Principies of

SME

Destined to become an industry standard, this comprehensive referenceexamines all aspeis of mineral processing, from the handling of rawmaterials to separation strategies to the remediation of waste producs. Thebook incorporates state-of-the-art developments in the fields of engineering,chemistry, computer science, and environmental science and explains howthese disciplines contribute to the ultmate goal of producing minerals andmetis economically from ores. With contributions from more than 20recognized authorities, this thorough reference presents the most currentthinking on the science and technology of mineral processing.

The book is an indispensable textbook for students of mineral processing andhydrometallurgy, and a practical reference for seasoned industry professionalsinterested in improving operational efficiencies. It presents the principies thatgovern various unit operations in mineral processing along with examples thatIlstrate how these principies apply to real-world situations.

The Society for Mining, Metallurgy, and Exploration, Inc. (SME), advances the worldwidemining and minerals community through information exchange and pwfessional development.

ISBN 0-87335-167-3

Principles ofMineralProcessingEdited by

Maurice C. Fuerstenau and Kenneth N. Han

Published by theSociety for Mining, Metallurgy, and Exploration, Inc.

All Rights Reserved. Printed in the United States of America.

Copyright 2003 Society for Mining, Metallurgy, and Exploration, Inc.

2002042938TN500.P66 2003622'.7--dc21

No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any formor by any means, electronic, mechanical, photocopying, recording, or otherwise, without the priorwritten permission of the publisher. Any statement or views presented here are those of the author andare not necessarily those of SME. The mention of trade names for commercial products does not implythe approval or endorsement of SME.

Information contained in this work has been obtained by SME, Inc. fram sources believed to bereliable. However, neither SME nor its authors guarantee the accuracy or completeness of anyinformation published herein, and neither SME nor its authors shall be responsible for any errors,omissions, or damages arising out of use of this information. This work is published with the under-standing that SME and its authors are supplying information but are not attempting to renderengineering or other professional services. If such services are required, the assistance of anappropriate professional should be sought.

:iSociety for Mining, Metallurgy, and Exploration, Inc, (SME)8307 Shaffer ParkwayLittleton, Colorado, USA 80127(303) 973-9550/ (800) 763-3132www.smenet.org

SME advances the worldwide mining and minerals community through information exchange andprofessional development. SME is the largest association of minerals professionals.

ISBN 0-87335-167-3

G~UIM~ ;" Library of Congress Cataloging-in-Publieation Data.r, ,.o" ! -- t, ::' I 'c..

. Principies of mineral processing / [edited by] Maurice C. Fuerstenaup. cm.

Includes bibilographical references and indexoISBN 0-87335-167-31. Ore dressing. 2. Hydrometallurgy. 1. Fuerstenau, Maurice C.

J:::. ,;e .O;:.J,? ~~~-,..' ~' 1',P1= 1 :

Contents

CHAPTER 1

CHAPTER 2

CHAPTER 3

CHAPTER 4

L1ST OF AUTHORS vii

PREFACE ix

INTROOUCTlON 1Maurice C. Fuerstenau and Kenneth N. Han

Goals and Basics of Mineral Processing 1Metallurgical Efficiency 1Economic Concerns 3Unit Operations 4Examples of Mineral Processing Operations 5Environmental Consequences of Mineral Processing 8

PARTlCLE CHARACTERIZATlON 9Richard HoggParticle Characteristics 9Mathematical Treatment of Particle Distributions 14Measurement of Particle Characteristics 29Comparison and Interconversion of Particle Size Data 53Appendix 2.1: Moment Determination and Quantity Transformation

from Experimental Data 54Appendix 2.2: Combination of Sieve and Subsieve Size Data 54

SIZE REOUCTlON ANO L1BERATlON 61John A. Herbst, Vi Chang Lo, and Brian Flintoff

Introduction 61Fundamentals of Particle Breakage 63

., . \Comminution Equiprnent 79Comminution Circuits 94Process Control in Cornminution 100Financial Aspects of Comminution 113Symbol Glossary 115

SIZE SEPARATION 19Andrew L. Mular

Introduction 119Laboratory Size Separation 121Sedimentation Sizing Methods 127

iii

alumWIIIIJIlUllIImI\lIlllll\\l---IIlIIIllIlIiUlllD-' ......-......'"'"'"-... .....-..--------------

CHAPTEFl 5

CHAPTER 6

CHAPTER 7

CHAPTER 8

CHAPTER 9

Industrial Screening :1.29Size Classification 148

MOVEMENT OF SOllDS iN UQUiDS 173I(enneth N. HanIntroduction 173Dynamic Similarity 173Free Settling 174Particle Acceleration 179Particle Shape 181Hindered Settling 183

GRAVITY CONCENTRATION 185Frank F. Aplan

Introduction 185The Basics of Gravity Separation 188Float-Sink Separation 195Jigs 202Flowing Film Concentrators, Sluices, and Shaking Tables 206Centrifugal Devices 212Pneumatic Devices 212Process Selection and Evaluation 214

MAGNETIC AND ELECTROSTATlC SEPARATION 221Partha Venkatraman, Frank S. Knoll, and James E. Lawver

Introduction 221Review of Magnetic Theory 221Conventional Magnets 228Permanent Magnets 232Superconducting Magnets 236Electrostatic Separation 239

FLOTATlON 245Maurice C. Fuerstenau and Ponisseril Somasundaran

Surface Phenomena 245Flotation Reagents 252Chemistry of Flotation 259Flotation Machines 292Column Flotation 296Flotation Circuits 299

L1QUID-SOLlD SEPARATION 307Donald A. Dahlstrom

Introduction 307Major Influences on Liquid-Solid Separation 309Liquid-Solid Separation Equipment 317Gravitational Sedimentation 317Filtration 322Basic Guidelines for Application 334

iv

CHAPTER 10

CHAPTER 11

CHAPTER 12

CHAPTER 13

CHAPTER 14

Gravity Sedimentation Applications 3

James E. LawverDeceased

Richard HoggProfessor EmeritusMineral Processing SectionThe Pennsylvania State UniversityUniversity Park, PA

Frank S. KnollConsultantCarpco Divisionoutokumpu Technology lnc.Jacksonville, FL

Matthew J. HreberProfessor EmeritusDepartrnent of Mining EngineelingColorado School of MinesGolden, CO

Yi ChangLoRetiredGS IndustriesKansas City, MO

John A. HerbstGeneral ManagerSvedala Minerals lndustry lnc.Kealakekua, HI

Kenneth N. HanRegents Distinguished ProfessorMetallurgical and Materials EngineelingSouth Dakota School of Mines and TechnologyRapid City, SD

vii

Stoyan N. GroudevProfessorDepartrnent of Engineering GeoecologyUniversity of Mining and GeologyStudenski grad-DurvenitzaSofa, Bulgaria

Brian C. FlintoffSenior Vice PresidentMetso MineralsKelowna, BC

Donald W. GentryPresident & CEOPolyMet Mining CorporationGolden, CO

Donald A. DahlstromProfessor EmeritusDepartrnent of Chemical EngineeringThe University of UtahSalt Lake City, UT

Maurice C. FuerstenauProfessor of MetallurgyMetallurgical and Materials EngineeringThe University ofNevadaReno, NV

Hendrick ColijnConsulting EngineerPisgah Forest, NC

Frank F. AplanDistinguished Professor EmeritusMineral Processing SectionThe Pennsylvania State UniversityUniversity Park, PA

Authors..............

Roben D. MorrisonTechnical DirectorJulius Kruttschnitt Mineral Research CenterIndoorroopilly, Queensland, Australia

Andrew 1. MularProfessor EmeritusMining and Mineral Processing EngineeringThe Unversity of British ColumbiaVancouver, BC

J. Mark RichardsonPresidentJ. K. Technology/Contract Support ServicesRed Bluff, CA

viii

Ross W. SmithProfessor EmeritusMetallurgical and Materials EngineeringThe University of NevadaReno, NV

Ponisseril SomasundaranVon Duddleson Krumb ProfessorMineral Engineering DepartmentColumbia UniversityNew York, NY

Partha VenkatramanResearch & Development ManagerCarpco DivisionOutokumpu Technology Inc,Jacksonville, FL

.

Preface

The world is faced with opportunities and chaIlenges that require ever-increasing amounts of rawmaterials to fuel various industrial sectors, and, at the same time, meet environmental constraints asso-ciated with excavating and processing these raw materials. In addition, gradual depletion of mineralresources and the necessity of handling more complex forms of resources, primary and secondary, haveled to chaIlenges in the development of state-of-the-art technologies that are metaIlurgically efficientand environmentaIly friendly. Unquestionably, technology advances are the key to sustaining a suffi-cient supply of necessary raw materials.

To advance the technology in the production of material resources, nations lcok to practicing andfuture engineers. Current and future mineral processing engineers must obtain sound and rigoroustraining in the sciences and technologies that are essential for effective resource development. Manyindustrial and academic leaders have recognized the need for more textbooks and references in thisimportant area. This was the driving force for writing a comprehensive reference book that coversmineral processing and hydrometaIlurgieal extraction.

This book was written first to serve students who are studying mineral processing and hydro-metaIlurgy under various titles. We also hope that the book will serve as a valuable reference tomany industrial practitioners in the mineral processing field. In the chapters that follow, you willfind first principles that govern various unit operations in mineral processing and hydrometaIlurgy,along with examples to illustrate how fundamental prncples can be used in real-world applieations.In general, the volume covers topies in the arder of the usual processing sequence. Comminution, thebreakage of rocks and other materials, is covered in such a way that the fundamental principles canbe used not only in mineral processing but also in other relevant are as such as chemieal engineeringand pharmaceutical fields.

Understanding the characteristics of particles and the separation of particulate materials fromone another is of ultimate impartance. Separation technologies based on properties such as magne-tism, electrieal properties, and surface properties of various minerals are present along with industrialexamples.

Because most mineral processing unit operations take place in water as a medium, understand-ing how solids can best be separated from water is of industrial importance. Efficiently using waterduring effective solid-Iqud separation is often vital to the success of the overall mineral beneficiationoperation.

With computer application technologies continuing to emerge rapidly, the mineral industry hasmade tremendous advances in its industrial production. Plant automation and control often playa vitalrole in the overall success of the plant operation. The chapter on comminution covers sorne of theseinnovations in automation.

ix

Once desired minerals are recovered from the unclesired portian of an ore deposir, chemical treat-ment to unlock the desired metal elements from various minerals is necessary. Hyclrometallurgicaltreatment for the chemical release of metal elements from various minerals is presented along withfundamental water chemistry ancl kinetic principIes.

We are fortuna te that many world-class authorities in various areas of mineral processing havejoinecl this endeavor, and we thank them for their participation. We would also like to take this oppor-tunity to thanlc the staff of the Society for Mining, Metallurgy, and Exploration, Inc., for their support inproducing this book.

x

CHAPTER 1

IntroductionMaurice C. Fuerstenau and Kenneth N. Han

The term mineral processing is used in a broad sense throughout this book to analyze and describe theunit operations involved in upgrading and recovering minerals or metals from ores. The field of mineralprocessing is based on many fields of science and engineering. Humanities and social science have alsobecome an integral part of this technology because mineral processing, like many other technologies, iscanied out to improve human welfare. In addition, environmental science and engineering have becomeinseparable components; the steps involved in mineral processing have to be founded not only on soundscientific and technological bases but on environmentally acceptable grounds as well.

GOALS AND BASICS OF MINERAL PROCESSING

In the traditional sense, mineral processing is regarded as the processing of ores or other materials toyield concentrated products. Most of the processes involve physical concentration procedures duringwhich the chemical nature of the minerales) in question does not change. In hydrometallurgicalprocessing, however, chemical reactions invariably occur; these systems are operated at ambient orelevated temperatures depending on the kinetics of the processes.

The ultimate goal in the production of metals is to yield metals in their purest formo Mineralprocessing plays an integral part in achieving this objective. Figure 1.1 shows a generalized flowdiagram for metals extraction from mining (step 1) through chemical processing. Steps 2 and 3 involvephysical processing and steps 5 and 7 involve low-temperature chemical processing (hydrometallurgy).All four steps are considered part of mineral processing. High-temperature smelting and refining (pyro-metallurgy), steps 4 and 6, are not included under the heading of mineral processing.

Table 1.1 specifies processing routes from ore to pure metal for a number of metals. Note thatprocessing routes can be quite different and that more than one route may be possible for many ofthese metals. For example, in the extraction of copper or gold from low-grade ores, dump or heapleaching is commonly practiced. The choiceiof this leaching practice is frequently driven by the overalleconomics of the operation. Because crushing and grinding of ores are quite expensive, leaching of oresin large sizes is attractive compared to the leaching of finely ground ores, even though the overallrecovery of metals from the leaching of fine particles is, in general, much greater than that obtainedwith large particles. The introduction of this innovative leaching process has made feasible the miningof many mineral deposits that could not be processed economically through conventional technologies.

METALLURGICAL EFFICIENCY

One of the most important and basic concepts in mineral processing is metallurgical efficiency. Twoterms are commonly used to describe the efficiency of metallurgical processes: recovery and grade.These phenomena are illustrated in the generalized process presented in Figure 1.2. In this example,100 tph of ore are being fed into a concentration operation that produces 4.5 tph of concentrate and

1

Low-temperatureRefining (7)

Pure metal (8)

FIGURE 1.1 Generalized flowchart of extraction of metals

TABLE 1.1 Processing sequence(s) for a number of selected metals5teps Involved in the Processing Route

(see Figure1.1)Metal Associated Major Minerals 1 2 3 4 5 6 7 8lron Hematite, Fe203; rnagnetite, Fe304 x x x x x xAluminum Gibbsite, AI203-3H20; diaspore, AI2203XH20 x x X x xCopper Chalcopyrite, CuFeS2; chalcocrte, Cu2S x x x x x x xZinc Sphalerlte, ZnS x x x x x x

x x x x x x x

Lead Galena, PbS x x x x x xGold Native gold, Au x x x x x x

x x* x x x

Platinum Native platlnum, Pt; platinum sulfides x x x x x xSilver Native silver, Ag x x x x x x

*Only crushing is practices; grindlng is usually omitted.

2 I PRINCIPLES OF MI~IERAl PROCESSiNG

ECONOMIC CONCERNS

Table 1.2 surnmarizes the total U.S. supply and recycled supply of selected metals in 1996. The totalsupply of iron and steel includes supply from primary and secondary sources as we11 as imports; thesetwo metals represent by far the largest of commodiries produced and consumed, fo11owed byaluminum, copper, and lead. Note that the recycled supply of these metals from processing scrap isstrikingly high. In addirion, the tonnage of precious metals consumed is rather small. However,because of the high prices of precious metals, their monetary value is substantial. For example, themonetaryvalue of 516 t of gold was $12.8 billion in 1996, compared to $10.7 billion for 5.3 million t ofcopper and lead.

Total Supply, Recycled Supply,Metal million t metal content million t metal content % Recycledlron and steel 183 72 39Aluminum 8.34 3.29 39Copper 3.70 1.30 35.1Lead 1.63 1.09 66.8Zinc 1.45 0.379 26.1Chromium 0.48 0.098 20.5Magnesurn 0.205 0.0709 35Gold 516 t* 150 t* 29

Source: U.S. Bureau of Mines (1997).*Value for 1995.

NYRODUGTiOill I 3

-----~. Concentrate20% A,80% B,4.5 tph

"------------> Tailings0.10% A,99.90% B,95.5 tph

Unit OperationFeed ---j;>1%A,99% B,100 tph

TABLE 1.2 U.S. total and recycled supply of selected metals in 1996

FIGURE 1.2 A simple material balance for a unit operation

95.5 tph of tailings. In upgrading this process, then, 1.0 tph of the desired material, A, is introducedinto the unit operarion and 0.9 tph (4.5 x 0.2) of this material reports to the concentrate, resulting in90% recovery (0.9/1.Ox 100). The grade of the mineral, A, has been improved from 1% to 20%. Theterm percent recovery refers to the percentage of the valuable material reporting to the concentrate withreference to the amount of this material in the feed. Note that obtaining the highest possible recovery isnot necessarily the best approach in a concentration process. High recovery without acceptable gradewi11lead to an unsalable product and is therefore unsatisfactory.

Mineral processing engineers are responsible for oprimizing processes to yield the highest possible\

recovery with acceptable purty (grade) for the buyers or engineers who wi11 treat this concentratefurther to extract the metal values. To achieve this goal, economic assessments of a11 possible techno-logical altemarives must be conducted.

UNIT OPERATIONS

4 I PRINCiPLES OF MINERAL PROCESSING

TAIBLE :1..3 Abundance of vartous elements ln the lEarth's crust compared to annualU.S. eonsumptlon

1.28 X 108

5.4 X 106

2.3 X 106

1.0 X 105

1.2 X 106

1134.52 x 103

U.S. Consumption, stjyear58.137 x 10-3

8 X 10-3

1.5 X 10-3

1.0 X 10-7

2.0 X 10-6

Relative Abundance, %FeAlCuzn .PbAuAg_

Element

Source: U.S. Bureau of Mines (1990).

Numerous steps, called unit operations, are involved in achieving the goal of extracting minerals andmetals from ores in their purest possible formo These steps inc1ude

Size reduction. The process of crushing and grinding ores is known as comminution. The pur-pose of the comminution process is threefold: (1) to liberate valuable minerals from the orematrix, (2) to increase surface area for high reactivity, and (3) to facilitate the transport of orepartic1es between unit operations.

Size separation. Crushed and ground products gene rally require c1assification by partic1e size.Sizing can be accomplished by using c1assifiers, screens, or water elutriators. Screens are usedfor coarse particulate sizing; cyclones are used with fine particulates.

Concentration. Physicochemical properties of minerals and other solids are used in concentra-tion operations. Froth flotation, gravity concentration, and magnetic and electrostatic concen-tration are used extensively in the indust:ry.

- Frothflotation. The surface properties of minerals (composition and electrical charge) areused in combination with collectors, which are heterogeneous compounds containing apolar component and a nonpolar component for selective separations of minerals. Thenonpolar hydrocarbon chain provides hydrophobicity to the mineral after adsorption ofthe polar portion of the collector on the surface.

- Gravity concentration. Differences in the density of minerals are used to effect separationsof one mineral from another. Equipment available inc1udes jigs, shaking tables, andspirals. Heavy medium is also used to facilitate separation of heavy minerals from lightminerals.

- Magnetic and electrostatic concentration. Differences in magnetic susceptibility and elec-trical conductivity of minerals are utilized in processing operations when applicable.

Table 1.3 lists the relative abundance of various metals in the Earth's crust, Most metals arepresent in extremely small concentrations in nature, and none of these metals can be recoveredeconomically at these concentrations. Rock that contains metals at these concentrations is not ore; oreis rock that can be processed at a profit. An average copper ore, for example, may contain 0.3% to0.5% copper. Even this material cannot be treated economically at high temperature without priorconcentration. There is no way that rock containing 10 lb of copper and 1,990 lb of valueless materialcan be heated to 1,300C and treated to recover this quantity of metal economically. Concentrating theore by froth flotation to approximately 25% or more copper results in a product that can be smeltedand refined profitably.

~HROi\)UGT!OI\1 5

EXAMPlES OF MINERAL PROCESSING OPERATIONS

FIGURE 1.3 Flowsheet for crushlng and grading roek

~ 2-2% in. Crushed Rock

~ %-% in. Crushed Rack

~ %-1 in. Crushed Rock%in.

%in.

VibratingScreens

36-in. Feedfrom Quarry

~S-in. Grizzly f,~~

Jaw Crusher Set at S in. r,J/J::J.--------------,

Vibrating Screen Set at 2% in.~ Gyratory Crusher SeI al 2~ ;0. 1 \JI

2~.~~----..~ 1Y2-2 in. Crushed Rock

1%in. y~---..~ 1-1% in. Crushed Rack

1in. y

~Sand to Dust

to Waste

'" Dewaterillg. Most mineral processing operations are conductecl in the presence of water. Solidsmust be separated from water for metal production. This is accomplished with thickeners andfilters.

I!J Aqueous dissoiution. Many metals are recovered from ores by dissolving the desired metal(s)-in a process termed leachillg-with various lixiviants in the presence of oxygen. Followingleaching, the dissolved metals can be concentrated by carbon adsorption, ion exchange, or sol-vent extraction. Purified and concentrated metals may be recovered from solution with a num-ber of reduction techniques, including cementation and electrawinning.

Figure 1.3 shows a typical flowsheet for crushing and sizing rack in a quarrying operation. Run-of-mineore can be present as lumps as large as 1.5 m (5 ft) in diameter. In this figure's example, 91.4-cm (3-ft)lumps of rack are fed to a crusher that reduces the material to 20.3 cm (8 in.) or less in diameter. Afterscreening to remove rack that is less than 57.2 mm (2114 in.) in size, rack between the sizes of 57.2 mm(2114 in.) and 20.3 cm (8 in.) is further reduced in size by a gyratory crusher. The praduct from this stepis then classified by screening to the desired product for sale.

Figure lA shows an integrated circuit demonstrating crushing, grinding, size separaton, andgravity concentration of a tin ore. Initial size separation is effected with a grizzly set at 11/2-in. Oversizematerial is fed to ajaw crusher set at 1112-in., and the crushed product is, then, further reduced in: sizeto 20 mesh by ball milling. The -20-mesh material is classified by hydrocyclones set at 150 mesh, andthe -ISO-mesh material is sent to shaking tables to concentrate the heavy tin mineral, cassiterite. Themiddlings in this pracess receive additional treatment. The concentrate from this operation is regroundand sized at 200 mesh. Two-stage vanning is used to praduce a fine tin concentrate.

FIGURE 1.4 Flowsheet for the gravity concentration of a tin ore

6 I IPRINCIPlES OF MINERAL PROCIESSING

"---:i:--P---:::i:+-'---:::i:f-l---' Tailings to WasteTin Concentrate

RougherShaking Tables

CI ~4 Regrindeaner I Mili Hydrocyclone SetShaking Table ~ 0----.. at 200 Meshs;:nConcentrating Vanner I~

Retreatment Vannerl~~-----.. Free TailingsFine Tin Concentrate to Waste

1Y2 in. Grizzly~Jaw Crusher S" at t ~ in. i~

t~O -'- !6IY~CIOne Set at 150 MeshBall Mili _ -oo' V ~~

The flowsheet describing the flotation processing of a copper ore containing chalcopyrite andmolybdenite is shown in Figure 1.5. After grinding and classification, pulp is fed to rougher flotation.The rougher tailings are thickened and sent to a tailings dam. The rougher concentrate is classified,and the oversize is reground. Cyclone overflow is fed to cleaner flotation, and the cleaner concentrateis recleaned. Cleaner tailings are recycled back to rougher flotation, and the recleaner concentrate isthckened and sent to the molybdenum recovery plant for further processing. In this operation, thefeed contains 0.32% Cu and 0.03% Mo. Rougher concentrate, cleaner concentrate, and recleanerconcentrate contain 7%-9% Cu, 18% Cu, and 25% Cu, respectively. Recleaner concentrate alsocontains 2%-3% Mo.

Figure 1.6 depicts a flowsheet for processing free-miliing oxidized gold ore. The kinetics of goldleaching is slow, and gold ores are frequently ground to less than about 75 urn before leaching. Eventhen, one day is usually required in the leaching step. In this process, run-of-mine ore is crushed andground. The ball mili discharge in subjected to gravity concentration to recover the larger particles offree gold. The tailings from this operation are thckened, and the underflow from the thickeners is thensubjected to cyanide leaching. In sorne instances, ores may contain oxygen-consuming minerals, suchas pyrrhotite and marcasite, and a preaeration step may be conducted ahead of cyanide leaching.

Heap leaching has revolutionized the gold mining industry. Low-grade oxidized ores containingapproximately 0.03 oz gold per short ton of ore can be processed with this technology, whereas theycould not be processed by the higher cost grinding/agitation leaching (miliing) process. Figure 1.7presents a simplified flowsheet of heap leaching. As the figure shows, run-of-mine ore mayor may notbe crushed. If crushing is done, the ore is generally crushed to

~Tailings Dam

Rougher Tailings

Tailings..

----------1I,IIIII,III

________J,,,

-t _

Preaeration

CyanideLeaching

'f'J{!"'~' t1Rougher R"Concentrate rr--Ih.

8 I PRINCIPI.ES or MINERAL PROCESSING

Run-oHvline Ore

1........- - .

Crushing

1

CHAPTER 2

Particle CharacterizationRichard Hogg

Particulate materials-dry powders as well as liquid or gas suspensions-play an increasingly importantrole in modern society. Most industrial processes involve particulates in sorne stage of the operation,perhaps as raw materals, as products, as unwanted by-products of wear, or simply as atmospheric dust.Particle systems are especially important in mineral processing-a field that deals almost exclusivelywith particulates, from run-of-mne ore to final concentrate. The objective of a mineral processing oper-ation is to take an input stream of particles with a given set of characteristics, modify those characteris-tics, and separate the material into product streams, each with its own set of specified characteristics.

Obviously, characterization is critical to the operation, assessment, and control of mineralprocessing unit operations and systems. The primary aims of this chapter are to address the goals ofparticle characterization for mineral processing applications in light of practical constraints, to discussgeneral schemes for representing particle characteristics, and to describe and evaluate the various tech-niques available for measuring particle characteristics.

Fine particle systems are a distinct class of materials whose behavior is often determined more bytheir particulate characteristics than by the bulle properties of the actual solids. Of these characterstics,the distributions of size, shape, and structure are especially important, and their evaluation is a vitalstep in process control and product specification. The characteristics are not usually single valued. Eachparticle has its own set of characteristics; the system of particles is described by the distributions of thedifferent characteristics. The use of average values may be appropriate in sorne cases; in others, it maybe quite inadequate. In addition to the individual particle characteristics, there are bulle properties thatbelong to the particle system. To sorne extent, these bulk properties are determined by the complete setof individual characteristics, but they may also depend on the relative arrangement of the particles inspace and on interactions among particles and with any intervening medium (air, water, etc.).

PARTlCLE CHARACTERISTICS

Two subsets of individual particle charactrstics can be considered: basic and derived. Basic character-istics represent a minimum set that, when taken together, completely define the particle. By defnition,the basic characteristics include

Size Shape Composition (chemical and mineralogical) Structure (single component or composite; arrangement of constituent phases including pores)

Examples of derived characteristics include

Density Optical characteristics: color, refractive ndex, reflectance

9

Distributions of Partiele Charaeteristies

:10 I PRiNCIPlES OFMiNERAL l"ROCESSING

(Eq.2.2)

(Eq.2.3)

(Eq.2.1)

(Eq.2.4)

(Eq.2.6)

(Eq.2.5)Pi

Pi+l

qi = f q(p)dp

P

Q(p) = fq(p)dpo

q(p)dp = the fraction for which p lies in the range p to p + dp

qi = the fraction of particles for which p has the specific value Pi

%Ic... = the fracton of particles for which p =Pi, r = fi, s = Sic, , etc.

or as a continuous variable:

Individual characteristics generally vary from particle to particle and can be represented by dstrbu-tions. In general, the distributions can be expressed as discrete values or continuous functons in eitherincremental or cumulatve formo For sorne characteristic p (e.g., sze, shape, composton), the incre-mental distribution can be defined as a set of discrete values:

" Electromagnetic characteristics: conductivity, magnetic susceptibility" Thermal characterstics: conductivty, heat capacityro Chemical characteristics: solubility, reactivryro Mechanical characteristics: strength, Young's modulus, Poisson's ratioDerived characteristics are-in prnciple, at least-fixed by and dependent on rhe set of basic char-

acteristics. In other words, all the characteristics just listed are determined by the sze, shape, cornposi-tion, and structute of an individual particle.

The bulle properties of a particle system includeJI Surface areaJI Reactivity" Toxicity

These are essentially determined by the set of basc, individual characteristics and by (1) bulle densityand porosity, (2) homogeneity, and (3) rheology. The latter features depend additionally on the spatalarrangement of the particles and on interactions among them. Bulleproperties are, by definition, singlevalued, but they may depend on the state of the system as well as on its contento

The cumulatve distribution is defined as the fraction for which p is less than sorne specific value.Thus, for the discrete case,

and the continuous equivalent is

Distributions of particle characteristics are, for the most part, inherentIy continuous; that is, notrestricted to specific values. It is often convenient, however, to consider discrete classes of particles, inwhich case

In practice, it is often necessary to consider variations in more than one characteristic; forexample, size andcomposition. Denoting these characteristcs by p, 1~ S, for example, the variations canbe described by using what is called the joint distribution:

Description of Particle Characteristics

The conditional and marginal distributions are related to the joint distribution, qy, through

(Eq.2.7)

(Eq.2.8)q (Pi) = qi = the fraction of all particles for which P =Pi regardless of the value of r

v (r) =Jj = fraction of particles with a given value of P for which r = 1)and the marginal distribution:

Although it becomes apparent that a11 four of the basic characteristics can vary in typical oresamples, it is normally practical to consider only two at the most (e.g., size and composition). In thiscase, a useful alternative is to introduce the conditional distribution:

For particle systems, various characteristics are commonly expressed relative to particle size. Thus,the particle size distribution is used as the marginal distributon, with the distributions of other charac-teristics (shape, campositian, etc.) as conditionals. Ores and coal can often be regarded as binarymixtures of values and gangueo Because these components typically vary signficantly in density, particledensity is widely used as an indicator of particle composition. This practice is especially appropriatewhen gravity separations are to be used for beneficiation. An example of a size/density distribution forcoal is given in Table 2.1 and in Figures 2.1 and 2.2. Figure 2.1 shows the overall size distribution(marginal) and the size distribution for 1.25 specific-gravity material (conditional). Table 2.1 andFigure 2.2 give the joint distribution.

qy= fp (1)) . q(p;)

To describe the characteristics of a particle, it is generally desirable to assign them numerical values.These values should be c1early defned, unique, and measurable in practice. Satisfying these criteria isnot simple, and problems arise for each of these various characteristics.

Particle Size and Shape. It is well known that the behavior of systems of fine particles is stronglydependent on the sizes of the individual particles and that the size effects become increasingly impor-tant as the particles become progressively smaller. Despite the obvious importance of particle size,however, the evaluation and even precise definition of particle size are far from simple tasks, In general,we want to express the size of a particle as a single, linear dimension and refer to, for example, a 6-ftboulder, a 1-n. pebble, and a 10-micron particle. The problem is, which linear dimension do we use?

Only in the case of simple shapes, such as spheres or cubes, can we identify a single dimensionthat adequately characterizes particle size. Note, however, that even in these cases we must specify the

TABLE 2.1 Example of a sizejspecific-gr.avity distribution for coal: Weight percent (qj) values

Specific Size (x)Gravity 6in.x 3in.x 1 5/s in. x 1j2 in. x 1/4 in. x 8x 14x 48 Mesh x

(Pi) 3 in. 1 5/s in. 1j2 in. 1/4 in. 8 Mesh 14 Mesh 48 Mesh O1.3 (float) 4.44 5.73 16.30 8.75 9.75 4.91 5.58 2.321.3 x 1.4 1.62 2.09 5.63 3.26 3.05 0.94 0.77 0.331.4 x 1.5 0.64 0.68 1.40 0.84 0.83 0.32 0.25 0.121.5 x 1.6 0.43 0.46 0.86 0.34 0.43 0.17 0.16 0.081.6 x 1.7 0.21 0.26 0.49 0.17 0.22 0.09 0.08 0.041.7 x 1.8 0.13 0.18 0.37 0.11 0.16 0.06 0.06 0.041.8 (sink) 4.13 2.79 3.55 0.82 1.26 0.51 0.40 0.28

Total 11.6 12.2 28.6 14.3 15.8 7.0 7.3 3.2Source: Data from Sokaski, Jacobson, and Geer 1963.

22 [ PHINCIPlES OF MINERAL PROCESSiNG

FIGURE 2.1 Example of size and specific-gravity distributions for coal: (A) overall size distribution(marginal); (8) size distribution for 1.25 specific gravity (conditional)

where r is the radial vector, at sorne angle 8, from the center of the image to sorne point on theperiphery. The complete set of coefficients (ao, al, az, ...), in effect, defines theshape of the image.Because each coefficient is likely to be different for each particle, applying this approach to real systemsis rarely practical.

(Eq.2.9)

100

100

10Particle 8ize, mm

10Particle 8ize, mm

r(8) =ao + al sin 8 + az sin 28 + ...

0.1

0.1

-

(A)

1-

r--1--

-

1-

I I

(8) -

1- r--r--

-

-1--

r--

I II I I

o

o

20

20

10

,.go

30

shape of the particle and which particular dimension is being used (diameter of sphere; side, face diag-onal, etc., of a cube; and so on). In general, we cannot define a particle's size without first describing itsshape. A more unique description of size could be obtained by considering the mass or volume of theparticle. However, because mass and volume can rarely be measured directly, at least for very fineparticles, and because the behavior of the particles depends on their shape, little advantage is gainedfrom this approach.

Describing particle shape is difficult; taking quantitative measurements is even more so. Regularshapes such as spheres, cubes, or tetrahedra can be described and quantified, but real particles veryrarely fati into such categories and are most commonly described as "irregular." In principIe, any shapecan be described by fitting a mathematical function to it. For example, a two-dimensional image can befitted to a Fourier's series. For example,

FIGURE 2.2 Example of a joint size specific-gravity distribution for coal

In sorne cases, associating particles with general, shape classes, such as ellipsoids, is a usefulmethod. The lengths of the major axes define the size and shape of the "partcle." This approach isappropriate for minerals that tend to occur as plates (e.g., clays) or needles (e.g., asbestos). For othershapes, the value of this method will often be outweighed by the difficulty in obtaining the necessarymeasurements.

One simple, practical solution to these problems is to combine the effects of size and shape and tocharacterize particles in terms of an equivalent, simple shape (usually a sphere) with a given dimen-sion. Thus, we say that a particle behaves ~s though it were a sphere of diameter d. There are severalimportant consequences of the use of this simplified approach. In the first place, we must recognizethat the definition of size will itself depend bn the method by which that size is determined. If an irreg-ular particle is sized using sieves, we can say that the particle acts as though it were a sphere whosediameter lies between two sieve opening sizes. However, in a sedimentation device, the same particlemay behave as a sphere of a quite different diameter. Thus, we must specify not only the "sze" of theparticle but also the method by which the size was obtained. Differences between these sizes can beascribed to particle shape; ratios of the sizes obtained by different methods are often called shapefactors.

The use of an equivalent spherical diameter involves the implied assumption that all particles in agiven system have essentially the same shape. Although this is often a reasonable assumption, therecan obviously be cases where it is not valido In these instances, variations in particle shape would mani-fest themselves as apparent variations in size.

From a practical standpoint, the uncertainty in the definition of particle size places sorne impor-tant restrictions on the choice of sizing methods:

2.2

-

18

16

14

12

10:J?o

~ 8

6

4

2

O

Plrti 10e/s s..

I

MATHEMATICAL TREATMENT OF PARTICLE DISTRIBUTIONS

Representation of Particle Characteristics

(Eq.2.10)

(Eq.2.11)

(Eq.2.12)3

V = TCX6

2A = TCX

P 4The volume, V; is given by

The projected area, Ap, is given by

1. Direct comparisons can be made only between sizes determined by the same kind of technique(sieving, microscopy, sedimentation, etc.).

2. For systems containing a broad range of sizes, we will want to use, as much as possible, thesame technique for all sizes. When this is not possible, we must pay particular attention to theevaluation of the appropriate conversion factors (e.g., sieve "size" to sedimentation "size").When more than one method is used, there should be as much overlap as possible in the rangescovered by each.

3. In choosing a sizing method, consideration should be given to matching the method to the par-ticular application for which the size information is desired. Thus, if we wish to characterizethe particles in a liquid suspension, we would try to use a method that evaluates the behaviorof the particles in a liquid; for exarnple, a sedimentation method. In this way, we can automat-ically compensate for sorne of the uncertainties in the meaning of size and shape.

Particle Coinposition and Structure. The chemical composition of a particle can be uniquelydefined and can sometimes be represented by the value of a derived characteristc, such as density,color, or magnetic susceptibility. The distribution of composition or its surrogates (e.g., density) can bedetermined by particle-by-particle analysis or by appropriate separation methods (gravity, magnetic,color sorting, etc.). Particle structure presents a more difficult problem. The same composition canarise in an infinite number of ways: homogeneous or composite (bnary, ternary, etc.). A binarycomposite, for nstance, can consist of two attached grains or one component dispersed within a matrixof the other. A dispersed component can have any number of possible grain size distributions withinindividual particles. Because of this complexity, there is little value in attempting to establish a generalscheme for describing the distributions of particle structure. One approach is to define a set of discreteparticle types that can be distinguished in practice and are useful indicators. The distributions of theother characteristics-size, shape, and overall composition-can then be evaluated for each type.

The distributions of particle characteristics are similar to and subject to the same constraints as proba-bility distributions. Many of the concepts and terminology used in probability and statistics can bedirectly applied to particle systems. The treatment presented here malees use of definitions and termi-nology established at the University of Karlsruhe (Rumpf and Ebert 1964).

To describe distributions of particle characteristics, we .must represent The value of the characteristic itself The relative amount of material that has that value

Particle size can be represented as a linear dimension, an area (surface area 01' projected are a) , avolume, 01' a mass. The relationships among these different representations depend on particle shapeand, in the case of mass, on density. Thus, for a sphere of diameter x and density p, the surface area,As, is given by

:1.4 I PRH~JC~PlES OF IVnNERJ.\l PROCESS!NG

Particle 5ize Distributions

(Eq.2.14)

(Eq.2.15)

(IEq.2.13)3

In = TCpX6

The mass, 111, is given by

where lc2 and k3 are shape factors defined by Eqs. 2.14 and 2.15, respectively.Partiele shape distributions require that shape be represented by a numerical factor. The factors lc2

and lc3 defined aboye are obvious choices. Other factors, such as aspect ratios ("length" to "width" ofelongated partieles), can also be used. When more than one factor is used to describe shape, the jointdistributions of all the factors must be considered.

Partiele composition can be described by using a set of composition variables. For nc chemicalconstituents, a minimum of nc - 1 variables must be specified. A similar approach can be applied topartiele structure by using appropriately defined structure "types." Again, the joint distributions ofcomposition and structure must be evaluated.

Representation of Particle Quantity

The quantity of particulate material that possesses specific values of the characteristics (size, shape,etc.) can be represented in various ways. For a system of partieles, there will generally be sorne numberni that are essentially identical-I.e., have the same size, Xi; the same shape factors.Ik-), and (/C3)i; thesame density, Pi, for example. The quantity of these partieles can be represented by

Total number: ni Totallength: tux; Total area: ni(/c2)iX? Total volume: ni(/c3)iXi3 Total mass: niPi(k3)iX?In general, the fractional quantities can be expressed as

ni(lcr)ix /(qr)i = (Eq.2.16)'Lni(lcr\Xi

r

with r = O, 1, 2, 3 corresponding to the number, length, area, and volume fractions, respectively, and kaand lcl = 1 by definition. The number, length, are a, and volume fractions are all elearly different whenthere are variations in partiele size. The ~rea and volume fractions depend on partiele shape. Thevolume and mass fractions differ only if there are variations in partiele density.

The distribution of partiele size is of major importance in mineral processing. The behavior of partielesin crushing and grinding circuits, concentration operations, and solid-liquid separations is stronglydependent on size. Furthermore, the range of sizes in a single process stream is typically very large andcan inelude partieles that vary in diameter from 1 m (3.3 ft) to less than 1 um (10-6 m).

It was noted in a previous section that partiele size can be expressed in a variety of ways: diam-eter, area, volume, or mass. However, regardless of how the size is measured, the almost universalpractice is to present size as a linear (very often, equivalent sphere) diameter x. Size is nherently acontinuous variable and data are cornmonly elassified into appropriate size intervals. In this chapter,we will define Xi as the upper boundary of size interval i and select Xl as the maximum size presento

For partieles of arbitrary shape, the relationships for area (A) and volume (V) can be written asfollows:

(Eq.2.35)

(Eq.2.34)

(Eq.2.33)

(Eq.2.32)

(Eq.2.31)

(Eq.2.30)

(Eq.2.29)

(1Er. 2.28)

PAlRnfiCtlE CIrAnJ.\CTEi:IZt\TIOI~ I 17

;-1

Xi

x

. dQr(x)qrlx)=~

Qr(X) = Jqr(x)dxo

The physical meaning of the density function is thatqr(x)dx = "x'" fraction whose size falls between x and x + dx

where Xi" refers to sorne size in the interval Xi to Xi+1. The dscrete, incremental distributions and thecontinuous density function are proportional to each other only when the interval widths are constant(linear intervals); they are not proportional for the more commonly used geometric intervals. Exam-ples of a distribution function and the corresponding density function and incremental distribution are

given in Figure 2.3.Transformations. Later in this chapter we will show that different methods for measuring

particle size distribution involve different quantity representations. Counting methods usually givenumber distributions (qo(x)), whereas gravimetric methods give mass 01' volume distributions (q3(x)),and sorne optical methods give area distributions (q2(X)). For comparison purposes, as well as in manyapplications, transforming from one type of distribution to another (e.g., from number to volume) isoften necessary. The general formula for transforming a distribution based on x fraction to one basedon JI is as follows (Leschonsld 1984): \

pCrXi-rqr(x)dxo

If the shape factors, ler, are independent of size, they can be eliminated from Eq. 2.34, leading to

and

It should be noted that the incremental distribution (qr)i is not directly equivalent to the densityfunction qr(x); instead, we have

and

j = 1The values of Qr(xa also represent points on the continuous form of the distribution function

Qr(X)' The latter can be used to define a particle size density function, qr(x), such that

and (Qr); can also be represented by

FIGURE 2.3 Example of particle size distributions: (A) continuous density and distributionfunctions; (B) discrete incremental distribution (histogram)

(Eq.2.36)

or, in discrete form,t-r

xi (qr\n

'\' t-rL..Xi (qr)i

i = 1

Equations 2.35 and 2.36 can readily be applied to real data. We should note that using Eqs. 2.35and 2.36 does not require that the shape factors be the same for all partic1es; instead, it requires merelythat there are no systematic variations with size. In other words, the average shape factors should bethe same for all sizes. Although there are obviously cases where this requirement is not satisfied-delamination of c1ays, for example-most systems of partic1es do not show significant variations inshape with size.

A more serious problem in the use of these transfarmations is the need, in effect, to extrapolate to"zero" in arder to integrate (Eq. 2.35) or sum (Eq. 2.36) over all sizes (Oto DO or i = 1 to n). The problemis usually not serious when t is greater than r (e.g., in transforming from number to volume), but it canbe critical in the reverse transformation (r < r; e.g., volume to number). Specifically, the problem lies indetermining the exact form of qr(x} for integration or in assigning an appropriate mean value of size inthe sink interval (i = n). An example illustrating this problem is given in Appendix 2.1. The only realisticsolution to the problem is to extend the range of reliable measurement to finer sizes.

18 I PRINCIPLES or iVlINEnAl P\'{OCESSING

1.0 ., 1.01111 ' .......c.."

(A) -:,0.8

.-0.8 ~~ ..9:g q(x) c-,0.6 , 0.6 oQ; Q(x) e(J)

e , :Ju; tr

, (J)e 0.4 , 0.4 LtoU , (J), >~ , ""LL 0.2 0.2 Qj

""

a:"

,

0.0............

0.00.1 1 10

Particle size (x), IJm

1.0

(B)0.15~>,oal 0.10::lrr[l:'

LL

0.05

0.000.1 1 10

Particle size (x), IJm

The value of the median depends on which distribution it refers to (r = 0, number; r = 1, length;etc.). In general,

Transforrnation of a volume distribution (q3(X)) to a mass distribution (Q3"(X) can be accom-plished by using the formula

(Eq.2.41)

(Eq.2.40)

(Eq.2.39)

(Eq.2.38)

(Eq.2.37)

X ms > Xmt for s > t

XSO,s > XSO,t for s > t

Ml,3 = volume mean diameter; i.e., particle diameter averaged with respect to the volumedistribution

k3M3,O = number mean volume; i.e., particle volume averaged with respect to the numberdistribution. In this case, the moment represents the mean value of x3; the shapefactor is necessary to convert to an actual volume.

The difference between values increases with increasing spread of the distribution.The mode of a distribution, xmr-sometimes referred to as the most frequent size-corresponds to

the peak in the density function qr(x). Again, the mode's value depends on whether r = 0, 1, 2, or 3. Inaddition,

Distributions with more than one maximum are said to be multimodal. Bimodal distributions (twomaxima) are quite cornmon. They occur in mixtures of particle systems (e.g., sand and grave!) and,under certain circumstances, can be generated in size reduction and agglomeration processes (Hogg inpress; Rattanakawin and Hogg 1998).

Mean sizes represent a group of ave(ages defined by the moments of a size d~stribution. The kthmoment of the size distribution qr(x) is defined by

MIc,r = fXlcqrex)dxo

and represents the quantity Jde averaged using the "t" distribution. For k = 1, 2, 01' 3 and using appro-priate shape factors, the moments correspond to mean diameter, area, 01' volume, respectively. Thus,for example,

P(X)Q3(X)

fP(x)Q3 (x)dxo

where p(x) is the average density of a partcle of sze x. If the density is independent of size, the mas sand volume distributions are identical. Variations in density with size can become significant when thedegree of liberaton of different minerals changes with size. This is a common occurrence in mineralprocessing systems where grinding is widely used for enhancing liberation.

Average Sizes. The use of average sizes can be convenient, but caution should be exercised andthe average should be carefully specified. Any average is an indicator of the location of the size distri-bution within the size spectrum. However, its value is also influenced by the width or spread of thedistribution. The nature and extent of this effect depend on the particular average being used.

A great many different average 01' characteristic sizes can be defined, such as median, mode, ormean. The specific surface are a, which will be discussed later in this chapter, also represents a kind ofaverage (but inverse) size. The values of these averages may vary widely depending on the particulardefinition and the form of the distribution they represent.

The median size in a distribution is that size that splits the distribution into two equal parts; that is,hall' of the material is finer and hall' is coarser. In general, the median size can be defined by XSO,r suchthat

20 I PRINCIPLES OF MINERAL PROCIESSING

(Eq.2.42)

(Eq.2.43)

(Eq.2.44)

(Eq.2.45)

(Eq.2.47)

(Eq.2.48)

(Eq.2.46)

S; = pSm

X/c,r s Xk+i,r+j for i andj both :::::: O

fle2QO(x )dxo

Sy =~------fle3x 3qo(x)dxo

If the shape factors are independent of size,

Applying the transformation formula, Eq. 2.35, to the moments in Eq. 2.47leads to

M _ M_I,32,O-~

-3,3

Specific surface orea, defined as the surface area per unit volume (Sy) or per unit mass (Sm), alsorepresents an average (but inverse) size. The volume and mass specific surface areas are relatedthrough the equation

Heywood (1963) defined several such mean sizes, all of which can be expressed as moments ofthe size distribution (Leschonsk 1984). The actual values of x/c,r depend on le and r and on the form ofthe distribution. In general, the values increase with increasing le or r. Specifically,

so that, for example,

X3,O = (M3,O)1/3 = number mean volume diameter; Le., the diameter corresponding to thenumber mean volume defined aboye. The shape factors appear implicitly on bothsides of Eq. 2.42 and cancel out.

Values of le are not, however, restricted to 1, 2, or 3. Other values, including negative numbers,are equally valid and are often encountered in practice. The zero'" moment (l( = O) is identically equalto unity, regardless of r, because Eq. 2.41 then expresses the fraction of particles that have any sizebetween zero and infinity; that is, al! of them. Negative values of le simply represent averages of 1/x,1/x2, and so on.

The integral in the denominator of Eq. 2.35 can be written as the moment M-;r. Useful relation-ships among the various moments are discussed in more detail by Leschonski (1984). The momentscan be expressed.as mean sizes (which are indicated with an overbar aboye the x term) via the equation

Sm has units of area/rnass (usually square meters per gram), whereas S; is an inverse size (e.g.,per micrometer [.Lm-IJ). The hybrid unit of square meters per cubic centimeter (m2/cm3) is numeri-cally equal to units of per micrometer.

For particles of uniform size,

where le23 is called the specific surface shape factor, defined as the ratio le2/le3 . For spheres, le2 = re and7(3 =re/6, so that 7(23 =6.

More generally, for systems with a distribution of sizes,

(Eq.2.52)

(Eq.2.51)

(Eq.2.55)

(Eq.2.53)

(Er.2.49)

(lEq.2.50)

(Eq.2.54)

_ le23X-13= --

, pS11l

If the shape factors and density are independent of size, X-l,3 can be expressed in terms of themass specific surface area; i.e.,

That is,

les = the size modulus, which locates the distribution in the overall size spectrurnex = the distribution modulus, which is an inverse measure of the spread of the distribution

Because le23 = 6 for spheres, an equivalent-sphere specific surface diameter can be defined by

The specific surface mean diarnerer, x-U is defined in the usual way (i.e., by using Eq, 2.42):

and1M 3,o = M_3 ,3

so that Eq. 2.47 can be replaced by the more convenient

S" = le23 M-1,3

This form is useful because specific surface area can be measured directly.Figure 2.4 shows an example of a fairly typical size distribution. Sorne of its associated averages

are given in Table 2.2. The more-than-tenfold range in the values for different averages for the samedistribution clearly illustrates the potential ambiguity involved in the unqualified use of average sizes.

Algebraic Forms. It is often useful (e.g., for application to process models) to fit specific algebraicfunctions to particle size distribution data. Typically, these functions have two parameters that can beadjusted to provide the best fit to a set of experimental data. The values of the parameters provide animproved means of summarizing the actual distribution as compared to using a single, average size.

It should be emphasized that, in general, there is no particular form that is expected, theoretically,ro describe size distribution data. For example, there are no equivalents to the binomial, Poisson, andnormal distributions of probability and statistics. However, sorne functional forms have been found togive a reasonable fit to sorne sets of data. These are simply equations that

Increase monotonically from Oto 1 Can fit data reasonably well, usually with only two adjustable parameters Are reasonably simple to applyThe distribution types discussed in the following paragraphs are (1) the Gaudin-Schuhmann

distribution, (2) the Rosin-Rammler distribution, and (3) the logarithmic-normal (or log-normal)distribution.

The Gaudin-Schuhmann distributioti expresses the mass (volume) distribution function by asimple power law

where

PP,'lTICLIE CW\RACTEHIZ'l\TiON I 23

(Eq.2.56)

(Eq.2.57)

(Eq.2.59)

(Eq.2.58)

10,000

ks =1,800 11m

1,000Particle Size, 11m

100

_(3) a (Xi + 1) a(Q3\ - le - k:s s

(~)(_X) a-1 for X::; lesQ3(x) = (s (sO for x> les

'"'

ID

~~lt

_ \\ll Schuhmann i /O Gaudin VV -/ Ojll

a 1./

./ Y

)J a=1

Ji..V- y

~/110

IDeu::;g

~ 10~:sE::l

From Eq. 2.55,

FIGURE2.5 Gaudin-Schuhmann size distribution

any set of data, the Gaudin plot is appropriate only for data arranged in a geometric series of sizeintervals.

It follows from Eq. 2.29 and 2.55 that the corresponding density function is given by

100

or

A log-log plot of frequency versus size would therefore yield a straight line of slope a-l.However, for materials that follow this distribution, the following special and more useful kind of

frequency distribution can be used. If the experimental data are given in the form of weight or volumefraction in discrete size intervals, and if the size intervals are arranged in a geometric progression(sieving data are normally generated in tls form, for example), the weight fraction in sorne interval Xito Xi+1 will be given by

where r s =xi/Xi+1'For size intervals arranged in a geometric progression, rs is constant and a log-log plot of the weight

fraction in the size interval versus sorne characteristic size in the interval should give a straight line ofslope a. These plots are often known as Gaudin plots; their major utility lies in their high sensitivity to

The density function passes through a ma:umum only if m > l.By expanding the exponential term as a power series, it can be shown that, for X ler

\

Q3(X) == (f)m for x k; (Eq.2.66)

(Eq.2.67)

(Eq.2.68)

10,0001,000

k, = 1,000 IJm

*

100

Particle Size, 11m

r(a + 1) = ftae-tdto

//

/

V

/

/1/

// Estimate m from:

m = 2/log(k,/x,)

/ x, = 1% passing size/ I 111111'1 1 I 1I

V 1I [111 11

X, = 10 IJm~

10

0.5

0.1

99.999

90

63.2150

IDe:

c/!.al 10>~"S 5E:J

FIGURE 2.6 Rosin-Rammler size distribution

where Xl is the 1% passing size (i.e., the size for which Q3 = 0.01, or 1%). Eq. 2.64 provides a simplifiedmeans of estimating m. The altemative-direct measurement of the slope of the line-often leads toconfusion and incorrect calculation. Special Rosn-Rammler graph paper is available commercially.

From Eqs. 2.29 and 2.62, the density function is given by

Q3(x) = (2)(f)m-1eXP[-(~rJ (Eq.2.65)

Thus, for the very fine sizes, the Rosin-Rammler distribution reduces to the same form as the Gaudn-Schuhmann distribution with ex =m and les = ler.

The moments can be obtained by using Eqs. 2.41 and 2.65. The general result is

le (le )M Ie , 3 = ler r ;:;:; + 1

where the gamma el) functon is defined by

PAIflT~ClE CH'\RI~(;TER!Zl\'m)\\1 I 27

In practice, the log-normal distrbution is found to apply reasonably we11 to a variety of particulatematerials, inc1uding fine c1ays and finely ground SO 11m),unc1assified powders. Furtherrnore, the math-ematically well-behaved nature of this distrbution gives it an advantage over the Gaudin-Schuhmann

It should be noted that transforraatons from number to volume and are a to volume, forexample, involve a translation of the curve in the Q,. direction rather than the x direction. Thus, asma11 deviation from log normal at, say, the fine end of the volume distribution can appear as asignificant deviation in the center of the number distribution (see Figure 2.8).

In contrast to the Gaudin-Schuhmann and Rosin-Rammler distributions, the moments MIc,,. of thelog-normal distribution are a11 finite, regardless of the values of le and r. The moments can a11 be deter-mined from

(Eq.2.75)

1,000100

Particle Size, IJm

10

E E E::J.. ::J.. ::J..t'-- N ~ ~

Ic [1 1 2JM Ic,,. = XSD,,. exp Z(lc n 0')

11.111 1 1'11 I1 I I I1I111

1 11 11 1 I111 I1 1 1

1 //

(J X,,/Xso

xsolx"(x,Jx,,)"

1/ 1111

I I I 11 I I 111 I 11 11

1 11111 1 111 ILI 11

99

0.011

90

(je 70i.I"goQl 50>~::;E 30:J

10

FIGURE 2.7 Lag-normal size distribution

99.99

99.9

Particle Shape Distributions

(Eq.2.76)

100

100

i I I--r-

__ Q,(x) ().. Qo(x)

-- Q3(X)--()._. Qo(x)

lc3(s)qo(s)

flc3(S)Qo(s)ds

I

10Particle Size, urn

10Particle Size, urn

(A)

99.9

99

90ID 70ei.L 50;Ro

30

10

11

99.9

99

90ID 70ei.L 50;Ro

30

10

11

and Rosin-Rammler equations for sorne applications, even though the latter rnay give a "better" represen-tation of the particle size data.

Description and rneasurernent of particle shape, especially quantitative evaluation of shape distribu-tions, are seldorn carried out in mineral processing applications. Nevertheless, the proceduresdiscussed previously for presenting, transforming, and rnanipulating particle size distributions can beequally applied to shape. Because rneasurernents are usually rnade on individual particle irnages, dataare generated as nurnber distributions, generallyat (approxirnately) constant size. If a rneasured shapeparameter s can be related to the volurne shape factor (/(3) as defined by Eq. 2.15, transforrnation frornthe nurnber distribution to the volurne distribution can be accornplished via

where the integration is carried out over all possible values of the pararneter s. Averages and rnornents,then, can be defined and evaluated as for size distributions. Because of the scarcity of inforrnation,standard forms have not been established for particle shape distributions. However, sorne data for

I

FIGURE 2.8 Transformation of volume distribution to number distribution: (A) log-norrnaldistribution; (8) log normal with deviation at the fine end of the volume distribution

28 I PRlNC!PlES OF i1lUNERALPROCESS!NG

(Eq.2.77)

ground minerals appear to conform quite closely to [he log-normal distribution (Kaya, Kumar, andHogg 1996).

Distribution of Partlcle cornposlton ami Structure

Formal presentations of the distribution of composition and structure are rarely encountered.However, density (specific gravity) distributions are widely used, especially in coal processing. Again,the procedures used for size distributions are generally applicable. The principal difference is that,because particle volume does not depend on compositon, the number and volume distrbutons (atfixed size) are identicaL The mass distribution q'3 (which is typically measured) does differ. Transfor-mation can be accomplished by using

q'3(P)/PJq'3(p)dp/pp

Although attempts have been made to use functional forms to fit density distribution data, stan-dard expressions analogous to those used for size dstrbutions have not yet been established.

MEASUREMENT OF PARTICLE CHARACTERISTICS

Before beginning to characterize a particle system, we must (1) obtain a representative sample of thematerial, (2) prepare the sample for analysis, and (3) select the most appropriate analytical procedure.Each of these steps can be critical in terms of the reliability and utility of the results obtained from theanalysis.

Sampling

The first step in any analytical procedure is to obtain an appropriate sample of the material to beanalyzed. Two important questions should be asked befare selecting the sample:

1. How large should the sample be?2. How can we be sure that the sample is truly representative of the material to be analyzed?AB quite distinct problems, these two should not be eonfused. Just beeause a sample is large

enough does not mean it is neeessarily representative; a representative sample may still be too small.Sample requirements for partieulate materials have be en diseussed in detail by Gy (1982). Sornesimple guidelines will be presented here.

Sample Size. For partieulate materals, the primary eriterion for establishing the requiredsample size is that all kinds (sizes, shapes, etc.) of particles should be adequately represented in thesample. Because particles are diserete entities, sample size is thus dietated by the number size distribu-tion of the material being sampled. Consider, for example, a system of quartz particles (specifie gravity2.65) that eontains 10% by weight of 1-em particles. Eaeh 1-em particle would weigh about 1.4 g, sothat a sample weight of 14 g would be neeessary to ensure a reasonable likelihcod of eontaining evenone of these particles. Obviously, this weight would be quite inadequate beeause there would be a highprobability of taking sueh a sample and finding none of those particles (0%) or two of them (20% byweight). If the sample size were inereased tenfold, to 140 g, we would expeet to find 10 of the 1-emparticles in any sample, and the errors introdueed by the ehanee inclusion of one extra or one fewerwould be eorrespondingly redueed; that is, a sample eontaining only 9 of them would analyze at 8.9%rather than 10%. Sample size, then, should be based on statistieal eriteria sueh that the errors intro-dueed by random variations in the numbers of different particles included are aeeeptably small.

where

30 I PFlINC~P'-ES OF IViINERlll PROCESSING

(Eq.2.78)

(Eq.2.79)

(Eq.2.81)

(Eq.2.82)

(Eq.2.80)

Corresponding Minimum Sample SizeMaximum Particle Size

(~-2)Wi+WE = 2[ M

Equation 2.82 then defines the 95% confidence interval on the measurements.

Based on analysis of the statistics of random mixtures, the required samp!e weight for partic!e sizeanalysis can be estimated by computing a set ofva!ues of the quantity Mvia

Mi = ~[(~-2)W+WJ

where Xm is the maximum size (in centimeters) presento Eq. 2.81 is based on the arbitrary designationof the maximum size as the average of the c!ass that contains the coarsest 5% (by weight) of the distri-bution. Sorne fairly typical examples of rninimum sample sizes for given maximum partic!e sizes aregiven in Table 2.3.

After the analysis, when the actual values of qi are known, Eq. 2.78 can be used to check theadequacy of the sample size used. Alternatvely, Eq. 2.78 can be inverted to evaluate the relative errors,Ei, associated with each measured qi value. Thus,

Equation 2.78 gives the sample weight Mi required to determine qi to an accuracy E at the 95%confidence leve!.

In general, there will be a minimum sample weight, Mi, for each c!ass in the distribution, and therequired sample weight, M, wil! be the largest of these Mi values. In practice, however, the values of qiare not known until the analysis is complete. It is necessary, therefore, to use an initial estimate todetermine the sample weight. If we recognize (1) that the largest partic!es wil! normal!y give themaximum value of Mi and (2) that for spherical partic!es,

Note: Calculations based on quartz (p = 2.65 g/cm3 ) with Gaudin-Schuhmann distribution (a. = 1), 10 welghtpercent in the top size interval, and allowable error of 5%.

100 um 40 rng1 mm 4g

1~ ~~10 cm 40 t

qi = the weight fraction in size c!ass iWi = the mean weight of a single partic!e in that c!ass

E = a specified tolerance on the estimated value of qi

The term w in Eq. 2.78 is the overal! mean partic!e weight and is given by

TABLE 2.3 Minimum sample size requirements

where xi and Piare the mean partic!e size and density in c!ass i, we can obtain an initial estimate from340pxmM""--2-

E

TABlE 2.4 Sample size requirements for semen analysls !.HU coal (specic gravlty = :ll..LJ.O)Weight Partlcle Required Expected Error

Size Range Mean Size Percent Weight l.lw, Sample tor 1-I

32 I PRINCIPLES OF MINERAL PROCESSiNGneeded for the undersize material is less, and an approprate subsample can be used. Two or three suchsteps will usually be sufficient to minimize the need to analyze large samples.

Sampling Procedures. Once the required sample size is deterrnined, the next step is to choose asampling procedure in which the different kinds of particles are selected entirely without bias. In prin-ciple, this simply requires complete mixing of the bulle material before the sample is taken. Unfortu-nately, complete mixing is often impractical, especially when very large quantities are involved.Furthermore, particulate materials are notoriously difficult to mix because of the tendency for differentkinds of particles to segregare, particularly for relatively free-flowing materials with wide variations inparticle size. Specific procedures for sampling from both batch and continuous-flow systems have beendescribed in detail by Gy (1982) and AIlen (1997).

In grab sampling, the simplest method of all, a scoop or shovel is used to take the appropriatequantity of material, essentially at randorn, from the bulle Grab sampling is satisfactory only if the bulkmaterial can be' thoroughly mxed, which may in fact be the case for reasonably small quanttes offairly cohesive (which usually means fine) powders. A series of grab samples taken from different loca-tions, or with intermediate mixing of the bulle, can offer significant improvement over the singlesample and is often the only practical alternative for very large populations. Care should be taken toavoid biasing the sample toward the surfaces of, for example, large piles. If each sample from a series isanalyzed separately rather than combined into a single analysis sarnple, information can be acquiredon relative homogeneity, and segregation, among other factors, in the material and on the extent towhich a sampling problem actually exists.

Cone-and-quarter sampling, which is widely practiced, presents certain advantages. The procedureinvolves mixing and turning the material over with a scoop or shovel and piling it into a conical heap.The heap is divided into roughly equal quarters; two opposite quarters are removed, and the remainingtwo are remixed. The procedure is repeated until the material is reduced in quantity to the desiredsample size. Advantages of this approach are (1) the entire batch is subject to the sampling procedurewith minimal operator bias, (2) no special equipment is needed, and (3) the method can be applied tovery large quantities (by using front-end Ioaders, for example). However, accumulation of fines as aresult of segregation during heap formation can lead to biasing of the sample. The procedure is rathertedious and time-consuming, and operators are often tempted to take shortcuts, which can increase bias.

Sample splitters (rifflesJ are mechanical devices used to divide a material into two or more parts ina random fashion. The simple chute splitter uses a series of alternately directed chutes to separate thematerial into two parts. Repeated applications can be used for further subdivision as required. Theprocedure is simple and effective but is usually limited to fairly small quantities. Loss of fines canpresent problems for "dusty" materials. Spinning rifflers in which the material is fed slowly, usuallyfrom a vibrating feeder, into a series of collection vessels on a rotating table are attractive for finepowders because the generation of a dust cloud can often be minimized. Before any kind of mechanicalsplitter or sample reducer is used, its design should be evaluated carefully to ensure that segregationdoes not lead to sample bias. An improperly designed sample splitter can easily become a c1assifierl

In sampling from very large populations (e.g., stockpiles), grab sampling is essentially the onlyoption. In these cases, several (as many as possible) individual samples should be taken. 1'0 minimizebias, the entire volume of material should be conceptually divided into a regular, three-dimensionalgrid. Sampling locations should then be selected at random from the grid points. AIthough it is notnecessary for each individual sample to satisfy the size requirement, the combination of all samplesmust. Obviously, each individual sample must be substantially larger than the coarsest partic1espresento

Sampling from slurries, particularly settling slurres, is especially difficult and frequently leads tobiased results. For small batches, it is sometimes best to fiIter and dry the entire batch and then use a drysampling procedure. Care must be taken to avoid loss of fines that may pass through a filter or be trappedin the filter medium. Nonsettling slurries can simply be mixed thoroughly and sampled as for homoge-neous lquds, Settling slurries can be subjected to vigorous agitation, then sampled in the same way.

For lsokinetic Sampling, u = v

1Sample

FIGURE 2.9 Isokinetic sarnpllng

Another approach is to circulate the slurry at high rate through a pump and sample from the flowstream. In sampling from flow streams-such as conveyors and slurry pipelines-the preferred method isto divert the entire stream for a short time rather than splitting off part of the stream. If repeated samplesare taken, time variations and cycling, among other factors, can be detected. For high-volume flows ofsuspended particles where diverting the entire stream is impractical, isolcinetic sampling should be used.The procedure for isolcinetic sampling is illustrated schematically in Figure 2.9. Samples are withdrawnfrom the stream through a probe located as shown. The rate of withdrawal is adjusted, by using a pump,so as to ensure that the inlet velocity is the same as the flow velocity in the main stream.

Measurement of Particle Size

The size distribution of a particulate material is a description of the relative abundance of the differentsizes presento Previously in this chapter we pointed out that, for the lcinds of irregular particles typicallyencountered in mineral processing systems, particle "size" cannot be uniquely defined. In most proce-dures for evaluating size distlibution, size is arbitrarily defined on the basis of response to sorne particular

\process, such as passage through an aperture, settling in a fluid, or scattering of light. The actualmeasurements involved in the analysis are of the relative quantities of material that give a specificresponse. The relationship between response and actual size is generally known only for spheres, so thatmeasurements give an estimate of the distlibution of equivalentspherical diameter. Because deviationsfrom the spherical shape will have different effects on the response to different processes, size dstribu-tion estimates obtained for the same material but by different techniques cannot be expected to agreeexactly, even in the absence of measurement error. Such discrepancies become especially important whenmore than one technique must be employed to span a broad range of sizes.The problems of data nterpre-tation in these cases will be discussed at sorne length in the final part of this chapter.

It is also important to recall that different analytical procedures use different measures of particlequantity, such as mass, volume, and number. Direct comparison is possible only on a common basis,and appropriate transformation of the distlibutions must be carried out where necessary. (See theprevious section entitled "Transformations.") The importance of specifying the basis of the distributionwhen reporting data cannot be overemphasized.

34 I PRINCIPLES or MINER/IL PROCiESSING

101Particle Size, JJm

-0- "True" Distribution--oQ-_. Apparent Dlstribution '

,,0,p'

,p'

i/pf

,p'/'

100

SO

Q; 60eu;

~o 40

20

O0.1

FIGURE 2.10 Graph demonstrating the effect of a lower detection Iimit on measurement of particlesize distribution

Limitations of Sizing Techniques. Essentially all methods for particle size measurement arelimited with respect to range of applicability. The nature of this Iimitation, however, is not the same forall techniques. In general, we can identify two principal kinds of size limitations:

A limit of measurement A limit of detectionIn the first case, measurement is limited to a specific range of sizes, but the existence of particles

outside that range is recognized and quantified. For example, conventional sieving gives data on mate-rial coarser than, for example, 400 mesh (37 um), but also specifies the amount of undersize presentoIn the second case (1imitof detection), on the other hand, material outside of the range is not detected;the method yields an apparent distribution that is based on the assumption that all particles presentfall in the measurement range. Thus, the measured (apparent) distribution refers only to an unspeci-fied fraction of the material and is not an estimate of the true distribution over a limited part of the sizespectrum. Figure 2.10 demonstrates the effects of detection limits. In this figure, the apparent distribu-tion does not account for particles below a certain size, although such particles are indeed presento

In general, sizing methods that are based on the collective response of an assemblage of particlesare subject only to measurement Irnits, although there may be effective detection limits in specificapplications. On the other hand, those techniques that involve the response of individual particles(e.g., the use of particle counters) are invariably subject to detection limits because there can always besorne particle that is too small to see. Clearly, measurement limits cause less serious problems thandetection limits. Data obtained from methods that involve detection limits should always be questionedwhen significant quantities are reported at sizes close to the limito

Resolution. Different pro cedures for size analysis vary in their ability to discriminate betweensizes. For example, standard sieves can distinguish between particles that vary in size by more than(2114 - 1); that s, 19%. Direct measurement methods such as microscopy can, in principie, detecteven smaller differences. Other techniques, however, have substantially lower resolution. As ageneral rule, methods based on individual particle measurement (e.g., the use of particle counters)have high resolution (but are prone to detection limits), whereas those that involve overall systemresponse (e.g., sedimentation and light scattering) have lower resolution (but are often subject onlyto measurement limits).

Resolution is especially important for very narrow size clistributions-Iow-resolution techniqueswill tencl to overstate the wiclth of the distribution. For broacl dstributions, errors at clifferent sizestencl to cancel each other out.

Dispersion of Fine Powders in Fluids. Virtually allmethocls for subsieve analysis requre thatparticles be completely dispersed in a fluid medium. Inadequate dispersion can lead to very seriouserrors in measured size distributions. Dispersion in air is usually quite difficult, and liquid dispersionmedia are generally preferred.

Dispersion of solid particles in a liquid can generally be considered as a three-step process (Parfitt1973): wetting, deaggregation, and stabilization.

Wetting of the solid surfaces by the liquid is a necessary prerequisite to dispersion. For hydrophilicsolids, such as quartz and most other oxide minerals, wetting is usually spontaneous ancl no specialprecautions are needed. Other solids such as coal, sulfide rnnerals, and many organics are less easilywetted by water, and it may be necessary to add a wetting agent to reduce the surface tension of thewater and promote wetting. Alternatively, a nonaqueous liquid (e.g., hydrocarbon) can be used insteadofwater.

Deaggregation is necessary to break up the small agglomerates that remain when a dry powder isincorporated into a liquido This objective can usually be accomplished by mechanical agitaton,although, of course, too much agitation could lead to breakage of the individual particles themselves. Abrief period of ultrasonic treatment is often found to be particularly helpful in brealdng up very smallagglomerates.

Stabilization is almost always necessary to prevent reagglomeration of the dispersed particles.Stabilization is usually accomplished by ensuring that there are adequate repulsive forces caused bysurface charges on the particles and solvation forces resulting from the presence of adsorbed films onthe particle surfaces. Dispersing agents generally function by controlling surface charges or byincreasing the solvation forces through adsorption. Electrolytes, especially those containing polyvalentions, have a serious effect on the electrical forces between particles. Other mpurities, such as organcs,can also tend to promote aggregation of the particles. Consequently, it is important to ensure thatglassware, mixer impellers, and other lab equipment are stricdy clean and to use only distilled waterand hgh-purity reagents.

The simplest procedure for evaluating the completeness of particle dispersion is to examine a dropof the suspension under a microscope. For submicron partcles, where the individual partcles or evensmall agglomerates may be difficult to resolve, the long-term stability of the suspension can be used asa criterion. Measuring supernatant turbidity after a fixed period of settling is one useful approach. Theoptimum combination of dispersion procedure and reagent addition is the one that gives the highestturbidity. Visual observation of relative clarity will often suffice if equipment for turbidity measure-ment is not available. An alternative approach is to use the actual sizing method results as the disper-sion criterion-the "best" dispersion will generally give the finest distribution.

Sieving. The sieving methods are \he most widely used means for sizing particles coarser thanabout 37 um (400 mesh). Conventional (woven wire) sieves are available with aperture sizes down toabout 25 um (see Table 2.5), and so-called micromesh sieves can be obtained with apertures as smallas 5 um.

The Tyler and U.S. Standard sieve series are most commonly used for size analysis. Both of theseuse a geometric progression of sieve apertures with a constant ratio of 2V4between adjacent members.It is common practice to omit the intermediate sieves, leaving a 2V2 ratio. Testing sieves manufacturedaccording to the two standards are essentially interchangeable, but it is important to note that the"mesh" designations are not always the same; for instance, a Tyler 12 mesh is equivalent to 14 mesh onthe U.S. scale; a Tyler 14 corresponds to a U.S. 16 (see Table 2.5).

Size analysis by sieving is normally carried out by using a stack of standard sieves with openingsizes that decrease progressively from top to bottom. The sample should be weighed accuratelybefore it is placed on the top (coarsest) sieve. The use of a mechanical shakng device is generally

36 I PRiNCiPLES or MiNERAL PROCESSINGTABlE 2.5 Size of standard test sleves

Tyler Series U.S. Series1.189 Ratio 1.4:1.4 Ratio 1.189 Ratio

Opening, Opening, Openlng,mm in. Mesh in. mm in. Mesh

26.67 1.050 1.050 26.9 1.06 1.06 in.22.43 0.883 22.6 0.875 7/8 in.18.85 0.742 0.742 19.0 0.750 3/4 in.15.85 0.624 16.0 0.625 5/8 in.13.33 0.525 0.525 13.4 0.530 0.530 in.11.20 0.441 11.2 0.438 7/ 16 in.

9.423 0.371 0.371 9.51 0.375 3/8 in.7.925 0.312 21j2 8.00 0.312 0/16 in.6.680 0.263 3 0.263 6.73 0.265 0.265 in.5.613 0.221 31/2 5.66 0.223 No. 31j24.699 0.185 4 0.185 4.76 0.187 43.962 0.156 5 4.00 0.157 53.327 0.131 6 0.131 3.36 0.132 62.794 0.110 7 2.83 0.111 72.362 0.093 8 0.093 2.38 0.0937 81.981 0.078 9 2.00 0.0787 101.651 0.065 10 0.065 1.68 0.0661 121.397 0.055 12 1.41 0.0555 141.168 0.046 14 0.046 1.19 0.0469 160.991 0.390 16 1.00 0.0394 180.833 0.0328 20 0.0328 0.841 0.0331 200.701 0.0276 24 0.707 0.0280 250.589 0.0232 28 0.0232 0.595 0.0232 300.495 0.0195 32 0.500 0.0197 350.417 0.0164 35 0.0164 0.420 0.0165 400.351 0.0138 42 0.354 0.0138 450.295 0.0116 48 0.0116 0.297 0.0117 500.246 0.0097 60 0.250 0.0098 600.208 0.0082 65 0.0082 0.210 0.0083 700.175 0.0069 80 0.177 0.0070 800.147 0.0058 100 0.0058 0.149 0.0059 1000.124 0.0049 115 0.125 0.0049 1200.104 0.0041 150 0.0041 0.105 0.0041 1400.088 0.0035 170 0.088 0.0035 1700.074 0.0029 200 0.0029 0.074 0.0029 2000.063 0.0024 250 0.063 0.0024 2300.053 0.0021 270 0.0021 0.053 0.0021 2700.044 0.0017 325 0.044 0.0017 3250.037 0.0015 400 0.0015 0.037 0.0015 400

0.031 0.0012 4500.026 0.0010 500

PAlRTiClE CiIARi\CTEmZJ.\TUON I 3'1recommended. Various sieve-shaking systems are available that can accommodate up to about 12 indi-vidual sieves. After the appropriate period of shaking (see the discussion of sieving kinetics that follows),the particles retained on each sieve are removed and weighed. Gentle brushing of the underside of eachsieve can aid in releasing particles trapped in the apertures. For analytical purposes, it isoften convenientto weigh the particles from the different sieves cumulatvely; that is, by adding particles from the secondsieve ro those previously weighed from the first and so on. The advantages of this approach are that errorsdo not accumulate and a mistaken reading for one sieve is autornatically corrected at the next.

Comparing the total weight collected (including that on the bottom pan) with the original sampleweight is a good way to check the overall procedure. Sorne weight loss is inevitable as a result of adhe-sion of fines to sieve surfaces and sticldng of particles in openings. These losses should not exceed 1%.Significant weight gain is an indication of weighing errors or the inclusion of material remaining fromprevious tests. In any case, such data should be discarded and the test repeated. Observed small weightlosses can be handled by expressing the individual weights as fractions of either (1) the actual finalweight or (2) the initial (sample) weight. The first approach, in effect, distributes the errors propor-tionately among all sizes. On the other hand, the use of the initial weight, with the discrepancyassigned to the "pan," meaning to sizes finer than the finest sieve used, involves the implicit assump-tion that alllosses are caused by fines adhering to surfaces or becoming airborne. The first approach ismost commonly adopted; the second may be appropriate for very dusty materials.

The particle size determined for sieving experiments is defined as the minimum square aperturethrough which the particle will pass. For irregular particles, size refers to the particle's smallest cross-sectional area. It is important to recognize that although a particle that has passed through a sieve isdefinitely smaller than that size, one that has not passed is not necessarily larger. Irregular, "near-sze"particles may require several attempts before their orientation is such that they can pass through theaperture. Thus, it is necessary to allow sufficient time for sieving to reach completion while recognizingt