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Powder Sampling and Particle Size Determination

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Powder Sampling and Particle Size Determination By

Terence Allen Formerly Senior Consultant E.I. DuPont de Nemours and Company Wilmington, Delaware, USA

2003

FI.SFVIFR AMSTERDAM - BOSTON - HEIDELBERG - LONDON - NEW YORK - OXFORD PARIS - SAN DIEGO - SAN FRANCISCO - SINGAPORE - SYDNEY - TOKYO

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© 2003 Elsevier B.V. All rights reserved. This work is protected under copyright by Elsevier B.V., and the following terms and conditions apply to its use: Photocopying Single photocopies of single chapters may be made for personal use as allowed by national copyright laws. Permission of the Publisher and payment of a fee is required for all other photocopying, including multiple or systematic copying, copying for advertising or promotional purposes, resale, and all forms of document delivery. Special rates are available for educational institutions that wish to make photocopies for non-profit educational classroom use. Permissions may be sought directly from Elsevier's Rights Department in Oxford, UK: phone (+44) 1865 843830, fax (+44) 1865 853333, e-mail: [email protected]. Requests may also be completed on-line via the Elsevier homepage (http://www.elsevier.com/locate/permissions). In the USA, users may clear permissions and make payments through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA; phone: (+1) (978) 7508400, fax: (+1) (978) 7504744, and in the UK through the Copyright Licensing Agency Rapid Clearance Service (CLARCS), 90 Tottenham Court Road, London WIP OLP, UK; phone: (+44) 20 7631 5555; fax: (+44) 20 7631 5500. Other countries may have a local reprographic rights agency for payments. Derivative Works Tables of contents may be reproduced for internal circulation, but permission of the Publisher is required for external resale or distribution of such material. Permission of the Publisher is required for all other derivative works, including compilations and translations. Electronic Storage or Usage Permission of the Publisher is required to store or use electronically any material contained in this work, including any chapter or part of a chapter. Except as outlined above, no part of this work may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without prior written permission of the Publisher. Address permissions requests to: Elsevier's Rights Department, at the fax and e-mail addresses noted above. Notice No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent verification of diagnoses and drug dosages should be made.

1^^ edition 2003

Library of Congress Cataloging in Publication Data A catalog record is available from the Library of Congress. British Library Cataloguing in Publication Data A catalogue record is available from the British Library.

ISBN:

0-444-51564-X

© The paper used in this publication meets the requirements of ANSI/NISO Z39.48-1992 (Permanence of Paper). Printed in The Netherlands.

Contents Acknowledgements

x vi i

Preface

xix

Editor's foreword

xxi

1 1.1 1.2 1.3

1.4

1.5

1.6 1.7 1.8 1.9 1.10 1.11 1.12 1.13

Powder sampling Introduction Sample selection Sampling stored material 1.3.1 Sampling stored non-flowing material 1.3.2 Sampling from heaps 1.3.3 Sampling stored bulk free-flowing powders 1.3.4 Sampling from sacks and drums 1.3.5 Sampling from trucks and railcars Sampling flowing streams 1.4.1 Sampling from a conveyor belt 1.4.2 Point samplers 1.4.3 Sampling from falling streams 1.4.4 Stream sampling ladles 1.4.5 Traversing cutters 1.4.6 Sampling dusty material 1.4.7 In-line sampling Sample reduction 1.5.1 Scoop sampling 1.5.2 Cone and quartering 1.5.3 Table sampling 1.5.4 Chute splitting 1.5.5 Spinning rifflers 1.5.6 Commercial rotary sample dividers 1.5.7 Miscellaneous sampling devices Slurry sampling Reduction of laboratory sample to measurement sample Number of samples required Theoretical statistical errors on a number basis Practical statistical errors on a number basis Theoretical statistical errors on a weight basis Practical statistical errors on a weight basis Experimental tests of sampling techniques

1 2 6 7 9 10 11 12 12 13 14 14 18 19 19 23 24 25 26 27 27 28 30 30 35 36 38 42 45 46 46 49

vi

Contents

1.14

Weight of sample required 1.14.1 Gross sample 1.14.2 Sampling by increments

2 2.1 2.2 2.3 2.4 2.5

Data presentation and interpretation Introduction Particle size Average diameters Particle dispersion Particle shape 2.5.1 Shape coefficients 2.5.2 Shape factors 2.5.3 Applications of shape factors and shape coefficients 2.5.4 Shape indices 2.5.5 Shape regeneration 2.5.6 Fractal dimensions characterization of textured surfaces 2.5.7 Other methods of shape analysis 2.5.8 Sorting by shape Determination of specific surface from size distribution data 2.6.1 Determination of specific surface from a number count 2.6.2 Determination of specific surface from a surface count 2.6.3 Determination of specific surface from a volume (mass) count Tabular presentation of particle size distribution Graphical presentation of size distribution data 2.8.1 Presentation on linear graph paper Standard forms of distribution fiinctions Arithmetic normal distribution 2.10.1 Manipulation of the normal equation The log-normal distribution 2.11.1 Relationship between number mean sizes for a log-normal distribution 2.11.2 Derived mean sizes 2.11.3 Transformation between log-normal distributions 2.11.4 Relationship between median and mode of a log-normal equation 2.11.5 An improved equation and graph paper for log-normal evaluations 2.11.6 Application Johnson's 5*^ distribution Rosin-Rammler-Bennet-Sperling formula Other distribution laws 2.14.1 Simplification of two parameter equations. 2.14.2 Comments The law of compensating errors

2.6

2.7 2.8 2.9 2.10 2.11

2.12 2.13 2.14 2.15

50 50 51

56 57 63 68 69 74 76 78 82 83 84 88 88 89 89 90 90 93 95 95 96 96 99 100 102 105 106 107 108 109 109 111 112 112 114 117

Contents

2.16

2.17

2.18 3 3.1 3.2 3.3

3.4 3.5 3.6 3.7

3.8 3.9 3.10 3.11

3.12 3.13

Evaluation of nonlinear distributions on log-normal paper 2.16.1 Bimodal intersecting distributions. 2.16.2 Bimodal non-intersecting distributions. 2.16.3 Other distributions 2.16.4 Applications of log-normal plots 2.16.5 Curve fitting 2.16.6 Data interpretation Alternative notations for frequency distribution 2.17.1 Notation 2.17.2 Moment of a distribution 2.17.3 Transformation from qjix) to q^{x) 2.17.4 Relation between moments 2.17.5 Means of distributions 2.17.6 Standard deviations 2.17.7 Coefficient of variation 2.17.8 Applications 2.17.9 Transformation of abscissae Phi-notation Particle size analysis by image analysis Introduction Standards Optical microscopy 3.3.1 Upper size limit for optical microscopy 3.3.2 Lower size limit for optical microscopy Sample preparation Measurement of plane sections through packed beds Particle size Calibration 3.7.1 Linear eyepiece graticules 3.7.2 Globe and circle graticules Training of operators Experimental techniques Determination of particle size distribution by number Conditions governing a weight size determination 3.11.1 Illustrative example of the calculation of a size distribution by weight Semi -automatic aids to microscopy Automatic aids to microscopy 3.13.1 Beckman Coulter RapidVUE 3.13.2 Micromeretics OptiSizer PSDA™ 5400 3.13.3 Oxford VisiSizer 3.13.4 Retsch Camsizer 3.13.5 Malvern Sysmex Flow Particle Image Analyzer 3.13.6 Sci-Tec PartAn - video Image Analyser

vii

117 117 122 122 123 123 125 125 125 126 126 127 128 129 130 130 132 136

142 144 145 145 146 147 151 151 153 154 154 156 157 15 8 160 162 164 167 167 167 168 168 168 169

via Contents

3.14

3.15 3.16

3.17 3.18 3.19 3.20

Quantitative image analysis 3.141 Calibration of image analyzers 3.14.2 Experimental procedures 3.14.3 Commercial quantitative image analysis systems 3.14.4 Confocal laser-scanning microscopy 3.14.5 On-line microscopy 3.14.6 Flatbed scanners 3.14.7 Dark field microscopy 3.14.8 Phase contrast microscopy 3.14.9 Polarized light microscopy (PLM) 3.14.10 Dipix 1440F power scope imaging microscope 3.14.11 Transmission wide field phase contrast microscopy Electron microscopy Transmission electron microscopy (TEM) 3.16.1 Specimen preparation for TEM 3.16.2 Replica and shadowing techniques 3.16.3 Chemical analysis Scanning electron microscopy Other scanning electron microscopy techniques Errors involved in converting a number to a volume count Evaluation of procedures

4 Particle size analysis by sieving 4.1 Introduction 4.2 Standard sieves 4.3 Tolerances for standard sieves 4.4 Woven-wire and punched plate sieves Electroformed micromesh sieves 4.5 4.6 Mathematical analysis of the sieving process 4.7 Calibration of sieves Sieving errors 4.8 4.9 Methods of sieving 4.10 Amount of sample required 4.11 Hand sieving 4.12 Machine sieving 4.13 Wet sieving 4.13.1 Manual 4.13.2 Wet sieving by machine 4.14 Air-jet sieving 4.15 The Sonic Sifter 4.16 The Seishin Robot Sifter 4.17 Automatic systems 4.17.1 The Rotex Gradex 2000 particle size analyzer 4.17.2 Labcon automatic sieve system 4.17.3 Gilson Compu-Sieve« analysis system

169 170 170 180 183 184 185 185 186 186 187 187 187 188 188 192 192 193 196 199 200

208 210 212 213 214 218 221 224 227 229 230 231 234 234 235 237 239 239 240 240 241 241

Contents

4.18 4.19 4.20 4.21 4.22 4.23 4.24

Ultrasonic sieving The sieve cascadograph Felvation Self organized sieves (SORSl) Shape separation Correlation with light scattering data Conclusions

5 5.1 5.2 5.3 5.4

Fluid classification Introduction Assessment of classifier efficiency Systems Counter-flow equilibrium classifiers in a gravitational field elutriators Theory for elutriators Water elutriators Air elutriators Counter-flow centrifugal classifiers; Zig-zag gravitational classifiers Zig-zag centrifugal classifiers The Warmain Cyclosizer Cross-flow gravitational classification 5.12.1 The Humboldt particle size analyzer TDS Cross-flow centrifugal classifiers 5.13.1 Analysette9 5.13.2 The Donaldson Acucut classifier Cross-flow elbow classifier Micromeretics classifier; Fractionation methods for particle size measurement Hydrodynamic chromatography Capillary hydrodynamic fractionation ; Capillary zone electrophoresis Size exclusion chromatography Field flow fractionation 5.21. r Sedimentation field flow fractionation (SFFF) 5.21.2 Centrifugal field flow fractionation 5.21.3 Time-delayed exponential SFFF 5.21.4 Thermal field flow fracfionation 5.21.5 Magnefic field flow fractionation 5.21.6 Flow field flow fractionation 5.21.7 Steric field flow fractionation 5.21.8 Multi-angle light scattering (MALS) The Matec electro-acoustic system EAS-8000 Continuous split fractionation Classification by decantafion;

5.5 5.6 5.7 5.8 5.9 5.10 5.11 5.12 5.13

5.14 5.15 5.16 5.17 5.18 5.19 5.20 5.21

5.22 5.23 5.24

ix

241 241 243 243 244 245 245

251 251 260 261 262 264 265 266 267 267 268 268 268 269 269 269 270 270 271 272 275 276 276 277 278 279 279 282 282 282 284 284 285 285 287

Contents

Interaction between particles and fluids 6.1 Introduction 6.2 Settling of a single homogeneous sphere under a gravitational force 6.2.1 Relationship between settling velocity and particle size 6.2.2. Calculation of particle size from settling velocity in the laminar flow region 6.3 Size limits for gravity sedimentation 6.3.1 Upper size limit 6.3.2 Lower size limit 6.4 Time for terminal velocity to be attained 6.5 Errors due to the finite extent of the fluid (wall effects) 6.6 Errors due todiscontinuity of the fluid 6.7 Viscosity of a suspension 6.8 Non-rigid spheres 6.9 Non-spherical particles 6.9.1 Stokes' region 6.9.2 Relationship between fiber diameter and Stokes diameter 6.9.3 Transition region 6.10 Relationship between drag coefficient and Reynolds number in the transition region 6.11 The turbulent flow region 6.12 Concentration effects 6.13 Hindered settling 6.13.1 Low concentration effects 6.13.2 High concentration effects 6.14 Electro-viscosity 6.15 Dispersion of powders 6.15.1 Dry powder dispersion 6.15.2 The use of glidants to improve flowability of dry powders 6.15.3 Wet powder dispersion 6.15.4 Role of dispersing agents 6.15.5 Wetting a powder 6.15.6 Determination of contact angle (0) 6.15.7 Deagglomerating wetted clumps 6.15.8 Suspension stability * 6.15.9 Tests of dispersion quality 6.16 Powder density 6.17 Liquid viscosity 6.18 Standard powders 6.19 National Standards 7 7.1 7.2 7.3

295 295 297 297 299 300 301 302 306 308 309 311 312 312 312 317 319 322 325 326 332 333 334 335 336 336 337 338 338 339 340 342 343 344 347 350 350 352

Gravitational sedimentation methods of particle size determination Introduction 359 Resolution of sedimenting suspensions 362 Concentration changes in a suspension settling under gravity 364

Contents

1.4

Homogeneous incremental gravitational sedimentation 7.4.1 The pipette method of Andreasen 7.5 Theory for the gravity photosedimentation technique 7.5.1 The Beer Lambert law 7.5.2 The extinction coefficient 7.5.3 Turbidity measurements (Turbidimetry) 7.5.4 The photosedimentation technique 7.5.5 Commercial photosedimentometers 7.5.6 Sedimentation image analysis 7.5.7 Transmission fluctuation spectrometry 7.6 Theory for concentration determination with the x-ray gravitational sedimentation technique 7.6.1 X-ray sedimentation 7.7 Relationship between density gradient and concentration 7.8 Hydrometers and divers 7.8.1 Introduction 7.8.2 Theory 7.8.3 Depth of immersion 7.8.4 Experimental procedure 7.8.5 Divers 7.9 Homogeneous cumulative gravitational sedimentation 7.9.1 Introduction 7.9.2 Theory 7.9.3 Sedimentation balances 7.9.4 Sedimentation columns 7.10 Line-start incremental gravitational sedimentation 7.10.1 Photosedimentation 7.11 Line-start cumulative gravitational sedimentation 7.11.1 Introduction 7.11.2 Methods 8 8.1 8.2 8.3 8.4

Centrifugal sedimentation methods of particle size determination Introduction Stokes' equation for centrifugal sedimentation 8.2.1 General theory Homogeneous, incremental, centrifugal sedimentation 8.3.1 General theory Variable time method (r and S constant, t variable) 8.4.1 General theory 8.4.2 The Simcar pipette disc centrifuge (r constant, S assumed constant, t variable) 8.4.3 Worked example 8.4.4 The Ladal x-ray disc centrifuge(r constant, S constant, t variable) 8.4.5 Discussion of the Kamack equation

xi

365 365 366 366 369 370 370 372 373 374 374 375 378 379 379 380 381 383 384 384 384 384 386 387 387 387 388 388 388

392 394 394 395 395 397 397 403 404 406 406

xii Contents

8.5

8.6.

8.7

8.8

8.9 8.10 8.11 8.12

8.13 8.14 8.15 8.16 8.17 8.18 8.19 9 9.1 9.2

Variable time and height method {S constant, both r and t vary) 8.5.1 Stokes diameter determination 8.5.2 Mass frequency undersize determination 8.5.3 DuPont/Brookhaven scanning x-ray disc centrifugal sedimentometer 8.5.4 Worked example Variable inner radius (Both S and / vary, r remains constant) 8.6.1 Stokes diameter determination 8.6.2 Ladal pipette disc centrifuge 8.6.3 Worked example 8.6.4 Mass frequency undersize determination Photocentrifuges 8.7.1 Introduction 8.7.2 Disc photocentrifuges. 8.7.3 Homogeneous mode Line-start incremental centrifugal sedimentation 8.8.1 Line-start, incremental centrifugal technique 8.8.2 Discussion of line-start theory 8.8.3 BI-DCP disc (photo)centrifuge particle size analyzer Cuvette photocentrifuges Homogeneous, cumulative, centrifugal sedimentation 8.10.1 General theory Variable time method (variation of P with t) Sedimentation distance small compared with distance from centrifuge axis 8.12.1 Hosokawa Mikropul Sedimentputer 8.12.2 Alpine long-arm centrifuge Variable inner radius (variation of P with S) 8.13.1 Alternative theory (variation of P with S) Variable outer radius (variation of P with R) Line-start cumulative centrifugal sedimentation 8.15.1 MSA analyzer Particle size analysis using non-invasive dielectric sensors Supercentriftige Ultracentriftige Conclusions Stream scanning methods of particle size measurement Introduction The electrical sensing zone method (the Coulter principle) 9.2.1 Introduction 9.2.2 Operating principle 9.2.3 Theory for the electrical sensing zone method 9.2.4 Effect of particle shape and orientation 9.2.5 Pulse shape

406 406 407 407 408 410 410 412 413 415 417 417 418 419 422 422 425 428 429 431 431 433 434 434 435 435 437 438 439 439 439 440 442 442

447 449 449 450 452 455 457

Contents xiii

9.2.6 9.2.7

9.3 9.4

9.5

9.6

9.7

9.8 9.9

Effect of coincidence Multiple aperture method for powders having a wide size range 9.2.8 Calibration 9.2.9 Carrying out a mass balance 9.2.10 Oversize counts on a mass basis using the Coulter Counter 9.2.11 Apparatus 9.2.12 Limitations of the method Fiber length analysis Optical particle counters 9.4.1 Light blockage 9.4.2 Optical disdrometer 9.4.3 Light scattering Commercial instruments 9.5.1 Aerometrics 9.5.2 Canty Vision Climet 9.5.3 Contamination Control Systems 9.5.4 9.5.5 Danfoss VisionSensor 9.5.6 Faley Status 9.5.7 Flowvision 9.5.8 Galai 9.5.9 Kane May Kowa 9.5.10 9.5.11 Kratel Malvern 9.5.12 Pacific Scientific Hiac/Royco, Met One 9.5.13 9.5.14 Particle Measuring Systems 9.5.15 Partikel Messetechnik Particle Sizing Systems 9.5.16 9.5.17 Polytec 9.5.18 Rion Spectrex 9.5.19 Dwell time Brinkmann 201 analyzer 9.6.1. Focused Beam Reflectance Measurement (FBRM) 9.6.2 Lasentec Messetechnik Optical Reflectance Method (ORM) 9.6.3 9.6.4 Procedyne Aerodynamic time-of-flight measurement 9.7.1 Thermo Systems Incorporated 9.7.2 Ancillary equipment Laser Doppler velocimetry (LDV) Laser phase Doppler principle

459 460 461 463 464 465 466 467 468 469 470 470 474 474 474 474 475 476 476 476 477 478 478 478 479 480 484 486 486 488 488 489 492 492 493 496 497 497 497 500 501 501

xiv Contents

9.10 9.11 9.12 9.13

9.14 9.15 9.16 9.17 9.18 9.19 9.20 10 10.1 10.2

9.9.1 TSI Aerometrics phase Doppler particle analyzer 9.9.2 Discusion 9.9.3 Differential phase-Doppler anemometry 9.9.4 Bristol Industrial Research Association 9.9.5 Dantec Particle Dynamic Analyzer Hosokawa Mikropul E-Spart Analyzer Shadow Doppler velocimetry Other light scattering methods Interferometers 9.13.1 Mach Zehnder type interferometer 9.13.2 The TSI LiquitrakTM interferometer Flow ultramicroscope. 9.14.1 ISPA image analysis system Measurement of the size distribution of drops in dispersions Dupont electrolytic grain size analyzer Light pressure drift velocity Impact size monitor Monitek acoustic particle monitors Erdco Acoustical Counter

Field scanning methods of particle size measurement Introduction Single point analyzers 10.2.1 Static noise measurement 10.2.2 Ultrasonic attenuation 10.2.3 (J-ray attenuation 10.2.4 X-ray attenuation and fluorescence 12.2.5 Counter-flow classifiers 10.2.6 Hydrocyclones 10.2.7 The Cyclosensor 10.2.8 Automatic sieving machines 10.2.9 Gas flow permeametry 10.2.10 Correlation techniques 10.3 Light scattering and attenuation 10.3.1 Introduction 10.3.2 Effect of extinction coefficient on turbidity 10.3.3 Transient turbidity 10.3.4 Holography 10.3.5 State of polarization of the scattered radiation 10.3.6 Forwardâckward intensity ratio (FBR) 10.3.7 Optical back-scattering 10.3.8 Transmission fluctuation spectroscopy 10.4 Light scattering theory 10.4.1 The Rayleigh region (dX) 10.4.2 The Rayleigh-Gans region (D < X)

502 502 503 504 504 505 506 507 508 508 509 510 510 510 512 512 513 513 514

524 525 525 526 527 527 527 528 528 529 530 531 5 31 531 532 535 536 537 538 539 539 539 539 540

Contents

10.5

10.6 10.7 10.8 10.9 10.10 10.11

10.12 10.13 10.14

10.15 10.16 10.17 10.18

xv

10.4.3 High order Tyndall spectra (HOTS) 10.4.4 Light diffraction 10.4.5 Early commercial light scattering equipment Multi angle laser light scattering; (MALLS) 10.5.1 Theoretical basis for MALLS instruments 10.5.2 Commercial instruments 10.5.3. Discussion Malvern (Insitec) Ensemble Particle Concentration Size (EPCS) Systems Optical incoherent space frequency analysis 10.7.1 Retsch Crystalsizer Pulse displacement technique (PDT) Small angle x-ray scattering (SAXS) Near infra-red spectroscopy (NIR) Ultrasonic attenuation 10.11.1 Introduction 10.11.2 Theoretical basis for ultrasonic instruments 10.11.3 Discussion Matec Acoustosizer (ACS) Ultrasonic attenuation and velocity spectrometry Photon correlation spectroscopy (PCS) 10.14.1 Introduction 10.14.2 Principles 10.14.3 Through dynamic light scattering 10.14.4 Particle size 10.14.5 Concentration effects 10.14.6 Particle interaction 10.14.7 Particle size effects 10.14.8 Polydispersity 10.14.9 The controlled reference method 10.14.10 Multi-angle measurements 10.14.11 Commercial equipment 10.14.12 Discussion 10.14.13 Spectral turbidity 10.14.14 Diffusion wave spectroscopy (DWS) 10.14.15 Photon migration

542 543 543 544 547 552 563

Turbo-Power Model TPO-400 in-line grain size analyzer Concentration monitors Shape discrimination Miscellaneous 10.18.1 Back-scatter intensity 10.18.2 Spectroscopy; photo-acoustic (PAS) and photo-thermal (PTS) 10.18.3 Transient electric birefringence

603 603 604 604 604

568 572 573 574 575 576 576 576 576 581 584 584 586 586 587 588 589 590 590 591 591 593 594 596 601 602 603 603

605 605

xvi Contents

10.18.4 10.18.5 10.18.6 10.18.7 10.18.8 Appendix

Crossed lasers Frequency domain photon migration Laser induced incandescence (LII) Spectral transmission and extinction Turbiscan multiple light scattering measurements

606 606 607 607 608

Manufacturers and suppliers

623

Author index

628

Subject index

653

Acknowledgements I would like to express my grateful thanks to Dr. Brian H. Kaye for introducing me to the fascinating study of particle characterization. After completing a Masters degree at Nottingham Technical College under his guidance I was fortunate enough to be offered a post at the then Bradford Institute of Technology (now Bradford University). At Bradford, Dr. John C. Williams always had time for helpful advice and guidance. John became a good friend and, eventually, my PhD supervisor. After 22 years at Bradford I retired from academic life and looked for other interests. It was then that I met Dr. Reg Davies who had been a postgraduate student with me at Nottingham. Reg was working for the DuPont company in America, who were in need of someone with my background, and I was fortunate to be offered the position. In my ten years with DuPont I have seen the development of the Particle Science and Technology (PARSAT) under Reg's direction. It has been my privilege to have been involved in this development since I consider this group to be pre-eminent in this field. Since my retirement Reg has also retired and the new Group leader is Dr. Arthur Boxman. My special thanks go to Dr. John Boughton who gave advice on the electron microscopy section. My thanks are also due to holders of copyright for permission to publish and to the many manufacturers who have given me details of their products. Terence Allen Cape St. Francis, 6312 Eastern Cape South Africa and Hockessin, DE, USA


Preface Although man's environment, from the interstellar dust to the earth beneath his feet, is composed of finely divided material his knowledge of the properties of such material is surprisingly slight. For many years scientists have accepted that matter can exist as solids, liquids or gases: Although the dividing line between the states may often be rather blurred; this classification has been upset by powders which at rest are solids, when aerated may behave as liquids and when suspended in a gas take on some of the properties of the gas. Superficially one would consider DuPont to be a chemical company but this is a misconception. Two-thirds of DuPont's products are sold in the form of powders and eighty percent, contain powders. It is therefore essential that employees of this and similar companies should have an understanding of powder properties and behaviour. It is now widely recognised that powder technology is a field of study in its own right. The industrial applications of this new science are far reaching. The size of fine particles affects the properties of a powder in many ways. For example, it determines the setting time for cement, the hiding power of pigments, the activity of chemical catalysts, the taste of food, the potency of drugs and the sintering shrinkage of metallurgical powders. Particle size measurement is to powder technology as thermometry is to the study of heat. In making a decision on which particle sizing technique to us, the analyst must consider the purpose of the analysis. What is generally required is not the size of the particles, but the value of some property that is size dependent. In such circumstances it is important whenever possible to measure the size dependent property, rather than to measure the "size" by some other method and then deduce the required property. For example, in determining the "size" of boiler (fly) ash with a view to predicting atmospheric pollution the terminal velocities of the particles should be measured; in measuring the "size" of catalyst particles, the surface area should be measured, since this is the property that determines the reactivity. The cost of the equipment as well as the ease and the speed with which the analysis can be carried out have then to be considered. The final criteria are that the method shall measure the appropriate property of the particles, with accuracy sufficient for the particular application at an acceptable cost, in a time that will allow the result to be used. Terence Allen


Editor's Foreword Particle science and technology is a key component of chemical product and process engineering and in order to achieve the economic goals of the next decade, a fundamental understanding of particle processes has to be developed. In 1993 the US Department of Commerce estimated the impact of particle science and technology to industrial output to be one trillion dollars annually in the United States. One third of this was in chemicals and allied products, another third was in textiles, paper and allied products and the final third was in food and beverages, metals, minerals and coal. It was Hans Rumpf in the 1950's who related changes in the functional behavior of most particle processes to be a consequence of changes in the particle size distribution. By measurement and control of the size distribution, one could control product and process behavior. This book is the most comprehensive text on particle size measurement published to date and expresses the experience of the author gained in over fifty years of research and consulting in particle technology. Previous editions have found wide use as teaching and reference texts. For those not conversant with particle size analysis terminology, techniques and instruments, the book contains basis information from which instrument selection can be made. For those familiar with the field, it provides an update of new instrumentation - particularly on-line or in - process instruments upon which the control of particle processes is based. Overall, the book continues to be the international reference text on particle size measurement and is a must for practitioners in the field. Dr. Reg Davies Particle Engineering Research Center, University of Florida


Powder sampling 1.1 Introduction There are many instances where estimates of population characteristics have to be made from an examination of a small fraction of that population and these instances are by no means confined to the field of powder technology. Regrettably, there are many workers in this field who still believe that sample selection procedure is unimportant. This results in the analyst being presented with hastily taken biased samples on which a great deal of energy is devoted to get precise results which do not reflect the characteristics of the bulk powder. Non-representative sampling results in incorrect analyses, process failure, unacceptable products and customer dissatisfaction. It is essential that the samples selected for measurement should be representative of the bulk in particle size distribution and the relative fractions of their various constituents, irrespective of whether a physical or chemical assay is to be carried out, since these characteristics are frequently inter-dependent. The magnitude of the problem may be realized when one considers that the characteristics of many tons of material may be assumed on the basis of analyses carried out on gram, or even milligram, measurement samples so the chances of carrying out the measurement on a non-representative sample are considerable. Two case studies illustrate problems that may arise. In case 1 a binary mixture of fine and coarse granules was blended to a satisfactory end-point. The blend was next emptied into a bucket elevator and thence to a hopper through a central feed point. The material was then fed, in 60 kg lots, to an extruder to produce an unacceptable finished product. The process started with a mixing operation followed by a segregating operation in the hopper. The central region (core) of the hopper contents was rich in fines and the outer regions rich in coarse, hence the initial feeds to the extruder contained an excess of fines whereas later feeds contained an excess of coarse. Since a bimodal mixture such

Powder sampling and particle size determination

as this has a high tendency to segregate it is better to weigh-feed the two streams concurrently into the extruder: This also has the advantage of eliminating process steps resulting in lower operating costs and a less expensive plant. The second case was a tabletting operation where a wellmixed fine powder was emptied into a core-flow hopper in such a way that segregation occurred. The resulting tablets were well outside specification. The solution was to replace the hopper with a properly designed mass-flow hopper. The total sampling error is made up of errors due to the primary sampling, subsequent sample dividing and errors in the analysis itself Sampling is said to be accurate when it is free from bias, that is, the error of sampling is a random variable about the true mean. Sampling is precise when the error variation is small irrespective of whether the mean is the true mean or not. The ultimate that may be obtained by representative sampling may be called the perfect sample; the difference between this sample and the bulk may be ascribed wholly to the expected difference on a statistical basis. Errors in particle size analysis may be due to: • •

instrument limitations; improper procedure e.g. inadequate dispersion, particle fracture during handling; • operator errors e.g. improper instrument set-up or poor calibration; • incorrect sampling. Two types of sampling errors are possible [1] •

Errors due to segregation of the bulk; this depends upon the previous history of the powder and can be minimized by suitable mixing and building up the sample from a large number of increments.

•

Statistical errors that cannot be prevented. Even for an ideal random mixture the quantitative distribution in samples of a given magnitude is not constant but is subject to random fluctuations. It is the only sampling error, which cannot be suppressed and occurs in ideal sampling. It can be estimated beforehand and reduced by increasing the sample size.

1.2 Sample selection Samples are withdrawn from a population in order to estimate certain characteristic of that population and to establish confidence limits for that

Powder sampling

3

characteristic. The characteristic may be particle size, composition or quality; a measure of the spread of the distribution may also be required. The objective may be to set up limits between which the quality of a final product is acceptable, to decide whether the characteristics of a given lot meets preset criteria, or it may be to estimate the variability within a lot or between lots. If the material comes in containers, or can be viewed as discrete units, the objective may be to estimate the number of units outside of specification. The value of the estimate is largely dependent on the sampling technique adopted. It is of little value, and could impart false information, if it is biased or imprecise. It is usually impossible to measure the size distribution of a bulk powder and so it is necessary to carry out measurements on a sample extracted from the bulk. This sample is itself frequently too large and has to be further sub-divided. The probability of obtaining a sample that perfectly represents the parent distribution is remote. If several samples are taken their characteristics will vary and, if these samples are representative, the expected variation can be estimated from statistical analysis. However, the sampling equipment will introduce a further variation, which may be taken as a measure of sampler efficiency. Imposed on this there may also be operator bias. The stages, in reducing from bulk to measurement samples may be conveniently divided into the five stages illustrated below: bulk or process stream (10" kg)

gross sample (>kg)

laboratory sample (•••*. «• • * ! 1 • • i*.l

'^-.•»1

lijjj|^j\ /l^:i;ljiil=!ij:i;k i^;ij!j!j!j|il;!!!iV /giisiii:M:!i!i!K

/ililinim

Fig. 1.14 (a) Cross-sectional sampler straight path action, in line (b) crossline (c) oscillating or swinging arc path

Powder sampling 19

1.4.5 Traversing cutters With large tonnages, samples taken from conveyors can represent large quantities of material that need to be further reduced. Often, a traversing cutter is used as a primary sampler, and the extracted sample is further cut into a convenient quantity by a secondary sampling device. It must be borne in mind that the secondary sampler must also conform to the golden rules of sampling. This equipment is satisfactory for many applications but it has limitations, which restrict its use. These are: • Although comparatively readily designed into new plant it is frequently difficult and expensive to retrofit an existing plant. The main reason for this is due to space requirements. • The quantity of sample obtained is proportional to product flowrate and this can be inconvenient when the plant flow-rate is subject to wide variations. On the other hand, where a plant's daily average is required, this is a necessary condition. • It is difficult to enclose the sampler to the extent required to prevent the escape of dust and fume when handling dusty powders. Commercial samplers are available which combine a traversing type sampler with an unacceptable table sampler. An alternative design is the radial cutter or Vezin sampler shown in Figure 1.15. These samplers vary in size from a 15 cm laboratory unit to a 152 cm commercial unit. 1.4.6 Sampling dusty material Figure 1.16 shows a sampler designed to sample a dusty material, sampling taking place only on the return stroke. This is suitable provided the trough extends the whole length of the stream and does not overfill. The sampler shown in Figure 1.17 was designed to extract a constant volume of homogeneous granular material for chemical assay and cannot be recommended when a physical assay is required [12]. The slide valve sampler (Figure 1.18) is suitable for collecting size-representative samples [13].

20


Primary cutter Traversing cutter drive Secondary sample hopper

Vezin sampler

Motor-driven rotary cutter

Discharge

Sample •:•••; ••.«'

Fig. 1.15 Schematic of a primary and secondary system based on Denver Equipment Company's type C and Vezin samplers. A variant of this problem is encountered in sampling from pre-weighed batches (Figure 1.19). The sampler is, essentially, a screw conveyor which extracts a sample continuously while the container is being filled. This system suffers from two drawbacks in that it limits the discharge opening thus reducing throughput and takes part of the stream for the whole of the time thus contravening the golden rules of sampling. Similar samplers are available to sample flowing streams continuously; these are essentially Archimedes screws (Figure 1.20). Many variations of this design are possible and the reader is referred to Cornish et. al [14] for a comprehensive treatment. Figure 1.21 shows an industrial sampler for free-flowing material including granules, powders and pellets. This can be mounted on screw conveyors, drag conveyors or angular gravity chutes with manual or automatic control.

Powder sampling 21

Normal position

Dischaiging sample

I Sample Fig. 1.16 Full-stream trough sampler Process stream

Sampling position Fig. 1.17 Constant volume sampler

Process stream

Dischaigc position

22


Process stream

Nonnal pdsiticm

Sampling position

Fig. 1.18 Slide-valve sampler • . • . A / . 4^. • . 4t ^ ^ f ^ ^^ € f ^ .

Hopper feed conveyor

Hopper dischai^ge device Sampling device Sampler receiver

Fig. 1.19 Sampling from a hopper

Hopper

Powder sampling 23

i

Fig. 1.20 Archimedian screw sampler "•

Direction of powder flow

Sliding gate mil

Sample

I HI I

dischaige ii*'i

Piston

Fig. 1.21 Sampler for screw conveyor. 1.4.7 In-line sampling The diverter valve shown in Figure 1.22a was used for extracting a sample from a pilot plant granulator operation. The sample was fed to a hopper and thence, by a vibratory feeder, to a low-angle laser light scattering instrument. The granules were then returned to the process stream. This system allowed on-line analyses every minute so that the process could be optimized. A more substantial design was required for continuous plant operation (Figure 1.22b) with a moving piston operated rectangular flap weighing around 50 lb. Heuer and Schwechten [15] designed a sampler for installation with a fluidized bed jet mill where the entire classifier product passed through a Sympatec laser measuring device. This system is suitable for use with

24


From granulator

To analyzer To dryer

; 1 To dryer

Fig. 1.22 Sampler for on-line particle size measurement: (a) diverter valve sampler (b) moving flap sampler low throughput pharmaceutical or food products but sample splitting is required for heavy throughputs. For this purpose a rotating pipe sampler was designed [16] which, in order to sample the whole process stream, rotated in a spiral path. Commercial samplers are available from Gustafson Intersystems and Forratechnic 1.5 Sample reduction The gross sample is frequently too large to be handled easily and may have to be reduced to a more convenient weight. ASTM C702 describes sample splitting and cone and quartering together with a miniature stockpile method designed for use with damp, fine aggregates only [17]. Obviously the method employed should comform to the two golden rules mentioned earlier. Usually the amount of material to be handled is small enough that getting it in motion poses no great difficulty. There is a natural tendency to remove an aliquot with a scoop or spatula and this must be avoided since it negates the effort involved in obtaining a representative sample from the bulk. To obtain the best results, the material should be made as homogeneous as possible by pre-mixing. It is common then to empty the material into a hopper and this should be done with care. A homogeneous segregating powder, when fed to a hopper from a central inlet, will segregate since, in essence, it is being poured into a heap. In a core flow hopper the central region, which is rich in fines.

Powder sampling 25

empties first, followed by the material nearer the walls that has an excess of coarse. The walls of the hopper should have steep sides (at least 70°) to ensure mass flow and should be filled in such a way that size segregation does not occur. This can best be done by moving the pour point about, so that the surface of the powder is more or less horizontal. Several sampledividing devices are available and these are discussed briefly below.

Fig. 1.23 Scoop sampling. 1.5.1 Scoop sampling The method consists of plunging a scoop into the powder and removing a sample (Figure 1.23). This is particularly prone to error since the whole of the sample does not pass through the sampling device and, since the sample is taken from the surface, where it is not representative of the bulk. It is sometimes stated that a satisfactory sample can be obtained by shaking the bottle containing the laboratory sample in order to mix it, then extracting the analysis sample with a scoop. Kaye used five modes of shaking and six operators in order that both operator and technique bias could be evaluated [5,18]. The samples consisted of coarse and fine particles mixed in known proportions. He found that moving the bottle rapidly around in a circle in a vertical plane, with a brisk horizontal shaking superimposed on this motion, introduced no bias. Similarly, if the bottle is placed in the palm of the hand and shaken in a circular motion in a plane at about 30° to the horizontal no bias occurs. The other three methods introduced a definite bias. Kaye concluded: (1) Shaking a bottle containing free-flowing particles of different sizes is not an effective method for mixing them. Anyone who doubts this should try the experiment of putting equal volumes of two powders of different sizes and colors into a bottle and examining the mixture after shaking. Pockets of segregated material form and cannot be

26


broken up by further shaking. In particular the surface region will be rich in large particles. A sample removed with a scoop will include the surface region, where the composition is very likely to be different from that of the body of the powder. (2) A further objection to the use of a scoop it is liable to be sizeselective favoring the collection of fine particles. The reason for this is that, when the scoop is removed from the material, some particles will flow down the sloping surface of the powder retained in the scoop; the finer particles tend to be captured in the surface craters and retained, whereas coarse particles are more likely to travel to the bottom of the slope and be lost. The effect is particularly important if a flat blade (such as a spatula) is used for the removal of the sample. 1.5.2 Cone and quartering This method consists of pouring the powder into a heap and relying on its radial symmetry to give four identical samples when the heap is flattened and divided by a cross-shaped cutter (Figure 1.5). The method would give reliable results if the heap were symmetrical about a vertical axis and if the line common to the two cutting planes coincided with the axis. In practice, the heap is unlikely to be symmetrical and symmetry of cutting would be very difficult to achieve without precision equipment. Departures from symmetry will result in differences in the amount of material in the four samples. Since severe size segregation will certainly occur with freeflowing powders in forming the heap, this departure from symmetry will generate differences in the size distributions of the four samples. Further, each operation reduces the sample size to a quarter if one sample is retained and a half if opposite quadrants are combined: In order to reduce the sample size still further it is necessary to repeat the exercise thus compounding the error. The method is very dependent on the skill of the operator and should be avoided. If coning and quartering is possible, this implies that the amount of material to be divided is such that it can be easily moved by hand, so that it is just as easy to feed it into the hopper of a device such as a spinning riffler in which increments are collected from a stream in an acceptable manner.

Powder sampling 27

Sample Fig. 1.24 Table sampling 1.53 Table sampling In a sampling table the material is fed to the top of an inclined plane in which there is a series of holes. Prisms placed in the path of the stream break it into fractions. Some powder falls through the holes and is discarded, while the powder remaining on the plane passes on to the next row of holes and prisms, and more is removed, and so on. The powder reaching the bottom of the plane is the sample (Figure 1.24). The objection to this type of device is that it relies on the initial feed being uniformly distributed, and a complete mixing after each separation, a condition not in general achieved. As it relies on the removal of part of the stream sequentially, errors are compounded at each separation; hence its accuracy is low. Its advantages are its low price and its lack of moving parts. 1.5.4 Chute splitting The chute splitter consists of a V-shaped trough along the bottom of which is a series of chutes alternately feeding two trays placed on either side of the trough (Figure 1.25). The material is repeatedly halved until a sample of the desired size is obtained. When carried out with great care this method can give satisfactory sample division but it is particularly prone to operator error, which is detectable by unequal splitting of the sample.

28


Fig. 1.25 Line diagram of chute splitter. In a review of factors affecting the efficiency of chute rifflers Batel [19] stated that there is little to be gained by increasing the number of chutes. Kaye and Naylor [5] point out that this is only true if the chute widths are kept constant; increasing the number of chutes by reducing the chute width increases the efficiency. They tested this hypothesis, using four rifflers of the same overall width having 4, 8, 16 and 32 chutes and found that the efficiency increased as the number of chutes increased. It follows that two narrow, oppositely directed chutes intercepting the flow of powder would provide a very efficient sampling device. An equivalent selection procedure is to have one narrow stream oscillating between two reception bins. This in fact forms the basis of the British Standards oscillating hopper sample divider and the I d oscillating paddle divider (section 1.5.6). 7.5.5 Spinning rifflers The rotary sample divider or spinning riffler was first described in 1934 [20] and conforms to the golden rules of sampling. The preferred method of using this device is to fill a mass flow hopper in such a way that segregation does not occur. The table is then set in motion and the hopper outlet opened so that the powder falls into the collecting boxes. The use of a vibratory feeder is recommended to provide a constant flowrate

Powder sampling 29 (Figure 1.26). In 1959 Pownall [21] described the construction and testing of a large laboratory spinning riffler and a year later Hawes and MuUer [22] described the construction of a small instrument. They also examined the riffler to find how various factors influence its efficiency using quartz and copper sulfate crystals of the same size.

Moss feed hopper,

Spinning riffler

Vibratory feeder

Fig. 1.26 Line diagram of the spinning riffler. Their conclusions were that the efficiency: (a)

is dependent on the relative proportions of the mixture increasing as the proportion of copper sulfate was raised from 1% to 5%.

(b)

increases with increasing particle size.

(c)

is reproducible under similar experimental conditions.

(d)

is not affected by a combination of variables.

(e)

is not affected by the number of presentations of the sample containers to the feed. (Since the minimum number of such

30 Powder sampling and particle size determination presentations was 100, this statement applies only to numbers greater than this). The mixtures in these experiments contained particles all the same size hence the main cause of segregation was not present and the conclusions may not apply in the more usual case where segregation occurs. Several commercial versions of this instrument are available, some of which were designed for free-flowing powders, some for dusty powders and some for cohesive powders. They handle quantities from 40 liters down to a few grams. 1.5.6 Commercial rotary sample dividers These are available from many manufacturers and can be divided into several groups; micro-dividers for small samples and larger versions for greater quantities; closed dividers for dusty and cohesive material (Figure 1.27) and open dividers for granular free-flowing material. Kaye and Naylor [5] suggest that samples of a given size may be obtained directly by controlling the amount of time, during each oscillation, that the feed is directed to each hopper or container. For efficient sampling the number of increments should be high (>35). In the Pascal turntable sampler the powder falls through a hopper outlet on to a cone whose position can be varied in order to alter the hopper outlet size. The powder slides down the cone into containers on a revolving table (Figure 1.28) 1.5.7 Miscellaneous sampling devices] In the oscillating hopper sample divider [23] ](Figure 1.29), the feed hopper is pivoted about a horizontal axis so that it can oscillate while emptying. Two collectors are placed under the hopper outlet so that the powder falls into them alternately so that at each step the sample is halved. The contents of one box are retained so that at each step the weight of the sample is halved. The oscillating paddle sample divider (Figure 1.30) works in a similar way. A similar unit may be installed in a feed pipe down which particulate material is flowing. In this unit the splitter cones are mounted within the feed pipe and the sample falling through the segmental slots passes out of a side pipe, while the remainder flows over the cone and continues down the feed pipe. Since the whole of the stream is not taken for many short intervals of time, non-representative sampling is possible.

Powder sampling 31

Fig 1.27 The Retsch sample divider

Lid provides handle 5 times diameter of laigest particle Rotation

Side plate bent over to form clip Fig. 1.28 Pascal turntable sample divider.

32


Fig. 1.29 Oscillating hopper sample divider

Fig.1.30 Oscillating paddle sample divider

Glen Creston rotary sampler incorporates this design (Figure 1.31) that contains an internal inclined pipe, which is rotated by a geared motor. The powder is fed into this pipe and with each revolution a proportion falls into the sample column. The opening in the sample column can be adjusted from outside the divider by means of a sliding plate. By this method it is possible to vary the ratio of sample to total throughput. Appropriate dividers are available for different particle sizes and sampling volumes. Retsch rotary divider PT 100 consists of a feeder unit with a dividing head that rotates at 110 rpm; this means that with a dividing head with 10 outlets the feed is split into 1100 aliquot parts per minute. The feed can range from a few grams to 5000 ml depending on the size of the collecting vessels. Retsch rotary divider PK 1000 (Figure 1.32) is designed for representative sampling and dividing of large quantities of bulk material. Depending on the lower cone, from one to six samples can be taken. The sample collecting vessels have a maximum capacity of 0.5 liters and the reject container 30 liters. Gustafson manufacture an automatic sampler for free flowîng materials from screw or drag conveyors (Figure 1.33).

Powder sampling 33

Fig. 1.31 The Glen Creston rotary sampler

Figure 1.32 Retsch PK 1000 laboratory rotary sample divider 1 feed hopper, 2 on/off switch, 3 time setting controls, 4 upper cone, 5 lower cone, 6 stand, 7 display, 8 start button, 9 sample slot adjustment, 10 sample vessel, 11 reject container

34


2

I ••90%

^

^

V

^

• KM]« • CZ3*

/0\

Fig. 1.33 Sampling from screw or drag conveyors.

|Motor| Sample

Sample

Fig. 1.34 (a)Sloping trough cutter (b) Vertical pipe cutter A sample splitter described by Fooks [24] consists of a feeder funnel through which the sample is fed. It passes on to the apex of two resting cones, the lower fixed and the upper adjustable by means of a spindle. Segments are cut from both cones and by rotation of the upper cone the effective area of the slits can be varied to vary the sampling proportion. Material, which falls through the segmental slots, is passed to a sample pot. The residue passes over the cone and out of the base of the unit.

Powder sampling 35

Pumped sample flow Rotating slot

Main flow

Fig. 1.35 (a) Osborne's rotating slot slurry sampler, sampling tank, (c) Cross's slotted pipe slurry sampler

(b) Osborne's

1.6 Slurry sampling Slurry process streams vary in flowrate, solids concentration and particle size distribution. Any sampling technique must be able to cope with these variations without affecting the representativeness of the extracted sample. For batch sampling, automatic devices are available where a sampling slot traverses intermittently across a free-falling slurry. Unfortunately it is difficult to improvise with this technique for continuous sampling, since such samplers introduce pulsating flow conditions into the system. Clarke [13] discusses the problems of sampling from liquid streams in open channels and describes a parallel-sided scoop for extracting a sample of a size that is proportional to flow rate.

36


Denver Equipment Company manufactures a sloping trough cutter (Figure 1.34a) and a vertical pipe cutter (Figure 1.34b) for sampling slurries. Osborne [25] described a sampler that consists of a narrow slot continuously rotated on an axis parallel to the sluny flow (Figure 1.35a). Cross [26] used a slotted pipe mounted vertically in the overflow compartment next to the vortex fmder of a hydrocyclone (Figure 1.35c). Since most continuous size analyzers require a constant volume flowrate, further subdivision is often necessary. Osborne's solution (Figure 1.35b) was to feed the sample stream to a well-agitated tank and withdraw a representative sample at a constant flow-rate. Hinde and Lloyd [27] are more interested in extracting samples for continuous on-line analysis. They state that the process streams from industrial wet classifiers can vary in volume flow rate, solids concentration and particle size distribution. Any sampling technique should be able of coping with these variations without affecting the representativeness of the extracted sample. Autometrics [28] analyze the whole of the extracted sample (Figure 1.36a). Gaps are left below the weirs to prevent sanding. Subdivision of the sample stream, for calibration purposes, can be accomplished very successfully by siphoning from a vertical fast- flowing stream (Figure 1.36b). Due to their simplicity side-wall samplers are superficially attractive (Figure 1.37) but serious errors in concentration and particle size distribution can arise unless the particles are fine, the concentration is high and a very high sampling velocity is used. A projection extending from the pipe wall only marginally improves sampling efficiency [29]. 1.7 Reduction of laboratory sample to measurement sample The methods described in section 1.5 are capable of reducing the gross sample of many kilograms down to a laboratory sample of up to about 1 kg and further reducing this to a few grams. In many cases this reduction is sufficient to generate a measurement sample whilst for some analytical techniques a further reduction is required. Coning and quartering a paste [30] has been used for preparing samples down to about 20 mg in weight but the efficiency of the method is operator dependent. Lines [31] compared three methods of sampling powder prior to Coulter analysis and found a spread of results of ±6% with cone and quartering, ±3.25% with random selection and ±1.25% with cone and quartering a paste.

Powder sampling 37

Pumped sample flow ^

Main flow Flow under weirs to prevent sanding (a)

(b) Overflow

Main flow

Fig. 1.36 (a) Autometrics'design for direct sampling (b) Autometrics' design for extracting a small sample

Stirrer Vent fill tube

Air supply Cover

PTFE seal ring QVF plastic flange Suspension

QVFpipe section

PTFE seal ring

Base plate Stainless steel capillary

10 ml centrifuge tubes

Plinth

4-

Fig. 1.37 Microscal suspension sampler, 100 ml to 10 ml.

38


In microscopy the required sample consists of a few milligrams of material. This may be extracted from the laboratory sample by incorporating it in a viscous liquid in which it is known to disperse completely. The measurement sample may then be extracted with a measuring rod [2]. Sampling an agitated suspension with a syringe or pipette is a good method for fine powders but, with coarse powders, concentration gradients and particle segregation due to settling and the centrifugal motion of the liquid due to the action of the stirrer may lead to selective sampling. Alternatively the measurement sample may be withdrawn with a sample divider as illustrated in [32]. The Microscal suspension sampler [33] (Figure 1.37) is designed to eliminate these errors. It consists of a glass cylinder closed at either end by stainless steel plates. Around the periphery of the base plate are ten equidistant holes leading to ten centrifuge tubes via stainless steel capillary tubes. The cover plate has a central hole through which passes a stirrer, a scalable inlet for introduction of the suspension, and a gas orifice which enables the gas pressure to be varied. In operation 100 cm^ of suspension is placed in the cylinder and, whilst undergoing agitation, gas is introduced under pressure in order to blow the suspension into the centrifuge tubes. Tests indicate that this system gives significantly less variation than syringe withdrawal, i.e. 1.0% standard deviation as opposed to 3.0%. 1.8 Number of samples required The basic assumption, in analyzing data statistically, is that the samples are representative of the populations from which they are withdrawn. However, there is an uncertainty associated with any measured property and this can be estimated using the confidence interval (CI). This can be expressed in general terms as: CI = estimate ± M (standard deviation of the estimate) where A/ is a multiplier that is determined by the chosen confidence interval (usually 90, 95 or 99%) and the amount of information available to calculate the standard deviation of the estimate. Given a random set of n samples with a measured mean x^ withdrawn from a powder whose standard deviation cr is known, the true mean will lie in the interval [34]: /^ = ^ ^ ± - 7 =

(1-1)

Powder sampling 39 At the 95% confidence level M= 1.96, i.e. there is a five in one hundred chance that the true mean lies outside these limits. The 90% confidence interval is smaller (M= 1.645) and is less likely to have the true value of x^ within its limits. The 99%) confidence interval is larger (M= 2.576) but is more likely to contain the true value of x^ within its limits. For pharmaceutical applications, a value / = 2 is used to denote working quality and a value / =^ 3 (99.9% confidence level) is used for total quality [35] The statistical reliability of analytical data can be improved by increasing the homogeneity of the sample (reducing a), increasing the size of the sample, or increasing taken (increasing n). In most instances the population standard deviation is not known and must be estimated from the sample standard deviation (s). Substitution of s for a in equation (1.1) with M = 1.96 does not result in a 95% confidence interval unless the sample number is infinitely large (in practice n>30). When s is used, multipliers, whose values depend on sample number, are chosen from the /^-distribution and the denominator in equation (1.1) is replaced by ^{n -1) . Assuming a normal distribution of variance, the number of samples required, to assume at the 95%) confidence level that the median is known to +.4, is given by ^ts^' n-

(1.2)

where ^ = |/i-x^| is the maximum allowable difference between the estimate to be made from the sample and the actual value. Example 1 In many sampling procedures, sub-samples are taken at different levels and locations to form a composite sample. If historical evidence suggests that the standard deviation between samples is 0.5, and it is necessary to know the average quality of the lot to within 0.3, the number of subsamples required, at the 95% confidence level, is given by equation (1.2) [36].

/; =

0.3

40


Example 2 16 samples, withdrawn at random from an unmixed powder, gave a median x^^ = 3.13 |iim by multi-angle laser light scattering (MALLS) with a standard deviation of ^ == 0.80 |Lim. Then, in microns, the median lies between the limits: // = 3.13±2.14- ^-^ V16-1 /i = 3.13±0.4 The multiplier ^ = 2.14 is obtained from a / table for n = 16-1 degrees of freedom at the 95% confidence level. Thus we are 95% confident that the median lies in the confidence interval (CI): CI = 2.69 < 3.13 < ± 3.57 Based on this data, the number of samples required to give an estimate to within ±0.10, (£' = 0.10) is: M=

^2.14x0.8^^ V

0.10

«=293 After mixing, 16 samples gave a median jc^ of 3.107 |a,m with a standard deviation s, of 0.052 fim. Then: /i = 3.107±2.14

^0.052^

//= 3.107 ±0.029 Thus we are 95% confident that the true median lies in the confidence interval: CI-3.078 < 3.107 < 3.136 i.e. the mixing step increases the measurement precision by a factor of fourteen.

Powder sampling 41 A single sample, run 16 times on a MALLS instrument gave a median of 3.11 jLim with a standard deviation of 5^ ^ 0.030 |Lim. The total variation Isf ] is the sum of the variation due to the measuring procedure Is^ ) and the variation due to the sampling procedure (^^). It is possible to isolate the sampling error from the measurement error. The standard deviation due to sampling (sj is 0.042 |Lim and the standard deviation of the measurement technique (s^) is 0.030 |Lim giving a total standard deviation (s^) of 0.052 |Lim. As can be seen from this example, there is little to be gained in using a measurement technique substantially more accurate than the sampling that preceded it. (1.3)

+ s: 0.042

Further, the number of samples required, after mixing, in order to assume at the 95% confidence level that the median is known to ±0.10 |Lim is: n-\

2.14x0.052 V 0.10

n = 2.2 i.e. 3 samples. Compounding increments from the unmixed powder whilst it is in motion, for example, by riffling in a spinning riffler can attain the same accuracy. Table 1.2 Means and standard deviations of active ingredients in simulated sampling trials Sample weight (g) 1 3 5 9

Mean (mg) 98.20 99.45 99.50 100.20

Standard deviation (mg) 5.78 4.78 3.74 3.10

Expected standard deviation (mg) 8.31 4.83 3.74 2.79

In the pharmaceutical industry there has been a tendency by federal agencies to request that blending validation be carried out using samples the size of a dosage unit. A theoretical experiment using random numbers was carried out to assess the effect of changing sample size [37]. A set of

42


assays of a dosage weighing 1 g containing 100 mg of drug substance was extracted from a bulk of 100 g. Four 1 g samples were averaged for each of 10 measurements and the standard deviation for each group of four determined. The experiment was repeated with 3 g, 5 g and 9 g samples. The means and averages for each set of 10 groups are given in Table 1.2. The expected standard deviation is based on the premise that it is inversely proportional to the square root of sample size. Assuming the 5 g sample gives accurate data it can be seen that the 9 g sample is slightly worse than expected and the 1 g sample is much better than expected. 1.9 Theoretical statistical errors on a number basis The ultimate that can be obtained by representative sampling may be called the 'ideal' sample. A powder may be considered as made up of two components A and B. The probability that the number fraction {p) of the bulk in terms of ^ shall be represented by the corresponding composition ip) of an ideal sample can be computed from the number of particles of .4 and B in the sample {n) and the bulk (N).

PO-P) r

1- n

(1.4)

where a; is the theoretical standard deviation, [the variance Var(o;) is defined as the square of the standard deviation]. For a normal distribution of variance, the spread of data about the mean is described by the probability equation d^ dp

1 CTIÎITT

exp

jp-p) 2a'

2\

Using the transformation a, j = {p- p) dip

1

dy

yjlK

(

y-^

(1.5)

(1.6)

(1.7)

Powder sampling 43

0.5 0.4 H

0.3 H

0.2 H

0.1 H 0.0

Fig. 1.38 Normal probability function (relative).

Fig. 1.39 Normal probability function (cumulative).

44


Table 1.3 Cumulative normal distribution

y -4.0 -3.0 -2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0 3.0 4.0

(%) 0.003 0.13 2.28 6.68 15.87 30.85 50.00 69.15 84.13 93.32 97.72 99.87 99.997

y -3.50 -2.50 -1.75 -1.25 -0.75 -0.25 0.00 0.25 0.75 1.25 1.75 2.50 3.50

d(t>IAy 0.13 2.15 8.60 18.38 29.96 38.30 39.89 38.30 29.96 18.38 8.80 2.15 0.13

Table 1.4 Variation in the number of black balls in samples taken from a bulk containing 4000 black balls and 8000 similar white balls.

Upper limit 3857 3873 3889 3905 3921 3937 3953 3969 3985 4001 4017 4033 4049 4065 4081

N umber of black balls Lower limit Median value (x) 3864 3872 3880 3888 3904 3896 3912 3920 3928 3936 3944 3952 3960 3968 3984 3976 3992 4000 4008 4016 4032 4024 4040 4048 4064 4056 4072 4080 4096 4088

Frequency of occurrence (A«) 0 1 6 10 43 103 141 195 185 160 90 37 17 11 1

Powder sampling 45

This differential equation is presented graphically in Figure 1.38 and the integrated version in Figure 1.39. From Table 1.3, 68.26% of all occurrences lie within ±1 a; (between y = -\ and j = +1) from the mean, 95.44% within ± 2 q and 99.94% within ±30;. Example 3 Consider a bulk made up of 8000 white balls and 4000 black balls from which 750 are extracted. Substituting in equation (1.4): 750 12,000, a; =0.0167 Hence: « ± wo; = (2/3 ± 0.0167) of 750 for the white balls

n ±na^ =n± 12.5 where n = 500 or 250 for the white and black balls respectively. Thus 68 times out of 100 there will be between 487.5 and 512.5 white balls in the sample; 95 times out of 100 there will be between 475 and 525 and 26 times out of 10, 000 there will be either more than 537.5 or less than 462.5. 1.10 Practical statistical errors on a number basis The mean, on a number basis, can be calculated from experimental data using the following equation: 3c=^^

(1.8)

The standard deviation can also be calculated using the following equation:

46


If the true mean is not known, the experimental mean, determined using equation (1.8), must be used. In this case the denominator in equation (1.9) is replaced by {n - 1), which has a negligible effect when n is large. Example 4 Assume that the sampling operation described in example 1 is carried out // ^^ 1000 times with the results presented in Table 1.4. Then, from equation 1.9, the standard deviation ir3

(2.76)

Data presentation and interpretation 105

The integration yields a value / = VTT giving: \nx^^^^\nXgj^+05\n^cTg

(2.77)

ln.x-;v^-lnx^;v'+10'n^^g

(2-78)

\ux^y = \nxg^ + 1.5 In^ a^

(2.79)

Similarly ln.v,;^v/ - In V + 2.0In^ cj^.

(2.80)

2.11.2 Derived mean sizes If the number size distribution of a particulate system is found to be lognormal, equations (2.77) to (2.80) can be used to determine other mean sizes. For example, the volume-moment mean size is the mean size of a volume (mass) distribution: r=oo

I 4AN, r=0

3

Therefore: In xyj^ - 4In x^j^ - 3.0In x^vj/

(2.81)

Substituting from equations (2.79) and (2.80): \\\xyj^^ = \nxg^j + 3.5In^ a^

(2.82)

Similarly, the mean size of a surface distribution is given by: \nxsy = Inx^^ +2.51n^ a^

(2.83)

106 Powder sampling and particle size determination

Using this equation, the volume specific surface of the particulate system may be determined since:

2.11.3 Transformation between log-normaldistrihutions If the number distribution is log-normal, the mass distribution is also lognormal with the same geometric standard deviation. Using the same treatment as was used to derive equation (2.77) gives, for a mass analysis: \nxyf,^ - \nx^y + 3.5In^ a^

(2.84)

Comparing with equation (2.82) gives: \x\x^y - Injc^yv +3.01n^ a^ Since the geometric expressed geometric

(2.85)

relations between the number average sizes and the number mean are known [equations (2.77) to (2.80)], these can now be as relationships between number average sizes and the weight mean jc^^ to give a similar set of equations.

\nx^f ^\nx^y-2.5\n^cj^

(2.86)

Injcy^.s =lnjc^j -2.01n^cr^

(2.87)

\nx^y=\x\Xgy-\.5\n^c7g

(2.88)

\nx^,f^\nx^y-\mn'-cj^

(2.89)

Other average sizes can be derived from the above using a similar procedure to that used to derive equations (2.84) and (2.85) to give: Inx^y - In x^y - 0.5In^ a^

(2.90)

Similarly, for a surface distribution, the equivalent equation to equation (2.80) is: InA-^^^ -IHA^V +2.01n2cr

(2.91)


Substituting this relationship into equations (2.86) to (2.89) yields the equivalent relationships relating surface average sizes with the surface geometric mean diameter. 2.11.4 Relationship between median and mode of a log-normal equation The log-normal equation may be written: d^_ 1 -exp(-X^) dx X v27r In cr, where

(2.92)

Incr =lnx-lnjc

so that: AX _ 1 dx X

V21n(

At the mode {x = x^) and: —y = 0 djc

I.e.

d^ -0 dx dx

— = 0 where Y = -j=^ expl-X^) Ax ^2%x\n(jg so that

dx

2(lnx:^

iexp(-X^) = 0

-\nxg)

V2Ir\Org

x^42\

r\Gr

J^--mv-«"-gy

In x^ = In Xg - In cj^

These relationships are summarized in Table 2.10.

(2.93)


Table 2.10 Relationship between average sizes for lognormal equations In %^ = In x^+ 0.5 In^cr^ In Xj^^ = In x^+ 1.0 In^cr^ \nxj^y=\nx^^+ 1.5 In^cr^ \n Xj^j^= \n x^r+ 2.0 \nâg In Xi^^^ In Xgyy-H 1.5 In^cr^ In x^,/= In Xgy^/+ 2.0 In^cr^ In x^^-= In x^-^ 2.51n2o^

In xî^ In x^-^ 2.5 In^cr^ In Xsj^= In Xgyv+ 3.0 In^cr^ In Xyj^= In Xg^+ 3.5 In^cr^ In Xgs= In Xgyv+ 2.0 In^cr^ Inx^^- lnXg^+3.01n2(T^ lnx^= '"^g~^ 10 ^^^^g

2.1 J. 5 An improved equation and graph paper for log-normal evaluations Using equation (2.59) and the relationship [94] : x-^ -exp(lnjc)

(2.94)

the log-normal equation can be written: d(f> _

dx

1

yf2nx^\na.

-exp

2 ' " ^^

exp

In

jx/x^)

(2.95)

V " 21nH y

This form of the log-normal equation is more convenient for use since the variable (x) only appears once. Equation (2.95) may be written in the simplified form:

dl - êxp -Z>ln^(x/x^)

(2.96)

dx The relationship between geometric mean and mode [equation (2.93)] takes the form: C

:exp

J_ lb

{1S)1)

Data presentation and interpretation 109 This modified form of the log-normal equation simplifies parameter determination from log-probability plots of experimental data. The graph paper may be furnished with additional scales of b and C both being determined by drawing a line parallel to the distribution through the pole (0.25 ^m, 50%). 2.11.6 Application Consider a log-normal equation with a geometric mean x^ = 6.75 jiim and a standard deviation a^ = 1.64. According to equation (2.63) the mode x^ will be 5.27 |Lim making: -^=0.1344exp -2.021n^(jc/5.27) djc

This form is particularly useful when further mathematical computation is envisaged, such as for grade efficiency GJx) since:

'

^ dFix)

G,(x) = Er^

(2.98) dx

where (d^^dx:) is the frequency of the coarse stream, (d^dx) is the frequency of the feed and E^ is the total efficiency (see Chapter 5). 2.12 Johnson's Sg distribution Johnson's S^ distribution is a bounded log-normal distribution, i.e. a truncated log-normal equation with a minimum and maximum size so that the whole size distribution is directly considered. This differs from conventional correlation analysis as summarized by Herdan [97]. It has been presented, in a series of papers by Yu and Standish [98-101] as a function that can represent all unimodal size distributions of particles. It may be written: dF(z)


y

_

V-^max

Q"z

-^min /

27c(^-^min)(^max-^)

exp

^ 1^

vV2y

,2^

//^+o-Jn

(^-^min)

(2.99) (^min ^f(x) to the form (/> ^fx/x^ 5), where JCQ 5 is the median. The result is an easy to handle function with a high degree of correspondence to the more complicated logarithmic function (t)j^ below:


(2.107)

^^ =(/>(bxr where ^ «2 1 u (p = 271 f——-a\ exp

(2.108)

Table 2.11 The properties of some commonly used density distribution functions Name

-00,+ OO]

1

Normal

Log-nomial

Range of x

Equation -exp

Ax

llncj

dlnx

1 -exp V27ilncr^,

V2 o0,+ 00]

V2ln(T„

Rosin-Rammler [104]

0,+ 00]

(^x Gates-Gaudin-Schumann [105-107]

^ = nb{bxr' Ax

^' ^maxl

Gaudin-Meloy[108]

(t> =

^' -^maxJ

Roller [109-111] Harris [113] Martin [115]

=C

\-{\-bx)" ^0.5

b '

^' -^maxJ

' = Cx

0,+ 00]

(/>^Cx'e-'"

Gamma function [116,117] Weinig[118] Heywood[119] Griffith [120] Klimpel-Austin [121] Beta function [122]

^' ^maxJ

+

0,+ 00]

(/> = Cx''-ê-'

0,+ 00]

= Cx'e-''"

0,+ 00]

(t>=^Cx-''e-'"''^

0,+ 00]

(t> = Cx^ \~n{\~Cxf

^' ^maxJ

p-\

^ so

oe

r-

•"

fo r*

^

V4

^/J

(6.93)

Bed expansion in particulate fluidization may be described by the Richardson-Zaki equation [38]. — = —^^-^

uu = U:e"

(6.94)

where u^ is the superficial gas velocity and n is an exponential parameter given by: -0.1 n = 4.45 + 1 8 ^ ^ Re-""'

(6.95)

for ]^r —

kK^LXog^îe)

c

where K^ is the mean value for the extinction coefficient. For non-reentrant (convex) particles, the ratio of the surface and projected area shape coefficients {a/k) is equal to 4. For re-entrant particles, the surface obtained by making this assumption is the envelope surface area. Making this assumption equation (7.16) simplifies to: S^^

^'^ """^^ K^L c

(7.18)

7.5,2 The extinction coefficient The extinction coefficient varies with the optical properties of the solid and liquid that make up the suspension. Knowing these properties, it is possible to generate a relationship between the extinction coefficient and particle size using Mie or a boundary condition theory. Since unit area of 0.2 |Lim Ti02 cuts of 10 times as much light as unit area of 0.1 |im Ti02, (Figure 7.9), if no correction is applied the measured distribution for submicron Ti02 will be heavily weighted towards the coarser particles. The relationships hold for an infinitely small solid angle between the detector and the suspension so a correction for the geometry of the analyzer may be


required. Alternatively, the instrument may be calibrated against some external standard. If no correction is made for variation in extinction coefficient {K =" \) the derived distribution is only a size dependent response and the method becomes a fingerprint method (i.e. useful for comparison purposes only).

0.2

0.4 0.6 Panicle size (JC) in microns

0.8

1

Fig. 7.9 Extinction curve for titanium dioxide in water for white light. 7.5.3 Turbidity measurements (Turbidimetry) Turbidimetry measurements, using monochromatic light, yields data that can be used to determine particle size distributions. It requires simple optical technology, but complex computational software to handle the Mie theory conversion. The sensitive diameter range for latex-water suspensions was found to be 0.1 to 10 |Lim. Different types of sensors have been conceived and applied to various experimental situations. The method is particularly useful in crystallization experiments. Other applications include agglomeration, attrition and nucleation studies. Applications of the equipment and software to studies of emulsions, fumes and aerosols are also envisaged [18]. 7.5.4 The photosedimentation technique The photosedimentometer combines gravitational settling with photoelectric measurement. The principle of the technique is that a narrow

Gravitational sedimentation methods

371

horizontal beam of parallel light is projected through the suspension at a known depth on to a photocell. Assuming an initially homogeneous suspension, the attenuation at any time will be related to the undersize concentration.

ac»6.0

Fig. 7.10 Polar light scattering diagrams [19]. The outer curve magnifies the inner by a factor of 10 in order to show fine detail, x = (nD/A) where D = particle diameter and A is the wavelength of light. Superficially, the attenuation is related to the random projected areas of the particles. The relationship is more complex than this however, due to the breakdown in the laws of geometric optics so that complex diffraction, scattering, interference and absorption effects have to be considered. For small particles, an amount of light flux, equal in magnitude to that incident upon the particle, is diffracted away from the forward direction (Figure 7.10), making their effective obscuration area twice their projected

3 72 Powder sampling and particle size determination

area. As the particle size increases, the diffracted light is contained in a decreasing solid angle in the forward direction. No matter how small the light detector, most of the diffracted light is accepted and the effective obscuration area becomes the same as the projected area. For partially transparent particles, some of the incident light is absorbed and some refracted to cause constructive and destructive interference in the transmitted beam. It cannot therefore be assumed that each particle obstructs the light with its geometric cross-sectional area. These effects are compensated for by inclusion of an extinction coefficient {K) in the equation, making the apparent area K times the geometric area. Early experimenters [20,21] were either unaware of, or neglected, this correction. Some research workers used monochromatic light and determined K theoretically [19,22] others used empirical calibration by comparison with some other particle sizing technique. Rose and Lloyd [23] attempted to define a universal calibration curve. Allen [24,25] designed a wide angle scanning photosedimentometer (WASP) which accepted forward scattered light so that K was constant down to a size of around 3 fim. Weichert [26] determined a relative extinction coefficient by the use of different wavelengths and speeded up the analysis by the use of different settling heights 7.5.5 Commercial photosedimentometers Kemsis K200 is a white light, narrow angle, scanning photosedimentometer. Caron et. al [27] described an application of this instrument for studies of turbidity and sedimentation measurements of solid-liquid dispersions. Sedimage 1000 uses a white light source and a linear sensor with 2048 detectors to determine the concentration gradient within a settling suspension. A linear image sensor, 28.6 mm in length, measures the transmitted light along this distance. The length of the image sensor limits the measurable height of particle sedimentation. The instrument continuously monitors the changing concentration of the settling suspension and the analysis is deemed complete when the smallest particle present passes the upper detector. After about 5 minutes the particle size distribution of a carborundum powder having a median size of 5 |Lim can be accurately determined [28] whereas conventional scanning photosedimentometers take around 20 minutes to carry out an analysis A method of determining the particle size distribution from a single measurement, with a digital image acquisition system, on a sedimenting


3 73

suspension has been presented [87]. Individual particles illuminated by a laser light sheet are tracked by a continuously operating CCD camera. The projected areas, shape factors and the centers of gravity are detected during the sedimentation process from a series of images with a constant time spread. As the algorithm is based on single particle tracking, the heterogeneity of the sample can be taken into account. From these measured particle characteristics particle sizes and settling rates are determined.

-6^

Reflected ?• \ /' Light

\-

Cuvet

Fig. 7.11 The Retsch Paar Lumosed, depths of beams are marked in mm. Retsch PAAR Lumosed (Figure 7.11), operates in the gravitational size range, with three light sources at different depths to speed up the analysis [29,30]. A range of cuvette photocentrifuges which also operate in the gravitational mode are also available commercially. With these instruments a K factor, obtained either theoretically or experimentally, can be inserted in the software algorithm. A photosedimentometer has also been described for measuring freefalling diameters up to 20 cm in size with an application using coke. The system has been used to accurately measure a range of materials [31]. 7.5.6 Sedimentation image analysis The basic idea of this method is the analysis of particles settling under gravity using a digital image acquisition system. A continuously operating CCD camera, with a frame grabber, tracks individual particles illuminated


by a laser light sheet. The laser light sheet is arranged vertically at about one-third from the bottom of the sedimentation vessel. Since the particles in the beam need to be representative of the whole sample, the analysis is over when the largest particles settle below the measurement volume. Consequently, short measurement times are not only desirable but necessary. The projected areas, shape factors, circularities and centers of gravity are determined, together with the settling velocities, during the sedimentation process from a series of images with constant time spread. The capacity of this method was demonstrated first for nearly spherical particles [32]. However it was found that problems arise when particles deviate strongly from spherical shape [33]. Later developments overcame these problems and allowed the determination of particle shape [34]. For every settling particle the projected area diameter is obtained from several images at different positions of the particle and a mean value calculated. Consequently the influence of particle shape on the sizing technique is greatly reduced. The number of detected particles is in the range of 3000 to give a reliable (number) distribution. The sizing technique was verified with BCR materials and applied to the particle size analysis of soils. 7.5.7 Transmission fluctuation spectrometry If an extinction measurement is made with high spatial and temporal resolution, the transmitted signal shows significant fluctuations. The strength of fluctuation is related to the physical properties of the suspension and the process of spatial and temporal averaging. This connection has been exploited to calculate the particle size distribution and particle concentration from transmission measurements. A theory of temporal transmission fluctuations has been developed for the case where the beam diameter was much smaller than the particle diameter [35]. This was expanded to provide an analytical solution to the use of a focused laser beam where the beam diameter is different for each monolayer according to the variation of the beam cross-section along the path length [36]. 7.6 Theory for concentration determination gravitational sedimentation technique

with

the

x-ray

A natural extension to the use of visible radiation is to use x-rays. In this case the x-ray density is proportional to the weight of powder in the beam. The Beer-Lambert law takes the form: I = IQQXP(-BC)

(7.19)


3 75

where J5 is a constant related to the atomic number of the powder in suspension, c is the powder concentration, / is the emergent flux with suspension in the beam and /Q is the emergent flux with clear suspension liquid. E, the x-ray density, is defined as: E = -\og{III^)

(7.20)

For powders of low atomic number, c needs to be high in order to obtain a large enough signal. Thus for silica powders, atomic number 13, a volume concentration of around 3% may be necessary and this can lead to hindered settling. 7.6.1 X-ray sedimentation Brown and Skrebowski [37] first suggested the use of x-rays for particle size analysis and this resulted in the ICI x-ray sedimentometer [38,39]. In this instrument, a system is used in which the difference in intensity of an x-ray beam that has passed through the suspension in one half of a twin sedimentation tank, and the intensity of a reference beam which has passed through an equal thickness of clear liquid in the other half, produces an inbalance in the current produced in a differential ionization chamber. This eliminates errors due to the instability of the total output of the source, but assumes a good stability in the beam direction. Since this is not the case, the instrument suffers from zero drift that affects the results. The 18 keV radiation is produced by a water-cooled x-ray tube and monitored by the ionization chamber. This chamber measures the difference in x-ray intensity in the form of an electric current that is amplified and displayed on a pen recorder. The intensity is taken as directly proportional to the powder concentration in the beam. The sedimentation curve is converted to a cumulative percentage frequency using this proportionality and Stokes equation. The introduction of more stable x-ray sources and detectors resulted in the development of simpler, commercially viable systems. Kalshoven [40,41] described an x-ray instrument which used a special program for scanning the sedimentation tank. As the concentration measurements by means of x-ray attenuation are very rapid, the scanning greatly speeds up the analysis, reducing the measurement time down to a few minutes. In this instrument it was done in such a way that the concentration, and hence the cumulative mass percentage undersize, is recorded as a function of Stokes diameter rather than time. An x-ray tube was used as a source and


a scintillation counter as a detector. The difference in intensity between a measurement beam and a reference beam, in which the emitted beam is alternately split, was measured by a rotating wedge that automatically set the difference to zero. Sub-micron particles can be measured if the sedimentation tank is spun in a centrifuge for some time. The time integral of the centrifugal force is measured and the tank is scanned after the centrifugation. The inventor claimed that volume concentrations in the range 0.01% to 1% could be used, depending on the atomic number of the analyzed material. Experimentally it was found that readings could be taken at short distances below the surface without seriously affecting the results. When the centrifuge was used the results were independent of the time of centrifugation but no comparison analyses were presented. Several commercial instruments utilizing these principles were developed. Oliver et. al. [42] patented a gravitational x-ray particle size analyzer that incorporated the absorption technique and improved the system described by Kalshoven. This instrument was described by Hendrix and Orr [43] and is available commercially as the Micromeretics' Sedigraph 5000 (Figure 7.12). The instrument automatically presents results as a cumulative mass percentage distribution , and the sedimentation tank is driven in such a way that the concentration is recorded directly as a function of Stokes diameter. An air cooled, low power x-ray tube is used for generation of x-rays. These are collimated into a narrow beam that passes through approximately 0.14 in, (3.6 mm) thickness of suspension. The sedimentation tank is only 1.375 in (35 mm) high, is closed at the top and, in use, completely filled with suspension. Filling and emptying of the tank is accomplished with a built-in circulating pump. The transmitted radiation is detected as pulses by a scintillation detector, these are amplified and discriminated to eliminate low energy extraneous noise. The pulses are next clipped to constant amplitude and fed to a diode pump circuit which, in conjunction with an operational amplifier with a diode feedback, gives a voltage proportional to the logarithm of the x-ray intensity and is therefore proportional to the powder concentration. The instrument can analyze powders with atomic numbers greater than 13, but rather high initial volume concentrations of powder have to be used for powders having low x-ray adsorption (0.5% to 3%). This is due for the need for the initial decrease in x-ray intensity to be greater than 20% of the intensity with clean liquid in order to obtain reasonable resolution of the resulting attenuation curve. An absolute system is used here, in which the initial intensity is first measured with clean liquid in the cell and the zero set; the suspension is then introduced. This assumes an excellent stability of the source that, in the case of x-ray tubes, may be unreliable. The authors

Gravitational sedimentation

methods

377

claim a good reproducibility and present several comparison analyses with microscopy etc. The range is claimed to be 300 to 0.1 |Lim for most powders. This lower limit is unreal since it is generally accepted that gravitational sedimentation is limited to particle coarser than around a micron due to the effects of Brownian motion. The acceptance by the manufacturers of this lower size has affected the instrument design in that only a 35 mm fall-height is possible and this restricts the upper size limit [44].

Outlet

SUt movement

X-iay tube

Detector Relative concentration signal

Inlet Digital position translator

Pump

Sample or

pure liquid

CeU positioning signal

Digital program computer

1 ^

50 5 0.5 Particle size in microns

Digital-to-position translator

Fig. 7.12 The Micromeretics Sedigraph. Sedigraph 5100 was later designed with three scanning speeds, slow, standard and fast. Micromeretics' Windows-compatible operating software permits automatic overlaying of plots, saves and recalls sets of run conditions, and can operate as many as four Sedigraphs from a single computer. The temperature of the suspension is controlled automatically with heaters located in the mixing chamber. The mixing chamber is situated outside the instrument, thus eliminating the need to switch off the x-rays when changing samples, which applies to the earlier version. The Sedigraph 5100 is also available with the MasterTech 051 Autosampler so that as many as eighteen samples can be analyzed unattended.


Quantachrome Microscan reduced the time for an analysis from about 45 to about 25 minutes. In this instrument the source and detector scanned up the sedimentation tank rather than the other way round; this was claimed to reduce vibration. In 1970 Allen and Svarovsky [45-47] developed an instrument in which the traditional x-ray tube was replaced by an isotope source. Allen and Svarovsky's design was incorporated in the Ladal x-ray scanning gravitational sedimentometer and the x-ray centrifugal sedimentometer, which are no longer commercially available. A later design of Allen's is available as the Brookhaven BI-XDC. This can operate in both the gravitational and centrifugal mode which greatly increases the size range covered. In the centrifugal mode the size range of nominal 0.2 |im titanium dioxide was found to be much narrower than in the gravitational mode and a 'phantom' bimodality appeared in the gravitational analysis which was ascribed to thermal diffusion i.e. Brownian motion. Many industries have large data banks on product size distributions by sieve analysis and want to continue using this form of presentation. In order to accommodate this need Cho et. al [48] converted Sedigraph data to sieve data using wet screened powder in the 38 to 53 \xm size range and fitted the data to a logarithmic distribution to give the slope and median size. This procedure must be use with caution since the conversion factors are shape dependent and a new calibration is required for each product. A review of these and other methods of size analysis is contained in a thesis by Svarovsky [49], papers by Svarovsky and Allen [50], and by Allen and Davies [51]. 7.7 Relationship between density gradient and concentration Following from equation 7.6: Let ^/z,/) be the density of the suspension at depth h and at time /. Then: {h,t)--

Ps^'s+Pf^f (/>(h,t) = ">"/


379

^^h,t)=^''-^^'^'^-'^>'^ V,+Vf (t>{h,t) = pf+Ps

^s+^f

Ps Also

Ps

Therefore

CXM.^.^Ml^ C(/z,o)

(7.21)

"^ (l>{KO)-Pf

where ^ /s the mass fraction undersize d^^. A plot of 100 xN^(/?,/)-py)/(^(/2,0)-/7y^M against d^^ gives a cumulative mass percentage undersize curve. 7.8 Hydrometers and divers 7.8.1 Introduction The changes in density of a settling suspension may be followed with a hydrometer, a method widely used in soil science and in the ceramic industry. A suspension of known concentration is made up and the hydrometer inserted. Some operators leave the hydrometer in the suspension throughout the analysis and some remove it after each reading and replace it slowly before the next. Objections can be raised to either procedure since, in the former, particles settle on the hydrometer causing it to sink to a lower level than it would otherwise sink whereas, in the latter,


the suspension is disturbed after each reading. To minimize errors some operators re-shake the container after each reading. 7.8.2 Theory With the hydrometer immersed, its weight W equals the weight of suspension displaced. Let the length of stem immersed in clear suspending liquid be Z, i.e. the same as would be immersed at infinite time; the length immersed at the commencement of the analysis be L^^ and the length immersed at time t be L^. Then, at the commencement of the analysis: W = V(f>{h,Q) + Lâp^

(7.22)

At time / = oo (clear liquid in the container) W = Vpj+Lapf

(7.23)

During the analysis, at time t W = V(t){h,j) + L,ap2

(7.24)

where Fis the volume of the hydrometer bulb, a the cross-sectional area of the stem and hj the depth of the hydrometer bulk at time t. Since the density of the suspension around the stem (pj, pj^ p^ varies negligibly compared with the variation in L equation (7.21) can be written: L,-L

^ (/>{h,,t)-py

LQ-L

(/>{h^,0)-pf

= ^^^LZJ^

(7.25)

WQ-W

where w is the specific gravity marked on the hydrometer stem. If the suspension is made up of W gram of powder making up 1 L of suspension, equation (7.25) can be written:


381

Fig. 7.13 Depth of immersion using a hydrometer.

^

1000 A ( A - / ^ / )

W

(7.26)

Ps-Pf

where p^ is the density of the suspension at time /. An equivalent formula fpr powders that are present as slurries in water removes the necessity for drying out the slurry. A specific gravity bottle is filled with water and weighed; the water is replaced with the slurry under test and the bottle is re-weighed; the difference in weights being /SW. The sample is then taken out of the bottle and used for the analysis. The equivalent formula is:

mQ(p,-pf)

^w

(7.27)

7.8,3 Depth of immersion With the hydrometer technique, both density and depth of immersion vary with each reading. If the temperature is maintained constant at the hydrometer calibration temperature the density may be read directly from the hydrometer stem otherwise a correction needs to be applied [52].


150-170

did

Fig. 7.14 Hydrometer (Calibration in gml"^ at20°C. All dimensions in mm). It is clear that a hydrometer with a long bulb does not measure density at a point; it only measures the average density of the suspension displaced by the hydrometer. The difficulty lies in determining the point of reference below the surface to which this density refers, for when the hydrometer is placed in the suspension the liquid level rises in the container, thus giving a false reference point (Fig. 7.13).


383

If the cross-sectional area of the container is A, the depth to be used in Stokes equation, from geometrical considerations, is [53]: L = L^+-\ L2--\ 2V Aj

(7.28)

Several workers, who claim that corrections have to be applied for the density gradient about the bulb, and the displacement of suspension by the stem, have challenged this simple formula. Johnson [54] for example, gives the sedimentation depth, in cm, as:

^ 2

-0.5

(7.29)

7.8.4 Experimental procedure The changes in density of a sedimenting suspension may be followed with a hydrometer (Figure 7.14), a method still used in the ceramic industry. In order to achieve sufficient accuracy in the specific gravity readings, it is necessary to use a concentration of at least 40 g L'^, which is well into the hindered settling region. The only justification for this that has been advanced is that the method gives reproducible results. Most hydrometers are calibrated to be read at the bottom of the meniscus and this is usually not possible when the hydrometer is immersed in a suspension. The readings are, therefore, taken at the top of the meniscus and an experimentally determined correction, which is usually of the order of 0.003 g ml"^ applied. It is usual to disregard calibration errors although these may be substantial. Good quality hydrometers are usually guaranteed to ±0.0005 g ml"^ which corresponds to an error of around ±1.5% under normal operating conditions. Johnson [54] recognized this error and suggested that it should be determined at several points by calibration in a series of dilute suspensions of common salt. A correction to meniscus reading error and density should also be applied if a wetting agent is used. The resolution, see Section 7.2, is particularly poor for the hydrometer method of size analysis where the height of the measurement zone is of the same magnitude as the depth of immersion in the suspension. Although the hydrometer cannot be recommended as an absolute instrument, it is useful for control work with wide size range continuous distributions. It is of little use with discontinuous size distributions since these give sharp


boundaries in the settling suspension, which lead to peculiar results. The method for carrying out a hydrometer analysis is given in BS 1377 [55]. 7.8.5 Divers Divers overcome many of the objections associated with the hydrometer technique. These miniature hydrometers were developed by Berg [56] for use with both gravitational and centrifugal sedimentation, but have never been widely used. Basically, divers are small objects of known density that are immersed in the suspension so that they find their density level. Berg's divers for example, were hollow glass containers that contained mercury to give the desired density. The density was then adjusted to the desired value by etching with hydrofluoric acid. Various modified divers were later developed, the final ones, by Kaye and James [57], being metal coated polythene spheres which were located with search coils. 7.9 Homogeneous cumulative gravitational sedimentation 7.9.7 Introduction The principle of this method is the determination of the rate at which particles settle out of a homogeneous suspension. This may be done by extracting the sediment and weighing it; allowing the sediment to fall on to a balance pan or determining the weight of powder still in suspension by using a manometer or pressure transducer. One problem associated with this technique is that the sediment consists both of oversize (greater than Stokes diameter) and undersize particles so that the sedimentation curve of amount settled (P) against time (0 has to be differentiated to yield the weight {W) larger than Stokes diameter. Several balance systems, based on this equation, have been described. 7.9.2 Theory The theory given below was developed by Oden [58] and modified by Coutts and Crowthers [59] and Bostock [60]. Consider a distribution of the form:

W=

\ d=dst

f{d)dd


385

where W is the mass percentage having a diameter greater than Stokes diameter. The weight percentage P, which has settled out in time t, is made up of two parts: One consists of all the particles with a falling speed equal or greater than u^^, the other consists of particles with a smaller falling speed which have settled out because they started off at some intermediate position in the fluid column (Figure 7.2). If the falling velocity of one of these particles is u, the fraction of particles of this size that will have fallen out at time t is ut/h, where h is the height of the suspension. Hence: "max

P= ^f{d)dd+

'^V

ljfid}dd

(7.30)

Differentiating with respect to time and multiplying by /:

'^'Ijm^

(7.31,

I.e. dP P = W-\-t— dt

(7.32)

Since P and / are known, it is possible to determine fusing this equation. It is preferable, however to use the equation in the following form [61]. dP P^W + -^^^ (7.33) din/ Several methods of applying this equation have been suggested. The most obvious is to tabulate / and P and hence derive dP, dt and finally W. Alternatively, P may be plotted against t and tangents drawn. A tangent drawn at point {P^^t^ will intercept the abscissa at W^, the weight percentage oversize dr. Another method is to tabulate P against t at times such that the ratio of {t/dt) remains constant, i.e. at time intervals in a geometric progression; a simple expression relating W and P then develops [62].


Weighing mechanism Thermal jacket \::::1

Adjustable balance clamp \ mmttrntrnM RMiMniiRRlMii

L-J

(a)

Pressure equalizing tube Sedimentatioii column Balance pan

I

(b)

Fig. 7.15 (a) Sedimentation balance with pan in the suspension, (b) Sedimentation balance with pan in clear liquid (Leschonski modification of the Sartorius balance) Many powders have a wide size distribution and, in such cases, the time axis becomes cramped at the lower end or unduly extended at the upper end; in such cases equation (7.33) should be applied. Evaluation proceeds from a plot of P against In t; tangents are drawn every half-unit of In t\ the point where the tangent cuts the ordinate line one In t unit less than the value at which it is tangential gives the weight percentage oversize W at that value [63]. 7.9.3 Sedimentation balances In the Gallenkamp balance [64,65] the pan is placed below a sedimentation chamber with an open bottom and the whole assembly is placed in a second chamber filled with sedimentation liquid so that all the powder falls on to the pan. The weight settled is determined from the deflection of a torsion wire, and either, the run continues until all the powder has settled out of suspension, or a second experiment is carried out to determine the supernatant fraction. Problems arise during the charging operation with leakage into the clear water reservoir and particle adhesion to the premixing tube.


387

In the Sartorius balance [66-68] the pan is suspended in the suspending liquid and a correction has to be applied for the particles which fall between the rim of the pan and the sedimentation vessel. In this instrument, when 2 mg of sediment has deposited, electronic circuitry activates a stepby-step motor which twists a torsion wire to bring the beam back to its original position. A pen records each step on a chart. The manufacturers suggest that about 8% of the powder does not settle on the pan. Leschonski [69] and Leschonski and Alex [70] reported losses of between 10% and 35%, depending on the fineness of the powder; the difference was attributed to the pumping action of the pan as it re-balances. Leschonski modified the instrument (Figure 7.15) by placing the pan at the bottom of a sedimenting column surrounded by a second column of clear liquid so that all the powder settled on to the pan. This eliminated powder losses and resulted in more accurate analyses [71]. The manufacturers of the Cahn micro-balance make available an accessory to convert it into a sedimentation balance [72]. The balance pan is immediately below the sedimentation cylinder in order to eliminate convection currents. Shimadzu also make a beam balance [73] that operates using a simple compensating system that is prone to considerable error. Yodshida et.al [74] describe an improved sedimentation balance. They compared the results using this balance with those from microscope counting, using three kinds of standard reference beads, and found good agreement. Fukui et al investigated data reduction and sedimentation distance for sedimentation balances [75}. 7.9.4 Sedimentation columns Sedimentation columns (ICI, BCURA) have also been described in which the sediment is extracted, dried and weighed. A full description of these and other sedimentation columns may be found in [1] 7.10 Line-start incremental gravitational sedimentation 7.70.7 Photosedimentation The Horiba cuvette photo(centri)fuge has been operated in this mode [76] but is not recommended since it is very difficult to make up a stable twolayer system in a cuvette.


7.11 Line-start cumulative gravitational sedimentation 7.11.1 Introduction If the powder is initially concentrated in a thin layer floating on the top of a suspending fluid, the size distribution may be determined by plotting the fractional weight settled against the free falling diameter. 7.11.2 Methods Marshall [77] was the first to use this principle. Eadie and Payne [78] developed the Micromerograph, the only method in which the suspending fluid is air. Brezina [79,80] developed a similar water based system, the Granumeter, which operated in the sieve size range, and was intended as a replacement for sieve analyses. The Werner and Travis methods [81,82] also operate on the layer principle but their methods have found little favor due to the basic instability of the system; a dense liquid on top of a less dense liquid being responsible for a phenomenon known as streaming in which the suspension settles en masse in the form of pockets of particles which fall rapidly through the clear liquid leaving a tail of particles behind. Whitby [83] eliminated this fault by using a clear liquid with a density greater than that of the suspension. He also extended the size range covered by using centrifugal settling for the finer fraction. The apparatus enjoyed wide commercial success as the (Mines Safety Appliances) MSA Particle Size Analyzer although it is less widely used today [84]. The MSA analyzer can be operated in the gravitational mode, although it is more usually used in the centrifugal mode. Several papers have been published on applications of this equipment. The line-start technique has also been used to fractionate UO3 particles by measuring the radioactivity at the bottom of a tube, the settled powder being washed out at regular intervals without disturbing the sediment [85]. References 1 Allen, T. (1990), Particle Size Measurement, Chapman & Hall, 4th ed., 360, 388 2 British Standard 3406, Determination of particle size distribution, Part 2 Gravitational methods, 360 3 British Standard 3406, Determination of particle size distribution, Part 6, Centrifugal methods, 360 4 DIN 66111, Particle Size Analysis, 360


389

5 DIN 66115 Particle Size Analysis, 360 6 NFX 11-681 Test Methods for Particle Size Analysis-Particle Size Analysis by Gravity Sedimentation in a Liquid Medium, 360 7 NF 11-683 Test Methods for Particle Size-Analysis-Particle Size by Variable Height Gravity Sedimentation in a Liquid-Method Using X-ray Adsorption Measurements, 360 8 ISO/WD 13317-1 Determination of Particle Size Distribution by Gravitational Liquid Sedimentation Methods- Part 1 (1996) General Principles and Guidelines, 360 9 ISO/WD 13317-3 Determination of Particle Size Distribution by Gravitational Liquid Sedimentation Methods- Part 3 (1996) The X-ray Gravitational Technique, 360 10 JIS Z8820 General Rules for the Determination of Particle Size distribution by Sedimentation in Liquid, 360 11 ASTM C958 (1997), Standard Test Method for Particle Size-Analysis of Alumina or Quartz by X-ray Monitoring of Gravity Sedimentation, 360 12 ASTM B761 (1998), Test Method for Particle Size-Analysis of Refractory Metals and their Compounds by X-ray Monitoring of Gravity Sedimentation, 360 13 Jillavenkatesa, A, Dapkunas, S.J. and Lum, L-S, H. (2001), Particle Size Characterization, NIST Sp. Publ. 960-1, 360 14 Robinson, G. W. (1922), J. Agr. ScL, (12)3, 305-321, 365 15 Andreasen, A.H.M. (1928), Kolloid Beith, 27, 405, 365 16 Andreasen, A. H.M. and Lundberg, J. J. V. (1953), Ber Dt. Keram. Ges., 11(5), 312-323,5^5 17 Allen, T. (1969), Powder Technology, 2(3), 132-140, 365 18 Crawley, G., Coumil, M. and Benedetto, D.D. (1997), Powder Technol 91(3), 197-208,577 19 Vouk, V. (1948), Ph D thesis, London University, 372, 373 20 Jarrett, B.A. and Heywood, H. (1954), Br. J. Appl Phys., Suppl. No 3, S21, 373 21 Morgan, V.T. (1954), Symp. Powder Metall, Iron and Steel Inst., Preprint Group 1,33-38, 575 22 Lewis P.C. and Lothian, G.F. (1954), Br. J. Appl. Phys., Suppl. No 3, S571, 373 23 Rose, H.E. and Lloyd, H.B. (1946), J. Soc. Chem. Ind, 65, 52, 373 24 Allen, T. (1968), Powder Technol., 2, 132-140,, 373 25 Allen, T. (1968), Powder Technol., 2, 141-153, 373 26 Weichert, R. (1981), Proc. Particle Size Analysis Conference, 303-310, ed. Allen, T. and Stanley-Wood, N.G. publ.John Wiley, 373 27 Caron, P., Fancompre, B., Membrey, F. and Foissy, A. (1996), Powder Technol., H9,9\-\00, 373 28 Ma, X. and Wan, M. (1997), Part. Part. Syst. Charact., 14, 267-271, 373 29 Staudinger, G., Hangl, M. and Peschtl, P. (1986), Proa Partec Nurnberg, publ. NiimbergMesse, 374


30 Staudinger, G., Hangl, M. and Peschtl, P. (1986), Part. Part. Syst. Charact., 3, 158-162,57^ 31 Marquard, H. von (1998), Aufbereit Tech., 39(9), 461-466, Eng., 374 32 Hubner, T., Will, S. and Leipertz, A. (1998), Proc. 7'^ European Symp. Particle Charact., Partec '98, Numberg, 243-253, 375 33 Hubner, T., Will, S. and Leipertz, A. (1999), Part. Part. Syst. Charact., 16, 8591,575 34 Hubner, T., Will, S. and Leipertz, A. (2001), Part. Part. Syst. Charact., 18, 7078, 375 35 Breitenstein, M, Krauter, U. and Riebel, U. (1999), Part. Part. Syst. Charact., 16, 249-256,, 375 36 Breitenstein, M, Riebel, U. and Shen, J. (2001), Part. Part. Syst. Charact., 18, 134-141,575 37 Brown, J.F. and Skrebowski, J.N. (1954), Br. J. Appl. Phys., Suppl No 3, S27, 376 38 Conlin, S.G. et. al. (1967), J. Scient. Instrum., 44, 606-610, 376 39 Nonhebel, G. (1964), ed. Gas Purification Processes, Newnes, 376 40 Kalshoven, J. (1965), British Patent, BP 1 158 338, 376 41 Kalshoven, J.(1967), Conf. Proc. Particle Size Analysis, Soc. Analyt. Chem., London, 376 42 Oliver, J.P., Hicken, G.K. and Orr, C. (1969), US Patent 3,449,567, 377 43 Hendrix, W.P. and Orr, C. (1970), Proc. Conf. Particle Size Analysis, Soc. Analyt. Chem., London, pp 133-146, ed. M.J. Groves and J.L. Wyatt-Sargent, 377 44 Borothy, J. (1975), Chimia, 29(5), 240-242, 378 45 Allen, T. (1970), Br. Patent, 1764/70 3, 379 46 Allen, T. and Svarovsky, L. (1970), J. Phys E, 3, 458-460, 379 47 Allen, T. and Svarovsky, L. (1970), Proc. Particle Size Analysis Conf. Bradford, Publ. Soc. Anal. Chem., 379 48 Cho, H., Yildirim, K. and Austin, L.G. (1998), Powder TechnoL, 95(2), 109117, 57P 49 Svarovsky, L. (1972), PhD thesis, Univ. Bradford, Yorkshire, UK., 379 50 Svarovsky, L. and Allen, T. (1973), Paper presented at Heywood Memorial Symp., Univ. Loughborough, UK., 379 51 Allen, T. and Davies, R. (1989), 4th European Symp. Particle Charact., Numberg, Germany, Preprints 1, publ. NumbergMesse 17-46, 379 52 Stairmand, C. (1947), Symp. Particle Size Analysis, J. Inst. Chem. Engrs., 25, 110,5^2 53 Edwald, P. (1942), Ind Engg Chem., Analyt. edn., 14, 66, 383 54 Johnson, R. (1956), Trans. Ceram. Soc, 55, 237, 384 55 British Standard 1377, The Hydrometer Method of Particle Size Measurement, 384 56 Berg, S. (1940), Ingen Vidensk., Skr. B., No 2, Phys. Suppl. No 3, S27, 385 57 Kaye, B.H. and James, G.W. (1962), Br. J. Appl. Phys., 13, 415, 385 58 Oden, S. (1916), KolloidZ., 18, 33-47, 385


391

59 Courts, J. and Crowthers, E. M. (1925), Trans. Faraday Soc, 21, 374, 385 60 Bostock, W. (1952), JScient. Instrum., 29, 209, 385 61 Gaudin, A. M., Schumann, R. and Schlechter, A. W. (1942), 1 Phys, Chem., 46, 903, 386 62 Kim, S. C , Schlotzer, G. and Palik, E. S. (1967), Powder Technol, 1, 54-55, 386 63 Bostock, W. (1952), JScient. Instrum., 29, 209, 386 64 Bostock, W. (1952), JScient. Instrum., 29, 209, 387 65 Cohen, L. (1952), Instrum. Pract., 13, 1036, 387 66 Gerstenberg, H. (1957), Chem. Eng. Tech., 8, 589, 387 67 Bachman, D. (1959), Dechema Monograph, 31, 23-51, 387 68 Gerstenberg, H. (1959), Dechema Monograph, 31, 52-60, 387 69 Leschonski, K. (1962), Staub, 22, 475-486, 388 70 Leschonski, K. and Alex, W. (1970), Proc. Int. Symp. Particle Size, Bradford, 236-254, publ. Soc Anal. Chem. ed Groves and Wyatt-Sargent, 388 71 Pretorius, S.T. and Mandersloot, W.G.B. (1967), Powder Technol., 1, 23-27, 388 72 Kaye, B.H. and Davies, R. (1970), Proc. Conf. Particle Size Analysis, Bradford, 207-222, publ. Soc Anal. Chem. ed Groves, M. W. G. and WyattSargent, J., 388 73 Suito, E. and Arakawa, M. (1950), Bull. J Chem. Res.,Kyota University, 23, 7, 388 74 Yoshida, H., Masuda, H., Futui, K, and Tokunaga, Y. (2001), Adv, Powder Technol., 12(1), 79-94, 388 75 Fukui, K., Yoshida, H., Shiba, M. and Tokunaga, Y. (2000), J Chem. Japan, 33(3), 393-399, 388 76 Hofftnan, R.L. (1991), J Colloid Interf. ScL, 143, 232, 388 11 Marshall, C.E. (1930), Proc. Royal Soc, A126, 427, 389 78 Eadie, F.A. and Payne, R.E. (1954), Iron Age, 174, 99, 389 79 Brezina, J.J. (1969), Sediment. Petrol, 16, 27-31, 389 80 Brezina, J. (1970), Proc. Conf. Particle Size Analysis Bradford, publ. Society Anal. Chem. ed. Groves, M.W.G. and Wyatt-Sargent, J.L, 389 81 Werner, D. (1925), Trans. Farad Soc, 21, 381, i^P 82 Travis, P.M. (\940),ASTMBull., 29, 102, 389 83 Whitby, K.T. (1955), Heat. Pip. Air Cond, Jan., Part 1, 231, June, Part 2, 139, 389 84 Whitby, K.T., Algren, A.B. and Annis, J.C. (1958), ASTM Spec Publ. 234, 111,389 85 Imris, P. and Landsperky, H. (1956), Silikaty, 1956, 9(4), 327, 389

8

Centrifugal sedimentation methods of particle size determination 8J Introduction Gravitational sedimentation particle size measurement techniques are of limited value for particles smaller than about a micron due to the long settling times required. In addition, gravitational sedimentation devices generate inaccurate data due to the effect of convection, thermal diffusion and Brownian motion. Centrifuging the suspension in order to speed up the settling process reduces these errors. As with gravitational sedimentation, various options are available (Table 8.1). These may be categorized as: • • • •

variable time (/ varies, all other parameters remain constant); variable measurement radius (r varies, all other parameters remain constant); variable surface radius (S varies, all other parameters remain constant) combinations of these variables e.g. variable time and height {/ and (r/S) var}', all other parameters remain constant}.

Centrifugal techniques may be classified as incremental or cumulative, homogeneous or line-start. Incremental, line-start techniques are restricted to photocentrifuges and the attenuation has to be corrected for the breakdown in the laws of geometric optics unless the data are being used solely for comparison purposes. This correction can be considerable for powders having a wide size range, for example a 0.10 |nm particle may cut off less than a tenth of the light that one would expect from its geometric size whereas a 1 |Lim particle may cut off more than one would expect.

Centrifugal sedimentation methods 393 Table 8.1 Commercial sedimentation particle size analyzers Homogeneous, incremental centrifugal sedimentation Simcar centrifuge Ladal pipette centrifuge I .adal x-ray centrifuge Brookhaven scanning x-ray centrifuge Brookhaven Bl-DCP, disc photocentrifuge Kaye disc photocentrifuge Coulter photofuge Technord photocentrifuge Horiba cuvette photocentrifuges LUM Lumifuge 114 Seishin cuvette photocentrifuge Shimadzu cuvette photocentrifuge

Homogeneous, cumulative, centrifugal sedimentation Alpine centrifuge Hosokaw a M i kropu 1 Sedimentputer Line-start, incremental, centrifugal sedimentation Joyce-Loebl disc photocentrifuge Brookhaven BI-DCP, disc photocentrifuge CPS disc photocentrifuge Line-start, cumulative, centrifugal sedimentation MSA analyzer

A further problem arises, due to the presence of a range of particle sizes in the light beam at any one time, so that, even excluding extinction coefficient problems, the response may not be proportional to the projected area of the particles, in homogeneous, incremental, centrifugal techniques, (Figure 8.1) matters are also more complex than for homogeneous, incremental, gravitational sedimentation. The particles move in radial paths, hence the number of particles smaller than Stokes diameter entering the measurement zone is less than the number leaving, so that the measured concentration of these particles is smaller than their original concentration. In this presentation, some of the methods for centrifugal sedimentation particle size analysis in current use are described. Although operating procedures are not covered here, it is stressed that two factors, more than anything else, lead to incorrect analyses. The first is incorrect sampling, since analyses are carried out on from a tenth of a gram up to a few grams and these samples must be representative of the bulk for the analyses to be meaningful. The second is dispersion; it has been said rightly that the most important factor in obtaining accurate sedimentation data is dispersion the second most important factor is dispersion and the third is also dispersion!


Fig. 8.1 Homogeneous, incremental, centrifugal sedimentation (the radial dilution effect). 8.2 Stokes* equation for centrifugal sedimentation S. 2. / Genera! theory A particle settling in a centrifugal field is acted upon by a drag force and a centrifugal force. The force balance in the laminar flow region is given by:

-,(Ps-

Pf yi ^^^ = l(Ps-

PfVWr

- ^ncl,,^

(8.1)

where: r dr/dt Pg,pr // d^, rfj a)

= radial distance from the axis of the centrifuge to the particle; = outward velocity of the particle; = density of the particle and suspension medium respectively; ^ coefficient of viscosity of the medium; = volume diameter of the particle; =^ drag diameter of the particle; =•'• speed of rotation of the centrifuge in rad.s ^

At the terminal velocity, the outward radial acceleration, d^r/dt- ^ 0 so that this equation becomes:

Centrifugal sedimentation methods 395

(ps~Pf)dWr dr — û,.^ d/ ' I8/7

(8.2)

Thus, the settling velocity is not constant as in gravitational sedimentation but increases with increasing radius. Comparing with Stokes equation for gravitational settling (settling velocity under gravity ^ 11^^: CO r u^. -^ - — u == 0 when r- ^ S for / > 0, i.e. the surface concentration falls to zero on start-up; with the additional condition that F{d) '-^ 0 when d^^ 0. Thus: Qsr ' • « s y ) - jexp(2fo/^/)dO

&

,

.2

dQ

(8.12)

0

d^f is the diameter of the particle that settles from the surface, radius .S', to the measurement radius r in time /. The expression was first developed by Berg [1] and later by Kamack [2]. Berg solved equation (8.12) graphically by plotting (r/S) against Q and determining the area under the curve, deriving the following formulae:

J F(d)dd - c, -f ^" " 5c, - 4F ^ --; 1/••(c/jdc/ - c\ f 1 +2x^

for x < -

for c^ < 0.15

(8,13)

(8.14)

where r =5'+jc and c^ is the concentration at radius x at the time required for a particle of diameter d^ to fall from the surface to measurement radius x. These equations are applicable for powders having a wide size range, losing accuracy for monodisperse powders where volume concentration changes rapidly with changing r. Equation (8.13) is used for the calculation of the smallest F(d), that is the smallest c,, and equation (8.14) for higher values of F(d). F{d) ^ 0 when J = 0, hence F(d/2) may be estimated by joining F{d) to 0 for a small value of d since most functions are linear towards the origin. The F(d) curve is then built up step-by-step. Berg determined concentration using pipette withdrawal or small divers.

Centrifugal sedimentation methods 399 Kamack offered the following solution to equation (8.12). If g is plotted as a iiinction of v^ - {rJSy with f^ o9^t as parameter, a family of curves is obtained whose shape depends on the particle size distribution function. The boundaiy conditions are that Q = 1 when / ' - 0 for all r^ (i.e. the suspension is initially homogeneous) and Q^^ 0 for r- = S when t'>0 (i.e. the surface region is particle free as soon as the centrifuge bowl spins). Hence all the curves, except for t'>0, pass through the point QÔ, r- - S, and they will all be asymptotic to the line /' =- 0, which has the equation ig ^ 1. Furthermore, from equation (8.12), the areas under the curves are each equal to F{d^f), Let Q\ be the smallest experimentally determined concentration so that h-^h^^r ^'^d let Q be determined at a fixed sampling distance r for various values of t. Then one point is known on each curve in addition to the common point j / - 1 , ^ = 0. Such a set of points is illustrated by the black circles in Figure 8.2. To each point corresponds a known value of d^^ obtained from equation (8.4). Further, the area included between each curve, the concentration axis and the ordinates g = 0 and g, / s =2.136{?2-0-626F, - 1.823(?2-1I4I^, Fy =2.136^3-1.008^2-0-200^1 ^4 = 2.136^4-1.008^3-0.174(;2-0.0.023(;, ^5 =2.136(^5-1.008^4-0.174^3+0.021 Q2^^S)A(>Q^ Ff, = 2A36Qg-\ .008^5-0.174^4+0.021 Qn^+O.Ol 1Q2+OM1 g, F, = 2.136^-1.008(^,_,-0.174(^,_2+0.021 g,„3+0.021 ^,,4+ Particle size is determined using equation (8.23) and measured concentration is converted to mass undersize using equation (8.21). The results are presented in Table (8.4). Material; quartz p^ = 2650 kg m~-^ pj= 1000 kg m^^ N= 1500 rpm ( « = 5Qn rad s ') 7= 0.001 Pa s , 9 X 0.001 X n(7/

f

^(2650-1000)x(507t)^/,


Table 8.4 Mass percentage iindersize determination for homogeneous, incremental centrifuge technique (variable time method)

I

1 2 3 4 5 6 7 8 9

Time (7) (min)

Size iixm)

Measured (%) cone.

dst 0.118 0.167 0.237 0.335 0.473 0.669 0.947 1.339 1.893

(0

256 128 64 32 16 8 4 2 1

Mass % Iindersize ' (Z^)

!

6.9 18.7 34.5 65.5 88.4 97.3 99.4 100.0 100.0

256 128 64 32 16 78.2 87.9 93.6 96.7

Table 8.5 6,7 ^6,8 ^6,9

66.10 ^6

L^6

5.3 6.1 6.5 6.7 6.8 6.9 6.9 6.9 6.9 3.8 6.9 86,9 91.8 94.4 95.9 78.2 97.3

22,3 QiA Ql.S

Qis Qij

22,8 22,9 22,10 22 ^2

27,8 27,9 27,10 27 ^7

14.2 16.3 17.5 18.1 18.4 18.6 18.7 18.7 10.8 18.7 93.0 96.2 97.8 87.9 99.4

23,4 23.5 23,6 23.7 23,8 23,9 23,10 23 ^^

27.2 30.6 32.4 33.5 34.0 34.1 34.4

28,9 28,10 28

96.6 98.3 93.6 100

^8

21.6 34.5

24,5 24,6 24,7 24,8 24,9 24,10 24

n

29,10 29 ^9

51.9 58.2 61.8 63.6 64.5 65.0 41.7 65.5

98.6 100 100

25.6 25,7 25,8 25,9 25.10 25 ^^5

2,0 ^10

74.2

m3 , 84.7 86.6 i 87.6 62.7 88.4

100 100


8.4.4 The Ladal x-ray disc centrifuge (r constant, S constant, t variable) This is an extension of the Allen and Svarovsky [13] x-ray gravitational sedimentometer. The x-rays are generated by an isotope source and, after passing through the suspension, they are detected by a scintillation counter. The signal from the counter passes to a pre-amplifier, and thence to a ratemeter, and a trace is recorded by a pen recorder. The attenuation of the x-ray beam is proportional to the mass concentration at the measurement radius that has to be converted to the size distribution using the Kamack equation. A size range of about 8:1 is covered in about an hour. 8.4.5 Discussion of the Kamack equation The Kamack treatment builds up the concentration gradients within the centrifuge for each of the measurement times. The Qjj values (Table 8.5) may be determined using equation 8.19 and, in combination with the y^ j values, generate the concentration gradients. In Figure 8.2 the black circles are the measured concentrations at a fixed measurement and surface radius and the white circles give the calculated concentration gradient for / ^ 7. In essence we are building up the concentration gradient between the measurement radius and the surface for each measurement time. For example, at / = 2 the concentration at the measurement radius is 3.8% and, from the first measurement we deduce that the concentration at a radius y, Q\2 "" '^ 5.3%. In figure (8.2) the concentration gradient for / "= 7 is displayed as an example 8.5 Variable time and height method {S constant, both r and / vary) 8.5.1 Stokes diameter determination. The approximation due to Kamack can be modified, for the scanning mode of operation, by replacing the constant {r/S) with the variable (r/S) where r^ is the position of the source and detector at time / i.e. equation (8.4) becomes:


97 In J,

'Isij

(8.27)

\{Ps'-pfW', C-..2

V. here y- =• (r/S)" and c/'.M+>Vi..|

/d, \

KÛ

/ = 1,2,3,...., w; V Q / ^ 1

(8.29)

8.5.3 DuPont/Brookhaven scanning x-ray disc centrifugal sedimentometer (BI-XDC) (S constant, r variable, t variable) This instrument was developed as a centritiigai version of the Allen and Svarovsky's [13] x-ray gravitational sedimentometer in order to reduce the analysis time and measure down to smaller sizes [14,15], The x-rays are generated by an air cooled low power x-ray tube and, alter passing through the suspension, they are detected by a scintillation counter. The signal is


then processed to generate the size distribution. The attenuation is proportional to the mass concentration at the measurement radius, which has to be converted to the size distribution using the Kamack equation. A size range of about 8:1 is covered in about an hour. This instrument was designed [16] to fill a need for fast, reproducible sedimentation analyses in the sub-micron size range. The heart of the instrument is a hollow, x-ray transparent, disc which, under nomial operating conditions, contains 20 ml of suspension at a concentration of around 0.2% by volume. The centrifuge speed is selectable in the range 750 to 6000 ipm. The default condition is for the source and detector to remain stationary for 1 minute at a radial position of 48.00 mm and then to scan towards the surface. Total run time is normally 8 min. A commercial version is available from Brookhaven as the Bl-XDC (Figure 8.17). The instrument can operate in the gravitational or centrifugal mode and the analyses can be blended to cover a total size range of 0.05 |im to over 100 jiim. A size range of 15:1 is covered in a standard 8 min analysis. Weiner et. al. [17] describe its application to accelerated size analyses down to 10 nm. A computer controlled particle size measurement device with very high resolution was presented by Foerdeneuther [18] based on the Brookhaven optical and x-ray disc centrifuges. 8.5.4 Worked example (a) Delermination of Ffactors for a centrifuge with scanning Hyperbolic scan - The analyzer is scanning upwards, at a vaiying rate, from an initial radius of 7 cm to just below the surface at S =• 4.304 cm so that the following relationship holds: 18.7 r,^ =

,0.24

/ in seconds, /-/ in |im. The hyperbolic scan gives similar resolution at all sizes hence is preferable for centrifugal particle size analysis. S - 4.304 cm 77 - 0.001 Pa s

p^ = 2650 kg m"^ pj-- 1000 kg m"^

N- 1500 rpm (^ - 5071 rad s"^)

Centrifugal sedimentation methods 409 From equation (8.15) y = (7/4.304)2 = 2.645 From equation (8.27):

/is X o.oobTR'irviysoi)"

dSlj

4130x(507r)'x/.

where y,- = (r/S)^ Applying equation (8.29): F, = F, = F3 = F4 = F5 = Ff, = Fj -

1.020^, 1.065^2-0.059^1 1.138^3-0.105F2-0.030F, 1.235e4-0.]55F3-0.061F2-0.017F, !.367e5-0.227F4-0.094F3-0.035F2-0.010F, 1.563^6-0.356F5-0.132F4-O.O51 F3-O.0109F2-0.005F, 1.940g7-0.683F6-0.169F5-O.O57F4--O.O21 F3-O.OO8F2-O.OO2F1

Ihus the F values as given in Table (8.6) and they, values given in Table (8.7) can be determined. Table 8.6 Centrifuge particle size analysis, homogeneous mode with scanning Time

Radius

(0

F(secs.)

rîixm)

7 6 5 4 3 2 1

60 120 180 240 300 360 420

1 5.927 5.377 5.019 4.757 4.553 4.388

Stokes diameter dfiiym) 1.893 1.086 0.740 0.532 0.384 0.263 0.143

Measured concentration

C>,(%) 95.6 88.6 75.7 55.8 32.5 11.5 0.9

Percentage undersize F,(%) 99.9 97.2 87.0 62.6 35.7 12.2 0.92

410 Powder sampling and particle size determination Applying equation (8.28) to the V/values given in Table 8.5, the following Vj j values are determined: Table 8.7 j - for centrifuge particle size analysis, homogeneous mode with scanning V|

1.038 >'2

Vi.2 I-OH \y\3 '-005 >i4 1.003 j;,5

1.001

1.119

>'3

1.222

yA

1.360

>'2.3

1.054

>'2,4

1.028

yÂ

1.110

>'2.5

1.014

J^3,5

1.055

3^4,5

1.175

1.007

>'3.6

1.025

>'4,6

1.077

1.002

>'3,7

1.008 y^.i

V,.6 "-001 yifi \y\j '-^00 yi.i

1.025

V,

1.561

3-6

1-896

.V,6 '-229 Vj 1.070

Vy

1.234

8.6. Variable inner radius (Both 5' and t varj% r remains constant) H.6.1 Stokes diameter determination Let the time of the first withdrawal be /j; the largest particle present in the withdrawn sample at this time will have fallen from the surface at radius ^S* to the measurement zone at radius r. Equation (8.6) will apply and may be written: JrA =Â:ln — ' ' S

(8.30)

The liquid level will then fall to S^ where: S

-S'^

nh

(8.31)

where v is the volume extracted (10 cm^) and h is the thickness of the centrifuge bowl (1.02 cm). The fall in the inner radius can therefore be determined:

AS^S^

-S

(8.32)


Let the time for the second withdrawal be ^2; the largest particle present in the withdrawn sample will have fallen from S to x^ 2 '^ ^'^^^ î' ^ distance Axi 2 due to the withdrawal of the first sample and from Xj 2+^^î 2 to r in a time t2-ti. Hence: dh^ - y t l n ^

(8.33)

dUt2-h)-k\n

-—

(834)

Adding equations (8.33) and (8.34) gives:

dhi = ^ l n ~ 1 . ^ . 2 ^V ^12 J

(8.35)

For the third withdrawal; in time t^ particles of size ^3 will fall from the surface at radius S to x^ 3, hence: d^t^=k\n^

(8.36)

These partiqles will then fall a distance ZVT, 3 due to the withdrawal of the first sample where, from equation (8.31): (^1,3 -^\3f

-^13 =3.1207

(8.37)

In the next time increment, particles of size d^ will fall from radius Xj 3 H-Axj 3 to radius X2 3 hence:

^3 0 2 - ^ i ) = ^>n

"^ ^2,3 ^ ^^2,3

(8-38)

412 Powder sampling and particle size determination These particles will then fall a distance A.\:2 3 due to the withdrawal of the second sample where: (8.39)

(.To 3 - A x o j f - x f 3 =3.1207 and dj {t2,-t\) = kIn

(8.40) ^23 + ^^^2.3

Adding equations (8.36), (8.38) and (8.40) gives:

dh, =A:ln —

Axi1,3

1+-

-V

Ax.2,3 1+-

M,3 )

'•2,3

V" (8.41) ;

The bracketed terms are the correction terms for the fall in level due to each extraction. Repeating this gives, for the «th withdrawal: A\-|l.n

./,^/„^^ln S

I+V

X2ji

J

Zix;n--\,n

(8.42)

^n-lM J

This differs from the variable time equation in that the Stokes diameter reduces more rapidly thus, effectively, making this technique into a scanning technique. 8.6.2 Ladal pipette disc centrifuge The Ladal pipette centrifuge was developed as a centrifugal version of the Andreasen gravitational pipette [11]. This pipette centrifuge (Figure 8,3) was designed by Allen and Svarovsky [19] to operate with a reduced volume of suspension (150 ml), as compared to the Simcar. The modified Kamack equation, given above, was derived to correct for the changing surface radius as samples were extracted. One of the consequences of this M\ in surface radius with time, is a reduction in the measurement time down to about an hour. At 500 rpm the measured size range for quartz is


approximately 8 |Lim to 0.8 |am in 1 h and these sizes are halved if the speed is doubled. 8.6.3 Worked example (a) Determination of ratio of Stokes diameters for constant r, variable S and t. Using a feed volume of 150 cm^ gives AS = 4.146 cm; hence equation (8.33) becomes: dh

=)tln(7/4.146)

letting (k/î) be equal to 1.91 makes d^ = unity. Second extraction, for withdrawals in a 2:1 progression in time,

Equation (8.33) gives:

d^ = 1.91/«(jCi2/4.146)

Fig. 8.3 Line diagram of the Ladal pipette centrifuge.


Equation (8.34) gives:

^ | -1.91ln[7/(x|2 ^ ^^n)[

Hence

x^2 + -^1.2^^1,2 = 29.022

and from equation (8.31):

(xj 2 + ^X\ 2 ) --Af2-3.1207

Solving simultaneously gives v, 2 =^5.244 and zitj 2 "^ 0.2896; hence cA ^0.670. Assuming that this progression in sizes continues then ili^'-'"-0.670^^0.449. Substituting this value in equation (8.36) gives: x,3-4.146exp(0.449^/1.9]) ,v, 3 - 4.608 Substituting in equation (8.37): A.v, 3=0.327 From equation (8.38):

^23 -4.935exp(0.449^/1.9lj .V2 3 - 5.484

From equation (8.39): z\x:2 3^0.278; Substituting these values into equation (8.41) gives a more accurate value for J3, (/^ --^ 4/,). ,2 1.91 '' 7 (. 0.327 V ' r . 0.278'^'^ a- =" In 1 + _: 1+ 4.146 4.608; I 5.484 4 V ^ , ^ , J c/3 = 0.440

This iteration is repeated until the assumed value and the derived value are sensibly equal. The derived diameter ratios are shown in Table 8.8. Table 8.8 Ratio of particle sizes for extraction in a 2:1 progression in time Ratio of times

1 2

Ratioof sizes

1

0.670

4

8

16

32

64

0.438

0.282

0.179

0.112

0.070

Centrifugal sedimentation methods 415 8.6.4 Mass frequency undersize determination I ,et the final sample withdrawn be of concentration (?, and let the surface be at .S'l immediately prior to this withdrawal, then: ^1=^(1+>'1 )Q^

(8.43)

where: Vj =

(8.44)

(8.45)

and:

(8.46)

y.^^y^^''^

Hence, by the trapezoidal rule: /'V

/î4(>'2+>i2)(a-a2) f

where y2 -

^2

r \î J

Substituting for (9, 2 gives:

^2^7(v2+Jî,2)ft +

>'2 + y\i + > 1,2

Proceeding in a like manner gives the general formula:

(8.47)

416 Powder sampling and particle size determination By successively eliminating the Q functions, this gives a general equation in recursive form as before: /-I 7=1

f

yi

y+yi-\,i

yj.\..+yj.,

.v+3^M,,

IF,

(8.48)

yj^+yn.,

\2

r

and y,.x.,=y,

/"'

\î J

A numerical solution is given below for a feed volume of 150 cm^ and a 2:1 progression in time, K , values are given in Table (8.5) and these are inserted into the general equation to give the F values presented in Table 8.9. The dimensions of the centrifuge bowl are such that)', ^1.364. Table 8.9 Tabulated V/, values for a pipette centrifuge J, 2 1.1654 y|3

1.0793

V2 3 1-2216

.F,.4 10391

y2.4 1-1057

y,,; 1.0195

^2,5 1 0 5 1 9 >'3.5 '••352 >'4.5 '-3616

>1.6 '0099 yiA "0263 >;|7 1.0053 >'2 7 1.0140

>'3 4 1.2863 V 3 , 1.0632 74.6 ''715 .)'.s.6 ' 4 6 0 y.^^ 1.0353 V4^ 1.0882 .V5,7 '-224 .V6,7 1-601

Calculation of F values; using a feed volume of 150 cm^ as before. Equation (8.43) gives

F, = l.l82e, Equation (8.47) gives F. =J-(1.494 + 1.1654)C?T + 1 - ^"^^^^ F, ^ 2^ ^^2 2.1654


F2 = 1.330g2-0.228F, Equation (8.48) gives: f;^:^ 1(1.651 + 1.222)^3 +

"2.873 2.873" /s + 2302 2.302 _

2.873 F, 2.079

^ 3 - 14366(^3- 0.249/^- 0.133Fi and so on, giving the general equations for the conditions / ^ 7, V "= 150 cm^ presented in Table 8.10. An alternative approach to the procedure outlined above has been presented by Dumm and Hogg [20]. Table 8.10 Table of F values for a pipette withdrawal centrifuge F,-1.18220^ F2-1.330g2-0-228Fi F3 - 1.4366e3-0.2486F2-0.133F, F4 - 1.5658eF4-0.3093/^V0-1509F2-0.0757F, F5=1.7255e5-0.3836F4-0.1950F3-0.0877F2-0.0427F, F^=^1.9373g5-0-4713F3-0.2595FF4-0.1200F3-0.0521Fv~0.0249F{ F7^2.2255|27-0.5755F6-0.3489F5~0.1717FF4-0.0760F3-0.0323F2 .0153F 8.7 Photocentrifuges 8.7.1 Introduction In the photocentrifuge method the concentration of a suspension is monitored using a light beam. The light can come from either a white light source (an incandescent bulb) or a monochromatic coherent source (a laser) and the detector may be either a photodiode or photomultiplier. The signal from the detector is usually digitized and converted to a size distribution via a computer. Photocentrifuges are available in both disc and cuvette configuration. The former are normally used in the line-start mode and the latter in the homogeneous mode. The line-start mode has a much higher resolution


than the homogeneous mode so that muhimodal distributions are closely defined. The homogeneous mode can be run using a gradient procedure, with acceleration over time, which greatly speeds up the analysis. Both modes suffer the disadvantage that the laws oi^ geometric optics do not apply, and the correction required can introduce large errors, especially with size distributions having a wide size range. For the examination of paint pigments, end-use properties may be more closely related to the attenuation curve than the derived size distribution. It is therefore arguable that the measured relationship between attenuation and Stokes diameter should be used to define the powder rather than size distribution. 8.7,2 Disc

photocentrifuges.

The first disc photocentrifuge was developed by Kaye [21]. In this instrument, concentration changes within a suspension are followed using a white light beam. The instrument is usually used in the line-start mode and intuitively, one would expect that attenuation would be proportional to the projected area of the particles in the beam so that the curve of attenuation against Stokes diameter would be a differential surface distribution. However, Treasure [22] derived a relationship which showed that the attenuation, due to the finite width of the light beam, was proportional to the volume (mass) of particles in the beam [23,24]. In the line-start mode it is necessary to use a spin liquid that is denser than the suspension, otherwise the suspension can break through the interface and settle en-rnasse in a phenomenon known as streaming. In order to eliminate this effect a buffer layer technique is often used (Figure 8.4). Fractionation Buffer layer

vSusjKnsion Fig. 8.4 The line-start technique.

Spin liquid


The spin liquid may consist of 15 ml of 10% aqueous glycerol on which is floated 0.5 ml of water. For a more viscous suspension the concentration of glycerol can be increased. It is however necessary- that the buffer layer be less dense than the fill liquid so that inversion does not occur. An interface forms between the two liquids and this may be broken up by a momentary change in the speed of the centrifuge although this is not always necessary to eliminate streaming. Typically, 0,25 ml of dilute suspension is then introduced; this tends to break through the air-water interface and spread out on the diffuse interface between the buffer liquid and the spin liquid, which is the starting radius for the subsequent sedimentation process [25]. If streaming persists it may be eliminated by using much smaller volumes of buffer liquid and suspension, e.g. 0.1 ml. Coll and Seartes [26] used 20 ml of sucrose solution topped by 1 ml of/?dodecane to prevent evaporation. Several injections of 0.25 ml of colloid sample, ^ (001%) concentration, were then injected through the oil layer. In the external gradient method, a hypodermic syringe is used to form the gradient. For spin conditions requiring 15 ml of aqueous spin liquid exactly 15 ml are drawn into a 25 ml syringe. Air bubbles are expelled and, with the needle pointing down, an additional 1 ml of methanol is drawn in and the entire volume injected into the disc. Finally 1 ml of a suspension containing < 0.5% by volume solids in an 80:20 water/methanol solution are added. An application of this technique is described by Devon et. al. [27]. A density method with correction for light scattering has also been published [28]. It must be stressed that the raw curves are not size distributions and calibration is required to convert to absolute values [29]. The importance of the correction for the breakdown in the laws of geometric optics is stressed by Weiner et, al. [30] who show excellent agreement between theot7 and experiment when this is done correctly. They also use the Brookhaven disc photocentrifuge to characterize ASTM carbon blacks.[31] This method has been used to characterize void-containing latex particles [32]. Commercial instruments are available from Joyce-Loebl, CPS Instruments and Brookhaven. 8.7.3 Homogeneous mode This is the preferred mode with cuvettes; it is still however necessar>^ to correct for radial dilution effects.

420 Powder sampling and particle size determination (a) Stokes diameter determination For a constant (centrifuge) speed operation equation (8.4) is applied. In the gradient mode (centrifuge speed increasing with time in order to speed up the analysis) co\s replaced by an expression relating centritiige speed with time. (h) Mass frequency undersize determination The light cut off by particles in the light beam is related to the concentration by the equation: \n^-± = b''f ^

where

K,.n,d^

(8.49)

r=min

/Q

is the intensity of the emergent light beam when no particles are present / is the intensity of the light beam at time / after the start of sedimentation b is a constant depending on the dimensions of the light beam, the geometry of the system and the shape of the particles n^ is the number of particles in the beam of diameter d^, dc^f is the diameter of the largest particle in the light beam K^, is the extinction coefficient for a particle of diameter d^.

Kamack's equation is applied with the assumption that the attenuation is proportional to the product of the extinction coefficient and the crosssectional area of the particles in the beam i.e. ln(/(//) is replaced by the optical density D where: D = \og^ I The data are then converted to a mass distribution by summating the product of AD and dd^dd

(8.58)

The solution to this integral, if higher order terms are neglected, gives the proportionality: Dx(^;-f/i)(/ + g ) 4 D^{f^g)dl

(8.59)

since {0 ^p) is a constant. The optical density of the suspension is therefore proportional to the total particle volume in the beam as before i.e. DocKsfn{dst)di

(8.60)

Other published solutions state that the amount of light cut off is proportional to particle volume [34,35] for line-start and to particle surface for homogeneous mode.


Têlson e/. al [36] challenged these derivations. Using a general series expansion, they stated that the solution to equation (8.53) takes the form

/(^) = (;0+0)Xg/i;'

(8.61)

Thus, the fraction of particles in the field of view varies in a complex fashion with 6/^^. The conclusion drawn from equation (8.56) is at odds with published data on polystyrene lattices and silver bromide, in which a volume proportionality is found [37,38]. However these distributions were narrow, and with narrow distributions the difference between volume and surface distributions is small. The conclusion is also at variance with data published on BCR 66 quartz powder, ranging in size from 0.3 to 3 )im. In this case, the median for the attenuation curve was 1.52 ^m which reduced to 1.14 \xx\\ with extinction factor correction [39] and a correction of this magnitude could hide the effect. Centrifugal photosedimentation yields an attenuation curve; particles at the fme end of the distribution, at say 0.1 \xm, obscuring the light by perhaps one twentieth of their geometric area whereas at the coarse end, say 1 |am, the ratio can be greater than two. The correction for extinction coefficient modifies the shape of the curve considerably, making decisions as to correct theory to apply difficult. The unmodified attenuation curve may be more relevant to end-use properties, for hiding power of pigments for example, than the derived size distribution. Since the introduction of a correction for extinction coefficient has such an enormous effect on the shape of the distribution curve for wide distributions the correction should be applied with caution. Putman et. al. [40] used a CONTIN-based approach towards data analysis of photosedimentometry using the Shimadzu SA-CP2~10 to accurately determine size distributions. Contin is a mathematical FORTRAN IV program, which analyzes an experimental signal consisting of a sum of exponential curves, and determines its individual functions each with a weight.

428 Powder sampling and particle size

determination

SjnMplflTTFV

n BaBB 1=1

BI-DCP

Fig. 8.7 The Brookhaven disc photocentrifuge. 8.8 J BI-DCP disc (photo)centrifuge particle size analyzer The technology that Brookhaven developed for the x~ray centrifuge has been transferred to their photocentrifuge (Figure 8.7). The revised software is for analysis by the homogeneous start technique plus a scanning head detector. The high-resolution line-start technique can be used, but this is not amenable to scanning since the low concentrations necessarily employed generate a noisy baseline. An additional benefit of the new software is that it allows for particles whose density is lower than that of the surrounding liquid, thus making it suitable for emulsion sizing.

Centrifugal sedimentation methods

429

Ceotrifofalcett

OfMian

Option

Fig. 8.8 Block diagram of the Horiba cuvette photocentrifuge. 8.9 Cuvette photocentrifuges In these instruments (Figure 8.8) the disc is replaced with a rectangular cell containing a homogeneous suspension. Unless corrections are applied for radial dilution effects and the breakdown in the laws of geometric optics, the derived data are suitable only for comparison purposes. For example, a 50:50 mixture of 0.25 and 0.60 |im spherical silica particles was recorded as 54.4:45.6 with no correction for radial dilution, and this increased to 70:30 without extinction coefficient correction with the smaller particles grossly under-counted. A computer program to correct for the light scattering of small particles reduces these errors [41]. Bowen et. al [42] report on a method of programming the Horiba CAPA-700 to generate accurate sub-|im measurement of alumina and quartz powders. Using the manufacturer's correction for light scattering was found unsatisfactory. It was also found that the light scattering correction was strongly affected by the shape of the particles [43]. An alternative procedure is to use several wavelengths and deconvolute the resulting set of linear equations that develop in order to find the size distribution. This procedure was applied by Niemann and Weichert who used a modified Phillips-Twomey algorithm [44,45]. Their photo-


centrifuge used white light from a short arc xenon high-pressure lamp. Two light beams are generated, one passing through the suspension at a depth of 2 mm and the other at a depth of 20 mm. Four cuvettes are used, two containing clear liquid and two containing suspension. The light beams are collected with fiber optic guides after passing through the cuvettes and then separated into four wavelengths. Additional photodetectors monitor the intensity of the lamp at the same four wavelengths. The speed of the wheel accelerates continuously over 20 minutes to a final speed of 3,000 rpm and maintained at this speed until all the particles have settled. This system enables a broad size distribution from below 0.05 |j.m to 10 |um (e.g. quartz in water) to be analyzed in 30 minutes. histruments are available from Horiba, Seishin, Shimadzu and LUM. They can be run in the gravitational, centrifugal, gravitational followed by centrifugal or gradient mode. In the gradient mode, the centrifuge accelerates over the analysis time to reduce the measurement time. The simpler instruments operate at constant speed and an analysis can take 45 min, which can be reduced to a few minutes in the more sophisticated versions. Horiha CAPA-700 covers the size range 0.01 to 300 |im, automatically selecting the best of five operating conditions, involving the three modes enumerated above, at centrifuge speeds from 300 to 10.000 rpm. Horiba CAPA-SOO is a more economical version covering the size range 0.04 to 300 |am. Seishin offer three versions of their micron photo-sizers covering the size range 0.1 to 500 |Lim, the SKC-2000, the SKC-3000 and the SKC5000. Shimadzu SA-CP3 operates at 120, 240 or 480 rpm and in any of four modes to cover the size range 0.02 to 150 |Lim. Shimadzu SA-CP4 operates in the range 500 to 11,000 rpm to cover the size range 0.01 to 500 jiim. L.U.M, LUMiFuge^^ 114 (Laboratoiy, Environmental, Medical Diagnostics & Technology) is a photocentrifuge covering the size range 0.1 to 300 jam with a sample volume of 0.1 to 2ml. No sample pre-dilution is required and eight simultaneous analyses can be carried out. The photocentrifuge operates in the gradient mode with at speeds accelerating from 300 to 3000rpm.

Centrifiigal sedimentation methods 431

Fig. 8.9 Berg's conoidal ceiitrifiige tubes 8.10 Homogeneous, cumulative, centrifugal sedimentation The shape of the centrifuge tube is immaterial for instruments operating in the incremental mode but the shape is important for instruments operating in the cumulative mode since particles travel in radial paths. The disadvantages of cylindrical tubes are that sedimenting particles strike the walls of the tube, agglomerate with other particles on the walls and reach the bottom more quickly than freely sedimenting particles. The oblique force of the suspension on the walls also set up convection currents within the suspension. The advantages of cylindrical tubes instead of sector or conoidal-shaped tubes (Figure 8.9) is that they are easier to construct and may be used in ordinary !aborator> centriliiges. 8.10.1 General theory Equation (8,6) may be written (for r. '-=--S and r ^= R): S -^ i?exp(- kdjî)

(8.62)

At time /, all particles greater than d^^^ will have reached the bottom of the container (r "= R). In addition, partial sedimentation will have taken place for particles smaller than d^^^. For each of these smaller sizes, a starting point XQ exists, beyond which all the smaller particles will have reached R where, from equation (8.62): .^Q

^Rcxp^-kd^tj

(8.63)

432 Powder sampling and particle size determination The volume fraction of suspension lying between R and XQ for shallow bowl or flat sector shaped tubes is equal to: /? -An

R'

R'-S'

R'-S'

1 ~ exp(--2fo/^/)

(8.64)

If the particle size distribution is defined such that the weight fraction in the size range d to d'rM is f{d)dd then the weight of particles with diameters greater than d^^ that have completely settled is: 00

W^

\f{d)Ad

(8.65)

The weight fraction of particles smaller than d^^^ which have completely settled is:

W,^ - ~ 2 — T j h ~ exp(~2yW'/) \f{d)Ad

(8.66)

The total weight fraction deposited is:

P^W + - ~ — y j[1 -exp(-2)tJ^/)J/(J)dJ

(8.67)

R —S n The weight fraction oversize can be determined if the weight traction deposited is measured for different values of the variables S, R and t. Similarly, the weight fraction of particles still in suspension at time / will consist of particles smaller than J^^that have originated in the volume between the surface S and radius XQ. By comparison with equations (8.66) and (8.67) this fraction is:

1 - P - - ^ — y 11 exp(-2^^^/) - Qxp{-2kdll)Y{d)dd

Cenlrifiigal sedimentation methods 433

^ M(M-2)(A/--4) 8 dJ 8

St

^

(8.69) where

l^ds,) = ~ ] Pd^'~'Ad d' 0

and

M

4[R^~S^) S^\n{RIS)

An exact solution to equation (8.67), as given by Kamack [50], is as follows:

Pids,)-

exp(a) -

(8.70)

ddP 2 M

where

q{d) = pid) +

and

h\ {x) = /2(x)exp( ~2x)


X ~\x\

^St

and the 'resolvent kernel' h{x) is a function that depends on the apparatus constant. Muschelknautz [51,52] designed a centrifuge in which the displacement of two diametrically opposed bodies floating in a dispersion was measured. The bodies are fixed on a common rod and are immersed at different depths in two chambers. The differential force yields the size distribution directly. Sokalov et. al [53] described a centrifugal sedimentometer with a float measurement system. 8.12 Sedimentation distance small compared with distance from centrifuge axis The simplest procedure with a homogeneous, centrifugal system is to make r-S small compared with S and assume that the particles fall with constant velocity. Equation (8.2) becomes:

(ps~Pf]4t 18/7

,(r^S

(8.71)

This approach has been used by several investigators and applies to the Alpine sedimentation centrifuge and the Mikropul. 8.12.1 Hosokawa Mikropul Sedimentputer The Hosokawa Mikropul Sedimentputer [54-56] has a sealed suspension in a cell that is rotated. As the particles settle, out the center of gravity changes, which creates an inbalance that causes the cell to vibrate (Figure 8.10). By detecting the amplitude and angular velocity of the vibration, the size distribution is obtained. Muschelknautz designed a centrifuge in which the displacement of two diametrically opposed bodies floating in a dispersion was measured [5760]. The bodies are fixed on a common rod and are immersed at different depths in two chambers. The differential force yields the size distribution.


, Angular vdodtygigiialg J 'ndtioaieter

Pkkiqp

>y lliOixa \ fofîlre 1 scn^Mr - ^ S i g n a l for revolution Rotor

îiLipjjprJ

-»1measuAsoicavi

P

Signal for an^plitude

(a) Center of gravity

B' tine particles. McCormick [78], for example, describes its use for determining the size distribution of polystyrene (0.088 < (^ < 0.511) |Lim and Brodnyan [79] uses it for determining emulsion particle size. It has also been used in combination with light scattering for polymer size distribution determination [80]. In a round-robin test series Mueller [81] found the ultracentrifuge to be the most satisfactory size analysis method for the sub-|Lim range; a sample containing nine monodisperse components with 10% diameter difference being resolved. An analytical ultra-centrifugation technique has been used in combination with a scanning optical absorption system for particle size distribution determination. The system was demonstrated for colloidal platinum (0,4 to 2) nm and unstabilized zinc (4-9) nrn during particle growth [82]. A review of examples of colloid analysis of nanosize particles by ultracentrifugation with a focus on multicomonent mixtures has been published.[83] This is claimed to be the first fractionating analytical technique with almost atomic resolution. 8.19 Conclusions Sedimentation techniques are widely used for particle size analysis since the determined size distribution relates to unit operations such as classification. The distribution also relates to many end-use properties


such as the hiding power and gloss of pigments. Thermal diffusion limits the use of gravity sedimentation to powders containing a limited amount of sub-micron material, but the technique is extended into the sub-micron size range with the use of centrifuges. Gravitational and centrifugal sedimentation using a pipette is attractive due to its versatility and low capital cost, but the analysis requires a skilled operator and is time consuming. Mass distributions are also determinable using x-ray systems that are available as gravity and centrifugal sedimentometers. These are essential where speed and running costs are more important than capital costs. Photosedimentometers are also available for gravitational and centrifugal sedimentation. Centrifugal photosedimentometers are available with disc and cuvette cells. The former are usually used in line-start mode which gives high resolution, whereas the latter operate in the homogeneous mode. These instruments need to be calibrated, or corrected, for the breakdown in the laws of geometric optics. The modified distribution greatly alters the raw curve for sub-micron powders with a wide size range, thus limiting the accuracy of this technique. Accuracy is not always important; detecting changes which may affect powder handling or final product may be all that is necessary. References 1 2 3 4 5 6 7 8 9 10

11

12

Berg, S. (1940), Ingen. Vidensk. Skr. B, number 2,398 Kamack, H.J. (1951), Anal Chem., 23(6), 844-850,i9r^ Kamack, H.J. (1972), Br. J. AppL Phys., 5, 1962-1968,^02 Svarovsky, L. and Friedova, J. (1972), Powder TechnoL, 5(5), 213-211,402 Hanis, C.C. (1969), AMIE Trans., 244, 187,402 Svarovsky, L. and Svarovska, J. (1975), J. Phys D, 5, 1962-1968,402 Svarovsky, L. and Svarovska, J. (1975), J. Phys £, 9, 959-962,402 Svarovsky, L. and Svarovska, J. (1976), Dechema Monogram, Nurnberg, Nos. 1589-1615,279-292,402 Alex, W. (1972), Dissertation, Univ. Karlsruhe, Germany,40J Lloyd, P.J., Scarlett, B. and Sinclair, I. (1970), Proa Symp, Particle Size Analysis, Bradford, 267-275, publ. Soc. Analyt. Chem., (1972), ed. M.!.Groves and J.L. Wyatt-Sargent,40J Allen, T. and Svarovsky, L. (1976), Dechema Monogratth Numbers 79(1589-1615), 279-292, Verlag Chemie, GmbH, Weinheim/Bergstrasse, Nurnberg, 403, 412 Slater, C. and Cohen, L. (1962), 1 Sclent, fnstrum., 39, 614,40J


13 14 15 16 17

18 19 20 o

22 23 24

25 26 27 28 29 30 31 32 33

34 35

36

Allen, T. and Svarovsky, L. (1974), Powder TechnoL, 10(1/2), 23-28, 406, 407 Allen, T. (1992), Centrifugal particle analyzer. US Patent 5,095,451J07 Allen, T. (1992), Proc Conf Particle Size Analysis, PSA 91 Loughborough University, Anal. Div. Chem Soc, publ. Heyden,^(?7 Allen. T. (1991), Proc. Int. Symp. Particle Size Analysis, publ. Royal Soc. Chem., ed. Stanley-Wood, N.G. and Lines, R., pp 498-513,'^0e) accept)

Fig. 9.16 The laser optic measuring system of the Brinkmann particle size analyzer Spectrex PCT-1 laser particle counter is a compact liquid particle counter designed specifically to monitor real-time particle counts in ultrapure water. The water to be monitored is fed to a rectangular cross-section Pyrex glass cell (Figure 9.15) and illuminated by a He-Ne laser at right angles to the direction of liquid flow. When a particle passes through the sensing zone, a pulse of scattered light is emitted and detected by a photomultiplier positioned at right angles to both the laser beam and liquid flow direction. Two channels of information are generated; total count coarser than 0.11 |Lim and counts above 0.2, 0.3, 0.5 or 0.7 jiim as set in a four-position switch. Measuring time is selectable from 1 second to 1 h. 9.6 Dwell time 9.6.1. Brinkmann 201 analyzer The Brinkman Model 2010 analyzer (figure 9.16) scans the sample with a shaped and focused laser beam using a rotating wedge prism. The time

Stream scanning methods

493

spent by the scanning beam on a particle is interpreted as particle size. A CCd TV microscope is incorporated into the basic unit, permiting the operator to observe the sample while it is being measured. The images can be enhanced, processed and analyzed to obtain an independent measure of size distribution, particle shape and state of dispersion. The instrument operates in the size ranges 0.7 to 150, 2 to 300, 5 to 600 and 12 to 1200 |Lim. The measuring cells are standard, a spectrophotometer cell, a microscope slide, a liquid flow-through cell, an aerosol flow-through cell and a thermo-electric cooled flow-through cell. This instrument has been compared with the Coulter and automated image analysis for the size determination of grain based products [121]. The electrical sensing zone and the image analysis techniques gave similar mean sizes whereas the time of transition gave significantly coarser 9.6.2 Lasentec Focused Beam Reflectance Measurement (FBRM) Lasentec 's Lab-Tec 100 uses a scanning infrared laser beam to measure the particle size distribution of particles in suspension. The beam is highly focused and illuminates individual particles in its path (Figure 9.17a). The back-scattered light pulses are picked up by a non-scanning stereoscopic detection system (Figure 9.17b). The size of each particle is determined by measuring the time that the particle is in the beam hence the size is recorded as a random chord length. The laser diode and detectors are stationary while the lens, which focuses the light beam, is vibrated normal to the laser detector plane. The vibrating action causes the beam spot (focal point) to be scanned up and down normal to the direction of fluid flow. The beam amplitude is 3 mm and measurements are carried out in the central 1.5 mm where the velocity is maintained constant; since the frequency is 400 Hz the scan rate is 1.2 m s~i. Since the focal point is only about a millimeter in front of the window it can operate with very high (40% by volume) concentration slurries [122]. Lasentec Labtec 1000 is a laboratory instrument that covers the size range from 0.7 to 250 \\m in 28 size channels. The data are generated as 'scanned counts' an empirical frequency distribution created from classification of chords from randomly oriented particles. Software can convert these chords to a spherical equivalent distribution on the assumption that the chords were generated from an assembly of spherical particles: this software contains a filter system to reject improbable data that would tend to skew the distribution to a coarser size. A discrimination

494 Powder sampling and particle size determination loop sorts impulse for short rise times; only pulses from particles that pass directly through the focal spot have short rise times and will be accepted. Spnni loaded „ l^500 m s"^; up to three components being measured using a combination of near forward scatter, near backward scatter and side scatter.


505

9.10 Hosokawa Mikropul E-Spart Analyzer Hosokawa Micron E-Spart Analyzer carries out simultaneous measurements of the size of a particle and its electrostatic charge. Characterization of electrostatic charge and aerodynamic size of particles is of critical importance in many electrokinetic processes [176]. A number of instruments are available that can characterize aerodynamic size distribution of particles. Likewise, instruments are available to estimate the net average electrostatic charge of particles. However, the choice of instruments for real-time simultaneous measurement of aerodynamic size and electrostatic charge distribution of particles on a single particle basis is limited. Electrical Single Particle Aerodynamic Relaxation Time (E-SPART) Analyzer simultaneously measures size in the range from submicron to 100 |Lim and particle charge distribution from zero to saturation levels [177,178]. The operating principle depends upon the phenomenon that when an airborne particle is subjected to an oscillatory external force, such as Particle inlet

Photomultiplier hb signal processor

Transducer Electrode

LDV signal

Acoustic or electric excitation Laser beam from laser Doppler velocimeter (LDV)

Particle outlet

Fig. 9.22 (a) Schematic view of E-Spart relaxation cell (b) principle of particle measurement. Individual particles are subjected to acoustic and/or electric excitation and the resultant response is measured by LDV to determine aerodynamic size and electrostatic charge.


acoustic excitation, the resultant oscillatory motion of the particle lags behind the external driving field. The particle vibrates at the same frequency as the acoustic field but with a phase lag due to particle inertia. The phase lag increases with increasing particle size and so can be related to particle size. To determine this phase lag, the analyzer uses a differential laser Doppler velocimeter (LDV) to measure the velocities of individual particles subjected to a combination of an acoustic and a DC electric field. Simultaneously a charged particle will have its vertical position shifted by the electric field by an amount related to the charge. The maximum count rate varies from 10 to 2,000 particles per second depending on particle size that, typically, can range from 0.3 |Lim to 75 |Lim. The particles are sampled in a laminar flow field through the LDV sensing volume. As each particle passes through the sensing volume it experiences the acoustic excitation and the superimposed DC electric field in a direction perpendicular to the direction of the laminar air flow. A typical sampling configuration is shown in Figure 9.22. A full description of the theoretical basis for the instrument has been published [179] together with some typical applications. In a further paper, the instrument was applied to the size and charge analyses of toners [180]. The size range for toner was limited to 2 \xm to 25|Lim; a lower acoustic frequency is used in the improved E-Spart that increases the upper size limit, for glass beads, to 50 |im [181]. 9.11 Shadow Doppler velocimetry The SDV is based on the imaging of a conventional LDV probe volume on to a linear photodiode array and has the advantage over other sizing methods in that it is tolerant of the optical misalignment and fouling which are inevitable when passing laser beams through windows in furnaces. The technique has been used for simultaneous size and velocity measurements of irregular particles in confined reacting fiows. [182]. The instrument was developed by Hardalupas et. al. [183]. The transmitting optics are identical to a conventional LDV but the receiving optics, in contrast, allow collection of the transmitting light beam so that an image of the LDV measuring volume is formed. This image is magnified by a microscope objective and projected on to a 32-element linear photodiode array. Particle size is derived by measuring the size of the shadows of the particle on the array and, because the presence or absence of the shadow is a binary phenomenon, the method is independent


507

of signal intensity. The analog outputs of the diode array are subsequently digitized and stored for processing and display. The accuracy of SDV was assessed by Morikita et. al. [184] who showed that the maximum difference between the arithmetic means of irregular particles by SDV and microscopic measurements was about 10%. Hishida et al. [185] recorded a maximum difference of 4% owing to beam wandering due to temperature gradients and concluded that the maximum error with increasing flame size cannot exceed 15%. 9.12 Other light scattering methods A simple light scattering photometer was designed, to measure the angular distribution of intensity of scattered, polarized, He/Ne light, by micron and sub-micron particles [186]. The photometer used an ellipsoidal reflector and simple optical components to collect the scattered light and focus it on a 512 element photodiode array. The intensity ratio method is based on measurements of light scattered at two angles and applies to the size range 0.1 to 10 |Lim [187]. Due to the possibility of large errors [188] the method has found little application. Using Scanning Flow Cytometry the size distribution of submicron spherical particles is determined from the scattered light intensity ratios at two angles. In one example the ratio at 6T and 15"^ was used to determine sizes between 1 and 15|Lim at a flow rate of 500 particles per second [189]. This was extended to 0.5 to 14 |im using a parametric solution based on analytical approximating equations [190]. The ratio of the polarized light intensity scattered from two different coaxial beams illuminating a particle can be used to determine particle size. Azzizy and Hess [191] used two coaxial beams of different wavelengths at 30*^ from the forward axis polarized in different directions. The ratio of these two parameters gives a unique curve that is a function only of particle size. They found errors of a similar magnitude to those found with intensity ratio methods. Hess [192] described an arrangement in which an LDV velocity system is positioned concentrically inside a larger sizing beam. The signal from the smaller beam is used to trigger the larger beam so that only particles passing centrally through the larger beam are counted. Wang and Henken [193] applied such a system for measuring particles in the 10 to 200 |Lim range.


9.13 Interferometers 9.13.1 Mach Zehnder type interferometer Unlike light scattering instruments, interferometers do not measure the scattered light, but the phase shift in light waves. They can distinguish between gas bubbles and particles because bubbles have a lower refractive index than the surrounding liquid and therefore produces phase signals of opposite polarity to those of particles. This makes these instruments particularly useful for examining the reagents used for semiconductor cleaning since these reagents often have high vapor pressures and tend to form bubbles that can be counted as particles using light scattering or obscuration instruments. Interferometers separate a laser beam into two beams and then recombine them to create a signal whose intensity depends on the phase difference between them. When a particle with a refractive index greater than that of the surrounding liquid passes through the beam the wave front is retarded and when a gas microbubble passes through it the wave front is advanced. The magnitude of the phase signal depends on particle size and the pulse can be calibrated with particles of known size.

Beani spHlier 1

M i d ssîaic

\1

^

J l%.Ho«Jcicil£>r

B

Fig. 9.23 Schematic diagram of a Mach Zehnder type interferometer. Figure 9.23 is a schematic of a Mach Zehnder type interferometer the design of which allows for a polarized light beam to be split in two. The incoming beam with intensity /Q is divided into two equally intense beam


509

with a beam splitter. The beam following path 1 undergoes a 90"" phase change and a second beam splitter combines beams 1 and 2 to form new beams A and B. The two beams are now 180° out of phase; when one is bright the other is dark. The intensities of the beams I^ and /^ are measured using separate detectors. Due to their inherent complexity, large size and susceptibility to vibration, interferometers have remained laboratory and research tools. The variation of the polarization ratio with time has been used to determine the droplet size distribution in fuel sprays. Although polarization ratio is generally applied to an assembly of droplets it can be used for single droplets provided the incident beam is circularly polarized [194] 9.13.2 The TSI Liquitrak^^ interferometer The TSI Liquitrak^^ interferometer [195-197] uses a dual beam interferometer to detect slight differences in the refractive indices of particles relative to the surrounding media. It is less sensitive to vibration than the Mach Zehnder type of interferometer because it does not separate the two light beams. Instead the beams are overlapped and polarized at 90° to each other, hence the beams do not interfere and are effectively separated from each other until combined by the second beam splitter. The narrow separation of the beams reduces vibration sensitivity dramatically because any vibration interference affects both beams equally. The dual beam interferometer also has the advantage of allowing flowrate measurement. The instrument's flow-rate ranges from 4 to 40 ml min"ôf which 1/200 is examined. Flow-rate is measured each time a particle signal is processed hence particle concentration can be measured in a fluctuating flow. The interferometer has several advantages over dark field scattering instruments. Because it is a bright field instrument it is less sensitive to the stray light scattered by interfaces between the instrument's capillary cell wall and the liquid medium. The instrument can also identify signals created by bubbles thus avoiding false counts. It can also measure flowrate and it is more sensitive to particles with a refractive index near that of the liquid than dark field instruments because it can look at forward light without noise interference from the incident laser beam. A drawback is that its inspected volume is small (0.5%) compared to the full flow stream since a highly intensive beam is required in the


viewing volume, rendering the instrument less useful for low concentration contamination measurements. An evaluation of this procedure is available in articles by Blackford and Grant [198] and Grant [199] and the instrument is available from Thermal Systems Inc. as Model 7750. 9.14 Flow ultramicroscope. In the flow ultramicroscope [200] dispersed particles are injected into a stream of liquid and hydrodynamically focused so as to pass through a laser beam. The scattering is detected by a photomultiplier and processed electronically as a series of pulse heights. The detector can be at right angles to the incident beam, with either a narrow or wide receiver angle, or forward angle scattering may be used. Since the scattered light intensity is highly dependent on particle size, the dynamic range of the photomultiplier can be exceeded by samples of relatively low polydispersity. The range is greatly increased by using a feedback system from the photomultiplier to the laser. The instrument measures number concentration and size distribution for spheres and particles of simple shapes in the size range 0.1 to 5 )Lim. Such an instrument has been used to measure particle flocculation [201]. 9.14.1 ISP A image analysis system ISPA 800 series characterizes the size and shape of 5 ju.m to 10 mm particles by measuring the projected areas of the particles in a strobed image. The unit, manufactured by Greenfield, can process 15 images per second and can analyze particles moving at 30 m sec~i. 9.15 Measurement of the size distribution of drops in dispersions The most common method is direct photography [202-205]. The method is simple, easy and accurate and covers a wide size range, controlled by microscope magnification, from a lower limit of 1 |Lim. The technique is only suitable for low concentration systems, particularly in the case of high opacity continuous phases. The procedure requires many pictures and lengthy analysis times. Direct image analysis of the data has met with limited success.


511

Light attenuation is a simple, widely used method for determining interfacial area, i.e. surface-volume mean diameter if droplet concentration is known but cannot be used for size distribution determination [206-208]. Light scattering has been used for measurement of small drop sizes below 10 |Lim in diameter [209] and also for drop sizes below 800 |Lim. Although on-line measurement is possible the technique is only suitable for volume concentrations smaller than 0.05 [210] Laser Doppler velocimetry (see section 9.6) has also been used for the measurement of a broad size range of drop sizes in solid-liquid and liquid spraying systems [211,212]. Drop size distribution in dilute suspensions of electrical conducting liquids may be determined using the Coulter principle but the need to add what may be undesirable conductive materials limits its applicability [213215]. The use of chemical means to measure interfacial area has been used extensively for gas-liquid dispersions. Chemical reaction methods for determining the interfacial area of liquid-liquid systems involve a reaction of a relatively unchanging dispersed-phase concentration diffusing to the continuous phase. The disadvantage of this approach is that the mass transfer can affect the interfacial tension, and hence the interfacial area [216-218]. Drop stabilization methods rely on the immediate stabilization of drops by encapsulation with thin polymer films or surfactants [219-221] a photomicrographic method has been employed usually after encapsulation of drops. However this method cannot always be used due to incompatibility of the encapsulating materials with some systems. The method also has the disadvantage of the influence of the chemical treatment on drop size. A special sampling apparatus has been developed to withdraw a sample of dispersed phase from the mixing vessel to stabilize drops with a surfactant and to force the dispersed sample through a capillary with a photometer assembly to measure both droplet size and concentration [222]. The capillary method employs a fine bore capillary of the order of the drop size for sampling from the liquid dispersion. As drops pass through the capillary, they are transformed into cylindrical slugs of equivalent volume. A laser beam is split into two rays using a beam splitter and a plane mirror and the rays pass directly through the capillary precisely 0.1 mm apart. The emergent beam is collected by a x 10 microscope objective lens and focused on to a photodiode. From the measurement of the passage time of a slug at one detector and its travel time between two detectors, the velocity and diameter of the drop can be calculated. The


method can be used to obtain broad drop size distributions in the range above 50 |am in real time and automatically [223-227]. The scintillation method uses short-range radioactive particles for measuring interfacial area. This technique is limited by the necessity of high immiscibility between the phases as well as the availability of suitable isotopes and target materials [228]. The Lasentec particle/droplet size analyzer can be used for laboratory and in-line analysis in the +1 \xm size range over a wide range of operating conditions. 9.16 Dupont electrolytic grain size analyzer The EGSA provides a rapid (4 to 10 min) absolute measurement of the charge required to electrolytically reduce/oxidize AgX crystals. This electrolytic decomposition is singularly effective as a basis for measuring the particle size distribution of photographic emulsion grains since the charge is directly related to grain volume. Grains are electrolytically reduced as they are rotated under a measuring electrode and the generated pulses are sorted according to their integral size and stored in 256 logarithmically distributed channels. Three size ranges cover an overall range of 0.05 |im to 2 |Lim [229]. 9.17 Light pressure drift velocity The motion of individual Brownian particles is observed using a confocal tracking microscope. Particles are trapped in a strongly focused laser beam. By evaluating light-pressure-drift-velocity and the back-scattered light intensity the particle size is determined to ±2%. The method was demonstrated on a mixture of seven polystyrene latices between 300 and 450 nm that were divided into six size classes. A discussion of the method is presented together with a suggestion as to potential applications [230]. Tuch et. al. [231] ran a Mobile Aerosol Spectrometer (0.1 to 2.5 |Lim) and an Electrical Aerosol Spectrometer (0.5 to 10 |im) side by side for 6 weeks and found both to be reliable with almost identical results. Total number counts agreed with results from a Condensation Particle Counter.


513

9.18 Impact size monitor Size distributions in pneumatic conveying systems are usually monitored by taking grab samples or by using the Malvem/Insitec optical diffraction particle size analyzer [232]. CSIRO has developed and patented a technique to measure particle size from measurements of the peak compression of an ultrasonic transducer subject to impact by the particles [233]. Each impact produces an independent pulse, the duration and amplitude of which conveys information about the particle size and velocity. The impact times for sub-mm particles is typically less than a |is; this short duration allows the measurement of tens of thousands of impacts per second. The instrument has been tested using several grades of ballotini from 50 to 165 ^m in size and was able to differentiate between particles of size 157 and 165 |Lim [234]. Fluid dynamic modelling of this instrument was carried out to determine the size range of particles which could be monitored and it was determined to be applicable to the size range 50 |Lim to 200 |Lim at load rates of up to 1 kg m"^ [235]. Impact sensors have been used previously to detect particles in streams such as sand in oil pipelines [236]. Quantitative measurements have also been carried out at low impact rates by dropping particles, in a vacuum, on to a sensor consisting of a hit plate in point contact with an ultrasonic transducer [237]. High-velocity air-laden dust has also been measured using transducers [238]. 9.19 Monitek acoustic particle monitors Monitek Micro Pure Systems acoustic particle monitorsusQs a focused acoustical beam to sense discontinuities in a flowing liquid and can detect the size and amount of suspended solids, entrained gases, fibrous material in any liquid, or oil droplets in water. The sensor mounts in-line without restricting the process flow and the acoustical beam is focused to a point approximately 0.8 to 1.5 in from the sensor tip. A piezoelectric crystal that acts both as a transmitter and receiver generates the high frequency. The transmitter emits hundreds of pulses per second and monitors the echoes; this high sampling rate makes the instrument insensitive to liquid flowrate. The amplitude of the echo is size sensitive so that a lower limit size threshold can be set; this limit can range from 0.2 |Lim to a few millimeters


9.20 Erdco Acoustical Counter Audible sounds may be produced by particles exiting from a high-velocity laminar flow tube into a low velocity tube. The phenomenon was first reported by Langer [239] and has since been investigated by Langer [240]. and others. The sensing zone of the Erdco counter is a capillary at the exit of a glass tube. As particles enter this section they interact with the boundary layer, resulting in a toroidal vortex that moves as a shock wave that is reflected back on the capillary. The pressure wave is detected by a transducer at the outlet of the capillary, whose displacement is measured by an optical probe. The displacement is proportional to particle size, which is measurable down to 4 \xm. References 1 2 3 4 5 6

I 8 9 10 II 12 13 14

15 16

Groves, M.J. (1991), Particle Size Distribution II, ed. T. Provder, Am. Chem. Symp. No 472,123-131,^^9 Coulter, W.H. (1953), US Patent No. 2,656,508. Appl., 1949, 449 Coulter, W.H. (1956), Proc. Soc. National Electronics Conf., 12, 1034, 449 Kubitschek, H.E. (1958), Nature, 182, 234-235, 449 Kubitschek, H.E. (1960), Research, 13, 128, 449 Coulter Counter and other scientific instruments, Industrial Bibliography January 1992, Coulter Electronics Ltd., Northwell Drive, Luton, Beds LU3 3RH, England, 450 Anderson, F.G., Tomb, T.F. and Jacobson, M. (1968), US Bureau Mines, R.L, 1\05,450 ASTM F751-83 (1997), Measuring particle size of wide-size range dry toners, 450 ASTM 577-83 (1997), Particle size measurement of dry toners, 450 ASTM C-690-86 (1997), Particle size distribution of alumina or quartz by electronic counting, 450 ASTM E 1772-95( 1995) (1997), Particle size distribution of chromatography media by electrical sensing zone technique, 450 ASTM 3451-92 (1992), Testing polymeric powders and powder coating, 450 ASTM C757-90 (1996), Nuclear grade plutonium dioxide powder, sinterable, 450 ASTM F-662-86 (1992), Measurement of particle count and size distribution in batch samples for filter evaluation using an electrical resistance particle counter, 450 ASTM D-4438-85 (1997), Particle size distribution of catalytic material by electronic counting, 450 ASTM F-660-83 (1993), Comparing particle size in the use of alternative particle counters, 450


17

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515

British Standard BS3405 Part 5, (1983), Determination of Particle Size Distributions, Recommendations for the Electrical Sensing Zone Method, The Coulter Principle, 450, 462, 463 ISO 13319 Determination of particle size distributions - Electrical sensing zone method, 450 Berg, R.H. (1958), ASTMSp. Publ. 234, 453 Batch, B.A. (1964), J. Inst, Fuel, 454, 462 Allen, T. (1967), Proc. Conf. Particle Size Analysis, Loughborough, Soc. Analyt. Chem. London, 454, 455, 462 Blois, R.W. de and Bean, C.P. (1970), Rev. Scient Instrum., 41(7), 909-916, 454 Gregg, E.L. and Steidley, K.D. (1965), Biophys. J., 5, 393, 454 Smythe, W.R. (1961), Phys. Fluids, 1, 633, 454 Talbot, J.H. (1974), J. Min. Vent. Soc. South Africa, 27(11), 161-167, 454 Grover, N.B. et. al. (1969), Biophys. J., 9, 1398, 1415, 454, 457 Andersen, J.L. and Quinn, J.A. (1971), Rev. Scient. Instrum., 42(8), 1257-1258, 454 De Blois, R.W., Bean, C.P. and Wesley, R.K.A. (1977), J Colloid Interf Sci., 61(2), 323-335, 455 Talbot, J.H. (1966), J. Scient. Instrum., 43, 744, 455 Eckhof, R.K. and Soelberg, P. (1967), Betontek. Publik, 7, 1, 455 Schrag, K.R. and Com, M.,(1970), Am. Ind Hyg Assoc. J., 446-453, 455 Simecek, J. (1967), Staub Reinhalt, Luft, 27(6), 33-37, 455 Michael, A. et. al. (1994), Part. Part. Syst. Charact. 11, 391-397, 455 Treweek, G.P. and Morgan, J.J. (1977), Environ. Sci. TechnoL, 11(7), 707-714, 455 Polke, R. (1977), Dechema Monogr., 79(1589-1615), PartB, 361-376, 455 Karuhn, R. et. al. (1976), Powder TechnoL, 11, 157-171, 455, 458, 463, 468 Marshall, K. (1969), M.Sc. thesis, Bradford University, UK, 455 Lloyd, P.J., Scarlett, B. and Sinclair, L (1972), Proc. Conf Particle Size Analysis, Bradford (1970), publ. Soc. Analyt. Chem. London, 455 Eckhoff, R.K. (1969), J. Phys. E, 2, 973-977, 455, 456 Harfield, J.G. and Cowan, M. (1986), Partec , Ntimberg, Germany, publ. NumbergMesse GmbH.51 \-5S0 455 Harfield, J.G., Wharton, R.T. and Lines, R.W. (1984), Part. Part. Charact., 1,32, 455 Kubitscheck, H.E. (1960), J. Res. Natl. Bur. Stand., A13, 128, 456 Thom, R von, Hampe, A. and Sauerbrey, G. (1969), Z Ges. Med, 151, 332, (1971), Ger. pat. 1 955 094, 457 Polke, R. (1977), Dechema Monogr., 79(1589-1615), PartB, 361-376, 457 Davies, R., Karuhn, R. and Graf, J. (1975), Powder TechnoL, 12, 157-166, 458, 468 Karuhn, R., Berg, R.H. and Davies, R. (1976), Powder TechnoL, 13, 193-202, 458, 468 Thom, R von, Hampe, A. and Sauerbrey, G. (1969), Z Ges. Med, 151, 332, 458


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10 Field scanning methods of particle size measurement 10.1 Introduction Field scanning methods are those in which the size distribution of an assembly of particles is inferred from the interaction between the assembly and a measurement probe. In the simplest systems, the powder (or slurry) is probed or classified in order to generate a single point on the distribution curve. For example, one might monitor the 100-mesh percentage oversize from a mill in order to control a continuous milling operation. If the percentage increases, the residence time in the mill is increased in order that the product size remains unchanged. It is commonly found that comminution shifts the whole distribution to a finer size distribution, to form a homologous family of curves, and plotting particle size against milling time on a log-log scale generates a straight line. Knowledge of two points on the distribution curve allows one to generate the whole distribution. An alternative method for plotting such distributions is the Gaudin-Schuman plot where the cumulative weight finer than a given size is plotted against that size, with each scale on a logarithmic basis. For the majority of milled material the relationship between the two variables is linear except at the coarse ends of the distributions. The distribution is characterized by two parameters; a distribution modulus, n (slope), and a size modulus, k. Again, n remains constant for consecutive grinding of the same material. Djamarani and Clark [1] state that many industrial processes are defined by a coarse ( Q and a fine fraction (F), for example, oversize and undersize. In their example they use sieve sizes of 1400 |um and 180 |um that they fit to a Rosin-Rammler distribution. They define a curve of C+F against C/F from which the Rosin-Rammler constants can be read. Field scanning instruments are ideally suited to on-line analysis. Rapid control of crystal size from a crystallizer; granule size from a granulator;

Field scanning methods

525

product size from a milling operation; particle size from a reactor etc. can yield enormous dividends in terms of less wastage (i.e. more material in specification) and superior product quality. One problem associated with implementing this technology is the need to build an interface between the process and the measuring instrument. This often requires a dilution step that may alter the size of the particles. In the case of crystallizer control, for example, it may be necessary to remove two streams from the crystallizer and filter one so that the mother liquor can be used as the diluent. Process streams often operate at high flowrates and these have to be split in order to obtain an acceptable flowrate in the measuring device. This reduction has to be carried out with care in order to minimize sampling errors. Ideally, the whole of the process stream should be examined. Inserting a probe directly into a process line is usually not feasible due to servicing and downtime problems. It is therefore preferable to use a side-stream that can be isolated from the process stream. A sophisticated on-line analyzer can cost around $100,000 and the interface can easily double this cost. However, a 1% increase in yield can pay back this investment inside a year, making on-line size analysis very attractive. At the present time retroactive fitting of size analyzers is often necessary and one is often faced with space limitations. Designing these units into new process lines greatly reduces cost and makes their introduction more attractive. 10.2 Single point analyzers 10.2.1 Static noise measurement This technique has been applied to the measurement of the average size of milled silica powder (size range 2 to 5 |Lim) suspended in air [2]. A continuous sample is drawn from the product stream into a sampling probe and diluted with an air injector that also provides the driving force. The sample stream is then passed through a 'uni-flow' cyclone that splits the sample into two streams; a low concentration 'fine' stream and a high concentration 'coarse' stream. As the relative mass flow rates of the two streams depend strongly on the size distribution of the feed (at a given flow rate), an average size may be found from a measure of the two concentrations. Most particles suspended in air carry an electric charge, particularly if they have passed through a highly turbulent process. A probe inserted into the stream will detect this charge as an AC voltage that is strongly


dependent on concentration. The system was calibrated by feeding in samples of known mean sizes and recording the signals these generated for comparison with signals from unknown samples. 10.2.2 Ultrasonic attenuation The attenuation of ultrasound by a slurry depends upon the particle size distribution and concentration of the solid phase. In order to separate these two variables it is necessary to carry out analyses at two different wavelengths, one of which is strongly dependent on concentration and the other on particle size distribution. The attenuation is also dependent on the spacing of the transmitter and receiver and other physical parameters in a predictable manner [3,4]. The first commercial ultrasonic on-line particle size analyzer was developed in the 1970's and was based on the measurement of ultrasonic attenuation at two frequencies with an empirical model to predict particle size and concentration [5]. Instruments based on this patent are available as the Denver Autometrics PSM-100, 200, 300 and more recently 400. These are pre-calibrated for the selected mesh size (100, 200, 300 and 400) and the mesh read-out is proportional to the mass percentage less than this. These instruments can operate at extremely high concentrations, up to 60% by weight, and have found their widest application in mineral processing plants for improved grinding circuit control. A major problem in the early development of the Autometrics' system was that traces of air could lead to substantial attenuation losses. The air must be stabilized or removed to allow accurate measurement of particle size. Removing the air with a device that utilizes a combination of centrifugal force and reduced pressure solved the problem. The need to remove air increases the cost of the overall system significantly and makes it an expensive instrument when compared with other instrumentation often installed in grinding circuits. Nevertheless, it appears to be perfectly compatible with other approaches when its inherent reliability and longterm stability as an accurate size analyzer is taken into account. Several articles have been written describing applications of the PSM systems [6-9]. The limitations of the system 100 are: (a) the percentage solids should be less than 60% by weight (b) the particle size distribution should be within the range 20% to 80% less than 270 mesh and (c) the slurry particles should not be magnetized. The PSM systems 200 and 400 are later instruments designed to overcome these limitations.


52 7

10.2.3 P-ray attenuation Instruments have been described that employ p-ray attenuation [10-15]. The accuracy of these devices is limited by their sensitivity to changes in feed density [16]. In order to calibrate it is necessary to re-circulate slurry samples in a closed loop at a number of dilutions for each slurry system, sieve analyses being carried on representative sub-samples. The signals from a scintillation counter can be used to control mill feed rate in order to compensate for changes in feed ore, grindability and feed particle size. Accuracies of 2% to 3% have been reported on Cornish granite, nepheline systems and copper and iron pulps in the size range 20 to 105 îm [17,18]. 10.2.4 X-ray attenuation and fluorescence This sensor is based on the comparison of the absorption of two x-ray beams, one of which is sensitive and the other insensitive to variations in particle size [19-21]. Each sensing head is specific to a particular system since the relationship between the two beams is dependent on the composition of the solids in the slurry stream. The technique is limited to x-ray opaque material. Von Alfthan [22] describes an on-stream x-ray fluorescence system that consists of two flow cells through which the slurry passes. In the classifying flow cell, the slurry flows in a straight path behind a window; it then strikes an obstacle that causes slurry mixing as it enters a turbulent flow cell. X-rays excite the slurry in both cells and the resulting fluorescent radiation is a measure of particle size. The system, sold as the Courier 300, measures both x-ray scattering and x-ray fluorescence and is intended primarily as a composition monitor. The measured data can be analyzed to give chemical composition, solids content and maximum particle size. 10.2.5 Counter-flow classifiers Two instruments have been developed for on-line measurement of flowing powders coarser than 100 |Lim in size [23-26]. In the first instrument a side stream of solid particles from a process line is fed into an air elutriator that separates it into an oversize and undersize stream. The particle flow rate into the elutriator is measured and the cut size for the elutriator adjusted so that the flow of oversize particles out equals 50% of the inlet flow. The elutriator cut size is then equal to the average size of the powder. In the second instrument the flow rate is varied and the signal ratio of the two


flowmeters is inputted as the j;-axis of an x-y recorder. The x-axis is reduced to the cut size for the elutriator. A sweep time of 40-60 s at flow rates of 2.4-3 g s"^ gives a cumulative distribution in the size range 100700 ^m. 10.2.6 Hydrocyclones An ideal on-stream sizing device would sample the whole of the stream and not include any special instrumentation. The nearest approach is to use a classifying hydrocyclone as these are easily installed and often form part of an industrial plant. By measuring the flow rates and pulp densities, and assuming a size distribution law for the feed, a computer program can be written to give the modulus and index of the feed. Under normal operating conditions the present state of the theory of cyclone operation renders this impracticable [27] although it can be used under favorable conditions [28]. Lynch et. al. [29] proposed that the percentage less than some chosen mesh size in the cyclone overflow could be related directly to the d^^^Q parameter of the cyclone, provided that the size distribution of the feed to the cyclone does not change appreciably. In closed production circuits there may be marked changes in the size distribution of the cyclone feed and an empirical relationship has been developed [30]. The application of this technique requires very thorough analysis of the circuit and repeated checking of empirical equations. An alternative approach has been to accept the inherent difficulties of sampling and install smaller, more precise, classifiers alongside the production classifiers [31]. Tanaka has also investigated the use of hydrocyclones for on-line analysis [32]. 10.2.7 The Cyclosensor This is a batch size analyzer [33] (Figure 10.1). An extremely dilute sample of milled ore is introduced, at a constant flow rate, to a coarse separator in the form of a tangentially fed cylindrical screen. The coarse fraction is allowed to settle and the fine fraction is further separated with an efficient hydrocyclone into a fine and a very fine fraction. The very fine fraction is discarded and the fine fraction is allowed to settle. The ratio of the times taken to fill the coarse and fine fraction collection vessels to indicated levels can be related directly to the particle size distribution. The cyclosensor has a sensitivity whereby a change of ±1.8% passing 100


Very fine

Metal core

Cylindrical screen

Coarse solids

529

Efficient cyclone

Fine solids

Fig. 10.1 The Cyclosensor. mesh can yield a 7% change in the ratio of the settling times. The reproducibility is such that for the same feed rate of the same solids the ratio of times remains constant to better than 1% and an increase in feed rate of 30% has no effect on the ratio. A patent has been issued on an instrument operating in a similar way [34]. It consists of a particle suspension sampler, a settler and a weight or volume sensor. Particle size distribution is determined from sensor output and the time for the settling particles to pass the sensor. 10.2.8 Automatic sieving machines This automatic wet sieving machine determines a single point on the size distribution curve in a few minutes without the need to dry samples [1,16]. The sieving vessel is first filled with slurry and topped up with water to a precise level to allow accurate determination of the mass of solids added (wi) by application of Archimedes' principle. The fine fraction is next removed from the vessel through a discharge valve. Screening is hastened by propeller agitation and with ultrasonics to maintain the sieve mesh free


of pegged material. The weight of the residue (^2) is determined by further application of Archimedes' principle, and the fraction coarser than the screen size is given directly by (W2AV1). It is interesting to note that the capacity of the rapid wet sieving device, expressed as screen charge mass per unit screen area, is more than an order of magnitude higher than that normally recommended for conventional dry test sieving [16]. A description has been given of a technique using a two-cell compartment divided by a screen [35]. The slurry density in the two compartments is determined using nuclear gauges to provide a single point on the distribution curve. A fully automatic sieving machine that can determine seven points on the size distribution curve has also been described [36]. In this technique a pulsating water column is used with the application of ultrasonics and the charges are dried and weighed automatically. 10.2.9 Gas flow permeametry Air, whose pressure varies sinusoidally with a specific amplitude and frequency, is forced through a moving bed of powder [37-40]. At a known height in the bed the attenuated and retarded air pressure is tapped by a pressure transducer, so that the amplitude and pressure drop are measured after being separated into the pulsating and steady flow components. The amplitude attenuation of the pulsating pressure is related to bed porosity and specific resistance. Using the Carman-Kozeny permeametry relationship the average size of particles can be evaluated. The bed is packed into a test cylinder and discharged by a vibratory feeder at the bottom after measurements have been taken. This enables a new bed to be packed and measured within minutes. Air permeability has also been used to determine the surface area of cement [41]. A porous piston compresses the sample of cement into a cell. Air is passed through a bottom porous plate, through the sample and porous piston, into the atmosphere. The inlet pressure is automatically adjusted and recorded to give a known air flowrate and the surface area is evaluated from the inlet pressure. The cell is emptied automatically becoming ready for the next test. Weiland [42] used a similar idea but based on the Blaine permeability method. An automatic weigher produced a packed bed of powder, of known voidage, in a standard cell. Air was drawn through the bed by the passage of water from one reservoir to another. After a certain volume of air had passed through the bed, measured by a certain volume of water flowing, the time required was converted to an electrical proportionality


531

signal. The measurements were repeated every 4 min and the signals used to control the feeder to a grinding mill. 10.2.10 Correlation techniques Correlation techniques can be used with signals from the attenuation of radiation, such as light, but these are mainly used for low concentration systems. The signals from two sensors in close proximity, situated in a flowing stream of slurry, are cross-correlated to give an autocorrelation function. Stanley-Wood et. al [43,44] found that this function obtained from alternating current transducers, initially designed to measure mass flow rate, gave a measure of the particle size of a sand/water mixture. A measure of mean size could be achieved by allowing the normalized signal from a correlator to be divided in two and passed through either high or low pass filters. This results in an inequality, due to variations in frequencies from large and small particles; the ratio of this inequality can be used to determine mean size after calibration. The particle size was between 70 and 2000 \\xxv with a concentration between 10% and 30% by weight. k Reflected

Vay Incident i^y\

Fig. 10.2 Interaction of a ray of light with a spherical particle. 10.3 Light scattering and attenuation 10.3.1 Introduction When light strikes a particle, some of it is absorbed, some is refracted, some diffracted and some transmitted (Figure 10.2). The amount absorbed depends upon the optical properties of the particles and surroundings and


is also a function of the cross-sectional area of the particles. The absorption, or turbidity, can be used to determine a mean particle size and, in conjunction with sedimentation, a size distribution. For very small particles however, the laws of geometric optics no longer hold and a correction has to be applied in the form of an extinction coefficient, K, which is defined as the effective particle cross-section divided by its geometric cross-section. This may be determined theoretically over the whole particle size refractive index domain using Mie theory or, over limited ranges, with modified theories. Other interactions between the particles and the incident radiation, such as state of polarization, light flux at a fixed angle to the direction of the incident beam and angular spectra can be used for particle size determination. Interaction between the incident and diffracted radiation gives rise to interference phenomena with characteristic maxima and minima in intensity [45]. In order to describe fully the scattering pattern it is necessary to assume that the particles are optically homogeneous and spherical and, in order to give independent, incoherent scattering, in a dilute random arrangement. 10.3.2 Turbidity measurements Turbidity has been widely used for determining the particle size distribution (PSD) of particles in suspension, since it is experimentally simple, can be used over a wide size range and does not disturb the system under investigation. It is also fast, reproducible and inexpensive. If a light beam falls on an assembly of macroscopic particles the attenuation is given by: /-/oexp(-aA2Z)

(10.1)

where / is the transmitted intensity when a light beam falls on a suspension of particles of projected area a and number concentration n and traverses it by a path of length L\ /Q is the transmitted intensity when no particles are present. Turbidity gives a measure of the attenuation of a beam of light passing through a suspension. More generally, equation (10.1) may be written: I = lQQxp{-KanL)

(10.2)

Field scanning methods 533 where the extinction coefficient, K, may be evaluated theoretically thus permitting the determination of particle size. If c^ is the volume concentration: (10.3)

^v=~^d^

where d^ is the mean volume diameter. The projected area (a= an) of an assembly in random orientation is: (10.4)

a = —nd} 4 ' where d^ is the mean surface diameter; Hence:

^

I = IQ exp

3KrO 2d,sv

(10.5)

J

where d,,, is the surface-volume mean diameter. Hence: / 1 \ / = /Q exp —Kc^S^L V

/ = /Q exp(-

(10.6)

4 TI)

(10.7)

where x is the turbidity. For a suspension of non-spherical, non-monosize non-adsorbing, isotropic particles, in the absence of multiple scattering: X --

2d„,

(10.8)

The surface volume mean diameter for a suspension of spherical particles is given by:

Y,d^f(d)M _ 0

Y^d^f(d)M 0

(10.9)


Equation (10.8) can be written in the form: 00

T = ~\Kd^f{d)dd 4o

(10.10)

This is a Fredholm integral equation of the first kind. The regularized solution to this equation has been applied to the measurement both for the moments and the size distribution of a wide range of latices [46]. K has been given by van de Hulst [45] in terms of particle size/refractive index domain. Mie theory applies to the whole domain but in the boundary regions simpler equations have been derived. For dilute suspensions of particles smaller than 0.04 |Lim in diameter, the turbidity can be calculated from the equation:

r=

3271^ ( w - i f ^ - ^ /

(10.11)

c is the mass concentration, A^ is the Avogadro Number and/is very nearly equal to unity [45 p396]. Turbidity measurements have been carried out on non-uniform latices and it is suggested that this is one of the most useful of the light scattering techniques for average size determination [47]. PSD can be estimated from the turbidity at different wavelengths provided the other variables are known. Kourti et. ah [48] assumed a log-normal PSD and observed that the parameters of the estimated distribution were so highly correlated that an infinite number of distributions could explain the data. However, all the alternative solutions were found to have the same weight average (surface-volume mean) diameter. With turbidity ratio, the ratio of the turbidities at two wavelengths, one of which is chosen as basis, is used. This has been successfully applied with large particles [49], (0.650.1 |Lim). Nicomp Model 380/L is designed for on-line monitoring either by itself or in combination with the Particle Sizing System AccuSizer. These instruments are available from Particle Sizing Systems. Otsuka Photal is a dynamic light scattering spectrophotometer that provides sub-micron sizing in the 3 nm to 3 |Lim size range and also provides information on the shape of polymers. Wyatt QELS is a compact add-on instrument that can be connected to a Dawn® Eos or a miniDawn Tristar in order to determine particle sizes and their distributions. The combined system gives the ability to size very small particles (under 10 nm), as well as very large particles on-line after they have been separated by some kind of fractionation. The Wyatt QELS comes with the ability to measure the correlation function at 18 angular positions. In the microliter batch mode, the system collects data from small samples that have been injected into special blackened cuvettes. Unfractionated samples produce complex correlation functions whose


601

departure from exponential decay arises from the presence of heterodisperse components in the suspension. In the flow mode, the instruments may be used to collect, simultaneously, the dynamic and classical light scattering data from which the molar mass and root mean square radii are calculated from each slice (See section on fractionation). For the mini-Dawn Tristar, dynamic light scattering signals are only collected from the 900 location, whereas for the Dawn Eros dynamic light scattering signals can be collected at any one of its 18 detector locations. 10.14.12 Discussion The basic theory and discussion of results are covered in papers by Thomas [308]. who uses a Brookhaven Instrument Fiber Optics QuasiElastic Light Scattering System (BI-FOQELS) with dynamic light scattering obtained using the BI-DLS and diluted samples. An autodilution unit has been described to analyze on-line particle growth during a polymerization process [309]. The results compared favorably with off-line dynamic light scattering and on-line turbidimetric data [310]. Several data analysis software packages are available and average sizes generated by these are not comparable [311-313] De Jaeger et.al. [314] carried out inter-laboratory tests using polystyrene lattices with particle sizes ranging from 30 nm to 2 \xm. They concluded that reliable particle sizes could be obtained for diameters less than 0.5 |Lim. In the range 0.5 to 1 |um this was only possible within a very narrow range of concentration. For the largest size investigated (about 2 jLim) the measurements were less reliable. In comparison tests, using standard latex particles, it was found that DLS gave data that is approximately 0.005 |im larger (0.064 |Lim) than electron microscopy and light turbidity methods, i.e. 0.059 ± 0.002 |Lim. In a comparison between the Coulter N4 and the Malvern Autosizer it was found that both instruments gave good results with monodisperse latices [315. For bimodal distributions, the two modes were not always detected and, if they were, the locations of the modes were incorrect [316]. Koehler and Provder [317] sized monodisperse PMMA latexes with a range of instruments: Disc centrifugal sedimentation (DCP), sedimentation field flow fractionation (SFFF), hydrodynamic chromatography (HDC), photon correlation spectroscopy (PCS), turbidimetry and transmission electron microscopy (TEM). TEM gave the smallest sizes, DCP and SFFF were in fair agreement in the center and PCS the highest sizes.


In an evaluation of a range of instruments Lange [318] stated that turbidimetry appeared the most reliable approach to average size determination, whereas ultracentrifugation, DCP and TEM with image analysis were superior for determining size distributions and polydispersity. Lee et. al. [319] compared FFF, PCS and TEM for sizing acrylic latexes. They stated that FFF is a useful tool for accurately determining mean size and size distribution due to its simplicity and ease of operation. PCS provided reasonable data for small diameters and narrow distributions but suggested that multi-angle analysis was needed for larger particles and broader distributions. Multi-angle PCS was also preferred by Bryant et.al [320,321] for measuring multi-modal PSD's, and they showed that accuracy increased with increasing number of angles. Four instruments were evaluated using eight polystyrene latexes, ranging in size from 0.039 to 0.804 |Lim, from Duke Standards [322]. The instruments were: Matec CHDC-2000, Brookhaven BI-DCP, Coulter N4Plus (PCS) and Hitachi H-7000 FA (TEM). TEM gave average sizes close to nominal values, CHDF gave values slightly larger than nominal except for the lowest size where it gave a value of 0.046 |Lim, PCS and DCP gave reasonable values. The measurement time was unduly protracted with the DCP for the finest fraction, the generated size being 0.70 jiim for a run time of 2 hours reducing to 0.056 |am when this was extended to 6 hours. It was found that both PCS and CHDC gave much broader distributions than DCP and TEM. For polydisperse (bi- and tri-modal) CHDF and DCP provided the most accurate distributions whereas PCS failed to capture the whole distribution. Photon correlation spectroscopy measurements for growth rate, together with a quartz crystal microbalance for mass deposition, have been integrated into a single platform to permit simultaneous in-situ real time measurement at times and temperatures representative to those found in aviation fuel systems [323]. Kroner et. a/.[324] compared DLS with static light scattering for determining soot particle size in a premixed flame. They concluded that static is preferable to dynamic since the latter procedure requires detailed information about the flame. 10.14.13 Spectral turbidity Beckman DU 7500 spectrophotometer has been used to determine the size distribution of "monodisperse" latex in the size range 200 to 800 nm, to


603

Study agglomeration during crystallization and attrition of potassium sulphate. A polychromatic light beam from a uv source passes through a silica fiber to a sensor immersed in the suspension. After crossing the measurement zone the transmitted light passes through a fiber to a holographic grating that splits the light into a spectrum that falls on to a 256 photodiode array. Using Mie theory the spectral turbidity yields the particle size distribution of the powder in suspension [325]. 10.14.14 Diffusion wave spectroscopy (DWS) DWS addresses dynamic light scattering in the multiple scattering concentration range. Pine et. al. [326] describe the theory for the technique and it has been applied to the determination of mean size and polydispersity [327,328]. The method has also been used for on-line measurement of concentrated suspensions [329]. 10.14.15 Photon migration In photon migration, an intensity-modulated light beam is launched into the sample and the photons diffuse through the sample and are detected [330]. The transmitted signal is attenuated and phase shifted relative to the incident beam. Measurement of the phase shift and attenuation as a function of modulation frequency yields scattering coefficient and particle size. Mie scattering theory is applied to generate particle size. The theory can be extended to higher concentration regimes. The particle size distribution is determined by determining the scattering coefficient at one wavelength over a range of known particle concentrations. 10.15 Turbo-Power Model TPO-400 in-line grain size analyzer Nisshin developed this instrument for the cement industry. At preset times it automatically samples a few kilograms of material and feeds it into a turbo classifier. The fines are fed into a micron line that determines the Blaine number for the powder. 10.16 Concentration monitors Monitek instruments measure the concentration of suspended solids in a liquid by shining a light through the stream and detecting the amount of light that is scattered by the suspended solids. The scattered light is seen as turbidity hence the name turbidimeters. Suspended solids scatter light


in all directions so that there are many potential viewing angles; 90° sidescatter instruments are called nephelometers. Forward scatter instruments sample a more representative cross-section of a process stream and can monitor as wide range of particle sizes from 0.01 to 100 \\m. For accurate measurement Monitek uses a ratiod forward scatter intensity where the scattered intensity is divided by the direct beam signal. 10.17 Shape discrimination The morphology of particles is an important characteristic that can seriously affect powder handling and end-use properties. Off-line techniques are often not suitable for monitoring industrial processes since extracting the particles from the process can alter their shape, e.g. particles taken from a crystallizer may fracture or aggregate. To resolve the problem an in-line camera, capable of imaging crystals produced in a commercial crystallizer, was developed [331]. The instrument is based on a borescope and video camera that fits inside the housing of a laser backscatter probe, which installs in a standard ball valve. A strobe light is used to "freeze" crystal motion. Crystal features seen directly include shape, surface roughness, inclusions and transparency. The information content of the uv/vis spectrum of sub-micron and micron size particles yields information on the size, chemical composition and shape of the particles [332]. The angular dependence of the scattered intensity is given by the Rayliegh-Gans-Debye (RGD) theory. The form factors for various particle shapes were calculated as a function of the angle of observation ând wavelength X of the incident light. Comparison of the scattering intensities for particles with different shapes showed that each differently shaped particle had a unique surface pattern thus suggesting the possibility of selecting combinations of X and ^to enable shape discrimination. 10.18 Miscellaneous 10.18.1 Back-scatter intensity A relationship has been derived between average particle size and concentration and the back-scatter intensity of a light beam entering a slurry of non-transparent particles [333]. Concentrations used were 0.3 to 2.5% and the average sizes were 40 to 1200 |im. The light source and detector were built into one tube that was mounted on the wall of a 1 liter

Field scanning methods 605

vessel that contained an agitated suspension of the particles under test. The derived relationship took the form: ib, -h.={hm-h.)exp(kcPd'^)

(10.45)

The suffixes refer to the back-scatter intensity from; b^ = total, b^ = container walls, b^ = maximum possible. For aluminum silicate, experiment yielded/? = 0.77, q = 1.67, k = 0.20. In a second paper [334] the technique was extended to on-line. A similar technique to the above, but using ultrasonics, has also been described [332,336]. 10.18.2 Spectroscopy photo-acoustic (PAS) and photo-thermal (PTS) The surface of a specimen may become heated through irradiation, the degree of heating depending on the material's absorption coefficient at the particular wavelength of radiation. A wavelength scan across a suitable part of the electromagnetic spectrum thus causes temperature changes that reflect the adsorption spectra at the point of illumination. In PAS the heating serves to increase the pressure inside a small chamber in which the sample is situated, being irradiated through a window. A recording of the pressure variations versus wavelength of illumination reflects the absorption spectrum of the material. In PTS one records the increase in thermal emission from the sample induced by irradiation. PAS techniques require samples to be placed in a spectrophone for analysis. PTS methods allow constant free on-stream inspection at a distance. Particle size may be inferred from the signal level. Since specific surface increases with decreasing size the signals also decrease. A discussion of these techniques has been presented by Kanstad and Nordal [337]. 10.18.3 Transient electric birefringence A dilute suspension of electrically and optically anisotropic colloidal particles becomes birefringent when subjected to an electric field. In random orientation the suspension is optically isotropic but when the grains align with a uniform electric field the suspension becomes anisotropic; in particular the effective refractive index of the ordered suspension parallel to the field direction differs from its refractive index


perpendicular to the field. This double refraction, or birefringence, has been used for evaluating the size distribution of sols in solution. Experimentally, a colloidal system, in random orientation, is illuminated with polarized light. The system is subjected to an electric field that aligns the particles due to the interaction between the field and any permanent dipole or electrical polarizability of the particles. The birefringence grows as the particles align; when the field is removed the birefringence decays as the particles revert to random orientation. For a monodisperse suspension the decay rate can be described by a first order rate equation. For a polydisperse suspension the decay rate is a sum of exponentials. Measurement of the decay rate permits computation of particle size [338]. Haseler and Hinds used this procedure to determine the size distribution of anisometric silver halide crystals using an instrument that they developed called an electric field birefringece. They found that the technique was capable of good accuracy and high precision [339]. 10.18.4 Crossed lasers A description of a crossed laser beam technique for particle sizing and its application to shock tube experiments was presented by Waterson and Chou [340]. 10.18.5 Frequency domain photon migration A method for on-line monitoring of particle size distribution and volume fraction in real time using frequency domain photon migration measurements (FDPM) has been described. In FDPM the time dependence of the propagation of multiply scattered light provides measurement of particle size distribution and volume fraction. The technique has been applied to a polystyrene latex and a titanium dioxide slurry at volume concentrations in the range 0.3 to 1% [341]. FDPM consists of launching sinusoidally modulated light into a scattering medium at a single point source and detecting the modulated light at another point some distance away from the source. The photon density wave is attenuated and phase-shifted relative to the incident signal as it propagates through the scattering medium due to the scattering and adsorption properties of the sample. These properties can be measured by fitting the phase shift and modulation to the appropriate solution of the optical diffusion approximation, which describes the transport of light in random media. Particle size distribution and volume concentration can be


607

determined from multi-angle measurements of FDOM and determination of the isotropic scattering coefficient. The technique is fast and the equipment relatively inexpensive. Moreover, since photon migration measures the time it takes for light to travel through the sample rather than the intensity of the detected signal, it is self-calibrating. 10.18.6 Laser induced incandescence (Lll) Aerosol particles are heated close to their evaporation temperature by a high energy laser pulse and start to emit thermal radiation. As the particles cool, mainly by heat transfer to the carrier gas, their thermal emission intensity decreases with time. Due to their higher heat capacity large particle cool more slowly than small ones and the cooling time can be used as a measure of particle size. Early applications mainly concentrated on average parameters for soot particles in flames [342] and this was extended to determination of a detailed characterization of the size distribution and internal structure of aerosols at room temperature [343]. The results showed good agreement with TEM and DMA data. Particle sizes were in the nanometer range at number concentrations down to 103 cm'^. Other accessible parameters are the diameters of agglomerates, the volume fraction and the mean number of primary particles [344]. LII has been applied ti in-situ measurement of primary particle size on the manufacture of carbon blacks [345]. The system has also been applied to the measurement of nanosize particles on soot and vehicle exhausts. Measurements on carbon black particles gave data which correlated well with product properties. Tests with titanium dioxide and metal powders were encouraging. [346] 10.18.7 Spectral transmission and extinction Cemi used spectral transmission and extinction using UV, visible and near IR to measure slurry particle size distributions with undiluted continuous flow [347]. The method uses multiple linear detector array spectrometers. It also uses multiple sample cells of different optical depths optimized for a specific spectral range, multiple optical paths and multiple linear detector arrays.


10.18.8 Turbiscan multiple light scattering measurements Turbiscan On-Line monitors and quantifies the effects of process variables in dispersed systems. It operates in the volume concentration range from 0 to 60% and the size range of 0.1 to 5,000 |am. The vertical scan macroscopic analyzer consists of a reading head that moves along a flat bottomed cylindrical cell to scan the entire sample length. The optical sensor consists of a pulsed near-infra-red light source and two synchronous detectors: The transmission detector monitors light transmitted through the suspension and the back-scattering detector receives the back-scattered light at 135°. The optical sensors acquire transmission and backscattering signals in from 0.1 to 10 seconds, every 40 |Lim along the sample tube for a maximum of 80 mm, and these signals are digitized and displayed by the software indicating real-time changes in transmission and backscattering intensities. These parameters are directly related to particle size and volume concentration. Turbiscan Lab measures concentrated suspensions at up to 95% by volume. The mean particle diameter can be calculated over the size range 0.05 to 1000 |im. Turbiscan Ma 2000 measures the destabilization of concentrated dispersions and determines the mechanisms driving it. The ma 2000 is used to improve formulations, document stability tests and shorten stability test time. Anisotropic particles have been measured using a variety of techniques and correlation between them was found to depend on particle morphology [348]. For cylindrical glass fibers photocentrifuge data gave good correlation with image analysis using the Turbiscan, whereas for platelets laser diffraction gave the best correlation. The correlation between Turbiscan for flake-like mica was found to be very good whereas the photocentrifuge gave better agreement with rod-like copper oxalate. In both cases some information from the image analysis data was required in order to make assumptions to simplify the deconvolution data of size and shape from the collected data. References 1 2 3 4

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Appendix

Names and addresses of manufacturers and suppliers Aerometrics Inc., (Now part of TSI) 550 Del Rey Avenue, Sunnyvale, CA 94086, USA, (408) 738 6688 aerometrics.com Alpine 89, Augsburg 2, Postfach 629, Germany (Air jJet sieve see Gilson) American Innovation (Now Oncor) Amtec, Alpes Maritimes Technologic, ler C.A., Avenue du Docteur, Julien-Lefebvre, 06270 Villeneuve-Loubert, France.(93 73 40 20) Analytical Measuring Systems, London Road, Pampisford, Cambridge, Cambs. CB2 4EF, England (01223) 836001 Analysette (see Fritsch) Armco Autometrics, 7077 Winchester Circle, Boulder, Colorado, 80301, USA, (303) 530 1600 Artec Systems Corp. 170 Finn Court Farmingdale, NY 11735, USA, (516) 293 4420, [France: Nachet, 106 Rue Chaptal, 92304 Levallois Peret-Cedex, 757 31 05] ATM Corporation, 645 S. 94th PI., Milwaukee, WI 53214, USA, (414) 453 1100 Autometrics, 4946N 63"^^ Street, Boulder. CO 80301, USA Bahco (see Gilson) Bausch and Lomb Inc., 820 Linden Avenue, 30320 Rochester, NY 14624, USA Beckman Coulter, Particle Characterization Div., P.O. Box 169015, ra/c 195/10, Miami, FL 33116-9015, 800-523-3713, www beckmancoulter.com Boekeler Instruments Inc., 3280 East Hemisphere Loop, #114 Tucson AR 85706-5024, 602 573 7100 Brinkmann Instrument. Company, 1 Cantiague Road, PO Box 1019, Westbury, NY 11590207, USA, (516) 334 7506 Bristol Equipment Company, Yorkville, IL, 708 553 7161 Bristol Industrial and Research Associates Ltd (BIRAL), PO Box 2, Portishead, Bristol, BS20 9JB, UK Brookhaven Instrument Corporation, 750 Blue Point Road, Holtsville, NY 11742, USA, (516)758 3200 Buckbee Mears Co., 245 East 6th Street, St Paul,, MN 55101, USA, (612) 228 6400 Buehler, Optical Instruments Division, 41 Waukegan Road, Lake Bluff, IL, USA, 60044 (312)295 6500 Canty Vision, Dublin, Ireland, 43 0732 7701 77 Carl Zeiss, 7082 Oberkochen, Germany Cilas 8 Avenue Buffon, BP 6319, 45163 Orleans, France (+33) 2 38 64 59 55164, http://www.cilasus.com/analyzer Climet Instruments Co., 1320 W. Colton Ave., PO Box 1760, Redlands, CA 92373,USA, (714)793 2788 Collimated Holes Inc.,Campbell, CA 95008

624 Manufacturers

and suppliers

Compix Inc. Imaging Systems, 230 Executive Drive, #102-E, Mars, PA 16046, 412 772 5277 Contamination Control Systems GmbH, D-8999 Munchen 2, Germany, (089) 18 80 06/07 Coulter Inc., (now Beckman Coulter)Box 2145, 590 W.20th Street, Hialieah, FL 33010, USA, (305) 885 0131 CSIR, South African Council for Scientific and Industrial Research, PO Box 395, Pretoria, South Africa

Dantec Measurement Technology A/S 16-18 Tonsbakken, 2740 Skovlunde, Denmark. (045) 44 57 80 00, e-mail [email protected], Web site www.dantecmt.com Denver Equipment Company, 1855 Blake Street, Suite 201, Denver CO 80202, 1-800-3211135 Dipix Technologies Inc. 1051 Baxter Road, Ottawa, Ontario K2C 3P1, Canada Tel: 613596-4942 Fax: 613-596-4914 Toll Free: 800-724-5929 E-mail: Vision Products Division: [email protected] Dispersion Technology Inc., 3 Hillside Avenue, Mount Kisco, NY 10549, (914)241-4791, email [email protected]. Donaldson Company Inc., 1400 W. 94'*^ Street, Minneapolis, MI 55431, USA Dow Chemicals, Midland, MI, USA Duke Scientific Corp., 1135D San Antonio Road, PO Box 50005, Palo Alto, CA 94303, USA, (415) 424 1100 DuPont Company (E.I. DuPont de Nemours), Materials Characterization Systems, Concord Plaza, Quillen Building, Wilmington, DE 19898, USA, 302 772 5488 Endecottes Ltd, 9 Lombard Road, London, SW19 3BR, UK, (081) 542 8121, [US Distributor, CSC Scientific Company] Erdco Engineering Corp., 136 Official Road, Addison, IL 60101, USA, (708)328 0550 Faley International Corporation, PO Box 669, El Toro, CA 92630-0669, Flowvision, Kelvin Microwave Corp., Charlotte, NC, USA, (704) 357 9849 FFFractionation 4797 South West Ridge Blvd., Salt Lake City, UT 84118-8429, www.fffract.com Fritsch, Albert & Co., Industriestrasse 8, D6580 Idar-Oberstein 1, Germany, 06784/70 0 Galai Production, PO Box 221, Industrial Zone, Migdal, Haemek 10500, Israel, 972-654 3369 Gallenkamp Ltd, Portrack Lane, Stockton on Tees, Co. Durham, UK Gelman Hawksley, 12 Peter Road, Lancing, Sussex, UK Gilson Co. Inc., PO Box 677, Worthington, OH 43085-0677, USA, (614) 548 7298 Glen Creston, 16 Dalston Gardens, Stanmore, Middlesex HA7 IDA ,UK, 0181 206 0123 Greenfield 01 866 296 3049 Gustafson Inc., 1400 Preston Rd., Piano, Texas, USA, (214) 985-8877 Hamamatsu Photonics France, 49/51 Rue de la Vanne, 92120 Montrouge, France, 46 55 47 58 Hiac/Royco Div., Pacific Scientific, 11801 Tech Road, Silver Springs MD 20904 (301) 680 7000

Manufacturers

and suppliers

625

Hitech Instruments, Inc., PO Box 886, 4799 West Chester Pike, Edgemont PA 19028, 215 353 3317 Horiba Instruments Inc. 17671 Armstrong Avenue, Irvine CA 92614-9782, 800-446-7422, 949 250 0924, e-mail; [email protected]/horiba Hosokawa Micron International Inc., 10 Chatham Road, Summit, NJ 07901, USA, (908) 598 6360 Insitec Inc., (now Malvern) 2110 Omega Rd., Suite D, San Ramon, CA 94583, USA, (510) 837 1330 InterSystems Industrial Products, 17330 Preston Road, Suite 105D, LB 342, Dallas, TX 75252, USA Joyce-Loebl, Marquisway, Team Valley, Gateshead NEl 1 OQW, UK, (0191) 482 2111 Kane May Ltd, Northey International Division, Nortec House, Chaul End Lane, Luton, Beds. LU4 8E2, UK, 01582 584343 Kemsis, 34830 Jacou, France Kowa Instruments, 355 Shoreway Road, South Vermont Avenue, Torrence CA 90502, (213) 327 Kratel SA, CH-1222 Geneve-Vesenaz, 64 Ch de St Maurice, Switzerland, 022 52 33 74 Labcon Ltd., 24 Northfield Way, Aycliffe Industrial Estate, Newton Aycliffe, CoDurham, DL5 6EJ, UK, 01325 313379. Lasentec, Laser Sensing Technology Inc., 15224 NE 95th Street, Redmond, WA, 98052, USA, (206) 881 7117) Lecotrac (see Horiba) Leeds & Northrup Instruments, Sunneytown Pike, PO Box 2000, North Wales, PA 19454, USA., (215) 699 2000 Leica Inc., 111 Deer Lake Road, Deerfield, IL 60015, USA LeMont Scientific, Inc. 2011 Pine Hall Drive, Science Park, State College, PA 16801, USA, (814)238 8403) L.U.M. Gmbh, (Gesellschaft fur Labor-Umweltdiagnosik & Medizentechnik mbH), Rudower Chaussee (OWZ), 12489 Berlin, Germany, 49-(0)30-67 19 81 88, e-mail [email protected], web www.lum-gmbh.de Malvern Instruments, Inc., 10 Southville Rd., Southborough, MA 01772, (508)480-0200, www.malvem.co.uk Matec Applied Sciences, 75 South Street, Hopkinton, MA 01748, USA, (508) 435 9039 Maztech Microvisions Ltd., 1306 Wellington Street, Suite 202, Ottawa, Ontario, Canada KlY 3B2, 613 596 5775, www.maztech.com Met One, (Pacific Scientific) 481 California Avenue, Grants Pass, OR 97526, USA. (503) 479 1248) Mettler (Switzerland) Instruments A.G., CH-8606, Greifensee-Zurich, Switzerland Micromeretics Instrument Corporation, One Micromeretics Drive, Norcross, GA 300931877. (770)662-3633, www.micromeretics.com Micron Powder Systems, 10 Chatham Road, Summit, NJ 07901, USA, (908) 273 6360 Micropul Corporation, 10 Chatham Road, Summit, NJ 07901, USA, (201) 273 6360 Microscal Ltd, 79 Southern Row, London WIO 5AL, UK, (0181) 969 3935

626 Manufacturers

and suppliers

Microtrac Inc., 148 Keystone Drive, MontgomeryviHe, PA 18936, (215)619 9920, www. microtrac.com Millipore (U.K.) Limited, Millipore House, Abbey Road, London NWIO 7SP, UK, (01965) 9611/4 Monitek Technologies, Inc., 1495 Zephyr Avenue, Hayward, CA 94544, USA, (415) 471 8300 MSA, (Mines Safety Appliances), 201 Braddock Ave., Pittsburgh 8, PA, USA Munhall Company, 5655 High Street, Worthington, OH, 43085, USA, (614) 888 7700 Nachet Vision SA, 125 Boulevard Davout B.P. 128, 75963 Paris Cedex 20, France, 33(1) 43.48.77.10 National Physical Laboratory, Middlesex, UK Newark Wire Cloth Company, Newark, New Jersey Nicomp Particle Sizing Systems, 75 Aero Camino, Suite B, santa Bardera, CA 93117, USA, (805) 968 1497 Nikon Corporation, Tokyo, Japan, www.nikon.co.jp Nisshin Engineering Co. Ltd., 1000 Corporate Grove drive, Buffalo Drive, IL 60089, 708 215 6865,708 520 9520 Nitto Denko, Tokyo, 1-12 Shimohozimi, Ibaraki, Osaka, 567-868, 81 726 22 2981 Oncor Instrument Systems, 9581 Ridgehaven Court, San Diego, CA 92123, 619 560 9584 Optomax, 9 Ash Street, Hollis, NH 03049, USA, (603) 465 3385 Otsuka(Munhall) Outokumpu Mintec Oy, PO Box 84, SF-02201, Espoo, Finland, 358 0 4211 Oxford Lasers Inc., 29 King Street, Littleton, MA 01460-1528, (978)742 9000, e-mail [email protected] Paar USA Inc., 340 Constance Drive, Warminster, PA 18974, USA, (215) 443 7570 Pacific Scientific Co., Hiac/Royco Div., 11801 Tech. Rd., Silver Springs, MD 20904, USA, (301)680 7000 Particle Measuring Systems Inc., 1855 South 57th Court, Boulder CO 80301, USA, (303) 443 7100 http://www.pmeasuring.com Particle Sizing Systems, 75 Aero Camino, Santa Barbara, CA 93117, (805)968-1497Pascall Ltd, Gatwick Road, Crawley, Sussex, RHIO 2RS, UK Pascal Ltd.,Gatwick Road, Crawley, Sussex, RHIO 2RS, UK PenKem(now Dispersion Technology) Pharma Vision Systems AB, Lund, Sweden PMT Partikel-Messtechnik GmbH, Carl Zeiss Str 11, Postfach 15 16, D-7250 LeonbergGebersheim, Germany, 7152 51008 Polymer Laboratories, 413 253 9554, www.polymerlabs.com/partsize Polytec GmbH & Co., D-7517 Waldbronn, Karlsruhe, Germany, (07243 6 99 44) Procedyne Corporation, 11 Industrial Dr., New Brunswick, NJ 08901, USA, (908) 249 8347 Proassist Company, 1614 East 5600 South, Salt Lake City, UT 84121, USA Procedyne Corporation, 221 Somerset Street, New Brunswick, NJ. USA Quality Control Equipment, PO Box 6010, Des Moines, Iowa 50309, 515 266 2289 Quantachrome Corp., 1900 Corporate Drive, Boynton Beach, FL 33426, USA, (407) 731 4999

Manufacturers and suppliers

62 7

Retsch, Inc., 17651 Armstrong Ave., Irvine, CA 92614, (949) 752-2004, e-mail [email protected] Rotex Inc., 1230 Knowlton Street, Cincinnati, Ohio 45223-1845, (513) 541-1236, www.rotex.com Rotex Inc., 1230Knowlten Street, CincinnatiOH 45223, USA 513 541 1236 Sartorius-Werke GmbH, Postfach 19, D-3400 Gottingen, Germany, (0551) 308-1 Sci-Tec-Inc, 6660 North High Street, Suite 2A, Worthington, OH 43085, 614 888 0285 http://sci-tec-inc.com Sedimage Photosedimentometer, www. cs.technion,ac.il/labs Seishin Enterprise Co. Ltd, Nippon Brunswick Bldg., 5-27-7 Sendagaya, Shibuya-ku, Tokyo 151, Japan, (03 350 5771) Shapespeare Corporation, 901 Park Place, Iowa City, lA 52240,USA, 319 351 3736 Shimadzu Scientific Scientific Instruments Inc., 7102 Riverwood Dr., Columbia, MD 21046, USA, (410) 381 1222, 1 800 477 1227, e-mail [email protected] Spectrex Corp., 3594 Haven Avenue, Redwood City, CA.94063, USA, (415) 365 6567 Stork Screens, 3201 North Interstate 85, Charlotte, NC 28213 Sympatec GmbH, 3490 U.S. Route 1, Princeton, NJ 08540-5706, (609)734-0404, e-mail [email protected] Technord photocentrifuge Thermo-Systems Inc (TSI), 500 Cardigan Road, Shoreview, MN 55126, USA, 651 490 2811, 1 800 874 2811, e-mail [email protected] Tracor Northern, 255 West Beltline Hishway, Middleton, WI 53562, USA 609 831 6511 Turbiscan, Fullbrook Systems Ltd., Unit 4, Boume End Mills Industrial Estate, Hemel Hempstead, Herts., HPl 2UJ. 01442 876 777, e-mail sales@fullbrook,com Turboscan, Woods Center, ANSTO, Technology Park, New lUawara Road, Lucas Heights, NSW 2234, Australia, Tel 02 9543 0477, e-mail [email protected] Tyler, W.S., Inc., E Hwy 12, Benson, MN 56215, USA, (216) 974 1047 Veco N.V. Zeefplatenfabrik, Eerbeck (Veluive), The Netherlands Vibrosonic sieves, Russel Finex Inc., Mount Vernon, NY Vorti-siv, 36135 Salem Grange Road, Salem, OH 44460, USA, (330) 332 4958, Fax: 1543 Warmain International Pty, Melbourne, Victoria, Australia Wyatt Technology Corp, 30 South La Patera Lane, 8-7, Santa Barbara, CA 93117, USA, (805) 681 9009, e-mail:[email protected] Zeiss, Carl Inc., One Zeiss Drive, Thomwood, NY 10594, USA, (914) 747 1800 Zeiss; Carl, 7082 Oberkochen, Postfach 35/36, Germany, (07364 20 3288)

Author index The unbracketed numbers give the reference locations in the text. The numbesr in brackets give the chapter number followed by the number of the reference in that chapter. Abbot, D., 550(10.143), 55/(10.149) Abeynayake, C , 502(10.321) Ackerman, L., 226(4.70) Adamson, A.W., 339(6.95) Adel, G.T., 144(3.10) Adeyayo, A., 572(10.205) Adjadj, L.P., 55J(10.262) Agar, A.W., 190(3.\62) Agarwel, J. K., 5/0(9.204) Aigal, A. K., 5d

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