURE 10.1 Three drying periods.
FIGURE 10.2 Drying characteristic curve.
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Drying Rate Because water evaporates on the material surface during the constant drying-rate period, the drying rate Rc is apparently equal to the evaporation rate from the free water surface: Rc ⫽⫺
h( H m ⫺ H ) W dw ⫽ k( Hm ⫺ H ) ⬇ A du CH
(10.3)
When the heat is supplied from the hot air only, the material temperature tm is equal to the wetbulb temperature of the air tw. Hence, Rc ⫽
h(t ⫺ t w ) rw
(10.4)
On the other hand, when the heat is supplied by convection, conduction, and/or radiation, the constant drying rate can be estimated by considering that the heat input to the material surface is consumed only for the evaporation of water. The material temperature sometimes deviates from the wet-bulb temperature in the case of other liquids, although a constant material temperature and drying rate are observed.1–3 The decreasing drying rate Rd is strongly affected by the material properties and drying method.3–5 This rate, Rd, is given as a function of water content w. The drying time ud during the decreasing drying rate period can be estimated by ud ⫽ ∫
5.10.2
w2 w1
w2 ⎛ dw du ⎞ ⫽ ∫ ⎜⫺ ⎟ dw w 1 ⎝ Rd dw ⎠
(10.5)
DRYER SELECTION AND DESIGN
Many types of dryers have been developed and operated, as listed in Table 10.1.4 First, dryers should be selected based on shape, size, water content of initial and final target wet material, the production rate, and the mode of drying operation. Then, the properties of the material (e.g., stickiness) are taken into consideration, and the selection of dryer type is made. Finally, the volume of the dryer selected should be roughly estimated. In the selection of dryer type, it is very important to estimate dryer volume. For dryers receiving convective heat from hot air, the rate of heat transfer q is given approximately by Batch operation: q ⫽ haV (t ⫺ tm )
(10.6)
q ⫽ haV (t ⫺ tm )lm
(10.7)
q ⫽ U k Ak (t k ⫺ tm )
(10.8)
Continuous operation:
For those of conductive-heating type:
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TABLE 10.1
Rough Estimation of Dryer Capacity4
(a) Convective-heating dryer (Batch mode) Type of dryer
ha[kJ/(h.K.m3-dryer volume)]
Critical water content.[%]
(t - tm) [K]
Inlet temperature of hot air[⬚C]
B o x ( p a r a l l e l - fl i o w )
800-1300 (h= 80-130)
>20
30-100a
100-150
Box(through-fliow)
13000-33000 (granules; layer 0.1- 0.15m)
thickness
a,b
100-150
>20
50
(b) Convective-heating dryer (Continuous mode) Type of dryer ha[kJ/(h.K.m3-dryer volume)]
Critical water content.[%]
(t - tm)lm [K]
Rotary
2-3
Countercurrent 80-150
200-600
Cocurrent 100-180
300-600
4000-13000 (mudlike, extruded pieces; layer thickness 0.1- 0.15m)
400-800
Inlet temperature of hot air [⬚C]
2-3
80-100
200-350
13000-25000
1-2
100- 180
400-600
Fluidized bed
8000-25000
2-3
Horizontal (single and 100-600 multicompartment) 50-150 Multistage (countercurrent) 200-350 80-100
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R o t a r y ( t h r o u g h - fl o w ) 1 3 0 0 - 6 0 0 0 Pneumatic conveying
Drying
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Spray
8 0 ( l a r g e p a r t i c l e ) - 4 2 0 ( fi n e p o w d e r )
T h r o u g h - fl o w v e r t i c a l
20000-54000
Tunnel(parallel-flow)
800-1300 (h= 80-130)
B a n d ( p a r a l l e l - fl o w )
170-330
B a n d ( t h r o u g h - fl o w )
3000-8000
30-50
(c) Conductive-heating dryer Type of dryer Uk [kJ/(h.K.m2-area contacting with material)] Agitated cylinder or rotary with steam-heated tubes
250-540d (small for very sticky materials)
2–3 ⬍20 ⬍20 2–3
Countercurrent 80-90
200-300
Coccurent 70-170
200-450
Countercurrent 100-150
200-300
Countercurrent 30-60 a,c
100-200
Cocurrent 50-7 0a,c
100-200
S a m e a s t u n n e l ( p a r a l l e l - 100-200 flow) a,c 4 0 - 6 0 a,b
Critical water content [%]
tk- tm [K]
2-5
50-100a
100-200
a
Values for the surface evaporation periods, and maximum drying rates will be obtained. Mean temperature difference between the hot air passing through the layer and the material temperature during the surface evaporation period. c Mean values at both ends of dryers for case a. d No change in the case of vacuum. b
Source: Toei, R., Kanso Souchi, Nikkan Kogyo Shinbun, Tokyo, 1966, pp. 9–11.
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conductive-heating type increases as drying proceeds, the average of the initial and final values is available to roughly estimate the dryer volume. One can increase the volumetric heat transfer coefficient and prolong the surface evaporation period by lowering the critical water content in order to minimize dryer volume. To do this, agitation or dispersion of particulate materials in hot air is very effective. The method of making a rough estimation of dryer volume introduced above can be useful only for the selection of dryer type suitable for the target material. Therefore, one should make a more accurate estimation considering static and transport properties within the material after a proper dryer selection by using the rough method. Some books can assist in accurate estimations.3,5,6 Several problems are solved for difficult materials (e.g., poor fluidizability) in fluidized-bed drying.7 A guideline for powder manufacture by spray-drying is proposed.8 Recently, some of other types of dryer (not listed in Table 10.1) are being used, for example, spouted bed, vibrated fluidized bed, and so on. 5,6,9,10 The spouted-bed dryer is essentially a modified fluidize done for larger particles. Whereas a single type of dryer is usually selected for a target material, the application of combinations of different types of dryer for the material will be more popular. The combination of a pneumatic conveying dryer and a succeeding fluidized-bed dryer allows a remarkable volume reduction of the fluidized-bed dryer in comparison with an independent use of another fluidized-bed dryer, because first removing the surface water of the particles in the pneumatic dryer prevents them from sticking to one another or to the dryer wall, and subsequent recycling of dried material becomes unnecessary. It is noted that one should apply the appropriate combination of dryers for the target material.11 Superheated steam is sometimes utilized as a heating medium instead of heated air and may have potential in the next decade.9,10 Advantages of superheated steam drying are reduced net energy consumption, no oxidative or combustion reactions, and higher drying rates, accompanied by the limitations of a more complex system. Highly moist slurries are sometimes predesaturated by electro-osmotic means for the purpose of reducing energy for the subsequent drying.9,10,12
Notation A Ak CH H Hm h ha k q R rw t tk tm tw Uk V W w w′ u
Drying area of material [m2] Heating area of material by conduction [m2] Humid heat of air [kJ/(kg-dry air K)] Humidity [kg-water/kg-dry air] Saturated humidity at tm [kg-water/kg-dry air] Film heat transfer coefficient [kJ/(m2.h.K)] Volumetric heat transfer coefficient [kJ/(m3.h.K)] Film mass transfer coeffiicient [kg/(m 2. h .ΔH)] Rate of heat transfer [kJ/h] Drying rate [kg-water/(h.m2)] Heat of evaporation of water at tw [kJ/kg] Hot-air temperature [˚C] Temperature of heat source [˚C] Material temperature [˚C] Wet-bulb temperature [˚C] Overall heat transfer coefficient [kJ/(h.K. m 2 -heating area by conduction)] Volume of dryer [m3] Mass of dry material [kg] Dry-basis water content [kg-water/kg-dry material] Wet-basis water content [kg-water/kg-wet material] Drying time [h]
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REFERENCES 1. Keey, R. B. and Suzuki, M., Int. J. Heat Mass Transfer, 17, 1455–1464, 1974. 2. Suzuki, M., in Drying ’80, Vol. 1, Mujumdar, A. S., Ed. Hemisphere, New York, 1980, pp. 116–127. 3. Suzuki, M., Imakoma, H., and Kawai, S., in Kagaku Kogaku Binran, 6th Ed., Maruzen, Tokyo, 1999, pp. 735–787. 4. Toei, R., Kanso Souchi, in Nikkan Kogyo Shinbun, Tokyo, 1966, pp. 9–11. 5. Keey, R. B., Drying of Loose and Particulate Materials, Hemisphere, New York, 1992. 6. Mujumdar, A. S., Ed., Handbook of Industrial Drying, 2nd Ed., Marcel Dekker, New York, 1995. 7. Filka, P. and Filkova, I., in Proceedings of the 11th International Drying Symposium, Halkidiki, 1998, pp. 1–10. 8. Masters, K., in Proceedings of the 13th International Drying Symposium, Beijing, 2002, pp. 19–27. 9. Mujumdar, A. S., in Proceedings of the 13th International Drying Symposium, Beijing, 2002, pp. 1–18. 10. Kudra, T. and Mujumdar, A. S., Advanced Drying Technology, Marcel Dekker, New York, 2002. 11. Mujumdar, A. S., Drying Technol., 9, 325–347, 1991. 12. Yoshida, H. and Yukawa, H., in Advances in Drying, Vol. 5, Mujumdar, A. S., Ed., Hemisphere, Washington, DC, 1992, pp. 301–323.
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5.11
Combustion Hisao Makino and Hirofumi Tsuji Central Research Institute of Electric Power Industry, Yokosuka, Kanagawa, Japan
5.11.1
INTRODUCTION
As shown in Figure 11.1, 1 combustion equipment for solid fuels is classified into the fixed-bed type, the fluidized-bed type, and the entrained-bed type, according to slip velocity between gas and particles. The fixed bed uses extremely large solid fuels of more than 10mm. The actual combustion equipment for particles is mainly the fluidized-bed type and the entrained-bed type. The bubbling bed, in which the gas velocity is 1–2 m/s, and the circulating bed, in which the gas velocity (4–8 m/s) is faster than that in the bubbling bed, were developed for the fluidized-bed type. In fluidized-bed combustion, the thermal NOx, which is generated from N2 and O2 in the combustion air, becomes lower because the combustion temperature is low. Pulverized coal combustion is one of the typical entrained-bed combustion materials. In this system, the pulverized coal particles of the order of 40 μm mean diameter are blown from the burner into the furnace to make the flame. Pulverized coal combustion is the most common method, because the combustion efficiency is higher than in the other systems and it is easy to scale up. In this section, the pulverized coal combustion system is reviewed as an example of combustion of solid fuels.
5.11.2
CONTROL OF THE COMBUSTION PROCESS
In order to utilize the system effectively, it is very important to achieve high combustion efficiency in the operation of combustion systems. In many cases, as the combustion efficiency is increased, concentration of fuel NOx increases, which is generated from the nitrogen in the coal and the O2 in the combustion air. It is, therefore, important to develop combustion technology, in which the formation of NOx is suppressed while improving combustion efficiency. Figure 11.2 2 shows the flame structure for this combustion technology. Figure 11.3 3 represents combustion profiles of the conventional method and the advanced method for low NOx combustion. In the conventional method, the combustion reaction near the burner is suppressed. In the advanced method, the combustion reaction at the low air ratio region near the burner is accelerated, and the NOx reduction region is formed effectively downstream of this combustion acceleration region. It is shown that the NOx emissions from the advanced method is lower than that from the conventional method. In order to achieve the further improvement of combustion efficiency and to make the NOx reduction flame wider, the new technology, in which the coal particles are pulverized into fine particles in the order of 10 μm mean diameter, was developed.
5.11.3
COMBUSTION BURNER
Figure 11.44 shows the structure of the advanced low-NOx burner for pulverized coal combustion. This burner was developed for the advanced method shown in the previous section. The
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FIGURE 11.1 Combustion equipment for solid fuels. [From Miyasaka, T., Thermal Nucl. Power Jpn., 32, 1056–1060, 1981. With permission.]
pulverized coal particles and the primary air are blown into the furnace from the primary air port. The amount of the primary air is about one fifth of the amount of the total air, which is approximately 1.2 times that of the theoretical air. The secondary air and the tertiary air are provided into the furnace from the outer region of the primary air port. A part of the total air is also provided from the furnace as two-stage combustion air, to suppress the formation of NOx near the burner. To accelerate the combustion reaction and to make a better NO x reduction flame, it is necessary that the residence time of the pulverized coal particles near the burner be lengthened by the recirculation flow. To achieve this concept, the flow rate and the swirl force of the secondary air and the tertiary air were optimized. Figure 11.54 shows the relationship between NOx emission and swirl vane angle of the secondary air when using this advanced burner and a conventional burner. For these two burners, NOx emission changes greatly with the swirl vane angle. NOx emission from the advanced burner is lower than from that of the conventional burner. Figure 11.64 shows relationship between NO x emission and the ratio of the secondary air flow rate to the sum of the secondary and the tertiary air flow rates. When decreasing the ratio of the secondary air flow rate, NOx emission becomes lower. It is well known that the flame stability of conventional burners decreases when lowering the coal feed rate to reduce the load of the furnace, because the coal particle concentration in the primary air decreases at low-load conditions. In order to improve the flame stability at low-load
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FIGURE 11.2 Concept of advanced low NOx technology. [From Makino, H., Tsuji, H., Kimoto, M., Hoshino, T., Kiga, T., and Otake, Y., in Ninth Australian Coal Science Conference, distributed on CD-ROM, 2000. With permission.]
FIGURE 11.3 Combustion profiles of conventional technology and advanced technology. [From Makino, H., Kimoto, M., Kiga, T., and Endo, Y., Thermal Nucl. Power Jpn., 48, 64–72, 1997. With permission.]
conditions, the new burner was developed.5,6 In the outer region of the primary air port of this burner, the coal particles are concentrated by use of the concentrated ring. Figure 11.76 shows the streamlined concentrated ring, which has been developed for the primary air flow without swirl force. At low-load conditions, the ring is moved to the burner exit to improve flame stability. When the load is increased, the ring is moved from the burner exit to unify the whole particle concentration in the burner.
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FIGURE 11.4 Structure of the advanced low NOx burner. [From Makino, H., Kimoto, M., Kagaku Kogaku Ronbunshu, 20, 747–757, 1994. With permission.]
FIGURE 11.5 Relationship between the swirl vane angle of the secondary air and NOx emission. [From Makino, H., Kimoto, M., Kagaku Kogaku Ronbunshu, 20, 747–757, 1994. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 11.6 Relationship between NOx emission and the ratio of the secondary air flow rate to the sum of the secondary and the tertiary air flow rates. [From Makino, H., Kimoto, M., Kagaku Kogaku Ronbunshu, 20, 747–757, 1994. With permission.]
FIGURE 11.7 Structure of the streamlined concentrated ring. [From Makino, H., Kimoto, M., and Ikeda, M., J. Soc. Powder Technol. Jpn., 37, 457–463, 2000. With permission.]
5.11.4
FURNACE AND KILN
Furnace In the utility boiler, about 10–40 burners are set up in a furnace. Figure 11.8 shows the schematic diagram of a pulverized coal-fired thermal power plant. Several burners are arranged in a row, and each burner makes a flame. Two or three rows are arranged in a furnace. The furnace is classified into the front-firing type, the opposed-firing type, and the corner-firing type. In the front-firing type, burners are © 2006 by Taylor & Francis Group, LLC
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FIGURE 11.8 Schematic diagram of a pulverized coal-fired thermal power plant.
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FIGURE 11.9 NOx emission and unburned carbon concentration in fly-ash when altering position of two-stage air port. [From Makino, H., Nensho Kenkyu, 111, 55–68, 1998. With permission.]
FIGURE 11.10 Structure of a rotary kiln. [From Akiyama, H., in Sangyou Nenshou Gijutsu, Energy Conservation Center, Japan, 2000, pp. 141–150. With permission.]
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set up on one side, and in the opposed-firing type, burners are set up on both sides. Burners are arranged at the four corners of a furnace in the corner-firing type. At the upper part of the burners, two-stage air ports are mounted to inject a part of the combustion air into the furnace. NOx concentration decreases in the region from the burners to the two-stage air ports, because a reduction atmosphere is formed in this region. Figure 11.97 represents NOx emission and unburned carbon concentration in fly-ash when altering the position of the two-stage air port. By increasing the distance from a burner to the two-stage air port, NOx emission becomes lower as the reduction atmosphere is expanded. However, unburned carbon concentration in fly-ash increases due to the decrease in combustion efficiency. In the actual furnaces, two-stage air ports are mounted at the optimized position.
Kiln The kiln is used in the cement, ceramic, and refractory material industry and others. The kiln for production of cement is the most common among them. The rotary-type kiln, shown in Figure 11.10,8 is mainly used as a cement kiln. Materials such as limestone and clay are supplied from the upper ports into the kiln, and fuels such as heavy oil are fed from the lower ports to heat and burn the materials. The temperature of materials reaches around 1500 °C. In some sites, refuse fuels are used instead of fossil fuels. While the kiln type is classified into the wet type and the dry type, the dry type is popular at present.
REFERENCES 1. Miyasaka, T., Thermal Nucl. Power Jpn., 32, 1056–1060, 1981. 2. Makino, H., Tsuji, H., Kimoto, M., Hoshino, T., Kiga, T., and Otake, Y., in Ninth Australian Coal Science Conference, distributed on CD-ROM, 2000. 3. Makino, H., Kimoto, M., Kiga, T., and Endo, Y., Thermal Nucl. Power Jpn., 48, 64–72, 1997. 4. Makino, H., Kimoto, M., Kagaku Kogaku Ronbunshu, 20, 747–757, 1994. 5. Makino, H., Kimoto, M., and Tsuji, H., J. Soc. Powder Technol. Jpn., 37, 380–388, 2000. 6. Makino, H., Kimoto, M., and Ikeda, M., J. Soc. Powder Technol. Jpn., 37, 457–463, 2000. 7. Makino, H., Nensho Kenkyu, 111, 55–68, 1998. 8. Akiyama, H., in Sangyou Nenshou Gijutsu, Energy Conservation Center, Japan, 2000, pp. 141–150.
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5.12
Dust Collection Chikao Kanaoka Ishikawa National College of Technology, Tsubata, Ishikawa, Japan
Various types of dust removal equipment are used to recover valuable materials or as emissions control equipment for air pollution. They adopt various mechanisms to remove particles, depending on particle and gas conditions. Dust removal equipment can be classified into dry and wet collectors. They are also classified into industrial dust collectors and air filters, depending on inlet dust concentration and the pressure drop. Industrial collectors are usually two or three orders of magnitude higher than those of air filters. When they are classified according to gas flow and dust collection mechanism, every dust collector belongs to one of the following categories, shown in Figure 12.1. (1) Flow-through-type dust collector: This type of collector utilizes the external force perpendicular to the mean gas flow to remove particles outward from the system. Hence the flow does not change with time, and thus removal performance and the pressure drop of the system are stable with time. Gravitational dust collectors, cyclones, and electrostatic precipitators belong to this type. (2) Obstacle-type dust collector: This type of collector captures dust on obstacles in the equipment. When solid particles are collected by a solid obstacle, captured particles remain on the obstacles, and thus it results in the increase in both collection performance and the flow resistance with operation time. This also makes the release of captured particles from the obstacles difficult, so that this type of dust collector is used as a disposable type of collector. Air filters, granular bed filters, louver-type dust collectors, and the venturiscrubber belong to this type. (3) Barrier-type dust collector: This type of collector uses permeable media for gas, not for dust, so that dust is captured and accumulates on the media surface. To maintain the continuous operation of this type of dust collector, accumulated dust has to be removed repeatedly when the pressure drop becomes high. Bag filters and ceramic filters are the typical dust collectors in this type. Table 12.l summarizes the lgeneral features of dust collectors. Among them, the bag filter shows the highest collection performance, but the collector size has to be large, as its filtration velocity is as low as 1 m/min. Furthermore, its highest operating temperature is limited to 250°C because of the poor thermal durability of the filter material. On the other hand, the electrostatic precipitator can treat gases moving as fast as 1 m/s at a very low pressure drop. However, its collection performance depends on the electrical property of dust particles. Figure 12.2 and Figure 12.3 show, respectively, the collection efficiency of dust collectors in relation to gas velocity and the initial capital cost of the equipment as a function of the handling gas volume. The total annual cost to operate a dust collector is the lowest at an economical gas velocity, which usually differs from the gas velocity at its best performance. Hence, one has to have a criterion for the selection of dust collectors.
5.12.1 FLOW-THROUGH-TYPE DUST COLLECTORS The collection efficiency of the collector in this type is expressed by E ⫽ vS/Q, if the air stream does not have turbulence, by E ⫽ 1 – exp(–vS/Q), if the particle concentration in a vertical cross section 637 © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.1 Classification of dust collectors according to gas flow and collection mechanism.
is assumed to be uniform because of turbulence, and by E ⫽ 1/(1 ⫹ Q/vS) – 1 if complete mixing is assumed in the collector. Here S is the total collection surface, Q is the total gas flow rate, and v is the migration velocity to the wall, and it is determined on the external forces acting on the particle such as gravity, centrifugal force, electrostatic force, and so on. The separation efficiency of this type is shown in Figure 12.4. As one can see from the figure, the existence of turbulence decreases the collection efficiency.
Gravitational Collectors The gravitational dust collector separates particles by the difference of gravitational settling velocity, which is suited for large particles. In most cases, it is used as a predust collector. Depending on the flow direction of the gas stream, the collectors are classified into single and multiplate types. In both cases, to improve the gravitational effect, the cross-sectional area of the collector is enlarged to operate at a low gas velocity. A Howard-type separator has many horizontal trays; hence, its separation efficiency is quite high because of the short settling distance. However, discharging separated particles and cleaning the equipment are difficult.
Centrifugal Collectors This type of dust collector separates particles by the difference in centrifugal force created by the change in flow direction and driving particles to the wall. The cyclone is the most widely used centrifugal dust collector because of its simple structure. It has no moving parts and the initial cost is inexpensive. It can be divided into two types: tangential and axial cyclones, according to the type of gas intake. Although the latter type of cyclone is customarily used in parallel so as to deal with a large volume of gas at a low pressure drop, the collection efficiency is not as high. The pressure drop in a cyclone of similar shape depends on the inlet velocity but not on the size. However, the collection efficiency becomes better with decreasing cyclone size. When the same volume of air is dealt with, both collection efficiency and pressure drop decrease in general at elevated temperatures. When dust loading becomes high, the pressure drop decreases, but the collection efficiency improves somewhat. Blowing some air down into the dustbin improves collection performance because it prevents or minimizes reentrainment of dust from the bin. Figure 12.5 shows typical cyclone dimensions, and Figure 12.6 shows the calculated correlation between cut size and cyclone diameter.
Electrostatic Precipitation An electrostatic precipitator (ESP) separates particles from the gas stream by utilizing Coulombic force acting on charged particles. Coulombic force acts directly on particles so that the structure of the ESP is very simple and different from that of other types of collectors. This feature makes © 2006 by Taylor & Francis Group, LLC
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TABLE 12.1 General Features of Dust Collectors Type
Flow through type
Pressure drop (Pa)
Maximum temperature (°C)
Inlet concentration Initial capital cost (g/m3)
Remarks
Gravitational collector
⬎ 20
50–150
1,000
50 ⬎
Low
Predust collector
Cyclone
⬎1
2,000 ⬎
1,000
500 ⬎
Middle
Convenient
Electrostatic precipitator
⬎ 0.02
300 ⬎
400
20 ⬎
Highest
Not for explosive material
Scrubber
⬎ 0.2
10,000 ⬎
1,000
100 ⬎
Middle
Require waste water treatment
Air filter
⬎ 0.01
500 ⬎
100
0.01 ⬎
Low
For building and house
Air cleaner
⬎1
1,000 ⬎
50
1⬎
Low
Engine and compressor
Granular bed
⬎1
10,000 ⬎
1,000
50 ⬎
High
Good for hot gas cleaning
Bag filter
⬎ 0.01
2,000 ⬎
250
20 ⬎
High
Not good for condensable material
Ceramic filter
⬎ 0.01
50,000 ⬎
1,000
50 ⬎
Highest
Good for hot gas cleaning
Obstacle type
Barrier type
Removable Particle Size (m)
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FIGURE 12.2 Dependence of collection efficiency on gas velocity.
FIGURE 12.3 Relation between initial capital cost and flow rate.
it possible to handle a large amount of gas at high temperature and/or at high humidity with a very low pressure drop. An ESP usually consists of particle charging and collection parts. Both charging and collection of particles are accomplished by applying a high voltage between a discharging wire and a plate or cylindrical electrode. Hence, there are two types of ESP: the one-stage type, in which charging and collection of particles take place simultaneously in one set of discharge and collection electrodes, and the two-stage type, in which charging and collection are carried out separately using two sets of electrodes in series. Figure 12.7 shows the structure of the wire and plate-type ESPs. Approximate collection efficiency of ESPs can be estimated by the following famous Deutsch equation, which is the same equation described before: ⎛ v S⎞ E ⫽ 1⫺ exp ⎜⫺ e ⎟ ⎝ Q ⎠
(12.1)
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FIGURE 12.4 Separation efficiency for flow-through-type dust collector.
where ve is the migration velocity of a particle to the collecting electrode, S is the area of the collecting electrode, and Q is the volumetric gas flow rate. Figure 12.8 shows the measured collection efficiency of an ESP as a function of particle size. Collection efficiency has a minimum around 0.5 μm because the predominant charging mechanism changes from diffusion to field charging around this size range, and the number of charges by diffusion charging is proportional to particle size but is proportional to the second power of size by field charging, whereas drag force is proportional to particle size. The decreasing trend of collection efficiency below 0.05 μm is also considered to be caused by the increase in the fraction of uncharged particles. The collection performance of ESPs also depends on the electric resistivity of the particle, r: 1. r ⬍ 5 ⫻ 102 ⍀ m: Charges on a particle are released immediately after arriving at the collection electrode so that particles are not retained firmly on the electrode, and thus reentrainment of particles happens frequently. 2. 5 ⫻ 102 ⬍ r ⬍ 5 ⫻ 108 ⍀·m: Charges on a particle are released at a reasonable rate so that captured particles do not reentrain but form a stable dust layer on the electrode. 3. 5 ⫻ 108 ⍀m ⬍ r: Because of the extremely high resistivity of dust, the accumulation rate of charges from a dust layer exceeds their release rate so that an electric field is formed between a thin dust layer and a collection electrode, and the field strength gets stronger with the accumulation of particles. Finally, back discharge occurs. Furthermore, accumulated dust particles are splashed from the collection electrode when back discharge occurs. © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.5 Size of standard tangential cyclone. [From Iinoya, K., Shujin-Sochi, Nikkan Kogyo Shinbun, Tokyo, 1965. With permission.]
For cases 1 and 3, the collection efficiency of an ESP decreases considerably. Hence, it is very important to control the electrical resistivity in the range between 5 ⫻ 102 and 5 ⫻ 108 ⍀m. Among many factors, humidity and the temperature of the gas are the most influential to the resistivity. Figure 12.9 shows a measured relationship between resistivity of fly-ash and gas temperature with humidity as a parameter. The resistivity decreases monotonically against temperature for dry gas, whereas it has a maximum in humid conditions, and it decreases as the water content increases. The appearance of maximum resistivity can be explained as follows. Although the resistivity due to the carriage of charges through a particle, which is referred to as volume resistivity, decreases as the temperature rises, the resistivity due to the carriage on a particle surface, which is referred to as surface resistivity and is thought to be related to the thin water layer absorbed on a particle, decreases as temperature decreases. Accordingly, it is clear that to attain favorable collection performance, an ESP has to be operated either higher or lower than at the temperature giving maximum resistivity. Several methods have been contrived to reduce the resistivity: spraying water, adding a small amount of chemicals to the gas, and so on. On the higher-temperature side, ESPs have not been used because high temperature can cause problems and can push up the construction cost. However, this is now being reevaluated from the energy-saving point of view, and pilot plants have been constructed for feasibility studies. © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.6 Cut size of standard cyclone. [From Iinoya, K., Shujin-Sochi, Nikkan Kogyo Shinbun, Tokyo, 1965. With permission.]
FIGURE 12.7 Parallel-plate electrostatic precipitator.
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FIGURE 12.8 Measured fractional collection efficiency of ESP; measured by EAA and electrical aerosol analyzer, and impactor.
Charging particles, especially charging high-resistivity particles, is one of the most important problems to be solved in the field of electrostatic precipitation if high collection performance is to be achieved.
5.12.2 OBSTACLE-TYPE DUST COLLECTORS Since particles are collected by obstacles in the equipment, collection efficiency for this type of collector is expressed by the following equation: ⎞ ⎛ A a hL ⎟ E ⫽ 1⫺ exp ⎜⫺ ⎝ vc 1⫺ a ⎠
(12.2)
where vc is the volume of one collection obstacle, a the packing density of the obstacle, and L the thickness of the obstacle. And A is the constant determined by the shape of obstacle, that is 4/(pDc) for cylindrical collector and 3/(2Dc). dc is the diameter of obstacle. In the equation, constants other than are determined from the equipment itself, so that E can be estimated if is given for a given collector and operation conditions, and thus the collection efficiency of obstacles with different shapes is calculated as shown in Figure 12.10. In the figure, Stk is the Stokes number defined by Stk ⫽
Cc rp dp2 u 9mDc
(12.3)
where, Cc is the Cunningham’s slip correction factor, p is the particle density, dp is the particle diameter, Dc is the representative length of a collecting body, u is the gas velocity, and is the gas viscosity. © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.9 Effect of gas temperature and humidity on electrical resistivity of dust.
Inertial Dust Collector The inertial dust collector separates particles by the difference in motion in a curvilinear flow field between the particle and the gas. Although the collection performance is much better than that of gravitational collectors, it is also used as a predust collector. It is classified into two groups, depending on the method of particle collection: separation of particles by many packed obstacles, and separation by flow channels with different curvatures. Figure 12.11 shows some examples of collection bodies and channels. The collection performance of inertial collectors improves with increasing particle size and gas velocity. Therefore, every inertial collector is designed to intensify the inertial effect by changing the flow direction as often as possible. Figure 12.12 shows a louver-type dust collector, which intensifies the inertial effect by a large inversion angle and a small gap between blades. © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.10 Calculated collection efficiencies of single collection obstacles.
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FIGURE 12.11 Shapes of packing materials for inertial-type dust collector.
In practical applications of inertial collectors, they are used mainly to separate mist rather than solid particles, as a discharge system for collected particles makes their structure complicated and costly.
Air Filters An air filter is one of the most reliable and efficient methods for dust collection, and particles are mainly used at a low dust concentration so that particles are captured inside the filter, and thus this type of particle collection is called a depth filter. An air filter is mainly used to purify the air in local environments such as hospitals, manufacturing lines, work spaces, and so on. In those environments, the dust concentration is usually lower than 10 mg/m 3 and the particle size is less than several micrometers. Hence, packed filter materials, especially fibrous mats, are used for this purpose. The porosity of a filter mat is generally higher than 85%; hence, particles are captured on individual fibers in the filter. Therefore, the filter efficiency E can be calculated from the single fiber collection efficiency h for a given filtration condition as follows: ⎛ 4 a L E ⫽ 1⫺ exp ⎜⫺ ⎝ p 1⫺ a Df
⎞ h⎟ ⎠
(12.4)
where, D f , a, and L are the fiber diameter, fiber packing density, and filter thickness, respectively. Although particles are collected on a fiber by a combination of effects, inertia, Brownian diffusion, interception, and gravity are the major factors in mechanical filtration. Electrostatic force becomes important when either fibers or particles or both are electrically charged, or an external electrical field is applied. © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.12
Shape of louver-type dust collector.
Many theoretical and empirical expressions have been proposed for each collection mechanism. Figure 12.13 shows a typical single-fiber collection efficiency chart calculated by taking inertia, Brownian diffusion, interception, and gravity into account simultaneously. It is clear from the figure that there exists a particle size or a filtration velocity that gives minimum efficiency, corresponding to the transition region among different mechanisms. There are two types of air filters: disposable and renewable. For the former type, fiber materials such as glass, metal, and natural and synthetic fibers are packed loosely in a frame or are formed into a matlike structure. Because captured particles accumulate inside the filter, its pressure drop increases with filtration time, so that a zigzag structure is adopted to increase the filtration area to lengthen its service life. It is finally replaced to avoid an increase in running cost when the pressure drop reaches a certain level. The latter type of filter renews part or all of a filtering surface periodically, also to avoid the pressure drop increase. High-efficiency particulate air (HEPA) and ultra-low-penetration air (ULPA) filters are types of paper filters composed of very fine fibers, less than 1 μm on average. They are defined as the filters that collect 0.3μm particles at the efficiency higher than 99.7% and 0.1 μm particles at higher than 99.9997%, with a pressure drop that does not exceed 12.7 mm H 2O at their specified flow rate, usually 2–5 cm/s. For this reason, HEPA and ULPA filters are used to create highly purified environments, which are necessary in the semiconductor industry, the precision machine industry, and so © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.13 Total single-fiber collection efficiency (Df ⫽ 10 μm, a ⫽ 0. 03).
on. Penetration curves of HEPA and ULPA filters are convex against particle size, and the maximum appears around 0.1μm (i.e., the collection efficiency of the filter is poorest around this particle size at the conventional filtration condition and decreases as filtration velocity increases). Hence, the filter collection performance can be roughly evaluated by measuring the efficiency in this size range. An electrostatic fibrous filter is another type of air filter that utilizes electrostatic force to remove particles. Electrostatic attractive force becomes effective when at least either fiber or particle is charged, except that both of them are charged in the same polarity. Therefore, several types are designed, depending on the charging state of fiber and particle. In general, an electrostatic fibrous filter is superior in collection performance to a mechanical fibrous filter but is difficult to operate stably. The electric filter is a stable electrostatic fibrous filter because it is composed of permanently charged fibers and is capable of collecting particles at a high efficiency in the initial filtration stage. However, the efficiency decreases with filtration time because of dust loading. Therefore, this drawback has to be eliminated in practical use.
5.12.3 BARRIER-TYPE DUST COLLECTORS The most pronounced feature of dust collectors in this type is the almost perfect particle collection regardless of size except just after the removal of accumulated particles on the barrier. This is because most of the particle collection is performed by the already captured accumulated particles by a sieving effect. This kind of collection is usually called surface or cake filtration.
Bag Filters A bag filter is usually made of fabric and in a cylindrical “bag” shape, hence the name “bag” filter. They are popularly called “fabric” filters, also. Although any type of fiber with a different weave can be used as a filter material, synthetic fibers such as polyester, polypropylene, nylon, and glass © 2006 by Taylor & Francis Group, LLC
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FIGURE 12.14 Schematics of bag filters.
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are often employed because of their strong resistivity to gas and dust. Their lifetime is usually more than 2 years. However, they cannot be used at a temperature higher than 250°C despite the efforts to develop a high-resistivity fiber. Even the most resistible of glass fibers cannot stand such a high temperature. Because of the small filtration area of each bag (10–300 mm in diameter and length less than 10 m for the cylindrical type and about 1 m ⫻ 2 m for the envelope shape), and low filtration velocity (0.5 to 5 m/min), many bags are assembled in a unit, called a bag house, when a large amount of gas must be handled. As mentioned before, collected dust particles accumulate on a fabric surface and thus raise the pressure drop. Hence, the accumulated dust has to be dislodged to maintain continuous operation. Bag filters are categorized into mechanical-shaking, reverseflow, and pulse-jet types, depending on the method of dislodging dust (i.e., mechanical-shaking and reverse-flow filters have to stop the gas flow during dislodging, thus the bag house of these types is divided into several compartments so that the entire house does not have to be stopped. A schematic drawing is shown in Figure 12.14. In the pulse-jet type, a filter can dislodge dust without stopping the house because it injects strong cleaning air for a short time. A new application of the fabric filter is the simultaneous separation of gas and dust. This technique is mainly applied to remove HCl and SOx gases from municipal incineration flue gas by injecting lime slurry or Ca(OH) 2. Particles upstream in the bag filter form solid particles. The resulting solid particles are separated by a filter.
5.12.4 MISCELLANEOUS Scrubbers are widely used as dust collectors because of their relatively high collection efficiency and low initial cost. The venturi scrubber, especially, operating with a very high pressure drop (about 1000 mm H 2 O), shows a collection efficiency as high as that of a bag filter. However, because a large amount of water is consumed and has to be treated afterward, it is not popular nowadays. Acoustic, magnetic, and thermal forces also have the potential to separate particles from a gas stream. However, they have not been put into practical use because of economic drawbacks. Other dust collection methods under development are a ceramic filter, and a granular bed filter for hot-gas cleaning. Although they show about the same performance level as that of a bag filter, there remain problems to be solved, especially durability to thermal shock and service life.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.
Iinoya, K., Shujin-Sochi, Nikkan Kogyo Shinbun, Tokyo, 1965. McCain, J. D., Gooch, J. P., and Smith, W. B., J. Air Pollut. Control Assoc., 25, 117–121, 1975. Sullivan, K. M., EPA Report, EPA-600/9/82–005b, 1982. Langmuir, I. and Blodgett, K. B., Army Air Forces Technical Report, 5418, 1946. Davies, C. N., Air Filtration, Academic Press, London, 1973. Yoshioka, N., Emi, H., and Fukushima, M., Kagaku Kogaku, 31, 157–163, 1967. Sell, W., VDI Forsch., 347, 1931. Kanaoka, C., Yoshioka, N., Iinoya, K., and Emi, H., Kagaku Kogaku, 36, 104–108, 1971. May, K. R. and Clifford, A., Ann. Occup. Hyg., 10, 83–95, 1967. Lewis, W. and Brun, R. J. NACA, TN3658, 1954. Masliyah, J. H. and Duff, A., J. Aerosol Sci., 6, 31–43, 1975. Brun, R. J. and Dorsch, R. G., NACA, TN3147, 1954. Dorsch, E. G., Brun, R. J., and Gregg, J. K., NACA, TN3587, 1955. Ushiki, K., Kubo, K., and Iinoya, K., Kagaku Kogaku Ronbunshu, 3, 172–178, 1977. Emi, H., Okuyama, K., and Yoshioka, N., J. Chem. Eng. Jpn., 6, 349–354, 1973.
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5.13
Electrostatic Separation Ken-ichiro Tanoue Yamaguchi University, Ube, Yamaguchi, Japan
Hiroaki Masuda Kyoto University, Katsura, Kyoto, Japan Electrostatic separation based on the difference of electric conductivity of particles has been widely utilized for the 200 μm to 3 mm size range. Figure 13.1 shows electric conductivity1 for many kinds of materials. Electrostatic force, which works on a charged particle, is given by F = qE
(13.1)
where q is the charge of particle and E is the strength of electric field. Electrostatic separation utilizes this force. Although there are many kinds of methods for particle charging, induced charging, corona charging, and tribocharging are mainly applied in particle separation. Table 13.1 shows the relationship between separating materials and charging method.
5.13.1
SEPARATION MECHANISM
Induction Charging As electric conductivity differs from the materials, the motion of a particle under an electrostatic field changes dramatically, as shown schematically in Figure 13.2. If a particle is conductive, the particle contacting the surface under an electrostatic field is charged up as high voltage is applied, and it becomes equipotential with the electrode surface. Then the particle jumps up from the surface because of the electrostatic repulsive force. On the other hand, a particle of insulating material needs a much longer time in order to get an induction charge. So, electrostatic force hardly works on it. The induction charge Q is given by2 ⎡ ⎛ st ⎞ ⎤ Q ⫽ C ⌬V ⎢1⫺ exp ⎜⫺ ⎟ ⎥ ⎝ C ⎠⎦ ⎣
(13.2)
where C is the capacitance of the particle, ΔV is the voltage difference between electrodes, and s is the electric conductivity of the particle.
Corona Charging and Induction Charging When negative high voltage is applied to a metal of large curvature, ionization of gas occurs around it, and then negative ions are accelerated to the grounded surface. Figure 13.3 shows the motion of particles under an electrostatic field. The particles, which are made of high or low electric conductivity, electrify negatively on the grounded surface. As the charge of the particle having high electric conductivity leaks easily to the grounded surface, it is electrified positively by induced electrification. It jumps from the grounded surface by electrostatic repulsive force. On the other 653 © 2006 by Taylor & Francis Group, LLC
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FIGURE 13.1 Electric conductivity for various materials.
TABLE 13.1
Relationship between Separating Materials and Methods
Separating material
Method
Apparatus for electrification
1) Powdered coal
tribo-charge
pipe, fluidized bed, cyclone, pulverizer
2) Farm produce
corona and induced charge
sliding belt, rotating drum
3) Wasted electric wires
induced charge
rotating drum
4) Plastics
tribo-charge
fluidized bed, vibrating transport
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hand, the particle with low electric conductivity hardly moves, because the charge of the particle does not leak; therefore, the particle remains on the plate by image force.
Tribocharging It is hardly practical to separate special plastics from mixed plastic materials or to separate ash and minerals from coal powder by using a corona charge or an induction charge, because the difference in electric conductivity is usually very small. In these cases, tribocharging is used on the basis of contact potential difference (CPD). The CPD that shows the work function difference between two materials is the most direct physical property in relation to particle electrification. Table 13.2 shows the typical work functions3,4 for many kinds of materials. In order to separate particles effectively, it is very important to select suitable materials for tribocharging. Charged particles were introduced into the electrostatic field and then separated according to the polarity of the particles.
Various Factors in Relation to Electrostatic Separation There are three processes in separation of particles by use of electrostatic force: selective charging of particles, charge transfer between particles and electrode, and dynamic motion of particles near
FIGURE 13.2 Motion of particles under an electrostatic field. C, particle made of conductive material; I, particle made of insulating material.
FIGURE 13.3 Motion of particles under corona charging. C, particle made of conductive material; I, particle made of insulating material. © 2006 by Taylor & Francis Group, LLC
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TABLE 13.2 Work Functions for Various Materials3,4 Material
Work function (eV)
Material
Work function (eV)
Material
Work function (eV)
Zn
3.63
BaO
1.1
Polyethylene
5.24⫾0.24
C
4
CaO
1.60⫾0.2
Polyethylene
6.04⫾0.47
Al
4.06–4.26
Y2O3
2
Polypropylene
5.43⫾0.16
Cu
4.25
No2O3
2.3
Polypropylene
5.49⫾0.34
Ti
4.33
ThO2
2.54
Polystyrene
4.77⫾0.20
Cr
4.5
Sm2O3
2.8
Polyvinyl chloride
4.86⫾0.73
Ag
4.52–4.74
UO2
3.15
Polycarbonate
3.85⫾0.82
Si
4.60–4.91
FeO
3.85
PMMA
4.30⫾0.29
Fe
4.67–4.81
SiO2
5
Polytetrafluoroethylene
6.71⫾0.26
Co
5
Al2O3
4.7
Polyimide
4.36⫾0.06
Ni
5.04–5.35
MgO
4.7
Polyethylene Terephthalate
4.25⫾0.10
Pt
5.12–5.93
ZrO2
5.8
Niron66
4.08⫾0.06
Au
5.31–5.47
TiO2
6.21
Pylex7740
4.84⫾0.21
the electrode surface. These processes depend strongly on relative humidity. If moisture adsorbs on the surface of a particle of insulating material, the apparent electric conductivity of the particle may increase. Especially, if ionic concentration in the moisture is high, the electric conductivity may dramatically increase. Generally, the relationship between the conductivity and the humidity is given by5 r ⫽ exp (⫺K1 ⫹ K 2 c )
(13.3)
where r is the relative value of electric conductivity, is the relative humidity, and K1 and K2 are constants. In Japan, electrostatic separation has not been widely introduced because it strongly depends on relative humidity. Electric conductivity of a material changes also with temperature. For metal materials, the conductivity is given by6 Met. ⫽
1 a ⫹ bT
(13.4)
where T is the ambient temperature, and A and B are constants. On the other hand, for semiconductors, the conductivity is given by ⎛ E ⎞ sSemi ⫽ s 0 exp ⎜⫺ ⎝ KT ⎟⎠
(13.5)
where 0, E, and K are constants. Therefore, the temperature dependence of electric conductivity differs considerably from material to material.
5.13.2
SEPARATION MACHINES
Electrostatic separation has been utilized in industries for refining mineral resources, treatment of pulverized coal, treatment of waste plastics, recycling of electrical appliances, and so on. In this section, some practical separation machines, which have been reported in recent years, are presented. © 2006 by Taylor & Francis Group, LLC
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Refining of Mineral Resources7–13 As fewer and fewer high-grade mineral ore deposits are discovered and developed and more become depleted, it becomes increasingly important to develop processes whereby the available low-grade ores may be mined profitably. The benefaction of mineral ores11 has been conducted by use of tribo- and induction charging, as shown in Figure 13.4. The apparatus consists of a horizontal aluminum plate with a vibrator, an inverted metallic roof sloped at a certain degree to the aluminum plate, an alternating voltage power supplier, and collecting bins. Figure 13.4a shows that ore particles were supplied on the vibrating aluminum plate, where the particles were charged by a combination of tribocharging and induction charging. They moved in the alternating current field. In Figure 13.4b, centrifugal force acts on the particles due to their complex circular motion induced by the force field, resulting in an outward movement of the particles. The highest charged particles move the furthest and tend to collect in the lateral bins.
Treatment of Pulverized Coal3,14–16 In order to use low-quality coals in a thermal power plant, it is important to remove the ash and mineral contents from pulverized coal. As charcoal contents and ash contents have high conductivity, electrostatic separation has been conducted by use of tribocharging.
FIGURE 13.4 Electrostatic beneficiation apparatus: (a) oblique view of apparatus showing inverted roof and collection bin locations; (b) end view showing trajectories of charged particles. © 2006 by Taylor & Francis Group, LLC
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Figure 13.5 shows the separation machine used to eliminate ash from pulverized coal. The coal was conveyed by high-speed gas flow to the tribocharging section. Charcoal contents were electrified positively while the remaining ash was charged negatively. In order to electrify them effectively, a spiral copper pipe was introduced in the charging section. Separation Technologies Inc.16 has developed a new electrostatic separation machine, as shown in Figure 13.6. Two parallel plates are set horizontally. The upper plate has a slit to introduce powdered coal. Positive and negative voltages are applied on the upper and lower plate, respectively. A meshed polymer belt was set between the plates and moved by drive rollers. The particles fed into the plates were tribocharged by the belt. Negatively charged particles with ash approached the upper electrode, while positively charged particles with carbon approached the lower electrode; each moved in different directions and then separated.
Treatment of Waste Plastics17–19 The development of a process to separate different types of plastics from each other can significantly improve the possibilities to recycle municipal waste. The waste plastics were crushed to pieces and separated to PMMA, ABS, PS, PE, PP, PET, and PVC materials by use of tribocharging. Hitachi Zosen Corp.19 has developed a new electrostatic separation machine for plastics, as shown in Figure 13.7. Plastics are fed into a mixer having many pins inside the vessel, where tribocharging of the plastics occurred. The charged plastics were supplied to a grounded rotating drum and were separated under an electrostatic field between the drum and the positive electrode where high voltage was applied.
Treatment of Waste Electrical Wires20 It is important to collect the copper and aluminum contents of waste electrical wires for recycling. In this case, waste wires must be stripped of their covering materials and then crushed to the desired size. They have been separated by use of induction charging or induced and corona combined charging.
Recycling of Electrical Appliances21 Waste electrical appliances were broken down, pulverized, and then sorted out into sizes of a few millimeters. The materials were composed of many kinds of compounds. Furthermore, these
FIGURE 13.5 Electrostatic separation machine for purified coal by use of tribocharging. © 2006 by Taylor & Francis Group, LLC
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compounds were separated by wind force, magnetic force, and eddy-current force and then collected. However, plastics having PVC and metal wires were not separated. Recently, a separation method for metal wires and PVC has been developed by Higashihama recycle center.21 The area enclosed by the solid line in Figure 13.8 shows the developed treatment method. The plastics with metal wires were pulverized again. Most metal wires were collected by use of a dry gravitational separator, while the powdered metal was collected by use of an electrostatic separator due to the induction charge. Finally, an electrostatic separator based on tribocharging was utilized for eliminating the PVC materials. The metal compounds were recycled as an ingredient of copper, while the plastics without PVC were utilized as a reducing agent in a blast furnace. Therefore, the amount of reclamation was reduced by one fifth to one tenth of the previous amount.
FIGURE 13.6 Electrostatic separation system for pulverized coal in Separation Technologies, Inc.5
FIGURE 13.7 Electrostatic separation machine for waste plastics. © 2006 by Taylor & Francis Group, LLC
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FIGURE 13.8 Application of electrostatic separation for treatment of plastics in recycling plant for electrical appliances.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.
Kelly, E. G. and Spottiswood, D. J., Miner. Eng., 2, 33–46, 1989. Ito, M. and Owada, S., Powder Sci. Eng., 35, 40–52, 2003 (in Japanese). Gupa, R., Gidaspow, D., and Wasan, D. T., Powder Technol., 75, 79–87, 1993. Chemical Society of Japan, in Kagaku Binran, Maruzen, 1984, pp. 493–494 (in Japanese). Chemical Society of Japan, in Kagaku Binran, Maruzen, 1984, pp. 494–495 (in Japanese). Kakovsky, I. A. and Revnivtzev, V. I., in Fifth International Mineral Proceedings Congress, London, 1960, pp. 775–786. Dance, A. D. and Morrison, R. D., Miner. Eng., 5, 751–765, 1992. Yongzhi, L., in Proceedings of the First International Conference on Modern Process Mineralogy and Mineral Processing, Beijing, 1992, pp. 385–390. Okur, E. and Onal, G., in Proceedings of the First International Conference n Modern Process Mineralogy and Mineral Processing, Beijing, 1992, pp. 391–396. Zhou, G. and Moon, K., Can. J. Chem. Eng., 72, 78–84, 1994. Celik, M. S. and Yasar, E., Miner. Eng., 8, 829–833, 1995. Inculet, D. R., Quigley, R. M., and Inculet, I. I., J. Electrostat., 34, 17–25, 1995. Li, T. X., Ban, H., Hower, J. C., Stencel, J. M., and Saito, K., J. Electrostat., 47, 133–142, 1999. Thomas, A. L., Rochard, P. K., Robert, H. E., and Nicholas, H. H., DOE-PETC-TR, 90–11, 23, 1990. Bouchillon, C. W. and Steele, W. G., Part. Sci. Technol., 10, 73–89, 1992. Tondu, E., Thompson, W. G., Whitlock, D. R., Bittner, J. D., and Vasiliauskas, A., Miner. Eng., 48, 47–50, 1996. Inculet, I. I. and Castle, G. S. S., Tribo-electrification of commercial plastics, in Air. Inst. Phys. Conf. Ser., No. 118, Section 4, Electrostatics ’91, Oxford, 1991. Inculet, I. I., Castle, G. S. S., and Brown, J. D., IEEE Trans., IAS, 1397–1389, 1994. Maehata, H., Inoue, T., Tsukahara, M., Arai, H., Tamakoshi, D., Tojyo, C., Nagai, K., and Sekiguchi, Y., Hitachi Zosen Technical Information, 59, 222–226, 1998 (in Japanese). Dascalescu, L., Iuga, Al., and Morar, R., Magn. Electr. Separ., 4, 241–255, 1993. Matsumura, T., Hyomengijyutsu, 52, 244–249, 2001 (in Japanese).
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5.14
Magnetic Separation Toyohisa Fujita The University of Tokyo, Japan
Many kinds of forces are used to separate or filter materials of different quality and size. Mineral processing especially includes the separation of solids, recovery of valuable solids from waste materials, and water or air purification. Materials have electric and magnetic properties; therefore, electromagnetic fields are employed to separate solids with different electric and magnetic properties. Consequently, for efficient separation it is important to increase the electromagnetic forces and decrease the interaction forces between different solid particles. Here various magnetic separation methods and unit operations to separate particles by virtue of differences in their magnetic properties are described.
5.14.1 CLASSIFICATION OF MAGNETIC SEPARATORS1 The magnetic separation technique is classified as shown in Table 14.1. In a narrow sense, magnetic separation is a separation technique whose goal is to concentrate a magnetic material, to remove magnetic impurity, or to extract valuable magnetic materials. This is accomplished by discharging the particles captured by a magnet at a position depending on their magnetic properties. Magnetic filtration is the method to separate magnetic particles by capturing them with a filter. When suspended particles are magnetic, the direct method, in which particles are captured directly on the filter, is employed. When the separation target is nonmagnetic or an ion, the magnetic reagent method,2 in which the separation target is sprinkled with a magnetic reagent, is employed. This magnetite seed method can also be used in the magnetic separation3 of various diamagnetic particles using the surface charge difference in water. Separation by electromagnetic induction is known as eddy-current separation. Magnetohydrostatic separation is a method used to separate nonmagnetic particles immersed in a magnetic fluid by adjusting the magnetic buoyant force on particles with a magnetic field gradient. Magnetic separators are classifid in terms of the difference of the separation mechanism (Table 14.2). A look at Table 14.2 shows that they are classified into static magnetic field type and AC magnetic field type. The first type is further classified into two main types on the basis of intensity, that is, a low-intensity and a high-power type. Moreover, the second type is classified into three further types, that is, a traveling magnetic field type, a vibrating magnetic field type, and an electromagnetic induction type. Drum-type, belt-type, and high-gradient-type separators have been extensively employed. The low-intensity type uses less than 0.2 T, the medium-intensity type uses from 0.3 to 1.0 T, while the high-intensity one employs more than 1 T. The magnetic capture condition is related to the magnetic field strength multiply by the gradient of magnetic field strength, and it is given by the following formula: H∇H or B∇B
(14.1)
The magnetic field strength, field gradient, and conditions are listed in Table 14.3. A look at Table 14.3 shows that the capture condition of a medium-intensity-type separator is about 10 times larger than that of a low-intensity type, while the magnetic capture condition of the high-intensity type is very large (i.e., is more than 10 to 104 times higher than that of a medium-intensity-type separator). 661 © 2006 by Taylor & Francis Group, LLC
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TABLE 14.1
Magnetic Separation1
Magnetic separation (in a broad sense) 1. Magnetic separation (in a narrow sense) Concentration, purification, extraction 2. Magnetic filtration 2.1 Direct method 2.2 Magnetic reagent method Magnetic seed method,2 ferrite coprecipitation method, iron flocculants method, adsorption method, organic reagent containing iron addition method 3. Separation by electromagnetic induction Eddy current separation 4. Magnetohydrostatic separation Separation using magnetic fluid
TABLE 14.2 Classification of Magnetic Separation top feed type
OS varnner* Ferrogum separator*
under feed type
Crockett wet separator*
top feed type
YS drum separator
belt type (low intensity type) drum type
under feed type belt type static magnetic field type
(medium intensity type)
drum type
GrÖndal drum separator Multistage drum separator*
top feed type
Eriez rare earth roll separator*
top feed type
Eriez rare earth drum separator PERMOS magnetic separator
wet high intensity magnetic separator WHIMS high intensity type
magnetic separator
(high intensity type)
high intensity magnetic separator (HIMS) high gradient type, HGMS
traveling magntic field type
a. c. magnetic field type
vibrating magneic field type
Frantz ferrofilter* Jones high-intensity separator* Carpco high-intensity wet separator*
Eriez wet high-intensity separator* NY drum separator* Superconductive suspended magnet Coupled pole separator High-gauss separator DESCOS (superconductive drum type magnetic separator) SALA-HGMS* DEM filter* Super conducting HGMS* OYB separator* AC spiral separator* OY table type separator* DS type separator*
electromagnetic induction type
*Matrixes like wire or balls are loaded between magnetic poles.
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TABLE 14.3 Magnetic Capture Conditions of Various Magnetic Separators Classification
Magnetic separator
Conventional low intensity Medium intensity
H (kA/m)
Magnetic condition H( H/ x)
H/ x 2
(MA/m )
(1012A2/m2)
Drum separator
40
4
0.16
Davis tube tester
320
32
10
360~720
10~100
3.6~72
Rare earth drum or roll separator
High intensity
Frantz ferrofilter NY type drum separator Jones separator DESCOS
800 1600 1600 2550
800 440 1600 10~50
640 700 2560 25~130
High gradient
Solenoid separator (Kolm-Marston) Superconducting HGMS
1600
16000~ 160000 16000~ 200000
25600~ 256000 64000~ 800000
High intensity
4000
5.14.2 STATIC MAGNETIC FIELD SEPARATORS Open Gradient Magnetic-Field-Type Magnetic Separation The main application of low-intensity magnetic separators is either removal of strongly magnetic impurities of tramp iron or concentration of a strongly magnetic valuable component. The Groendal drum wet separator has been widely used for the concentration of different materials. A permanent magnet is used instead of electromagnets, and raw material is fed from the lower part, as shown schematically in Figure 14.1. 4 The drum with fixed magnets inside rotates in contact with the floating or suspended particles in compartment a1. Then magnetic particles are removed as they are attracted and captured on the drum surface by the magnetic force. Nonmagnetic particles are carried over to the next compartment, a2. It was reported that the highest capturing efficiency of particles on the drum surface is obtained when the peripheral velocity of the drum is equal to the feeding velocity. Figure 14.2 illustrates a drum separator of the dry type, in which raw material is fed on the top of the rotating drum with a fixed permanent magnet or electromagnet inside. Particles are classified by the balance in magnitude between magnetic and centrifugal forces. When rare-earth permanent magnet blocks are used in the drum, they are fixed inside the drum in such a way that different poles are placed alternately. In this case the magnetic flux density on the surface of the drum is 0.7 T at the maximum. This separator is used for purification of zircon, quartz, feldspar, glass cullet, coal, and so forth.5 It is effective in separating relatively large particles (from 75 μm to 25 mm). The drum-type magnetic separator with a superconducting magnetic system6 is shown in Figure 14.3. The magnetic flux density on the drum surface is approximately 3 T. The drum, which as an exterior diameter of 1.2 m and an external length of 1.5 m, is made of carbon-reinforced plastics. The drum rotates at a speed of 2 to 30 rpm. The evaporating helium is led back to the liquefying plant, which is connected to the drum. For example, the magnetizable gangue serpentine could be completely separated from magnesite in the 1 to 100 mm range size. Also the results of separation of hematite and goethite from refractory crude bauxite (1–10 mm) and the separation of clay from potassium raw salts (1–4 mm) were reported, indicating a very effective process. On the other hand, in China, a suspended superconductive magnet is employed to remove iron impurities from coals. The magnetic flux density is 1.7 T on the surface of the bottom magnet and 1 T at 300 mm from bottom.7 The coupled-pole dry separator, high-gauss separator, Dings-induced roll separator, and Wetherilll–Rowand separator belong to the High-Intensity Magnetic Separation (HIMS) type. © 2006 by Taylor & Francis Group, LLC
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FIGURE 14.1 Groendal drum wet magnetic separator. [From Gaudin, A. M., Principle of Mineral Dressing, McGraw-Hill, New York, 1939, p. 450. With permission.]
FIGURE 14.2 Drum magnetic separator of the dry type.
FIGURE 14.3 High-intensity drum-type magnetic separator with superconductive magnetic system (DESCOS).
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A coupled-pole dry magnetic separator is shown in Figure 14.4. It is designed to generate a magnetic flux at the gap between two rotors, creating a high-intensity magnetic field of more than 2 T. This type of separator is used for the concentrated recovery and the purification of material.
Matrix-Loaded-Type Magnetic Separation When matrixes such as ferromagnetic wire or balls are set between magnetic poles, a high-gradient magnetic field can be produced. The Frantz Ferrofilter separator (Figure 14.5), Jones high-intensity wet magnetic separator (Figure 14.6), Carpco high-intensity wet magnetic separator (Figure 14.7), and New York (NY) drum magne separator (Figure 14.8) belong to the Wet High Intensity Magnetic Separator (WHIMS) type. The Frantz Ferrofilter separator, which is the oldest model, is equipped with metal screens or grids as the matrix. Magnetic particles are separated as they are captured by
FIGURE 14.4 A coupled-pole dry magnetic separator (Nippon Magnetic Dressing Co., Ltd.).
FIGURE 14.5 Frantz Ferrofilter (U.S. Patent 20,704,085). © 2006 by Taylor & Francis Group, LLC
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FIGURE 14.6 Jones high-intensity wet magnetic separator (U.K. Patents 768,451 and 767,124).
FIGURE 14.7 Carpco high-intensity wet magnetic separator.
the matrix and placed in the magnetic field during the processing of pulp through the matrix. This type of separator has been employed to remove iron particles from clay. The Jones-type separator, which has a matrix of corrugated iron sheets, has been used to dress hematite ore. Balls, cubes, and twisted rods are employed as the matrix in the Carpco-type separator, while expanded metal or steel wool is used in the Eriz-type separator. The NY drum separator is a Yashima (YS) magnetic separator with many steel balls on the drum surface. Magnetic particles are captured by the high-intensity magnetic field created around the contact points of the balls. The high-gradient magnetic separator has been known since 1968 when the device was first installed at a Georgia kaolin plant to remove weakly magnetic particles less than 2 μm in size from © 2006 by Taylor & Francis Group, LLC
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FIGURE 14.8 NY drum separator.
clay. When the matrices are arranged between the solenoid coil and the slurry, flow is parallel to the magnetic field direction, and the captured amount is much larger when compared to the device with the matrices between the magnetic poles.8 The continuous high-gradient magnetic separator (SALA-HGMS)9 is shown in Figure 14.9. Expanded metal or steel wool compose ferromagnetic matrices that are filled in the area under the influence of the magnetic field generated by a solenoid coil with a yoke. Moreover, it is desirable that matrices be corrosion and abrasion resistant. Matrices, filled in the magnetic field, increase sharply the magnetic force, that is, H∇H in Equation 14.1. The separation box with matrices rotates continuously, with the magnetic flux density in the separator less than 2 T (Figure 14.9). This high-gradient magnetic separator has been applied to the separation of iron ores, purification of industrial minerals, separation of rare-earth minerals, water treatment, and so forth. Also, the continuous high-gradient magnetic separation, which uses highquality permanent magnets (Fe-Nd-B), has been produced to capture magnetic particles while the magnetic flux density between magnetic poles is about 0.4 T. On the other hand, a high-gradient magnetic separator using superconducting magnets has been developed. Compared to the ordinal HGMS, the magnetic force condition is very much larger. Therefore, finer and lower magnetic susceptibility paramagnetic particles can be captured. In addition, the high magnetic flux density enables an increase of the separation capacity as the flow rate also increases. One example is the superconducting high-gradient magnetic separator that is shown in Figure 14.10. 5 T of magnetic flux density is generated in the space of 500 mm height and 100 mm diameter, which is filled with steel-wool-type matrices. The superconducting coil is placed in liquid helium and is connected to a liquefying plant of the vaporized helium gas. The slurry is fed from the bottom of the canister and discharged from the top. This separator has been employed to remove ferric impurities of micron © 2006 by Taylor & Francis Group, LLC
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FIGURE 14.9 Continuous high-gradient magnetic separator (SALA-HGMS).
FIGURE 14.10 Superconducting high-gradient magnetic separator (Eriez).
size from kaolin clay. The 2 T separator using a superconducting magnet can operate on 80 to 90% of the power required for conventional HGMSs.
5.14.3 MAGNETOHYDROSTATIC SEPARATION A colloidal solution of surfactant-coated ferromagnetic particles of 10 nm size, which remains uniform and stable even under a magnetic or centrifugal field, is called a magnetic fluid or ferrofluid. When nonmagnetic materials are immersed in the magnetic fluid under a magnetic field gradient, a magnetic force acts on the nonmagnetic materials toward the direction of low magnetic intensity. © 2006 by Taylor & Francis Group, LLC
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If the magnetic field gradient is directed in the gravity direction, the net force on a nonmagnetic material F is given by F = V [(p ⫺ )g ⫺ I ∇H ]
(14.2)
where V is the volume of nonmagnetic material immersed in the fluid, rs and r represent the densities of the nonmagnetic material and magnetic fluid, respectively, I is the average magnetization of magnetic fluid, ∇H is the magnetic field gradient positioned nonmagnetic material, and g is the gravitational acceleration. The nonmagnetic material sinks if Fg < 0 and floats if Fg > 0 when Fg is the direction of g. This separator enables even a nonmagnetic material of density 20 g/cm3 to float; thus materials can be separated by adjusting the magnitude of Fg. For example, I = 0.025 T and ∇H = –8 × 106 A/m2 give Fg/V = 2 × 105 N/m3, which is enough to float a nonmagnetic particle of specific gravity of 20. For the magnetohydrostatic separator, a full equipped pilot plant for sink–float separation for recovering nonmagnetic metal scraps has been developed. The separator that recovers aluminum as a float product is shown in Figure 14.11. The use of rare-earth permanent magnets serves the purpose of reducing capital investment and energy consumption, while the use of waterbased magnetic fluid is nontoxic, easy to wash out from the products, and easily reused. 10 Shredded automobile nonferrous metal scraps can be separated effectively; for example, 99.9 % recovery of aluminum at a grade of 98% at 500 kg/h capacity can be achieved. The distance between magnetic poles can be regulated to float zinc from a mixture of zinc, copper, and lead. This separator is convenient for recovering particles of 0.5 to 100 mm size. For industrial operation, De Beers Consolidated Mines Ltd. uses a sink–float separator, which employs a kerosene-based magnetic fluid to separate diamond minerals.11 On the other hand, the Mag-sep Corporation (MC) process12 with a magnet-hydrodyamic separator has been developed to separate finer particles ranging from about 5 μ m to 1 mm in size by using a more diluted magnetic fluid of low cost. The mineral particles are fed through the duct as a slurry in a magnetic fluid. As the slurry and duct are rotated, low- and high-density particles are discharged from the inner and outer cylinders respectively. Large oppositely directed force can be applied to particles as light as 0.3 g/cm3, leading to rapid and precise separation.
FIGURE 14.11 Sink–float separator using magnetic fluid (Nittetsu Mining Co., Ltd., Japan): © 2006 by Taylor & Francis Group, LLC
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FIGURE 14.12
Eddy-current separator.
5.14.4 ELECTROMAGNETIC-INDUCTION-TYPE SEPARATION Eddy-current separation is used for the recovery of nonferrous metals from material mixtures whose magnetic fractions are removed by a magnetic separator. A time-dependent change of a magnetic field exerts repulsive forces on electrically conductive particles, which can be used to separate metallic particles from nonmetallic particles. Many eddy-current separators were developed for the recovery of nonferrous metals from shredded car scrap, granulated power cables, municipal solid waste, and so on. The alternating magnetic fields can be generated by the utilization of a linear motor or permanent magnet configurations. Two types of eddy-current separators with permanent magnets are shown in Figure 14.12. By using permanent magnets, the generation costs of eddy-current separation can be reduced considerably and the construction of equipment can be less complex. Figure 14.12a shows a static separator where particles are moving through a field distribution generated by the configuration of permanent magnets mounted on stable ramps or walls. Figure 14.12b shows a dynamic separator where the magnetic field is generated by machinery with moving magnets and the repulsive forces can be enhanced by the motion of the magnetic fields. Two belt drums support the conveyer belt. One drum is driven, and inside the other the magnet system rotates at an appreciably greater speed, for example, 2500 rpm. The repulsive force acting on the particle can become larger as the value of conductivity divided by density increases. Also, the separation condition depends on particle dimensions and shape and the field distribution in the separation area.13 In particular, recent use of high-quality rare-earth permanent magnets in eddy-current separation has improved the separation efficiency. For example, 3 to 100 mm sizes of aluminum from a mixture can be separated effectively at 5 m3/hr.14
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8.
Yashima, S. and Fujita, T., J. Soc. Powder Technol. Jpn., 28, 318–328, 1991. Fujita, T. and Mamiya, M., J. Mining Metall. Inst. Jpn., 103, 35–40, 1987. Fujita, T, Wei, Y., and Mamiya, M., J. Mining Metall. Inst. Jpn., 103, 513–518, 1987. Gaudin, A. M., in Principle of Mineral Dressing, McGraw-Hill, New York, 1939, p. 450. Marinescu, M., IEEE Trans. Magn., MAG-25, 2732–2738, 1989. Wasmuth, H. D., Aufbereitungs-Technik, 30, 753–760, 1989. Homma, T., Proc. Fall Annual Meeting MMIJ Ube, C2-18, 2003. Kim, Y. S., Fujita, T., Hashimoto, S., and Shimoiizaka, J., Proc. 15th Int. Mineral Processing Congress (IMPC), Cannes, 1, 381–390, 1985.
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9. Oberteuffer, J. A. and Wechsler, I., Proc. Fine Part. Process., Las Vegas, 1178–1216, 1980. 10. Fujita, T., in Magnetic Fluids and Applications Handbook, Berkovski, B., Ed., Begell House, New York, 1996, pp. 755–789. 11. Sbovoda, J., Proc. MINPREX, Melborne, 297–301, 2000. 12. Walker, M. S., Proc. 15th Int. Miner. Process. Congr. (IMPC), Cannes, 1, 307–316, 1985. 13. Fujita, T., Sotojima, Y., and Kuzuno, K., J. Mining Mater. Process. Inst. Jpn., 111, 177–180, 1995. 14. Warlitz, G., Aluminium. 65, 1125–1131, 1981.
Acknowledgment The author gratefully acknowledges Dr. K. Ushiki’s work for writing articles 3.4 and 5.13, which appeared in the first edition.
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5.15
Gravity Thickening Eiji Iritani Nagoya University, Chikusa-ku, Nagoya, Japan
Sedimentation is a separation of solid particles from a suspension due to the effect of a body force, which may be either gravity or centrifugal, on the buoyant mass of the particle. The sedimentation operation is often referred to as either thickening or clarification. If the main purpose of the operation is to concentrate solids into a denser slurry, the operation is generally called thickening, whereas, if the major concern is to produce a relatively clear liquid phase, the operation is usually called clarification. Because both thickening and clarification occur in any sedimentation basin, both functions have to be considered in the thickener design.
5.15.1
PRETREATMENT
The sedimentation rate of fine particles can be artificially increased by the addition of coagulants or flocculants, which causes precipitation of colloidal particles and the formation of flocs. For these to be effective, it is often necessary to adjust the pH of the slurry. Coagulants are usually inorganics that neutralize the surface charges on the particles, thus allowing them to collide and adhere. Common examples include aluminum sulfate, ferric chloride, and ferric sulfate. Flocculants are organic polyelectrolytes or long-chain polymers, which cause a physical linkage by bridging and enmeshment, and sometimes by particle–charge interaction. Examples include polyacrylamide and polyacrylate. Performance of coagulants or flocculants proposed for a given case can be determined beforehand in jar tests. A very useful test for filtration applications is the capillary suction time test, which measures the permeability of a flocculated suspension by using the uniform capillary suction of a filter paper.1 The chemicals are combined with the slurry by rapid mixing, then allowed to settle at a lower shear rate consistent with the plant unit.
5.15.2
IDEAL SETTLING BASIN
A slurry of discrete particles settling in an ideal basin will be considered. The paths followed by the particles are straight lines determined by the vector sums of the two velocity components, as illustrated in Figure 15.1. The horizontal velocity component of the particle, due to the fluid flow across the basin, is equal to v. A particle with settling velocity u0, starting at the top of the inlet, will reach the bottom at the outlet. Consequently, particles with a settling velocity less than u0 will be only partially removed. A particle with settling velocity us will reach the bottom at the outlet if it enters the settling basin at height h from the bottom. Only particles initially at heights less than h will be removed. Therefore, the fraction hp of particles with settling velocity us that are removed from an ideal basin is given by hp ⫽
u h us ⫽ ⫽ s H u0 Q / A
(15.1)
where H is the liquid depth, Q is the volumetric flow rate of slurry, and A is the surface area of the basin. 673 © 2006 by Taylor & Francis Group, LLC
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FIGURE 15.1 Settling behavior of discrete particles in ideal settling basin.
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The analysis of sedimentation in an ideal basin clearly demonstrates that the slurry depth should be as shallow as practical. Gravity settlers that use inclined surfaces in the form of flat plates (lamellae) or tubes have the effect of increasing the settling rate of the particulates.2 When flat plates or tubes are inclined at a steep angle, the settled sludge slides downward. This induces an upward flow of the clarified effluent, and the solids can be effectively collected at the bottom.
5.15.3
SETTLING CURVE
A typical batch settling curve is illustrated in Figure 15.2. The height of the interface between the settling particles and the clear supernatant liquid is plotted as a function of time. In zone settling, the slurry settles as a mass with a clear interface between the slurry and supernatant because of the relatively high solids concentration. In this stage, the slurry moves downward at a uniform velocity, which is dependent on the initial solids concentration. As the settling solids approach the bottom of the basin, they develop a layer with a higher concentration of suspended solids. After the subsiding interface reaches the rising layer of higher concentration, the sedimentation rate of the interface decreases gradually. During this transition zone, the subsidence rate of the interface is closely related to the instantaneous concentration at the interface. As the concentration continues to increase at the bottom, a zone of compacting sediment forms where solids are supported partially by the solids beneath them. The interface continues to subside in compression until an equilibrium condition is attained. The sedimentation rate in the consolidation region is given approximately by
⫺
dH ⫽ k ( H ⫺ H⬁ ) du
(15.2)
where H is the height of the sediment at time θ, H ⬁ is the final height of the sediment, and k is the empirical constant. Integrating Equation 15.2, the time taken for the slurry line to fall from a height
FIGURE 15.2 Batch settling curve for settling slurry. © 2006 by Taylor & Francis Group, LLC
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Hc, corresponding to the critical settling point where the compression zone begins, to a height H can be obtained by ln ( H ⫺ H ∞ ) ⫽⫺ku ⫹ ln ( H c ⫺ H ∞ )
(15.3)
Consequently, if ln(H–H∞) is plotted against u, a straight line with slope –k is obtained.
5.15.4
KYNCH THEORY
3
Kynch has analyzed the behavior of concentrated suspensions during batch sedimentation using a continuity approach. The theory is mainly based on the postulate that the settling velocity of solids is a function only of the local solids concentration. When the settling velocity is u at some horizontal level where the volumetric concentration of particles is C, the volumetric sedimentation rate G per unit area (i.e., the solids flux) is given by G⫽Cu
(15.4)
Consider a material balance of an elemental layer between a height H above the bottom and a height H ⫹ dH. Since in a time interval du the accumulation of particles in the layer is given by the difference in flux, between G(H ⫹ dH) into the upper layer and G(H) out through the lower layer, one obtains ∂C ∂G dG dC ⫽ ⫽ ⋅ ∂u ∂H dC dH
(15.5)
Because the concentration of particles is generally a function of both position and time, one obtains dC ⫽
∂C ∂C dH ⫹ du ∂H ∂u
(15.6)
Constant concentration conditions (dC ⫽ 0) are therefore defined by ∂C ∂C ⎛ dH ⎞ ⫽⫺ ⎜ ⎟ ∂H ∂u ⎝ d u ⎠
⫺1
(15.7)
Substituting Equation 15.7 into Equation 15.5, one obtains ⫺
dG dH ⫽ ⫽V dC du
(15.8)
Because Equation 15.8 refers to a constant concentration, dG/dC must also be constant. Consequently, V ( ⫽ dH/dθ) is constant too for any given concentration and is the propagation velocity of a zone of constant concentration C. While during sedimentation the interface between the clear liquid and the settling solids is moving downward, layers of constant concentration appear to move upward at the base of the vessel. This result is widely employed to obtain the relation between solids flux and concentration needed for thickener design from a single batch sedimentation test. It is possible to obtain the plot © 2006 by Taylor & Francis Group, LLC
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of G versus C over the concentration range C0 to Cmax (concentration at the compression point) from only a single experiment of a slurry with an initial solids concentration C0. The sedimentation curve has a decreasing negative slope reflecting the increasing concentration of solids at the interface after point B, as illustrated in Figure 15.3. Line OP represents the locus of points of some concentration C (C0 ⬍ C ⬍ Cmax). It corresponds to the propagation of a wave at the upward velocity V from the bottom of the sediment. When the wave propagates the interface of point P, all the particles in the slurry must have passed through the plane of the wave. Therefore, one obtains C (u ⫹V ) u1 ⫽ C0 H 0
(15.9)
By drawing a tangent to the settling curve ABPD at P, the intercept T of the tangent on the H axis is located. Then, uθ1 is equal to QT because ⫺u is the slope of a tangent to the settling curve at P, and Vθ1 is equal to OQ because V is the slope of line OP. Consequently, the concentration C corresponding to the line OP is given by C ⫽ C0
OA OT
(15.10)
and the corresponding solids flux is obtained from G ⫽ Cu ⫽ C0
OA u OT
(15.11)
By drawing the tangent at a series of points on the curve BPD and measuring the corresponding slope ⫺u and intercept OT, it is possible to calculate the solid flux G for any concentration C (C0 ⬍ C ⬍ Cmax).
FIGURE 15.3 Construction of Kynch theory. © 2006 by Taylor & Francis Group, LLC
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DESIGN OF A CONTINUOUS THICKENER
A thickener is a sedimentation basin that is employed to concentrate a slurry prior to filtration or centrifugation. Normally a larger fraction of the total liquid is removed in thickening than in subsequent operations. In most cases, the particles in a thickener settle collectively in the zone settling regime. As shown in Figure 15.4, the bottom part of a thickener is filled with a layer of settled solids, which increases in concentration with greater depths. A clarified liquid separates from the thickened liquid at the interface and is taken off at the top. A thickener therefore fulfills the dual function of providing a concentrated underflow and a clear liquid overflow. In designing a thickener, the areas needed for clarification and thickening are therefore examined separately. The larger of the two areas determines the size required to achieve the specified performance.
Clarification Area The cross-sectional area needed for clarification can be calculated from the initial sedimentation velocity obtained from a batch sedimentation test. The area must be large enough so that the rising velocity of overflow liquid is less than the batch sedimentation velocity. The minimum area required in a thickener for clarification is, therefore, determined by
Ac ⫽
Qe us
(15.12)
where Ac is the surface area needed for clarification, Qe is the overflow rate of the clear liquid, and us is the initial sedimentation rate of slurry at the feed concentration.
FIGURE 15.4 Schematic view of continuous thickener. © 2006 by Taylor & Francis Group, LLC
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Coe and Clevenger Method In the thickening region, solids move toward the underflow both by gravity sedimentation and by bulk movement resulting from underflow withdrawal. The total flux G of solids at concentration Ci can, therefore, be given by G ⫽ Ci ui ⫹ Ci ub
(15.13)
where ui is the sedimentation velocity at concentration Ci, and ub is the downward velocity of slurry arising from the removal of the underflow. The form of solids flux terms is shown in Figure 15.5. The flux of solids by bulk transport is a linear function of solids concentration with slope ub. The gravity flux of solids goes through a maximum as the concentration increases, as obtained from batch sedimentation tests for various concentrations. For most zone settling slurries, the combination of gravity and bulk flow flux terms produces a total flux curve with a maximum and minimum. The minimum in the total flux curve exists at some limiting concentration between the feed and underflow concentrations and represents the limiting solids-handling capacity of the slurry. The total solids contained in the feed must be less than the solids-transmitting capability of this limiting concentration layer. Because the gravity flux becomes zero at the bottom of the thickener, all solids are taken off by bulk flow only. Consequently, if a horizontal line is drawn tangent to the minimum in the total flux curve, it intersects the bulk flux line at the underflow concentration Cu. Assuming that all solids in the feed leave in the underflow, the cross-sectional area required for thickening can be obtained from4 A⫽
Cf Qf CuQu ⫽ Gl Gl
(15.14)
where Cf is the solids concentration of the influent, Qf is the flow rate of the influent, Gl is the limiting solids flux at concentration C l , and Q u is the flow rate of the underflow. An alternative approach employs the batch flux curve directly. 5 The settling velocity ui by gravity is the slope of line from the origin to any point on the batch flux curve, as illustrated in Figure 15.6
FIGURE 15.5 Total solids flux curve in continuous thickener. © 2006 by Taylor & Francis Group, LLC
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FIGURE 15.6 Solids flux in continuous thickener based on batch flux curve.
(often known as a Kynch flux plot). If this line intersects a tangent to the batch flux curve at the point of tangency, the intersection corresponds to the limiting solids concentration C l , and gravity flux G g . By using Equation 15.14, the bulk velocity is given by
ub ⫽
Qu Gl ⫽ A Cu
(15.15)
Thus, the bulk downward velocity is the slope of a tangent operating line connecting solids flux G l on the flux axis to the corresponding underflow solids concentration C u on the concentration axis. The distance from G g to G l is the solids flux by bulk downward transport when solids are removed at concentration Cu. Consequently, the limiting solids-handling capacity Gl in Equation 15.15 can be determined for a given underflow concentration Cu.
Talmage and Fitch Method The procedure of Talmage and Fitch6 requires data from only a single batch settling curve, on the basis of the method of Kynch. On the basis of the material balance, the values of H2 and Hu shown in Figure 15.7 are given by C0 H 0 ⫽ C2 H 2 ⫽ C u H u
(15.16)
where C 2 is the concentration at the compression point u 2 . An “underflow” line is drawn parallel to the time axis at H ⫽ Hu on a plot of height H versus sedimentation time u. By drawing a tangent to © 2006 by Taylor & Francis Group, LLC
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FIGURE 15.7 Talmage and Fitch method.
the settling curve through point H2 on the H axis, the value of uu can be determined from the intersection of the tangent and the underflow line. Consequently, the required thickener area A can be found from ⎛ u ⎞ A ⫽ Cf Qf ⎜ u ⎟ ⎝ C0 H 0 ⎠
(15.17)
REFERENCES 1. 2. 3. 4. 5. 6.
Gale, R. S., Filtr. Separ., 8, 531–538, 1971. Nakamura, H. and Kuroda, K., Keijo J. Med., 8, 256–296, 1937. Kynch, G. J., Trans. Faraday Soc., 48, 166–176, 1952. Coe, H. S. and Clevenger, G. H., Trans. Am. Inst. Mining Eng., 55, 356–384, 1916. Yoshioka, N., Hotta, Y., Tanaka, S., Naito, S., and Tsugami, S., Kagaku Kogaku, 21, 66–74, 1957. Talmage, W. P. and Fitch, E. B., Ind. Eng. Chem., 47, 38–41, 1955.
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5.16
Filtration Eiji Iritani Nagoya University, Chikusa-ku, Nagoya, Japan
Filtration is the operation of separating a dispersed phase of solid particles from a fluid by means of a porous filter medium, which permits the passage of the fluid but retains the particles. Filtration is probably one of the oldest unit operations. The old forms of filtration by straining through porous materials were described by the earliest Chinese writers. A gravity filter used in a chemical process industry was described in an Egyptian papyrus which has its origin in about the third century a.d. In recent years, many developments have increased the application of filtration. Filtration steps are required in many important processes and in widely divergent industries. The importance of filtration technique has been emphasized by the increased need for protection of the environment. Recently, membrane filtration of colloids has become increasingly important in widely diversified fields. The operations are divided into two broad categories: “cake” and “depth” filtration. From the viewpoint of a driving force, cake filtration is further divided into pressure, vacuum, gravity, and centrifugal operations. In cake filtration, particles in a slurry form a deposit as a filter cake on the surface of the supporting porous medium while the fluid passes through it. After an initial period of deposition, the filter cake itself starts to act as the filter medium while further particles are deposited. In depth filtration (sometimes called filter medium filtration or clarifying filtration), particles are captured within the complex pore structures of the filter medium, and the cake is not formed on the surface of the medium. In many processes, a stage of depth filtration precedes the formation of a cake. The first particles can enter the medium, and with very dilute slurries, there can be a time lag before a cake begins to form. Smaller particles enter the medium, whereas larger particles bridge the openings and start the buildup of a surface layer. Depth filtration is generally used to remove small quantities of contaminants. Cake filtration is primarily employed for more concentrated slurries. In practice, cake filtration is employed in industry more often than depth filtration. The following discussion will be concerned with cake filtration. For purposes of mathematical treatment, cake filtration processes are classified according to the variations of both pressure and flow rate with time. The pumping mechanism generally determines the flow characteristics and serves as a basis for division into the following categories: (a) constantpressure filtration (the actuating mechanism is compressed gas maintained at a constant pressure, or a vacuum pump), (b) constant-rate filtration (positive displacement pumps of various types are employed), and (c) variable-pressure, variable-rate filtration (the use of a centrifugal pump results in the rate varying with the back pressure on the pump). The relation between filtration pressure p and time u for the three types of filtration is illustrated in Figure 16.1. The constant-pressure curve is represented by a horizontal line. The pressure increases with time linearly for the constant-rate filtration of an incompressible cake. For a compressible cake formed in constant-rate filtration, the p versus u curve is concave upward. Variable-pressure, variable-rate filtration is conducted by using a filter actuated by a centrifugal pump. Depending on the characteristics of the centrifugal pump, widely differing curves might be encountered. The structure of the cake formed and, consequently, its resistance to liquid flow depend on the properties of the solid particles and the liquid-phase suspension, as well as on the conditions of filtration. The cake structure is first established by such hydrodynamic factors as cake porosity, mean particle size, size distribution, particle-specific surface area, and sphericity. It is also strongly influenced by some factors that can be denoted conditionally as physicochemical. The influence of 683 © 2006 by Taylor & Francis Group, LLC
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FIGURE 16.1 Relation between filtration pressure and time for different operations.
physicochemical factors is closely related to surface phenomena at the solid–liquid boundary. For fine particle suspensions, colloidal forces control the nature of the filter cake. The repulsive electrostatic forces will vary with the surface charge of the suspended particles, which varies with the solution environment. Therefore, the filtration behaviors of the colloids are affected significantly by the solution properties, including pH and electrolyte strength.1
5.16.1
BASIS OF CAKE FILTRATION THEORY
Equations of Flow through Porous Media Basic laws governing the flow of liquids through uniform, incompressible beds serve as a basis in developing formulas for more complex, nonuniform, compressible filter cakes formed on the filter medium during cake filtration. Darcy’s law can be expressed in the form dpL m u dx Kp
(16.1)
where pL is the local hydraulic pressure, x is the distance from the filter medium, m is the viscosity of liquid, Kp is the permeability, and u is the apparent liquid velocity relative to the solids. This velocity is expressed as the volumetric flow rate per unit area, which is defined by u q
r q er 1
(16.2)
where q is the apparent flow rate of liquid, is the local porosity, r is the apparent migration rate of solid particles, and e is the local void ratio. In filtration, it is customary to use the volume v of © 2006 by Taylor & Francis Group, LLC
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dry solids per unit medium area instead of the distance x from the medium. Thus, the incremental volume dv is given by d v (1 ) dx
(16.3)
Substituting Equation 16.3 into Equation 16.1, one obtains dpL m u mrs au dv K p (1 )
(16.4)
where r s is the true density of solids and a is the local specific filtration resistance.
Drag on Particles When suspended solids are deposited during cake filtration, liquid flows through the interstices of the compressible cake in the direction of decreasing hydraulic pressure. The solids forming the cake are compact and relatively dry at the filter medium, whereas the surface layer is in a wet and soupy condition. Thus, the porosity is minimum at the point of contact between the cake and the filter medium where v 0 (Figure 16.2) and maximum at the cake surface (v v0, where v0 is the solids volume of the entire cake per unit medium area) where the liquid enters. The drag imposed on each particle is communicated to the adjacent particles. Consequently, the net solid compressive pressure increases as the filter medium is approached, thereby accounting for the decreasing porosity. On the assumption that inertial forces are negligible, the force balance over the increment dv can be described by pL ps 0 v v
(16.5)
pL ps p
(16.6)
or upon integration,
where ps is the local solid compressive pressure. Equation 16.5 and Equation 16.6 clearly state that the solid compressive pressure increases as the hydraulic pressure decreases, as shown in Figure 16.2.
Porosity and Specific Filtration Resistance It is generally assumed in compressible cake filtration theory that the local porosity and specific filtration resistance are unique functions of the solid compressive pressure. These relations can be determined accurately by use of the compression-permeability cell.2 Such relations can be also determined directly from the filtration experiments in which a filter is subjected to a sudden reduction in its filtration area.3 As a fair approximation of compression-permeability cell data for a number of substances, the power functional relations among the porosity, specific filtration resistance, and solid compressive pressure can be employed as follows: i ,
ps pi l s
0 p
,
ps pi
a ai ,
ps pi
a bp a0 p p ⬇ a0 p p , n s
n s
Ps Pi
(16.7)
(16.8)
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FIGURE 16.2 Schematic diagram of compressible filter cake.
where n is a compressibility coefficient, which is equal to zero for incompressible substances. The local specific filtration resistance a is related to the local porosity in the form a
kS0 2 (1 ) rs 3
(16.9)
where S 0 is the effective specific surface area per unit volume of the particles and k is the Kozeny constant which normally takes the value of 5.
Average Specific Filtration Resistance Combining Equation 16.4 and Equation 16.5, the basic flow equation can be rewritten as dp dpL s mrs au dv dv
(16.10)
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Integration of Equation 16.10 is carried out between the limits (see Figure 16.2) of v 0 at the filter medium and v v0 at the cake surface. In addition, ps is taken as p – p m at the filter medium and 0 at the cake surface. The pressure p m at the exit of the filtrate from the cake is related to the resistance R m of the filter medium by pm mu1 Rm
(16.11)
where u1 is the filtration rate. Integration of Equation 16.10 on the postulate that u is constant ( u ) 1 throughout the cake leads to dv p u1 du m ( aav w Rm )
(16.12)
where v is the filtrate volume per unit medium area, u is the filtration time, w ( rSv0) is the solids mass of the entire cake per unit medium area, and a av is the average specific filtration resistance, which is defined by aav
p pm
∫
p pm
0
(1 / a) dps
(16.13)
The total cake resistance changes as the mass of cake grows with time. Many analyses of filtration start with Equation 16.12. Recently, the cake filtration theory can be also used to analyze membrane separation such as ultrafiltration of protein solutions, and it has the potential for analyzing the membrane fouling during ultrafiltration. 4 The above-mentioned derivation is not rigorous in that it has been assumed that u and the filtration area are constant throughout the cake. Shirato et al.5 presented a more complex equation, which took the variation of u with distance into account. Equation 16.13 is only approximately correct; hence, the average specific filtration resistance must be modified for a general expression. This is apparent, especially for thick slurries, as
aav J s aav
(16.14)
where a' av is the average specific filtration resistance that accounts for the internal flow variations in filter cake, and J s is the correction factor. The factor J s depends on both the filtration pressure and the slurry concentration. Although the pressure has relatively little effect, Js can change remarkably for the concentrated slurry of compressible materials, as illustrated in Figure 16.3.5 Whenever the filtering area varies, as in radial flow filtration, Equation 16.12 must be modified. In the external two-dimensional filtration onto the cylindrical element, the filtration rate under constant-pressure conditions can be given by
{
}
p (ro / ri ) 1 dv du 2 maav w ln (ro / ri ) 2
(16.15)
where ri is the radius of the medium surface and ro is the radius of the cake surface. © 2006 by Taylor & Francis Group, LLC
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FIGURE 16.3 Effect of slurry concentration s on correction factor Js.
Material Balance From an overall viewpoint of filtration, a material balance can be written on a unit medium area basis in the form w mw rv s
(16.16)
where s is the average mass fraction of solids in the slurry, m is the ratio of wet to dry cake mass, and r is the density of the filtrate. This material balance assumes that all feed slurry is filtered to form a cake. Solving for w in Equation 16.16, one gets w
rs v 1 ms
(16.17)
Frequently, the quantity m is related to the average porosity av of the cake by m 1
rav rs (1av )
(16.18)
It is important to relate the cake thickness L to the filtrate volume v per unit medium area. For the entire cake, Equation 16.3 in combination with Equation 16.17 yields w
5.16.2
rs v rs (1av ) L 1 ms
(16.19)
CONSTANT-PRESSURE AND CONSTANT-RATE FILTRATION
Constant-Pressure Filtration An accurate solution of Equation 16.12 requires numerical techniques. However, on the postulate of constant-pressure filtration that a av is constant during filtration and a function of p alone, a simple © 2006 by Taylor & Francis Group, LLC
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relation between v and u can be obtained. Combining Equation 16.12 with Equation 16.17, one obtains du 2 ( v vm ) dv K v
(16.20)
where vm is the fictitious filtrate volume per unit medium area required to form a filter cake of resistance equal to the medium resistance Rm, and Kv is Ruth coefficient of constant-pressure filtration6 defined by Kv
2 p (1 ms ) maav rs
(16.21)
Integration of Equation 16.20 yields
( v vm )2 K v (u um )
(16.22)
where um is the fictitious filtration time corresponding to the medium resistance. Equation 16.22 yields a parabolic relation between v and u. Constant-pressure filtration has long been the favorite method for obtaining experimental data in the laboratory because of its simplicity. Interpretation of data of constant-pressure filtration test is generally based on Equation 16.20. It might appear easier to use Equation 16.22 in the form u 1 2 v vm v Kv Kv
(16.23)
Thus, plotting u/v against v produces a straight line. Equation 16.20 is, in fact, better because it does not require identification of the precise time at which u 0. Plotting the reciprocal filtration rate (du/dv) against v may lead to a straight line having the slope of 2/Kv in accordance with Equation 16.20. Knowing the value of Kv, it is possible to obtain the value of aav from Equation 16.21 and then to construct a logarithmic graph of aav versus p. The medium resistance can be calculated from the intercept. Empirically, aav can be represented as functions of p by aav a0 ( p pm ) ⬇ a0 p n ’ n’
(16.24)
Constant-Rate Filtration Substituting Equation 16.11 and Equation 16.17 into Equation 16.12, one obtains u1
1 ms ( p pm ) maav rsv
(16.25)
In constant-rate filtration, the filtrate volume is related to the time by the simple relation v u1u
(16.26)
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Substituting Equation 16.26 into Equation 16.25 and solving it with respect to time u, one obtains the relation between p and u at constant-rate filtration as u
1 ms ( p pm ) maav rsu12
(16.27)
If aav is represented by Equation 16.24, Equation 16.27 leads to 1 ms 1n
p pm ) 2 ( ma0 rsu1
u
(16.28)
In constant-rate filtration, the pressure-time relationship is measured. A logarithmic plot of the pressure drop across the filter cake against time yields a straight line. When the cake is incompressible (n′ 0), Equation 16.28 implies a linear relationship between filtration time and pressure drop, as illustrated in Figure 16.1.
5.16.3
INTERNAL STRUCTURE OF FILTER CAKE
In filter cakes, the variation of porosity with distance from the cake surface is important from both theoretical and industrial viewpoints. In the development of filtration theory, the porosity plays a fundamental role in its relation to flow rates, pressure, and other parameters involved in the differential equations of flow through compressible, porous material. Porosity variation determines the average porosity and liquid content of the filter cake in commercial operation. Equation 16.10 is integrated over a portion of the cake and then the entire cake, assuming that u is constant ( u1) throughout the cake. Also, it is assumed that the medium resistance is negligible. Integrating from v to v0 and 0 to v0 yields respectively 1 ps (1 / a) dps mu1 ( v0 v) rs ∫0 1 p (1 / a) dps mu1v0 rs ∫0
(16.29) (16.30)
Equation 16.30 provides a relationship among the filtration rate u1, the solids volume v0, and the pressure drop p across the cake. Dividing Equation 16.29 by Equation 16.30, one obtains
∫ (1 / a) dp ∫ (1 / a) dp ps
s
0
p
0
s
1
v v0
(16.31)
To convert v/v0 in Equation 16.31 into x/L, the following equation can be employed: v / v0 d ( v / v ) 0 x ∫0 1 1 d (v / v ) L 0 ∫0 1
(16.32)
Consequently, Equation 16.31 with aid of Equation 16.32 gives the fractional distance through the cake as a function of the upper limit ps of integration. It indicates that ps versus x/L curves are independent of © 2006 by Taylor & Francis Group, LLC
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flow and total thickness. In turn, pL (equal to p – ps) and can be related to x/L through their functional relationships to ps. Equation 16.10 with aid of Equation 16.3 can be used in an analogous manner to relate ps to the distance through the cake directly by the form dps a (1 ) x 1 p dps L ∫0 a(1)
∫
ps
0
(16.33)
Because u1 varies markedly throughout the cake for highly concentrated slurries, Equation 16.31 and Equation 16.33 must be modified. The power relation can be used to approximate the porosity versus solid compressive pressure data as follows:
1 E ps l
(16.34)
Substituting Equation 16.34 and Equation 16.8 into Equation 16.33, the integral in Equation 16.33 can be replaced by approximate formulas to yield7 ⎛ pL ⎞ ⎜⎝ 1 p ⎟⎠
1nl
1
x L
(16.35)
FIGURE 16.4 Relation between hydraulic pressure and normalized distance. © 2006 by Taylor & Francis Group, LLC
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FIGURE 16.5 Relation between local porosity and normalized distance.
In Figure 16.4, the fractional hydraulic pressure drop is shown as a function of the fractional distance through the cake for a number of substances.8 As the compressibility increases, the hydraulic pressure drop becomes large near the filter medium. In Figure 16.5, the porosity is plotted against the fractional distance through the cake. 9 Under constant-pressure filtration, the local porosity depends on the normalized distance x/L alone.
5.16.4
NON-NEWTONIAN FILTRATION
In filtration of non-Newtonian fluid–solid mixtures, an empirical relation known as a power law is widely used in representing the rheological behavior of the filtrate in the form t Kg&
N
(16.36)
. where t is the shear stress, K is the fluid consistency index, g is the shear rate, and N is the flow behavior index, which is a measure of the degree of the non-Newtonian behavior of the fluid; the greater the departure from unity of N, the more remarkable is the non-Newtonian behavior of the fluid. On the basis of Equation 16.36, the equation for describing the filtration rate of power-law non-Newtonian fluids can be obtained in the form10 K gav rs ( v vm ) ⎛ 1⎞ ⎛ du ⎞ ⎜⎝ u ⎟⎠ ⎜⎝ dv ⎟⎠ p (1 ms ) 1 N
N
(16.37)
where g av is the average specific filtration resistance for power-law fluids. In Figure 16.6, du/dv and (du/dv) N are plotted against v for constant-pressure filtration. The curve of du/dv versus v is concave upward, whereas the plot of (du/dv)N versus v shows a linear relationship in accordance with the theory indicated by Equation 16.37. From the analysis for the internal structure of filter cake, it is derived that the cake formed by non-Newtonian filtration of pseudoplastic fluids (N 1) is denser in structure than that formed by usual Newtonian filtration. © 2006 by Taylor & Francis Group, LLC
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N
FIGURE 16.6 du/dv and (du/dv) versus v in non-Newtonian filtration.
5.16.5
FILTRATION EQUIPMENT
A wide variety of filters is commercially available in many industries. The operating principles and important features of typical filtration equipment are described below. 11,12
Filter Press The filter press has been the best known and perhaps the most widely used of all batch pressure filters. The common filter press is the plate-and-frame design, which is held together as a pack in a strongly constructed framework, as illustrated in Figure 16.7. A varying number of filter chambers are assembled, which consist of medium-covered plates alternating with frames that provide space for the cake. The chambers are closed and tightened by a screw or hydraulic ram, which forces the plates and frames together, making a gasket of the filter cloth. The slurry feed is introduced into the chambers under pressure and fills each chamber approximately simultaneously. The liquid passes through the filter medium, which in turn retains the solids. The filtrate is removed at a discharge outlet. The filter cake forms until the frames are full. Completion of the cake formation is judged by the filtering time, decrease in the rate of the feed, or rise in back pressure. Once the frames are full © 2006 by Taylor & Francis Group, LLC
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FIGURE 16.7 Schematic view of plate-and-frame filter press.
of cake, filtration is stopped and a wash liquid is applied if the solids are to be recovered. This can be followed by deliquoring of the filter cake.
Rotary Drum Filter The rotary drum filter is the most common continuous filter. A standard drum filter consists of three main parts: a drum mounted horizontally with an automatic filter valve; a slurry reservoir with a stirring device; and a scraper for cake discharge, as illustrated in Figure 16.8. The cylindrical drum, having a porous wall and covered on the outside with the filter media, rotates about a horizontal axis with a portion immersed in the slurry. The filter operates continuously through stages of cake formation due to filtration, washing, drying with air or steam, and discharge with an air blower. The normal driving force is vacuum, although enclosed pressurized units are sometimes built. The drum is usually subdivided into a number of separate compartments so that the various stages can be performed. The principal advantage of this filter is the continuity of its operation.
Thin-Cake or Dynamic Filter The thin-cake or dynamic filter has appeared on the market in recent years. Crossflow filtration operates by recirculating the feed flow parallel to the filter medium. The velocity of the feed in recirculation sweeps away particles deposited on the filter medium, thereby limiting the cake thickness. Crossflow filtration is especially effective for membrane filtration of colloids. The continuous pressure filter shown in Figure 16.9 is a kind of filter thickener, consisting of fixed filtering plates alternating with thin rotating disk. The filtering plates are covered with a filter medium on both sides, and the thin disks are rotated at a high speed sufficient to prevent the cake growth on the filter medium. The feed slurry is dewatered as it flows through the stages, and it emerges continuously from the discharge valve as a concentrate of a pastelike consistency without the danger of filter blocking because the slurry is continuously moving. Little cake formation occurs in the intensive shearing induced by high slurry velocities parallel to the filter medium. As a result, high filtration rates are accomplished. Very thin cake formation allows such a filter to be designed of relatively small size. © 2006 by Taylor & Francis Group, LLC
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FIGURE 16.8 Schematic view of rotary drum filter.
FIGURE 16.9 Schematic view of thin-cake filter.
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REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Iritani, E., Toyoda, Y., and Murase, T., J. Chem. Eng. Jpn., 30, 614–619, 1997. Grace, H. P., Chem. Eng. Prog., 49, 303–318, 1953. Iritani, E., Nakatsuka, S., Aoki, H., and Murase, T., J. Chem. Eng. Jpn., 24, 177–183, 1991. Iritani, E., Hattori, K., and Murase, T., J. Membrane Sci., 81, 1–13, 1993. Shirato, M., Sambuichi, M., Kato, H., and Aragaki, T., AIChE J., 15, 405–409, 1969. Ruth, B. F., Ind. Eng. Chem., 27, 708–723, 1935. Tiller, F. M. and Cooper, H. R., AIChE J., 8, 445–449, 1962. Tiller, F. M. and Shirato, M., AIChE J., 10, 61–67, 1964. Shirato, M., Aragaki, T., Ichimura, K., and Ootsuji, N., J. Chem. Eng. Jpn., 4, 172–177, 1971. Shirato, M., Aragaki, T., Iritani, E., Wakimoto, M., Fujiyoshi, S., and Nanda, S., J. Chem. Eng. Jpn., 10, 54–60, 1977. 11. Cheremisinoff, N. P., in Liquid Filtration, Butterworth-Heinemann, Woburn, MA, 1998, pp. 88–117. 12. Shirato, M., Murase, T., Iritani, E., Tiller, F. M., and Alciatore, A. F., in Filtration: Principles and Practices, Matteson, M. J. and Orr, C., Eds., Marcel Dekker, New York, 1987, pp. 309–324.
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5.17
Expression Eiji Iritani Nagoya University, Chikusa-ku, Nagoya, Japan
Expression is the operation of squeezing the liquid from solid–liquid mixtures by compression under conditions that allow the liquid to pass while the solid particles are retained between the supporting media. Expression is distinguished from filtration in that the pressure is applied by movement of the retaining walls such as pistons, membranes, rollers, or belts rather than by pumping a slurry into a fixed-volume chamber. In filtration, the material is sufficiently fluid to be pumpable, whereas in expression, the material can appear to be either entirely semisolid or a slurry. Operational expressions have become increasingly important in such widely divergent fields as food processing and fermentation industries, wastewater treatment, and chemical process industries because the energy required to express liquid from cakes is negligible compared to the heat required for thermal drying.
5.17.1
BASIS OF EXPRESSION
Figure 17.1 shows the schematic view of the compression-permeability cell employed to investigate the deliquoring mechanism of the compressed cake during an expression operation. In the experiments, the solid–liquid mixture is introduced into a cylinder, and a constant mechanical load is applied through the piston. The liquid squeezed from the mixture is allowed to drain from both the top and bottom media. The variation with time u of the thickness L of the mixture or compressed cake is measured by the dial gauge. The expression is made up of two stages, filtration and consolidation, according to the flow mechanism within the porous materials. When the original mixture in the cylinder is able to flow as a slurry, applying a load to the piston causes a sudden, uniform increase of hydraulic pressure in the slurry, and the resulting hydraulic pressure is equal to the applied pressure, and consequently deliquoring proceeds on the principle of filtration. Slurry filtration terminates in cake consolidation when the whole slurry forms a filter cake. In the consolidation stage, the bulk volume of the cake is reduced. On the basis of a mass balance, the maximum thickness L 1 of the filter cake can be calculated from ⎛ m 1 1 ⎞ ⎟ rs v0 L1 ⎜ rs ⎠ ⎝ r
(17.1)
where m is the ratio of the mass of wet cake to the mass of dry cake, r is the density of liquid, rs is the true density of solids, and v0 is the net solid volume per unit medium area. Deliquoring due to consolidation is distinguished from that due to filtration in that the hydraulic pressure distribution through a compressed cake varies significantly. In contrast, the hydraulic pressure distribution through a filter cake during constant-pressure filtration essentially does not change. Provided the original concentration of the mixture is larger than a limiting value, deliquoring of the mixture proceeds from the beginning on the principle of consolidation.
5.17.2
MODIFIED TERZAGHI MODEL
The equations for describing the consolidation behaviors are derived from the basic equation of flow through porous media. It is convenient to use a solid particle distribution represented by v divided by the cross-sectional area, where v is a moving plane that represents the solid volume 697 © 2006 by Taylor & Francis Group, LLC
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FIGURE 17.1 Schematic view of compression-permeability cell.
between the plane and the medium. The apparent velocity u of the liquid viewed from this moving material can be described by u
1 pL 1 ps ⋅ ⋅ mars v mars v
(17.2)
where m is the viscosity of liquid, a is the local specific flow resistance, pL is the local hydraulic pressure, and ps is the local solid compressive pressure. The mass balance of liquid in the infinitesimal layer leads to the equation of continuity in the form e u uc v
(17.3)
where e is the local void ratio and uc is the consolidation time. Combination of Equation 17.2 with Equation 17.3 leads to ⎧⎪ 2 p 1 d a ⎛ ps ⎞ 2 ⎫⎪ ps Ce ⎨ 2s ⋅ ⎬ uc a dps ⎜⎝ v ⎟⎠ ⎭⎪ ⎩⎪ v
(17.4)
where C e is the modified average consolidation coefficient defined by 1
⎛ de ⎞ ⎪⎫ ⎪⎧ Ce ⎨mars ⎜ ⎟ ⎬ ⎝ dps ⎠ ⎭⎪ ⎩⎪
(17.5)
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Combining Equation 17.2 with Equation 17.3 on the assumption that Ce is constant, one obtains the well-known form of the diffusion equation ps 2 ps Ce uc v2
(17.6)
This equation is similar to the basic consolidation equation presented by Terzaghi1 in the field of soil mechanics with spatial fixed coordinates. It is important to know the initial condition, specifically distributions of the hydraulic pressure (or the solid compressive pressure) in the cake, in order to obtain the solution of Equation 17.6. The hydraulic pressure distributions within filter cakes with moderate compressibility can be generally approximated by a sinusoidal curve. Thus, the solution of Equation 17.6 for a constant-pressure expression of filter cake is given by2 Uc
⎛ p 2Tc ⎞ L1 L 1 exp ⎜ 4 ⎟⎠ L1 L∞ ⎝
(17.7)
where Uc is the average consolidation ratio, and L1, L, and L∞ are the thickness of the cake at uc 0, uc, and , respectively. The dimensionless consolidation time Tc is defined by Tc
i 2Ce uc v0 2
(17.8)
where i is the number of drainage surfaces. This equation clearly indicates that the consolidation time u c required for attaining a specified value of the degree of consolidation is directly proportional to v02 and is inversely proportional to i2. The solution of Equation 17.6 for a constant-pressure expression of homogeneous semisolid materials is given by ∞ ⎧⎪ ( 2 N 1)2 p 2 ⎫⎪ L1 L 8 Uc exp ⎨ 1 ∑ Tc ⎬ 2 2 4 L1 L∞ N1 ( 2 N 1) p ⎪⎩ ⎪⎭
(17.9)
Figure 17.2 compares the experimental data for consolidation of a homogeneous semisolid of the Korean kaolin1 with calculations using Equation 17.9. The agreement is rather poor, whereas the calculated curve is similar in shape to the experimental curve. In addition, although Equation 17.4 can be solved numerically by considering variations of Ce, the agreement is still poor. The practical and important quantity Ce can be determined by the “fitting method” by consideration of the similarity in shape of the theoretical curve of Uc versus c ., and experimental observations. In constant-pressure expression of the filter cake, the time u90 for attaining 90% of Uc can be obtained from the experimental data, as shown in Figure 17.3, and thus the value of Ce is calculated from Ce
0.933v0 2 i 2u90
(17.10)
The quantity Ce can also be estimated using the compression-permeability cell data shown in Figure 17.4. The specific flow resistance a is plotted against the solid compressive pressure ps, at the left in the figure, and the void ratio e is plotted against ps, at the right, for a number of materials with widely different compressibility. Compression-permeability cell data can be represented by a a0 a1 ps n
(17.11)
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FIGURE 17.2 Constant-pressure expression of semisolid material.
FIGURE 17.3 Fitting method for determining Ce from experimental data.
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e e0 Cc ln ps
(17.12)
On the basis of the empirical constitutive equations 17.11 and 17.12, the modified average consolidation coefficient Ce can be calculated from Ce
(
ps,av
mrsCc a0 a1 ps,av n
)
(17.13)
where ps.av is the average solid compressive pressure, which is often approximated by the arithmetic mean value of the initial average solid compressive pressure ps(e1.av) within the cake and the applied pressure p. The symbol e1.av denotes the initial average void ratio. Since this equation essentially affords an estimation of Ce, the consolidation process can be theoretically predicted from calculations based on compression-permeability cell data.
5.17.3
SECONDARY CONSOLIDATION
Secondary consolidation has to be taken into account in order to obtain more rigorous equations for describing the consolidation behavior. Because the variation of the void ratio e within the cake is caused by both the change in the local solid compressive pressure ps and the simultaneous creep effect of materials, e is a function of both ps and the consolidation time uc. Assuming that the rheological behavior of secondary consolidation is described by the Voigt element shown in Figure 17.5, the average consolidation ratio Uc is given by Uc
⎧⎪ ⎛ p 2 Tc ⎞ ⎫⎪ L1 L (1 B ) ⎨1 exp ⎜ ⎬ B 1 exp (huc ) L1 L∞ 4 ⎟⎠ ⎪⎭ ⎝ ⎪⎩
{
}
(17.14)
FIGURE 17.4 Compression-permeability cell data. © 2006 by Taylor & Francis Group, LLC
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for constant-pressure expression of filter cakes, and by Uc
⎡ ⎧⎪ ( 2 N 1)2 p 2 ⎫⎪ ⎤ L1 L 8 (1 B ) ⎢1 ∑ Tc ⎬ ⎥ exp ⎨ 2 L1 L∞ 4 ⎢⎣ N 1 ( 2 N 1) p 2 ⎪⎩ ⎪⎭ ⎥⎦
{
(17.15)
}
B 1 exp (huc )
for constant-pressure expression of semisolid materials.3 The quantity B is an empirical constant defined by B vsc. max / vc. max
(17.16)
where vsc.max and vc.max are the total liquid volumes squeezed by the secondary consolidation and until final equilibrium, respectively. The quantity h is the empirical constant due to creep of the materials. Because –p 2Tc/4 h for the large values of uc, Equation 17.14 and Equation 17.15 become approximately Uc
L1 L ≈ 1 B exp (huc ) L1 L
(17.17)
FIGURE 17.5 Schematic view of Terzaghi–Voigt combined model. © 2006 by Taylor & Francis Group, LLC
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Consequently, the values of both B and h can be determined from the later stage of the plot of ln(1 – Uc) versus uc. In Figure 17.2, the prediction based on Equation 17.15 is compared with the experimental data. The data are in relatively good agreement with the predictions. However, with biological and cellular materials, the consolidation mechanism is much more complex because the liquid is stored mainly in cells.4
5.17.4
SIMPLIFIED ANALYSIS
The solutions based on the modified Terzaghi model give less satisfactory values, especially in the later stage of consolidation. Sophisticated solutions based on the Terzaghi–Voigt combined model coincide very well with the experimental results. However, the solutions might not be useful in normal industrial practice because they include the three empirical constants Ce, B, and h, which have to be determined from two graphical plots. It is clear from Figure 17.2 that the relation between Uc and uc , calculated by Equation 17.9, is linear and nearly coincides with experimental data in the early stage of consolidation of semisolid material under constant-pressure conditions. However, agreement is poor in the later stage, and the theoretical curve approaches the final equilibrium more rapidly than the experimental results. Because Uc is directly proportional to uc for small values of uc and asymptotically approaches unity for large values of uc, one of the simplest equations, instead of Equation 17.9, can tentatively be derived as5 n 1 / n ⎤ ⎡4 ⎧ L1 L 4 ⎪ ⎪⎫ ⎛ ⎞ ⎥ Uc ⎢ Tc ⎨1 ⎜ Tc ⎟ ⎬ L1 L∞ ⎢ p ⎩⎪ ⎝ p ⎠ ⎭⎪ ⎥ ⎣ ⎦
1/ 2
(17.18)
where n is the consolidation behavior index which takes secondary consolidation effects into account. When Uc is plotted against Tc , the maximum percentage of error calculated by Equation 17.18 with n 2.85, compared to Equation 17.9, is only 0.60. It was previously indicated that Equation 17.9 gives less satisfactory values. However, it can be seen from Figure 17.2 that Equation 17.18 with n < 2.85 gives quite satisfactory results, and the effect of secondary consolidation is considered in this range.
5.17.5
EXPRESSION EQUIPMENT
Various types of equipment for expression are available in industry. 6 The automated filter press with a compression mechanism was developed in the 1960s in order to reduce the cake moisture content. A lateral, recessed plate and an expression plate with the impermeable membrane comprises a filter chamber, as shown in Figure 17.6. After feed slurry is forced into the chamber under pressure, filtration occurs and the membrane is pressed against the expression plate. Once the chamber is filled with filter cake, the membrane is expanded by pumping compressed air or water into the cavity between the membrane and the plate. As the membrane moves toward the opposite plate, deliquoring of the filter cake due to consolidation takes place. On completion of expression, the chamber is automatically opened and the cake is discharged. As shown in Figure 17.7, the belt press as an example of continuous expression equipment has a horizontal gravity dewatering zone of belt preceding the pressure dewatering zone. The slurry is sandwiched between the carrying belt and the cover belt at the end of the gravity dewatering zone. From this point to the cake discharge point, the sludge is compressed by compressive and shear forces that are applied through the belt. In recent years, a screw press, used in the oil milling industry as an expeller, has been widely employed in the chemical industry and sewage sludge treatment. In the screw press, materials are fed to a perforated barrel containing a rotating worm. Materials trapped between the worm and the inside of the cylindrical barrel pass through a gradually reducing flow area, experiencing an increasing pressure. The compressed cake leaves the unit through an adjustable discharge port. © 2006 by Taylor & Francis Group, LLC
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FIGURE 17.6 Schematic view of automatic filter press.
FIGURE 17.7 Schematic view of belt press.
REFERENCES 1. 2. 3. 4. 5.
Terzaghi, K., in Theoretical Soil Mechanics, John Wiley, New York, 1948, pp. 265–296. Shirato, M., Murase, T., Kato, H., and Fukaya, S., Kagaku Kogaku, 31, 1125–1131, 1967. Shirato, M., Murase, T., Tokunaga, A., and Yamada, O., J. Chem. Eng. Jpn., 7, 229–231, 1974. Lanoisellé, J. L., Vorobyov, E. I., Bouvier, J. M., and Piar, G., AIChE J., 42, 2057–2068, 1996. Shirato, M., Murase, T., Atsumi, K., Aragaki, T., and Noguchi, T., J. Chem. Eng. Jpn., 12, 51–55, 1979. 6. Shirato, M., Murase, T., Iritani, E., Tiller, F. M., and Alciatore, A. F., in Filtration: Principles and Practices, Matteson, M. J. and Orr, C., Eds., Marcel Dekker, New York, 1987, pp. 371–377.
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5.18
Flotation Hiroki Yotsumoto National Institute of Advanced Industrial Science and Technology, Tsukuba, Ibaraki, Japan
Flotation is one of the most important methods for mineral separation. Because of its capability of fine particle processing, flotation has been applied to low-grade ores that other separation methods cannot handle economically. In flotation, ore is finely ground and is mixed with water to make a suspension called pulp. After the addition of flotation reagents such as a collector, frother, and so on, the pulp is subject to violent agitation with aeration in a flotation cell for sufficient collision between minerals and bubbles. The targeted minerals coated with hydrophobic collector film are captured at the air–water interface of bubbles and are lifted to the surface of the pulp to form a froth layer. The froth is skimmed off from the flotation cell as a concentrate, whereas unwanted minerals remaining in the pulp are discharged as tailing. Plain flotation was first practiced to obtain one type of concentrate from simple sulfide ores. Differential flotation was later developed to recover progressively more than two kinds of concentrates from complex sulfide ores bearing many valuable minerals. With the development of flotation reagents as well as flotation machines, flotation has been applied successfully to recovering many kinds of minerals including soluble-salts minerals. Recently it has also been applied to recovering various substances such as chemically precipitated colloids, ions, microorganisms, and so on in chemical and food industries and in wastewater and seawater treatment.
5.18.1
PRINCIPLES OF FLOTATION
Flotation is a complex physicochemical process and its theory is not completely understood. One of the theoretical approaches is to correlate the contact angle of mineral with its floatability. The static state in which a mineral particle is attached to a large bubble in water is similar to that of a particle at a free surface as illustrated in Figure 18.1.1 If the particle is assumed to be a cylinder at a free water surface as shown in Figure 18.1b, the lifting force, f, acting on the particle is expressed in the form f ⫽ 2prT sin u ⫹ prhrg
(18.1)
where r is the radius of the cylinder, T is the surface tension of water, r is the density of water, g is the acceleration of gravity, h is the depth of the dimple, and u is the contact angle as shown in Figure 18.1b. On the right-hand side of Equation 18.1, the first term is associated with the force due to the surface tension of water, and the second term is the force equal to the weight of the cylindrical mass of liquid displaced by air. The contribution of both terms to the lifting force, f, varies with the mineral size, r. For large particles, the lifting force can be given only by the second term. For small particles, on the other hand, the second term becomes negligible, and the lifting force is regarded as f ⫽ 2prT sin u
(18.2)
When T is constant, Equation 18.2 shows that the lifting force increases with the increase of u. The contact angle at the solid–water–air interface is a measure of the adhesion force between bubble and particle and also is a measure of the hydrophobicity of particulate solid. u in Equation 18.2 705 © 2006 by Taylor & Francis Group, LLC
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FIGURE 18.1 Suspension of a mineral particle at an air–water interface.
TABLE 18.1 Relationship between Contact Angle and Floatability for Various Kinds of Minerals Mineral Galena
Contact Angle (deg) 70–75
Floatability (%) 90
Sphalerite
71–72
95–98
Pyrite
58–73
89–92
Quartz
55–58
78–79
Calcite
45
11–56
Slate
13
6
Sandstone
0
1
is closely related to the contact angle of the suspended particulate solid. Such minerals as fresh galena, native sulfur, graphite, talc, and so on are generally hydrophobic, with very large contact angles. On the other hand, quartz, calcite, dolomite, corundum, and so on are considered to be hydrophilic, which means that their contact angles are relatively small. Peterson2 presented the relationship between contact angle and floatability for several kinds of minerals under certain experimental conditions, as shown in Table 18.1. Another approach is to explain the bubble–particle attachment with DLVO theory,3 which predicts the coagulation of two solid particles considering the balance of opposing surface forces, namely, electrostatic repulsion and molecular (van der Waals) attraction between the particles. In a slow coagulation condition, an energy barrier is formed at a certain interparticle distance by the interaction of repulsive and attractive potential energies (see Figure 18.2a). If the relative kinetic energy of the two particles is larger than the magnitude of the energy barrier, the particles are supposed to collide with each other. Assuming that a bubble is one of the particles, one can calculate the interaction energy between the bubble and particle using DLVO theory. In the case of bubble–particle attachment, molecular force is always repulsive, unlike the case of particle–particle attachment. When the bubble and particle are unequally charged, for instance, ⫹100 mV (bubble) and ⫹50 mV (particle), electrostatic repulsion turns into attraction at a very short bubble–particle distance.4 As a result, a very large, sometimes infinitely large, energy barrier is formed between bubble and particle. The potential curve Vt ⫽ Ve ⫹ Vd in Figure 18.2b shows this large energy barrier. Such a barrier is supposed to prevent babble–particle attachment, whereas flotation occurs even if an infinitely large barrier is predicted. To solve this discrepancy, some researchers assume the existence of “hydrophobic attraction” between a bubble and a particle. The force was found by Israelachvili and Pashley in 1982 between
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FIGURE 18.2 Concept of DLVO theory for coagulation and flotation.
mica surfaces coated with a surfactant monolayer.5 The hydrophobic attraction was of a long-range nature and was much larger than the van der Waals attraction. If this force exits between a bubble and a particle, it overcomes the van der Waals attraction and electrostatic repulsions and reduces the energy barrier to a certain magnitude, which corresponds with the floatability of mineral. The potential curve Vt′ ⫽ Ve ⫹ Vd ⫹ Vs in Figure 18.2b shows this small barrier. The hydrophobic attraction may be the major driving force of flotation. However, there are arguments about its existence in a bubble–particle system.
5.18.2
CLASSIFICATION OF MINERALS ACCORDING TO THEIR FLOTATION BEHAVIOR
In general, minerals can be classified into several groups depending on their surface property such as hydrophobicity or hydrophilicity. There are some minerals which have high natural floatability. Minerals of this type are graphite, sulfur, molybdenite, diamond, coal, and talc, as shown in group A in Table 18.2. They are sometimes called nonpolar minerals. The surfaces of nonpolar minerals are characterized by strong hydrophobicity depending on their chemical composition and bonding type. The nonpolar minerals are composed of atoms bonded with nonpolar covalent bonds or covalent molecules held together by van der Waals force. Minerals with polar covalent or ionic bonding are known as polar types. These minerals are naturally hydrophilic. Minerals belonging to the polar group are roughly divided into various subclasses B to F, as listed in Table 18.2, depending on the magnitude of hydrophilicity. Minerals in group B are sulfides and native metals. As can be seen in Table 18.2, the degree of hydrophilicity increases from sulfide minerals, through sulfates in group C, to carbonates, phosphates, and so on in group D, oxides in group E, and, finally, silicates and quartz in group F.
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TABLE 18.2 Group A B
Classification of Minerals
Surface Property Nonpolar
Minerals Graphite, sulphur, molybdenite, diamond, coal, talc
Polar
Sulphides: bornite, chalcopyrite, galena, pyrite, sphalerite stibnite, argentite, arsenopyrite, etc. Native metals: Au, Pt, Ag, Cu
5.18.3
C
Polar
Sulfates: barite, anhydrite, gypsum, anglesite
D
Polar
Carbonates: cerrusite, malachite, azurite, smithsonite, siderite, calcite, witherite, agnesite, dolomite Other minerals: fluorite, apatite, monazite, scheelite
E
Polar
Oxides: hematite, magnetite, goethite, chromite, ilmenite, rutile, corundum, pyrolusite, limonite, wolframite, cassiterite
F
Polar
Silicates: zircon, willemite, hemimorphite, beryl, feldspar, garnet, sillimanite, quartz
FLOTATION REAGENTS
Although a great number of chemicals have been known to be flotation reagents so far, no more than 50 chemicals seem to be employed in practical flotation plants at present. Flotation reagents can be classified into four categories based on their action in flotation: collectors, frothers, activators, and depressants, as indicated by Wark.6 A reagent can belong to more than one of these categories. For example, soluble organic compounds, besides being frothers, are collectors for some minerals as well. Lime and soda ash used to regulate the pH of the pulp are often called pH regulators. Further, a sufficient flotation of a desired mineral sometimes requires a good dispersion of gangue slime in the flotation pulp. The reagents used for this purpose are called dispersants.
Collectors A collector is generally an organic reagent that adsorbs on a mineral surface and makes it hydrophobic. Thus, it serves as a bridge between the mineral and air bubble. Common collectors are listed in Table 18.3.
Frothers Frothers are used to help the dispersion of air bubbles throughout the pulp in the flotation cell, to promote the stability of attachment between mineral and air bubble, and also to stabilize the froth formation at the surface of pulp. Table 18.4 shows the types of chemical reagent used as frothers in flotation. The type and molecular structure of frothers affect their frothing power, the stability of froth, and the physical characteristics of froth layer. Pine oil, whose main constituent is terpineol, is one of the most popular frothers in the flotation of various minerals.
Depressants When ore contains more than two valuable minerals to separate as individual concentrates, it is necessary to regulate the floatability of minerals. The reagents used to depress the flotation of desired minerals are called depressants. Reagents such as lime, caustic soda, and sodium carbonate are often used to regulate the pH of flotation pulp in the alkaline range. They are also depressants for sulfide minerals, because most sulfide minerals are depressed in high-pH solutions. Sodium cyanide, sodium sulfide, sodium sulfite, and sulfur dioxide gas are well-known depressants in sulfide flotation. Sodium silicate, fluoric acid, starch, tannin, lignite, and so on are depressants in the flotation of nonsulfide minerals. © 2006 by Taylor & Francis Group, LLC
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TABLE 18.3 Classification of Common Collectors Class Oils
Typical Collectors Coal tar oil, fuel oil, creosote
OR
Acids containing a hydrocarbon group and the salts of their acids
Alkyl dithiocarbonates:
S=C SM NR2
Dialkyl dithiocarbamates:
S=C SM OR
Dialkyl dithiophosphates:
S=P
OR SM
(R: hydrocarbon chain, C ⫽ 2–5) Fatty acids: RCOOM
Alkyl sulfonates: R–SO3M (R: hydrocarbon chain, C ⱖ 8)
Bases containing a hydrocarbon group and the salts of their bases
Amines derivatives: RNH3X (primary) R2NH2X(secondary) R3NHX (tertiary) R4N–X (quaternary) (R: hydrocarbon group, C ⱖ 8)
Aimed Minerals Natural sulfur, graphite, molybdenite, coal, etc. Sulfide minerals: chalcopyrite, chalcocite, convellite, sphalerite, galena, pyrite, etc. Phosphate ores: barite, chromite, fluorite, scheelite, hematite, calcite, etc. Iron oxides: kyanite, talc, garnet, chromite, barite, rhodochrosite, etc. Silica, mica, vermiculite, smithsonite, sylvite, feldspar, etc.
Activators Activators promote the adsorption of a collector onto a mineral surface by altering the chemical nature of the mineral surface. Copper sulfate is one of the most important activators for spharlerite when a sulfydril collector is used. It is also an activator for arsenopyrite, cobaltite, and stibnite. Sodium sulfide is, in general, an activator for several nonsulfide minerals, such as oxides, carbonates, and sulfates of heavy metals. For such minerals, sodium sulfide reacts with metal constituents at the solid–water interface to form a kind of metal sulfide film at the mineral surface. Then the minerals thus treated are amenable to flotation by using conventional collectors for sulfide minerals. Instead of sodium sulfide, sodium hydrosulfide and ammonium sulfide can be used. Lead nitrate is used as an activator for stibnite in flotation with sulfydril collector, and also as an activator for halite in the fatty acid flotation of this salt in the presence of sylvite. There are some other activation instances: the activation of mica by lead salts for flotation by an oleic acid collector and the activation of feldspar by hydrofluoric acid for flotation by an amine collector. © 2006 by Taylor & Francis Group, LLC
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TABLE 18.4
Classification of Common Frothers
Chemical Type Alcohols
Type Formula R–OH (R: hydrocarbon group)
Representative Frothers Aliphatic alcohols: R is a straight or branched hydrocarbon chain (C ⫽ 5–8) Cresylic acids: R is a benzene ring with shortchain alkyl substituents Pine oils: R is a terpene ring structure
Ethers
R–O–R’ (R, R’: hydrocarbon groups)
Alkoxy substituted paraffins (triethoxybutane) Polypropylene glycols Methoxytripropylene glycol
Ketones
R–C–R’ | O
Camphor
(R, R’: hydrocarbon groups)
5.18.4
FLOTATION MACHINES
Although many different machines have been developed and discarded in the past and many machines are currently being manufactured, it can be said that there are two groups: pneumatic machines and mechanical agitation or subaeration machines.
Pneumatic Machines Pneumatic machines use air blown in by means of pipes, nozzles, or perforated plates, in which case the air must be dispersed by baffles to create a great deal of bubbles in the pulp and to give sufficient aeration or agitation. Figure 18.3 shows one of the old types of pneumatic machines, called the SW cell. It consists of a long tank with a V-shaped base into which many vertical pipes deliver compressed air. Internal baffles provide sufficient agitation and aeration. The Davcra cell is shown in Figure 18.4. The cell consists of a tank segmented by a vertical baffle. Air and feed slurry are injected into the tank through a cyclone-type dispersion nozzle. The flotation column 7 has been developed for the flotation of fine particles, a possible configuration of which is shown in Figure 18.5. Air is introduced from the lower part of the column while feed slurry enters from the upper part of the column. Therefore, mineral particles countercurrently contact a rising swarm of air bubbles in the section below the feed point. Floatable particles collide with and adhere to the bubbles and are transported to the washing section above the feed point. Wash water cleans the froth and releases particles entrained in the water lifted by rising bubbles. Nonfloatable particles are removed from the bottom of the column as tailing. High-grade concentrates are obtained by the use of wash water. Industrial use of flotation columns has increased in recent years.
Mechanical Agitation or Subaeration Machines A mechanical agitation machine is equipped with a specially designed agitating device, called an impeller or rotor, which agitates the pulp violently and at the same time introduces natural air into the pulp by centrifugal pressure. This type of machine can either be self-aerating or have air blown in. The flotation is a continuous process. Flotation cells are arranged in series, forming a bank. There are two basic modes of flotation machine operation: cell-to-cell flotation and free-flow flotation. In the second of these modes, as there is no weir between cells; pulp is free to flow through the cells without interference. Figure 18.6 shows schematically the Denver Sub-A cell, which is of the cell-to-cell operation type. The flotation takes place in an individual square cell separated from the adjoining cell by an adjustable weir. The feed from the weir of the preceding cell enters the inlet pipe and is carried directly to the next © 2006 by Taylor & Francis Group, LLC
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FIGURE 18.3 SW flotation cell.
FIGURE 18.4 Davcra cell.
cell, the flow being aided by the suction action of the impeller. The pulp then falls on top of the rotating impeller while air is also drawn down the standpipe. This design assures a favorable air–pulp mixture. The mineral-laden bubbles go upward to form froth. Removal of the froth is accomplished by froth paddles. Tailings that do not float pass on from cell to cell, being subjected over and over to the flotation process. The Denver DR free-flow machine shown in Figure 18.7 is designed to handle larger tonnages © 2006 by Taylor & Francis Group, LLC
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FIGURE 18.5 Flotation column.
FIGURE 18.6 Denver Sub-A cell.
in the flotation circuit. In individual cells, feed pipes are eliminated. The pulp level is controlled by a single tailing weir at the end on the trough. This machine requires pressurized air. The cell that is most commonly used in industry is of a mechanical agitation type: the Agitair, Fargergren, Warman, and Denver cells. They all have their own special designs in agitation mechanism, air introduction, flow pattern of pulp in the cell, and collision efficiency between particles and bubbles. © 2006 by Taylor & Francis Group, LLC
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FIGURE 18.7 Denver DR flotation machine.
TABLE 18.5 Critical pH Values for Various Sulfide Minerals by Using 25 mg/l of Potassium Ethyl Dithiocarbonate Mineral Sphalerite Pyrrhotite Arsenopyrite
Critical pH Value (room temperature) — 6.0 8.4
Galena
10.4
Pyrite
10.5
Marcasite
11.0
Chalcopyrite
11.8
Covellite
13.2
Cu-activated sphalerite
13.3
Bornite
13.8
Tetrahedrite Chalcocite
13.8 ⬎14.0
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DIFFERENTIAL FLOTATION
Separating a complex ore into two or more valuable minerals and gangue by flotation is called differential flotation or selective flotation. The pH of flotation pulp is one of the most important parameters in view of selective flotation. For example, at a fixed addition of a collector, there is a pH value below which any given mineral floats and above which it does not float. This critical flotation pH value depends on the nature of the mineral, the type of collector, the collector concentration, temperature, and the type of existing chemical species, if any, and its concentration. Wark and Cox8 determined critical pH values for various types of sulfide minerals in terms of collector type, collector concentration, and temperature. One of the results is presented in Table 18.5. This table clearly shows the importance of controlling the pH of the pulp in the selective flotation of sulfide minerals. Besides controlling pulp pH, a successful differential flotation is generally achieved by the use of suitable depressants and activators.
5.18.6
PLANT PRACTICE OF DIFFERENTIAL FLOTATION
As a typical example of the differential flotation, the flow sheet of the Matsumine mill plant in the Hanaoka mine in Japan is shown in Figure 18.8. The mill was closed in 1995 because of economic problems, but a number of technological developments were achieved with respect to complex sulfide ore processing. The treated ore was complex sulfide ore, locally called “Kuroko” (black ore), which was composed of fine-grained minerals. The main sulfide minerals are chalcopyrite, pyrite, galena, and sphalerite, whereas some nonsulfide minerals are barite, gypsum, sericite, chlorite, kaolinite, montmorillonite, and quartz. After ore was crushed down to less than 100 mm in size at the underground crushing site, the run-of-mine ore was conveyed up to the mill plant. The ore was washed to remove slime and clay adhering to the ore,
FIGURE 18.8 Flowsheet of Matsumine mill plant. © 2006 by Taylor & Francis Group, LLC
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using a washing screen. As in Figure 18.8, the washed lamp ore was transported to the secondary crushing section where the size of the ore was reduced to less than 25 mm. Material put through the washing screen was classified into sand (less than 25 mm) and slime fractions. The sand fraction and the product of the secondary crushing section were combined as the feed of the grinding stage. At the grinding section, material was ground down to around 0.05 mm in size by two-stage grinding, namely the rod–ball mill grinding circuit, to produce flotation feed. In the flotation section, the flotation circuit consisted of five subcircuits to recover the following: 1. Bulk Cu-Pb concentrate in the bulk flotation of copper sulfide minerals and galena using diethyldithiophosphate as a collector after conditioning the flotation feed with sulfur dioxide gas and lime at pH 4–5.5 2. Cu concentrate in the copper mineral flotation by heating the bulk Cu-Pb slurry to about 70°C to depress galena, the tailings being the Pb concentrate 3. Zn concentrate in the activated sphalerite flotation with copper sulfate as an activator and xanthates as collectors when the bulk flotation tailings were used as the feed 4. Py concentrate in the flotation of pyrite with xanthate after adjusting the pH of pulp to 3.5 5. Ba concentrate in the flotation barite with alkyl-sulfonate as a collector at pH 6.0 Generally, each subcircuit consisted of rougher, scavenger, and cleaner steps. The slime obtained at the washing screen stage was thickened in a thickener and then treated through a specially designed slime-process circuit including cyclones, decanters, and flotators. The products of the slime-processing circuit were fed to appropriate sections of the principal concentration circuit, the tailing being discarded at the dam. As can be seen in Figure 18.8, five kinds of concentrates were recovered by differential flotation at the Matsumine mill plant.
REFERENCES 1. Gaudin, A. M., in Flotation, 2nd Ed., McGraw-Hill, New York, 1957, pp. 157–159. 2. Peterson, W., in Schwimmaufbereitung, Vertag von Theodor Steinkopff, Dresden, 1936, pp. 48–49. 3. Verwey, E. J. W. and Overbeek, J. Th. G., in Theory of the Stability of Lyophobic Colloids, Elsevier, New York, 1948, pp. 106–115. 4. Derjaguin, B. V., Churaev, N. V., and Muller, V. M., in Surface Forces, Plenum Publishing, New York, 1987, pp. 198–202. 5. Israelachvili, J., in Intermolecular and Surface Forces, 2nd Ed., Academic Press, San Diego: 1991, pp. 282–287. 6. Wark, I. W., in Principles of Flotation, Australian Institute of Mining and Metallurgy, Melbourne, 1938, p. 9. 7. Finch, J. A. and Dobby, G. S., in Column Flotation, Pergamon Press, New York, 1990, pp. 1–4. 8. Wark, I. W. and Cox, A. B., Trans. Am. Inst. Mining Metall. Eng., 112, 189–302, 1935.
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5.19
Electrostatic Powder Coating Ken-ichiro Tanoue Yamaguchi University, Ube, Yamaguchi, Japan
Hiroaki Masuda Kyoto University, Katsura, Kyoto, Japan Electrostatic powder coating has been widely utilized for surface finishing in many industrial applications.1,2 Recently, the coating has been making inroads in the automotive industry. Since the coating process does not use solvents, it is becoming increasingly attractive in light of possible new clean air standards for volatile organic compounds. The coating system consists of powder coating guns, powder feeding machine, powder coating booth, and powder collecting machine. It is important to combine effectively the equipment for various objectives.
5.19.1
COATING MACHINES
Different configurations for coating guns have been proposed, and they may use two distinct techniques for powder charging. One is corona-charging and the other is tribocharging. Furthermore, other coating machines have been developed.
Corona-Charging Guns Figure 19.1 shows a corona-charging gun. Powder particles are assembled with air and fed from a nozzle of the gun as a gas–solid two-phase flow. A needle electrode is set in the tip of the nozzle. Negative high voltage is applied on the needle electrode in a range of about –60 kV to –100 kV. At this time, a high electric field is formed between the needle and grounded objects. The powder is exposed to ionized air flow, and corona charging occurs. With continued increase in charge collection by particles, electric field strength E will increase until a certain value is reached corresponds to a situation where no further transport of ions on the particle is possible. This limiting value of surface charge, the Pauthenier limit,3 is expressed mathematically by the following relationship; ⎛ − 1⎞ 2 q = 4p 0 ⎜ 1+2 r rP E r + 1 ⎟⎠ ⎝
(19.1)
where, 0 is the permittivity of free space, 8.854 × 10–12 Fm–1, r is the relative permittivity of the powder particle, r p is the radius of the particle, and E is the electric field strength. Particles are transported toward the objects by coulomb force and drag force and then deposited. The coating gun has been utilized for almost all powder particles. However, as there are free ions in the booth, the “orange peel phenomenon” will occur sometimes on the surface of a thick powder layer. New corona guns with an earth ring to collect free ions have been developed to prevent the effect. 717 © 2006 by Taylor & Francis Group, LLC
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FIGURE 19.1 Corona-charging gun.
FIGURE 19.2 Tribocharging gun.
Tribocharging Guns Figure 19.2 shows a tribocharging gun. Powder particles are assembled with air flow and they collide with the inner wall of the gun. At this time, tribocharging of particles occurs. As the inner wall of the gun is commonly made of polytetrafluoroethylene, the powder particles are mainly charged positively on a basis of contact potential difference. Then the charged powder particles are ejected from a nozzle of the gun. As a gun of this kind needs no high-voltage generator, the coating system is very simple and has the following merits: 1. Even if we touch the tip of the nozzle, we don’t receive an electric shock. 2. As an electric field hardly occurs between guns and objects, the system can uniformly coat objects with some concave parts. 3. There are no free ions in the coating booth. Therefore, the orange peel phenomenon seldom occurs. However, it is difficult to control the charge freely because the system is influenced strongly by relative humidity and room temperature,4 contact potential difference5,6 between particle and inner wall of the gun, particle size,7 and air velocity8 in the gun. Furthermore, the tribocharge depends strongly on the tangential impact velocity of the powder particles.9–10 © 2006 by Taylor & Francis Group, LLC
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Disc-Type Coating Machine (a Kind of Corona-Tribocharging Gun) Figure 19.3 shows the disc-type coating machine. In this coating machine, the disc plate, which is made of resin, is installed at the outlet of the powder. The powder particles transported by the powder feeding machine with air flow are spouted out uniformly from the edge of the disc. Work piece objects are carried by a loop-type conveyor around the disc, as shown in Figure 19.3. The coating machine is expected to have six times the ability of the above-mentioned guns.
5.19.2
POWDER FEEDING MACHINE11
In order to feed powder particles from a vessel to the gun, a pneumatic conveying system with an air injector is mainly utilized as an easy and low-priced method. However, it is difficult to keep a stable feed of the powder particles using only an air injector. Recently, a few kinds of powder feeders, which are able to supply particles with a constant feed rate, have come onto the market.
Screw Feeder This type has been mostly applied to the combination with powder coating, as shown in Figure 19.4. The apparatus consists of a powder tank, powder-feeding section with screw feeder, and an air injector. The powder particles are fed to the air injector with high accuracy (about 2%) by controlling the rotational speed of a motor. In this case, the injector has a role that only conveys the powder particles to the coating machine. Therefore, the powder feed rate does not vary even if air flow rate to the air injector is changed. To prevent changing the concentration of the powder in a hopper, the powder tank and the powder feeding section are separated through a control valve. The amount of powder particles is monitored by a level meter and controlled by the gate valve.
Table Feeder The construction of this apparatus is almost the same as for the screw feeder. The feeder section consists of a turntable, motor, hopper, and air injector. The powder particles on the table are conveyed by air force or a mechanical scraper to the injector. The flow is smoother than in the screw feeder.
FIGURE 19.3 Disc-type coating machine. © 2006 by Taylor & Francis Group, LLC
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FIGURE 19.4 screw feeder.
Powder feeding machine with
FIGURE 19.5 Powder coating booth using cartridge filters.
5.19.3
POWDER COATING BOOTH
As there are many combinations of powder coating booths and powder collectors, it is necessary to select the booth carefully according to configurations and dimensions of the objects, ensuring space, change of colors, and protection of the working area. In this section, characteristics of representative coating booths are introduced.
Cartridge Filter Type In Figure 19.5, cartridge filters are set on a wall of the coating booth, and suspended powder particles in the booth are collected directly by suction fan. In order to change colors of powder, it is necessary to prepare the same number of cartridge filters and suction fans as there are colors. Although the coating booth needs no large space, it is difficult to conduct the multicolor powder coating.
Cyclone after Filter Type In Figure 19.6, precipitation of powder particles are conducted in two stages: cyclones and afterfilters. First, most of oversprayed powder particles are collected by the cyclones. At the second stage, small particles passed through the cyclones are collected by after-filters. This coating booth is available for multicolor coating. © 2006 by Taylor & Francis Group, LLC
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FIGURE 19.6 filters.
5.19.4
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NUMERICAL SIMULATION FOR ELECTROSTATIC POWDER COATING12–17
Recently numerical calculation for the electrostatic powder coating has been investigated. The equation of motion of a particle is given by
mp
pDP2 rCD v u ( v u) dv mp g SFE 8Cc dt
(19.2)
The right-hand terms in the equation show drag force, gravity force, and electrostatic force, respectively. Charged particles are affected by coulomb force (FE,C qE ) and other electrostatic forces, which are described in Section II 5.1.
Air Flow Field To calculate the aerodynamic forces acting on a particle is complex. The Navier–Stokes equations are used in evaluating fluid flow conditions and normally include three scalar equations for momentum, one for mass conservation, and one for energy conservation. These equations are given by,18
r
r
Du p [ ⋅ t ] rg Dt
(19.3)
D ru 0 Dt
(19.4)
Def ( ∇fe ) ( ∇ ⋅ pu) ( ∇ ⋅ [ t ⋅ u]) Dt
(19.5)
where r is the fluid density, p is the fluid pressure, t is the viscous stress, ef is total fluid energy, and f e is the energy flux. Furthermore, an equation of state is required in order to couple the mass conservation with momentum conservation via fluid density. r
PM RT
(19.6)
where M is the molecular weight of fluid gas, R is the gas constant, and T is the fluid temperature. For common electrostatic powder coating, the air flow field can be treated as incompressible. © 2006 by Taylor & Francis Group, LLC
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Furthermore, physical properties in the equations are assumed to be constant. Therefore, the air flow field can be solved using Equation 19.3 and Equation 19.4. As the momentum Equation 19.3 includes both the pressure field and velocity field, the fields must be solved severally. SIMPLE19 and SMAC20 schemes can be utilized as the representative algorithm in order to calculate the Navier–Stokes equations.
Electric Field The electric field that is generated due to the imposed voltage between the emitting electrode and the grounded plate can be described by a following Laplace equation, F 0
(19.7)
where the potential F is related to the electricfiled E according to E –∇ F. Since a high voltage is applied between the two electrodes, a corona discharge may take place. The presence of a space charge generated by the corona complicates the electric field solution in the computational domain. Such an electric field with a space charge is governed by Poisson’s equation:
E F— 0
(19.8)
where E is the space charge density and 0 is the electrical permittivity of the gas phase. The Laplace and Poisson equations can be solved by the successive overrelaxation method. A numerical technique proposed by Elmoursi21 is applicable to obtain a self-consistent solution for an ionic space charge field. The Lagrangian particle-tracking model is coupled with the electric field during the iteration. The procedure is described in the following steps: 1. Solve the electric field under Laplace condition (Ε ). This provides the initial estimation of the electric field. 2. Calculate particle trajectories with discrete phase model using Equation 19.2. 3. Determine the charge density distribution due to the charged powder in the ion drift region. 4. Solve the electric field with space charge. 5. Repeat steps 2–4 until the solution of the space charge field is convergent. This procedure can be utilized for the numerical simulation of the electrostatic powder coating with corona charging. On the other hand, for tribocharging, the image force works mainly on the charged particles. If many charged particles are deposited on the work piece object, apparent coulomb force decreases with time and then the particles hardly deposit further. Tanoue et. al have investigated the fact experimentally and numerically.17,22
REFERENCES 1. Reddy, V. and Dawson, S., Powder Coating Applications, Society of Manufacturing Engineers Powder Coating Institute, Alexandria, VA, 1990. 2. Boochi, G. J., Powder Coating, 4, 14–18, 1993. 3. Pauthenier, M. M. and Moreau-Hanot, M., J. d’Physique Radium, 7, 590–613, 1932. 4. Nomura, T., Taniguchi, N., and Masuda, H., J. Soc. Powder Technol. Jpn., 36, 168–173, 1999 (in Japanese). © 2006 by Taylor & Francis Group, LLC
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723
Itakura, T., Masuda, H., Ohtsuka, C., and Matsusaka, S., J. Electrostatics, 38, 213–226, 1996. Tanoue, K., Morita, K., Maruyama, H., and Masuda, H., AIChE J., 47, 2419–2424, 2001. Masuda, H. and Iinoya, K., AIChE J., 24, 950–956, 1978. Masuda, H., Komatsu, T., Mitsui, N., and Iinoya, K., J. Electrostatics, 2, 341–350, 1976/1977. Ema, A., Sugiyama, S., Tanoue, K., and Masuda, H., J. Inst. Electrostatics Jpn., 26l, 130–136, 2002. Ema, A., Yasuda, D., Tanoue, K., and Masuda, H., Powder Technol., 135–136, 2–3, 2003. Yanagida, K., Morita, T., and Takeuchi, M., J. Electrostatics, 49, 1–13, 2000. Woolard, D. E. and Ramani, K., J. Electrostatics, 35, 373–387, 1995. Chen, H., Sims, R. A., Mountain, J. R., Burnside, G., Reddy, R. N., Mazumder, M. K., and Gatlin, B., Part. Sci. Technol., 14, 239–254, 1996. Adamiak, K., J. Electrostatics, 40, 395–400, 1997. Ye, Q., Steigleder, T., Scheibe, A., and Domnick, J., J. Electrostatics, 54, 189–205, 2002. Ang, M. L. and Lloyd, P. J., Int. J. Multiphase Flow, 13, 823–836, 1987. Tanoue, K., Inoue, Y., and Masuda, H., Aerosol Sci. Technol., 37, 1–14, 2003. Bird, R. B., Stewart, W. E., and Lightfoot, E. N., in Transport Phenomena, 3rd. Ed., John Wiley, New York, 1960, p. 322. Patanker, S. V., Numerical Heat Transfer and Fluid Flow, McGraw-Hill, New York, 1980. Amsden, A. A. and Harlow, F. H., The SMAC Method: A Numerical Technique for Calculating Incompressible Fluid Flows, Los Alamos Scientific Laboratory Report LA-4370, 1970. Elmoursi, A. A., IEEE Trans. Ind. Appl., 28, 1174–1181, 1992. Tanoue, K., Yamamoto, M., Ema, A., and Masuda, H., Kagaku Kogaku Ronbunsyu, 28, 196–201, 2002 (in Japanese).
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5.20
Multipurpose Equipment Jun Oshitani and Kuniaki Gotoh Okayama University, Okayama, Japan
Shigeki Toyama* Nagoya University, Aichi, Japan
5.20.1
FLUIDIZED BEDS
If a gas passes upward through a bed of particles, the bed is fixed at low gas velocities, but when the velocity is increased further, the particles separate from each other and are supported by the gas, then the bed starts fluidizing. In the fluidized bed, the particles contact with the gas at high efficiency, and temperature is kept uniform because of good mixing of the particles. The features have been utilized for industrial applications 1; the first use of the fluidized bed as a large-scale reactor was Winkler’s coal gasifier in 1926. Since the 1940s the fluidized bed had played an important role for fluid catalytic cracking using a combination of a reactor and a catalyst regenerator. Recently a circulating fluidized bed, as shown in Figure 20.1, has been widely used as a boiler for solid fuel and waste. Figure 20.2 illustrates a typical diagram of a pressure drop ΔP versus gas velocity u0. If the gas velocity is increased, the pressure drop gradually increases with the gas velocity and reaches a maximum in the fixed bed. When the gas velocity is increased further, the pressure drop falls down slightly because of the bed expansion, and then becomes constant independent of the gas velocity in the fluidized bed. The gas velocity at which fluidization occurs is called the minimum fluidization velocity u mf. During the fluidization, the apparent particle weight is equal to the pressure drop as expressed by P L (1 e)( rp rf )g
(20.1)
FIGURE 20.1 Schematic drawing of circulating fluidized bed. * Retired
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FIGURE 20.2 Typical diagram of a pressure drop versus gas velocity.
where L is the bed height, e is the void fraction, and rp and rf are the densities of the particles and gas, respectively. The minimum fluidization velocity can be estimated by combining Equation 20.1 and the Kozeny–Carman equation as follows: fc2 g( rp rf ) Dp e3 ⋅ ⋅ 180 (1 e) m 2
umf
(20.2)
where fc is the shape factor, Dp is the particle diameter, and m is the gas viscosity, respectively. If the gas velocity is increased from umf, the bed starts bubbling, and the gas velocity at which bubbling first occurs is called the minimum bubbling velocity u mb. Figure 20.3 shows a bubble, which is consistent with regions of low solid density in the fluidized bed. The bubble rises, accompanying the surrounding particles in the wake, and grows by coalescing with other bubbles. If the bubble diameter is almost equal to the bed diameter, the state is called slugging. When the gas velocity is much larger than the terminal velocity ut, the particles are conveyed by the gas. Gerdart classified the particles into four groups through careful observations of fluidization using various kinds of particles.2 The groups A–D are shown in Figure 20.4 by using the particle diameter Dp and the density difference between the particle and gas (ρp – ρf). Geldart C particles are difficult to fluidize because they are fine and cohesive powders. Vibration is sometimes used to help the fluidization of such particles. Geldart A particles having low density (ρ p < 1400 kg/m3) and/or small mean diameter easily fluidize. When the particles are fluidized, the bed expands before bubbling occurs; in other words, umb/umf > 1. Geldart B particles (1400 kg/m3 < ρp < 4000 kg/m3 and 40 μm < D p < 500 μ m) are sandlike, and for these particles bubbling starts at incipient fluidization (umb/umf > 1). Geldart D particles are dense and/or large, and bubbles rise more slowly and grow to large size with rapid coalescing. The fluidized beds of Geldart A, B, and D particles behave like a liquid, having their own density and viscosity.3,4 Thus, the gas–solid fluidized bed can be used as a separator of different density objects. In the separation technique, the fluidized bed acts like a dry separation medium in a similar manner to the wet medium of dense medium separation.5,6 When objects are immersed in the fluidized bed, the objects with a density smaller than the apparent density of the fluidized bed float, whereas those of larger density settle in the fluidized bed. Fraser and Yancey first applied fluidizedbed medium separation (FBMS) to coal cleaning.7 Joy et al. developed a unique separator combining the effects of a fluidized bed and a vibrating table for continuous separation,8 Coal cleaning with the FBMS has also been investigated in China.9,10 The FBMS was applied to the separation of agricultural products in the 1980s.11–13 Oshitani et al. developed a continuous separator and investigated the separation performance of the FBMS for silicastone and pyrophyllite,14,15 automobile shredder residues,16,17 and coal cleaning.18 The apparent density of the fluidized bed can be controlled by mixing the proper particles into the bed. However, segregation takes place in some combinations of binary particle mixture. © 2006 by Taylor & Francis Group, LLC
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FIGURE 20.3 Bubble observed in fluidized bed.
FIGURE 20.4 Particle classification by Geldart (fluidization by air).
Lighter particles (flotsam) float while heavier particles (jetsam) settle in the fluidized bede 19 Mixing/ segregation of a binary particle mixture and the mechanisms of segregation in the gas–solid fluidized bed have been investigated extensively.20–25
5.20.2
MOVING BEDS
Type and Application Iron and Steel A blast furnace to produce pig iron is shown in Figure 20.5a, and the moving grade shown in Figure 20.5b is applied to the ore sintering as a pretreatment process and also to the dry quenching © 2006 by Taylor & Francis Group, LLC
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of coke to recover heat. The simple moving bed shown in Figure 20.5c is also applied to the dry quenching of coke. The type in Figure 20.5d has been developed as a dust collector at a high temperature and is capable of recovering heat simultaneously by inserted pipes. Cement and Lime The type in Figure 20.5e is an old standard in which limestone and cokes are alternatively charged. The type in Figure 20.5f has been developed to burn liquid fuels. The cross-current type in Figure 20.5g permits the design of a large scale. Environments The cross-current type in Figure 20.5h is used to remove contaminants from the effluent gas by a moving bed of lime or activated carbon. The Herschoff type, in Figure 20.5i has been developed as a calcinator of iron sulfate and applied to an incinerator of swage mud. Coal The types in Figure 20.5j,k,l are developed as coal gasifiers.
FIGURE 20.5 Various types of moving beds. [From Association of Powder Process Industry and Engineering, Ed., Kilns for Processes, Nikkan-KougyoShinbun, Tokyo, 1985, pp. 6–7. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 20.5 (Continued)
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FIGURE 20.5 (Continued)
Others The types in Figure 20.5m,n are used in the petroleum industry.
Treatment of Particulate Materials Feeding The formation of a uniform bed is of primary importance because the segregation of particles is apt to take place caused by the difference in size and density. The following methods are recommended to avoid this: (1) multiple legs (Figure 20.5m), a distributing cone (Figure 20.5j,l,m), (3) a rotating bed (Figure 20.5f) or trough feeders (Figure 20.5h), and (4) a vertical pipe having multiple outlets. When isolation from the atmosphere is requested, double or triple dumpers (Figure 20.5j,l,m) or screw types are used. Discharge of Particles A uniform moving down is also desired to attain a uniform reaction and to expand the effective volume. The uniformity of the moving bed is broken at any contraction part. To minimize the © 2006 by Taylor & Francis Group, LLC
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broken region, the following methods are recommended: (1) installation of a diamond cone or guide vanes (Figure 20.6a,b), (2) insertion of a grizzly sieve or a perforated plate (Figure 20.6c,d), and (3) multiple legs (Figure 20.6e) or multiple plates (Figure 20.6f).
5.20.3
ROTARY KILN
Type and Functions The standard type is shown in Figure 20.7, and a variation is shown in Figure 20.8. These are quoted from patents for preheaters of cement raw materials having bypaths or dividers.28 The type
FIGURE 20.6 Techniques to attain uniform discharge. [From Association of Powder Process Industry and Engineering, Ed., Kilns for Processes, Nikkan-Kougyo-Shinbun, Tokyo, 1985, pp. 44–48. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 20.7 A standard rotary kiln.
FIGURE 20.8 Various designs of rotary kilns. [From Association of Powder Process Industry and Engineering, Ed., Kilns for Processes, Nikkan-Kougyo-Shinbun, Tokyo, 1985, p. 9. With permission.]
in Figure 20.8a, is a coaxial dual-cylinders system with different rotating speeds, and the type in Figure 20.8b is a design of outside heating to isolate the calcining material from the heating gas. Multiple burners in Figure 20.8c are for attaining a uniform temperature distribution.
Retention Time and Bed Depth Several equations to predict the retention time of particle flow through a rotary kiln have been correlated as shown in Table 20.1.29 The depth distribution of a particle bed is predicted from the graph shown in Figure 20.9.30 A uniform depth is attained when the parameter hL/RNφ = 0.193.
Calcining Process The advantage of rotary kilns is that they are capable of intensive heating by direct radiation from the flame and uniform calcinations by mixing through kiln action. On the other hand, the © 2006 by Taylor & Francis Group, LLC
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TABLE 20.1 Equations to Predict the Retention Time of a Particle Flowing through Rotary Drums Reported by Sullivan et al.
Bayard
Kramers et al.
Equation ⎛ L8 ⎞ t1 0.03102 ⎜ ⎝ DSN ⎟⎠ 1/ 2
t2
0.00308 (6 24 ) ⎡ LS 6H (Hh1/2 ) H ⎤ 12H ⎢ ⎥ 2 NS D h 2 D 2H h ⎣ ⎦
h /RN 0.193 L X h0 1 h 0.193 ln 0 h/RN 0.193 L RH RN N NK N
Vahl et al.
Remark Simple equation
With rings
Thickness distribution of particle bed
F sin b R cosb N r S HR 3 tan a1 K tan a1
⎛ D⎞ ⎛ L⎞ t 4 0.91 ⎜ ⎟ ⎜ ⎟ ⎝ D⎠ ⎝ hf ⎠ 2
0.57
tan b N
No inclination (S 0)
where D/hf is given by ⎛h ⎞ F 0.74 ⎜ f ⎟ ⎝ D⎠ Baranovskñ
Zablotony
t 5 0.431
t 6 0.433
2.05
D 4 N cotb fs L
L sin b ⎛ 2f − sin f ⎞ x⎜ ⎝ sin 3 f ⎟⎠ DN a L ⎛ b⎞ ⎜ ⎟ 0.85 DN ⎝ a ⎠
Including f
Simple equation
D, inner diameter of rotary drum (m); F, feed rate (kg/min); g, gravitational acceleration (m/min2); h, thickness of particle bed (m); h0 = h at outlet (m); H, height of ring (m); L, length of rotary drum (m); N, rotational speed (rpm); R, rotation radius (m); S, inclination (m/m); X, volume fraction of particle hold-up; α, inclination of rotary drum (deg); β, angle of repose of particles (deg); x, axial distance (m); ρs, bulk density of particles (kg/m3); φ, circumferential angle of particle bed (deg); τ, retention time (min). Source: Toyama S., Chem. Eng., 11, 533, 1966 (in Japanese).
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FIGURE 20.9 Diagram to predict a thickness distribution of particle bed. [From Association of Powder Process Industry and Engineering, Ed., Kilns for Processes, Nikkan-Kougyo-Shinbun, Tokyo, 1985, pp. 65–72. With permission.]
FIGURE 20.10 Axial and cross sectional distributions of temperature and reaction yield in a rotary kiln for calcining lime. [From Kramers, H. and Crookewit, Chem. Eng. Sci., 1, 259–267, 1952. With permission.]
composition of gas in the kiln cannot be changed to a large extent. Typical temperature distributions in a lime calciner are shown in Figure 20.10.31 Mathematical models to simulate their reactions and heat have been abundantly developed.32,33
REFERENCES 1. Kunii, D. and Levenspiel, O., in Fluidization Engineering, 2nd Ed., Butterworth-Heinmann, Boston, 1991, pp. 15–60. 2. Geldart, D., Powder Technol., 7, 285–292, 1973. 3. Schmilovitch, Z., Zaltzman, A., Wolf, D., and Verma, B. P., Trans. ASAE, 35, 11–16, 1992. 4. Bakhtiyarov, S. I. and Overfelt, R. A., Powder Technol., 99, 53–59, 1998. © 2006 by Taylor & Francis Group, LLC
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5. Wills, B. A., in Mineral Processing Technology, 6th Ed., Butterworth-Heinmann, Boston, 1997, pp. 237–257. 6. Burt, R., Miner. Eng., 12, 1291–1300, 1999. 7. Fraser, T. and Yancey, H. F., Coal Age, 29, 325–327, 1926. 8. Joy, A. S., Douglas, E., Walsh, T., and Whitehead, A., Filtration Separation, 9, 532–538 and 544, 1972. 9. Chen, Q. R., Yang, Y., Yu, Z. M., and Wang, T. G., in Proceedings of Eighth Annual International Pittsburgh Coal Conference, Pittsburgh, 1991, pp. 266–271. 10. Luo, Z. F. and Chen, Q. R., Int. J. Miner. Process., 63, 167–175, 2001. 11. Zaltzman, A., Feller, R., Mizrach, A., and Schmilovitch, Z., Trans. ASAE, 26, 987–990 and 995, 1983. 12. Zaltzman, A. and Schmilovitch, Z., Trans. ASAE, 29, 1462–1469, 1986. 13. Zaltzman, A., Verma, B. P., and Schmilovitch, Z., Trans. ASAE, 30, 823–831, 1987. 14. Oshitani, J., Kondo, M., Nishi, H., and Tanaka, Z., J. Soc. Powder Technol. Jpn., 38, 4–10, 2001. 15. Oshitani, J., Tani, K., and Tanaka, Z., J. Chem. Eng. Jpn., 36, 1376–1383, 2003. 16. Oshitani, J., Kajiwara, T., Kiyoshima, K., and Tanaka, Z., J. Soc. Powder Technol. Jpn., 38, 702–709, 2001. 17. Oshitani, J., Kiyoshima, K., and Tanaka, Z., Kagaku Kogaku Ronbunshu, 29, 8–14, 2003. 18. Oshitani, J., Tani, K., Takase, K., and Tanaka, Z., J. Soc. Powder Technol. Jpn., 41, 334–341, 2004. 19. Rowe, P. N., Nienow, A. W., and Agbim, A. J., Trans. Inst. Chem. Eng., 50, 310–323, 1972. 20. Rowe, P. N. and Nienow, A. W., Powder Technol., 15, 141–147, 1976. 21. Rice, R. W. and Brainovich, J. F., Jr., AIChE J., 32, 7–16, 1986. 22. Hoffmann, A. C., Janssen, L. P. B. M., and Prins, J., Chem. Eng. Sci., 48, 1583–1592, 1993. 23. Wu, S. Y. and Baeyens, J., Powder Technol., 98, 139–150, 1998. 24. Marzocchella, A., Salatino, P., Pastena, V. D., and Lirer, L., AIChE J., 46, 2175–2182, 2000. 25. Formisani, B., De Cristofaro, G., and Girimonte, R., Chem. Eng. Sci., 56, 109–119, 2001. 26. Association of Powder Process Industry and Engineering, Ed., Kilns for Processes, Nikkan-KougyoShinbun, Tokyo, 1985, pp. 6–7. 27. Association of Powder Process Industry and Engineering, Ed., Kilns for Processes, Nikkan-KougyoShinbun, Tokyo, 1985, pp. 44–48. 28. Association of Powder Process Industry and Engineering, Ed., Kilns for Processes, Nikkan-KougyoShinbun, Tokyo, 1985, p. 9. 29. Association of Powder Process Industry and Engineering, Ed., Kilns for Processes, Nikkan-KougyoShinbun, Tokyo, 1985, pp. 65–72. 30. Kramers, H. and Crookewit, P., Chem. Eng. Sci., 1, 259–267, 1952. 31. Takatsu, M., Yogyokyokai-si, 73, C359–363, 1965. 32. Elgeti, K., Chem. Eng. Technol., 25, 651–655, 2002. 33. Martins, M. A., Oliveira, L. S., Franca, A. S., Zement-Kalk-Gips, 55(4), 76–87, 2002.
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5.21
Simulation Charles S. Campbell University of Southern California, Los Angeles, California, USA
Ko Higashitani Kyoto University, Katsura, Kyoto, Japan
Yutaka Tsuji Yokosuka Research Laboratory, Central Research Institute of Electric Power Industry, Yokosuka, Kanagawa, Japan
Toshitsugu Tanaka Osaka University Suita, Osaka, Japan
Shinichi Yuu Kyushu Institute of Technology, Kitakyushu, Fukuoka, Japan
Kengo Ichiki The Johns Hopkins University, Baltimore, Maryland, USA
Yoshiyuki Shirakawa Doshisha University, Kyoto, Japan
5.21.1 COMPUTER SIMULATION OF POWDER FLOWS This review will describe computational techniques for simulating the flows of powders or granular materials. These techniques simulate systems consisting of discrete particles within which each individual particle is followed as it interacts with other particles and with the system boundaries. In effect this involves simultaneously integrating all the equations of motion for all the particles in the system. This may appear to be a massive task but has been accomplished for several million particles. This technique is especially useful for studying granular flows due to the difficulty of obtaining detailed results from direct experimentation. That difficulty lies in the relative lack of instrumentation that can operate in a dense flowing particle environment. For example, as a result of the large concentration of particles, granular flows are opaque to optical probing. Ultrasonic probing has been suggested as a possible alternative but has not yet been developed into workable instruments. But even the intrusive probes that have been developed give only limited information. The computer simulation techniques described in this chapter are used to set up simulated flowing powder systems upon which “experiments” are performed. These experiments take the form of averages over the properties of the system. The advantage of a computer simulation is that literally everything about the simulated system is known and is accessible to the computer experimenter. The techniques are fairly basic, and the art of using these computer simulations lies not in setting up the programs but in knowing how to interrogate the systems to provide whatever information is of interest. 737 © 2006 by Taylor & Francis Group, LLC
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These simulations are sometimes called the discrete or distinct element method (but always DEM), a term coined by the originator of the technique, Peter Cundall. That term has come to be used for many different numerical techniques (and was not very descriptive in the first place), so they will simply be referred to here as “computer simulations.” The origins actually lie in the field of molecular dynamics, which was the first field to mechanistically follow the motions of molecules under the influence of external forces (e.g., work by Alder and Wainwright1). Cundall2 was the first to use this type of model to study granular flows. Since then the use of these models has become widespread—too widespread to include an exhaustive bibliography of all applications in the space of this review. The simulations described here will assume that the flow occurs in the absence of an interstitial fluid. The interstitial fluid is ignored because it is not yet possible to model the motion of the fluid with the same degree of accuracy as it is to model the motions of the particles that only respond to contact and body forces. As such, these techniques are applicable to cases in which the interstitial fluid does not play a significant role in determining the overall mechanics of the system. Such is the case when the majority of the forces experienced by the particles are due to the solid–solid contacts with their neighbors and with the system boundaries, and are not strongly influenced by any fluid that might fill the interparticle gaps. Some simulations have added interstitial fluid effects in an approximate manner (e.g., work by Tsuji and Kawaguichi,3 Haff et al.,4 and Potapov et al.5), but these will not be discussed in detail here. These computer simulations can roughly be divided into two types, rigid- and soft-particle models, which because of the differences in the ways that the particle interactions are modeled, follow different computational algorithms. Other reviews of computer simulation techniques can be found in work by Campbell6 and Herrmann and Luding.7
Rigid-Particle Models The stress levels in a most granular flows are usually small enough so that the particle surfaces will not elastically deform to any significant degree. It is then a reasonable approximation to assume that the particles are perfectly rigid and cannot deform at all. But perfectly rigid particles have infinite elastic moduli, so that any collision between such particles must occur instantaneously. At their essence, rigid-particle models are synonymous with assuming that all particle interactions are instantaneous collisions. (Similar assumptions are made in all theoretical calculations of granular flow in the so8 called rapid-flow regime; see the review by Campbell. ) A convenient byproduct of this assumption is that simultaneous collisions between three or more particles will occur with zero probability so that only two-particle or binary collisions need be accounted for. Examples of rigid-particle simulations can be found elsewhere.9–17 The assumption of instantaneous collisions dictates the way that the simulation evolves through time and consequently determines the way that the simulation algorithm is structured. Between collisions the particles follow simple kinematic trajectories, which in gravity-free conditions are just straight lines but become parabolic when gravity is present. These trajectories only change as the result of collisions, and a particle’s trajectory between collisions is explicitly defined by the result of the particle’s last collision. Thus time in the simulation is most efficiently updated from collision to collision, as the state of the system between collisions can be easily determined if needed. (Thus, these are sometimes referred to as “event driven” simulations.) The algorithm is shown in Figure 21.1 and proceeds as follows: When the simulation is started, the time at which the first collision occurs is computed from the particle trajectories. The positions and velocities of all the particles are updated to that time. The collision result is computed, the time of the next collision to occur is found, and the process is repeated. The collision result is determined from a standard center-of-mass collision solution of the type learned in elementary physics classes. (An excellent implementation of a rigid-particle collision solution is described in detail by Walton.18) As the particle is assumed to be infinitely rigid, elastic properties do not appear in the equations, and the collision solution is determined solely from © 2006 by Taylor & Francis Group, LLC
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FIGURE 21.1 A flowchart illustrating the rigid-particle algorithm.
momentum and energy considerations. Inelasticity is typically introduced through a coefficient of restitution, which may be constant or a function of the collisional velocity. The particle velocities in the tangential direction are typically coupled frictionally. The rate at which time progresses in a rigid-particle simulation is inversely proportional to the collision frequency, making this procedure extremely efficient especially at low solid concentrations where collisions are infrequent. As the concentration becomes large, collisions occur so frequently © 2006 by Taylor & Francis Group, LLC
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that rigid-particle models become very inefficient. As a further consequence, rigid-particle models cannot be applied to any situation involving stagnant zones or other situation where particles are in contact for long periods of time. Such regions are common in real granular systems, and when they occur, rigid-particle simulations will try to model the long-duration contacts as a number of rapidly occurring instantaneous collisions; as a result, the collision frequency goes to infinity and the time between collisions goes to zero. At that point, the simulation time, which progresses from collision to collision, cannot change, and the simulation effectively stops. Furthermore due to the particle inelasticity, particles form clusters,19 which eventually may lead to situations where particles in the clusters come into continuous contact. This process has recently been called “inelastic collapse,”20 and there are approximate methods for avoiding its occurrence. For example, Campbell9 noted these events occurring for particles that were trying to roll along bounding walls. He solved the problem by designating a particle as “rolling” when collisions with the wall became too frequent. Luding and McNamara21 used a similar technique, which removed dissipation from that collisions that became to frequent; while this eliminated the inelastic collapse, it in no way approximates the dynamics of a long-duration contact.
Soft-Particle Models The most realistic solution to this problem is to simply allow long-duration contact between particles. To do this in a consistent manner means that the particles cannot be perfectly rigid, and therefore any contact will be of finite duration. This means that an allowance must also be made for a particle to be in contact with several particles simultaneously. These “soft-particle” simulations were originally developed by Peter Cundall2,22 and applied primarily to quasi-static situations. (Although to demonstrate its utility, the model was used 2 to simulate flow problems such as the emptying of a hopper.) For reasons that will soon become clear, this is the most common simulation technique and has been used by many other investigators.23–36 As before, the structure of the simulation is controlled by how the particle’s material is modeled. Most flowing systems are not subject to tremendous stresses (certainly not when compared to geological systems). Consequently, the interparticle forces are generally too small to cause significant changes in the shapes of the particles, and for computational ease, the particle shape in a soft-particle simulation is not allowed to change. Instead when particles come into contact, their surfaces are allowed to overlap somewhat, and the degree of overlap determines the magnitude of the elastic force at the contact. Generally, the force generated normal to the contact point is modeled as a simple spring. (For cohesionless particles, the spring is not allowed to support any force in tension and is eliminated as soon as the particles lose contact.) For most models, a linear Hookean spring model is used, although it presents few problems to use a nonlinear spring. In granular flows, each collision must dissipate energy. In soft-particle models, this is usually accomplished by connecting a viscos dashpot in parallel with the spring. A constant dashpot coefficient coupled to a linear spring can be shown to be the equivalent to a constant coefficient of restitution. To represent the forces generated in the direction tangential to the point of contact, a spring (occasionally coupled with a dashpot) is also employed. In addition, a frictional coupling is used to link the surfaces of the two contacting particles together in the tangential direction that allows the surface to slip tangertially when the frictional strength is overcome. The ubiquity of this particular configuration is probably due to its use in the original model of Cundall.2 Each contact exerts both a force and moment on each of the particles involved. The total force and moment on a particle are the sum of those applied by all its contacts possibly combined with a body force such as gravity. The subsequent motion of the particle is governed through Newton’s second law by an ordinary differential equation, and the motion of the entire bulk material is determined by the simultaneous solution of all the differential equations for all of the constituent particles. The solution is readily accomplished using any numerical method that is capable of solving systems of ordinary differential equations; however, the integration must be carefully performed, © 2006 by Taylor & Francis Group, LLC
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FIGURE 21.2 A flowchart illustrating the soft-particle algorithm.
as the forces on each particle and consequently the differential equations they obey will vary drastically as contacts with other particles are made or broken. The soft-particle algorithm is shown in Figure 21.2. In essence it is simply a numerical integration; however, at each time step, it is necessary to check for new and broken contacts and to compute the interparticle forces. Furthermore the stimulation properties are sampled and averaged at each time step to yield whatever quantities © 2006 by Taylor & Francis Group, LLC
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are of interest. The size of the time increment by which the integration routine proceeds should be chosen so that a typical binary collision would occur in 30 to 50 time steps. (On top of pure numerical inaccuracies, there are other sources of inaccurate collision results if very large time steps are chosen. For example, if two particles come into contact with a large relative velocity, too large a time step may allow erroneously large initial overlaps to occur within that first time step, which results in an exaggerated particle response and a net addition of energy to the granular system.) But on the positive side, the soft-particle algorithm also permits contacts to exist for arbitrarily lone duration, allowing the simulation of static situations; little computational effect is expended in the actual integration of the equations of motion, hence there is little to be gained in using elaborate integration techniques. (Typically second-order algorithms are used.) Most of the time is taken up in updating the positions and forces on the particles at each time step and, in particular, checking for new or broken contacts between particles. In binary collisions of linear spring contacts, the collision time will vary as the square root of the ratio of spring constant to particle mass (and is independent of the impact velocity). As mentioned in the last paragraph, the size of the time step is a specified fraction of the binary collision time and is thus controlled by the smallest mass and stiffest spring in the granular system. The computational time required to run a given simulation is not dependent on the collision rate (as it is for rigid-particle models) and is therefore independent of the particle packing. Hence, soft-particle models are computationally inefficient at small concentrations where collisions are infrequent, and most of the computer time will be spent in updating the particle positions as they move unimpeded along their kinematic trajectories. But the simulations become more efficient at larger when collisions are frequent . Ironically, it is also at the largest densities that the soft-particle models may run into trouble. This is because an improved computational efficiency is often obtained by choosing an extremely soft spring in the interparticle force model (one that is much softer and permits larger overlaps than would be found in common granular materials). This lengthens the average collision time and with it the time step required to accurately represent a given collision. At small particle concentrations, this should not be a problem, but at larger concentrations, the “excluded volume” taken up by the particles causes a singular behavior in many of the material transport properties within an actual granular material. If the simulated particles are inordinately soft, the overlaps will be large, causing a reduction in the excluded volume and a dampening of the singular behavior, resulting in significant reductions in the measured values of the transport properties. Furthermore, the extended contact time permits a greater number of simultaneous multiparticle contacts to occur than would be encountered in a system with a more realistic particle stiffness. Typically, the stiffness is made large enough so that the particle overlaps do not exceed a specified value (usually 1% of a particle diameter). Recently, another limit on the stiffness has become apparent. Campbell24,25 has shown that in shear flows, materials can undergo a regime transition to elastic-inertial at either large shear rates g, or small stiffness, k. The transition occurs at small values of a parameter k/pd 3y 2 (where d is the particle diameter and p the solid density). Thus, inordinately soft particles may operate in a different flow regime than particles with more realistic stiffnesses. However, flow maps given by Campbell 24,25 indicate the regime transition point so that they may be avoided. Shear flows at fixed concentration were studied by Campbell. 24 Realizing that the majority of granular flows occur in situations where the stress and not the volume is fixed (for example, by a material overburden) 25 repeats those studies for fixed-stress cases where the volume is allowed to change slightly to keep the stress fixed. The results show that fixed-stress flows were shown to behave very differently from fixed-concentration flows and follow a different set of flowmaps. Thus simulators must be careful to use the appropriate flowmaps to avoid unintentional flow regime changes. One advantage of the soft-particle method is that there is no limitation on how the particle’s material is modeled or in the nature of the forces to which the particle is subject. This may be very important in future simulations of systems with very soft plastic materials or for fine powders where long-range electrostatic or van der Waals forces are important. Langston et al.30 describe a model © 2006 by Taylor & Francis Group, LLC
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with a continuously varying normal force that actually extends slightly beyond the outer edges of the particles; but this is done mostly to enhance the computational efficiency of the simulations.
Rigid- vs. Soft-Particle Models It has long ago become apparent to those that work in the field that soft-particle simulations are preferential to rigid-particle simulations. The major reason is not just because soft-particle codes are capable of handling static assemblies of particles and of incorporating more elaborate collision models, but rather a result of the way that the execution time scales with the number of simulated particles. Time in a rigid-particle simulation updates from collision to collision, and the rate of time progression is dependent on the collision rate. However, in soft-particle simulations, time progresses with the integration time step, which is determined by the particle stiffness, making the time progression independent of the collision rate (although that time step is a fraction of the particle contact time). Rigid-particle models had definite advantages when computers were slow and the number of particles that could be simulated were small. In such cases the time between collisions is large (especially when the concentration is small), and a rigid-particle simulation will progress much more rapidly than a soft-particle simulation. However, as the number of particles becomes large, the time between successive collisions goes to zero. In contrast, the progress of a soft-particle simulation depends on the integration time step and is independent of the number of particles in the simulation (although the computational demands that need be performed at each time step rise roughly proportionally to the number of particles). The break-even point between the two simulation types occurs when approximately one collision will occur in the entire simulated system in one soft-particle integration time step. It is clear that at the large concentrations that are typically encountered in industrial simulations (usually more than 50% by volume), the break-even point will be less than a thousand particles. As the future of granular simulations lies in simulating large systems of particles (a million and more), the soft-particle method is clearly preferential. However, the work by Campbell24,25 has added another reason to abandon rigid-particle models, at least for flows at concentrations above 50% by volume, where most common granular flows occur. At such concentrations stress is transmitted through the material through “force chains,” highly loaded structures that form internal to the flow and carry the majority of the stress (see, e.g., work by Cundall and Strack,22 Drescher and De Josselin de Jong,37 Mueth et al.,38 Howell et al.,39 and Howell et al.40). Force is carried along the chain by elastic deformation of the particle contacts, and the force generated is proportional to the stiffness, k, of the contacts. Such behavior cannot be modeled by rigid-particle models, as they implicitly assume that k and thus would always predict infinite stresses in any elastically deformed force chains. Campbell24,25 has shown that the stiffness, k, is an essential rheological parameter, thus putting solid properties into the rheology of granular solids. Such rheological effects can only be handled by soft-particle models. As a result, soft-particle simulations are required to accurately model the vast majority of common granular flows.
Approximate Simulation Techniques There have been several approximate algorithms (ones that do not exactly follow the motion of the particles and/or the interparticle forces) employed to study powder shows, usually with the goal of improving the efficiency of the simulation. Hopkins19,41 developed a hybrid technique that attempts to exploit the advantages of the both rigidand soft-particle models. In essence, it employs a rigid-particle interaction model but marches time with a fixed time step, allowing collisions to occur whenever overlaps are detected at the end of a time step. This increases efficiency, as the time step is not limited by material properties and can be much larger than for a soft-particle simulation. However, only one collision between any two particles is permitted at each time step, and thus the time step effectively sets an upper limit on the collision rate. As a result, such a model is only advantageous at small solids concentrations. Where the collision rates are small a large time step is permissible. © 2006 by Taylor & Francis Group, LLC
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The earliest molecular computer simulations (e.g., work by Metropolis et al.42) were not true dynamical simulations but made use of the apparently random nature of the molecular motion in gases to approach the problem from a statistical point of view. The goal was to derive distribution functions from which transport properties could be determined from integral relations derived for the theory of nonequilibrium gases (see, e.g., work by Chapman and Cowling43). Implicit in this method is Boltzmann’s “Stosszahlansatz” or assumption of molecular chaos. Briefly this is the assumption that the velocities of individual particles are independently distributed within a velocity distribution function without regard to history or to the behavior of neighboring particles. This will clearly not be the case in dense concentrations where particles will experience long-duration contacts with their nearest neighbors, and there will be strong correlations between the positions and velocities of neighboring particles. Thus Monte Carlo methods have limited usefulness in granular flow studies. Nonetheless, molecular chaos is an essential assumption in the theory of rapid granular flows, 8 so that Monte Carlo simulations do provide a forum 44,45 for evaluating rapid-flow theories within the context of their basic assumptions. Lattice gas models have been a popular computational technique for studying gas flows, and it should not be strange that they should be adapted for granular flows. There have been several realizations of this technique (e.g., work by Baxter and Behringer,46 Gutt,47 and Désérable48), so that only a generalized picture will be given here. Basically the particles in a “lattice grain” model are confined to fixed and regularly spaced lattice points in space. Only one particle may occupy any given point. At each “time” step, particles may move into any of the neighboring lattice points, provided that they are vacant. However, the probability of a given movement is limited by potentials acting on the systems; for example, under gravity a particle is much more likely to move to a lattice point located below it, than to ones located to the side or upward. Walls and flow obstructions can be inserted by simply eliminating lattice points along the boundaries and thus prohibiting the motion of particles in those directions. The best that can be said about this type of model is that it is kinematically correct. There are no dynamics in the simulation scheme, no forces act on the particles (as described above, gravity acts only to affect the directional probability of motion and is not an acceleration), and the particles exert no forces on each other. Consequently, no information about forces can be obtained, and it is not clear that the simulation is responding in an accurate manner to the interaction with other particles or the bounding walls of the system. In addition, velocities are limited by the lattice spacing and the time step in the sense that the fastest a particle can move is one lattice spacing in one time step. In that sense, the velocities of the particles are really meaningless. Thus the information that can be obtained from these simulations is very limited.
Nonround Particles Most of these simulations have been performed for round particles. This is largely because it is easy to detect contacts between round particles, as contact occurs when the particle centers are the sum of their radii apart. When particles are not round, the orientation and local shape of the particle become important in the contact decision, and that is computationally difficult to handle. Yet particle shape certainly has a strong effect on granular flows, both due to the ability of angular particles to interact and form strong internal structures and simply by the added resistance to rolling. Convincing evidence can be easily seen in the angle of repose. For round particles, it is difficult to obtain angles of repose of more than a few degrees and much less than the 30° angles of repose typical of granular materials. One solution to this problem is simply to prevent the particles from rolling so that they only interact frictionally in the direction tangential to the point of contact. This allows one to build sand piles with realistic angles of repose but also can produce significant nonphysical effects such as stable columns of particles that would clearly collapse were the particles capable of rotation. Also the surface friction values required to produce realistic angles of repose are, themselves, unrealistically small. For example, it requires a surface friction coefficient of only about 0.2 to produce a 30° angle of repose. Thus, even though the angle of repose may be realistic, the internal particles interactions are not. © 2006 by Taylor & Francis Group, LLC
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The most natural solution for this problem is to use particles that are not round. Several such models have been proposed. In general they can be divided into two classes. The most general technique is to choose particles that are polygonal (e.g., work by Hopkins,49 Hogue and Newland,50 and Kohring et al.51; those utilized in fracture simulation simulations of Potapov et al.52–54; and for three-dimensional polyhedral particles, Cundall,55 Ghaboussi and Barbosa,56 and Potapov and Campbell57). Fairly realistic looking particles may be created in this manner, but at tremendous computational costs. Polygonal or polyhedral simulations can take orders of magnitude more computer time than their round counterparts. The majority of the computer time is spent in determining contact between particles and in calculating intersections of the sides of contacting polygons, which is a necessary part of the overlap determination. A computationally more efficient technique is to use shapes for which an analytic expression exists. For example, algorithms have been developed for particles with elliptic or ellipsoidal shapes.58–60 These can be generalized to a large variety of shapes known as superquadrics (e.g., work by Mustoe and DePooter61). These are described by equations similar to ellipsoidal particles except with exponents larger than 2 and can have shapes that approach polygons although the corners stay rounded. Due to their angular nature, such particles can interlock, but they do not resemble natural materials. Also, aspect ratios for elliptical particles must be significantly different form unity in order to obtain realistic angles of repose, which may present other problems, as the majority of natural granular materials have aspect ratios that are close to one. For ellipsoidal or superquadric particle shapes, the majority of the computer time is spent in iterative solutions of the nonlinear equations used to calculate the particle overlap. More computationally efficient techniques have recently been developed. Potapov and Campbell62 built two-dimensional particles out of circular sections. Particles of arbitrary convex shapes can be constructed in this way, and the particle overlap and contact determination with an edge is essentially the same as for circular particles. Additional complications arise from determining which circular segment is involved in a contact, but the execution times are only about twice as long as for their round counterparts. Unfortunately the technique could not be generalized to threedimensional particles. Ellipsoidal particles were built31 out of four overlapping spherical particles. As in Potapov and Campbell’s62 work, the contact and overlap determinations are essentially the same as for spherical particles, although one must potentially check four sphere contacts for each quasi-elliposidal particle. Also there are complications in determining the moment of inertia of the composite particles, since the spheres overlap and thus their properties are not the same as four separate spheres. Another possibility is to create agglomerates63,64 of circular or spherical particles “glued” together at the contacts of their surfaces. This give less choice in particle shape but is easy to implement and requires no special calculation of moment of inertia and so forth; the intraparticle force transmission is accomplished by the forces transmitted across the glued joint between the particles. It is unlikely that a completely accurate model of particle shape could ever be incorporated into computer simulations. Not only is it difficult to model the many complex shapes of real granules, but one would also have to take into account the way that the shapes change due to handling (e.g., as angular corners are broken off ). So one should attempt to create an adequate rather than an exact model, one that captures the essential physics and is easy to implement. This is still an open question, but hopefully, some of the similar, more computationally efficient models will surface in most cases.
Particle Interaction Models From the above discussion, it is clear that the soft-particle models are by far the most useful, and as the power of computers continually improves, most of the advantages of the rigid-particle and other approximate models will disappear. All soft-particle models are built upon assumptions about how particles interact with one another and with solid boundaries of the system, and how they react to any driving force such as gravity. All the approximations in the models lie in these basic assumptions, and thus interaction models deserve a place in any discussion of granular simulations. © 2006 by Taylor & Francis Group, LLC
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All particle interaction models must contain three basic components: (1) a mechanism for generating a normal force at the contact point that works to separate the particle surfaces, (2) some mechanism for dissipating the collisional energy, and (3) some frictional interaction that acts tangential to the particle surfaces. But the appropriate way of modeling these forces still involves a great source of controversy (see, e.g., work by Walton,18 Thornton and Randall,65 and Tüzün and Walton66). As mentioned previously, the forces applied in the normal direction of soft-particle models are usually assumed to act as linear springs and generate a force proportional to the overlap between the particles. Generally, that spring is connected in parallel to a linear dashpot to provide the energy dissipation. Constant spring and dashpot coefficients can be shown to correspond to constant coefficients of restitution. Nonlinear dashpots have also been proposed. Walton and Braum 34,35 used a “latched spring” model that loads with one spring constant and unloads with another as a way of incorporating the energy dissipation; they found this to be closer to the results of elastic–plastic modeling of the impact of round particles.18,67 Hertz68 solved for the interaction between linearly elastic particles with arbitrary radii of curvature and found a nonlinear response that stiffens as the particle deforms. Because Hertz produced a nearly exact solution, one is tempted to believe that it should be the correct model for interparticle contacts. But there is a great deal of evidence that this is not the case for many granular solids. One such source lies in the elastic-particles analyses18,67 mentioned in the last paragraph. The interaction of the elasticity and the plasticity generated a nearly linear response to displacement, at least when the particle’s yield strength had been exceeded. But there is some indication that the Hertz theory may fail in granular materials even when the particles are behaving elastically. The evidence comes from measurements of the contact stiffness69 and of sound speed in granular materials (see the discussion by Goddard70). From a Hertzian point of view, the bulk elastic modules of a granular material should increase with the applied pressure. This occurs because the higher the contact force, the more the particle is deformed, and the stiffer the contacts between particles. Hertz predicts a repulsive force that varies as the displacement to the three-halves power. That corresponds to a bulk elastic modulus that varies as the cube root of the applied pressure. But sound speed measurements indicate that the modulus varies more like the square root of the pressure, indicating non-Hertzian behavior. In other words, the experimental results are not in agreement with the theoretical, although the reason for this discrepancy is not certain. Goddard70 has attributed this to the fact that the applied forces are often too small to deform the macroscopic shape of the particle (which is assumed to be the case in Hertzian theory), but instead interact across and only deform the asperities that project from the surface. Only for heavily loaded contacts will the asperities be flattened so that the macroscopic shape of the particles deforms and Hertz theory becomes valid. From Goddard’s observations, one would anticipate using a spring where the generated force is quadratically dependent on the displacement for small displacements but demonstrate Hertzian behavior at large displacements. Direct measurements69 support this general notion, although they demonstrate a more complicated behavior at small loadings, but also approach Hertzian behavior at large loadings. Some of the tangential force measurements support the idea of contact through asperities. To further confuse the issue, Drake and Walton71 (Walton is one of the authors of the article by Mullier et al.69) reinterpret the large stress data in the article by Mullier et al.69 and argue that it plots better as a linear spring. It should be noted that it is not difficult to place a nonlinear spring in any existing model, the only difficulty is deciding which nonlinear spring to use and whether a nonlinear spring is justified. In many cases, however, the choice of a normal force model is moot. For example, in a true rapid granular flow, only binary collisions occur, and the collision solution can be completely determined from momentum and energy considerations (as is done with a rigid-particle simulation). Thus, the collision result is independent of any assumptions about the nature of the particle stiffness. This seems to be validated by flow simulations 34 that utilized both a linear spring and approximate Hertzian contacts in their model but found little disagreement between the two choices. On the other hand, if the concern is the propagation of elastic waves through a granular material, then the stiffness model is critical in correctly modeling the wavespeed and, given the discrepancy between the © 2006 by Taylor & Francis Group, LLC
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theoretical and experimental results alluded to above, would probably have to be determined experimentally. Campbell24 showed that the contact model was important in his force-chain-dominated “elastic” flows and suggested a preliminary scaling behavior for nonlinear springs. How important the contact model is for the areas lying between these extremes has yet to be determined. To sum up, we have three conflicting models for the particle interactions. Hertz’s model, which may only be good for large deformations if that, Goddard’s model, which predicts a different behavior at small deformations, and Walton’s elastic–plastic model, which shows linear behavior. On top of all that, it is not clear under what flow conditions the differences between these models are significant. With the current confusing state of affairs, it appears that the usual linear spring is as good as any for studying flowing systems unless there are careful measurements that indicate the contrary for the particular material involved. In much the same way, the linear dashpot commonly employed to introduce inelasticity into the simulation is clearly not a representation of the true physics of any contact. It has been used largely because it is a simple way to add inelasticity into the contact model, and a linear dashpot coupled with a linear spring produces a constant coefficient of restitution which can be easily compared to theoretical rapid-flow models which make similar assumptions. But it has also been shown that the coefficient of restitution is far from a constant but, instead, is a steadily decreasing function of impact velocity.72,73. This reflects the increasingly larger plastic damage and plastic energy dissipation that accompanies larger impact velocities. Walton and Braum34,35 utilized an approximation to this inelastic behavior in their computer model. While it is clear that this effect may be important, especially for rapid granular flows in which large relative velocities might be encountered, it is also clear that assuming an invariant from for the energy loss as a function of velocity introduces its own source of error. A clue to this can be found in an article by Raman,72 who states that in order to obtain repeatable results it was necessary to polish his test spheres between measurements. The implication of this remark is that the plastic damage from earlier tests alters the surface properties and changes the particle’s response to subsequent impacts. If such effects were to be incorporated into computer models it would necessary to follow the impact history of every location on the particle surface—and even then one would have to have a good model for characterizing how each impact affects the local surface properties. The case is similar, but even more difficult, for incorporating frictional models and tangential compliance into the computer models. The equivalent to Hertzian contact theory would be the very detailed work of Mindlin and Dereiewicz,74 which assumes Hertzian behavior in the normal direction and predicts complex hysteretic behavior in the tangential direction whenever the frictional limit has been exceeded (i.e., when there is slip between the particle surfaces). Thornton and Randall65 have incorporated an approximation to Mindlin’s theory into their model. But even more complicated models have been proposed.66 However problems with using Mindlin’s theory appear from the very start. Measurements by Sondergaard et al.75 indicate that the particle surface friction may not be a well-defined property, but vary statistically about some mean value. This is easy to understand if one assumes that the friction is strongly affected by microscopic surface asperities that are not evenly distributed about the particle’s surface. Except under very intense loadings, the contact area between two particles will be very small and involve only few asperities; consequently the frictional behavior detected by Sondergaard et al. reflects the significant local frictional variations about the surface of the particle. Conversely, friction measurements on different materials76 obtained a constant friction coefficient. Along the same lines, measurements by Maw et al. 77 seem to demonstrate a well-defined frictional behavior for steel, but a large degree of scatter for similar experiments performed on rubber. None of this should be surprising, as surface friction is, after all, a material property, and whether it has a constant value or is statistically distributed over the surface also appears to vary from material to material. In any case, it would seem to be fruitless to incorporate a complicated theory like Mindlin’s into the model if one cannot even be certain of the value of the surface friction coefficient. Compounding the confusion are the surface frictional measurements by Mullier et al.69 These found strong inconsistencies between the experiments and Mindlin’s theory. The odd measurements were traced to observations of plastic alteration of the surface characteristics that accompanied © 2006 by Taylor & Francis Group, LLC
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frictional yielding; in fact the asperities that strongly effect the frictional behavior were observed to have been sheared off the surface. (This can be seen quite clearly in before and after SEM photos that are presented in the paper.) As a result the frictional behavior has changed between the loading and unloading phases. The same effect may be qualitatively experienced by rubbing large glassbeads in one’s hands after they have been run through a flowing experiment a few times and comparing the feel to that of fresh glassbeads; it is clearly apparent that surface properties have changed even long before there is any visible change in the surface characteristics. This would seem to indicate that a truly accurate simulation would somehow have to incorporate these changes and implies that once again it would necessary to follow the detailed loading history of each portion of the particle surfaces and to have some model for how that history affects the surface frictional properties in addition to how it affects the normal response. The upshot of this whole section is that it is currently unrealistic to incorporate exact particle interaction into the simulations, especially considering the lack of knowledge of the effects of plastic deformation working on the particle surface properties. Given that, it seems unnecessary to include complex interaction models into these simulations—at least until the problems are better understood, or that detailed experiments exist that show that current simple interaction models lead to inaccurate results.
Conclusions This chapter has several methods, both approximate and nearly exact, for creating computer simulations of granular flows. These days, the approximate models are of limited usefulness, and only the soft-particle models should be seriously considered as an investigative and design tool. This is for a variety of reasons, including an increase in computational efficiency when modeling dense flows and large systems of particles. However, the most important reason for using soft-particle models is that the interparticle stiffness has recently been shown to be an important rheological parameter, which cannot be modeled by rigid-particle or other approximate techniques. How good are even these computer simulations? In the section on particle interaction models, I may have left the reader with the impression that these models have inherent inaccuracies arising from approximations in the material modeling, and that it is nearly hopeless to expect the development of contact models that accurately mimic natural particles. That is the best view of the future as seen from the present, but again the same is true of most models and techniques that are commonly employed for engineering analysis and design. Consider for example an industrial material such as coal which usually has particles of sizes ranging from submicron to several centimeters in size, that are angular in shape and are brittle and thus will break into smaller pieces while being handled. The flow of coal is thus very complex, especially as the bulk material properties will change when the particles break up due to handling. But I think I can say with confidence that the soft-particle computer simulations described herein, even those that make the most gross material approximations such as linear springs and dashpots, describe the flow of coal as accurately as laboratory experiments performed on “nice” materials such as glassbeads that share few of the properties of coal. Furthermore, because these simulations at least approximate the actual mechanics of the system, they must be much more accurate than other commonly used engineering models such as K– models for turbulence, or two-fluid models for multiphase flows. In such company, this type of computer simulation stands out as a most promising technique for guiding industrial design, especially as the rapid increase in computer power allows the simulation of progressively larger systems.
5.21.2
BREAKAGE OF AGGREGATES
When a three-dimensional large aggregate of arbitrary shape composed of N spherical particles of radius a and density r p is placed in a flow, the hydrodynamic drag force and torque act on the outside particles exposed directly to the flow and are propagated into the inside particles through interactions © 2006 by Taylor & Francis Group, LLC
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between constituent particles. This will result in the deformation and breakup of the aggregate. In the present model, the total force and torque on each particle is evaluated at time t, and then the trial displacement at t + t is estimated using the discrete element method (DEM). Repeating this procedure for all the constitutive particles, the kinetic behavior of the whole aggregate is simulated. The translational and rotational motions of a particle i in the aggregate are expressed by the following equations: m I
du pi
dpi dt
Fhi ∑ Fmij
(21.1)
M hi a ∑ Fmij nij
(21.2)
dt
j
j
where m ( (4/3)pa3rp) and I ( (8/15)pa5rp) are the mass and moment of inertia of a particle, respectively, upi and vpi are the velocity and angular velocity of particle i, Fhi and Mhi are the hydrodynamic drag force and torque, respectively, Fmij is the mutual interaction force imposed on the particle i by the particle j, nij rij/rij, rij (xi – xj), and xi is the position vector of the center of particle i. Because the flow field around aggregates is extremely complicated in general, it is almost impossible to evaluate the values of Fhi and Mhi rigorously. In conventional models, the drag force on the constituent particle is assumed to be given by the Stokes law for a single particle, neglecting the disturbance due to neighboring particles. This is called “free-draining approximation.” In the present model, the drag force is assumed to act only on the particle surface exposed directly to the flow. This exposed area Si, illustrated schematically in Figure 21.3, is determined as follows. The surface of a particle is divided into 2592 sections such that the angle between the grid lines is p/36. Then a straight line is drawn to a corner of a given section from the particle center. If all the lines for four corners do not intersect with the surface of the other particles within the distance of 6a, the section is assumed to be exposed directly to the flow. Repeating this procedure for all the sections, the exposed area Si for particle i is determined, as illustrated as the dark sections in Figure 21.3. Supposing that a single particle of velocity upi and angular velocity vpi is in an applied homogeneous flow of velocity u 0 and angular velocity v0, the fluid velocity u f(x) and the corresponding pressure field p(x) at an arbitrary position x around the particle is given by the following equation 78,79: 5
3
⎛ a⎞ 5 ⎛ a⎞ ⎛ a2 ⎞ r r E r ⎛ a ⎞ uf ( x) ( E ) x - ⎜ ⎟ E ri ⎜ ⎟ ⎜ 1 2 ⎟ i i 2 i + ⎜ ⎟ 2 ⎝ ri ⎠ ⎝ ri ri ⎠ ⎝ ri ⎠ ⎝ ri ⎠
(
pi
)
0 ri
(
3a ⎛ a2 ⎞ 1+ 2 ⎟ upi u0 + ⎜ 4ri ⎝ 3ri ⎠
(
)
)
a 2 ⎞ ri upi - u0 ri 3a ⎛ 1 2 ⎟ ⎜ 4ri ⎝ ri ⎠ ri 2 p(x)
3
(21.3)
5 mf a 3 ri E ri 3 mf a 3 upi u0 ri ri 3 ri 2 2ri
(
)
(21.4)
where ri = x – xi, ri = 1/2ri1/2, µf is the fluid viscosity, E is the rate of strain tensor, and V is the vorticity tensor. The stress tensor t for a Newtonian fluid around a particle is given by
(
pI mf ∇uf ∇uf T
)
(21.5)
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FIGURE 21.3 Schematic drawing of the simulation model of an aggregate and the coordinate system. Dark sections on the particle i indicate sections which are regarded as exposed directly to the flow. Lines are drawn to 4 corners of each section from the particle center in order to determine whether the section is exposed to the fluid or not.
Then the force and torque acting on the area Si are calculated by the following equations, respectively:
Fhi ∫ ⋅ n
ri a
dS
(21.6)
Si
M hi ∫ ri ⋅ n
ri a
dS
(21.7)
Si
where n is the unit vector normal to the surface. It is known that the velocity field around a particle is influenced by the neighboring particles. This effect is taken into account as follows. The local velocity around a particle in the particle bed of porosity v is given by v f(v)u0, where f(v) is a porosity function and 0 f(v) 1. Hence we assume that the velocity u0 in Equation 21.3 and Equation 21.4 may be replaced by v f(v)u0, using the local porosity v around particles. We use the following Steinour’s equation for f(v)80: f ( ev ) 10
1.82 (1 ev )
(21.8)
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TABLE 21.1 Parameters of Aggregates Employed in the Simulation and Estimated Values of c and p Flow Field Shear
Elongation
Aggregate I
N[–] 512
Type of Aggregate pc
Dfr[–] 2.47
aou aav 109[m] 100
II
256
pc
2.44
100
0.4
61.6
0.981
59.1
0.875
d109[m] 0.4
III
1024
pc
2.48
100
0.4
IV
512
pc
2.46
100
2.0
c 10-3[–] 64.1
5.70 12.5
P[–] 0.936
0.945
V
512
pc
2.45
500
0.4
VI
512
pc
2.43
500
2.0
0.850
0.946 1.04
VII
512
cc
1.74
100
0.4
7.52
0.725
VIII
512
pc
2.31
100 (sst = 3.336)
0.4
4.21
0.881
Exp
—
—
2.2
70
2.58
IX
512
pc
2.47
100
0.4
21.7 111
0.879 1.60
The local porosity v around a particle is defined as the porosity for the spherical space between the inner radius a and the outer radius 2a. The hydrodynamic force and torque given above will be propagated to inside particles through the interparticle interactions. Two kinds of propagation mechanisms are considered. When particles are not contacting, they interact through the interaction forces given by the DLVO theory. Here only the van der Waals force for equal spheres is considered for the sake of simplicity, although the electrostatic repulsive force is able to be taken into account, if needed. On the other hand, when the particle surfaces contact or overlap each other because of the trial displacement by the DEM, a repulsive force acts because of their volume exclusion effect. The interaction due to the volume exclusion is calculated by the conventional method of DEM.81,82 The quantitative validity of the present model was confirmed by comparing the dynamic shape factor simulated by the present model with the experimental one for well-defined rectangular aggregates composed of chromium spherical particles in quiescent silicon oil.83 Values simulated by the present model are found to be in a good agreement with the experimental data. Hence, the model is now applied to simulate the behavior of aggregates in flows. All the aggregates I~IX employed are listed in Table 21.1, which may be classified into two kinds in terms of fractal dimension: a particle–cluster (pc) aggregate of rather compact structure whose value of fractal dimension Dfr is 2.4, and a cluster–cluster (cc) aggregate of rather loose structure whose value of Dfr is 1.7. Figure 21.4 shows a series of snapshots of the deformation and breakup of the pc-aggregate I composed of monodispersed particles in the shear flow of µfgs = 500 Pa, where gs is the shear rate. It is found that the aggregate is rotated, elongated into the flow direction, and then split into smaller fragments, but not eroded one by one to single particles from the aggregate surface as in the case of the breakup by ultrasonication.84 This splitting breakup is consistent with the photographic observation given by van de Ven,85 and also with the rupturing process of highly viscous droplets. 86,87 Repeating the similar computation for shear flows of various intensities, the relation between the average number of particles in the final fragments
and the shear stress µfgs can be obtained. It is worth noting that a power-law relation holds between and µfgs, as follows: i c ( mf gs )
P
(21.9)
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FIGURE 21.4 Snapshots of the fragmentation of an aggregate I of in simple shear flow.
where values of c and P are listed in Table 21.1. It is clear that the value of P is nearly constant irrespectively of the aggregates. This implies that aggregates I~VI and VIII are fragmented essentially in the same fashion. On the other hand, the value of c varies with values of a, the number of particles in an aggregate N, the gap between particles d, and the standard deviation of size distribution sst. It is especially sensitive to the value of d; the final size of fragments is very sensitive to the minimum gap between particles. It is clear that the power-law relation also holds for cc-aggregates, though the value of P is smaller than that of pc-aggregates. These results indicate that P depends mainly on Dfr, but not on N, a, d, and sst; the aggregates with the same Dfr will be broken in a similar fashion. This is consistent with the report by Yeung and Pelton88 that the strength of aggregates does not vary with the size but rather with the fractal dimension. It is examined whether the fragmentation process of aggregates may follow any scaling law. It is plausible to assume that the final size of fragments is determined by the balance between the adhesive force between particles and the hydrodynamic drag on particles. The ratio of the magnitudes of these forces, N DA , is defined by the following equation: 2 ⎛ Aa ⎞ 72p/mf ags d N DA 6p /mf a 2gs ⎜ ⎝ 12 d2 ⎟⎠ A
(21.10)
where A is the Hamaker constant. All the data of are plotted against NDA in Figure 21.5. It is important to note that almost all data for the aggregates I ~VIII fall around a single line, illustrated © 2006 by Taylor & Francis Group, LLC
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FIGURE 21.5 Dependence of the average number of particles in a fragment on dimensionless parameter NDA (Exp indicates the experimental relation given by Sonntag, R. C. and Russel, W. B., J. Colloid Interface Sci., 113, 399–413, 1986; the solid line indicates the best fit for simulated results; and the thin chain line indicates the experimental data for dmin = 0.65 nm.).
by a solid line, although the data for the pc-aggregate IV of large minimum separation and for the cc-aggregate VII of loose structure tend to deviate from the line. This line is expressed by the following equation: i 27.9 N DA 0.872
(21.11)
This equation gives us a good tool to estimate the average size of flocs that exist stably in the shear flow. The present simulation was compared with quantitative experiments in which effects by reaggregation among broken fragments are carefully avoided. Sonntag and Russel carried out experiments of the breakage of aggregates with D fr = 2.2 in the simple shear flows, which are composed of monodi sperse latex particles of a = 7.0 × 10–8m.89 It is found that a power-law relation with c = 2.11 × 104 and P = 0.879 holds between and µfgs. The comparison with Equation 21.11 is made by the dashed line in Figure 21.5. We consider that the agreement between the simulation and experiments are satisfactory, because the slope is nearly the same, that is, aggregates are fragmented in the similar fashion. As for the absolute magnitude of , the experimental value is greater. Sonntag and Russel estimated that d is 2.58 nm, using their model. This value is much larger than the minimum surface separation widely employed, that is, 0.4 nm. But, if d = 0.65 nm is assumed, their data are expressed by the thin chain line drawn in the figure and coincide extremely well with Equation 21.11. We consider this value of d to be much more reasonable. Simulation by the present model is carried out also for elongational flows. It is found that aggregates do not rotate but are elongated to the flow direction and then split into smaller fragments in the same manner as the shear flow. The value of is also expressed by Equation 21.9, replacing gs by the elongation rate ge. Values of c and P are listed in Table 21.1. We cannot find any quantitative data to compare. This is probably because the experiment for purely elongational flows is difficult. As for qualitative observation, the breakage process was observed using a four-roller device.90 It was found that aggregates of irregular shape are split into a few fragments followed by erosion of much smaller fines, but spherical aggregates tend to be broken by erosion. The present simulation indicates that aggregates are broken by splitting, but not by erosion, which is essentially consistent with the above-mentioned observation for aggregates of irregular shape. It is important to know which flow is more adequate to break up aggregates, shear or elongational flows. Values of for the aggregate I are plotted against the dissipation energy dis in Figure 21.6 under the flow conditions that appear in usual industrial processes. The value of for © 2006 by Taylor & Francis Group, LLC
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FIGURE 21.6 Comparison between the fragment size of aggregates I in shear flow and that in elongational flow.
the elongational flow is always smaller than that of the shear flow in this range. This indicates that the elongational flow is more effective and preferable to disperse coagulated particles. This result is consistent with the observation by Kao and Mason, 91 who claimed that the elongational flow is more effective for floc breakup, because the energy of flow is consumed to break up the aggregates but not to rotate them.
5.21.3
PARTICLE MOTION IN FLUIDS
Trajectories of particle motion in fluids are obtained by integrating the Newtonian equation of motion considering the fluid force. There are two approaches concerning how to give fluid force to the equation of motion. The first one is to make use of established expressions of the force, which are shown in monographs and handbooks. If the particle Reynolds number is sufficiently small, theoretical expressions such as Stokes drag and Suffman lift forces are applied. If the particle Reynolds number is not sufficiently small, the fluid forces are expressed by the use of the drag and lift coefficients. The Newtonian equation of motion taking into account the fluid forces and other forces acting on a particle is called the Basset–Boussinesq–Oseen equation (BBO equation). The monograph by Crowe et al. 92 describes the details of the BBO equation. The simplified form of the BBO equation is called the Langevin equation. The second approach to give the fluid forces is based on calculated results of flow around moving particles. The second approach has been developed in the last 10 years, while the first one has been used in most simulations in the past. Expressions of drag and lift described in monographs and handbooks are precise when a single particle meets the fluid with constant velocity. If the particle concentration becomes higher, those expressions should be modified. Wen and Yu’s modification 93 is well known. In the second approach, the flow around each particle is solved based on the full Navier–Stokes equation 94–96 or the Lattice–Boltzmann simulation, 97 as shown in Figure 21.7. The fluid forces acting on particles are estimated by integrating stresses on the surface of particles that are the solutions of the basic equation. In this method, particles are regarded as moving boundaries. In principle, the fluid forces acting on the particle can be estimated precisely in any circumstances, but the computation load in this method is much heavier than the first one. Thus, as the number of particles increases, it © 2006 by Taylor & Francis Group, LLC
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FIGURE 21.7 Flow around particles.
FIGURE 21.8
is difficult to use the second approach. In a recent simulation, cases of more than 2000 particles were calculated by the second approach. The following part of this section focuses on the first method. If the particle concentration is sufficiently small, particle motion does not affect fluid motion and particle–particle interaction can be neglected. In that case, as long as the flow field of the fluid is given, numerical integration of the BBO equation or Lagevin equation is straightforward. The problem is how to give the fluid motion, particularly in the case of turbulent flows. One method is to express instantaneous fluid motion empirically or semiempirically by assuming turbulent intensity and eddy life time. The other method is to use the results of LES or DNS of the Navier–Stokes equation. Many studies of turbulent diffusion of small particles have been done along this line. Apart from particle diffusion, some researchers are interested in heterogeneity in particle concentration caused by turbulent vortices.98,99 In most industrial processes, particle concentrations are not so low. In such cases, the fluid motion is affected by the presence of particles, and thus not only forces acting on particles but forces acting on fluid due to particles must be taken into account. This interaction of particle with fluid is sometime called two-way coupling. It is necessary to solve fluid motion and particle motion simultaneously. In numerical analysis of the two-way coupling, a flow field is divided into cells, as shown in Figure 21.8. The size of the cell should be larger than the particle size and smaller than the size of the flow system. The effects of the presence of particles on the fluid are taken into account by the void fraction of each phase and momentum exchange through drag force. This approach has been developed by Anderson and Jackson100 and called the “local averaging approach.” © 2006 by Taylor & Francis Group, LLC
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FIGURE 21.9
In addition to two-way coupling, particle–particle interaction should be considered at higher concentrations. Concerning particle–particle interaction, two phenomena are known: collision and contact. If particles are in a dispersed phase, binary collision between particles is dominant. As the particle concentration becomes much higher, particles keep in contact with each other. In the foregoing part of this section where the particle–particle interaction can be neglected, it is not necessary in calculation to treat all individual particles; a certain number of sample particles are enough to represent all actual particles if the number of sample particles is large enough to obtain statistically stable values. In cases where the particle–particle interaction should be considered, calculation is made for all particles if collision or contact partners are found from trajectories in a deterministic way. The deterministic method of finding partners is inevitable in dense phase flows where particles are in contact. In collision-dominated flows, a method based on ample particles is available: the direct simulation Monte Carlo method, which makes, use of the collision probability. As an example to which this method was applied, a gas–solid flow in a vertical channel is shown in Figure 21.9. 101 It is found that particles form clouds or clusters in the channel by repeated collision. The contact forces in contact-dominated flows are modeled by using mechanical elements such as springs, dashpots, and friction sliders, as shown in Figure 21.10.102 This model was originally proposed in the field of soil mechanics. Combining particle motion based on this model with fluid motion, simulation has a wide range of applications such as dense-phase pneumatic conveying 103 and dense-phase fluidized beds.104,105 Figure 21.11 shows a calculated bubble rising in a fluidized bed.
5.21.4
PARTICLE METHODS IN POWDER BEDS
Powder flow controls the performance of many engineering operations and is also important in natural processes. What kind of method can be used to reveal the flow mechanism of innumerable powder particles? It would be a continuum method for which the constitutive equation for the stress–strain relation and the equation of the state for the bulk density stress relation are necessary. We present numerical simulation results of the velocity and stress fields for a flowing powder using the smoothed particle hydrodynamics (SPH) method106,107 based on the constitutive equations obtained by DEM. The SPH method, which has been used in astrophysics and hydrodynamics, is a Lagrangian continuum model. In this method the Lagrangian equation of motion is solved for hypothetical particles constituting the continuum. The Lagrangian trajectories of hypothetical particles are calculated, and therefore it is possible to describe the discrete characteristics of particulate matters. © 2006 by Taylor & Francis Group, LLC
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FIGURE 21.10
FIGURE 21.11 Bubble rising in a fluidized bed. [From Kawaguchi, T., Yamamoto, Y., Tanaka, T., and Tsuji, Y., Proc. 2nd Int. Conf. on Multiphase Flow, Kyoto, FB2–17, 1995. With permission.] © 2006 by Taylor & Francis Group, LLC
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Numerical Method The computational domain is divided into many particles (element pieces) which overlap with each other, and the trajectories of particles with mutual interaction are calculated to describe the motion of the continuum, for example, an approximated powder bed. The physical quantity of the particle is distributed by the kernel function, and the quantity at a fixed point is obtained by integration over all overlapping particles. The particle concentration shows the continuum density which decides the interaction range of the kernel function, since the computational domain at high density is occupied by small particles and that at low density is occupied by large particles. In the SPH method, the physical property f is determined from the kernel function W. Smoothed f(r) is expressed as
( ) (
)
f (r ) i ∫ f rj W ri rj , h drj D
(21.12)
The integral interpolant is given by N
fj
j1
rj
f (r ) i ∑ m j
(
W ri rj , h
)
(21.13)
where W(ri –rj, h) is an interpolating kernel with
∫ W (r r , h ) 1 i
(21.14)
j
and mj is the mass of an imaginary particle and rj is bulk density of the powder. The Gaussian function was used as an interpolating kernel. To satisfy the principle of action and reaction between particles, the kernel function should be symmetric, that is, W(ri−rj)=W(rj−ri). Then the kernel function takes the form 1 ⎧⎪ e W rij ⎨ 2p ⎩⎪
( ri rj )2 / h2i
( )
h 2i
+
( ri rj )2 / h2j
e
h 2j
⎫⎪ ⎬ ⎭⎪
(21.15)
h is given as
,
⎛ m0 ⎞ hi j ⎜ ⎟ ⎝ rb i ⎠
⎛ m0 ⎞ hj j ⎜ ⎟ ⎝ rb j ⎠
(21.16)
where j = 1.4 and subscripts i and j show the reference and the surrounding particles. The effects of other particles with centers which are within the radius 3hi are taken into account to obtain the smoothed value. Substitution of velocity into vj = fj into Equation 21.13 gives v i ∑ j
mj rj
(
v j W rij , h
)
(21.17)
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The gradient of f(r) may be obtained as N
fj
j1
rj
f ( r ) i ∑ m j
(
W rij , h
)
(21.18)
The Rankine static stress model was used for the initial stress distribution of the powder in the tank. To determine the pressure generated by bulk density differences due to motion in powder, the following equation is applied: ⎛ r ⎞ P K r gH ⎜ 1⎟ ⎝ r0 ⎠
(21.19)
where H is the depth of powder bed. The constant for the powder is not clear but should be chosen so as to be in agreement with experimental results. The bulk density is calculated using the continuity equation transformed as dr ( rv) v r dt
(21.20)
Yuu et al.108 have calculated the forces acting on each powder particle in the powder bed using DEM and obtained the shear and the normal stresses by locally averaging these forces on the plane in the powder bed that assumed continuum. This is essentially the same method of molecular dynamics that gives the stress field in the fluid. The experimental relationships were also obtained under the same conditions. The comparison of calculated and experimental stress–stain rate relationships shows good agreement, and both dynamic shear and dynamic normal stresses are expressed as linearly dependent on strain rates over a fairly wide strain rate region. The following equations obtained by Yuu et al. show the stress–strain rate relationships in the particulate matter: s x s x 0 s x20 s y20 A1Dxx
(21.21)
s y s y 0 s x20 s y20 A1Dyy
(21.22)
t x y t x y 0 s x20 s y20 A2 Dxy
(21.23)
and
where Dxx, Dyy and Dxy are the deformation rates. The coefficient A2 is much larger than coefficient A1. This means that shear deformation occurs more easily than normal deformation in the particulate matter. To calculate the stagnant zone on the powder bed, the Bingham-type plastic model is used. Momentum equations for powder beds are expressed as 1 P 1 ⎛ s x t x y ⎞ du rb x rb ⎜⎝ x y ⎟⎠ dt
(21.24)
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and dv 1 P 1 ⎛ t x y s y ⎞ g dt rb y rb ⎜⎝ x y ⎟⎠
(21.25)
We have calculated Equation 21.21 through Equation 21.24 for the velocity field of the powder bed using the SPH method. Figure 21.10(a) shows results obtained with the SPH method using Lagrangian equations by Sugino and Yuu.109 At 0.6 s, the powder bed height has decreased further, and at 1.0 s, the discharge of powder has diminished. The stagnant zone of powder in the corner of the tank was found to be the same as indicated by experiment. Figure 21.10b shows experimental results for powder in the tank. The calculated flow patterns and shape of the stagnant zone and those measured are basically the same.
5.21.5 TRANSPORT PROPERTIES To evaluate numerically transport properties such as the diffusion coefficient and shear viscosity for dispersed systems, we have two methods: direct and indirect simulations. In the former, we simulate the experimental situation on the computer and measure the quantity directly, while in the latter, we calculate time-correlation functions in the equilibrium state and interpret them into the transport properties by the Green–Kubo formula, which is widely used for simple liquids.110 The main difference between simple liquids and dispersed systems such as colloidal suspensions is the existence of hydrodynamic interaction by the fluid surrounding the particles. 111 Here we focus on the contribution of the hydrodynamic interaction.112 Other contributions such as interparticle forces and Brownian motion are briefly commented on later.
Diffusion Coefficients The self-diffusion coefficient is defined by the mean-square displacement of a tracer particle as 1 d x (t ) x (0) Ds 6 dt
2
(21.26)
where x(t) is the position of the tracer particle at time t. The brackets <> denotes the average of many time sequences. It has two asymptotes: short- and long-time diffusion coefficients. The splitting timescale is a2/D0, where a is the particle radius and D0 is the Stokes–Einstein diffusion coefficient of a single isolated particle in the fluid defined by D0
k BT 6pha
(21.27)
Here k B is the Boltzmann constant, T is the temperature, and h is the viscosity of the fluid. The longtime self-diffusion coefficient is also written as 〈| x(t ) x(0) |2 〉 1 ∞ ∫ 〈U (0)U (t ) t →∞ 6t 3 0
D∞s lim
(21.28)
where U is the velocity of the tracer particle. The last form with the time-correlation function of the velocity is called the Green–Kubo formula. According to Hansen and McDonald110 or Phillips © 2006 by Taylor & Francis Group, LLC
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et al., 112 the self-diffusion coefficients are calculated from many series of dynamical simulations; this is the indirect way. The direct way to calculate diffusion coefficients is to calculate the mobility of the tracer particle. The factor 1/6πha in Russel et al.111 is the mobility of a single isolated particle. For dispersed systems, on the other hand, the mobility is not just 1/6πha because of the hydrodynamic interaction among particles. Under the Stokes approximation, the hydrodynamic interaction is expressed by the resistance equation ⎛ F ⎞ ⎛ RFU ⎜⎝ S ⎟⎠ ⎜⎝ R SU
RFE ⎞ ⎛ U − m ∞ ⎞ RSE ⎟⎠ ⎜⎝ E ∞ ⎟⎠
(21.29)
where F is the force and torque vector with 6N components, S is the stresslet, U – m is the translational and rotational velocities relative to the imposed flow, and E is the strain imposed to the system, for N particles in the system. The whole resistance matrix with 11N × 11N elements depends only on the configuration of the particles and is calculated by the Stokesian dynamics method. 113 From Brady and Bossis, 113 the particle velocity U under no external flow is given by U RFU . F
(21.30)
Then, the Stokes–Einstein relation is extended for dispersed systems as
( )
−1 D0s K BT 〈 RFU
aa
〉
(21.31)
where D0s is a matrix with 6 × 6 elements and R[ 1/Fu]aa denotes the self part of the inverse of the resistance matrix with 6 × 6 elements. The diagonal elements of the translational part of D0s is the short-time self-diffusion coefficients D s0 . Figure 21.12 shows the numerical results of D0s /D0 with experimental results.112
Shear Viscosity The shear viscosity is usually calculated by the direct way rather than the indirect way. The reason is that the interparticle force of hydrodynamic interaction needed for the Green–Kubo formula of the shear viscosity is complicated. In the direct way, the bulk stress p is calculated under the shear flow. Neglecting Brownian motion and interparticle forces, the bulk stress is given by 〈 p 〉1 2h〈 E ∞ 〉 n〈 S 〉
(21.32)
where p is the pressure and n is the number density of particles.114 From Brady and Bossis,113 the stresslet for force-free particles is given by
(
)
1 S RSU . RFU . RFE RSE . E ∞
(21.33)
After taking the average of the stresslet S, the shear viscosity of dispersed system hr scaled by the fluid viscosity h is obtained as n〈S〉 2h(hr (f) 1)〈 E ∞ 〉
(21.34)
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FIGURE 21.12 Short-time self-diffusion coefficients obtained by Stokesian dynamics methods (solid symbols), a theoretical calculation (solid line), and experimental results (open symbols). [From Phillips, R. J., Brady, J. F., and Bossis, G., Phys. Fluids., 31, 3462–3472, 1988. With permission.]
where f is the volume fraction of particles. Note that in the dilute limit, we have the Einstein’s result 5 hr 1 f 2
(21.35)
Figure 21.13 shows the numerical results of hr with experimental results.112
Remarks Using the Stokesian dynamics method, we can calculate the microstructures of dispersed systems, as shown in Brady and Bossis,113 under arbitrary circumstances and evaluate any quantities such as sedimentation velocity and normal stresses as well, by the direct way. To do this, however, we need to know the quantity to calculate, as shown by Batchelor.115 The Brownian contributions are shown for the diffusion coefficient 116 and the bulk stress. 115 The general form of the bulk stress has been shown, 117 and thorough numerical analysis on the Péclet number dependence of diffusion coefficients and shear viscosity has been shown.118
Notation a D0
Particle radius (m) Diffusion coefficient of an isolated particle (m2/s)
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FIGURE 21.13 Shear viscosities obtained by Stokesian dynamics methods (solid symbols), theoretical calculations (lines), and experimental results (open symbols). [From Phillips, R. J., Brady, J. F., and Bossis, G., Phys. Fluids., 31, 3462–3472, 1988. With permission.]
Ds Ds Ds 0 Ds0 E F kB n p R S t T U(t) U – u x(t)
Self-diffusion coefficient (m 2 /s) Long-time self-diffusion coefficient (m 2 /s) Short-time self-diffusion coefficient (m2/s) Short-time self-diffusion matrix Rate-of-strain tensor imposed to the system (1/s) Force and torque vector of particles Boltzmann constant (J/K) Number density of particles (1/m3) Pressure (Pa) Resistance matrix Stresslet of particles (N m) Time (s) Temperature (K) Velocity of particle at time t (m/s) Translational and rotational velocities relative to the imposed flow of particles Position of particle at time t (m)
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h hr f S
Viscosity of a fluid (Pa5s) Effective viscosity of the dispersed system Volume fraction of particles Bulk stress (N m)
Acknowledgment The author would like to acknowledge support by DOE grant FG02–99ER14966.
5.21.6
ELECTRICAL PROPERTIES OF POWDER BEDS
Percolation Theory Electrical conductivities are very sensitive to the structure of powder beds. Simulations of electrical conductivity can be utilized for the structural evaluation or analysis of powder beds. For example, a simulation of the electrical conductivity for a mixture of conducting and insulating particles can be performed as an application of percolation concepts. If an arrangement of the particles in the mixture is described as a lattice model, the electrical conductivity can be derived as a function of concentration of the conducting particle. In this simulation, a drastic change of the electrical conductivity with changing concentration especially relates to a critical point or percolation threshold.119–121 Percolation thresholds of some lattice models are tabulated in Table 21.1. The lattice model should be selected in taking into account the packing density of the powder bed.
Calculation by Applying Kirchhoff’s Law There is another approach to analysis of powder bed structures by simulation. The simulation of electrical resistance for binary systems leads to understanding the change of particle arrangements or the degree of mixing. Particle contact resistances were calculated by using an electrical equivalent circuit based on Kirchhoff’s law. Fujihara and coworkers122,123 performed the simulation of a binary particles system consisting of stainless steel and lead particles and compared the simulated results with experimental ones in a milling process. In the simulation, the randomness or degree of mixing was estimated from a standard deviation s concerning the concentration,
2
N
∑ (C C ) i
s
TABLE 21.2
i1
Three dimension
N
Percolation Thresholds for Several Lattice Models Lattice
Two dimension
0
Honeycomb
Coordination Number 3
Threshold 0.6962
Square
4
0.592745
Triangle
6
0.5
Diamond
4
0.428
Cubic
6
0.3117
Body center cubic
8
0.2460
Face center cubic
12
0.119
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where N is the number of sample spaces, Ci concentration of sampled particles in i space, and C0 is the average concentration in the mill. Then, they discussed their simulation model to reduce the calculation load.
DEM Simulation of Charged Particles Flow dynamics of charged particles is very important for the formation of powder beds. Yoshida and his coworkers performed a DEM simulation of polymer and toner particles in terms of a charge site model. In this model, a particle was divided into several independent parts, as shown in Figure 21.14.
FIGURE 21.14 Illustration of the simulation model for charged particles. This particle is divided by 200 charged sites.
FIGURE 21.15 Time dependence of the total charge of polypropylene particles. © 2006 by Taylor & Francis Group, LLC
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Charge transfer occurs from one site to another site at intersite contact in a probability accompanied by the charge difference between the sites. The simulation of polymer and toner particles was compared with a multiple-impact charging experiment in a metal container. In these simulation and experiments, the particles collided many times in a vibration field. Figure 21.15 shows the time dependence of the total charge of the polypropylene particles. In this comparison, the space charge effect has a large contribution in the powder charging by multiple contacts. Charged particles need to be controlled for required formation of powder beds.
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Walton, O. R. and Braun, R. L., J. Rheol., 30, 949, 1986. Walton, O. R. and Braun, R. L., Acta Mech., 63,: 73–86, 1986. Zhang, Y. and Campbell, C. S., J. Fluid Mech., 237, 541–568, 1992. Drescher, A. and De Josselin de Jong, G., J. Mech. Phys. Solids, 20, 337, 1972. Mueth, D. M., Jaeger, H. M., and Nagel S. R., Phys. Rev. E., 57, 3164–3169, 1998. Howell, D. W., Behringer, R. P., and Veje, C. T., Chaos, 9, 559–572, 1999. Howell, D. W., Behringer, R. P., and Veje, C. T., Phys. Rev. Lett., 26, 5241–5244, 1999. Hopkins, M. A., Particle Simulation, Vol. 1, Rep. No. 87-7, Department of Civil Engineering, Clarkson University, Potsdam, NY, 1987. Metropolis, N., Rosenbluth, M., Teller, A., and Teller, E., J. Chem. Phys., 21, 1087–1092, 1953. Chapman, S. and Cowling, T. G., The Mathematical Theory of Non-Uniform Gases, 3rd Ed., Cambridge University Press, 1970. Hopkins, M. A., M.S. Thesis, Clarkson University, Potsdam, NY, 1985. Hopkins, M. A. and Shen, H. H., J. Fluid Mech., 244, 477–491, 1992. Baxter, G. W. and Behringer, R. P., Phys. Rev. A, 42, 1017–1020, 1990. Gutt, G. M., Ph.D. thesis, California Institute of Technology, Pasadena, CA, 1989. Désérable, D. and Martinez, J., in Powders and Grains 93, Thornton, C., Ed., A. A. Balkema, Rotterdam, 1993, pp. 345–350. Hopkins, M. A., The Numerical Simulation of Systems of Multitudinous Polygonal Blocks, U.S. Army Cold Regional Research Engineering Laboratory, USACRREL Rep. CR 99-22, 1992. Hogue, C. and Newland, D. E., in Powders and Grains 93, Thornton, C., Ed., Balkema, Rotterdam, 1993, pp. 413–420. Kohring, G. A., Melin, S., Puhl, H., Tillemans, H. J., and Vermohlen, W., Comput. Methods Appl. Mech. Eng., 124, 273–281, 1995. Potapov, A. V., Hopkins, M. A., Campbell, C. S., Int. J. Mod. Phys., C6, 371–398, 1995. Potapov, A. V., Campbell, C. S., and Hospkins, M. A., Int. J. Mod. Phys., C6, 399–425, 1995. Potapov, A. V. and Campbell, C. S., Int. J. Mod. Phys., C7, 155–180, 1996. Cundall, P. A., Int. J. Rock Mech. Mining Sci., 25, 107–116, 1988. Ghaboussi, J. and Barbosa, R., Int. J. Num. Anal. Methods Geomech., 14, 451–472, 1990. Potapov, A. V. and Campbell, C. S., Int. J. Mod. Phys., C7, 717–730, 1996. Rothenburg, L. and Bathurst, R. J., Geotechnique, 1, 79–85, 1992. Rothenburg, L. and Bathurst, R. J., Polym.-Plast. Tech., 35, 605–648, 1996. Ting, J. M., Comp. Geotech. 13, 175–186, 1992. Mustoe, G. G. W. and DePooter, G., in Powders and Grains 93, Thornton, C., Ed., A. A. Balkema, Rotterdam, 1993, pp. 421–427. Potapov, A. V. and Campbell, C. S., Granular Matter, 1, 9–14, 1998. Thornton, C. and Kafui, K. D., Powder Technol., 20,109–124, 2000. Kafui, K. D. and Thornton, C., in Powders and Grains 93, Thornton, C., Ed., A. A. Balkema, Rotterdam, 1993, pp. 401–406. Thornton, C. and Randall, C. W., in Micromechanics of Granular Materials, Satake, M. and Jenkins, J. T., Eds., Elsevier, Amsterdam, 1988, pp. 133–142. Tüzün, U. and Walton, O. R., J. Phys. D, 25, A44–A52, 1992. Thornton, C., J. Appl. Mech., 64, 383–386, 1997. Hertz, H., J. Math Crelles J., 92, 156–171, 1882. Mullier, M., Tüzün, U., and Walton, O. R., Powder Technol., 65, 61, 1991. Goddard, J., Proc. Roy. Soc., 430, 105–131, 1990. Drake, T. G. and Walton, O. R., J. Appl. Mech., 62, 131–135, 1995. Raman, C. V., Phys. Rev., 12, 442–447, 1918. Goldsmith, W., Impact, Edward Arnold, London, 1960. Mindlin, R. D. and Deresiewicz, H., J. App. Mech. Trans. ASME, 20, 327–344, 1953. Sondergaard, R., Chaney, K., and Brennen, C. E., J. Appl. Mech., 57, 694–699, 1990. Foerster, S. F., Logue, M. Y., Chang, H., and Allia, K., Phys. Fluids, 6, 1108–1115, 1994. Maw, N., Barber, J. R., and Fawcett, J. N., J. Lub. Technol., 103, 74–80, 1981. Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge University Press, 1989.
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Powder Technology Handbook Dhont, J. K. G., An Introduction to Dynamics of Colloids, Elsevier, Amsterdam, 1996. Steinour, H. H., IEC, 36, 618–624, 1944. Cundall, P. A. and Strack, O. D. L., Geotechnique, 29, 47–65, 1979. Society for Powder Technology, Japan, Ed., Funtai Simulation Nyumon, Sangyo-Tosho, Tokyo, 1998. Niida, T. and Ohtsuka, S., KONA, 15, 202–211, 1997. Higashitani, K., Yoshida, K., Tanise, N., and Murata, H., Colloids Surf. A, 81, 167–175, 1993. van de Ven, T. G. M., in Colloidal Hydrodynamics, Academic Press, London, 1989, p. 532. Torza, S., Cox, R. G., and Mason, S. G., J. Colloid Interface Sci., 38, 395–411, 1972. Williams, A., Janssen, J. J. M, and Prins, A., Colloids Surf. A, 125, 189–200, 1997. Yeung, A. K. C. and Pelton, R., J. Colloid Interface Sci., 184, 579–585, 1996. Sonntag, R. C. and Russel, W. B., J. Colloid Interface Sci., 113, 399–413, 1986. Pandya, J. D. and Spielman, L. A., J. Colloid Interface Sci., 90, 517–531, 1982. Kao, S. V. and Mason, S. G., Nature, 253, 619–621, 1972. Crowe, C., Sommerfeld, M., and Tsuji, Y., in Multiphase Flows with Droplets and Particles, CRC Press, 1997, pp. 81–88. Wen, C. Y. and Yu, Y. H., Chem. Eng. Prog. Symp. Ser., 62, 100–111, 1966. Hu, H. H., Int. J. Multiphase Flow, 22, 335–352, 1996. Takiguchi, S., Kajishima, T., and Miyake, Y., JSME Int. J. Ser. B, 42, 411–418, 1999. Pan, T. W., Joseph, D. D., Bai, R., Glowinski, R., and Sarin, V. J., J. Fluid Mech., 451, 169–191, 2002. Qi, D., Int. J. Multiphase Flow, 26, 421–433, 2000. Crowe, C. T., Chung, J. N., and Troutt, T. R., Prog. Energy Combust. Sci., 14, 171–194, 1988. Squire, K. D. and Eaton, J. K., Phys. Fluids, A3 5, 1169–1178, 1991. Anderson, T. B. and Jackson, R., Ind. Eng. Chem. Fundam., 6, 527–539, 1967. Yonemura, S., Tanaka, T., and Tsuji, Y., ASME/FED Gas-Solid Flows, 166, 303–309, 1993. Cundall, P. A. and Strack, O. D. L., Geotechnique, 29, 47–65, 1979. Tsuji, Y., Tanaka, T., and Ishida, T., Powder Technol., 71, 239–250, 1992. Tsuji, Y., Kawaguchi, T., and T. Tanaka, Powder Technol., 77, 79–87, 1993. Kawaguchi, T., Yamamoto, Y., Tanaka, T., and Tsuji, Y., in Proceedings of the Second International Conference on Multiphase Flow, Kyoto, 1995, pp. FB2–17. Monaghan, J., J. Comput. Phys., 110, 399–406, 1994. Monaghan, J., Phys. D, 98, 523–533, 1996. Yuu, A., Hayashi, M., Waki, T., Umekage, J. Soc. Powder. Technol. Jpn., 34, 212–220, 1997. Sugino, S., Yuu, Chem. Eng. Sci., 57, 227–237, 2002. Hansen, J. P. and McDonald, I. R., Theory of Simple Liquids, Academic Press, London, 1986. Russel, W. B., Saville, D. A., and Schowalter, W. R., Colloidal Dispersions, Cambridge University Press, 1989. Phillips, R. J., Brady, J. F., and Bossis, G., Phys. Fluids., 31, 3462–3472, 1988. Brady, J. F. and Bossis, G., Annu. Rev. Fluid Mech., 20, 111–157, 1988. Batchelor, G. K., J. Fluid Mech., 41, 545–570, 1970. Batchelor, G. K., J. Fluid Mech., 83, 97–117, 1977. Batchelor, G. K., J. Fluid Mech., 74, 1–29, 1976. Brady, J. F., J. Chem. Phys., 98, 3335–3341, 1993. Foss, D. R. and Brady, J. F., J. Fluid Mech., 407, 167–200, 2000. Ottavi, H., Clerc, J., Giraud, G., Roussenq, J., Guyon, E., and Mitescu, C. D., J. Phys. C Solid Phys., 11, 1311–1328, 1978. Stauffer, D. Introduction to Percolation Theory, Taylor & Francis, London, 1988. Odagaki, T., Introduction to Percolation Physics, 2nd Ed., Shokabo, Tokyo, 1995. Fujihara, Y. and Yoshimura, Y., J. Soc. Mater. Sci. Jpn., 36, 1198–1204, 1987. Yoshioka, T., Kaneko, T., Fujihara, Y., and Yoshimura, Y., J. Soc. Mater. Sci. Jpn., 34, 1255–1259, 1985. Yoshida, M., Shimosaka, A., Shirakawa, Y., Hidaka, J., Matsuyama, T., and Yamamoto, H., Powder Technol., 135–136, 23–34, 2003.
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Part VI Process Instrumentation
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6.1
Powder Sampling Hiroaki Masuda Kyoto University, Katsura, Kyoto, Japan
6.1.1
SAMPLING EQUIPMENT
When powder is poured into a heap, size segregation will occur: fine particles tend to remain at the center of the heap and coarse ones congregate at the periphery. Even when the powder is slowly transported by a belt conveyor, coarse particles tend to float on the surface of the powder bed. It is known that size segregation occurs more frequently for powders having higher flowability. The following are said to be the golden rules of sampling: 1. A powder should be sampled while in motion. 2. The entire stream of powder should be taken out for a long series of short time intervals. The first rule recommends sampling from flowing powder, such as a discharging flow from a belt conveyor or feeding flow from one storage vessel to another. Even in these cases, size segregation may occur. Therefore, the entire stream of powder should be sampled by traversing the stream, and the sampling should continue for a long series of short time intervals. If the sampling speed v (m/s) is low, the mass of the sampled powder may be too large. The sampled mass w (kg) is calculated from the relation
w⫽
Wb LW b⫽ v L v
(1.1)
where L (m) is the width of the powder stream, W (kg/s) the powder flow rate, and b (m) is the width of the sampler’s mouth (i.e., sampling width or cutter width). Samplers obeying the rules above are called full-stream samplers. One of the full-stream samplers, named a cutter sampler, is shown schematically in Figure 1.1. There are several types of full-stream samplers, such as car, belt, and rotary. The sampling width (or cutter width) and the sampling speed are adjustable in these samplers. Therefore, it is possible to take out the desired mass of powder according to Equation 1.1. Random sampling is also possible by randomly changing the starting time of the sampler. Other types of samplers are the snap sampler and the slide spoon sampler. These samplers take a small amount of powder out of several positions of a powder stream or powder bed using a specially designed spoon. The mass of the sampled powder, called the increment, is much less than that of the full-stream sampler, and the final sample collected of all increments can be treated easily in a laboratory measurement. A scoop is used to take a sample from powder loaded on a truck. In this case it is recommended that the sample be dug out from 30 cm below the surface, because the surface region is usually affected by segregation. Sampling positions are determined so as to divide the powder bed into equal areas. Sampling spears can be used to take samples from deep inside the powder bed. A sampling device of the air-suction type (Figure 1.2) is also useful to get vertical samples of the powder bed in which particles are mixed with air at the sampler nose and the resulting gas-solid mixture is sucked into a gas–solid separator such as a cyclone or bag filter. 771 © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.1
Full-stream sampler (cutter sampler).
FIGURE 1.2
Suction sampler.
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When continuous sampling from a storage vessel is necessary, a screw conveyor may be used. A table feeder with multiscrapers developed in the author’s laboratory is useful to take small samples during continuous feeding of powder to a process.1 Particles flowing through the table feeder do not suffer from breakage, whereas a screw conveyor often causes particle breakage. A special constant-volume sampler or slide-valve type of sampler may be adopted in the sampling of powder sliding on an inclined chute. These samplers are inserted into the powder stream very rapidly, so that mechanical parts of the samplers could cause problems. Moreover, if the powder stream is only partly sampled, the effect of size segregation is unavoidable. The powder assembly of all increments is called the gross sample. The total mass of the gross sample is usually too much to be used as a sample for laboratory measurements of chemical composition, particle size distribution, and so on. It is necessary to reduce the mass of the sample before use in the laboratory. The procedure used to reduce a sample mass is called sample division. Sample division should be done without segregation. The principles involved here are the same as those already discussed regarding collection of the gross sample. Figure 1.3 shows a sample divider called a spinning riffler, which satisfies the golden rules of sampling. This spinning riffler consists of a turntable with many chutes and bottle-receivers under the table. The performance of the sample divider depends on the rotational speed of the table. A higher speed (as high as 300 rpm) gives a better result.
FIGURE 1.3 Spinning riffler. © 2006 by Taylor & Francis Group, LLC
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There is another type of divider which consists of a rotary feeder and several receivers. A sample divider of this type is called a mechanical distributor or sample splitter. Similar but simpler are an oscillating hopper sample divider and an oscillating paddle sample divider.2 These two sample dividers are not as effective as the spinning riffler or mechanical distributor in reducing the sample mass. All of the sample dividers mentioned above has moving parts, which are apt to cause mechanical problems in a dusty atmosphere. On the other hand, a sample divider called a chute riffler or chute splitter has no moving parts. It has a series of two chutes directing right and left, one after another. The number of chutes on the right is equal to that on the left. Powder poured onto a chute riffler is divided by the series of the two chutes. Sample dividing by the chute riffler is affected by particle segregation, although the effect becomes insignificant as the number of chutes is increased. The segregation effect may be diminished by use of a procedure called Carpenter’s method. A chute riffler requires repeated dividing to reduce the sample size (mass). A sample divider called a table sampler or multicone divider can sufficiently reduce the sample size by only one trial. The table sampler has a series of holes and prisms on an inclined table. Powder is fed to the top of the sampler (divider). Prisms placed across the path of the powder stream divide it into many fractions. Some particles fall through the holes and are discarded, while the powder remaining on the table proceeds to the next row of prisms and holes, some powder is removed, and so on. The powder reaching the bottom of the table is the final sample. Sample dividers similar to the table sampler but with no holes are called a 1/16 divider, 1/32 divider, and so on, according to the number of final samples obtained.
6.1.2 ANALYSIS OF SAMPLING Various sorts of data can be obtained by analyzing the sampled powder. The mass median diameter (MMD) of the sample, for example, is obtainable from the particle size measurement. The value of MMD will change for every sample. An entire set of these values is called a population. Let the value x1 be a first sample extracted from the population with a certain probability. Such a variable (e.g., the MMD) is called a random variable. The random x variable is characterized by a probability density function or frequency distribution f(x). The probability that the sampled value is found in the range x~x⫹dx is given by f (x)dx. The following relation is applicable for various types of random variables in natural phenomena and industrial processes: f ( x ) dx ⫽
⎛ ( x − )2 ⎞ exp ⎜⫺ ⎟ dx 2 2 ⎠ 2ps ⎝ 1
(1.2)
where is the population mean and σ2 is the population variance. The function f (x) is called the normal distribution or Gaussian distribution. The population mean µ and the population variance σ2 are unknown parameters in Equation 1.2, which should be estimated from the following sample mean x– and sample variance s2: x⫽
1 m ∑ xj fj n j ⫽1
(1.3) 2
s2 ⫽
1 m ⎛ ⎞ ⎜⎝ x j ⫺ x ⎟⎠ f j ∑ n ⫺ 1 j ⫽1
2 ⎞ ⎤ 1 m ⎡ 2 1⎛ m xj fj ⎟ ⎥ ⫽ ∑ ⎢ x j f j ⫺ n ⎜⎝ ∑ n ⫺ 1 j ⫽1 ⎢ ⎠ ⎥⎦ j ⫽1 ⎣
(1.4a)
(1.4b)
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where xj is the representative value of the j th class, fj is the frequency of xj (j=1 to m), and n is the m sample size n = Σ fj . The denominator of Equation 1.4a is (n−1) in place of n to make the variance j=1 unbiased. The confidence interval of the population mean with the confidence level is given by the following relation:
(
)
s
x ⫺ t ( v, p )
n
⬍ μ ⬍ x ⫹ t ( v, p )
s n
(1.5)
Some of the numerical values of t(v,p) are listed in Table 1.1.3 The definition of t is t⫽
x ⫺m n s
(1.6)
This is also a random variable and follows Student’s t−distribution with degrees of freedom v=n−1. If the population mean is known, one can obtain the bias d=x−m. On the other hand, the confidence interval of the population variance with confidence level (1- p ) is given by ns 2 ⎛ 1 x 2 ⎜ v, ⎝ 2
⎞ p⎟ ⎠
⬍ 2 ⬍
ns 2 1 ⎛ x 2 ⎜ v,1 ⫺ ⎝ 2
⎞ p⎟ ⎠
(1.7)
The definition of x 2 is x 2⫽
ns 2 2
(1.8)
This is also a random variable obeying the x 2 distribution. The precision of the sampling is usually represented by 2σ. The standard deviation σ is called the average error and (2/3) σ is the probable error. The quantity CV (coefficient of variation) is more often utilized in expressing the precision of the sampling. CV ⫽
TABLE 1.1
s ⫻ 100 (%) m
(1.9)
Student’s t-Distribution t(v,p) 0.25
0.05
0.01
3
1.4226
3.1825
5.8409
4
1.3444
2.7764
4.6041
5
1.3009
2.5706
4.0321
6
1.2733
2.4469
3.7074
7
1.2543
2.3646
3.4995
8
1.2403
2.3060
3.3554
9
1.2297
2.2622
3.2498
n/p
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On the other hand, the accuracy of the sampling is represented by the bias d. Now we will consider, as an example, whether a snap sampling is accurate or not with confidence level of 95% by assuming that the full-stream sampling is correct. Let the data obtained from the full-stream sampling be denoted by x´i and that from the snap sampling be denoted by x´i. The procedure is as follows: 1. 2. 3. 4. 5.
Calculate di = xi−xi⬘ for i = 1,2,…n Calculate its variance from Equation 1.4. – – Calculate t from Equation 1.6 with d = x − . Find the value of t(v,0.05) from Table 1.1. Compare t with t(v,0.05). If t>t(v,0.05), the snap sampling has a significant bias with – confidence level of 95%. The bias is given by d.
The next problem is how many particles should be sampled so as to obtain, for example, the particle size distribution by sizing and counting. In case that the log-normal size distribution is applicable, distribution of sample mean particle size can be analytically obtained.4 Based on the sample mean distribution, the minimum number of particles n* can be estimated by the following equation: n ∗⫽
v d2
(1.10)
where d is an error admissible in the evaluation of particle size analysis, and v is a parameter defined by ⎛1 ⎞ v ⫽ zc 2 m 2 s 2 ⎜ m 2 s 2 ⫹ 1⎟ ⎝2 ⎠
(1.11)
where z c is the standardized normal variable for the stated level of confidence which is determined by Table 1.2, m is a parameter in the evaluation function x— given by Equation 1.13 below, and s is m,0 a standard deviation of log-normal particle size distribution which is given by the natural logarithm of geometric standard deviation sg as follows. s ⫽ ln sg
(1.12)
⎛1 ⎞ xm,0 ⫽ x1,0 exp ⎜ ms 2 ⎟ ⎝2 ⎠
(1.13)
— The evaluation function xm,0 is
TABLE 1.2
Parameter zc in Equation 1.11
Confidence level(%)
90 1.64
95 1.96
97.5 99 99.5 2.24 2.57 2.81
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For particle size measurements, the following evaluation functions are recommended. x6,0 ⫽ x1,0 exp 3s 2
( )
(1.14)
Sauter diameter:
⎛5 ⎞ x5,0 ⫽ x1,0 exp ⎜ s 2 ⎟ ⎝2 ⎠
(1.15)
Mean volume diameter:
⎛3 ⎞ x3,0 ⫽ x1,0 exp ⎜ s 2 ⎟ ⎝2 ⎠
(1.16)
Mass median diameter:
The minimum number of particles can be estimated if one of these evaluation functions is adopted.5 The procedure under the mass median diameter as the evaluation function is as follows: 1. Assume the estimation probability (confidence level). If it is assumed as 95%, then Table 1.2 gives u = 1.96. 2. Comparing Equation (1.13) with Equation (1.14), the parameter m = 6. 3. Calculate the parameter ω by Equation (1.11).
(
)
(
)
v ⫽ 36u 2 s 2 18s 2 ⫹ 1 ⫽ 138.3s 2 18s 2 ⫹ 1
(1.17)
4. Set the admissible error δ. If it is assumed as δ = +−0.05 (within +−5% error), Equation (1.10) gives
(
)
(
)
n ∗ ⫽ 138.3s 2 18s 2 ⫹ 1 ⁄ ␦2 ⫽ 55320 s 2 18s 2 ⫹ 1
(1.18)
5. Determine the geometric standard deviation sg of the measured particle size distribution. If sg for the sample powder is 2.0, for example, then s = ln sg = ln 2.0 = 0.693. Therefore, Equation 1.18 gives n* = 256228. This is the minimum number of particles which should be sampled so as to obtain the particle size distribution whose mass median diameter is within 5% of the corresponding population value with 95% confidence level. The above procedure is applicable to other cases such as the analysis of specific surface area, chemical element of particles, and so on, as far as these variables depend on particle size. The evaluation function should be determined appropriately based on the mean particle diameter suitable for each variable.
REFERENCES 1. 2. 3. 4. 5.
Masuda, H., Kurahashi, H., Hirota, M., and Iinoya, K., Kagaku Kogaku Ronbunshu, 2, 286–290, 1976. Allen, T., in Particle Size Measurement, 3rd Ed., Chapman & Hall, London, 1981, chap. 1. Hoel, P. G., Elementary Statistics, Wiley, New York, 1966, appendix. Masuda, H. and Iinoya, K., J. Chem. Eng. Jpn., 4, 60–66, 1971. Masuda, H. and Gotoh, K., Adv. Powder Technol., 10, 159–173, 1999.
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6.2
Particle Sampling in Gas Flow Hideto Yoshida Hiroshima University, Higashi-Hiroshima, Japan
Hisao Makino Central Research Institute of Electric Power Industry, Chiyoda-ku, Tokyo, Japan Some problems related to the sampling of particles are reviewed and discussed here. In the particle sampling processes, a so-called sampling loss occurs at the entrance and inner wall of the sampling probe, so that the measured value can be incorrect. The subjects to be discussed here are (1) anisokinetic sampling errors, (2) sampling in stationary air, and (3) practical applications of particle sampling.
6.2.1 ANISOKINETIC SAMPLING ERROR If the sampling velocity u is different from that of the main stream velocity uo, the particle concentration in the sampling tube becomes different from that in the main stream. This type of discrepancy in particle concentration is referred to as anisokinetic sampling error. Figure 2.1 shows a computer simulation of particle trajectories near the sampling probe under anisokinetic sampling conditions. When the sampling velocity u is higher than that of the main stream velocity uo, the measured concentration Co becomes lower than the main stream concentration Co. Conversely, when u < uo, C becomes greater than Co. Table 2.1 lists some typical proposed expressions for anisokinetic sampling errors. Figure 2.2 shows the relation between the concentration ratio and the velocity ratio. The use of Davies’ equation1 is in good agreement with the numerical results.2,3 Watson’s equation4 underestimates the sampling error. Figure 2.3 shows the comparison between several empirical or semitheoretical equations listed in Table 2.1 and the results obtained by the numerical calculations for both potential and viscous flow. The error estimated by a numerical calculation for potential flow is the highest among them, whereas C/Co, as estimated by Davies’ equation, is intermediate between the numerical results of potential flow and viscous flow. From Figure 2.2, it can be seen that the concentration ratio C/C o approaches unity when the velocity ratio uo/u is less than 0.25. Figure 2.4 shows the relation between the concentration ratio and Levin’s parameter k for a velocity ratio uo/u of less than 0.25. The experimental data of Gibson and Ogden7 and Yoshida et al.8 are compared with Levin’s equation9: u C ⫽ 1⫺ 0.8k ⫹1⫺ 0.8k 2 , k ⫽ 2 P o u Co
(2.1)
The experimental data agree well with Levin’s equation and with the numerical calculation for point sink flow. 779 © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.1
6.2.2
Particle trajectories for anisokinetic sampling conditions.
SAMPLING IN STATIONARY AIR
It is impossible to carry out isokinetic sampling in stationary air. Therefore it is important to estimate the error of anisokinetic sampling. This sampling error has been investigated by Levin,9 Davies,1 Kaslow and Emrich,10 Breslin and Stein,11 Gibson and Ogden,7 Yoshida et al.,8 and Agarwal and Liu. 12 Figure 2.5 shows the calculated fluid stream lines near the sampling probe for the case of viscous flow and potential flow. The fluid stream lines for both cases are nearly the same except for the probe inlet. Figure 2.6 shows one of the calculated trajectories for particles in the viscous flow field. Sampling efficiency was also calculated using the point sink approximation. The analytical solution of Levin 9 agrees with the numerical calculations for the point sink flow up to Levin’s parameter k = 0.5. For larger values of k, some particles orbit about the point sink, as shown in Figure 2.7. Experimental data by Yoshida et al. 8 agrees well with their theoretical calculations based on the viscous flow. Figure 2.8 shows the calculated C/Co as a function of inertia parameter P and gravitational parameter G. The solid lines represent upward sampling and the dashed lines show downward sampling. The results obtained by Agarwal and Liu12 and Davies’1 region of perfect sampling are also shown in Figure 2.8. When the probe is facing downward, the concentration ratio C/Co is less than unity. However, when the probe is facing upward, the ratio C/Co can become larger than unity because of © 2006 by Taylor & Francis Group, LLC
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TABLE 2.1
781
Expressions for Anisokinetic Sampling Error
Researchers
Empirical or Semiempirical Equation
(
)
Watson (1954)
u0 ⎡ 1 + F u ⎣
Badzioch (1959)
( −2 L ) ⎤ , L = 6 − 1.6D PD ⎡ ⎞ ⎛u 1 ⫹ a ⎜ 0 ⫺1⎟ , a ⫽ ⎢1 − exp ⎥ ⎠ ⎝u 2L ⎣ PD ⎦
Davies (1968)
u0 ⎛ P ⎞ ⎛ 0.5 ⎞ ⎜ ⎟ ⫹ ⎜⎝ ⎟ u ⎝ P ⫹ 0.5 ⎠ P ⫹ 0.5 ⎠
Zenker (1971)
u0 ⎛ u ⎞ + K ⎜ 1 − 0 ⎟ , K 1 + exp (1.04 + 2.06log ( P/ 2 )) ⎝ u u⎠
{
Belyaev and Levin (1974)
1⫹
}
−1
P (1⫹ 0.31 uⲐu 0 ) ⎛ u0 ⎞ ⫺1⎟ ⎠ 1 ⫹ P (1⫹ 0.31 uⲐu 0 ) ⎜⎝ u
0.18 ⬍
FIGURE 2.2
2 u⁄u 0 − 1 ⎤ , F is a function of P ⎦
u ⬍ 6.0, u0
0.36 ⬍ P ⬍ 4.06
Calculated anisokinetic sampling errors.
the direct settling of larger particles. The effect of gravity is clearly shown in Figure 2.9, where the sampling efficiency Cm/Co is represented by Upward sampling: Cm C 1 ⫽ Co Co 1⫹ G
(2.2)
Cm C 1 ⫽ Co Co 1⫺ G
(2.3)
Downward sampling:
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FIGURE 2.3 Relationship between concentration ratio and inertia parameter (numerical results compared with other equations).
FIGURE 2.4 ones.
Experimental results of high-speed sampling errors compared with theoretical
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FIGURE 2.5
783
Calculated stream lines near sampling probe.
The maximum value of Cm/Co is unity, as shown by the particle mass balance. A rapid fall in the efficiency of the downward sampling as the inertia parameter increases is found. Particles with a diameter of less than about 10 μm are regarded as harmful to humans because small particles are easily deposited in the human lung. In Japan, particle concentration measurements with a diameter of less than 10μm are regulated as a 10 μm cut. The particle deposition efficiency on the lung surface increases sharply for particle diameters of less than about 2 μm. The concentration measurement of particles with a diameter less than 2.5 μm is referred to as PM-2.5 sampling. Figure 2.10 shows U.S. regulations regarding a PM-2.5 sampler. In this case, the sampling flow rate and particle density are regulated to 16.7 l/min and 1,000 kg/m 3 , respectively. Classification performance of the sharp-cut cyclone (SCC) inlet cyclone does not satisfy this criterion and further research will be needed to determine the sharp cut PM-2.5 sampler.
6.2.3 PRACTICAL APPLICATIONS OF PARTICLE SAMPLING Sampling Probe and System Used in Practical Application Figure 2.11 shows the sampling probes that are commonly used for measurement of dust concentration. They are referred to as static-pressure-balance-type sampling probes and standard-type sampling probes. The inner static pressure must be equal to the outer static pressure for isokinetic sampling in the case of static-pressure-balance-type sampling probe. However, if the inside crosssectional area of the sampling probe is constant, the sampling velocity is lower than the main flow velocity because of the pressure drop from the nozzle inlet to the static-pressure detecting hole in the sampling probe. Tamori13 has proposed an advanced static-pressure-balance-type sampling probe, in which the inner diameter expands. Whitely and Reed14 investigated the effect of the shape of the sampling probe on measurement of dust concentration. They measured dust concentration using the simplified probe and the standard probe. For the simplified probe, the distance between the probe tip and the bend is short. They concluded that the difference in measured concentration between these two sampling probes is small. Measured concentration also varies with the angle between the sampling probe and main flow. Watson4 and Raynor15 examined the sampling error by changing the angle between the sampling probe and main flow. They found no significant difference at an angle less than ±5º. Belyaev and © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.6 Simulated particle trajectories in stationary air (viscous flow).
Levin16 investigated the effect of the thickness of the probe tip and concluded that the sampling error increases when the probe diameter is decreased.
Measuring Method for Particle Size Distribution by Anisokinetic Sampling The sampling efficiency during anisokinetic sampling is dependent on the particle size when the main flow velocity and the diameter of the sampling probe are fixed. Consequently, it should be noted that the anisokinetic sampling performs the classification of the particles. Particle size distributions can be measured by using this “classification effect” during anisokinetic sampling. 17 © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.7 Calculated particle trajectories in stationary air (point sink approximation).
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FIGURE 2.8 Constant sampling efficiency curves in stationary air.
The sampling efficiency of monodisperse particles has a linear relation with the velocity ratio, according to Davies’ equation, so that the anisokinetic sampling efficiency can be easily estimated. The slope a of this linear line varies with the particle size, as shown in Figure 2.12. In the case of polydisperse particles, the concentration ratio also has a linear relation with the velocity ratio. The slope A of this relation for polydisperse particles is obtained by integrating the product of the slope a and the particle size distribution function f(x): P ( x) ⎞ ∞ ∞ ⎛ A ⫽ ∫ o af ( x ) dx ⫽ ∫ o ⎜ ⎟ f ( x ) dx P ⎝ ( x ) ⫹ 0.5 ⎠
(2.4)
Equation 2.4 can be used to determine f(x) when the slope A is experimentally obtained because the inertial parameter can be easily calculated. However, when a large number of variables in f(x) exist, a solution of the above equation cannot be obtained. Some assumptions are necessary to calculate the particle size distribution. For example, the particle size distribution function is required. Figure 2.13 shows the particle size distribution of fly-ash particles calculated by this method. Flyash particles properly follow a log-normal distribution, and the geometric standard deviations of these particles are nearly 2.5, so that the median diameter x50 can be calculated. In this figure, the results measured by the Andersen stack sampler (ASS), which is one of the most common methods used in particle size distribution measurements, are also shown. It is concluded that the measured values by two different methods fully coincide with each other is impossible, because the actual © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.9 Sampling efficiency curves for downward and upward probe directions.
FIGURE 2.10
Regulation of PM-2.5 in USA and data of PM-2.5 SCC inlet cyclone.
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FIGURE 2.11
Sampling probes for dust measurement.
FIGURE 2.12
Change of the slope “a” in Davies’ equation.
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FIGURE 2.13 Comparison of size distributions measured by anisokinetic sampling and ASS.
particle size distribution does not follow the ideal log-normal distribution. However, the differences between this method and a conventional method are very small. On the other hand, to measure complicated particle size distributions, the new method proposed by Tsuji et al.18,19 uses plural sampling probes of different diameters. The inertia parameter can be changed by altering the diameter of the sampling probe. It, therefore, is possible to obtain plural equations such as Equation 2.4, which show the relationship between slope A and f(x) when plural sampling probes of different diameters are used. If more than two sampling probes are used, these equations can be solved by use of Twomey’s method.20 Figure 2.14 shows the particle size distributions of fly-ash measured by this method (the three-probe method) and ASS. While there are small differences in the measured values by two methods, these methods are roughly in agreement. Since the size distributions of droplets and melted particles may change after sampling and these particles flow on the plates in ASS, measurements of these particles by use of conventional methods becomes very difficult. On the other hand, the method shown in this section is suitable for these particles, because the result is obtained from only the measurement of particle concentration.
Measuring Method for Particle Size Distribution by Backward Sampling The method using anisokinetic sampling is capable of measuring the size distribution of particles whose size is in the order of 10 μm. the accuracy of this method becomes lower, because the slope a in Figure 2.12 is small and does not change significantly with particle size. The reason for this is that the fine particles easily follow the changes of the stream lines near the probe during the anisokinetic © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.14 Particle size distribution of fly-ash measured by the three-probe method and ASS.
sampling due to their small inertia forces. The backward sampling, in which the probe is aligned at 180 o to the flow stream, is effective for the classification of fine particles. 21,22 In backward sampling, the stream line near the sampling probe changes substantially compared to anisokinetic sampling. Large particles cannot follow these stream lines and are not aspirated into the probe. Fine particles, which easily follow the stream line, enter the probe. The next equation was proposed to express the classification efficiency during backward sampling.21,22 ⎧ ⎫ Ci U ⫽ exp ⎨⫺5.09 0 P ( x )⎬ C0 Ui ⎩ ⎭
(2.5)
In order to measure particle size distribution by use of classification during backward sampling, the backward sampling method and the combined method are proposed. The backward sampling method involves classification during backward sampling, and the combined method involves classification during both backward sampling and anisokinetic sampling. The accuracy of these methods was compared using a computer simulation.23 Figure 2.15 shows one of the results. In this figure, it is assumed that the true distribution in the main flow is the bimodal distribution whose peaks are 1 μm and 10 μm. The value by the three-probe method is also shown in this figure. While the three-probe method cannot estimate the two peaks, the backward sampling method is able to estimate these peaks. However, there are small differences between the true distribution and the value by the backward sampling in the 10 μm region. The combined method is the most accurate of these methods. The reason for this is that the combined method utilizes both © 2006 by Taylor & Francis Group, LLC
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FIGURE 2.15 Comparison of the accuracy of the three-probe method, backward sampling method, and the combined method.
backward sampling, which is effective for fine particles less than several μm, and anisokinetic sampling, which is effective for particles of the order of 10 μm. The particle size distribution of fly-ash in a coal combustion exhaust gas measured by the backward sampling method and the combined method are shown in Figure 2.16. This fly-ash has a typical size distribution whose peak is at 5 μm. The distributions measured by these two methods nearly agree with the result by ASS. The accuracy of the methods by use of the backward sampling is similar to ASS.
Notation a A C/Co D, D1 F(x) G
Slope of linear relation between concentration ratio and velocity ratio for monodisperse particles Slope of linear relation between concentration ratio and velocity ratio for polydisperse particles Particle concentration in a sampling probe and main stream (kg/m3) Sampling efficiency in stationary air Probe diameter and duct diameter (m) Particle size distribution function (μm-1) Gravitational parameter in duct flow and in stationary air (= ρ Px2g/18μuo, = ρPx2g/18μu)
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FIGURE 2.16 Particle size distribution of fly-ash measured by backward sampling and the combined method
k p R Re Repo r(=r/R), z(=z/R) rc(=rc/R) u, u uo vt m Pp x x50
Levin’s parameter Inertia parameter in duct flow and in stationary air ( = ρ Px2uo/18μR, = ρPx2u/ 18μR) Probe radius (m) Flow Reynolds number Free-stream particle Reynolds number ( = xuoρ/μ) Dimensionless coordinates Radial coordinate of critical particle trajectory Sampling velocity of duct flow and in stationary air (m/s) Main flow velocity (m/s) Particle settling velocity (m/s) Viscosity of fluid (Pa s) Density of fluid (kg/m 3) Particle density (kg/m3) Particle diameter (μm) Mass median particle diameter (μm)
REFERENCES 1. Davies, C. N., J. Appl. Phys., 1, 921–929, 1968. 2. Yoshida, H., Ohsugi, T., Masuda, H., Yuu, S., and Iinoya, K., Kagaku Kogaku Ronbunshu Japan, 2, 336–340, 1976. 3. Masuda, H., Yoshida, H., and Iinoya, K., J. Powder Technol. Jpn., 18, 177–183, 1981. 4. Watson, H. H., AIHA Quart., 15, 21–29, 1954. 5. Badzioch, S., Br. J. Appl. Phys., 10, 26–29, 1959. 6. Zenker, P., Staub. Rein Luft, 31, 30–38, 1971. © 2006 by Taylor & Francis Group, LLC
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7. Gibson, H., and Ogden, T. L., J. Aerosol Sci., 8, 361–369, 1977. 8. Yoshida, H., Uragami, M., Masuda, H., and Iinoya, K., Kagaku Kogaku Ronbunshu Japan, 4, 123–128, 1978. 9. Levin, L. M., Bull. Acad. Sci. USSR Geophys. Ser., 7, 87–94, 1957. 10. Kaslow, D. E. and Emrich, R. J., Technical Report 23, Lehigh University, 1974. 11. Breslin, J. A., and Stein, R. L., AIHA J., 36, 576–586, 1975. 12. Agarwal, J. K. and Liu, B. Y. H., AIHA J., 41, 191–199, 1980. 13. Tamori, Y., Funtai Kogaku Kenkyu Kaishi Japan, 11, 3–13, 1974. 14. Whitely, A. B. and Reed, L. E., J. Inst. Fuel, 32, 316–325, 1959. 15. Raynor, G. S., Am. Ind. Hyg. Assoc. J., 31, 294–299, 1970. 16. Belyaev, S. P. and Levin, L. M., J. Aerosol Sci., 3, 127–136, 1972. 17. Makino, H., Tsuji, H., Kimoto, M., Yoshida, H., and Iinoya, K., Kagaku Kogaku Ronbunshu Japan, 21, 896–903, 1995. 18. Tsuji, H., Makino, H., Kimoto, M., Yoshida, H., and Iinoya, K., J. Aerosol Sci., 26 (Suppl. 1), S113– S114, 1995. 19. Tsuji, H., Makino, H., Yoshida, H., and Iinoya, K., J. Aerosol Sci., 28 (Suppl. 1), S681–S682, 1997. 20. Twomey, S., J. Comput. Phys., 18, 188–200, 1975. 21. Tsuji, H., Makino, H., Yoshida, H., Ogino, F., Inamuro, T., and Fujita, I., Kagaku Kogaku Ronbunshu Japan, 25, 780–788, 1999. 22. Tsuji, H., Makino, H., and Yoshida, H., Powder Technol., 118, 45–52, 2001 23. Tsuji, H., Makino, H., and Yoshida, H., J. Powder Technol. Jpn., 36, 810–818, 1999.
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6.3
Concentration and Flow Rate Measurement Hiroaki Masuda and Shuji Matsusaka Kyoto University, Katsura, Kyoto, Japan
6.3.1
PARTICLE CONCENTRATION IN SUSPENSIONS
The concentration of particles suspended in gas or liquid is expressed in several ways. The various definitions of particle concentration are summarized as follows: 1. Mass flow ratio m (–): m⫽
W Q
(3.1)
where W and Q are the mass flow rates of particles and fluid, respectively. 2. Flow concentration c (kg/m3): c⫽
W ⫽ ram Qv
(3.2)
where Qv is the volumetric fluid flow rate and r a the density of fluid. If the concentration c is divided by the particle density rp, it becomes dimensionless and is called the volumetric concentration. 3. Holdup concentration ch (kg/m3): ch ⫽
dw Wdl W ⫽ ⫽ V nV nA
(3.3)
where dw is the mass of suspension, V the volume occupied by the suspension, dl the length, v the mean velocity of particles, and A the cross section. The above-mentioned three types of concentration are often called the particle concentration for simplicity. The holdup concentration ch is related to the flow concentration c by the following equation: u ch ⫽ c
(3.4)
where u ( = Q v /A) is the velocity of fluid. In gas–solids suspension flow, the particle velocity n is usually smaller than the gas velocity u, so that the holdup concentration ch is higher than the flow concentration c. The number concentration is also utilized instead of the mass or volumetric concentration. The optical particle counter, for example, measures the number concentration. Table 3.1 lists the methods for the measurement of relatively high particle concentration in powderhandling processes. A gamma-ray densitometer or density gauge is used especially for the continuous measurement of slurries with a high solids concentration. Figure 3.1 shows this instrument, which
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FIGURE 3.1
TABLE 3.1
Gamma-ray density gauge.
Powder Concentration Meter
Principle Electric capacitance increase
Notes Electric condenser, parallel-plate condenser
Gamma-ray attenuation
Gamma-ray densitometer, Lambert-Beer law
Statistical correlation of various noise
Autocorrelation
Microwave reasonance
Resonant frequency decrease
TABLE 3.2 Dust Concentration Meter Principle Photo extinction
Notes Photometer, transmissometer, Lambert-Beer law
Light scattering
Photometer, nephelometer, Optical particle counter
Laser radar
Lidar (long-range measurement)
Nucleated condensation
Condensation nuclei counter (ultrafine particles)
Electrostatic charging
Charge transfer by impact
consists of a radioactive source, radiation detector, amplifier, and analyzer. For a radioactive source, 137 Cs (E = 662 keV) and 241Am (E = 60 keV) are usually utilized. The gamma rays produced by these sources pass through the pipe line, and the output is manipulated to get the slurry concentration. The slurry or pulp density can be measured by various methods, including the above-mentioned nuclear density meters, hydrometers, balanced flow vessels, differential air bubblers, ultrasonic density meters, and so on.1–4 The methods listed in Table 3.2 are applicable to fine particles. Table 3.3 lists the methods for detecting the mass of fine particles sampled on a filter or sampled in the sensing zone of an instrument. © 2006 by Taylor & Francis Group, LLC
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797
Dust Monitor (Sampled Particles)
Principle Beta-ray attenuation
Notes Beta gauge, Lambert-Beer law
Light attenuation
Lambert-Beer law
Electrically forced vibration
Mechanical resonance (frequency decrease)
Piezoelectric vibration
Piezobalancea (frequency decrease)
Electrostatic charging
Charge measurement after unipolar charging
a
Trade name.
The various electrical phenomena and the attenuation technique can be used in measurement as follows.
Electric Capacitance Change Electric capacitance of a parallel-plate capacitor increases with increasing particle concentration in a gas–solids suspension. The measured concentration is the volumetric concentration. As the area of the parallel-plate capacitor is made as small as 1 cm2, the sensor is utilized to measure the local concentration of particles in powder-handling equipment such as a fluidized bed. 5
Electrostatic Charging The electric current generated by particle impact is proportional to the powder flow rate. Therefore, the flow concentration c of a gas–solids suspension will be obtained if the suspension is sampled with a constant velocity. Particles in sampled gas impact on the sensor target placed in the sampled flow or impact on an inside wall of the sensor tube. Any sensitive materials can be used as the sensor.6
Induction Charging The electroconductive particles will be polarized in an electric field. If the polarized particle contacts with a target electrode, negative or positive charge will be neutralized by the opposite charge supplied from the electrode; the particle obtains net charge after the contact. Therefore, the current required for the charging will be found between the electrodes as a voltage pulse.7 The number of pulses found per unit time can give the number concentration.
Piezoelectric Effect A piezoelectric sensor can be applied to the measurement of particle concentration in a flowing suspension. The sensor detects the number frequency of impact. At the same time, the particle mass is measured so that one can transform the frequency into the corresponding mass concentration. In this kind of sensor, the impact efficiency should be correctly taken into consideration. 6,8
Electrically Forced Vibration (Resonance Technique) This method is based on the electrically forced vibration of collected particle mass. Suspended particles are sampled isokinetically onto a fibrous filter that is connected to a spring. The sensing element is electrically vibrated. If the frequency of the forced vibration coincides with the natural © 2006 by Taylor & Francis Group, LLC
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frequency, resonance will be found. Therefore, if the resonance frequency is detected, the mass m will be calculated. As the particles deposit on the filter, the mass m increases from m to m + Δ m , and the resonance frequency changes. If the frequency shift is measured, the mass of the particles will be obtained. 9 Then the flow concentration c is calculated by c⫽
⌬m Qv ⌬t
(3.5)
where Δt is the duration of sampling and Q v is the sampled flow rate.
Quartz Crystal Microbalance A mass sensor utilizing a quartz oscillator is based on the same principle as discussed above. The sensor is called the quartz crystal microbalance (QCM). Suspended fine particles are sampled by use of a suction pump and are collected on the quartz crystal.10 The resonance frequency decreases as discussed above. The frequency shift Δf (Hz) for the quartz crystal with the natural frequency of 5 MHz is given by ⌬f ⫽⫺ 5.65 ⫻ 106
⌬m S
(3.6)
where S is the particle-collection area (m2). The mass sensitivity (Cf = 5.65 × 106) is independent of the physical properties of the deposited particles. A frequency shift as small as 0.1 Hz is measured by use of a simple frequency counter. Therefore, Equation 3.6 shows that the QCM can detect a small amount of particles of 1.8 × 10–8 kg/m2 ( = 1.8 ng/cm2).
Attenuation Technique The energy of waves such as light (electromagnetic wave), sound, or radioactive rays will be dissipated through suspended particles or a powder bed because of wave absorption or scattering. If a light beam passes through a uniformly suspended aerosol layer, the intensity of light will decrease as follows: I ⫽ exp (⫺ hm Ch L ) I0
(3.7)
where I0 is the initial incident intensity of light (i.e., intensity of the transmitted light without particles), c h is the holdup concentration, h m is the mass absorption coefficient, and L is the total thickness of the aerosol layer. Equation 3.7 implies that the intensity of the light beam decreases exponentially with the thickness of the layer. This fact is known as the Lambert–Beer law. The maximum attenuation will be found for a certain particle size between 0.1 and 2 μm. On the other hand, the mass absorption coefficient for very short wavelengths such as UV light (0.01 to 0.4 μm) or X-rays (less than 0.01 μm) is approximately independent of the particle size. For the mass measurement of particles sampled on dust filters, gamma rays used in the density gauge are too strong, and beta rays from 14C, 147Pm (promethium), or63 Ni are utilized. The concentration meter based on the beta rays is applicable to the measurement of higher concentration than the QCM. The mass absorption coefficient hm is a function of the maximum energy Emax of the beta rays, and it is almost independent of the kinds of materials deposited. Therefore, the meter is easily calibrated by use of a plastic film such as a Mylar film. 11 © 2006 by Taylor & Francis Group, LLC
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POWDER FLOW RATE
There are two types of powder flow: bulk solids flow and suspension flow. The principles of powder flowmeters for bulk solids flow and suspension flow are listed in Table 3.4 and Table 3.5, respectively. The weighing method in Table 3.4 belongs to direct measurement, but others belong to inferential measurements. The powder flow rate W (kg/s) in suspension flow is the product cvA (kg/s) of particle concentration c (kg/m3), particle velocity v (m/s), and cross-sectional area A (m2). Therefore, the measurement of the powder flow rate requires a combined measurement of particle concentration and particle velocity. Table 3.6 lists the principles of the particle velocity measurement. The mean velocity of powder flow across the cross-sectional area should be measured to obtain the powder flow rate from the combined concentration-velocity measurement.
Weighing Method The weighing method is the most reliable and accurate in the measurement of the powder flow rate in the bulk solids flow. 1 The hopper scale (weigh hopper) and the belt scale (belt weigher or conveyor scale) are the typical flowmeters of this kind. The hopper scale receives powder in the hopper, weighs it, and then discharges. The frequency of receiving and discharging is 100–400 cycles per hour, which can be changed according to the process requirement. A combination of high speed in the early stage of the feeding and low speed in the final stage allows a short cycle time with high accuracy. 6 A hopper scale which weighs the scale hopper after the feeding is perfectly stopped is called the net-weight mode hopper scale. For continuous powder feeding, the hopper scale is operated in the loss-in-weight mode, in which the powder is added before the hopper becomes empty and the discharge flow is kept continuous. The powder adhered on the hopper wall does not cause a weigh error of the hopper scale in this mode. The belt scale is, however, more suitable than the hopper scale for the continuous feeding of powder, where the flow rate is obtained as a product of the weight of powder on the belt of unit
TABLE 3.4
Powder Flowmeter for Bulk Solids Flow
Principle Weighing
Notes Belt scale, hopper scale
Impulsive force
I m p a c t fl o wm e t e r
Coriolis force
Massometer,a etc.
Back pressure of air jet
Gas purge typ e of flow meter
Volume displacement
Turbine type of flow meter
a
Trade name.
TABLE 3.5 Powder Flowmeter for Suspension Flow Principle Pressure drop
Notes Venturi-type flowmeter, Venturi-orifice flowmeter
Electrostatic charging
Particle electrification by impact
Microwave
Microwave-type flowmeter (microwave resonance)
Statistical phenomena
Correlation-type flowmeter (concentration ⫻ velocity)
Heat absorption
Metallic particles
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TABLE 3.6
Particle Velocity Meter
Principle Statistical Phenomena
Notes Correlation technique (electrostatic current, electric capacitance, light extinction
Spatial filtering
Microwave type, laser velocimeter (lattice window)
Piezoelectric phenomena
Piezocrystal (momentum detector)
Doppler effect Time of flight
Laser Doppler velocimeter Double-beam laser
FIGURE 3.2 Impact flowmeter.
length and the belt speed. When the flow rate is controlled so as to feed a constant powder mass, the belt scale is called a constant-feed weigher. Belt speed control is more preferable than gate opening control in the operation of the constant-feed weigher.12 A mechanical unit or load cell is used to weigh the powder.
Impulsive Force Flowmeter The flowmeter shown in Figure 3.2, which is based on the impulsive force, is called an impact flowmeter. Powder falls down by gravity and impacts on an inclined plate. The force acting on the plate is calculated from the impulse versus momentum relation, and it is expressed by the horizontal and vertical components. The impact flowmeter has been able to detect the vertical component since the early stage of its development,13 which is the sum of the impulsive force and the weight of the powder adhered on the plate. On the other hand, a recently developed one detects the horizontal component and it is not affected by the adhered powder.6 The horizontal component of the impulsive force is given by Fh ⫽
1 (e ⫹ ) K sin 2u 2 ghW 2
(3.8)
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where e is the coefficient of restitution of particles, m is the coefficient of friction, K is the air-drag coefficient, h is the free-falling height of the powder, u is the inclination of the plate, and W is the powder mass flow rate. The flowmeter is very compact and its dynamic response is very fast. Further, dust intrusion into the mechanical parts is easily shut out. The principle has also been applied to gas–solids suspension flow. 14 In this case, the gas and particle velocities should be measured by a suitable method. Also, the particle inertia must be large enough so that all particles impact on the detector.
Coriolis Force Flowmeter When the powder flow is received on a rotating disk and thrown off from the periphery of the disk, the powder suffers the Coriolis force in the direction counter to the rotation. The resulting total torque T on the rotating axis is given by T ⫽ vr 2 W
(3.9)
where v is the angular velocity of the table, r is the table radius, and W is the powder mass flow rate. Equation 3.9 contains no calibration constant, which is the most important feature of the Coriolis force flowmeter.
Differential Pressure Method Mass flowmeters based on the differential pressure or pressure drop are applied mainly to suspension flow. However, the principle is also utilized in the measurement of bulk powder flow.15 One feature of the method is that the electric parts of the flowmeter can be isolated from the main process where there might be radioactive rays. When particles are transported by a pneumatic conveyor, the pressure drop along the pipeline usually becomes larger than the corresponding airflow only. The additional pressure drop depends on the mass flow ratio m. In the measurement of powder flow rate, it is necessary to know the relationship between the mass flow ratio and the pressure drop, besides 16 the gas flow rate in the gas–solids mixture flow. The venturi meter or the orifice meter is applicable for this purpose. The powder flow rate W is calculated from W⫽K r
⌬P ⫺ k ⌬Pa ⌬Pa
(3.10)
where ΔP is the pressure drop at the straight pipe section, ΔPa is the pressure recovery at the diffuser, r is the air density, and K and k are the calibration constants. The calibration constant k is adjustable to unity if the distance between pressure taps in the straight pipe section is designed so that the pressure drop for airflow only is equal to the pressure recovery at the diffuser. The pressure drop of a venturi nozzle is utilized in the venturi-orifice flowmeter. 17 A venturi with a long throat is also used to increase the pressure drop and attain higher meter sensitivity.18 Another version of the flowmeter is composed of a straight pipe with a core inserted so as to make an annular flow between them. The particles will be accelerated by high-speed annular flow, resulting in a pressure drop. The flowmeter is insensitive to upstream disturbances and does not require a long straight inlet section as in the case of the venturi flowmeter. The above-mentioned flowmeters can be used in gas–solids mixture flow with mass flow ratio m below 10. For a higher mass flow ratio, the relation between the pressure drop ratio (ΔP/ΔP a) and the mass flow ratio m becomes nonlinear. © 2006 by Taylor & Francis Group, LLC
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Electrical Method The electric currents generated by particle impact can be applied to the measurement of the powder flow rate in gas–solids mixture flow. However, the concentration should be rather small, because the relation between the current and the flow rate becomes nonlinear for a higher concentration.19 The sensitivity depends on the material of the detector.20 If the detector is covered with particles, impact electrification becomes ineffective; therefore, the air velocity should be higher to prevent the particles from adhering. Also, there is a measuring system that uses two detectors that can measure both the powder flow rate and the average electrical charge of particles in aerosol pipe flow. 21,22 The electromagnetic flowmeter is based on Faraday’s law of electromagnetic induction. Figure 3.3 shows an electromagnetic flowmeter, which is truly obstructionless and is widely used for slurry flow measurement. The pipe is made from a nonmagnetic material such as stainless steel, through which the magnetic field can pass. The inside wall is covered by a nonconductive plastic or ceramic. A pair of electrodes is set on the wall perpendicularly to the magnetic field. The potential difference induced between the electrodes as the slurry moves across the magnetic field is given by V ⫽ kBDu
(3.11)
where k is the proportional constant (unitless), B is the magnetic flux density (T), D is the inside diameter of the pipe (m), and u is the average flow velocity (m/s). The volumetric flow rate Qv (m3/s) is given by Q⫽
pD V 4 kB
(3.12)
The electromagnetic flowmeter is not affected by variations in density, viscosity, pH, pressure, or temperature. If the concentration is higher than 10%, the proportional constant can become larger because of the increase in magnetic flux density, and recalibration of the flowmeter will be necessary. Also magnetic solids such as magnetite can cause an erroneous reading, and particle deposition on the wall will affect the pickup of potential difference.
FIGURE 3.3
Electromagnetic flowmeter.
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There is another method based on electrostatic induction. By monitoring the electrostatic flow noise generated while charged particles are conveyed through the pipeline, the mass flow rate could be calculated.23 The piezoelectric method is used only to detect local flow.24 The sensor is made of a piezoelectric crystal or ceramics. If the dielectric crystal is deformed by an impulsive force, displacement of ions in the crystal takes place (polarization), which induces an electric pulse between the upper and lower surfaces of the crystal. The pulse can be detected if the surfaces are coated with metal and connected to a pulse analyzer. The particle mass should be known to measure the local mass flux or concentration by use of the piezoelectric sensor. A strain gauge can be used as a sensor of the impact force instead of the piezoelectric crystal.25 The microwave (i.e., electromagnetic wave with frequency range 109–1012 Hz) method utilizes a circular tube resonator lined with a dielectric material.26 If the gas–solids mixture flows through the resonator, the resonance frequency changes so that the volumetric concentration cv can be obtained. On the other hand, there exists a standing wave in the resonator, and it works as a lattice window for the velocity measurement (i.e., spatial filtering). The powder flow rate is obtainable as a product of concentration and velocity.
Statistical Method A pair of sensors is set along a conveyor line a distance L apart. The signals detected by the sensors will fluctuate because of the flow turbulence. If the autocorrelation is calculated for various values of the delay-time parameter , it takes a maximum at a certain value of = 0. 0 gives the time interval for the particle cloud to travel between two sensors. Hence the particle flow velocity n is determined as n⫽
L t0
(3.13)
The powder flow rate W will be determined as the product of the particle velocity by particle concentration and the cross-sectional area of the flow. 27 The statistical method has no obstruction to the particle flow and hence is preferable for process instrumentation. The method can be applied to the bulk solids flow in hoppers or chutes by use of a sound noise pickup (i.e., microphone). Fiber optics can be used in the measuring system for particles flowing on a chute or moving in a vessel.
REFERENCES 1. Perry, R. H. and Green. D. W., in Perry’s Chemical Engineers’ Handbook, 6th Ed., McGraw-Hill, New York, 1984, p. (5) 8–20. 2. Mitchel, J. W., in Coal Preparation, American Institute of Mining, Metallurgical, and Petroleum Engineers, New York, 1979, p. (19) 42. 3. Wills, B. A., in Mineral Processing Technology, Pergamon Press, Oxford, 1979, p. 45. 4. Weiss, N. L., in Mineral Processing Handbook, American Institute of Mining, Metallurgical, and Petroleum Engineers, New York, 1985, p. (10) 173. 5. Bakker, P. J. and Heertjes, P. M., Br. Chem. Eng., 3, 240, 1958. 6. Iinoya, K., Masuda, H., and Watanabe, K., in Powder and Bulk Solids Handling Processes, Marcel Dekker, New York, 1988, pp. 65–130. 7. Keily, D. P. and Millen, S. G., J. Meteorol., 17, 349, 1960. 8. Mann, U. and Crosby, E. J., Ind. Eng. Chem. Process Des. Dev., 16, 9, 1977. 9. Wang, J. C. F., Kee, B. F., Linkins, D. W., and Lynch, E. Q., Powder Technol., 40, 343–351, 1984. 10. Lu, C. and Czanderna, A. W., Application of Piezoelectric Quartz Crystal Microbalances, Elsevier, New York, 1984. © 2006 by Taylor & Francis Group, LLC
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11. Sem, G. J. and Borgos, J. A., Staub Reinhalt. Luft, 35, 5, 1975. 12. Grader, J. E., Control Eng., 15 (3), 60, 1968. 13. Iinoya, K., Yoneda, T., Kimura, N., Watanabe, K., and Shimizu, T., J. Res. Assoc. Powder Technol. Jpn., 3, 424–431, 1966. 14. Barth, W., Chem. Ing. Technol., 29, 599, 1957. 15. Kagami, H., Maeda, M., and Yagi, E., Kagaku Kogaku Ronbunshu, 1, 327–329, 1975. 16. Masuda, H., Ito, Y., and Iinoya, K., J. Chem. Eng. Jpn., 6, 278–282, 1973. 17. Farber, L., Trans. ASME, 75, 943, 1953. 18. Iinoya, K. and Gotoh, K., Kagaku Kogaku, 27, 80, 1963. 19. Masuda, H., Mitsui, N., and Iinoya, K., Kagaku Kogaku Ronbunshu 3, 508–509, 1977. 20. Matsusaka, S., Nishida, T., Gotoh, Y., and Masuda, H., Adv. Powder Technol., 14, 127–138, 2003. 21. Masuda, H., Matsusaka, S., and Nagatani, S., Adv. Powder Technol., 5, 241–254, 1994. 22. Masuda, H., Matsusaka, S., and Shimomura, H., Adv. Powder Technol., 9, 169–179, 1998. 23. Gajewski, J. B., J. Electrostat., 37, 261–276, 1996. 24. Heertjes, P. M., Verloop, J., and Willems, R., Powder Technol., 4, 38–40, 1970. 25. Raso, G., Tirabasso, G., and Donsi, G., Powder Technol., 34, 151–159, 1983. 26. Kobayashi, S. and Miyahara, S., Keisoku Jido Seigyo Gakkai Ronbunshu, 20, 529, 1984. 27. Beck, M. S., Drane, J., Plaskowski, A., and Wainwright, N., Powder Technol., 2, 269–277, 1969.
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6.4
Level Measurement of a Powder Bed Hiroaki Masuda and Shuji Matsusaka Kyoto University, Katsura, Kyoto, Japan
6.4.1
LEVEL METERS AND LEVEL SWITCHES
Level meters are classified into two types: a continuous type and a discrete type. Discrete level meters are also called level switches and are usually used to maintain the powder level between the prescribed upper and lower limits by feeding or discharging powder.1 Table 4.1 illustrates the principles of continuous level meters. The electric capacitance method is also utilized in measurement of the powder level. Table 4.2 lists the principles of various level switches. Level meters or level switches should be located at suitable positions because the powder bed does not take a horizontal surface. Further, the profile of the surface will gradually change during feeding or discharging. For free-flowing particles, the profile may be estimated from the angle of repose in general. If the estimation is difficult, experiments should be carried out to determine suitable locations for level meters.
TABLE 4.1
Powder Level Meter (Continuous Type)
Principle Weighing
Notes Load cell
Electric capacitance Electric resistance Sounding
Wire rope with a sinker
Attenuation
Radiation, ultrasonic wave
Reflection time (reflectometry)
Ultrasonic wave, electromagnetic pulse, pulsed microwave
TABLE 4.2
Powder Level Switch Principle
Notes
Back pressure increase
Gas purge or gas injection
Elastic deformation (powder pressure)
Diaphragm with mercury switch
Mechanical blockage of motion by powder
Reciprocation, rotation, swinging motion
Electrically forced vibration
Piezoelectric vibration, tuning fork
Attenuation
Microwave, electromagnetic wave
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MECHANICAL METHOD
Sounding Level Meters A sounding level meter mounted at the top of a storage vessel is shown in Figure 4.1. A weight connected with wire rope is lowered slowly from a wire drum of the level meter. When the weight reaches the surface of the powder bed, the wire rope becomes tensionless. Hence the torque acting on the wire drum decreases. The change of torque is detected mechanically, and the wire rope is rolled up by reversing the motor rotation. The powder bed level is measured as the length of rope is released, which can be determined from the number of rotations of the drum. Level meters of this type can measure up to a depth of 50 m from the top with 0.1 m accuracy. The meter is applicable to a high-temperature vessel such as a blast furnace. The level of solid materials immersed in liquid can also be measured.
Piston Level Switches Figure 4.2 is a schematic illustration of a piston level switch. A shaft having a disk at the end is moved reciprocally by a motor and crank mechanism. If the disk is blocked by particles from the left-hand side in the figure, the spring is compressed by motor action, and the link changes to a V shape. Then the link pushes a microswitch and the motor is switched off. If the disk is released from the particles, the link is returned to the original shape by spring action. Then the motor rotates again. Bearings are isolated by bellows so that dust does not intrude into the mechanical part. However, adhesion of particles on the disk, bellows, and the shaft can cause problems.
FIGURE 4.1
Sounding level meter.
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Rotating Blade Level Switches A level switch of this type has a blade at the end of a shaft, which is rotated slowly (1–3 rpm) by a synchronous motor. Mechanical blockage of the rotating blade by particles is effectively detected by a spring and two microswitches. If the blade is blocked by particles, the motor housing rotates in the counter direction and pushes a microswitch. Then the motor stops and generates an electrical signal showing the powder level. Rotating blade level switches are widely used because of their simplicity. The blade can, however, be broken by a firm blockage of powder. If the blade is buried in powder flow, it can be forced to rotate, resulting in an erroneous signal.
Swing Level Switches Figure 4.3 shows a swing level switch. A blade swings from side to side through a motor rotation and mechanical connectors. Leaf springs are inserted between the blade shaft and the motor. Mechanical parts are isolated from particles by a diaphragm. If the blade is blocked by particles, the leaf springs will bend, and the motor rather than the blade is moved along a guide. Then the motor pushes a microswitch, as in the case of the piston level switch and the rotating level switch. If the blade is released from the particles, the motor returns to the normal position by spring action. The blade is set vertically in a storage vessel. Therefore, the drag force caused by powder flow is not as large as in the case of rotating blade level switches. Thus, the above-mentioned erroneous signal caused by powder flow can be avoided in swing level switches.
Diaphragm Level Switches Level switches of the diaphragm type have no motors. A diaphragm is connected with a microswitch or a mercury switch. If powder pressure acts on the level switch, the diaphragm deforms elastically and turns the switch on. After the powder level goes down, the diaphragm returns to the normal position and the switch goes off. This type of level switch has no parts protruding into storage vessels, in contrast to the other level switches mentioned above. Diaphragm level switches can be strong against mechanical damage in this sense.
Level Switches Based on Vibration A weak vibration of a rod or a tuning fork causes deformation of a piezoelectric crystal, giving an electric output. The output is amplified and is supplied to the other piezoelectric crystal as its input.
FIGURE 4.2
Piston level switch.
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FIGURE 4.3
Swing level switch.
Then the latter crystal is electrically excited and, finally, vibrates at the resonance frequency. The resonance frequency decreases when the rod or the tuning fork is constrained by particles. Thus, the level of powder is detected. This type of level switch is not affected by the electrical properties of particles (although the vibration is electrically forced, the vibration itself is purely mechanical). The level of powder immersed in liquid can also be detected. However, adhesion of fine particles to the rod or the tuning fork causes a frequency decrease, resulting in erroneous signals.
6.4.3
ELECTRICAL METHOD
Electric Capacitance Level Meters Electric capacitance increases with particles existing between two electrodes. For continuous measurement of the powder level, a wire electrode is vertically stretched in a storage vessel. The counter electrode is the vessel itself. Electric capacitance level switches, which have a rod- or bar-type electrode, are also available. Some of them have an earth electrode at their base. Level switches of this type have no moving parts and are easily adapted to high-pressure or high-temperature conditions. It is, however, necessary to adjust their sensitivity at the installation site, because the sensitivity also depends on the shape of the storage vessel. The sensitivity also depends on the dielectric constant of the powder materials. Therefore, readjustment is necessary when the kind of powder changes. Fine powders adhered to the electrodes can increase the electric capacitance and cause erroneous signals.
Electric Resistance Level Meters Constant voltage is applied between a rod electrode inserted in a storage vessel and the vessel wall. As the specific resistance of particles is smaller than that of air, electric current through the powder layer becomes larger as the powder level increases. The current also depends on the shape of the vessel. © 2006 by Taylor & Francis Group, LLC
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The specific resistance of the powder layer depends on the packing density, types of material, moisture content, and temperature. Therefore, the sensitivity is affected by these variables.
Microwave Level Meters Microwave level meters can detect a 20-m distance with a resolution of 1 cm. If a microwave is emitted to a powder bed, a part of the wave is reflected on the surface of the powder bed and detected by a sensor of the level meter. The reflection time of a pulsed microwave is utilized to measure the powder level based on the velocity of light. Emission and detection of the wave are carried out by a single antenna. The microwave can pass through plastics, and the powder level in a vessel closed by plastics can also be measured. Microwave level meters need no mechanical contact with the powder, and their installation is easy. Their sensitivity is less affected by suspended dust or water vapor than ultrasonic level meters.
6.4.4
ULTRASONIC WAVE LEVEL METERS
An ultrasonic wave is a sound wave with a frequency higher than 20 kHz, and ultrasonic level meters can detect a 50-m distance with 1–2% accuracy. If the ultrasonic wave is emitted from the level meter to a powder bed, a part of the wave is reflected on the surface of the powder bed and detected by a sensor of the level meter. The reflection time of a pulsed ultrasonic wave is utilized to measure the powder level based on the velocity of the wave in air, which is given by u ⫽ 331 ⫹ 0.6T
(4.1)
where u is the sound velocity (m/s), and T is the temperature of the air (°C). Emission and detection of the wave are carried out by a single element such as a piezoelectric vibrator. The element is excited periodically by a pulsed electric power supply. Then the surrounding air vibrates and an ultrasonic pulse wave is emitted. The wave reflected on the powder bed surface is detected by the same element and is transformed into an electric signal. Ultrasonic level meters need no mechanical contact with powder, and their installation is easy. Their sensitivity is, however, affected by suspended dust or water vapor in storage vessels because of the sound attenuation.
6.4.5
RADIOMETRIC METHOD
Radiation from high-energy gamma-ray sources such as cesium-137 or cobalt-60 can be used. In most cases, detectors are placed opposite the radiation source in order to perceive a change in the intensity of transmitted radiation. A Geiger-type detector is usually used for simple on-off level switches. For continuous level measurement, more stable scintillation counters are preferable. Random decay of radioisotopes can cause short periodic fluctuations in output signals. Therefore, the output signals will be averaged for a certain period of time (which can be selected by varying the time constant of the detector) in order to avoid the error caused by the fluctuations. Usually, a larger radioisotope source is necessary for larger vessels or thicker-walled vessels. Special windows for sources and detectors installed in the wall with dead space will substantially reduce source size. However, the alignment of the windows is critical. The use of these level meters on empty vessels must be carefully considered to ensure that the operating and maintenance personnel will not receive excessive radiation. Human tolerance for safe exposure is about 100 mrem/week. A reflectance level switch should be considered, instead of the ordinary transmission unit, for large-scale vessels.2 © 2006 by Taylor & Francis Group, LLC
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PNEUMATIC METHOD AND OTHERS
Pneumatic Method Pneumatic level switches utilize air (or gas) nozzles. Air is injected into storage vessels through a nozzle equipped at the side walls. The air flow rate is held constant by a flow regulator. If the air jet is blocked by the powder bed, the back pressure in an air nozzle increases, and that is detected by a pressure transducer. Air must be injected continuously even if the level measurements are not required, because the nozzles can be blocked by particle intrusion. For adhesive particles, a cave can be formed in the powder bed after the particles near the nozzle are blown off. Hence, no increase in the back pressure can be obtained, and the level switch fails to detect the powder bed.
Weighing Method If the purpose of level measurements is to determine the content of powder in a storage vessel, a weighing method is preferable. Load cells are applied in weighing level meters as in the case of a hopper scale. 1 It should be noticed that the weighing will be affected if air or gas flows through the vessel. Numerical calculation must be done based on the bulk density of powder and the geometrical shape of the vessel in order to estimate the level of powder bed according to the weighing method. Powder bridging or rat holing can also cause errors in the estimation.
Optical Method A high-power laser is applied to obtain the surface profile of a powder bed in a storage vessel. The laser beam scans the surface of the powder bed, and the trajectory of light on the surface is recorded by a highly sensitive TV camera, as shown in Figure 4.4. Data obtained are analyzed by a microcomputer based on the triangulation. The surface profile can be estimated by the optical method within ±50 mm. The measuring time is about 10 s.3
FIGURE 4.4
Laser level meter.
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REFERENCES 1. Iinoya, K., Masuda, H., and Watanabe, K., in Powder and Bulk Solids Handling Processes, Marcel Dekker, New York, 1988, pp. 131–149. 2. Rowe S. and Cook, Jr, H. L., Chem. Eng., 76 (2), 159–166, 1969. 3. Yamasaki, H., in Jido Seigyo Handbook-Kiki Ouyou Hen (Handbook for Automatic Control Instruments and Applications), Ohmsha, Tokyo, pp. 842–843
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6.5
Temperature Measurement of Powder Hiroaki Masuda Kyoto University, Katsura, Kyoto, Japan
Kuniaki Gotoh Okayama University, Okayama, Japan
6.5.1 THERMAL CONTACT THERMOMETERS Thermocouples Thermocouples are composed of two different kinds of metal wires that are connected at both ends, making a loop. One of the ends is called the hot junction (i.e., sensing point) and the other the reference junction (i.e., cold junction). The hot junction is usually welded. If there is a temperature difference between these two junctions, electromotive force (or thermo-electromotive force) will be generated in the loop, which is almost directly proportional to the temperature difference. Table 5.1 illustrates common thermocouples, their composition, and their operating temperature range. The numerical values of the electromotive force are found in standard calibration tables.1 The reference junction is usually kept 0ºC in an ice bath, or an electrical cooler (i.e., electrical compensator) can be used for this purpose. The sensing parts of the thermocouples are protected against corrosion and mechanical damage by use of ceramic or metal tubes filled with alumina or magnesium oxide powder.
Resistance Thermometers Electric resistance thermometers utilize the fact that the electric resistance of conductors (i.e., metals) increases with increasing ambient temperature. The electric resistance of semiconducting materials also changes with the ambient temperature. In the latter case, the concentration and mobility of the charge carrier in the materials increase with increasing temperature, and therefore the resistance decreases. The temperature dependence of the electric resistance of semiconducting materials is expressed by ⎛ c⎞ R ⫽ aT ⫺b exp ⎜ ⎟ ⎝T⎠
(5.1)
where R is the resistance (⍀), T is the temperature (K), and a, b, and c are calibration constants. Resistance thermometers constructed of semiconducting materials are known as thermistors (i.e., thermally sensitive resistors). Thermistors are less stable and accurate than metal thermoresistors but offer the advantages of a lower manufacturing cost and a much higher resistivity than that of a metal thermoresistor.2 Table 5.2 lists the types of resistance thermometers and their operating temperature ranges. Industrial resistance thermometers are usually constructed of platinum. The platinum wire (diameter 0.03 to 0.05 mm and length 1 m) is wound on a mica plate or glass rod and is then inserted into a protecting tube or enclosed in tempered glass. The electric resistance of platinum resistance 813 © 2006 by Taylor & Francis Group, LLC
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TABLE 5.1 Thermocouples Material
Abbreviation
Temperature Range (°C)
EMF (µV/°C)
Accuracy (°C)
Copper-constantan
CC
⫺180–300
50
2–5
Iron-constantan
IC
0–500
60
3–10
Chromel-alumel
CA
0–1000
40
2–10
Platinum-platinum 10% rhodium
PR
100–1400
10
TABLE 5.2 Material Platinum
0.5–5
Resistance Thermometers Temperature Range (°C) ⫺180–500
Accuracy (%) 0.5–2
⫺50–120
0.5–2
0–120
0.5–2
⫺50–200
0.5–2
Nickel Copper Thermistor
thermometers changes 0.4% for each temperature change of 1ºC (temperature coefficient = 0.4%/ºC). The temperature coefficient of thermistors is about 10 times larger than that of the platinum resistance thermometer. The temperature versus resistance relation of thermistors is, however, nonlinear, as one can see from Equation 5.1; and the calibration constants a, b, and c will be different for each thermistor. Although thermistors have this defect, their sensing elements are very small (0.3 to 2 mm diameter), and the dynamic response can be made faster than in platinum resistance thermometers. The time constant is 0.3 to 2 s.
6.5.2
RADIATION THERMOMETERS
Every object emits thermal radiation energy, depending on its temperature. Therefore, the temperature of particles can be estimated by detecting their thermal radiation energy. Radiation thermometers need no thermal contact with the particles. The thermal radiation energy of a perfect emitter (i.e., blackbody) is given by Planck’s law. The radiation energy is concentrated mainly in the visible and infrared regions with wavelengths below 10 μm. There are three definitions of the temperature of solid materials: 1. Radiation temperature, TR. If the emissive power of an object is equal to that of a perfect emitter of temperature TR, TR is called the radiation temperature of the object. 2. Luminance temperature, TS. If the monochromatic luminance of an object is equal to that of a perfect emitter of temperature TS, TS is called the luminance temperature of the object. 3. Color temperature, TF. If the color of an object is the same as that of a perfect emitter of temperature TF, TF is called the color temperature of the object. Following is the relationship between the luminance temperature TS and the true temperature T: 1 1 l ⫽ ⫺ ln l T Ts c2
(5.2)
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where λ is the monochromatic emissivity (i.e., blackness) of the object. Further, the relationship between the color temperature TF and the true temperature T is given by 1 1 1 ln ( l1 Ⲑ l2 ) ⫽ ⫹ T TF c2 (1Ⲑ2 ) ⫺ (1Ⲑ1 )
(5.3)
where λ1 and λ2 are the emissivities at wavelength of λ1 and λ2, respectively. From Equation 5.2 and Equation 5.3, the following relationship can be derived: TS ⬍ T ⬍ TF
(5.4)
The emissivity of a powder bed is usually approximated by unity (i.e., blackbody), and therefore, T ⬵ TS ⬵ TF. This is incorrect for a particle and suspended particles. Table 5.3 illustrates the ordinary thermosensor (i.e., detector), measuring temperature ranges, and accuracy of radiation thermometers. Radiation pyrometers measure the total energy E(T), collecting the radiation by lenses and focusing it on a detector such as a thermopile, silicon cell, thermistor bolometer, or photocell (PbSe, PbS). Thermopiles are composed of thermocouples connected in series so as to increase the sensitivity. Accuracy and stability are, however, not as good as in other detectors. Portable radiation pyrometers are also available. Two-color pyrometers are more expensive than radiation pyrometers. The radiation from particles is collected by lenses and projected to an interference filter made from, for example, indium phosphate (InP). The radiation reflected by the filter and that transmitted through the filter are measured by two different photocells, respectively, and the ratio of the output signals of each photocell is calculated. As the wavelength of reflected radiation is different from that of transmitted radiation, luminance temperatures at two different wavelengths (i.e., two colors) are obtained. The measured temperature is insensitive to variations in the emissivity of particles and is less affected by several disturbances caused by aerosols such as water vapor or fumes. Two different optical filters may also be utilized to get radiation intensities at two different wavelengths. Temperature measurement in gas–solids suspension flow can be done by use of a two-color pyrometer.3 A more economical thermometer for the measurement of temperature ranging from 700 to 3000ºC is an optical pyrometer, which is a type of brightness pyrometer. The pyrometer determines the temperature of an object by comparing the luminance of the object with that of a reference filament of a standard electric bulb. The accuracy of an optical pyrometer is about ±5ºC at 1000ºC. It is easy to control the electric current supplied to the filament so as to get the same luminance as the object. Such a pyrometer is called an automatic optical pyrometer.
TABLE 5.3
Radiation Thermometers
Thermometer Radiation pyrometer
Detector Thermopile Silicon cell Thermistor bolometer
Two-color pyrometer
Temperature Range (°C) 200–1000 600–3000
Accuracy (%) ±1–2 ±5–1
0–500
±0.5–1
PbSe, PbS
100–1000
±0.5–1
PbS
300–1000
±0.5–1
Silicon cell
700–2000
±0.5–1
1000–3500
±0.5–1
Photomultiplier
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The measuring points of radiation thermometers can be changed by use of a rotating or vibrating mirror. These instruments, called scanning thermometers, are applicable to obtain the temperature profile of large equipment such as a cement kiln. Commercially available infrared scanning thermometers can be applied in measurement of the temperature range of 10 to 1000ºC with sensitivity 0.5ºC at 25ºC and resolving power 3% of full scale. The focal length of the instruments is between 1.2 m and 600 m.
REFERENCES 1. West, R. C. and Astle, M. J., CRC Handbook of Chemistry and Physics, 61st Ed., CRC Press, Boca Raton, Fla., 1980, p. E-108. 2. Gardner, J. W., Microsensors—Principle and Application,: John Wiley, New York, 1994. 3. Themelis, N. J. and Gauvin, W. H., Can. J. Chem. Eng., 40, 157–161, 1962.
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6.6
On-Line Measurement of Moisture Content Satoru Watano Osaka Prefecture University, Sakai, Osaka, Japan
6.6.1
INTRODUCTION
Powder handling processes are mainly categorized into wet and dry processes.1,2 In wet processes, the degree of wetness (moisture content) seriously affects physical properties (size, shape, density, etc.) of the final products. In order to produce the desired product efficiently and continuously without difficulty, monitoring and control of moisture content are required. Even in dry processes, moisture content sometimes causes problems. Generally, powder materials contain moisture, to some extent as water of hydration, bound water, and so forth, which exists on the powder surface or forms liquid bridges between powders. Moisture on the powder surface greatly changes the powder’s properties such as flowability and internal friction, causing handling problems. The liquid bridges generate agglomeration of powders and adhesion to the vessel wall. In all cases, control of moisture content is necessary in order to avoid problems.
6.6.2
ELECTRICAL METHODS
Table 6.1 lists moisture sensors and their measuring principles. In powder handling processes, there are no sensors which can measure moisture content directly. Therefore moisture content is determined by using the correlation between indirect parameters (absorbance of spectrum, dielectric constant, etc.) and moisture content that have been previously measured by a direct method (such as the drying method). So far, chemical, electrical, and optical methods, as well as methods applying radioactive rays and nuclear magnetic resonance spectrometry, have been available for powder handling processes. Among them, electrical and optical methods have been widely used in industrial practical application. In the following, electrical and optical methods for measuring moisture content of powder materials are briefly explained. Figure 6.1 illustrates a schematic diagram of an electrical resistance–type moisture sensor. In this sensor, moisture content is determined by measuring the electrical resistance change due to moisture increase/decrease. As seen in the figure, powder samples are packed between two electrodes, and apparent specific resistance R is determined by measuring the applied voltage V and the electric current I as
R
VS Id
(6.1)
where S and d indicate areas of electrode and distance between the two electrode, respectively. The apparent specific resistance R varies with moisture content W as log R a log W b
(6.2)
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TABLE 6.1
List of Moisture Sensors
Methods
Drying method
Principle of measurement
Continuous measurement
Detach/contact to powder bed
Wide application
Normal drying Infrared heating Measuring weight loss due to drying of High frequency induction heating wet samples
×
Contact
{
Distillation method
×
Contact
{
×
Contact
{
{
Contact
{
{
Contact
{
Range 0.1∼100%
0.01∼20% Chemical method
Perform distillation and measure water volume
Water is titrated with KF reagent Karl Fischer (KF) titration method containing iodine, sulfur dioxide and pyridine in methanol. Infinitesimal moisture content can be determined. Electric resistance method (DC)
Electric method
Method applying high frequency
Measure electric resistance change (direct current) due to moisture increase/decrease. Measure electric current (high frequency) change due to moisture increase/decrease. Smaller quantity of water can be detected than the DC method.
0.001∼100%
3∼50%
0.1∼60%
Measure dielectric constant change due to moisture increase/decrease.
{
Contact
{
2∼25%
Method applying infrared ray
Measure absorbance of infrared –ray at absorption band (1.43, 1.94μm) of water near infrared.
{
Detach
{
1∼90%
Method applying microwave
Measure attenuation of microwave due to water
{
Detach
{
1∼90%
Resonance (NMR)
Sample is dissolved in a solvent and its NMR
×
Spectrometer
is measured.
Optical method
Nuclear Magnetic
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Method applying dielectric constant (electrostatic capacity)
Measure moderation/ scattering of neutron/γ-
scattering
ray due to water.
radioactive rays
Method measuring attenuation strength Measure attenuation strength of β or γ- rays
2∼80% {
Detach
×
{
Detach
{
due to water. Other method
Balanced relative humidity
Calculate moisture content by relative humidity, which is balanced with circumstances.)
{ : Possible ×: Impossible A: Very good B: good C: Poor
1∼80%
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FIGURE 6.1
Schematic diagram of electrical resistance–type moisture sensor.
where a and b are constants depending on the shape of electrode, properties of powder materials, and circumstance conditions (humidity and temperature). By using Equation 6.2, moisture content can be determined. For the electric current, direct current (DC) or high-frequency alternating current is used. The electrical resistance–type moisture sensor is widely used since its electrical circuit is simple, the measurement principle is easy, and the measurement time is short. However, it has some drawbacks: (1) temperature correlation is required since the electrical resistance generates heat, (2) it cannot be used for a powder bed that moves during the measurement since powder packing conditions affect the resistance, and (3) measurement error may occur if the powder sample contains electrolyte. Figure 6.2 shows a schematic diagram of a dielectric constant–type moisture sensor. Assuming that the electrical capacity when powder materials exist inside between two electrodes or inside a cylindrical electrode is C1, and the one without any powder materials is C0, the apparent dielectric constant a is obtained as a 0
C1 C0
(6.3)
By using the apparent dielectric constant, moisture content can be determined, since the apparent dielectric constant ea shows an approximately linear relationship to moisture content as a j kw
(6.4)
where j and k are constants. These constants require previous calibration, since they vary with different powder materials. The dielectric constant–type moisture sensor has merits in that it is less affected by temperature and packing conditions than the electrical resistance–type moisture sensor. However, its circuit is complicated, and it cannot measure moisture content of powder materials containing electrical conductors, such as metal and coal powders. © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.2
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Schematic diagram of a dielectric constant–type moisture sensor.
A microwave-type moisture sensor is able to measure moisture content continuously without touching the objective material. Figure 6.3 investigates absorbance characteristics of microwaves in the case of flour and cornstarch powders. When the microwaves are transmitted to the powder material containing water, water absorbs the microwaves and the microwaves attenuate. Since the degree of attenuation has a correlation to moisture content, as shown in Figure 6.3, moisture content of powder materials can be determined. However, the system is complicated to avoid the microwave leakage. In addition, the measured value may vary to some extent, depending on the leakage/refraction, powder surface condition, and temperature.
6.6.3
INFRARED MOISTURE SENSOR
In general, an optical sensor utilizes the principle that physical values (concentration, distance, etc.) are indirectly detected by using the absorbed/reflected energy of spectra when the spectra are radiated to the object. The so-called spectra are categorized into ultraviolet, optical, and infrared rays, depending on the wavelength, and each sensor uses a specific wavelength, which fits characteristics of the measured object. Figure 6.4 shows infrared (IR) absorption characteristics.3,4 Water absorbs infrared spectra markedly at wavelengths of 1.43, 1.94, and 3.0 μm, due to the resonance with atomic vibration between oxygen and hydrogen in water molecules. Since energy of IR spectrum is absorbed remarkably at these wavelengths in proportion to moisture content, moisture content can be detected by measuring the difference between the irradiated energy and the reflected one. Since the absorption spectrum measurement is easily disturbed by granule surface conditions and fluctuation in beam © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.3 content.
Relationship between degree of microwave attenuation and moisture
length, a reference method which calculates the reflected energy ratio of the absorbed spectrum (1.94 μm) and of the bilateral unabsorbed spectrum (reference spectrum of 1.78 and 2.14 μm) has been adopted. This cancels out any disturbance, since the disturbance affects the absorbed and unabsorbed spectra equally. Details of the IR moisture sensor are illustrated in Figure 6.5. The continuous light from the source is condensed by a condenser, then changed to a chopper beam by passing through a rotating sector of four optical filters composed of the different selective spectra of 1.94 μm for absorption, the bilateral 1.78 and 2.14 μm for unabsorbed references, and a visible spectrum for targeting. When these portions of the spectra are irradiated to the object, the absorption spectrum is absorbed in proportion to the moisture content of the object, while the reference spectra are fully reflected. 4 These spectra are transformed into electrical signals, and the degree of absorptivity, defined as the ratio of energy absorbed to the supplied energy of the spectrum, is calculated. Based on the degree of absorptivity, the absorbance X, the output from the moisture sensor, is calculated using the following equation: ⎪⎧ ( K 2 V2 V3 ) 2 ⎪⎫ X k0 In ⎨ ⎬ K1 V1 ⎩⎪ ⎭⎪
(6.5)
where, V1, V2, and V3 show absorption at 1.94, 1.78, and 2.14 μm in the IR spectra after temperature compensation, respectively, and k0 , k1, and k2 indicate the calibration coefficients, respectively. © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.4 IR absorption characteristics of powder material (lactose and cornstarch mixed 7:3 by weight) under various moisture contents.
Although the infrared moisture sensor is able to measure moisture content continuously without touching to the object and is not influenced by powder density and packing conditions very much, it measures moisture content on the surface or just below the surface of the object due to the long wavelength. Normally, moisture is distributed inside the powder material, and there are variations in moisture content between the surface and the core. Therefore, in order to measure moisture content by using the infrared sensor, a calibration curve between sensor output (surface moisture content detected by IR sensor) and total moisture content measured by a drying method is used. Figure 6.6 indicates the calibration curves when the lactose and cornstarch mixing ratio was varied.3 The calibration curves are different depending on the powder materials.3 It is because the powder’s physical properties such as water-absorbing potential, surface conditions, IR refraction characteristics, and moisture distribution are different.
6.6.4 APPLICATION OF MOISTURE CONTROL TO POWDERHANDLING PROCESSES Wet granulation is one of the most sensitive operations to moisture content. Wet granulation is defined as a size enlargement that adheres or sticks powders together by using liquid binders to produce granules having the desired size, shape, and density. The properties of the final product are significantly influenced by the moisture content. 5,6,7 Therefore, monitoring and control of moisture content are required to produce the desired product continuously and with high accuracy. © 2006 by Taylor & Francis Group, LLC
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FIGURE 6.5
FIGURE 6.6
Details of IR moisture sensor.
Calibration curves under various lactose and cornstarch mixing ratios.
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FIGURE 6.7 content.
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Relationship between granule mass median diameter and moisture
FIGURE 6.8 Moisture control system in fluidized-bed granulation.
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Figure 6.7 illustrates a system which measures and controls moisture content by using an IR moisture sensor in a fluidized-bed granulation.4,8,9 The IR detector and fluidized bed vessel are connected by optical fibers, and heated purge air is blown at the extremity of the sensor (the place where the optical fibers are connected at the vessel wall) to prevent powder sticking. Figure 6.8 indicates the relationship between granule mass median diameter and moisture content when powder samples composed of lactose, cornstarch, and hydroxypropylcellulose (HPC) are granulated by a purified water (binder).3 Figure 6.8 shows the granule mass median diameter has a linear relationship to the moisture content, implying that granule growth can be controlled by measuring and controlling the moisture content. In the Japanese pharmaceutical industry, automatic manufacturing of drugs (granules) is commonly conducted by fluidized-bed granulation with moisture control, and uniform properties can be maintained easily.10
REFERENCES 1. Gotoh, K., Masuda, H., and Higashitani, K., Eds., Powder Technology Handbook, 2nd Ed., Marcel Dekker, New York, 1997. 2. Society of Powder Technology Japan, Ed., Huntaikogaku binran, 2nd Ed., Nikkan Kogyo Shinbun, Tokyo, 1998. 3. Watano, S., Hideo, H., Sato, Y., Miyanami, K., and Yasutomo, T., Adv. Powder Technol., 7, 279–289, 1996. 4. Watano, S., Sato, Y., and Miyanami, K., J. Chem. Eng. Jpn., 28, 282–287, 1995. 5. Watano, S., Morikawa, T., and Miyanami, K., J. Chem. Eng. Jpn., 28, 171–178, 1995. 6. Watano, S., Morikawa, T., and Miyanami, K., Chem. Pharm. Bull., 44, 409–415, 1996. 7. Schaefer, T. and Worts, O., Arch. Pharm. Chem. Sci., 5, 178–193, 1977. 8. Watano, S., Yamamoto, A., and Miyanami, K., Chem. Pharm. Bull., 42, 133–137, 1994. 9. Watano, S., Takashima, H., and Miyanami, K., J. Chem. Eng. Jpn., 30, 223–229, 1997. 10. Watano, S., Takashima, H., Miyanami, K., Murakami, T., and Sato, T., Chem. Pharm. Bull., 42, 1302–1307, 1994.
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6.7
Tomography Richard A. Williams University of Leeds, West Yorkshire, United Kingdom
Recent developments in the design and application of noninvasive tomographic sensors now allow direct observation of the behavior of powders and particulate suspensions inside process equipment. This offers the possibility of acquiring measurements in situ. The methodology can be used for several purposes: process auditing, fault detection, process control; development of advanced models; and to enable verification of computational fluid dynamics predictions. The main sensing methods are summarized and illustrated with selected application examples.
6.7.1
INTRODUCTION
It is widely recognized that little is known about the behavior of particles during their passage through process equipment. To date, the lack of fundamental data has hindered the development of generic and reliable models. Tomographic measurements seek to address the widespread need for the direct analysis of the internal characteristics of process equipment. Ideally, the measuring instruments for such applications must use robust, noninvasive sensors that can operate in the proximity of aggressive and fast-moving flowing powders and multiphase mixtures. Tomographic technology involves the acquisition of measurement signals from sensors located on the periphery of a process vessel or the pipeline through which it is conveyed. This reveals information of the nature and distribution of components within the sensing zone. Most tomographic techniques are concerned with abstracting information to form a cross-sectional image. This can, for example, utilize a parallel array of sensors place so that their sensing field interrogates a projection of a suitable radiation across the subject. In order to explore the entire cross section, it is necessary to obtain other projections by rotating the subject in the direction normal to the direction of the sensor field or, preferably, to rotate the measurement sensors around the subject. It might not always be possible to adopt such a methodology because it might be impractical to physically rotate the subject, or the sensors, and the time required to rotate the assembly might take too long compared to changes occurring within the subject under investigation. Process tomography instruments have been devised to overcome these limitations. The basic components of process tomographic instruments are shown in Figure 7.1,1 embodied in hardware (sensors, signal/data control) and software (signal reconstruction, display and interpretation facilities, generation of output control signals to process hardware). In this figure sensors are mounted around the exterior of the (nonmetallic) pneumatic conveying pipe. Data are collected and processed to yield a reconstructed image of the profile of the particulate bed at a given instant of time. The image can be analyzed, for instance, to provide measurements of the local solid concentration C(x,y) at any desired point. The procedure can be repeated by acquiring additional frames with time. It is, however, necessary to emphasize that obtaining a good-quality computed “image” may only represent the first stage of information input for the powder technologist. Sometimes an image is not even required since measurement data can be fitted directly to a model of the process, such as reported by West et al.2 The ultimate goal is quantitative (numerical) interpretation of an image or, more likely, many hundreds of data sets and/or images corresponding to different spatial and temporal conditions. In most circumstances, a visual diagnosis based on glancing at the images will be 827 © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.1 Components of a process tomographic sensing system, showing here an electrical capacitance system controlling a pneumatic conveyer.
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insufficient, except perhaps to diagnose gross malfunction occurring in a process. Therefore, there is a major and fundamental difference between the general philosophy of medical tomography and that of process tomography.
6.7.2
SENSOR SELECTION AND SPECIFICATION
The heart of any tomographic technique is the sensor system that is deployed. The basis of any measurement is to exploit differences in the properties of the process being examined. A variety of sensing methods can be employed based on measurements of transmission, diffraction, or electrical phenomena.3 Table 7.1 shows a selection of the most widely used sensing methods and their principal attributes. Whereas most devices employ a single type of sensor, there are a number of opportunities for multimode systems using two (or more) different sensing principles. The choice of sensing system will be determined largely by the following: •
•
• •
The nature of components contained in the pipeline, vessel, reactor, or material being examined (principally whether they exist as a solid, liquid, gas, or a multiphase mixture, and if so in what proportions) The information sought from the process (steady state, dynamic, resolution, and sensitivity required) and its intended purpose (laboratory investigation, optimization of existing equipment, process monitoring or control) The process environment (ambient operation conditions, safety implications, ease of maintenance, etc.) The size of the process equipment and the length scale of the process phenomena being investigated.
High-resolution tomography can be expensive, rather slow, and, in some cases, impose special requirements for safe operation. Nevertheless, such methods can be extremely valuable for detailed laboratory studies.4 In contrast to this, the lower-resolution methods are less expensive, offer a fast dynamic
TABLE 7.1
Examples of Sensors for Process Tomography
Principle Electromagnetic radiation
Typical Spatial Resolution (percentage of diameter of cross section)% Practical Realization 1 Optical
Comment Fast, optical access required
X-ray and gamma ray
Slow, radiation containment
Positron emission
Labelled particle, not on-line
Magnetic resonance
Fast, expensive for large vessels
Acoustic
3
Ultrasonic
Sonic speed limitation complex to use
Measurement of electrical properties
10
Capacitive Conductive Inductive
Fast, low cost, suitable for small or large vessels
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response, and generally impose fewer safety constraints. Therefore, the process need for image resolution must be carefully considered. For the particular case of electrical tomography, considered later, the spatial resolution is determined by the nature of the measurement and associated reconstruction method. Imaging frequency (frame rates) from a few up to several hundred frames per second are attainable. To date, frame rates up to 1000 Hz have been demonstrated.5 The use of more than one plane of image sensors offers the possibility of three-dimensional imaging. This has led to the adoption of voxel–voxel correlation methods from which velocity information can now be routinely deduced.6 Related methods such as electrical resistance tomography can be used to image solid concentration profiles in aqueous slurries (e.g., in mixers, separators, etc.), as described in detail elsewhere.7
6.7.3
MEXAMPLES OF POWDER-PROCESSING APPLICATIONS
Pneumatic Conveying The transport of dry and damp minerals via hoppers and in conveying lines is of importance, as plant operability is often poor due to pluggage of conveying lines due to subtle changes in the feedstock (e.g., moisture content, size distribution, particle shape). Electrical capacitance tomography (ECT) can be used to image the dielectric constant inside conveying lines for dilute- and dense-phase processes, as reported in detail elsewhere.8–10 Analyses of the changes in the image with time can be used to identify the prevailing flow regime. For instance, Figure 7.2 shows a real time visualization of powder flow for two conditions in dense phase conveying. Analysis of the data also reveals characteristics such as slug length and velocity that can be computed from the images, and also slugging characteristics can be recovered from statistical analysis of void age fluctuation data (Figure 7.3). These methods have been used in a variety of conveying systems, and in some cases schemes for on-line control have been proposed.11,12 For example, in an earlier work on coal conveying estimation of mass flow rate using a simplified fuzzy logic approach, each membership function was adapted by means of a genetic algorithm.13 In this way, the control function did not drift from the ideal due to plant wear or changes in the nature of the transported materials. The genetic algorithm was used to minimize energy consumption. In principle, the controller could be tuned to optimize any fitness function, such as plant wear or product degradation, providing that these aspects can be measured on-line. Commercial equipment is enabling routine measurements to be obtained with significant benefit to those wishing to gain detailed analysis of flow structure.14
Fluidized Beds An obvious extension of the measurements described above is the use of ECT to follow the complex dynamic interactions between gas and solids in fluidized-bed reactors, dryers, and
FIGURE 7.2 Real-time visualization of granular flows in pneumatic conveying using electrical capacitance tomography. [From Ostrowski, K. L., Luke, S. P., Bennett, M. A., and Williams, R. A., Powder Technol., 102, 1, 1999. With permission.] © 2006 by Taylor & Francis Group, LLC
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risers.1,5 Figure 7.4 shows experimental measurements of the solid volume fraction at the base of a 150-mm-diameter bed adjacent to the distributor plate.15 Images can be acquired for different flow regimes, determined by the superficial gas velocity (u). For both bubbling (u ⬍0.2 m/s) and slugging (0.2 ⬍u ⬍0.73 m/s), fluidization bubbles appear near the wall of the pipe. The bubble diameter increases with increasing air velocity, but the flow patterns are not markedly different. In the transition from slugging to the turbulent regime (u ⬎ 0.73 m/s), the pipe cross section is largely occupied by gas, and bubbles are not observed attached to the walls. Detailed time transients can be obtained which allow the bubble size and transit velocity to be estimated. ECT enables the dynamic characteristics of the bed to be assessed and then analyzed, using chaos theory approaches or other statistical methods. The correlation fluctuations in time and space can be deduced either within a plane and between multiple planes spaced up the fluidized bed. These have found widespread application, as reviewed above.6
Powder Mixing For laboratory mixing and scale-up studies, it is possible to label a particle with a positron-emitting tracer and to follow the trajectory of an individual particle in three dimensions using a positronemission tomography camera.16 The method of positron-emission particle tracking (PEPT) relies on detecting the pairs of back-to-back gamma rays produced by the annihilation of positrons, emitted from a radioactive tracer, with electrons. Typically, the volume that can be proved is up
FIGURE 7.3 Examples on measurements of fluctuations of the relative volume fraction of solids across at a single plane during dense phase conveying (slugging) with time, for three different dense-phase flows. [From Ostrowski, K. L., Luke, S. P., Bennett, M. A., and Williams, R. A., Chem. Eng. J., 3524, 1, 1999. With permission.] © 2006 by Taylor & Francis Group, LLC
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FIGURE 7.4 Isometric solid volume fraction plots (upper) and corresponding concentration maps (lower) obtained using real-time ECT in the vicinity of the distributor of a silica-gas fluidized bed for three flow regimes: (a) bubbling fluidization, (b) slugging regime, and (c) in the transitional regime from the slugging to turbulent regime.
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to 500 × 500 × 250 mm3 with a voxel resolution of approximately 0.5 × 1.5 × 1.5 mm3.17 Figure 7.5 shows a sample result of a two dimensional slice through a laboratory bead mill), from which detailed information of bead velocities and slip effects can be deduced.18 Similar methods were also used to follow the size-dependent segregation of particles in a dense suspension4; particle motion in granulators, extruders, dryers, conveyors, and so forth and for validation of Distinct Element Modeling (DEM) predictions.19
FIGURE 7.5 PET studies of a laboratory stirred bead mill (left) showing bead distribution and occupancy changes on increasing stirring speed (200, 600, 800 rpm).
FIGURE 7.6 X-ray microtomograms of granules, showing (a) transmission image and (b–d) sections showing particle positions. [After Golchert, D. J., Moreno, R., Ghadiri, M., Litster, J., and Williams, R. A., Adv. Powder Technol., 15, 447, 2004. With permission.] © 2006 by Taylor & Francis Group, LLC
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Powder and Granule Structure The use of bench-top X-ray microtomography systems is now commonplace, delivering spatial resolution down to a few microns for three-dimensional scanning of small samples (a few mm3) over a few hours.7 These methods can provide detailed information on granule microstructure and packing of assemblies such as granules, 20 as shown in Figure 7.6, and structural modifications such as drying.21 Further development in higher-resolution systems can be anticipated.
REFERENCES 1. Williams, R. A. and Beck, M. S., Eds., Process Tomography–Principles, Techniques, and Applications, Butterworth-Heinemann, Oxford, 1995, p. 581. 2. West, R. M, Jia, X., and Williams, R. A., Chem. Eng. J., 77, 1–2, 31, 2000. 3. Williams, R. A., and Xie, C. G., Part. Part. Syst. Charact., 10, 252, 1993. 4. Scott, D. M. and Williams, R. A., Eds., Frontiers in Industrial Process Tomography, AlChE/Engineering Foundation, New York, 1995. 5. Dyakowski, T., Jeanmeure, L. F. C., and Jaworski, A., Powder Technol., 112, 174, 2000. 6. Mosorov, V., Sankowski, D., Mazurkiewicz, L., and Dyakowski, T., Meas. Sci. Technol., 13, 1810, 2002. 7. Williams, R. A. and Jia, X., Adv. Powder Technol., 14(1), 1, 2003. 8. McKee, S. L., Bell, T., Dyakowski, T., Williams, R. A., and Allen, T., Powder Technol., 82, 105, 1995. 9. Ostrowski, K. L., Luke, S. P., Bennett, M. A., and Williams, R. A., Powder Technol., 102, 1, 1999. 10. Ostrowski, K. L., Luke, S. P., Bennett, M. A., and Williams, R. A., Chem. Eng. J., 3524, 1, 1999. 11. Jaworski, A. and Dyakowski, T., in Proceedings of the Second World Congress on Industrial Process Tomography, Virtual Centre for Industrial Process Tomography, Leeds/Manchester, 2001, p. 353. 12. Jaworski, A. and Dyakowski, T., Powder Technol., 125, 279, 2002. 13. Neuffer, D., Alvarez, A., Owens, D. H., Ostrowski, K. L., Luke, S. P., and Williams, R. A., in Proceedings of the First World Congress on Industrial Process Tomography (Buxton), Virtual Centre for Industrial Process Tomography, Leeds/Manchester, 1999, p. 71. 14. Hunt, A., Pendleton, J. D., and White, R. B, in Proceedings of the Third World Congress on Industrial Process Tomography, Virtual Centre for Industrial Process Tomography, Manchester/Leeds, 2003, p. 281. 15. Wang, S. J., Dyakowski, T., Xie, C. G., Williams, R. A., and Beck, M. S., Chem. Eng. J., 56, 95, 1995. 16. Parker, D. J., Hawkesworth, M. R., Broadbent, C. J., Fowles, P., Fryer, T. D., and McNeil, P.A., Nucl. Instrum. Methods, A348, 583, 1994. 17. Parker, D. J. et al., Nucl. Inst. Meth. Phy. Res. A, 477, 540, 2002. 18. Conway-Baker, J., Barley, R. W., Williams, R. A., Jia, X., Kostuch, J. A., McLoughlin, B., and Parker, D. J., Minerals. Eng., 15, 53, 2002. 19. Seville, J. P. K., in Proceedings Particulate Systems Analysis , Harrogate, UK, 2003. 20. Golchert, D. J., Moreno, R., Ghadiri, M., Litster, J., and Williams, R. A., Adv. Powder Technol., 15, 447, 2004. 21. Leonard, A, Blacher, P, Marchot, P., Pirard, J.-P., and Crine, M., Proceedings of the Third World Congress on Industrial Process Tomography, Virtual Centre for Industrial Process Tomography, Manchester/Leeds, 2003, p. 730.
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Part VII Working Atmospheres and Hazards
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7.1
Health Effects Due to Particle Matter Isamu Tanaka and Hiroshi Yamato University of Occupational and Environmental Health, Kitakyushu, Fukuoka, Japan
7.1.1
INTRODUCTION
There are many kinds of dusts and other airborne particles in the total environment in which humans work and live. They vary from time to time in number and chemical and physical properties. In the work environment, the health hazards resulting from the exposure to them is of particular importance because they are a common feature and their biological effects are often insidious. In order to prevent or minimize impairment of the health of workers, it is essential to know the concentrations and diameters of harmful substances, their chemical and physical characteristics, and their biological actions. We must be aware not only of the biological effects but also of the mechanisms underlying and controlling their occurrence. This chapter discusses principles relating to the lung response to exposure to particulate matter in the work environment.
7.1.2
RESPIRATORY SYSTEM
The primary purpose of the respiratory organ is to act as a gas exchange mechanism. Figure 1.1 shows a schematic anatomy of the lung. Air inspired through the nose or the mouth enters the larynx and then the trachea, which is divided into two main bronchi. The bronchial tubes repeatedly divide and diminish in diameter with bifurcation, the smallest tubes being called bronchioles. The diameter of the main bronchi and terminal bronchioles are 1.2 cm and 0.5 mm, respectively. These dimensions and numbers have major implications with respect to deposition and clearance of particles entering the lung while entrained in respired air. Terminal bronchioles lead into the respiratory bronchioles and then into the respiratory space which is composed of a number of alveoli. A very thin layer of venous blood is circulated through the pulmonary capillaries, which are arranged in a network over the surface of approximately 300 million air-containing alveoli. The total effective gas exchange surface of the alveoli has been estimated at 100 m2. The alveolocapillary wall separating the blood from the gas phase is on average 0.55 m in thickness. During the passage of venous blood over the surface of this barrier, the rapid movement of oxygen from the air into the blood and of carbon dioxide out of the blood into the air phase is carried out by gas diffusion. The trachea and the bronchial tubes are covered with ciliated cells on the epithelium. A carpet of mucus is continuously moved upward by ciliated action, where the mucus is then swallowed or expectorated. This system, known as the mucociliary escalator, is the mechanism whereby particles deposited on the airway surface are removed. Beyond the terminal bronchioles the surface lining of the airways is no longer ciliated. Each alveolus is lined mainly by type I pneumocytes. Type II pneumocytes apparently secrete surfactant, a material that affects the surface tension of the ultrathin liquid layer on the epithelium. Alveolar macrophages, large free cells 10–50 m in diameter, are located within the alveolar
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FIGURE 1.1
Schematic structure of the respiratory tract.
region. These are large mobile cells that are the principal phagocytic cell. They also are involved in immunologic reactions.
7.1.3 PENETRATION AND DEPOSITION OF PARTICLES IN THE RESPIRATORY TRACT Particles penetrate into the human body through the respiratory system, causing a local effect on the lungs and other parts of the system and then passing into the blood stream and causing systemic effects. Very soluble substances are easily absorbed from all parts of the respiratory tract. Therefore the site of deposition is of less importance for systemic effects. For insoluble particles, the site of deposition in the respiratory system is of fundamental importance. It depends on the aerodynamic properties of the particle, its shape, dimensions of the airways, and the pattern of breathing. Particles vary markedly in shape and density, and both these factors play a role in the behavior of particles in the respiratory tract. For our purposes we consider all particles as being spheres of unit density, with the understanding that there could be some variation between particles with respect to speed of settling, depending on their shape and density. This equivalent spherical diameter of unit density is called the aerodynamic diameter. The dynamics of inhaled particles in the respiratory tract consider particles in terms of the aerodynamic diameter, to reach a reasonable agreement between theoretical predictions and experimental observations. When considering the deposition of particles, the respiratory system is divided into three regions: l. Nasal–pharyngea1: This begins at the nose and goes to the level of the larynx or epiglottis. 2. Trancheobronchia1: This next region consists of the trachea and the bronchial tubes down to the terminal bronchioles. 3. Pulmonary: The third region is regarded as the gas exchange space of the lungs. © 2006 by Taylor & Francis Group, LLC
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FIGURE 1.2 Regional deposition fraction in the respiratory tract. Each shaded area indicates the variability of deposition for a given mass media aerodynamic diameter (m) in each region.
Figure 1.2 indicates the range of variability of deposition for a given aerodynamic diameter in each lung compartment. The particles are deposited in the lung by the following four mechanisms: l. Sedimentation: T\he deposition is proportional to the particle speed of settling (proportional to aerodynamic diameter squared) and to the time for settling. 2. Internal impaction: The deposition is also proportional to the particle speed of settling and air velocity, especially where the flow line changes direction. 3. Brownian displacement: The deposition of small particles of less than 0.1 m in size is governed by diffusional force. 4. Interception: The deposition is provided by the geometric dimensions of a particle following an air stream line.
7.1.4
FATE OF DEPOSITED PARTICLES
The fate of the particles deposited in the respiratory tract is dependent on their diameter and solubility and the site of their deposition, following the defense mechanism in the respiratory system for inhaled particles. l. Nasal filtration: Particles larger than 10–l5 pm are mainly trapped in the nose by filtration by nasal hairs and in the throat and upper airways by impaction. Ciliated epithelium-covered © 2006 by Taylor & Francis Group, LLC
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2.
3.
4.
5.
7.1.5
mucus in the nose contributes to the removal of particles, which are then swallowed. These swallowed particles are significant when dealing with materials that cause systematic intoxication. Mucociliary clearance: The trachea and bronchi, down to the terminal bronchioles, are lined with a ciliated epithelium covered by a mucous layer. The cilia are in continuous and synchronized motion, which causes the mucous layer to have a continuous upward movement at a speed usually in the range of 10–30 mm/min. Particles deposited on the ciliated epithelium are moved toward the epiglottis by the mucociliary escalator, and then swallowed into the gastrointestinal tract or expectorated from the organism within a relatively short time. Coughing and sneezing: These reactions transport the particles toward the upper respiratory tract. Coughing and sneezing create strong currents of air, which help to clear the airways. Phagocytosis: The epithelium of the pulmonary spaces is not ciliated. However, insoluble particles deposited in this region are engulfed by alveolar macrophage cells. The engulfed particles move up to the ciliated epithelium and then are removed upward by the mucociliary escalator and out of the respiratory system. They may also remain in the pulmonary space or enter the lymphatic system. Particles containing free crystalline silica are cytotoxic; that is, they destroy the alveolar macrophage cells. Dissolution: The particles deposited in nonciliated airways are cleared by dissolution.
HEALTH EFFECTS OF INHALED PARTICLES
Some particles cause various biological effects. l. Pneumoconioses: The pneumoconioses constitute lung diseases that result from the inhalation of certain types of particles deposited in the pulmonary spaces. The alveolar macrophages are part of the defense system in this region. They ingest particles but may be killed or damaged in the process, which results in fibrogenic responses in lung tissue. Some particles introduce such responses, which lead to loss of elasticity in the lung and impair gas diffusion. There is a long interval known as the latent period between first exposure and development of the disease. Silicosis is the most important of the pneumoconioses. Its risk is linked to the percentage of free crystalline silica of inhaled particles. The typical pathological changes are determined by the cytotoxic and fibrogenic capacity of the silica particles. Other pneumoconiosis may be produced by deposition in the nonciliated airways of excessive amounts of, for example, asbestos, coal mine dust, beryllium, kaolin, barium, tin, iron oxide, talc, and graphite. 2. Systemic poisoning: The inhaled particles penetrate into the circulation and internal organs of the body after dissolution. Manganese, lead, cadmium, and the other heavy metals are examples of toxic systemic agents occurring in particulate form. 3. Cancer: Arsenic, chromate, policyclic aromatic hydrocarbons, nickel, radioactive particles, and asbestos have a carcenogenic potential leading to lung cancer. Soluble carcinogens may be a risk to both lungs and other organs. 4. Irritation: Exposure to irritant particles may lead to tracheitis and bronchitis, pneumonitis, and pulmonary edema. There are many irritant particles in the work environment, such as fumes of Cd, Be, V, and Zn, pesticides, acid mists, and fluorides. 5. Allergic reactions: The deposition and retention of inhaled sensitizing substances may produce allergic reactions or other sensitivity responses. Synthetic organic compounds (amine compound, plastics), many vegetable dusts (bagasse, corn, cotton, flax, flour, straw, tea, tobacco, wood), and metal dusts (nickel, chromium) may produce inhalant allergy, asthma, hay fever, or urticaria. © 2006 by Taylor & Francis Group, LLC
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6. Metal fume fever: This is a condition resulting from exposure to freshly generated metal fumes (zinc oxides, magnesium oxide). The main symptoms include chills, fever, muscle pains, nausea, and weakness. It is an acute condition, of brief duration, and with no after effects. 7. Infection: Particles containing fungi, viral, or bacterial pathogens may play a role in the transmission of infectious diseases.
7.1.6 THRESHOLD LIMIT VALUE Regarding chemical substances present in inhaled air as suspensions of particles, the potential hazard depends on particle size as well as mass concentration. The threshold limit value (TLV) refers to the airborne concentration of particles and represents conditions under which it is believed that nearly all workers may be repeatedly exposed day after day without adverse health effects. The TLV is based on available information from industrial experience and from experimental human and animal studies. Health impairments include those that shorten life expectancy, compromise physiological function, impair the capability for resisting other toxic substances or disease processes, or adversely affect reproductive function or developmental processes. The size-selective TLV of particles has been recommended for crystalline silica for many years in recognition of the well-established association between silicosis and respirable mass concentration. The size-selective TLV of particles is usually divided into two types: l. Inhalable particles: These materials are hazardous when deposited anywhere in the respiratory tract. The aerodynamic diameter of the inhalable particles is less than 200 m. The concentration of the inhalable particles is measured as the total dust by an air filter sampler with the inlet air speed of 0.5–0.8 m/s. 2. Respirable particles: The materials are hazardous when deposited in the gas exchange region. The aerodynamic diameter of respirable particles is less than 10 m. The mass concentration of the respirable particles is measured by the air sampler with an elutriater or cyclone. The Japan Society for Occupational Health has recommended TLVs for airborne particles in the work environment by using the occupational exposure limits as shown in Table 1.1.
TABLE 1.1 I
Occupational Exposure Limits for Dusts
Dust with more than 10% crystalline silica Respirable dust:
M⫽
2.9 mg/m 3 , 0.22Q ⫹ 1
M: OEL, II
Q:
Total dust:
M⫽
12 mg/m 3 0.23Q ⫹ 1
content of crystalline silica %
Dust with less than 10% of free crystalline silica Class 1:
OEL: Respirable dust 0.5 mg/m3, total dust 2 mg/m3 Activated charcoal, alumina, aluminum, bentonite, diatomite, graphite, kaolinite, pagodite, pyrites, pyrite, cinder, talc
Class 2:
OEL: Respirable dust 1 mg/m3, total dust 4 mg/m3 Bakelite, carbon black, coal, cork dust, cotton dust, iron oxide, grain dust, joss stick material dust, marble, portland cement, titanium dioxide, wood dust, zinc oxide
Class 3:
OEL: Respirable dust 2 mg/m3, total dust 8 mg/m3 Limestone, inorganic and organic dusts other than classes l and 2
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The toxicity of all particles generated in the work environment cannot evaluated. For new particles developed and produced, the nature of the database associating exposure and disease is inconclusive. If a disease is observed in laboratory animals exposed to new particles, this would need to be viewed as a positive indicator of the potential health hazard for humans. Especially, inhalation animal studies are credible because of the similarity of human exposures. The health effects of airborne particles in the work environment should be investigated in order to prevent impairment.
REFERENCES 1. Task Group on Lung Dynamics, Health Phys., 12, l73–207, 1966. 2. Japan Society for Occupational Health, J. Occup. Health, 44, 267–282, 2002. 3. World Health Organization, WHO Offset Publication, No. 80, WHO, Geneva, 1984, p.75.
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7.2
Respiratory Protective Devices for Particulate Matter lsamu Tanaka and Hajime Hori University of Occupational and Environmental Health, Kitakyushu, Fukuoka, Japan
7.2.1
INTRODUCTION
Respiratory protective devices (respirators) are used for protecting the respiratory system from inhalation of particulate matter. However, the use of respirators is the last choice for occupational hygiene management. The first step for management is to improve working environments, for example, to enclose dust-generation sources, to change the method or process of work, and to use local and general ventilation systems. Respirators should be used in an environment where the workplace is still hazardous even though improvements have been made, or where improvements are difficult or impossible.
7.2.2 TYPES OF RESPIRATORS In principle, respirators are classified into two types: air-purifying respirators and atmospheresupplying respirators. The air-purifying respirator removes contaminants from the ambient air, and the atmosphere-supplying respirator provides air from a source other than the surrounding atmosphere. Both types can be further subclassified by the type of inlet covering and the mode of operation. Figure 2.1 shows the subclassifications of respirators for particulate matter.
7.2.3 AIR-PURIFYING RESPIRATORS Classification of Air-Purifying Respirators Air-purifying respirators are further classified as nonpowered and powered respirators. Nonpowered Respirators Air in the atmosphere enters the filter in a manner that corresponds to workers’ breathing. Only tight-fitting coverings are available for nonpowered respirators, because the pressure in the covering may be negative and contaminants could then come through the covering during inhalation. Nonpowered respirators are generally small and are easily maintained. They restrict the wearer’s movements the least and may present little physiological strain to the wearer. They should not be used in atmospheres containing less than 18% oxygen nor in atmospheres immediately dangerous to life or health. Powered Air-Purifying Respirators The powered air-purifying respirator (PAPR) uses a blower to pass contaminated air through an element that removes the contaminants and supplies the purified air to a respiratory inlet covering. The covering may be a facepiece, helmet, or hood.
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FIGURE 2.1 Classification of respirators.
This type of respirator comes in several different configurations. One configuration consists of the air-purifying element(s) attached to a small blower which is worn on the belt and is connected to the respiratory inlet covering by a flexible tube. Another type consists of an air-purifying element attached to a stationary blower powered by a battery or an external power source and connected by a long flexible tube to the respiratory inlet covering. The third type consists of a helmet or facepiece to which the air-purifying element and blower are attached. Only the battery is carried on the belt.
Filtration Mechanisms and Lifetime All particulate filtering respirators use a filter to remove the contaminant. Many kinds of filters are available, such as nonwoven fiber material, natural wool or synthetic fiber felt, compressed natural wool or synthetic fiber felt, resin-impregnated felt, loose-packed fibrous glass, and a high-efficiency filter made of a flat sheet of material. Particulate filters are of two types, that is, absolute and nonabsolute filters. Absolute filters use screening to remove particles from the air; that is, they exclude the particles that are larger than the pores. However, this type of filter has large breathing resistance due to small pores. Therefore, most respirator filters are nonabsolute filters. In nonabsolute filters, pore sizes are larger than the particles to be removed, but the particles are captured and removed by combinations of interception, sedimentation, inertial impaction, diffusion, and electrostatic force. The contribution of each factor to the filtration mechanism depends on the flow rate through the filter and the size of particle. The airflow resistance of a particulate-removing respirator filter element increases as the quantity of particles retained in the filter increases. This resistance increases the breathing resistance offered by a nonpowered respirator and may reduce the rate of airflow in a powered respirator. Filter element plugging by retained particles may also limit the continuous use time of a respirator. Between uses, reusable respirators should be packaged to reduce exposure to conditions that cause filter degradation, such as high humidity. In powered air-purifying respirators, since there is a constant flow through the air-purifying element instead of a flow only during inhalation, the useful service lifetimes of the elements are shorter than those of comparable ones attached to a negative pressure respirator. In order to overcome this problem, some powered air-purifying respirators have a spring-loaded exhalation valve assembly.
Filter Efficiency For air-purifying respirators, the filtration efficiency (FE) of a filter is defined as follows. FE ⫽
Co ⫺ C i C ⫽1⫺ i Co Co
(2.1)
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TABLE 2.1 Class of Filter N-series
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Efficiency (%)
N100
99.7
N99
99
N95
95
R-series R100
99.7
R99
99
R95
95
P-series P100
99.7
P99
99
P95
95
Test Agent NaCl
Maximum Loading (mg) 200
Type of Contaminant Solid and waterbased particulates (i.e., nonoil aerosols)
DOP oil
200
Any
DOP oil
Stabilized efficiency
Any
Service Time Nonspecific
One work shift
Nonspecific
Note: The three categories of resistance to filter efficiency degradation are N (not resistant to oil), R (resistant to oil), and P (oil proof).
Where C i is defined as the concentration of a contaminant inside the respirator facepiece cavity, and Co is defined as the contaminant concentration outside the respirator facepiece. The filtration efficiency is applied only to component parts of the respirator system. Table 2.1 shows the filter efficiency for nonpowered, air-purifying, particulate-filter respirators certified under 42 CFR Part 84 in the United States.1
7.2.4 ATMOSPHERE-SUPPLYING RESPIRATORS Supplied-Air Respirators Airline Respirators Airline respirators use compressed air from a stationary source delivered through a hose under pressure. The respiratory inlet covering may be a facepiece, helmet, hood, or complete suit. These are available in demand, pressure-demand, and continuous-flow configurations. In a demand or negative pressure–type regulator, air flows into the facepiece only on demand by the wearer. On the other hand, a pressure-demand or positive pressure regulator has a modified regulator and special exhalation valve to maintain positive pressure in the facepiece at all times, so that any leakage should be outward. Therefore, a pressure-demand airline respirator provides very good protection. The airline respirator may be used for long continuous periods. The disadvantage of an airline respirator is the loss of the source of respirable air supplied to the respiratory inlet covering. Also, the trailing air supply hose of the airline respirator severely restricts the wearer’s mobility. Hose Masks Hose masks supply air from an uncontaminated source to the respiratory inlet covering through a strong, large-diameter hose. Two types are available. One has a hand- or motor-operated air blower that pushes © 2006 by Taylor & Francis Group, LLC
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low-pressure air through the hose to the respiratory inlet covering. The other type of hose mask has no blower and requires the wearer to inhale through the hose. The hose mask without blower has many advantages, namely, its long use periods, simple construction, low bulk, easy maintenance, and minimal operating cost. An advantage of the hose mask with blower is its minimal resistance to breathing. The contaminated air may leak in if the covering fits poorly, because the air pressure inside the respiratory inlet covering of the hose mask with no blower is negative during inhalation. Therefore, hose masks, with and without blower, are certified only for use in atmospheres not immediately dangerous to life or health. The trailing air supply hose of the hose mask also severely limits a worker’s mobility.
Self-Contained Breathing Apparatus The distinguishing feature of a self-contained breathing apparatus (SCBA) is that the wearer need not be connected to a stationary breathing gas source, such as an air compressor. Therefore, it allows comparatively free movement over an unlimited area. Instead, enough air or oxygen for up to 4 hours, depending on the design, is carried by the wearer. This limited service life makes SCBAs unsuitable for routine use for long continuous periods. Two types are available. Open Circuit An open-circuit SCBA exhausts the exhaled air into the atmosphere. A cylinder of high-pressure compressed air supplies air to a regulator that reduces the pressure for delivery to the facepiece. This regulator also serves as a flow regulator by passing air to the facepiece on demand. Most open-circuit SCBAs have a service life of 30 to 60 minutes. Two types of open-circuit SCBA are available: demand and pressure demand. These are very similar in basic operation to airline respirators, except that the air is supplied from a portable air source instead of a stationary one. Closed Circuit The exhaled air is rebreathed after carbon dioxide has been removed and the oxygen content restored by a compressed or liquid oxygen source or an oxygen-generating solid. These devices have a longer service life than the open-circuit SCBA, that is, they are designed for l to 4 hour use in oxygen-deficient atmospheres. Because negative pressure is created in the facepiece of a nonpositive pressure apparatus during inhalation, there is increased leakage potential. Therefore, negative pressure closed-circuit SCBAs should be used in atmospheres not immediately dangerous to life or health only. For use in oxygen-deficient atmospheres over long periods, closed-circuit SCBAs are also satisfactory.
7.2.5
PROTECTION FACTOR
The protection factor (PF) of a respirator is defined as follows: PF ⫽
Co Cin
(2.2)
where, C in is defined as the concentration of contaminant the inside of the covering. Co is defined as the contaminant concentration outside the respirator facepiece. PF assessments are made almost exclusively on man/respirator systems. It is important to recognize that on a man/respirator system, Cin depends on a complicated function of many individual sources of penetration (e.g., air-purifying element penetration, exhalation valve penetration, face seal penetration, and other inboard © 2006 by Taylor & Francis Group, LLC
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TABLE 2.2 Protection Factor Values against Particulate Exposures Assigned by NIOSH Type of Respirators Air purifying respirators Powered air purifying respirators
Assigned PF 5–50 25–50
Atmosphere-supplying respirators Supplied air/continuous flow
25–50
Supplied air/SCBA/demand
10–50
Supplied air/SCBA/pressure demand
1000–10000
penetrations) and those environmental conditions that would affect penetration. To deal with the multiple methods for determining and applying protection factors, a number of definitions have been proposed.2 Table 2.2 shows protection factors assigned by NIOSH.3
7.2.6
NOTES FOR USING RESPIRATORS
A worker should be aware of the following: l. Air-purifying respirators cannot be used if the oxygen concentration in the atmosphere is less than 18%. 2. Respirators should be selected on the basis of hazards to which the worker is exposed. Coverings should be fitted for the worker’s face. 3. Respirators should be regularly inspected. 4. Worn and deteriorated parts should be replaced. 5. Respirators should not be allowed to leak contaminants through gaps between the worker’s face and the coverings. 6. Respirators should be kept clean and disinfected. 7. Respirators should be stored in a dry, clean place. 8. Used filters should be disposed of by a method that does not scattering the dust. 9. The worker should be instructed and trained in the proper use of respirators and their limitations. 10. An assigned person who is responsible for mask management should instruct workers how to use respirators.
REFERENCES 1. National Institute for Occupational and Safety and Health, NIOSH Guide to Industrial Respiratory Protection, Bollinger, N. J. and Schutz, R. H., Eds., U.S. Department of Health and Human Services, Cincinnati, OH, 1987. 2. National Institute for Occupational and Safety and Health, NIOSH Guide to the Selection and Use of Particulate Respirators Certified Under 42 CFR 84, NIOSH Publication No. 96-101, U.S. Department of Health and Human Services, Cincinnati, OH, 1996. 3. Myers, W. R., Lenhart, S. W., Campbell, D., and Provost, G., Am. lnd. Hyg. Assoc. J., 44, B25, 1983.
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7.3
Spontaneous Ignition and Dust Explosion Tatsuo Tanaka* Hokkaido University, Sapporo, Japan
7.3.1 SPONTANEOUS IGNITION OF POWDER DEPOSITS In spite of the absence of fire sources, an ignition or subsequent dust explosion can occur in a coal or grain dust deposit. This is called spontaneous ignition and takes place as the result of an accumulation of heat of reaction, raising the local temperature, which, in turn, promotes the rate of reaction to give off more heat. When the deposit temperature exceeds a certain limit, the powder deposit is ignited after a certain time. The former is the ignition temperature, which is defined as the highest ambient temperature at which no ignition can take place, and the latter is called the induction time. These two characteristics should be estimated for safety purposes for a specific powder deposit.
Basic Differential Equation Consider a cylindrical powder deposit of an infinite length placed at time 0 under the ambient temperature T a . According to thermal theory, a combustible dust deposit has two heat fluxes: one is the heat generation caused by the chemical reaction, and the other is the heat loss dissipated to the surroundings. If the reaction rate is expressed by the zero-order equation of Arrhenius, the heat balance is taken as ⎛ ∂T ⎞ ⎛ E ⎞ k ⎡ ∂ ⎛ ∂T ⎞ ⎤ rC ⎜ ⎟ ⎛ ⎞ ⎢ ⎜ r Qf exp ⎜ ⎝ ∂t ⎠ ⎝ r ⎠ ⎣ ∂r ⎝ ∂t ⎟⎠ ⎥⎦ ⎝ RT ⎟⎠
(3.1)
where r is the radial distance (m) in the cylindrical coordinates, t is the storage time, rC is the volumetric specific heat (J/m3K), k is the thermal conductivity of powder (W/mK), T(r, t) is the temperature of the deposit (K), Q is the heat of reaction (J/mol), f is the frequency factor (mol/s/m3), E is the activation energy (J/mol), and R is the universal gas constant (J/mol K). This equation is solved numerically under the following initial and boundary conditions by using the specific data of cork as I.C. T = Tr ( = 293 K), at t = 0 B.C. −kT/r = h(T − Ta) at r = R1 where Tr is the room temperature, R1 is the radius of the cylinder, and h is the heat transfer coefficient from the side wall of the deposit. For example, the calculated temperatures1 are compared with Leuschke’s2 experimental data of the same conditions under Ta = 414 K, as in Figure 3.1a and 3.1b. We see that good agreement is obtained between the theory and the experiment where R1 and the length are 0.16 m, and the subscripts m, h, and b are at the center, halfway, and near the surface of the deposit, respectively. The temperature profiles immediately before the ignition vary with the ambient temperature, as indicated in Figure 3.2 and Figure 3.3, the calculation and the experiment being compared at 414 K and 444 K, respectively. When * Retired
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FIGURE 3.1 (a) Heat propagation simulated in cork dust deposit in the lapse of time at Ta = 414 K. [From Liang, H., Tanaka, T., and Nakajima, Y., J. Soc. Powder Technol. Jpn., 23, 326, 1986. With permission.] (b) The corresponding experimental data by Leuschke. [From Liang, H., Tanaka, T., and Nakajima, Y., J. Soc. Powder Technol. Jpn., 23, 326, 1986. With permission.]
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FIGURE 3.2 Temperature profile simulated immediately before ignition (Ta = 414 K). [From Liang, H., Tanaka, T., and Nakajima, Y., J. Soc. Powder Technol. Jpn., 23, 326, 1986. With permission.]
FIGURE 3.3 Temperature profile simulated immediately before the ignition (Ta 444K). [From Liang, H., Tanaka, T., and Nakajima, Y., J. Soc. Powder Technol. Jpn., 23, 326, 1986. With permission.]
the ambient temperature is set to 408 K, the calculated deposit temperature maintains constant. The measured ignition temperature has been determined to be 412 K. Leuschke3 varied the volume of the cork dust deposit to examine the ignition temperature, which is compared with the calculation by Equation 3.1, as shown in Figure 3.4. Good agreement suggests that simulation is possible by use of Equation 3.1 to examine various factors affecting the ignition © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.4 Simulated dependence of self-ignition temperature on the dust volume compared with experimental data. [From Liang, H. and Tanaka, T., Kagaku Kogaku Ronbunshu, 13, 63–70, 1987. With permission.]
temperature as well as the induction time. For this purpose, Equation 3.1 is rewritten in a dimensionless form, together with the initial and boundary conditions: ∂ 2 u 1 ∂u ∂u G ⎞ ⎛ SG exp ⎜ ⎟ 2 ⎝ z ∂z ∂t 1 uG ⎠ ∂z
(3.2)
I.C. u = GD at t = 0 B.C. u / z Bu at z 1 where u E (T Ta ) ⁄RTa2 , z r ⁄R1 , t kt ⁄rCR12 , B hR1 ⁄ k (= Biot number), S QfR12 /kTa , G E/RTa , and D (Ta Tr ) /Ta .
Evaluating Ignition Temperature and Induction Time From the above definitions, the dimensionless induction time ti is related to t i by ti = kt i / rCR 12 by corresponding to a reasonably high value of the dimensionless temperature u, and the fire spreads immediately in all directions once the bed is ignited. ti can thus be regarded independent of u and z. Therefore, ui becomes a function of S, G, D, and B. The numerical calculations give interrelationships between ui and for G various values of the parameters S, D, and B and represent that the critical value of Gc, to give ti → becomes a function of only S in the cases where B is larger than 25. This means that the heat generated equals the heat lost to the surroundings ( i.e., the steady state), where u/t is equal to 0 in Equation 3.2. This is the critical condition that has been dealt with by past workers who tried to solve the basic equation in an analytical way, namely, the right-hand side in Equation 3.2 was approximated at steady state by Frank-Kamenetskii,4 because for u/G << 1 ⎡ G ⎞ u ⎞⎤ ⎛ ⎛ SG exp ⎜ 艐 SG exp ⎢G ⎜ 1 ⎟ ⎥ ⎝ 1 u / G ⎟⎠ ⎝ ⎠⎦ G ⎣ SG exp (G) exp (u) d exp(u)
(3.3)
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FIGURE 3.5 Nomogram for calculating the ignition temperature. [From Liang, H. and Tanaka, T., Kagaku Kogaku Ronbunshu, 13, 63–70, 1987. With permission.]
where d SG exp (G). Therefore, in the steady state we have d 2 u ⎛ 1 ⎞ du ⎜ ⎟ 艐 dexp (u) dz 2 ⎝ z ⎠ dz
(3.4)
As far as the above equation for the cylindrical coordinates is concerned, it has an analytical solution u of for all d up to dc, the latter of which was given by Frank-Kamenetskii to be 2exp(-2/B), B being the Biot number. According to our calculations, where no approximation be made, there is obtained dc= 2.14 exp (2/B). If dc is exceeded, there is no steady-state solution of Equation 3.2, suggesting the runaway reaction. After all, we have an equation including all original variables concerned for obtaining the ignition temperature, Tsi. ⎛ QfR12 E ⎞ ⎛ ⎛ 2k ⎞ E ⎞ ⎜⎝ kRT 2 ⎟⎠ exp ⎜⎝ RT ⎟⎠ exp ⎜⎝ hR ⎟⎠ 2.14 si 1 si
(3.5)
Likewise, the Frank-Kamenetskii number for spherical coordinates and for the plane slab are respectively 3.32 and 3.51, as mentioned later. A nomograph is presented in Figure 3.5,5 which makes it possible to obtain the ignition temperature of a given cylindrical deposit based on Equation 3.5. Starting from the right ordinate of the calculated value, draw a horizontal line 2 which intersects the reference curve at point a. Draw a horizontal line 1 from the left ordinate of the activation energy and obtain an intersecting point b with the vertical line passing the point a, and connect b with the original point, which can be extended straight forward to the temperature axis as denoted by line 4, where we can get the corresponding ignition temperature Tsi. © 2006 by Taylor & Francis Group, LLC
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On the other hand, the dimensionless induction time is somewhat complicated but is correlated in the following empirical expression: ⎛ G⎞ ti K ⎜ ⎟ ⎝ S⎠
m
(3.6)
where K and m depend on the dimensionless groups, D and W [=R(TaTsi)/E]. For instance, when G/S is greater than 10 0.2/(ba), ti 10 0.4(G/S)a; when G/S is smaller than 10 0.9/(ba), then ti 101.1(G/ S)a; when G/S is in between, then ti 10 0.2(G/S) b, wherein a 2.32 1.55logD 3.93logW, b 2.02(1logD) 4.77logW and c 3.67 1.59logD 4.20 logW. The calculated induction time by use of Equation 3.6 together with the known values of cork dust deposit are shown in Figure 3.6a against the volume of the deposit. Unfortunately, there are no data for cork, but similar data are available for tobacco. The estimation seems to be reasonable from Figure 3.6b.
Examination of the Steady-State Solution If d is smaller than dc, the system might be safe; that is, no spontaneous ignition would take place. More rigorous insight can be given for a plane slab of the length L. The steady-state equation corresponding to Equation 3.4 becomes d 2u d exp(u) 0 dz 2
(3.7)
where z is the distance from the bottom of the slab divided by L. The analytical solution is obtained, under z = 0 and 1, 0, and z 1/2, u um (the maximum temperature), as 2 ln
(
exp um exp um 1 exp um
)=
d 2
(3.8)
The left-hand side of the above equation becomes maximum when um 1.1868, and, accordingly, d on the right-hand side is equal to 3.51. This is the critical value dc, beyond which there is no solution of Equation 3.7; that is, the slab becomes a runaway system. However, if d is less than dc (e.g., d=2), then Equation 3.7 has two solutions, um 0.32895 and 2.8955, the latter of which, however, is an unstable solution, and the system might become a runaway state according to the circumstances. Consequently, even when d is less than d c, the system is not necessarily safe and stable. In fact, the temperature profile within the slab is much higher for the latter case, and if the profile be exceeded for some reason, the system cannot hold a thermal stability. In either way, aeration quenching appears an effective countermeasure against the disaster.
Effect of Oxygen Diffusion on Thermal Stability in a Powder Bed Before discussing air quenching, the thermal stability influenced by the diffusion of oxygen in a plane slab should be considered. So far, the basic equation has been solved under the assumption that the oxygen concentration is constant everywhere due to the sufficient makeup. However, if the diffusion of oxygen is not enough, then the aeration of fresh air will enhance the rate of reaction, leading eventually to a hazardous situation. The heat balance equation for a one-dimensional plane slab can be written as6 n
⎛ c⎞ ∂2T ⎛ E ⎞ ⎛ ∂T ⎞ rC ⎜ ⎟ k 2 Q ⎜ ⎟ f exp ⎜ ⎝ RT ⎟⎠ ⎝ ∂t ⎠ c ∂x ⎝ 0⎠
(3.9)
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FIGURE 3.6 (a) The induction time simulated for different overtemperatures varied with the deposit volume. [From Liang, H., Tanaka, T., and Nakajima, Y., J. Soc. Powder Technol. Jpn., 23, 326, 1986. With permission.] (b) The experimental induction time to ignition by Leuschke for tobacco. [From Liang, H., Tanaka, T., and Nakajima, Y., J. Soc. Powder Technol. Jpn., 23, 326, 1986. With permission.]
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where x is the distance from the bottom of the slab, c is the concentration of oxygen as a function of x and t, c0 is the oxygen concentration outside the powder bed, and n is the order of reaction. In regard to the oxygen diffusion through the bed, we can write n
⎛ ∂2 c ⎞ ⎛ c ⎞ ⎛ E ⎞ ⎛ ∂c ⎞ ⎜ ⎟ D∗ ⎜ 2 ⎟ ⎜ ⎟ f exp ⎜ ⎝ RT ⎟⎠ ⎝ ∂t ⎠ ⎝ ∂x ⎠ ⎝ c0 ⎠
(3.10)
where is the porosity of the bed and D* is the effective diffusivity of oxygen. Equation 3.9 and Equation 3.10 are rewritten in the nondimensional forms by utilizing the dimensionless terms quoted in Equation 3.2: ⎛ rCL2 ⎞ ∂u ∂ 2 u ⎛ Gu ⎞ n ⎜⎝ k ⎟⎠ ∂t ∂z 2 df exp ⎜⎝ G u⎟⎠
(3.11)
⎛ L2 ⎞ ∂ f ∂ 2 f ⎛ ⎞ n ⎛ Gu ⎞ ⎜⎝ D∗ ⎟⎠ ∂t ∂z 2 ⎜⎝ G⎟⎠ df exp ⎜⎝ G u ⎟⎠
(3.12)
where the length L is used in place of R1 for z and d, and f c/c0, an important dimensionless group Λ (kTa /D*c0Q), and d is the Frank-Kamenetskii number defined by (QL2G/kTa)f exp (G). At steady state, combining Equation 3.11 and Equation 3.12, we have 2 d2f ⎛ ⎞ ⎛ d u⎞ = −⎜ ⎟ ⎜ 2 ⎟ 2 ⎝ G ⎠ ⎝ dz ⎠ dz
(3.13)
The following boundary condition is assumed here that the bottom of the bed is closed to air flow and the ambient temperature is Ta;u 0, df/dz0; at z 0; and u0, f1 at z1, from which we get n
⎡ ⎛ ⎞ ⎤ d 2u ⎛ Gu ⎞ d ⎢1 ⎜ ⎟ {u (1 z ) u ’0 }⎥ exp ⎜ 2 ⎝ G u ⎟⎠ dz ⎣ ⎝ G⎠ ⎦
(3.14)
where u'0 du dz at z 0. The boundary condition is u 0 at z 0 and 1. Figure 3.7 shows the calculated u at z = 1 as a function of u'0 for some sets of parameters, from which we see that there is a single steady-state solution for d = 4, but when d 4.5, three solutions exist. At d 5.0387, the two solutions come closer to give a multiple solution, beyond which there is obtained only one steady-state solution. Therefore, three different solutions are possible in a certain range of d. Of the solutions, the ones labeled B turn out to be unstable solutions; as stable solutions, we have the lower-temperature mode (A) and the higher-temperature mode (C) in general, as shown in Figure 3.8. The arrows indicate the direction of the distribution shift as d increases. On the other hand, the oxygen distributions are depicted in Figure 3.9, in which for the lower-temperature mode, the oxygen concentration is higher, whereas for the higher-temperature mode, the oxygen is extremely low in the major part of the bed. Here, the smoldering combustion can be assumed. When d is smaller than dc = 5.0387, the lower-temperature mode certainly exists, but this mode coincides with the unstable solution at d = 5.039, being likely to move toward the higher-temperature mode, as is seen from Figure 3.8 (A3, B3 → C3 ). At any rate, the first type of critical state can be called the Frank-Kamenetskii type, and the second type called the oxygen-deficient type. © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.7 The calculated u at z 1 as a function of u'0. [From Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 247–254, 1993. With permission.]
Stability Criteria and Type Discrimination Assumption is made here that the thermal stability is secured in a powder bed if (1) the lowertemperature mode exists and/or (2) the maximum temperature rise in the higher-temperature mode is smaller than 0.2Ta. The latter number means that about 60°C can be allowed for the temperature rise. As the result of the numerical integration of Equation 3.14 for a variety of the parameters concerned, , G, and n, Figure 3.10 shows the critical dc values thus determined, in which the bottom broken line refers to the Frank-Kamenetskii theory (dc = 3.51). Note that a remarkable difference between dc and the bottom value is due to the pronounced effect of the limited rate of oxygen diffusion on the thermal stability. The critical state of the FrankKamenetskii type lies on the left portion of the curve of a constant G divided by an open circle. Once dc is exceeded, the system is runaway toward the higher-temperature mode and eventually results in smoldering or even explosion if the bed remains uncontrolled. On the other hand, the right portion of the curve indicates the oxygen-deficient type, in which the temperature rises up gradually and continuously from 0.2Ta if d is greater than dc. From the theory, however, the maximum temperature rise never exceeds Ta/. For these reasons, aeration quenching is effective for preventing spontaneous ignition if the bed is of the Frank-Kamenetskii type, but rather dangerous for the oxygen-deficient type. The discrimination can be done by use of values of G, n, and , as shown in Figure 3.10. It is fortunate that the scale of the bed and the frequency factor of reaction are not necessary for the discrimination. © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.8 Temperature distributions for solutions given in Figure 3.7. [From Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 247–254, 1993. With permission.]
Effect of Aeration Quenching on Criticality for Spontaneous Ignition As discussed above, aeration is expected to be effective for the Frank-Kamenetskii type. A quantitative analysis is briefly given in this section for the aeration contributing to the thermal stability of the powder bed and a new risk arising from the aeration will be mentioned.7 The heat balance equation in a one-dimensional plane slab with aeration is written with x being the distance from the bottom of the slab: ⎛ ∂2T ⎞ ⎛ ∂T ⎞ ⎛ ∂T ⎞ ⎛ E ⎞ rC ⎜ ⎟ k ⎜ 2 ⎟ Ga Ca ⎜ ⎟ Qf exp ⎜ ⎝ ∂t ⎠ ⎝ ∂x ⎠ ⎝ RT ⎟⎠ ⎝ ∂x ⎠
(3.15)
where G a is the mass velocity of air through the slab and Ca is the specific heat of air. This equation can be reduced to a dimensionless expression as u 2 u u ⎞ ⎛ u ⎞ ⎛ 2 b ⎜ ⎟ d exp ⎜ ⎝ z ⎠ ⎝ 1 u / G ⎟⎠ t z
(3.16)
wherein t tk/rCL2, z x/L, u (TTa)G/Ta, and b GaCaL/k (the characteristic term for aeration). As the results of dynamic characteristics analysis, the following condition holds for the thermal stability: ⎛ b2 ⎞ (G C L / k ) d p ⎜ ⎟ p2 a a 4 4 ⎝ ⎠
2
2
(3.17)
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FIGURE 3.9 Oxygen concentration distributions for solutions given in Figure 3.7. [From Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 247–254, 1993. With permission.]
The above condition should be consistent with that proposed by Frank-Kamenetskii when b 0 (Ga 0). The critical value calculated is equal to 9.87 (p2) which is three times greater than that calculated with dc 3.51. Taking dm at the maximum temperature Tm, in place of d at Ta, gives 2
⎛ G ⎞ ⎛ Gum ⎞ ⎛ b2 ⎞ 2 dm d ⎜ exp p ⎜⎝ 4 ⎟⎠ ⎜⎝ G u ⎟⎠ ⎝ G um ⎟⎠ m
(3.18)
where um is the maximum dimensionless temperature rise [ i.e., G(Tm−Ta)/Ta]. When uc is the critical temperature rise, um should be replaced by uc in Equation 3.18 to give ⎡ ⎛ Guc ⎞ ⎛ b2 ⎞ ⎤ ⎡ G uc ⎤ dc ⎢p 2 ⎜ ⎟ ⎥ ⎢ ⎥ exp ⎜⎝ G u ⎟⎠ G 4 ⎝ ⎠ ⎦ c ⎦⎣ ⎣ 2
(3.19)
Calculating dc by using uc1.187 of Equation 3.8 when b = 0, we have dc = 3.49 (if G = 25 is assumed), which agrees with 3.51, the criticality constant. To obtain um and uc for various values of G, b and d, Equation 3.16 is numerically integrated at steady state, ut 0 with the boundary conditions that u 0 at z 0, and du/dz 0 at z 1. Figure 3.11 shows the calculated u0 at z = 0. The values of u1 orresponding to u0 = 0 are the proper values to fulfill the steady-state solution of Equation 3.16. Within a limited range of d, there are two values, the lower one of which is stable and the other is unstable. There seems to be a critical value of dc = 215, beyond which no equilibrium solution can exist for this condition (G 20, b 200). © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.10 Critical values of varied with G, n, and . [From Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 247–254, 1993. With permission.]
Obviously the lower value of u1 in Figure 3.11 is equal to um, the maximum temperature rise obtained for a single set of the parameters, G, b, and d. The calculated results are summarized by the following empirical equation as a function of G, b, and d: ⎛ d exp(1.9 / G ⎞ ⎛ 3.8 ⎞ um ln ⎜ 1 exp ⎜ ⎟ ⎝ G ⎟⎠ ⎝ ⎠ b
(3.20)
When um uc, d dc in the above equation, so that dc and uc are solved using Equation 3.18 and Equation 3.20 simultaneously, as shown in Figure 3.12 and Figure 3.13. It can be noted that dc is of the same order of b, which amounts to a few hundred, assuming the aeration superficial velocity of 1 cm/s with a slab a few meters thick. This outstanding difference in dc from 3.51 without aeration suggests an essential improvement in the thermal stability of the powder layer.
Risk Accompanied by Aeration Because of the unstable solution, there is a possibility for spontaneous ignition to occur when some disturbances cause the temperature to exceed the stable solution. For protection, the following restorable temperature distribution should be maintained even if d is smaller than dc: G (T Ta ) uˆ ln (1 az ) ; a 1 exp (uc ) Ta
(3.21)
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FIGURE 3.11 Calculated bottom temperature u0 plotted against u1. [From Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 483–489, 1993. With permission.]
where u is the restorable temperature. uc can be found in Figure 3.13. It is recommended for safety purposes to monitor and regulate the temperature distribution along the axis of the slab in response to any unexpected disturbance, so that the temperature may be kept well below the restorable limit. Finally a remark will be given about the physicochemical properties needed for the calculations above. The order of reaction (oxidation), n, is reported to be from 0.4 to 0.8 for coal and 0.7 for sawdust. G ranges from 10 to 100: 10 for coal, 40 for sawdust and cork, 60 for cellulose, and 50–100 for nitrocompounds. D* can be calculated from 2/3 Do where Do is the diffusivity of oxygen by 1.8 × 105(T/273)1.75 (m2/s). Accordingly, ( kTa /D*c0Q) ranges from 1 to 1.3 for sawdust, 2–2.5 for cork, 2.5–3 for coal, and the highest is 5. The heat evolution Qf(J/m3 s) is plotted against the activation energy E (kJ/mol) in Figure 3.14 for various materials.8
7.3.2 DUST EXPLOSION MECHANISM AND PREVENTION A lot of combustible solid materials are handled in industries, in which extreme care should be taken for fine powder against dust explosion, which is a phenomenon of the ignition of a particle followed by flame propagation between solid particles. The ignitability and the explosibility are represented by measurements for ignition temperature, minimum ignition energy, flame propagation velocity, pressure development in a closed vessel, and so on. Although a number of experiments have been conducted with various equipment, theoretical analysis remains difficult and insufficient, which is due to the extreme complexity of factors involved in the mechanism of dust explosion. Here, a simple dust cloud model will be described, aiming at the theoretical prediction of various phenomena of dust explosion, which is compared with experimental data, leading to a sufficient understanding of the mechanisms. © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.12 Critical d as a function of G and b for aeration quenching. [From Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 483–489, 1993. With permission.]
Dust Cloud Model The uniform dispersion postulated is illustrated in Figure 3.15, where the explosion begins with ignition of the central particle of diameter Dp, and the flame propagates radially with spherical symmetry. If the flat flame propagation in ducts is to be considered, a one-dimensional configuration is appropriate. Particles are distributed on each spherical surface with one particle at the center. Distances between two neighboring surfaces and between the center and the second surface are all L; L is the representative distance, burned or unburned, in the radial direction. Therefore, ⎛ rs ⎞ L⎜ ⎟ ⎝c ⎠
1/ 3
Dp
(3.22)
d
where cd is the dust concentration and rs is the density of the solid. The number of particles on the nth spherical surface, N(n), can be calculated from N ( n ) 24 n 2 48n 26
(3.23)
Flame propagation from burning particles to the next unburned ones is assumed to occur when the unburned particles are heated by convection and radiation from the flame and the temperature of © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.13 Critical u as a function of G and for aeration quenching. [From Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 483–489, 1993. With permission.]
particles reaches the ignition point. Suppose the flame has just propagated to the (n–1)th spherical surface. The flame front radius b is assumed to be b ( n 1) L Rb
(3.24)
where R b is the flame radius inherent to a single burning particle. For liquid droplets, R b is given by Miesse10 as ⎛ 2k ∗ ⎞ ⎤ ⎛ Dp ⎞ ⎡ Rb ⎜ ⎟ ⎢1 1 ⎜ ⎟⎥ ⎝ 4 ⎠⎢ ⎝ Dp ⎠ ⎥ ⎦ ⎣
(3.25)
where k* is an equilibrium constant ranging from 1 to 10 cm for liquids. Similar values of k* are assumed for solids. For calculating the temperature of unburned particles in the nth surface at time t, a dynamic heat balance is taken as 1 ⎛ dTs ⎞ ⎛ p⎞ 3 ⎡ ⎤ pDp2 ⎢h Tg Ts F p f sTf4 ⎥ ⎜⎝ ⎟⎠ Dp rsCps ⎜⎝ ⎟ dt ⎠ 6 2 ⎣ ⎦
(
)
(3.26)
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FIGURE 3.14 Relationship between heat generation rate and the activation energy for some materials. [From Liang, H. and Tanaka, T., Funsai (Micromeritics), 34, 42, 1990.]
where Cps is the specific heat of particles, h is the heat transfer coefficient, T s and T g are the temperatures of particles and gas at a distance nL from the center at time t, respectively, F is the view factor, p and f are the emissivity of the solid and flame, respectively, and σ is the Stefan–Boltzmann constant. Tf is the flame front temperature. Integrating the above equation gives ⎤ ⎡ ⎛ B⎞ Ts exp (At ) ⎢ A ∫10 exp ( At ) Tg ( nL, t )dt ⎜ ⎟ [exp ( At ) 1] + Ts 0 ⎥ ⎝ ⎠ A ⎦ ⎣
(3.27)
where A = 6h/(CpsρsDp) and B 3Fpf T f4/Cps rsDp. Ts0 is Ts at time 0. Solving the thermal diffusion equation with respect to the spherical coordinates, we have the temperature of gas Tg(r,t) using the boundary and initial conditions: Tg = Tf at r = b and Tg = Ti0 = constant at t = 0. ⎛ r b ⎞ ⎛ b⎞ Tg (r , t ) (Tf Ti0 ) ⎜ ⎟ erfc ⎜ Ti 0 ⎝ r⎠ ⎝ 2 kt ⎟⎠
(3.28)
where k is the heat diffusivity. Substituting Tg into Equation 3.27, the temperature of particle can be calculated. Let Dt be the time interval for the flame to propagate from the (n–1)th surface to the nth surface, calculated from t and Ts = ignition temp. using Equation 3.27. Figure 3.16 shows the calculated Dt against n, whereas Dt plotted against cd, the solid concentration, is depicted in Figure 3.17, indicating that Dt is of the order of 10−3 s, from which one can estimate the burning velocity to be less than 1 m/s as well as the flame propagation velocity for deflagration to be less than 100 m/s, all of which roughly agree with our past experience. The latter is calculated based on the gas expansion due to an increase in moles and heat release as a result of particle combustion.9 © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.15 Dust cloud model. [From Nomura, S. and Tanaka. T., Kagaku Kogaku Rombunshu, 5, 47–51, 1979. With permission.]
Assuming that the rate of combustion of a single particle is controlled by oxygen diffusion, it is reported by Yagi and Kunii11 that Dp(t), the particle diameter at time t after ignition at t = 0, is given by Dp ( t ) t 1 t Dp 0
(3.29)
Accordingly, the mass of a particle burned by time t, m(t), is given by ⎧⎪ ⎡ ⎛ t ⎞ ⎤ 3 ⎫⎪ m ( t ) m0 ⎨1 ⎢1 ⎜ ⎟ ⎥ ⎬ ⎝ t⎠⎦ ⎪ ⎩⎪ ⎣ ⎭
(3.30)
where t is the time needed for complete combustion of a particle and is proportional to the square of the initial diameter as t K d 0 Dp20
(3.31)
where Kd0 is called the burning constant, which is about 2000 s/cm2 for solids on the average, according to Essenhigh and Fells.12 When the flame propagates to the nth spherical surface at time t after ignition at the center at t = 0, the total mass of particles burned, M(t), is obtained as the sum of the mass of particles burned in each spherical surface in Figure 3.15, namely, M ( t ) m0
n
i 1
3 ⎡ ⎛ t (i 1) t ⎞ ⎤ ⎢ N (i ) 1 ⎜ 1 ⎟⎠ ⎥ t ⎢⎣ ⎝ ⎥⎦
(3.32)
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FIGURE 3.16 Calculated time interval variation with the number of particles for flame propagation. [From Nomura, S. and Tanaka, T., Kagaku Kogaku Rombunshu, 5, 47–51, 1979. With permission.]
Substituting Equation 3.23 into Equation 3.32 and approximating gives M (t )
M (t ) M0
⎛ 8m ⎞ ⎛ t ⎞ ⎜ 0 ⎟ ⎜ ⎟ ⎝ M 0 ⎠ ⎝ t ⎠
3
(3.33)
where M0 is the maximum mass of particles burned in a closed vessel, which is proportional to the vessel volume V0 as M0 aV0
(3.34)
where a is a proportionality constant depending on the dust and oxygen concentrations. These terms should be used later. © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.17 Calculated time interval between the neighboring particles for flame propagation varied with dust concentration. [From Nomura, S. and Tanaka, T., Kagaku Kogaku Rombunshu, 5, 47–51, 1979. With permission.]
Upper and Lower Explosible Limit Concentrations In order for a flame to propagate throughout a dust cloud, the dust concentration must lie within an explosible region bounded by the upper and lower limits. These limits are defined respectively as the minimum and maximum dust concentrations to sustain flame propagation. For safety, combustible powders should be handled in a region outside the two limits. The lower limit can be considered to be the concentration calculable from Equation 3.22 by substituting the maximal distance that is obtained from the condition that the temperature of particle 2 © 2006 by Taylor & Francis Group, LLC
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reaches its ignition point when particle 1 is just burned out. Equation 3.27 through Equation 3.31 are simultaneously solved by the use of b Rb and r L to give the lower-limit explosible concentration as shown in Figure 3.18. Calculations are carried out for several k* values ranging from 1 to 10, and the experimental data appear reasonable against the particle size, so long as the appropriate value of k* can be specified. Near the upper limit, however, oxygen is short, and the flame propagation is strongly affected by the oxygen diffusion. For simplicity of calculation, a situation is assumed where the total oxygen present in a dust cloud is equally allocated to all of the particles in the system. Accordingly, the flame propagation from a burning particle to the neighboring unburned particles should be made before the allocated oxygen to the first particle is entirely dissipated. The critical condition is mathematically given by the above derivation of the lower limit, taking the role of oxygen into consideration. The time t1 needed for exhausting the allocated oxygen is given by13
⎡ ⎛ cg M p ⎞ ⎤ t1 1 ⎢1 ⎜ ⎟⎥ t ⎢⎣ ⎝ cd N1 ⎠ ⎥⎦
1/ 3
(3.35)
where cg is the oxygen concentration in the air, Mp is the molecular weight of the particle, and N1 is the number of moles of oxygen by combustion of 1 mole of the particle. Figure 3.19 compared the calculation (solid line) with experiment (dotted line) for some materials. General agreement is obtained, in which we see that the concentration of oxygen lower than 10% would sustain no flame propagation, suggesting the basis for inerting to prevent explosion.
Ignition Temperature and Minimum Ignition Energy As measures of explosion sensitivity, both ignition temperature and ignition energy are determined experimentally by use of a specific furnace and an apparatus preparing the ignition spark electrode. Cassel and Liebman15 discussed theoretically the ignition temperature using the rates of heat generation on a particle surface and of heat loss, as denoted by H and U, respectively, and the ignition temperature defined as the minimum air temperature for initiating combustion is obtained by solving simultaneously the two relationships H = U and dH/dTs=dU/dTs, where Ts is the solid temperature, and they neglected the effect of radiation to give rise to disagreement with experimental data. Analysis by Mitsui and Tanaka,13 taking the radiant heat from particles to the wall of apparatus in addition to the convective heat into account, is carried out by the same procedure to give Figure 3.20, in which the ignition temperatures are plotted against the particle diameter for some metals and plastics. The different trends of the curves arise from the different emissivities of the particles. With regard to the minimum ignition energy, Kalkert and Schecker16 proposed the following theoretical equation, ra being the air density:
W ( 4p k )
3/2
⎡⎛ ln 2 ⎞ ⎛ rsCps ⎞ ⎤ ra Ca ⎢⎜ ⎟⎜ ⎟⎥ ⎢⎣⎝ 12 ⎠ ⎝ k ⎠ ⎥⎦
3/2
Tf Dp3
(3.36)
where W is the minimum ignition energy (J) and Tf is the flame temperature. Proportionality between the measured W and the cube of particle size Dp is found for polyethylene to prove the above equation. © 2006 by Taylor & Francis Group, LLC
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FIGURE 3.18 Comparison between calculated and experimental data of minimum explosible limit for various materials. [From Nomura, S. and Tanaka. T., Kagaku Kogaku Rombunshu, 5, 47–51, 1979. With permission.]
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FIGURE 3.19 (a) Comparison of the calculated and experimental upper and lower explosible limits in relation to oxygen concentration for aluminum. [From Nomura, S., Torimoto, M., and Tanaka, T., J. Soc. Chem. Eng. Jpn., 45, 327, 1981. With permission.] (b) Comparison of the calculated and experimental upper and lower explosible limits in relation to oxygen concentration for terephthaic acid. [From Nomura, S., Torimoto, M., and Tanaka, T., J. Soc. Chem. Eng. Jpn., 45, 327, 1981. With permission.]
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FIGURE 3.20 Calculated and experimental ignition temperatures plotted against particle diameter for some materials. [From Mitsui, R. and Tanaka, T., IEC Process Des. Dev., 12, 384, 1973.]
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Pressure Development on Closed and Vented Vessels One of the most important explosion characteristics is the rate of pressure rise in a closed vessel. Using spherical vessels, a scale-up relationship between the maximum rate of pressure rise (dP/ dt)max and the vessel volume Vo, which is called the “cubic law,” was obtained experimentally: ⎛ dP ⎞ V01 / 3 ⎜ ⎟ K st ⎝ dt ⎠ max
(3.37)
where Kst is a constant. However, no theoretical background is given to the above empirical equation about whether Kst is a universal constant or dependent on some experimental variables, without which wide applicability cannot be expected. Adiabatic change is assumed in a closed vessel of volume V0. After ignition at the vessel center, the reduced pressure P ( P / P0 ) starts to rise up due to the gas mole increase and the heat released by combustion. The rate of pressure rise at time t is
⎛ d M (t ) ⎞ ⎛ dP ⎞ 1(1 / g ) 1/ g P m 1 ⎜ ⎟ ⎜ dt ⎟ g P ⎝ ⎠ ⎝ dt ⎠
)
(
(3.38)
where Pm Pmax / P0 with Pmax RTg0 nc0 /V0, wherein Tg0 is the maximum gas temperature, nc0 is the resultant total number of gas moles, and g is the ratio of the specific heats. This equation can be integrated to give
(
)
P ⎡⎣ Pm1 / g 1 M ( t ) 1⎤⎦
g
(3.39)
The relationship between P and t for spherical vessels calculated from Equation 3.39 is shown in Figure 3.21 by using Equation 3.33. The maximum rate of pressure rise is obtained at the point (Pmax,t0) where t0 is the time when the total mass of particles burned M(t0) becomes unity. From Equation 3.33 ⎛M ⎞ t0 ⎜ 0 ⎟ ⎝ 8m0 ⎠
1/ 3
t
(3.40)
and d M ( t ) ⎛ 24 m0 ⎞ ⎛ t ⎞ ⎜ ⎜ ⎟ dt ⎝ M 0 Δt ⎟⎠ ⎝ t ⎠
2
(3.41)
Combining Equation 3.34 and Equation 3.38 through Equation 3.41, we have the following relation: ⎛ gDp 0 ⎞ ⎛ 36prs ⎞ ⎛ dP ⎞ V01/3 ⎜ ⎟ ⎜ ⎝ dt ⎠ max ⎝ t ⎟⎠ ⎜⎝ a ⎟⎠
1/ 3
(
1 / g
Pm 1 Pm
)
(3.42)
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FIGURE 3.21 Calculated pressure development in the lapse of time. [From Nomura, S. and Tanaka. T., Kagaku Kogaku Rombunshu, 5, 47–51, 1979. With permission.]
This equation proves the validity of the empirical cubic law given by Equation 3.37, where Kst is obviously represented by the right-hand side of Equation 3.42 and is constant for a fixed dust concentration and particle size. Figure 3.22 compares the theory with the experimental data of Ishihama and Enomoto17 for starch dust. The predicted maximum rate of pressure rise is somewhat greater than the experiment, as the theory is based on the assumption of adiabatic change. The most common and economical approach to explosion protection is to incorporate relief vents. The area of the vent for a particular vessel must be large enough to discharge the combustion products quickly to prevent pressures exceeding the strength of the plant vessel. Suppose the dust explosion takes place in a vented vessel with volume V0 and vent area S. The macroscopic energy balance is applied to the system and the following approximate equation is given for the rate of pressure rise: ⎛ d M ( t ) ⎞ ⎛ g ⎞ 1(1-g ) ⎡ 1 / g ⎛ wb ⎞ ⎛ wn ⎞ ⎤ 1(1 / g ) 1/ g dP gP P m 1 ⎜ ⎢Pm ⎜ ⎟ ⎜ ⎟ ⎥ ⎟ ⎜ ⎟ P dt ⎝ rm ⎠ ⎝ r0 ⎠ ⎦ ⎝ dt ⎠ ⎝ V0 ⎠ ⎣
(
)
(3.43)
The first term on the right-hand side is the rate of pressure rise due to the combustion with the same form as that of Equation 3.38, and the second term is the reduction of the developed pressure by discharge of burned and unburned gases through the vent. w denotes the mass flow rate, and the sub© 2006 by Taylor & Francis Group, LLC
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FIGURE 3.22 Calculated maximum rate of pressure rise compared with experimental data by Ishihama and Enomot. [From Nomura, S. and Tanaka. T., Kagaku Kogaku Rombunshu, 5, 47–51, 1979. With permission.]
scripts b and u denote “burned and unburned.” When the pressure becomes maximum, as denoted by Pv, dP / dt 0 . The first term could be replaced by the maximum rate of pressure rise in a closed vessel and the second term is approximated using the coefficient of discharge and the vented area S as follows: ⎛ dP ⎞ ⎛ S ⎞⎛P ⎞ 0⎜ ⎜ ⎟ ⎜ 0 ⎟ ⎟ ⎝ dt ⎠ max ⎝ V0 ⎠ ⎝ r0 ⎠
1/ 2
( )
f (g ) g P v
(3.44)
where f(g) and g( Pv ) are given by Nomura and Tanaka18 for P 1/0.53 ⎛ 2g ⎞ f (g ) g ⎜ ⎝ g 1⎟⎠
1/2
( )
1(1 / g )
, g Pv Pv
(P
1(1 / g ) v
)
1
1/ 2
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FIGURE 3.23 Comparison of calculated s/v 02/3 with experimental s/v 02/3. [From Nomura, S. and Tanaka, T., J. Chem. Eng. Jpn., 13, 309–313, 1980. With permission.]
⎡ ⎛ 2 ⎞ (g1) / (g1) ⎤ ⎥ f (g ) g ⎢g ⎜ ⎟ ⎥⎦ ⎢⎣ ⎝ g 1⎠
1/ 2
( )
, g Pv Pv
(3g1) / 2g
(3.45)
Substituting the cubic law in Equation 3.37 leads to a new concept of a vent ratio:
S V0 2 / 3
1
⎤ ⎡⎛ P ⎞ ⎛ P ⎞ 1 / 2 ⎢ ⎜ 0 ⎟ ⎜ 0 ⎟ f (g ) g P v ⎥ ⎥⎦ ⎢⎣⎝ K st ⎠ ⎝ r0 ⎠
( )
(3.46)
– The new dimensionless vent ratio, S/V02/3, is estimated from Kst for a given material Pv and the predetermined maximum pressure related directly to the strength of the plant. It seems dubious that the conventional vent ratio, S/V0 (i.e., the dimensional parameter) provides a sound basis for designing a vent area for various scales of equipment. As to the vented explosions, many experiments were car– ried out with various scales of test apparatus and various materials.19–21 A set of reported Pv and Kst © 2006 by Taylor & Francis Group, LLC
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values are substituted into Equation 3.46 to calculate the theoretical S/V02/3 which is plotted against the experimental one as depicted in Figure 3.23. The theoretical data, being a little above the corresponding straight line, are favorable for the purpose of the safety design. For all of the calculations throughout this section, the SI unit system should be used, although most equations are dimensionally sound. For instance, P0 and Pmax are in Pa, V0 in m3, S in m2, Kst in Pa/m s, and so on.
Measurements of Dust Explosion Characteristics Explosibility tests have been developed in many countries, particularly in Europe and the United States. Although a number of similarities are present in the tests, a current lack of standardization has resulted in the variations in the test procedures in each country.22 In Japan, the Association of Powder Process Industry and Engineering standardized a test method for dust explosibility in 1991,23 developing two types, which are mainly used for determining the explosibility of a test powder and the lower explosible limit concentration. The first type is called the Vertical Tube Apparatus, which is characterized by the dust cloud formed by means of blown-up compressed air. The second type, called the Tapping Sieve Type Apparatus, features the sieve tapping mechanism for obtaining more homogeneous dust dispersion. As to the determination of the severity of dust
FIGURE 3.24 Estimating Kst from other characteristics in comparison with experimental data for a number of materials. [From Tanaka, J., Soc. Powder Technol. Jpn., 32, 237–239, 1995. With permission.] © 2006 by Taylor & Francis Group, LLC
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explosion, investigation is being made for the standardization with a 1-m3 cylindrical apparatus having a length-to-diameter ratio close to unity. The cubic law described in the preceding section turns out to hold for the spherical volume greater than 20-l. Therefore, a 20-l sphere test apparatus has also been investigated for use as an alternative for determining the maximum pressure as well as the maximum rate of pressure rise. The Research Institute of Industrial Safety, Japan,24 published a guide to test a method for explosion pressure and the rate of pressure rise for combustible dusts, in which are recorded a number of experimental data for wood, coal, agricultural products, plastics, chemicals, metals, and so on. The documented parameters are particle size, lower explosible limit, maximum pressure, Kst value, and others. These data can be used for the confirmation of the theoretical derivation, Equation 3.42, as indicated in Figure 3.24, in which Dp0, Pm and rs are utilized along with cdm (minimum explosible limit) in place of a ( effective combustible dust concentration defined in Equation 3.34). Dt is a function of D p0 as shown in Figure 3.16. Except for metals and alloys, the experimental Kst appears to lie along a straight line corresponding to the calculated Kst,25 because the densities of most materials range from 0.5 to 1.5. On the other hand, the rs of metal and alloy ranges from 2 to 8 (g/cm3), so that the data are placed a little above, in proportion to (rs)1/3 according to Equation 3.42. The unit of Kst in Figure 3.24 is expressed conventionally by bar m/s, as indicated in the original table.
Explosion Suppression and Protection Reduction of oxygen concentration is significant for dust explosion prevention as discussed in relation to the explosible limits in Equation 3.36. This is termed “inerting,” and the oxygen is frequently replaced by nitrogen, carbon dioxide, and flue gas. Inerting is also made by adding inert dusts such as CaCO3, CaSO4, NaHCO3, and so forth, which are called diluents, reducing the explosibility of the combustible dust. Relief venting is also described theoretically in Equation 3.46. The vent should be located in the vicinity of an expected ignition position and various types of diaphragms for vent cover made of flexible plastics, metal foil, and so on are available. An automatic explosion suppression system should also be considered, which consists of a detector responding to the pressure developed to transfer the signal to a controller unit, which actuates the extinguisher supply to the fire to suppress explosion.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
Liang, H., Tanaka, T., and Nakajima, Y., J. Soc. Powder Technol. Jpn., 23:326, 1986. Leuschke, G., Int. Symp. Loss Prevent. Safety, 2, 6471980. Leuschke, G., Inst. Chem. Eng. Symp. Ser., 68, 1981. Frank-Kamenetskii, D. A., in Diffusion and Heat Transfer in Chemical Kinetics, 2nd Ed., Transl. Appleton, Plenum Press, New York, 1969, p. 388. Liang, H. and Tanaka, T., Kagaku Kogaku Ronbunshu, 13, 63–70, 1987. Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 247–254, 1993. Nakajima, Y., Furusawa, S., Liang H., and Tanaka, T., J. Soc. Powder Technol. Jpn., 30, 483–489, 1993. Liang, H. and Tanaka, T., Funsai (Micromeritics), 34, 42, 1990. Nomura, S. and Tanaka. T., Kagaku Kogaku Rombunshu, 5, 47–51, 1979. Miesse, C. C., in Sixth Symposium (International) on Combustion, Combustion Institute, Pittsburgh, 1957, p. 732. Yagi, S. and Kunii, D., in Fifth Symposium (International) on Combustion, Combustion Institute, Pittsburgh, 1955, p. 231. Essenhigh, R. H. and Fells, H., Discuss. Faraday Soc., 30, 208, 1960. Nomura, S. Torimoto, M., and Tanaka, T., J. Soc. Chem. Eng. Jpn., 45, 327, 1981. Mitsui, R. and Tanaka, T., IEC Process Des. Dev., 12, 384, 1973. Cassel, H. M. and Liebman, I., Combust. Flame, 3, 467, 1959. Kalkert, N. and Schecker, H. G., Chem. Ing. Tech., 51, 1248, 1979.
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17. Ishihama, W. and Enomoto, H., in Fifteenth Symposium (International) on Combustion, Combustion Institute, Pittsburgh, 1974, 479. 18. Nomura, S. and Tanaka, T., J. Chem. Eng. Jpn., 13, 309–313, 1980. 19. Hartmann, I. and Nagy, J., Ind. Eng. Chem., 49, 1743, 1957. 20. Donat, C., CEP Tech. Manual Loss Prevent., 11, 87, 1977. 21. Schwab, R. S. and Othmer, D. F., Chem. Process Eng., (April), 165, 1964. 22. Firld, P., Dust Explosion, Elsevier, Amsterdam, 1982, p. 46. 23. Association of Powder Process Industry and Engineering, Test Method for Dust Explosibility, ASP 002-1991, 1991, p. 1. 24 Research Institute of Industrial Safety, Japan, Guide to Test Method for Explosion Pressure and Rate of Pressure Rise for Combustible Dusts, RIIS-TR-94-1, 1994, p. 24. 25. Tanaka, J., Soc. Powder Technol. Jpn., 32, 237–239, 1995.
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