Particle Breakage, Volume 12 (Handbook of Powder Technology)

Particle Breakage HANDBOOK OF POWDER TECHNOLOGY Advisory Editors J.C. WILLIAMS School of Powder Technology, Universit...

Author: Agba D. Salman | Mojtaba Ghadiri | Michael Hounslow

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Particle Breakage

HANDBOOK OF POWDER TECHNOLOGY Advisory Editors J.C. WILLIAMS School of Powder Technology, University of Bradford, Bradford, West Yorkshire, England and T. ALLEN Senior Consultant, E.I. DuPont de Nemours & Co., Inc., Newark, Delaware, U.S.A. The Handbook presents, in convenient form, existing knowledge in all specialized areas of Powder Technology. Information that can be used for the design of industrial processes involving the production, handling and processing of particulate materials so far did not exist in a form which it is readily accessible to design engineers. Scientists responsible for characterizing particulate materials, specifying the requirements of industrial processes, operating plants, or setting up quality-control tests all have similar problems in their fact-finding missions through the scattered and scanty literature. The aim of this handbook is to remedy this deficiency by providing a series of thematic volumes on various aspects of powder technology Vol. Vol. Vol. Vol. Vol.

1. 2. 3. 4. 5.

Particle Size Enlargements (C.E. Capes) Fundamentals of Gas-Particle Flow (G. Rudinger) Solid-Gas Separation (L. Svarovsky) Dust Explosions (P. Field) Solid Liquid Separation Processes and Technology (L. Svarovsky) Vol. 6. The Packing of Particles (D.J. Cumberland and R.J. Crawford) Vol. 7. Dispersing Powders in Liquids (R.D. Nelson) Vol. 8. Gas Fluidization (M. Pell) Vol. 9. Powder Technology and Pharmaceutical Processes (D. Chulia, M. Deleuil and Y. Pourcelot, Eds) Vol. 10. Handbook of Conveying and Handling of Particulate Solids (A. Levy and H. Kalman) Vol. 11. Granulation (A.D. Salman, M.J. Hounslow and J.P.K. Seville) Vol. 12. Particle Breakage (A.D. Salman, M. Ghadiri and M.J. Hounslow)

Particle Breakage Edited by Agba D. Salman Department of Chemical & Process Engineering, The University of Sheffield, Sheffield, United Kingdom Mojtaba Ghadiri Institute of Particle Science and Engineering, School of Process, Environmental and Materials Engineering, University of Leeds, Leeds, United Kingdom Michael J. Hounslow Department of Chemical & Process Engineering, The University of Sheffield, Sheffield, United Kingdom

Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney Tokyo

Elsevier Radarweg 29, PO Box 211, 1000 AE Amsterdam, The Netherlands Linacre House, Jordan Hill, Oxford OX2 8DP, UK

First edition 2007 Copyright r 2007 Elsevier B.V. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the Publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively, you can submit your request online by visiting the Elsevier web site at http://www.elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material 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 Library of Congress Cataloging in Publication Data A catalogue record for this book is available from the Library of Congress British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN: 978-0-444-53080-6 ISSN: 0167-3785 For information on all Elsevier publications visit our website at books.elsevier.com

Printed and bound in The Netherlands 07 08 09 10 11 10 9 8 7 6 5 4 3 2 1

CONTENTS Contributors

ix

Foreword

xi

I. Fundamental 1.

Breakage of Single Particles: Quasi-Static Luı´s Marcelo Tavares

3

2.

Impact Breakage of Single Particles: Double Impact Test Kam Tim Chau and Shengzhi Wu

69

3.

Particle Breakage due to Bulk Shear John Bridgwater

87

4.

The Principles of Single-Particle Crushing Georg Unland

117

II. Milling 5.

Rotor Impact Mills Roland Nied

229

6.

Wet Grinding in Stirred Media Mills Arno Kwade and Jo¨rg Schwedes

251

7.

Roller Milling of Wheat Grant M. Campbell

383

8.

Air Jet Milling Alain Chamayou and John A. Dodds

421

9.

Breakage and Morphological Parameters Determined by Laboratory Tests Meftuni Yekeler

10.

Selection of Fine Grinding Mills Toyokazu Yokoyama and Yoshiyuki Inoue

v

437

487

vi

CONTENTS

11.

Fine Grinding of Materials in Dry Systems and Mechanochemistry Qiwu Zhang, Junya Kano and Fumio Saito

509

12.

Comminution Energy and Evaluation in Fine Grinding Yoshiteru Kanda and Naoya Kotake

529

13.

Enabling Nanomilling through Control of Particulate Interfaces Marc Sommer and Wolfgang Peukert

551

14.

Analysis of Milling and the Role of Feed Properties Mojtaba Ghadiri, Chih Chi Kwan and Yulong Ding

605

III. Modelling 15.

Monte Carlo Method for the Analysis of Particle Breakage Barada Kanta Mishra

16.

Numerical Investigation of Particle Breakage as Applied to Mechanical Crushing Chunan Tang and Hongyuan Liu

661

The Cohesion of Fractal Agglomerates: An Elementary Numerical Model Emile Pefferkorn

741

The Linear Breakage Equation: From Fundamental Issues to Numerical Solution Techniques Margaritis Kostoglou

793

17.

18.

637

19.

Analysis of Agglomerate Breakage Mojtaba Ghadiri, Roberto Moreno-Atanasio, Ali Hassanpour and Simon Joseph Antony

837

20.

Modelling of Mills and Milling Circuits Petya Toneva and Wolfgang Peukert

873

IV. Applications 21.

Particle Strength in an Industrial Environment Gabrie M.H. Meesters

915

CONTENTS

vii

22.

The Strength of Pharmaceutical Tablets Iosif Csaba Sinka, Kendal George Pitt and Alan Charles Francis Cocks

23.

Crystal Growth and Dissolution with Breakage: Distribution Kinetics Modelling Giridhar Madras and Benjamin J. McCoy

971

Liberation of Valuables Embedded in Particle Compounds and Solid Waste Wolfgang Schubert and Ju¨rgen Tomas

989

24.

941

25.

Attrition in Fluidised Beds Renee Boerefijn, Mojtaba Ghadiri and Piero Salatino

1019

26.

A Mechanistic Description of Granule Deformation and Breakage Yuen Sin Cheong, Chirangano Mangwandi, Jinsheng Fu, Michael J. Adams, Michael J. Hounslow and Agba D. Salman

1055

27.

Descriptive Classification: Failure Modes of Particles by Compression Ian Gabbott, Vishal Chouk, Martin J. Pitt, David A. Gorham and Agba D. Salman

28.

A New Concept for Addressing Bulk Solids Attrition in Pneumatic Conveying Lars Frye

Subject Index

1121

1149

1219

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CONTRIBUTORS Michael J. Adams Simon Joseph Antony Renee Boereﬁjn John Bridgwater Grant M. Campbell Alain Chamayou Kam Tim Chau Yuen Sin Cheong Vishal Chouk Alan Charles Francis Cocks Yulong Ding John A. Dodds Lars Frye Jinsheng Fu Ian Gabbott Mojtaba Ghadiri David A. Gorham Ali Hassanpour Michael J. Hounslow Yoshiyuki Inoue Yoshiteru Kanda Junya Kano Margaritis Kostoglou Naoya Kotake Arno Kwade Chih Chi Kwan Hongyuan Liu Giridhar Madras Chirangano Mangwandi Benjamin J. McCoy Gabrie M.H. Meesters Barada Kanta Mishra Roberto Moreno-Atanasio Roland Nied Emile Pefferkorn Wolfgang Peukert Kendal George Pitt Martin J. Pitt Piero Salatino Agba D. Salman Wolfgang Schubert

1055 837 1019 87 383 421 69 1055 1121 941 605 421 1149 1055 1121 605, 837, 1019 1121 837 1055 487 529 509 793 529 251 605 661 971 1055 971 915 637 837 229 741 551, 873 941 1121 1019 1055, 1121 989

ix

x

Jo¨rg Schwedes Marc Sommer Iosif Csaba Sinka Fumio Saito Chunan Tang Luı´ s Marcelo Tavares Ju¨rgen Tomas Petya Toneva Georg Unland Shengzhi Wu Meftuni Yekeler Toyokazu Yokoyama Qiwu Zhang

CONTRIBUTORS 251 551 941 509 661 3 989 873 117 69 437 487 509

FOREWORD It is indisputable that Particle Breakage is an important issue for a number of industries involving processing, crystallisation, granulation, transportation or storage of particulate materials. Breakage arises as a consequence of particle–particle or particle–equipment interactions and can be undesirable or intentional depending on the nature of an application. For instance, attrition of particulate products during storage and transportation is undesirable as it leads to product quality degradation. On the other hand, different comminution techniques such as grinding and milling have been developed for the purpose of intentional particle size reduction. In all these cases, better understanding of particle failure is essential in order to control breakage as desired. This handbook attempts to provide a full overview of the current state of the art and our understanding of particle breakage. This is from the small scale of a single particle, to the study of whole processes for breakage; both by experimental study and mathematical modelling. Despite a history going back over the centuries, particle breakage is still a lively technical ﬁeld. The book is divided in to four sections, with each covering a different aspect of particle breakage: Fundamentals, Milling, Modelling and Applications. We think it is particularly important that the ﬁrst three sections are related directly to product formation, rather than simple size reduction, showing that studies of breakage can be found at the heart of modern particle technology. We would like to thank all contributors for the quality of their work and speed of response. Agba D. Salman and Michael J. Hounslow The University of She⁄eld, UK Mojtaba Ghadiri University of Leeds, UK

xi

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Part I: Fundamental

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CHAPTER 1

Breakage of Single Particles: Quasi-Static Luı´ s Marcelo Tavares Department of Metallurgical and Materials Engineering, Universidade Federal do Rio de Janeiro, Brazil Contents 1. Introduction 2. Single-particle impact testing 3. Drop weight testing 4. Pendulum testing 5. Compression testing 6. Hopkinson pressure bar and impact load cell 6.1. Description 6.2. Principle of the measurements 6.3. Signal deconvolution 6.4. Comminution energy and coefﬁcient of restitution 7. Particle breakage characteristics 7.1. Particle fracture energy and fracture probability distribution 7.2. Particle strength, PLT strength and KIC 7.3. Particle stiffness 7.4. Energy-speciﬁc progeny size distribution 7.5. Energy utilization 8. Inﬂuence of selected variables on particle breakage characteristics 8.1. Type of stressing 8.2. Stressing intensity 8.3. Stressing and deformation rate 8.4. Particle size 8.5. Particle shape 8.6. Moisture content 9. Application to comminution Acknowledgements Appendix: Deﬁnition of terms References

3 7 9 12 18 21 21 22 25 28 31 33 36 37 40 44 45 45 47 49 51 57 58 59 62 63 66

1. INTRODUCTION Particle breakage in comminution and degradation processes is the result of a number of poorly understood microprocesses. Complex interactions among the Corresponding author. Tel.: 55(21) 2562-8538; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12004-2

r 2007 Elsevier B.V. All rights reserved.

4

L.M. Tavares

contributions of material characteristics, stressing conditions and the environment will determine the outcome of these industrial processes. Material characteristics relevant to particle breakage are the fracture strength and the deformation behaviour. Fracture strength can be deﬁned in terms of the energy required to cause fracture (or critical tensile stress). Material deformation behaviour can be classiﬁed as elastic (brittle) or inelastic. Inelastic behaviour includes semi-brittle, plastic [1] and quasi-brittle, the latter characterized by gradual accumulation of crack-like damage [2]. Stress conditions can be classiﬁed by type of stresses applied (compressive or shear), number of loading points, stressing intensity and stressing rate. Contributions of the environment are generally associated with the presence of moisture or surface-active agents. A complete understanding of the interactions of these several variables is not generally possible unless they are analysed in the most elementary breakage micro event, which is a single particle subjected to stresses. During a breakage micro event, two main modes of comminution have been identiﬁed. In the major mode, a particle is subjected to compressive stresses, resulting in disintegrative fracture. In the minor mode, called attrition or abrasion, the particle suffers gradual wearing of its surface leaving the parent particle largely intact but usually more rounded, the result of stress concentration at some surface sites on the corners or protrusions, leading to abrasion. Only a comparatively limited number of investigations have dealt with this mode of breakage [3–5]. In industrial comminution processes, particles are mainly fractured by compressive loading and the fundamental properties of the fracture process can be studied most effectively by well-controlled experiments on single particles. Singleparticle fracture studies have provided the basis for most of the fundamental particle comminution research that started in 1960s, with important contributions from researchers in the United States [6–9], Germany [1,10–12] and Japan [13–16]. Single-particle breakage tests have been used to elucidate a number of phenomena in particle breakage, including: 1. fracture phenomena [9–11]; 2. energy utilization in the comminution process and a measure of the different types of losses [1,10,12,17–19]; 3. effect of particle size, shape, material physical properties and modes of loading on particle breakage characteristics [16,20]; 4. energy-size reduction relationships [8,10,21,22]; 5. breakage characteristics of materials for modelling comminution and degradation processes [22–25]; 6. material deformation response under applied stresses [1,26]. The mechanism of the particle fracture process is largely understood from single-particle fracture studies and there is a general agreement in the literature

Breakage of Single Particles: Quasi-Static

5

on the basic model description [20]. From the instant of initial contact by the tools (such as a drop weight in free fall and a hard anvil) the particle is stressed and energy is stored in it as strain energy. The small amount of plastic deformation at and around the loading points is not generally considered to be signiﬁcant for most materials. The deformation of the particle is generally considered to be described by the Hertzian model during this initial period [20]. Beyond a certain point and for a number of materials [2], crack-like damage starts to accumulate within the particle and when the criteria for the failure are met, a macrocrack grows unstably and increasingly rapidly as an advancing crack, which may or may not branch [2]. When the crack or cracks emerge from the particle, several progeny particles are formed. The number and size of the progeny particles depends on the size and location of the initiating ﬂaw, on the material microstructure and on the extent of crack branching. The initiation of unstable crack growth is governed by the Grifﬁth criterion but conventional fracture mechanics techniques cannot be used in any quantitative way because of the great difﬁculty in calculating the stress ﬁeld inside an irregularly shaped particle. Fracture mechanics dictates that if the crack grows unstably the energy release rate must be greater than the crack resistance which is twice the surface speciﬁc fracture energy of the material [27,28]. The essential feature of this model of particle fracture is that the energy required to drive the crack comes entirely from the stored strain energy, which is available at crack initiation, which is called particle fracture energy. This idea is based on the observation that no other energy source is available to deliver energy at the required rate during crack growth, which is generally several orders of magnitude larger than the rate at which the energy is delivered to the particle during a loading event. A necessary consequence of this model is that the stored particle fracture energy must be at least as large as the total energy that is dissipated at the growing crack tip. In general the particle fracture energy will exceed the dissipated energy by signiﬁcant amounts and the unused energy is dissipated after fracture is complete, mostly as kinetic energy of the progeny fragments, which in turn can result in further breakage depending on the physical conﬁguration of particle and stressing tools [20]. The single-particle fracture process does not terminate after ﬁrst failure at a ﬂaw because kinetic energy may still be available either from the tools that apply the stresses or from the ﬂying fragments of the particles. This remaining energy must be dissipated during the second stage of the process, which results in secondary fracture of the initial progeny and possibly several further stages of sequential fracture as well. A variety of testing methods have been used to measure the breakage characteristics of single particles subject to compression, each of which allowing investigation over a restricted range of deformation rates. These tests can be

6

L.M. Tavares

Drop test

Pneumatic gun

Rotary impact tester

Single impact

Drop weight

Pendulum Press

Impact load cell

Double impact

Point-load tester

Rigidly-mounted roll mill

Slow compression

Fig. 1. Different types of single-particle breakage tests.

classiﬁed according to the mode of application of stresses and the number of contact points in (Fig. 1): 1. Single impact, 2. Double impact, 3. Slow compression. Single impact can be performed by drop tests [25,29,30] or by propelling particles against a target, using a pneumatic gun, for example [17] (Fig. 1). In either one of these cases the speciﬁc impact energy is given directly by half of the square of the particle velocity at the instant of collision. Energy beyond which is necessary just to break the particle by a single fracture event is still in the fragments from crushing as kinetic energy. This energy can cause secondary breakage, and so on, especially since the main vector of the fragments velocity is directed towards the surface. Double impact tests correspond to those where a test specimen is crushed between two hard surfaces at a moderate deformation rate (Fig. 1). The ﬁrst test of this type is the drop weight test, where a particle resting on top of a hard surface is struck by a falling weight. The second type is known as the pendulum test. In this, a particle is hit by one or two moving hammers. In both tests, the amount of energy that is available to the particle at the instant of impact is known precisely from classic mechanics. Further, with proper instrumentation, these devices (or modiﬁed versions of them) can also be used to determine the fraction of the available energy that is actually absorbed by the particle during impact. Slow compression tests are conducted using uniaxial compression presses or using the rigidly mounted roll mill (Fig. 1).

Breakage of Single Particles: Quasi-Static

7

With only the exception of high-velocity ballistic impact tests, all other experimental techniques listed and analysed in the present chapter deal with loading particles at rates such that the duration of the contact is sufﬁcient to allow the stress to propagate and equilibrate throughout the particle. As a result, these techniques are called ‘‘quasi-static’’. Comprehensive reviews that cover the earlier contributions to single-particle breakage can be found elsewhere [31,32]. This chapter covers in detail the different methods of testing particles by compression, with emphasis on some of the most recent contributions in the ﬁeld. A number of measures of special interest of the comminution result are deﬁned, including the particle fracture energy, the fracture probability or proportion of broken particles, the particle strength, the size distribution of the progeny, as well as the energy utilization, among others. The effects of several variables on these measures of particle breakage are analysed and applications of data from single-particle breakage tests to comminution are reviewed brieﬂy.

2. SINGLE-PARTICLE IMPACT TESTING Since single-particle impact testing is the subject of Chapter 2 of this handbook, experimental techniques will be reviewed in this section brieﬂy and only for the sake of completeness, emphasizing on those techniques that load particles under quasi-static conditions. Single-particle impact tests can be classiﬁed into low-impact velocity and highimpact velocity tests. The ﬁrst group is represented by drop, drop shatter or freefall tests (Fig. 1). These are the simplest type of single-impact testers, which consist of a release system for the particle, and a thick metal plate, against which the particle falls under gravity. Details of a convenient apparatus to study this mode of fracture can be found elsewhere [9]. Since free-fall conditions are met in such tests, the input energy Ei and the speciﬁc input energy Eis may be calculated by E i ¼ mp gho and E is ¼ gho

ð1Þ

where mp is the particle weight, ho is the distance from the bottom of the particle to the impact plate and g is the acceleration due to gravity. If one considers about 20 m or so as the limiting drop height that can be used in such a test, then the maximum speciﬁc impact energy that can be reached is only about 200 J kg–1. Considering the particle size dependence on the fracture strength of particles [20], then it is evident that this test may only be used to study fracture of particles of reasonably coarse size or low strength. The second group of single-impact tests is represented by higher impact velocity tests (Fig. 1). In general, these use compressed air for propelling the particle against a solid surface. Indeed, Dan and Schubert [17] used a pneumatic

8

L.M. Tavares

gun to investigate both the breakage probability distribution and the relationship between input energy and the progeny size distribution and energy utilization. Impact velocities as high as 60 m s–1, which correspond to input energies as high as 1800 J kg–1 were reached in the tests, which were also used to study the inﬂuence of impact angle. In these tests the input energy and the speciﬁc input energy are given by 1 E i ¼ mp v 2o 2

and

E is ¼

1 2 v 2 o

ð2Þ

where vo is the velocity at the instant of impact. No instrumentation is typically used in single-impact tests. An exception is the system used by Vervoorn and Scarlett [33], who instrumented the target plate with force transducers. The result was that the force–time proﬁle could be recorded during the test, and the particle strength measured. Another type of impact tester has been used by Ghadiri and co-workers [4] to study breakage by abrasion/attrition of ﬁne particles. Impact velocities of up to 50 m s–1, which correspond to input energies of up to 1200 J kg–1, have been reached using this experimental setup, which used a continuous ﬂow of compressed air coupled to a vacuum system positioned near the target plate. Still another single-particle fracture apparatus is the rotary impact tester (Fig. 1), which has been used as a convenient alternative to traditional highvelocity impact testers [34]. In the device, a vibration feeder transports single particles from the feed chamber to the disk-shaped rotor. There, the particles enter the rotor centrally and are accelerated in one of the radial channels by the centrifugal force. As the grinding chamber is evacuated and friction during the acceleration can be neglected, the impact velocity of the particles on the test ring is given by the ejecting velocity from the rotor as a function of the rotational speed n [35]. vo ¼

pﬃﬃﬃ 2p D r n

ð3Þ

where Dr is the outer diameter of the rotor and n is the frequency of rotation. Devices similar to this have been used to relate the product size distribution to the various testing variables [36,37]. More recently, this type of device has also been used to calculate the breakage probability as a function of impact velocity [34,35]. Impact velocities from 60 to 140 m s–1 have been used, so that speciﬁc impact energies of up to about 10,000 J kg–1 can be reached, even allowing for the fracture of ﬁne particles [35]. It is not uncommon to combine low-velocity and high-velocity impacts in order to properly describe the breakage probability distribution of particles over a range of impact energies [38].

Breakage of Single Particles: Quasi-Static

9

Drop weight

Collection box

Guide ho

Particle

Anvil

Fig. 2. Schematics of a drop weight tester.

3. DROP WEIGHT TESTING The drop weight test (Fig. 2) is one of the simplest and most commonly used methods of investigating breakage characteristics of materials. Most of the earlier experiments have been conducted to establish a form of energy-size reduction relationship [21,39–41]. In these tests, the input energy was related to the surface area created, or to a characteristic size of the product. The test consists of dropping a weight (striker), from a known height, against a particle positioned on top of a hard anvil (Fig. 2), so that the input energy is given by E i ¼ mb gho

ð4Þ

where mb is the mass of the drop weight and ho is the net drop height, that is, the distance between the bottom of the drop weight and the top of the particle. For a more precise estimation, it may be required to add the distance between the top of the particle and the remaining height of the particle after impact [42]. In this case, precise computation of the input energy becomes only possible upon completion of the test. The release of the drop weight can be performed by an electromagnet [43,44], a pneumatic system [24,45] or with the aid of pulleys and strings [46,47]. Some researchers have even equipped the simple drop weight tester with a system for arresting the falling weight [9,38] that is used to avoid (or, at least, limit) the contribution of secondary fracture. Arbiter et al. [9] used a spring-loaded

10

L.M. Tavares

device that operated immediately after primary fracture which was not only able to arrest the falling mass, but also record its residual kinetic energy. The system proved to be effective with spheres [9], but probably suffers from limitations in the case of irregularly shaped particles. Equation (4) is valid when free-fall conditions prevail. When guiding systems are used [23,24,44] to control the drop of the falling weight, a loss of momentum may occur due to friction, so that the input energy is more appropriately calculated by Ei ¼

1 mb v 2o 2

ð5Þ

where vo is the velocity of the striker in the instant of collision. Equations (4) and (5) are equivalent for the free-fall conditions, where the impact velocity is given by pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ v o ¼ 2gho ð6Þ A comparison is shown in Fig. 3 between the expected value of impact velocity from free fall and the value calculated from measurements made with a laserphotodiode system for a drop-weight tester that uses linear guides to control the motion of the falling weight [44]. The good correspondence demonstrates that well-designed guiding systems, which have low frictional losses, result in impact velocities between 95% and 99% of free-fall velocity, so that equation (4) may be used to estimate the energy input with reasonable accuracy. This investigation [44] also demonstrated that no noticeable difference exists between the size 6

Impact velocity - vo (m/s)

5

4

3

2

1

0 0.0

Mean free-fall velocity on the interval Δh Measured mean impact velocity on the interval Δh

0.2

0.4

0.6 0.8 Drop height - ho (m)

1.0

1.2

1.4

Fig. 3. Comparison between the measured (mean) drop velocity (over a distance Dh of 20 mm) in the drop weight tester and the free-fall velocity for a drop weight tester equipped with a linear guiding system.

Breakage of Single Particles: Quasi-Static

11

Cumulative passing (%)

100

10

1 Ball-ball Ball-flat Flat-flat 0.1 0.01

0.1 Particle size (mm)

1

Fig. 4. Effect of loading geometry on breakage of 2.8–2.0 mm apatite particles at 874 J kg–1 (0.24 kWh t–1) impact energy (ball diameter: 25.4 mm).

distributions of progeny from free-falling or guided drop weights, as long as geometry of the falling mass remained unchanged. Different geometries of the falling masses and anvils have been used in the drop weight: Krogh [23] used a cylinder with a surface of a particular curvature; several workers [46,47] used steel spheres of different diameters; while more recently researchers have favoured drop weights with a ﬂat contact surface [24]. The inﬂuence of drop weight geometry – at a constant impact energy – has been investigated [47] and is illustrated in Figs. 4 and 5 for the cases of impact using a ﬂat drop weight with the particle placed on a ﬂat anvil (ﬂat–ﬂat), a sphere with a ﬂat anvil (ball–ﬂat) or with an anvil of semi-circular shape (ball–ball). Figure 4 shows that when the impact energy is low the loading geometry only marginally affects fragmentation. This is because a greater proportion of the input kinetic energy is consumed in causing primary fracture of the parent particle and little energy is left for subsequent breakage of the fragments. At the higher impact energy (Fig. 5), signiﬁcantly different progeny size distributions resulted from different loading geometries. A narrow size distribution with smaller proportions of ﬁnes resulted from ﬂat–ﬂat loading, whereas a broader size distribution with larger proportions of ﬁnes resulted from loading with ball–ball geometry. Such differences are explained by recognizing that at high impact energies primary fracture of the parent particle consumes only a small fraction of the input energy, leaving a considerable amount of energy for subsequent fracture events. Depending on material properties and loading geometry, a fraction of the fragments produced by primary breakage will escape laterally from the active breakage zone and avoid further breakage. The larger active breakage zone for ﬂat–ﬂat

12

L.M. Tavares

Cumulative passing (%)

100

10

1 Ball-ball Ball-flat Flat-flat 0.1 0.01

0.1 Particle size (mm)

1

Fig. 5. Effect of loading geometry on breakage of 2.8–2.0 mm apatite particles at 6488 J kg–1 (1.80 kWh t–1) impact energy (ball diameter: 25.4 mm).

loading ensures that a greater proportion of the fragments will be positioned between the falling drop weight and the anvil. For a brittle material such as apatite, the fragments will be dispersed so that the coarser fragments are reselected for breakage. This continues progressively until either all the input energy has been dissipated or until the reactive force from the pile of fragments is equivalent to the acting force, resulting in a rebound of the drop weight or its ﬁnal rest. On the other hand, the smaller active breakage zone in ball–ball loading results in a smaller fraction of fragments being subject to further breakage. The high concentration of energy in those fragments results in the generation of a large proportion of ﬁnes, evident in Fig. 5. Perhaps the greatest limitation of drop weight tests lies in the fact that they do not allow direct measurement of the comminution energy [48] – see appendix. However, this limitation is overcome by the instrumented version of the device, called impact (or ultra fast) load cell, which is analysed later in the chapter.

4. PENDULUM TESTING The pendulum is a good alternative to the drop weight test, as it conveniently allows the use of variable impact energies in a safe and convenient manner. Different types of pendulum devices have been used in testing single particles, having been used in a number of early investigations of single-particle breakage [49–51]. Perhaps one of the most popular early conﬁgurations used until this day is the one standardized by Fred Bond [50] to determine the resistance of rock to crushing,

Breakage of Single Particles: Quasi-Static

13 Counterweight

θ Hammers

ho

Collection box

Fig. 6. Schematic diagram of the pendulum tester to determine Bond crushing work index.

and the corresponding work index. The crushing work index is determined from a test where two pendulum mounted hammers are dropped simultaneously (Fig. 6) on each side of a particle. When at rest, the hammers are separated by a 5 cm gap. The centreline of each hammer, measuring approximately 5 5 70 cm, is positioned 41.3 cm away from the axis of rotation. In fact, Bond even suggested attaching each hammer to the rim of a 2200 bicycle wheel, in order to avoid interference from the weight of the arm. Alternatively, this can be accomplished by appropriate balancing of the system using counterweights (Fig. 6). The test begins by weighing individual irregularly shaped particles contained in the 75–50 mm size fraction for testing, and then placing them, one at a time, on the pedestal between the two hammers. Each particle should be mounted on this pedestal (with the aid of a piece of modelling clay) so that its smallest dimension is between the two hammers. The particle dimension in the direction of impact (D) is measured and then the hammers are released (using, perhaps, an appropriate rope and pulley system) from a starting angle of 10 degrees to free-fall and strike the particle. The particle is then checked for cracking or breakage. If it remains unbroken (breakage is here deﬁned as loss of at least 10% of the particle original weight), then it is again mounted on the pedestal and hit again by the hammers, now placed at an angle 5 degrees larger than the previous impact. This is repeated until the particle is broken. In this moment the last release angle is noted. This procedure is repeated for 10–20 particles and, after completion of the test, the fragments are collected for size analysis. In this test the kinetic energy of each pendulum can be calculated on the basis of the difference in height ho between the initial position of the centre of gravity of the pendulum and its position at rest (Fig. 6), E i ¼ m1 gho where m1 is the mass of each pendulum (13.6 kg).

ð7Þ

14

L.M. Tavares

The difference in height ho is controlled by the value of the angle y between the hammer and the vertical (Fig. 6), so that ho ¼ L L cos y

ð8Þ

where L is the distance from the axis of rotation to the centre of gravity of the hammer, proposed as 41.3 cm by Bond. Replacing equation (8) in (7) with the appropriate values, the impact energy resulting from the combined strike of both hammers is given in Joules by: E i ¼ 117 ð1 cos yÞ

ð9Þ

Bond [50] deﬁned the impact energy per unit of thickness CB required to break each particle by dividing the impact energy by the thickness of each particle D, so that CB ¼ 117 ð1 cos yÞ=D

ð10Þ

–1

which is given in J mm . From equation (10) and the material density rp (given in g cm–3), Bond proposed to estimate the crushing (or impact) work index by Wi ¼

53:49 CB rp

ð11Þ

given in kWh t–1. Typical experimental results are given in Fig. 7, which show the signiﬁcant variability of the measurements. A more detailed description of the experiment and the analysis of the data can be found elsewhere [52]. Values of the impact work index were found to vary from as low as 1 to as high as 40 kWh t–1 for different rocks. In analogy to what happens in the drop weight tester, in the pendulum not all energy provided by the striker is used for particle breakage, with some residual energy remaining available for restitution (rebound) of the hammers. One advantage of the pendulum over drop weight testing is the potential to measure the fraction of the energy that is actually absorbed by the particle by measuring the residual energy of the strikers. Attempts have been made by Awachie [18] to record the coefﬁcient of restitution of the striker using a four-piece pendulum apparatus, later replaced by a two-piece apparatus. The two-piece (twin) pendulum was constructed to overcome the problems of determining the coefﬁcient of restitution of the four-piece device during single-particle breakage tests because the number of collisions on the latter. Also, high system energy losses were observed at input energy levels higher than about 25 J as a result of increased twisting and rotation of the pendulum pieces on impact, thus making the use of the four-piece pendulum prohibitive under these conditions [48]. Narayanan [53] introduced a modiﬁcation of the twin pendulum by monitoring the motion of the impact pendulum with the aid of appropriate instrumentation

Breakage of Single Particles: Quasi-Static

15

99

95

Cumulative distribution (%)

90

70 50 30

10

Sossego copper ore Cantagalo limestone Conceição itabirite

5

1

1

10

100

Crushing Work Index - Wi (kWh/t)

Fig. 7. Distribution of values of impact (crushing) work index for selected rocks from Brazil, showing the log-normal ﬁt to the data.

and a computer. This so-called computer-monitored twin-pendulum device, illustrated in Fig. 8, consists of a metallic sphere (typically manganese steel) which falls, by gravity, in a swinging motion against a particle that is attached to a steel cylinder, called rebound pendulum [22]. The metallic sphere, called impact pendulum, is elevated up to an established height (ho), using a rope that passes through a pulley, guaranteeing smooth liberation. The rebound pendulum presents three ﬁns, two in one side and one in the other. The motion of the rebound pendulum is monitored by a computer, through measurement of the time required for the triple ﬁn arrangement to cross a narrow laser beam. An average period is computed from a number of periods (typically six) that correspond to the passage of six ﬁn edges. A total of 25 swings of the rebound pendulum are monitored to determine the period losses per swing due to uneven motion or friction, from which the corrected period for the ﬁrst swing is computed. This corrected period, T, exhibits a linear relationship with the angle (y) subtended by the rebound pendulum relative to its equilibrium position. This relationship is expressed by [22] T ¼ aT þ bT y2

ð12Þ

where aT and bT are constants, determined by calibration of the equipment, which consists of letting the rebound pendulum swing from known angles and recording the periods of vibration using the triple ﬁn arrangement.

16

L.M. Tavares

ho

Rebound pendulum

Input pendulum

Particle Collection box

Fin

Laser detector

Fig. 8. Schematic diagram of the computer-monitored twin pendulum.

The energy losses during the impact, which includes the energy absorbed by the particle, Ec, is given by [22] E c ¼ E i E R1 E R2

ð13Þ

where ER2 is the energy transmitted to the rebound pendulum, and ER1 is the residual energy of the impact pendulum after the collision. The kinetic energy of the impact pendulum (Ei) is calculated by equation (7). The energy absorbed by the rebound pendulum is given by E R2 ¼ m2 gh

ð14Þ

where m2 is the mass of the rebound pendulum. The height h at which the pendulum is elevated after collision is given by equation (8), where y is the angle swept by the pendulum during rebound, calculated from the period of oscillation of the system of ﬁns and the laser beam (equation (12)). The residual energy of the impact pendulum is given by E R1 ¼

1 m1 v 21 2

ð15Þ

From the deﬁnition of coefﬁcient of restitution e as the ratio between the restitution and impact impulses (and given that the rebound pendulum is at rest

Breakage of Single Particles: Quasi-Static

17

before collision), e¼

v2 v1 u1

ð16Þ

where v1 and v2 are the velocities of the impact and rebound pendulums after impact and u1 is the velocity of the impact pendulum at collision. Equation (15) may be rewritten as E R1 ¼

1 m1 ðv 2 eu1 Þ 2

ð17Þ

wherepuﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 is ﬃthe velocity of the input pendulum in the instant of collision, given by u1 ¼ 2gho . The coefﬁcient of restitution e is calculated substituting equation (16) in the equation of conservation of linear momentum for the experiment with the pendulum, given by

so that

m1 u1 ¼ m1 v 1 þ m2 v 2

ð18Þ

m1 þ m2 v 2 1:0 e¼ m1 u1

ð19Þ

Now replacing the values of input energy, as well as the kinetic energies of the pendulums after collision for a particular test, Ec may be calculated by equation (13). Ec, which corresponds to the energy losses in the system, mainly comprised of the energy used in particle breakage, is sometimes called comminution or breakage energy (see appendix). The twin pendulum is thus appropriate to measure the amount of energy that is actually absorbed by particles (comminution energy) under the dominant condition found in most comminution equipment, that is, loading at moderate speeds. Comparing the fragmentation resulting from different levels of input or comminution energy, it is possible to establish the relations between energy and fragmentation for the material, as discussed later in the chapter. Energy transfer efﬁciency is deﬁned as the ratio between the amount of energy that is actually used in breakage of the particle (breakage or comminution energy) and the input (impact) energy, Energy transfer efficiency ¼ 100

Comminution energy ð%Þ Input energy

ð20Þ

Figure 9 demonstrates that it is generally found to be about 50%, decreasing with an increase in input energy. Thus, the breakage energy increases in a lessthan-proportional relationship with input energy. It is also evident from the ﬁgure that the scatter in the measurements is very low. Some of the disadvantages of the instrumented twin pendulum are associated with the time consumed in the test, which requires sticking each particle to the

18

L.M. Tavares

Energy transfer efficiency (%)

100 Mount Lyell copper ore (5.6-4.75 mm) Ensham coal (16.0-13.2 mm)

80

60

40

20

0 1

2

3 4 Input energy - Ei (J)

5

6

7

8 9 10

Fig. 9. Relationship between input energy and energy transfer efﬁciency for testing single particles of Mount Lyell ore [22] and Ensham coal [48] in instrumented twin-pendulum devices. Error bars show the standard deviations of the measurements.

rebound pendulum and recording the various rebounds of the pendulum, and also with the secondary motion of the rebound pendulum that often occurs after collision. Further, it is restricted in its energy and particle size ranges of application (although these can be addressed by using multiple pendulum devices of different sizes).

5. COMPRESSION TESTING Different types of uniaxial compression-testing machines have been used over the years in testing single particles. Examples are some reasonably crude mechanical presses [8], simple point-load testers (PLT) [54], sophisticated presses used for testing micron-size particles [26,55] and large-scale hydraulic presses for testing particles of sizes approaching one metre [56]. A signiﬁcant advantage of compression testing over traditional drop weight and pendulum (instrumented or not) devices is that the applied forces, and often the deformations, can be recorded during the test in order to determine the load– deformation proﬁle, and thus several strength-related measures. In this regard, measures of particular interest are the energy at primary fracture, called particle fracture energy, and the total energy absorbed by the particle during the test (called comminution energy, Ec – see appendix). Both can be calculated from direct numerical integration of the load–deformation proﬁle Z Dc E¼ FdD ð21Þ 0

where F is the load, D the deformation and Dc is a critical deformation.

Breakage of Single Particles: Quasi-Static

19

1800 1500

Force (N)

1200 900 600 300 0

0

1

2 3 Deformation (mm)

4

5

Fig. 10. Force–deformation proﬁle from testing of a 16.0–13.2 mm Paragominas bauxite particle by slow compression.

Figure 10 shows a typical result from the test, where the stored strain energy is given by the area below the curve (equation (21)). It also shows that, after primary fracture, many more fracture events occur due to secondary fracture of progeny fragments. Some mechanical presses and PLT only allow measurement of the load corresponding to fracture and, thus, only measures such as particle strength and point load strength can be obtained, besides the size distribution of the progeny fragments. PLTs, for example, have been used to determine the maximum loads expected in the toggle of primary and secondary jaw crushers [54]. Dial deformation gauges have been incorporated in some of these devices, which are then more appropriately called point load–deformation devices [54,57]. One important question in single-particle compression testing is related to how particle deformations are actually measured. While no difﬁculty exists in measuring forces, which can be done using precision load cells, measurement of deformations require greater attention. In testing materials with low values of elastic modulus and stiffness it may be possible to estimate particle deformations on the basis of the velocity at which the piston moves during the test, since deformations on the tools (plates) may be neglected. On the other hand, when testing particles of materials having moderate to high stiffness, direct measurement of the particle deformation by appropriate transducers becomes necessary. Evidently, this contributes even further for an increase in the time and cost of the already tedious test. Some setups that have been used in testing particles by slow compression in the extremes of the particle size spectrum of interest to the industry are worth analysing in greater detail. For example, Unland and Szczelina [56] described the

20

L.M. Tavares

construction of a test apparatus that is capable of applying forces and deformations as high as 4000 kN and 0.5 m, respectively. The apparatus allows testing particles from 20 to 600 mm, covering the entire size range of interest in industrial crushing. PTL, on the other hand, apply loads by two rounded cone points, instead of ﬂat plates normally used in most compression-testing equipment (Fig. 1) [54,58]. The result is the PLT strength that is calculated from particles with approximately the same linear dimensions, as is described in a later section. At the ﬁne end of the size spectrum, some micropresses used along with scanning electron microscopes allowed testing particles of sizes as ﬁne as 1 mm and observing important phenomena in particle breakage [26,55]. Tests in these devices offered the experimental conﬁrmation of a brittle-plastic transition size for particles (even those of very brittle materials) on loading by compression [55]. It is worth noting that such valuable information on particle breakage behaviour, which was then only available using these highly sophisticated custom-made presses, can now be obtained by using nanoindentation machines [59], now in widespread use advanced materials science research. With an appropriate ﬂatended indenter and an automation routine, these machines will certainly become a valuable tool for testing particles down to submicron sizes. The greater control of the loading process provided by compression-testing machines, when compared to impact testers, made them a valuable tool in some of the most fundamental studies of particle breakage. Particle size effects on strength were studied by direct measurements of strength of particles of a wide range of sizes by a number of investigators [1,8,14,16]. Detailed studies on the relationship between the energy actually required to fracture single particles and the resulting fragment size distribution by Baumgardt et al. [38] demonstrated the validity of Rumpf’s similarity principle [10], and are discussed in later sections in the chapter. With the aid of appropriate devices (polariscopes) attached to simple presses and two-dimensional models, isochromatic fringe patterns could be recorded, thus allowing the study of the stress distributions in particle fracture [9,58]. Compression-testing experiments have also been used, along with additional instrumentation, to measure precisely the kinetic and sound energy resulting from fracture of single spherical particles [7,13]. Yashima et al. [13] found that conversions of particle fracture energy into kinetic energy varied from as high as about 39% for borosilicate glass to 2% for limestone. Conversion of particle fracture energy into sound energy was found to be in the range of 104%. Testing single particles by compression consists of placing individual particles, one at a time, between two ﬂat parallel plates mounted on a uniaxial compression loading machine and subjecting them to increasing loads up to fracture. This is certainly a tedious and, particularly in the case of slow compression, very time consuming operation. An alternative has been to automate the test, such as with a robotic compression tester [60]. Another alternative is the use of a rigidly mounted high-pressure roll mill (Fig. 1). It allows testing a stream of particles,

Breakage of Single Particles: Quasi-Static

21

while the torque, and thus the power, is recorded as a function of time. The result is that the energy can be accurately measured and appropriate energy-size reduction relationships established [61–63].

6. HOPKINSON PRESSURE BAR AND IMPACT LOAD CELL 6.1. Description The split Hopkinson pressure bar (SHPB) is a device originally developed by Hopkinson [64] to measure the properties of materials during dynamic loading. It consists of two steel bars, between which a test sample is positioned; a system to generate dynamic stresses in one extreme and another system (such as a pendulum) whose purpose is to absorb the residual impulse applied to the opposite extreme. The bars are maintained precisely aligned horizontally, smoothly sliding over bearings. The deformation waves that travel through the bars are monitored with the aid of strain gauges. After conditioning and storage using the appropriate data acquisition system, the transient signals are then used to calculate the energy balance of the entire system and the load–deformation proﬁle of the test sample. Different systems are used for producing the strain waves, which include explosives [65] and compressed air [66], thus allowing the investigation of dynamic loading under a wide range of strain rates. The application of the original SHPB to comminution research is fairly limited, in part due to the tedious nature of the experiments and the great variability encountered in the fragmentation behaviour of geological materials. A more effective and convenient alternative to the SHPB is the impact load cell (ILC). It is essentially a hybrid between the traditional drop weight tester and the Hopkinson pressure bar, more speciﬁcally the Davis pressure bar [67]. The original development of the impact load cell (originally called ultra fast load cell) and application to comminution investigation is credited to Reiner Weichert and took place at the University of Utah [68]. Twenty years later, the device has today reached its maturity in both testing individual particles [20,43,69] and beds of particles [70,71] by impact, with detailed calibration procedures having been presented in several publications [45,69,71,72]. A schematic diagram of the ILC is presented in Fig. 11. It consists of a long steel rod equipped with strain gauges on which a single particle or a bed of particles is placed and impacted by a falling weight. The compressive wave resulting from the impact travels down the rod and is sensed by the solid-state strain gauges. The strain gauges are glued to the rod in pairs positioned in opposite sides on the face of the rod, in order to limit any contributions due to ﬂexion in the measurements. The passage of the strain wave through the gauges results in a voltage change in a Wheatstone bridge (signal conditioner), which is

22

L.M. Tavares

Drop weight release system Data acquisition board Laser source

Collection box

Photodiode

Strain gauges Computer

Signal conditioner

Fig. 11. Outline of the impact load cell.

then recorded as a function of time using appropriate digital storage systems, such as an oscilloscope or a data acquisition board.

6.2. Principle of the measurements The ILC allows the calculation of both loads and deformations that a particle undergoes during an impact. The load applied to the top of the rod by the particle during an impact is calculated from the law of proportionality of the strain gauges and Hooke’s law. Assuming that no dispersion or attenuation of the wave takes place from the point of contact to the measuring station (strain gauges) and that the bulk deformations inside the rod are predominantly elastic (which is commonly valid, given the low level of the stresses that are measured), the load is given by F r ¼ Ar Y

ð22Þ

where Ar, Y and e are the cross-section, the modulus of elasticity and the unit deformation in the rod, respectively. The unit deformation of the rod is related to the response of the Wheatstone circuit response (Fig. 12) by V ¼ GFl ð23Þ V in

Breakage of Single Particles: Quasi-Static

23

Fig. 12. A typical conﬁguration of the Wheatstone conditioning circuit, where R1, R2, R3 and R4 are strain gauge resistances, Ra and Rb are ﬁxed resistances and Rv is the variable resistance, obtained by a potentiometer, used in conditioning the ILC signal.

where l is the proportionality constant of the Wheatstone bridge circuit, which depends on the bridge conﬁguration used and must be adjusted by calibration [45], GF is the gauge factor, provided by the manufacturer of the strain gauges used, Vin is the excitation voltage of the bridge circuit and V are the output voltages measured during the test and later deconvoluted. Replacing equation (23) in (22) the expression for calculating the force exerted on the rod is given by Ar Y V Fr ¼ ð24Þ GFl V in Thus, given the physical and mechanical properties of the rod, the constant of proportionality of the bridge circuit and the gauge factor, the volt-time history that is recorded during each test can be individually transformed in a force–time history. The compression that a particle that is positioned on top of the rod undergoes is not directly measured using the ILC. It is calculated from the momentum balance of the drop weight in free fall, as well as from the rod deformation. The motion of the drop weight during impact can be determined from the equation of conservation of linear momentum mb

d2 ub dt 2

¼ F b þ mb g

ð25Þ

where ub is the position of the centre of gravity of the drop weight, mb is its weight and Fb is the load applied by the particle against the drop weight. Integrating equation (25) subjected to the initial conditions at the instant of contact (t ¼ 0) given by dub =dt ¼ v o and F b ¼ 0 gives Z t dub 1 ¼ v o þ gt F b ðtÞdt ð26Þ mb 0 dt

24

L.M. Tavares

where vo is the velocity of the drop weight at the instant of contact. Since in most cases it is possible to guarantee that free-fall conditions predominate during the test, v0 may be calculated by equation (6). During collision, different types of waves propagate inside the ILC rod, which include longitudinal, transverse and Rayleigh waves [67]. Given the type of contact and the long length of the steel rod in comparison to the diameter, it is possible to consider only the longitudinal waves in the computations. In this case, the loads and deformations on top of the rod in the ILC are related by dur 1 ¼ F r ðtÞ rAr C dt

ð27Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃ where C is the wave propagation velocity inside the rod, given by Y =r, with r being the density of the rod. Considering that the loads in the surfaces in contact are in equilibrium (Fr ¼ Fb ¼ F), then equations (27) and (26) are subtracted (Fig. 13), which, upon integration, gives aðtÞ ¼ v o t þ

gt 2 1 mb 2

Z tZ

t

Fð^tÞd^t dt 0

0

1 rAr C

Z

t

FðtÞ dt

ð28Þ

0

where a ¼ ub ur and t^ is an integration variable. Equation (28) enables calculation of the approximation between the centre of gravity of the drop weight and a point in the rod distant from the contact point. Therefore, a corresponds to the overall deformation in the vicinity of the contact point, resulting predominantly from the particle compression, but also with the added contributions of the local indentations of the rod and the falling mass. Therefore, the deformation a is calculated (with the aid of numerical integration)

ub

ur

At initial contact

During impact

Fig. 13. Illustration of the principle used to calculate deformations experienced by a particle during impact on the ILC.

Breakage of Single Particles: Quasi-Static

25

from the initial impact velocity, the mass of the drop weight and the force–time history of the experiment. Equations (24) and (28) are valid until the arrival of the reﬂected waves at the measuring stations (where the sensors are positioned). When this happens, a deconvolution of the signal [19,45] is used, which is the subject of the following section. Equation (28) is strictly valid for spherical drop weights. In the case of strikers of different geometry, such as ﬂat-ended cylinders, a different procedure must be used [72]. Different diameters and lengths of Hopkinson pressure bars and ILCs have been used in single-particle breakage testing (Table 1). Diameters ranging from 9.5 to 100 mm and lengths from 0.5 to 6.3 m have been reported. While the diameter is associated with signal resolution, the length is associated with the ability to resolve a greater or smaller portion of the signal without the need for deconvolution. Particle sizes ranging from 0.2 to about 100 mm have been tested in these ILCs.

6.3. Signal deconvolution The force–time history of the entire event at the contact surface is required for calculating the various measures obtained in the ILC. In practice, however, this force–time history is actually measured at the strain gauges (Fig. 14), which are placed at a distance dr from the contact surface. If we consider that, although the load is initially concentrated on the contact surface, the wave becomes Table 1. Rod characteristics of documented impact load cells

Institution

Rod length Rod diameter (m) (mm) Orientation Reference

University of 4.9 Utah, USA CSIRO-DM, 5.7 Australia JKMRC, 6.3 Australia 1.5 4.0 COPPEUFRJ, Brazil 1.0

19, 51

Vertical

Weichert and Herbst [68]

20, 60, 100

Vertical

Frandrich et al. [43]

25.4

Horizontal

Briggs [73]

20 19, 63, 100

Vertical Vertical

Bourgeois and Banini [45] Tavares and Lima [74]

9.5

Vertical

Tavares and Lima [74]

26

L.M. Tavares

dr

2l C 2l - 2dr C

l

Fig. 14. Scheme of the ILC rod and deconvolution (modiﬁed from Ref. [45]).

essentially planar within a few rod diameters travelled, then the measured trace simply corresponds to the time-delayed signal. In reality, given the very high propagation velocity of the strain wave inside the rod (about 5200 m s–1 in steel), the measured signal is a convoluted version of the actual force–time history experienced by a particle sitting on the top of the rod. This is particularly critical for short rods and whenever the experimenter wishes to describe the entire force–time history of the impact, and not only the force–time history until the instant of primary fracture. Thus, in this case, a deconvolution of the signal may be necessary. If one regards both ends of the ILC rod as free, so that the reﬂection of the wave is perfect in both ends (no signal attenuation), it gives [19] X X 2il 2d r 2il V ðtÞ ¼ V m ðtÞ þ V t V t ð29Þ C C i i where V is the deconvoluted voltage, Vm is the output voltage measured from the bridge. Voltage terms are zero for negative time arguments, l is the length of the ILC rod, and dr is the distance from the contact surface to the strain gauges on the ILC rod (Fig. 14). The shorter the ILC rod, the larger the number of deconvolutions required to completely resolve the measured traces. For example, for a 4.9 m long rod, usually 2–4 deconvolutions are required (summations of up to i ¼ 4 usually sufﬁce in equation (29)) to resolve the entire signal. These may correspond to a time span of up to 10 ms. The application of equation (29) in the deconvolution of a typical voltage–time proﬁle measured during the impact of a particle using the ILC is illustrated in

Breakage of Single Particles: Quasi-Static

27

Fig. 15. Force–time proﬁle measured in the ILC resulting from impact of a 2.4 mm Bingham Canyon copper ore particle.

Fig. 16. Deconvoluted force–time proﬁle of data given in Fig. 15.

Figs. 15 and 16. Despite the introduction of some high-frequency noise, deconvolution satisfactorily allowed the resolution of the entire signal. In some cases, equation (29) does not allow an appropriate description of the entire event. This may be due to damping of the signal, which can be

28

L.M. Tavares

incorporated in equation (29). Thus, a modiﬁcation of equation (29) has been proposed to account for this effect [45] X X 2il 2d r 2il i i i i1 V ðtÞ ¼ V m ðtÞ þ r sV t rs V t ð30Þ C C i i where r and s are damping parameters, so that rZ0 and sr1. Values of 0 stand for complete damping of the signal and values of 1 stand for no damping at all, in which case equation (30) is transformed into equation (29). Bourgeois and Banini [45] observed that appropriate values of these parameters were about 0.98–0.99, which correspond to very little damping, which is characteristic of alloy steels.

6.4. Comminution energy and coefﬁcient of restitution The energy absorbed by a particle during a test in the ILC can be calculated directly using equation (21), which may be rewritten as Z Df Ec ¼ FdD ð31Þ 0

Replacing the deformation on the particle D by the approximate value a, given by equation (28), it gives Z t f 2 Z tf Z tf Z tf 1 1 Ec ¼ vo FðtÞ dt þ g FðtÞt dt FðtÞ dt F 2 ðtÞ dt 2m rA C b r 0 0 0 0 ð32Þ where F(t ) is the force–time history of the impact event and tf is the ﬁnal time of contact. Similar to the pendulum, Ec represents the total energy losses, including the energy consumed in particle breakage, and other losses such as friction, plastic deformation and heat. The coefﬁcient of restitution can also be calculated from an ILC test. Given that the bulk motion of the rod is negligible if compared to the motion of the striker during an impact, the coefﬁcient of restitution of the system, deﬁned as the ratio of the magnitude of the restitution impulse to the impact impulse, is given by 1=2 v1 E1 e ¼ ð33Þ vo Ei which is in direct analogy to equation (16) for the pendulum. While the input energy of the striker is given by equation (7), the residual energy of the striker (E1) requires and energy balance to be established during an impact, given by Ei ¼ Ec þ Er þ E1

ð34Þ

Breakage of Single Particles: Quasi-Static

29

where Er is the energy absorbed in bulk deformation of the rod, calculated by integration of equation (27), giving Z tf 1 Er ¼ F 2 ðtÞ dt ð35Þ rAr C 0 Replacing equations (32), (33) and (35) in (34) gives " Z t f 2 #1=2 Z tf Z tf 1 1 e ¼ 1=2 E i v o FðtÞ dt g FðtÞt dt FðtÞ dt mb 0 0 0 Ei

ð36Þ

A value of e close to one implies a nearly elastic impact, whereas a lower value denotes an inelastic collision, which may either result from plastic deformation of the tools in the case of steel-on-steel impacts or in particle breakage or compaction in impacts on fragments. It is possible to validate equation (36) by comparing its estimates of coefﬁcient of restitution to those obtained by direct experiments. An experimental measurement of the coefﬁcient of restitution essentially requires an estimate of the striker rebound velocity (equation (33)), since its velocity at the instant of impact can be calculated precisely (equation (6)). This rebound velocity of the striker can be estimated by measurement of the maximum height (h1) reached by the ball after bouncing, pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð37Þ v 1 ¼ 2gh1 Figure 17 presents a comparison between experimental measurements of the coefﬁcient of restitution and values given by equation (36) from the measured

Calculated coefficient of restitution

1.0

0.8

Steel-on-steel Bingham Canyon copper ore Karlsruhe quartz Soda-lime glass

0.6

0.4

0.2

0.0 0.0

0.2

0.4

0.6

0.8

1.0

Measured coefficient of restitution

Fig. 17. Comparison of coefﬁcients of restitution measured using high-speed video and calculated from the force–time proﬁle (equation (36)) in the ILC. Data from Ref. [19].

30

L.M. Tavares

force–time curve recorded during an ILC test. Measurements of rebound height for a number of different contact conﬁgurations were conducted with the aid of an EKTAPROs EM motion analyser (Kodak, Inc.) [19] using high-speed video. Experiments were conducted without particles (steel-on-steel impacts) and then with quartz and copper ore particles, as well as glass spheres. Figure 17 shows the very good agreement, which demonstrates that the energy balance calculated using data from ILC experiments is accurate and can be used to satisfactorily estimate the coefﬁcient of restitution. Figure 17 also shows that the coefﬁcients of restitution for steel-on-steel impacts are signiﬁcantly higher than those for impacts involving particles, given that deformations were mainly elastic in the former. Data from ILC tests and equations (32–36) also allow for an energy balance throughout the entire impact test. This is illustrated in Figs. 18 and 19 for quartz and copper ore particles, respectively. The ﬁgures show force–time proﬁles from impact, along with the corresponding energy partitions during the entire tests. Both particles fractured at comparable loads, but quartz suffered little re-breakage in comparison to copper ore because the fragments from primary fracture were ejected at high velocities outside of the crushing zone. As a result, few fragments were nipped between the drop weight and the anvil, mostly resulting in steel-on-steel collision. The less brittle copper ore particle, on the other hand, suffered successive breakage events until all the kinetic energy of the ball was dissipated. Figure 18 shows that a signiﬁcant part of the energy during impact of quartz was either consumed in bulk deformation of the rod (equation (35)) or was recovered by the striker during rebound. As a result, the quartz particle absorbed only 67% of the input energy, in contrast to the 99% absorbed by the copper ore particle under identical loading conditions. Given the deﬁnition of energy transfer efﬁciency (equation (20)) it is also relevant to analyse how these measures compare with those obtained in the twin pendulum, shown previously in Fig. 9. These results are shown in Fig. 20 as a function of the input energy divided (normalized) by the corresponding median particle fracture energy of the material. This normalization is used in order to allow plotting data of materials of very different strengths in a single graph. A value of this ratio equal to one means that the input energy was just high enough to fracture half of the particles present in the sample. Figure 20 shows that in some instances – particularly in impacts at high energies and of highly brittle materials such as quartz and glass – a signiﬁcant fraction of the input energy is recovered in the restitution of the striker. The low energy-transfer efﬁciencies and high coefﬁcients of restitution of elastic and highly brittle materials are due to the fact that their fracture is often associated with the fragments ejecting at high speeds outside the crushing zone. Only a few fragments are then nipped between the ball and the anvil, resulting mostly in steel-on-steel collision, which is, predominantly, elastic. In contrast to that, less brittle and often polycrystalline

Breakage of Single Particles: Quasi-Static

31

400

1.1 mm Quartz

Impact of the ball against the anvil Descending motion of the ball without breakage of fragments

Force (N)

300 Particle primary fracture

200

100

0 0

tc

300

600

900

1200

1500

1800

Time (μs)

tf

2100

4 Ball rebound Input energy: 3.86 mJ

Energy (mJ)

3

2 Particle fracture energy

Rod deformation

1

Comminution energy

0

0

300

600

900

1200

1500

1800

2100

Time (μs)

Fig. 18. Force–time (top) and energy partition (bottom) from impact of a 1.1 mm Karlsruhe quartz particle in the ILC with a 28.3 g ball from a 1.5 cm drop height.

materials, such as iron and copper ores, typically suffer successive breakage events during impact until all the kinetic energy of the ball is consumed either in particle breakage or in agglomeration of the fragments.

7. PARTICLE BREAKAGE CHARACTERISTICS Single-particle breakage tests can yield a number of measures that are useful for characterizing all relevant characteristics of irregularly shaped materials for

32

L.M. Tavares 100 2.4 mm Copper ore

Force (N)

80

Particle primary fracture

60

Rebreakage of the fragments

40

20

0

0

200

400

600 800 Time (μs)

1000

1200

1400

4

3 Energy (mJ)

Input energy: 3.51 mJ Particle fracture energy

2

Comminution energy Rod deformation

1

0 0

200

400

600 800 Time (μs)

1000

1200

1400

Fig. 19. Force–time (top) and energy partition (bottom) from impact of a 2.4 mm Bingham Canyon copper ore particle in the ILC with a 28.3 g ball from a 1.5 cm drop height.

comminution. These measures can be classiﬁed in two major groups: ﬁrstly those that are based on measures taken up to the instant of failure and secondly those based on post-failure of the particle. In the case of regularly shaped test samples, such as cylinders, cubes, spheres and other regular shapes, a number of other measures commonly used in rock mechanics and materials science can also be determined from test samples including compressive, tensile and ﬂexure strength, Young’s modulus and Poisson’s ratio. However, description of these measurements is beyond the scope of this chapter.

Breakage of Single Particles: Quasi-Static

33

Energy transfer efficiency (%)

100

80

60

40

1.0-1.18 mm Karlsruhe quartz 2.0-2.8 mm Bingham Canyon copper ore 2.0-2.8 mm IOCC iron ore

20

0 1

10

100

1000

Input energy/mean particle fracture energy

Fig. 20. Efﬁciency of transfer of energy from the ball to breakage energy for selected materials tested in the ILC as a function of relative impact energy. Error bars correspond to the standard deviations of the observations.

7.1. Particle fracture energy and fracture probability distribution The (primary) particle fracture energy corresponds to the strain energy stored in the particle up to the instant of failure and corresponds to the area below the load–deformation curve, that is calculated by Z Dc E¼ FdD ð38Þ 0

in which D is the particle deformation and Dc is the deformation in the instant of failure. Equation (38) can only be used to calculate the fracture energy of particles tested in compression testers, Hopkinson or ILC devices. In the case of the latter, it is calculated by replacing D in equation (38) with the overall deformation in the vicinity of contact a (obtained by equation (24)), so that Z t c 2 Z tc Z tc Z tc 1 1 E ¼ vo FðtÞ dt þ g FðtÞt dt FðtÞ dt F 2 ðtÞ dt 2mb rAr C 0 0 0 0 ð39Þ where tc is the time of primary fracture of the particle. More often a mass-speciﬁc fracture energy is used, given by E m ¼ E=mp , where mp is the particle weight. Whenever the fracture characteristics of individual particles in a sample in a given size range of a material are measured, a large scatter of the data appears. This is due to the distribution of cracks in size and quantity that exists in each

34

L.M. Tavares

individual particle, as well as due to its shape, in the case of irregularly shaped particles. Such variability must be described in great detail – beyond simple descriptions of mean and standard deviation of the data – by using order statistics. This consists in ranking the test results in ascending order and then assigning i ¼ 1,2,y, N to the ranked observations, where N is the total number of valid tests performed. The cumulative probability distribution for the speciﬁc particle fracture energy is approximated by

PðE m;i Þ ¼

i 0:5 N

ð40Þ

The fracture (also called breakage or crushing) probability describes the likelihood at which particles of a given material and size show characteristic breakage events as a function of the stressing intensity. In this context those irregularly shaped particles which have lost at least 10% of their original mass by breakage are considered as broken [12]. Whenever the experimental apparatus does not allow identifying, and thus, weighing particles individually after the test, they may be considered broken when they become smaller than the smallest size of the initial particles in the sample [35]. Breakage probability distributions are found by stressing a sufﬁciently large number of initial particles (typically at least 100), one by one, of a given material and size with ﬁxed type of stress and determining the ratio of broken to stressed particles as a function of stress intensity (input energy). This tedious process was used by Hildinger [75], Baumgardt et al. [38] and Krogh [23] with the drop weight tester and by other investigators [17,38] with single-impact testers. More recently, Vogel and Peukert [35] proposed the use of slowly fed rotary impactor that rotates at a controlled frequency in order to measure this even more quickly (Fig. 1). Since the breakage probability describes the distribution of the random variable particle fracture energy, the correspondence between the breakage probability distribution and the particle fracture energy distribution becomes evident. This is particularly useful when using equipment that lacks instrumentation and where energy losses are not signiﬁcant, such as the drop weight tester and various single-impact testers (the signiﬁcant energy losses and momentum transfer to the rebound pendulum are likely to limit such measurement using the twin-pendulum apparatus). The validity of this lies in the fact that, at one point during loading, nearly all the stressing energy is available as strain energy of the particle. In the case of the ILC, only a relatively minor proportion has been used in bulk deformation of the rod (Figs. 18 and 19). In the case of using the drop weight test, one has to be aware that an additional source of error is the fact that the speciﬁc energy input represents only an average value, since typically the energy input is maintained constant and the weight of the particle (even those contained in a narrow size fraction) varies in a lot of material contained in a narrow size range.

Breakage of Single Particles: Quasi-Static

35

100 Cumulative broken or distribution (%)

Specific fracture energy Fracture probability

80

60

40

20

0 10

100

1000

Specific input energy or specific fracture energy (J/kg)

Fig. 21. Fracture probability and particle fracture energy distribution of 4.75–4.00 mm Karlsruhe limestone particles. Vertical error bars represent the 90% conﬁdence interval (estimated from the binomial distribution) and horizontal error bars the standard deviations of the speciﬁc input energy, due to variations in particle weight.

A comparison between the distribution of particle fracture energies and the probability of fracture for a limestone sample is shown in Fig. 21, which shows very good correspondence. For some materials, a less-than ideal agreement may be due to either difﬁculty in estimating the fracture probabilities or in the measurement of particle fracture energies. Difﬁculties in the former may be associated with the fact that, for some materials, the integrity of a particle may be lost during an impact but cracks may not have been able to split the particle into pieces, leading to an underestimation of fracture probability. In the case of the latter, difﬁculties may be associated with signiﬁcant subjectivity that is required to identify the point of primary fracture for some materials, which may not be as evident as shown in Figs. 18 and 19. Particle fracture energy (pairs [Em,i, P(Em,i)]) or fracture probability data can be ﬁtted to an appropriate statistical distribution using standard least-squares analysis. Several statistical distributions have been proposed to describe the data. The Weibull distribution has been used by Weichert [76] to describe particle fracture energy data of glass beads. Multiple Weibull distributions have been used to describe particle fracture energy and particle strength data of nearly spherical cement clinker particles [77]. The log-logistic has been used with success to describe particle fracture energy data of irregularly shaped particles [78]. However, the log-normal and the upper-truncated log-normal distribution have been the most commonly used distributions to describe particle fracture energy (or fracture probability) data of a variety of irregularly shaped brittle

36

L.M. Tavares

materials [17,20,38,69]. The log-normal distribution is given by 1 ln E m ln E m50 pﬃﬃﬃ 1 þ erf PðE m Þ ¼ 2 2sE

ð41Þ

where Em50 and sE are the median and the standard deviation of the distribution, respectively.

7.2. Particle strength, PLT strength and KIC The internal state of stresses in individual irregularly shaped particles resulting from the application of loads cannot be calculated directly, so that stress-based measurements of particle strength are only strictly valid for particles of regular shapes. Given the good correlation between tensile strength and crushing behaviour found by Bearman et al. [42], seeking an approximate measure from single-particle breakage data is worthwhile. The strength of a particle cannot be unequivocally deﬁned, as its internal state of stress is not known a priori. Using photoelastic methods, Hiramatsu and Oka [79] showed that the stress states of a sphere, a prism and a cube subject to a pair of concentrated loads are similar. They analysed the stresses of an elastic sphere subject to point-load compression and, after simpliﬁcations, obtained an expression for the tensile strength, called particle strength, which is given by sp ¼

2:8F c pD2

ð42Þ

where Fc is the load responsible for fracture and D is the distance between the loading points. For convenience, D in equation (42) is estimated as the geometric mean of the sieves used to prepare the monosize, that is, the representative size d. The validity of this expression was veriﬁed by Hiramatsu and Oka [79] through comparison of strengths calculated using equation (42) from compression of irregularly shaped specimens and tensile strengths estimated using the Brazilian test, and good correspondence was observed. Another approximate measure of strength is the point-load test (PLT) strength. In the test (Fig. 1), irregularly shaped particles of any shape and size, but with approximately the same linear dimensions, are loaded and the force corresponding to fracture Fc and the equivalent diameter D of the minimal cross-section of the sample are determined. The PLT strength is then calculated by [54] sPLT ¼

Fc D2

ð43Þ

Comparisons between the distributions of particle strengths and PLT strengths for irregularly shaped particles to tensile strengths measured with the Brazilian

Breakage of Single Particles: Quasi-Static

37

99

Compressive strength Tensile strength Particle strength PLT strength

Cumulative distribution (%)

95 90

70 50 30

10 5

1

1

10

100

Strength (MPa)

Fig. 22. Comparison of different measures of strength of particles of Vigne´ sienite (compressive strength on 50 mm diameter and 100 mm length cylinder; tensile strength on 50 mm diameter and 25 mm length cylinder and particle strength and PLT on irregular particles contained in 63–53 mm size fraction).

method show that the variability in strengths measured from irregularly shaped particles is signiﬁcantly greater than that from tests on regular particles (Fig. 22). Measures of particle strength were also found to underestimate the actual tensile strength of the material, whereas measures of PLT strength tend to overestimate. Another strength-based measure that found application in comminution, given its direct correlation with power consumption in crushers [80], is the fracture toughness. Bearman [81] showed that the mode I fracture toughness can be estimated empirically by K Ic ¼

26:56F c ðwDÞ3=4

ð44Þ

where KIc is the mode I fracture toughness (MN m–1.5) and w is the width of the specimen (mm). These various strength-based measures can be determined using compression testers, the SHPB or the ILC.

7.3. Particle stiffness The particle stiffness [20] is determined on the basis of the Hertzian contact theory [82]. The relationship between force and deformation for an elastic

38

L.M. Tavares

spherical or nearly spherical particle compressed between ﬂat platens (such as in a slow compression tester or a double impact tester) presents an apparent work hardening behaviour, and is given by FðtÞ ¼

Kd 1=2 aðtÞ3=2 3

ð45Þ

K is the local deformation coefﬁcient of the Hertzian contact, given by K¼

kpk kp þ k

ð46Þ

where k is the stiffness of the tools (platens in a compression tester, the drop weight and the anvil in the ILC), given by k ¼ Y =ð1 m2 Þ, and kp is the particle stiffness, given by k p ¼ Y p =ð1 m2p Þ where Y is the Young’s modulus and m is the Poisson’s ratio. If the modulus of elasticity and the Poisson’s ratio of the particle are known, the particle stiffness can be directly calculated using this last equation. Otherwise a simple procedure, described as follows, can be used. Substituting equation (45) in (38) and integrating, the particle fracture energy can be related to the deformation at fracture and the local deformation coefﬁcient E¼

2 1=2 5=2 d Kac 15

ð47Þ

Alternatively, K can be related to the critical load and the particle fracture energy by substituting equation (44) in (47) and rearranging, !1=2 F 5c K ¼ 0:576 3 ð48Þ dE then the stiffness of the particle can be calculated by rearranging equation (46) giving kp ¼

Kk kK

ð49Þ

Equations (48) and (49) show that the local deformation coefﬁcient of the Hertzian contact and the particle stiffness can be estimated simply using the critical load and the particle fracture energy. Replacing equation (42) in (48), given that E m ¼ E=ðbrp d 3 Þ, where b is the shape factor and rp is the particle density, then it is given by sp ¼ ðE m brp Þ3=5 K 2=5

ð50Þ

which is valid for slow compression and double impact tests. Tavares et al. [78] demonstrated that the equivalent relationship between particle fracture energies and particle strengths for one-point contact loading

Breakage of Single Particles: Quasi-Static

39

(single impact) is given by sp ¼ ð2 E m brp Þ3=5 K 2=5

ð51Þ

Equations (50) and (51) establish that the relationship between the three measures – the particle speciﬁc fracture energy, the particle strength and the particle stiffness – is independent of particle size, depending only on the number of loading points. Lines representing equation (50) and with a slope 3/5, as well as data for a variety of materials are presented in Fig. 23. This plot gives an indication of the inherent resistance of materials to mechanical size reduction by impact. The ﬁgure shows that the variation in strengths of particles of different materials is very signiﬁcant. This method used to estimate the particle stiffness applies strictly to elastic spherical particles. For irregularly shaped particles, a precise constitutive equation is not deﬁned a priori. Testing of a large number of non-spherical but relatively uniform aspect particles showed that discrepancies to the theory were generally limited to the initial portion of the proﬁles, particularly due to surface crushing. This suggests that a relatively isometric particle can be modelled as a hard brittle core surrounded by a thin crumbly skin. The degree of surface

E'/E

3 0.5 0 0.782 0. 2 0 .9 6 0. 9 8 0. 9 9 0.9 1.0

Particle strength - p (MPa)

100

Particle stiffness kp (GPa) 10

2 00 1000 5 20 10 5

2 1

1 0.01

0.1

1

Em p (J/cm3)

Fig. 23. Relationship between median particle fracture energy and median particle strength for 2.8–2.0 mm particles of various materials measured in the ILC. Diagonal lines, which characterize materials with constant particle stiffness, are calculated from equation (50).

40

L.M. Tavares

roughness has been shown to have a signiﬁcant effect when comparing the stiffness of rough and cubic particles of the same size. Provided that the particles are approximately isometric, equation (48) can be used to estimate the stiffness of irregularly shaped particles. For some materials, however, inelastic deformation immediately before fracture gives rise to a signiﬁcant discrepancy. In that case equation (48) must be modiﬁed to account for the accumulation of damage during deformation of the particles [2]. Particle stiffness measurements have been successfully used to estimate crack-like damage caused by thermal pre-treatment [20]. Measurements of particle stiffness are useful to estimate the extent of the overall deformation that is measured during a stressing event (a) that actually corresponds to deformation of the tools. The extent of that deformation will depend on the mechanical properties of the solids in contact. Assuming perfect elasticity and lubrication of the bodies in contact the ratio between the compression of the particle D and the overall deformation in the vicinity of the contact a can be estimated using Hertz contact theory, which gives [20] D k ¼ a k þ kp

ð52Þ

Equation (52) is particularly relevant to a number of force–deformation tests as it can be used to estimate the fraction of the measured particle fracture energy (E) calculated using, for example, equation (39) that actually occurs in the particle (E0 ). Replacing equation (52) in (47) and rearranging it gives 1 0 E ¼E ð53Þ k p =k þ 1 The ratio E0 /E, which represents the fraction of the measured energy consumed by the particle relative to the tools, calculated with equation (53), is shown in Fig. 23, being constant along the lines of constant particle stiffness. It is evident that, with a few exceptions, generally less than 18% of the total strain energy during the stressing event is consumed by the tools, conﬁrming that the assumption of negligible indentation in the tools used to derive equation (39) is valid under most circumstances for the ILC.

7.4. Energy-speciﬁc progeny size distribution Energy-size reduction relationships can be conveniently investigated using the various single-particle breakage techniques described in this chapter by measuring the size distribution of the progeny fragments at different levels of input energy (Ei). These progeny size distributions, also called energy-speciﬁc breakage functions, can also be related to the amount of energy actually absorbed by the particle during loading, the comminution energy (Ec). In this case,

Breakage of Single Particles: Quasi-Static

41

Cumulative passing (%)

100

Mean particle fracture energy 382 J/kg 750 J/kg 1214 J/kg 1762 J/kg 2387 J/kg 3087 J/kg

10

1 0.1

1 Particle size (mm)

10

Fig. 24. Progeny size distribution from breakage of 5 mm glass spheres under slow compression (data from Ref. [38]).

compression testers, the instrumented pendulum, the SHPB or the ILC may be used to provide the data. A less common, but important, energy-size reduction relationship is obtained by comparing the progeny size distributions to the amount of energy actually required for primary fracture, the particle fracture energy (Em). In this case, progeny fragments from breakage of particles of a given material and size range need to be carefully collected immediately after primary fracture and their size distribution measured while separated in groups of increasing values of particle fracture energy. Figure 24 shows that the progeny size distribution from primary fracture, even for particles of the same material, particle size and shape, varies, given the individual distribution of ﬂaws present in each particle. The greater the strain energy stored in the particle prior to primary fracture the ﬁner the size distribution of the progeny. As previously discussed, in single-particle breakage tests, as in comminution processes, particles are often loaded beyond the point of primary fracture (Figs. 10, 18 and 19). One should, therefore, distinguish between two states: the ﬁrst one limited by the fracture point, at which the primary fragments were created (Fig. 24). In order to analyse how the overall fragmentation process is inﬂuenced by input or comminution energy, the size distribution of progeny fragments of particles subjected to the same (mean) input or comminution energy should be investigated. By conducting experiments at variable energy levels the energysize reduction relationships may be determined. A set of experimental results from such a test is shown in Fig. 25. Besides the evident increase in ﬁnes with the increase in input or comminution energy, these curves are often characterized by the presence of a commonly variable slope with increasing impact energy, which can make data ﬁtting a non-trivial task.

42

L.M. Tavares 100 t10

Cumulative passing (%)

Specific input energy

10 51837 J/kg 5814 J/kg

1773 J/kg

1 623 J/kg 379 J/kg

0.1 0.01

0.1

1

10

Particle size (mm)

Fig. 25. Progeny size distribution from breakage of 4.75–4.00 mm Bingham Canyon copper ore particles at variable energy inputs in a drop weight tester.

Several equations with two, three, four or even more parameters have been proposed to describe the size distribution from single-particle breakage resulting from the use of different impact (or comminution) energies [6,48,83]. Gutsche et al. [62] demonstrated that the progeny distributions of singleparticle breakage in the rigidly mounted roll mill are self-similar, that is, when the cumulative passing is plotted as a function of a re-scale size axis (sieve size divided by the 50% passing size), the curves superimpose. This 50% passing size is then related by a logarithmic relationship to the input energy. It has been shown, however, that this is not applicable to data produced by double impact loading [84]. Klotz and Schubert [85] described progeny size distributions from singleparticle slow compression breakage data using sums of truncated log-normal distributions [83]. This procedure allowed very precise description of the data, but required ﬁtting a large number of parameters. The complexity associated with the varying slope of the progeny size distribution curves found with increasing input or comminution energy led Narayanan and Whiten [46] to propose an alternative description, which became very popular in describing double impact breakage data [24]. Instead of relating mathematically the cumulative passing as a function of progeny size, Narayanan and Whiten proposed to describe the relationship between a number of parameters taken from the size distribution curve. These parameters, called tn, represent the

Breakage of Single Particles: Quasi-Static

43

100

80 tn parameters (%)

t2 60

t4 40

t10 t25

20

t50 t75

0

0

10

20 t10 parameter (%)

30

40

Fig. 26. Relationship between t10 and tns for a copper ore. Symbols are experimental data and lines represent ﬁtting with splines.

fraction smaller than 1/nth of the parent size. One particularly important value is the t parameter corresponding to n ¼ 10, called t10, which is deﬁned as the fraction of the progeny that is smaller than 1/10th of the parent particle size and is taken from the approximately linear portion of the progeny size distribution. Each progeny size distribution has a unique value of t10 and all tn values are related to that individual distribution. At varying levels of input (or comminution) energy these parameters are related, resulting in the so-called family to t curves, illustrated in Fig. 26. These curves are often interpolated using splines [22], but also by incomplete beta functions [86] and truncated Rosin-Rammler or logistic functions [83]. The ﬁnal step to describe the progeny size distribution is establishing the relationship between input or comminution energy and product ﬁneness. This is illustrated in Fig. 27 for the data in Fig. 24. The relationship between the input (or comminution) energy and the t10 parameter has been described by [24] t 10 ¼ A½1 expðbE cs Þ

ð54Þ

with t10 given in percent and Ecs given in kWh t–1 (where 1 kWh t–1 ¼ 3600 J kg–1). Although equation (54) was ﬁrst proposed for use with data from the twin pendulum, Napier-Munn et al. [24] proposed to use it also with data from the drop weight tester, assuming that in this device the input energy may be considered to be equivalent to the comminution energy. Indeed, this is discussed later in this chapter. Parameters A and b characterize the material’s fragmentation behaviour, with A characterizing the limiting value of t10. A*b is the slope of the curve of zero input energy (equation (54)), and can be used to characterize the material’s

44

L.M. Tavares

t10 (%)

100

10

1

0.1

1 1000

10 10000

kWh/t J/kg

Specific input energy - Eis

Fig. 27. Relationship between the parameter t10 and input energy (or comminution energy) for 4.75–4.00 mm Bingham Canyon copper ore particles (1 kWh t–1 ¼ 3600 J kg–1).

amenability to fragmentation by double impact. Values of A*b were found to vary from as low as 10, for materials with very high resistance to fragmentation by impact, to as high as 250, for very weak materials. Figure 28 shows that this product has an approximate inverse relationship with the impact work index (equation (11)). Progeny fragments have been analysed not only regarding their distribution of sizes, but also their shapes. When subjected to different input or comminution energies Unland and Szczelina [57] showed that the proportion of lamellar particles increased with decreasing particle size and increasing comminution energy.

7.5. Energy utilization Energy utilization is deﬁned as the ratio of the new surface area and the energy, which can be the particle fracture energy, the comminution energy or the input energy. The energy utilization is a very good measure for comparing different loading conditions, including type of stressing and stressing intensity and also for comparing breakage of single particles to breakage in particle beds and then to industrial comminution processes. Indeed, it has been shown that single-particle breakage presents higher energy utilization than particle bed breakage, which, in turn have greater energy utilization than industrial comminution. Therefore, single-particle breakage offers a basis for establishing the energy efﬁciency in comminution processes [1].

Breakage of Single Particles: Quasi-Static

45

300 250

A*b

200 150 100 50 0

0

2

4

6 8 10 12 14 16 Impact work index - Wi (kWh/t)

18

20

22

Fig. 28. Comparison of Bond impact work index and Ab for a variety of materials. Values of Ab obtained from testing particles contained in ﬁve narrow size ranges from 13.2 to 63 mm.

Energy utilization presents some important characteristics: Baumgardt et al. [38] demonstrated, using data from experiments in Fig. 24, that the energy utilization for spheres on the basis of the particle fracture energy, is a material constant, even independent of size. Different methods have been used to estimate the surface area of progeny fragments, including estimates from particle size distributions, gas adsorption and permeametry.

8. INFLUENCE OF SELECTED VARIABLES ON PARTICLE BREAKAGE CHARACTERISTICS 8.1. Type of stressing A reasonable number of experimental results exist in the literature dedicated to the comparison of the different stressing methods, that is, single impact, double impact and slow compression [8,9,12,29,38]. However, such comparisons are very often difﬁcult because of the method used for calculating the energy (input or comminution) and because of system-speciﬁc differences, such as different values of hardness and elastic modulus of the tools [38]. Nevertheless, important conclusions can be drawn from these investigations. Initially, it has been shown by several researchers that the different stressing methods give very similar product size distributions [8,9,38]. Further, it has been

46

L.M. Tavares 99

95

Cumulative distribution (%)

90

70 50 30

10

Single impact Double impact Slow compression

5

1

10

100

1000

Specific input energy (J/kg)

Fig. 29. Breakage probability distributions of 12.5 mm ﬁred clay pellets under different stressing conditions (data from Ref. [12]).

observed that breakage probability distributions are inﬂuenced by stressing method. Typical results are shown in Fig. 29 for ﬁred clay pellets (12.5 mm) stressed under different conditions. It is found that breakage probability increases from slow compression to double impact and then to single impact, so that slow compression yielded the highest fracture energies and thus lowest breakage probabilities for a given input energy [12]. Similar trends have also been reported for glass spheres [38], while results showing the opposite behaviour were reported by Arbiter et al. [9] for sand-cement and glass spheres. These differences between double impact and slow compression can be explained on the basis of deformation rate effects, discussed in greater detail later in this section, as well as on differences in hardness and elastic modulus of the tools used in the different devices. On the other hand, differences between single impact (one-point loading) and double impact/slow compression (two-point loading) may be explained on the basis of a number of effects, which include: The state of stresses in one-point and two-point loading differs signiﬁcantly.

From Hertz contact theory (equations (50) and (51)) it is demonstrated that twice as much energy is required to reach the same maximum level of tensile stresses within the particle in two-point loading than in one-point loading, if one considers elastic deformations only [78]. For a uniform distribution of ﬂaws in the particle, two-point loading would require twice the amount of energy for breakage than one-point loading.

Breakage of Single Particles: Quasi-Static

47

At a given stressing level, the probability of ﬁnding a ﬂaw of critical size in two-

point loading is expected to be higher when compared to a single loading point, when the distribution of ﬂaws in the particle is not uniform and the former presents a larger number of contact points. This would lead to increased breakage probabilities for double impact and slow compression in comparison to single impact. When loading particles of irregular shapes, still another effect contributes to a reduction of the differences found: in two-point loading tests particles are positioned in their most stable position, so that their smallest dimension is in the direction of the application of stresses. From the known effect of particle shape on breakage characteristics (seen later in this section) these particles are likely to require less energy to fracture in this direction. On the other hand, in the case of single impact, particles rotate during ﬂight so that no guarantee exists of which will be the particle position when it reaches the target. As a result, particles may appear stronger, and thus have a lower breakage probability on single impact than on double impact and slow compression. Finally, a different trend from the one observed in breakage probability is found in regard to energy utilization for the testing methods, with slow compression being generally found to yield higher energy utilization than single impact [1,12]. This is ﬁrst explained by the different methods used to determine energy consumption. For compression testing, it is calculated from integration of the force–deformation curve, whereas in single impact and double impact tests the input energy is generally used. In single impact, progeny fragments ﬂy off with a certain velocity and therefore only a part of the initial input energy is consumed by the particle breakage. The small chance for using the remaining part of the kinetic energy by secondary impacts results in lower efﬁciency of single impact in comparison to double impact and slow compression. From the discussion above, the signiﬁcance of the type of stressing on singleparticle breakage is such that the investigation method should be suited to the industrial application sought. Use of single-particle breakage data from one type of stressing method to describe comminution processes where the predominant type of stressing is different can lead to poor results.

8.2. Stressing intensity The stressing intensity, given by the input or comminution energy, is the most signiﬁcant variable controlling the progeny size distribution from single-particle breakage and has been the variable most intensively investigated. In the case of single impact, the energy beyond which is necessary just to break the particle (primary fracture) by a single fracture event is still contained in

48

L.M. Tavares

the fragments as kinetic energy. This energy can cause secondary breakage, especially since the main vector of the kinetic energy in the fragments is directed towards the surface. In the case of double impact, the remaining kinetic energy in the drop weight or the impact pendulum beyond which is necessary for primary fracture continues to crush the fragments, causing secondary breakage, until all the kinetic energy of the striker is dissipated or until the applied load is equilibrated by the response force from the pile of fragments. Figure 26 illustrates the inﬂuence of impact energy on the progeny size distribution of a copper ore. Increasing degrees of ﬁneness are obtained by raising the magnitude of the input energy, however, particles of inﬁnite ﬁneness are not produced in a single loading event by increasing indeﬁnitely the stressing energy, so that a limiting progeny size distribution is reached. This limiting size distribution is found to depend on the spatial distribution of fragments from the ﬁrst few fracture events, loading geometry, loading rate and on the material’s amenability to agglomerate. The relationship between the input (or comminution) energy and the energy utilization is such that at low input or comminution energies, insufﬁcient energy may be provided to the particle, not causing fragmentation; at high energies, the same inefﬁciency observed in Fig. 26 is observed, mainly due to losses as friction, etc. (Fig. 30). The result is that a maximum energy utilization exists, which typically corresponds to input energies responsible for about 100% fracture probability [12,17]. However, more recently Tavares [87] demonstrated that the optimum input energy considering multiple stressing events (breakage of the parent particles, followed by selection and re-breakage of progeny fragments)

Energy utilization (cm2/J)

100

80

60

40

20

0 10

100 1000 10000 Specific input energy -Eis (J/kg)

100000

Fig. 30. Energy utilization as a function of input energy for 4.8 mm quartz particles broken using a drop weight tester (data from Ref. [10]).

Breakage of Single Particles: Quasi-Static

49

required to reach a given ﬁnal product size has optimum conditions at a breakage probability from 75% to about 95%, depending on material. The difference between this and the 100% fracture probability associated with maximum energy utilization is due to the fact that in multiple-event breakage, stressing events that were not able to fracture the particle, only damage it, may be responsible to make it more amenable to fracture in a future stressing event [88]. It is worth noting that this reduction in energy utilization with the increase in comminution or input energy has been observed with all stressing methods, including single impact, double impact [10] and slow compression [12], being more signiﬁcant for the former.

8.3. Stressing and deformation rate Deformation (or stressing) rate can inﬂuence the stress ﬁeld within a particle and crack propagation behaviour, and thus particle breakage, due to the effect of elastic waves and by strain rate sensitivity of deformation and crack propagation behaviour. The effect of elastic waves is associated with the propagation of waves inside the solid, since, although loads are applied locally, they are distributed inside the particle as deformation waves. In these conditions the stress ﬁeld within the particle may change due to interference of internal reﬂections. This effect is noticeable only when the impact time is shorter than the time required for a wave to travel through the particle, that is, when the ratio between wave propagation velocity and loading velocity is larger than one. Scho¨nert [1] observed that in order to satisfy this condition, the impact velocity must be higher than about 100 m s–1, as one can calculate from the Hertzian equations. Literature in rock mechanics states that effects of loading rate are also generally associated with the crack propagation behaviour: at low loading rates only the larger (and more critical) cracks are responsible for failure of the solid; on the other hand, at high loading rates, several cracks are responsible for the simultaneous propagation (even coalescence) of cracks, given the incapacity of a single crack – which presents a limited velocity of propagation – of relieving the tensile stresses [89]. This effect may be observed when the ratio between the crack propagation velocity and the loading velocity is larger than one. It is important to analyse the validity of these various arguments from the standpoint of particle breakage. In fact, Gildemeister and Scho¨nert [90] demonstrated that the onset of crack propagation occurred at high impact velocities of spherical particles. However, they also demonstrated that for irregularly shaped particles and within the interval of deformation rates of interest in most singleparticle breakage tests and comminution equipment (typically smaller than 100 m s–1) it is very unlikely that wave reﬂection effects will inﬂuence crack

50

L.M. Tavares

propagation and, thus, change particle breakage. On the other hand, it is important to note that crack propagation effects associated with strain rates may be relevant in single-particle breakage. Tavares and King [2] suggested that some materials show inelastic response due to gradual damage accumulation – growth of a network of cracks – prior to fracture, exhibiting lower (subcritical) net crack growth velocity. The result is that stressing rate effects may be present for this type of material. Still, in the quasi-static conditions used in the single-particle tests of interest in the present chapter, it is unlikely that wave effects play any signiﬁcant role on particle breakage behaviour. Strain rate effects are often also associated with the deformation response of solids to applied loads. Indeed, inelastic behaviour, especially in the case of viscous inelasticity, depends heavily on strain rate. The inelastic part of the deformation and the relaxation is reduced with increasing strain rate. During stressing for very short time periods, the material reacts almost as a brittle solid. For such materials, which is the case of synthetic polymers, the deformation velocity has a great inﬂuence on particle breakage [1]. These materials can be ground much more effectively only with impact mills and whenever possible at very low temperatures [1]. Inelastic response due to accumulation of damage is also critically inﬂuenced by deformation rate [89]. There have been a number of experimental investigations on deformation rate effects in particulate materials. Yashima and co-workers [91] carried out the most comprehensive investigation on the subject to date. By using a number of singleparticle breakage testing equipment (slow compression press, instrumented drop-weight tester and Hopkinson pressure bar) they were able to measure the response of spheres of various materials to loading at different rates, from low speed (0.025 kg s–1) to dynamic loading (500 kg s–1) under two-point loading. They observed that materials respond more rigidly (with higher stiffness) when subject to higher loading rates. It was also found that materials present higher particle strength at higher rates. The combination of these effects – evident from equation (50) – resulted in variable inﬂuences of loading rate on particle fracture energy: for glass (silica and borosilicate) particle fracture energies decreased with loading rate, whereas for quartz and feldspar they remained relatively constant, increasing for limestone, marble, gypsum and talc. Figure 31 show experimental results from slow compression and the ILC on the effect of deformation velocity on particle fracture energy. It is observed, as was previously shown by Krogh [23] and Tavares and King [20] that, within the range of conditions commonly covered in double impact testing, breakage characteristics are not likely to affect particle breakage response. Therefore, within the range of deformation velocities covered in the single-particle breakage tests discussed in this chapter, it may be concluded that the deformation rate effect is probably of limited importance, only requiring to distinguish between slow

Breakage of Single Particles: Quasi-Static

51

Median specific fracture energy (J/kg)

1000

100

Slow compression

Impact load cell

10 0

1

2

3

4

5

6

7

Impact velocity - vo (m/s)

Fig. 31. Inﬂuence of deformation velocity on median mass-speciﬁc fracture energy for a sample of 16.0–13.2 mm Paragominas bauxite particles tested in the ILC. Error bars denote the 90% conﬁdence interval.

compression and double impact. Any distinction in material response between these and single-impact tests may be explained not only on the basis of deformation rate effects, but also on the type of loading, as discussed earlier.

8.4. Particle size Particle size affects a number of materials breakage characteristics. Figure 32 shows that particle fracture energy distributions are strongly affected by particle size, so that a decrease in the particle size resulted in a shift of the distributions to higher values. Similar results are observed for particle strength and PLT strength, but results are omitted for brevity. This increase in strength with the decrease in size is commonly observed in brittle materials and is due to the fact that ﬂaws, pores and grain boundaries are embodied in any solid material and especially in geological materials. These structural inhomogeneities cause stress concentrations that result in inelastic deformations and cracks [1]. Thus, they determine particle strength, particle fracture energy and the size distribution of the progeny. The larger the ﬂaw, the smaller the stress needed for crack release. Since size and number of ﬂaws decrease with decreasing particle size, the stress has to be increased in order to break ﬁner particles. Eventually the yield strength will

52

L.M. Tavares 99.9 90.0 - 75.0 mm 45.0 - 37.5 mm 16.0 - 13.2 mm 5.60 - 4.75 mm 2.83 - 2.36 mm 0.70 - 0.59 mm

Cumulative distribution (%)

99

90 70 50 30 10

1 0.1

1

10

100

1000

10000

Specific particle fracture energy (J/kg)

Fig. 32. Breakage probability distributions of particles of different sizes of Vigne´ sienite.

become greater than the fracture strength, thus reaching the brittle-plastic transition region. Particle fracture energy distributions of a variety of materials over a range of sizes have been measured [16,20,43] using different single-particle breakage testing equipment. Tavares and King [20] report results on the effect of particle size, in the range of 0.3 to about 15 mm, while Frandrich et al. [43] in the range of 2 and 90 mm in the ILC. On the other hand, Scho¨nert [1], Yashima et al. [14,16] and Unland and Sczelina [56] presented results on the inﬂuence of particle size on fracture characteristics of materials over the range of a nearly half a metre to a few microns by slow compression. Typical results that illustrate the effect of particle size on particle fracture energy are presented in Fig. 33. It is evident that data for ores at ﬁner sizes and for minerals can be generally well described by a power law. Power law relationships between particle fracture energy and particle size were derived by Yashima et al. [16] and by Weichert [76] based on Hertzian contact theory and Weibull’s weakest link criterion [92]. Indeed, recently Vogel and Peukert [35] considered Weibull’s criterion and Rumpf’s similarity law of fracture mechanics [10], and thus the validity of Rittinger’s law, to suggest that the energy corresponding to 50% fracture probability (median particle fracture energy) varies inversely with particle size (E m50 / 1=d). Careful inspection of Fig. 32, however, shows that as particle size increases, the measured energies often tend towards a constant, materialspeciﬁc value. This limiting constant value of fracture energy at coarser sizes is

Mean specific fracture energy - Em50 (J/kg)

Breakage of Single Particles: Quasi-Static

53

1000

100

10

1 0.1

Karlsruhe quartz Apatite Galena Paragominas bauxite Santa Luzia gneisse Sossego copper ore

1

10

100

Particle size- d (mm)

Fig. 33. Variation of median particle fracture energy with particle size for various materials measured in the impact load cell.

consistent with the validity of Kick’s law of comminution. A model based on reliability theory [47] that describes the data from Fig. 33 well is " f # do E m50 ¼ E m;1 1 þ ð55Þ d dp with Em,N, do, dp and f being material constants, where Em,N represents the residual fracture energy of the material at coarse sizes and do is a characteristic size of the material microstructure and dp is the particle size below which deformation becomes predominantly plastic. Equation (55) has been ﬁt to particle fracture energy data on a variety of materials over a range of sizes and results are summarized in Table 2. Values of f were found to vary below one and about 2.5. The residual particle fracture energy Em,N of minerals is generally found to be consistently lower than that of ores and rocks, which indicates the higher toughness of the latter. Also, the characteristic size do is signiﬁcantly coarser for minerals than polycrystalline materials, such as ores and rocks, which reﬂects the ﬁner microstructures of the latter. A discussion on the role of microstructure on particle breakage characteristics is presented elsewhere [93]. Scho¨nert [1] stated that increasing inelastic deformation in the contact volume is partially responsible for the strong size inﬂuence on fracture strength of particles below a few millimetres. Thus, there is a transition size range, below which the particles deform mainly inelastically. This change in deformation behaviour becomes evident from the force–deformation curve [1,26]. The transition range

54

L.M. Tavares

Table 2. Parameters describing the effect of particle size on breakage characteristics of selected materials

Equation (55)

Material Apatite Kalsruhe quartz Uintah Basin gilsonite Paragominas bauxite Bingham Canyon copper ore Cyprus Sierrita copper ore IOCC iron ore Santa Luzia gneisse Vigne´ sienite Utah marble

Em,N (J kg–1)

f

Size range (mm)

A (%)

b0

19.3 3.48 7.07 14.6 1.17

1.62 1.61 1.60 0.91 1.26

0.25–8.00 0.25–4.75 1.18–10.0 0.50–75.0 0.25–15.8

45.4 38.8 – – 44.8

0.0115 0.0176 – – 0.0263

170.9

1.37

1.41

0.50–12.5

58.9

0.0204

47.3 48.7 26.5 45.9

1.08 2.78 101. 0.882

2.30 1.82 0.67 1.76

0.25–15.0 0.50–90.0 0.50–90.0 0.35–10.0

65.4 54.8 66.2 76.3

0.0932 0.0274 0.0146 0.0792

1.05 43.4 5.50 70.3 96.1

do (mm)

Equation (56)

Table 3. Materials and brittle-plastic transition range from slow compression [1]

Material

Particle size range (mm)

Boron carbide Quartz Limestone Cement clinker Marble Coal

2–3 4–6 5–10 10–22 15–30 30–40

has been measured for several materials and Table 3 shows a summary for selected materials. It shows that this size approximately increases with a decrease in material’s hardness. This brittle-plastic transition size range is important because if one wants to activate mechanically a material, grinding must progress below this size in order to ensure destruction of its crystalline structure [1]. Figure 34 shows the effect of particle size on the mean particle stiffness for a number of materials. Unlike particle strength and particle fracture energy, particle stiffness data for all sizes fall within a narrow band, suggesting that stiffness is a material property and is essentially independent of particle size. This independence of size on particle stiffness has also been observed for other materials [20]. In general, the data shows a slight decrease in stiffness as particle size increases

Breakage of Single Particles: Quasi-Static

55

Median particle stiffness - kp (GPa)

100

10

1 Karlsruhe quartz Apatite Paragominas bauxite Utah gilsonite Vigné sienite

0.1

0.1

1

10

100

Particle size - d (mm)

Fig. 34. Variation of median particle stiffness with particle size for various materials measured in the impact load cell.

although this trend appears to be reversed for the copper ore that was studied. At a micro scale level, the modulus of elasticity (and stiffness) depends on the atomic and molecular structure and is an intrinsic property of the material [94]. At the macroscopic level, however, the stiffness depends upon the cumulative compliance of structure elements and therefore would be affected by microstructural features such as pores, cracks, grain boundaries, parting planes, etc. Summarizing, particle fracture energy and particle strength are structuresensitive properties as they are strongly affected by the presence of critical ﬂaws and cracks in zones of high stress in the material. As particle size decreases cracks progressively disappear which results in increases in both strength and fracture energy. Particle stiffness, on the other hand, is not a structure-sensitive property, as it depends only on the cumulative effect of the deformations in individual portions of the particle. As a result it can also increase or remain constant with a reduction in particle size. In addition to fracture strength, particle size also inﬂuences signiﬁcantly progeny size distribution and energy utilization in single-particle breakage. Fig. 35 shows a relationship between t10 and Eis (or Ecs) for different particle sizes of apatite, besides different input energies. The difference observed among the curves is expressed in the parameter b in equation (54). It is then proposed to rewrite equation (54), considering that the parameter b varies with particle size ðb ¼ b0 E m50 Þ, so that b0 E cs t 10 ¼ A 1 exp ð56Þ E m50

56

L.M. Tavares 100

t10 (%)

2.80-2.00 mm 4.75-4.00 mm 8.00-6.70 mm

10

1 0.01

0.1

1

10

Specific input energy - Eis(kWh/t)

Fig. 35. Variation of t10 as a function of particle size an impact energy for apatite.

t10 (%)

100

10

2.80-2.00 mm 4.75-4.00 mm 8.00-5.60 mm

1

10

100

Input energy / median particle fracture energy

Fig. 36. Variation of t10 as a function of the ratio between the impact energy and particle fracture energy for apatite.

This effect of particle size on size distribution of the progeny may be described by plotting the data in Fig. 35 as a function of a rescaled x-axis, given by the ratio between the input (or comminution) energy and the median particle fracture energy (Fig. 36). A summary of the parameters for a number of materials is shown in Table 2.

Breakage of Single Particles: Quasi-Static

57

8.5. Particle shape Particle shape has a marked effect on a number of measures of particle breakage, since it directly inﬂuences the state of stresses inside the particle. A number of authors demonstrated that the energy utilization varies with particle shape [10,12]. It has been found that the energy utilization for glass spheres is independent of comminution energy, being equal to about 0.003 m2 J–1 [12]. This constant energy utilization has been considered to be a demonstration of Rumpf’s law of similarity of fracture mechanics [10]. With irregularly shaped glass particles loaded by slow compression, energy utilization has been found to vary, decreasing with an increase in comminution energy [12]. This reduction in energy utilization with irregularly shaped particles at higher comminution energies is attributed to be the result of friction work due to the contribution of superﬁcial fractures (surface abrasion) [12]. Kenny and Piret [95] investigated the strength of glass as a function of particle shape by slow compression. Tavares and King [20] investigated the inﬂuence of particle shape on the fracture characteristics of quartz particles tested in the ILC. These results (Table 4) demonstrate that particle strength and particle stiffness decrease signiﬁcantly as particle shapes become more irregular, whereas particle fracture energy is not signiﬁcantly affected. This reduction in particle strength may be partially explained by the fact that it is customary to determine the distance from the loading points (D) in equation (42) as the geometric mean sieve sizes (d) of the initial particle. For the case of lamellar particles this results in an overestimation of D, and thus an underestimation of the particle strength. The variation of particle stiffness appears to result from the limitation of the Hertzian contact theory to describe deformation of non-isometric particles. This sensitivity of particle stiffness to shape shows that it has only comparative signiﬁcance for samples with statistically similar shapes. Particle shape also inﬂuences progeny size distribution from single-particle breakage. It has been demonstrated [24] that ﬁner progeny results from breakage Table 4. Effect of particle shape on fracture characteristics of 1.00–1.18 mm quartz

Particle fracture energy

Particle strength

Particle stiffness

Particle shape

Shape Em50 factor b (J kg–1)

s2E

sp50 (MPa)

s2s

kp50 (GPa)

s2k

Rounded Isometric Flaky

0.476 0.373 0.305

0.364 0.345 0.433

63.5 39.0 27.4

0.164 0.268 0.196

57.7 34.2 12.1

0.262 0.540 0.563

381.1 328.1 364.8

58

L.M. Tavares 100

t10 (%)

Flaky

10

1 0.1

Nonflaky

1

Specific input energy - Eis (kWh/t)

Fig. 37. Effect of shape on breakage characteristics of basalt (data from Ref. [24]).

of ﬂaky (lamellar) particles in the drop weight tester when compared to isometric (non-ﬂaky) ones (Fig. 37).

8.6. Moisture content A very limited number of investigations have dealt with the inﬂuence of the environment on the breakage behaviour of single particles. It is known that the formation of cracks in glass may be signiﬁcantly inﬂuenced by humidity. Given sufﬁcient time, hydrolytic stress corrosion reactions occur, which greatly favour cracking. Scho¨nert et al. [96] demonstrated that the crack extension energy varies with environment. The experiments were carried out in water, humid air and in high vacuum. At very low crack propagation speeds a much lower value of crack extension energy than in a high vacuum was sufﬁcient to propagate the crack. In water there may be still other effects, such as capillary action. Therefore, that study demonstrated that at low crack propagation speeds, signiﬁcant effects of environment on crack extension energy are to be expected. However, at the higher crack propagation velocities expected in comminution equipment and in the single-particle breakage tests described in the present chapter, Scho¨nert concluded that it is unlikely that environment effects would be able to inﬂuence signiﬁcantly crack propagation behaviour. Yashima et al. [97] observed that both the particle strength and the particle fracture energy decrease when the medium is changed from vacuum to air, and then further when it is changed to water.

Breakage of Single Particles: Quasi-Static

59

Table 5. Effect of moisture content on breakage characteristics of bauxite particles (16.0–13.2 mm) tested in the ILC at a loading velocity of 3.2 m s–1

Speciﬁc fracture energya

Particle strength

Particle stiffness

Moisture content (%)

Em50 (J kg–1)

s2E

sp50 (MPa)

s2s

kp50 (GPa)

s2k

0.0 2.1 8.1

186.7 256.4 269.1

0.504 0.535 0.602

9.58 10.85 11.66

0.140 0.142 0.140

2.46 2.25 2.10

0.500 0.930 1.193

a

Dry basis.

Experiments have also been conducted by the author in the ILC with bauxite particles in order to assess particle breakage behaviour under different conditions: oven dried, superﬁcially wetted by rapid immersion in water immediately prior to testing, and tested after immersion in water for a period of one day. A summary of the results is given in Table 5. A signiﬁcant increase in moisture content is evident, which is not expected for most hard-rocks and minerals but is not uncommon for bauxite, given its large internal porosity. Table 5 shows that the increase in moisture content results in increases in both median strength and median particle fracture energy. This signiﬁcant increase, which contradicts data from Yashima et al. [97], may be explained by the combined effect of the high porosity of the material and the high proportion of clay materials, which intensiﬁes the plastic response of the ore, when wet. The moisture content also probably inﬂuences the post-primary fracture behaviour and, therefore, the progeny size distribution. Very dry or materials embedded in water are less likely to agglomerate than particles containing just a small moisture content.

9. APPLICATION TO COMMINUTION Recently, interest has been renewed in the investigation of single-particle breakage as the basis of comminution processes. Some of the attempts to apply data generated in these experiments to industrial comminution are discussed as follows. Single-particle breakage data can be applied almost directly to crushers, as particles are stressed mainly individually or in single layers. This is the case of jaw and gyratory crushers, where a throw is applied on each particle no matter how much energy is required or how high the crushing forces necessary [56]. As a result, single-particle breakage tests have been used in the design of these

60

L.M. Tavares

types of crushers. Examples are the use of PLT strength [56] and of Bond crushability work index [52] in crusher selection and calculation of power draw. Point load–deformation data, along with a simple beam model of the swing plate motion, have been used by Dowding and Lytwynyshyn [57] to determine the maximum force exerted by the toggle of jaw crushers. This allowed them to match the stiffness of the plates to the rock to be crushed, thus reducing energy consumption in crushing. Energy-size reduction relationships from single-particle breakage tests using the instrumented twin pendulum, and more recently, the drop weight tester have been successfully used in the Julius Kruttschnitt Mineral Research Centre (JKMRC) to calculate the breakage function for modelling crushers and also to estimate their power draw [24]. Data from these tests have also been used to calculate the breakage function in vertical shaft impact crushers, from the velocity at the rotor tip [24]. Yashima and co-workers [16] used single-particle fracture data obtained from slow compression and, with the assumption of full conversion of kinetic energy to elastic strain energy and no wave propagation effects, attempted to predict velocities for breakage in jet and impact mills. In the case of crushers and impact or (to a lesser extent) jet mills, where particle interaction is not so signiﬁcant, single-particle breakage data can be used almost directly. In several comminution machines particles are stressed in assemblies so that particle interactions become relevant. Particle–particle interactions are particularly signiﬁcant in high-pressure roll grinding, where particles are loaded in conﬁned beds. In the ball mill as in other tumbling mills, particle interactions are not as severe, as comminution occurs primarily in unconstrained particle beds. In fact, Ho¨fﬂer [71] observed that during impact most of the material is ejected from the bed volume so that the active breakage zone is actually very limited. Single-particle breakage data have been directly used to predict the energyspeciﬁc breakage function of a number of mill types, including rod, ball, autogenous and semi-autogenous mills at the JKMRC. The population balance model formulation, implemented in the mineral processing plant simulator JKSimMets, and the use of particle breakage data, are partially responsible for greater popularity achieved by single-particle breakage testing today in the minerals industry. These models, however, still rely heavily on empirical data for scale-up of the breakage rate functions. A comparatively smaller number of studies have dealt with the calculation of not only the breakage function but also the breakage rates (selection function) of comminution equipment from single-particle breakage data. Researchers at the University of Utah [83,98] used single-particle and particle-bed breakage data, along with media motion simulations using the discrete element method – DEM – [99] to predict ball milling. Breakage distribution functions were calculated from the spectrum of impact energies and the progeny size distributions from single-particle breakage in the ILC or a drop weight tester. Breakage rate

Breakage of Single Particles: Quasi-Static

61

functions were calculated using a combination of the distribution of particle fracture energies from ILC tests and the impact energy spectrum from DEM. In both functions, it was taken into account the energy distribution among single particles within a particle bed [83]. In spite of their great potential, this approach has not yet been validated using either laboratory or plant data. Additionally, ongoing work in the author’s laboratory [30] aims to describe the combined use of ILC, drop tests and media motion using DEM [99] to predict selfbreakage of lumps for modelling autogenous and semi-autogenous mills. Several investigations have dealt with the application of single-particle breakage tests to predicting ore degradation during handling. Weedon and Wilson [100] used the instrumented twin pendulum to predict iron ore degradation, whereas Sahoo [48] used it to predict coal degradation due to handling. The results of these tests, along with simple models [101] built in simulators such as JKSimMets can be used to predict degradation when the material is dropped from a signiﬁcant height, usually during transportation and handling, as well as by abrasion due to gravity ﬂow through bins. Data from testing single particles in the rigidly mounted roll (Fig. 1), along with data from bed breakage tests have been used to describe comminution in highpressure grinding rolls (HPGR) [61–63]. However, establishing the link between single-particle breakage and mill performance is more difﬁcult in high-pressure roll grinding, given the severe interaction effects that exist in the conﬁned bed, which results in waste of as much as 50% of the mill energy in interparticle friction and agglomeration of the product. Even so, Morrell et al. [102] demonstrated that single-particle impact-breakage data from the drop weight tester, combined with piston-press particle-bed breakage data can be successfully used to model highpressure roll grinding. In that case, single-particle breakage data are used to describe areas in the HPGR where the breakage is of a single-particle nature, in particular to describe edge effects. Single-particle breakage also offers useful data for calculating the energy efﬁciency of comminution processes. Measurements of energy utilization have been used in a number of studies to compare not only the different single-particle stressing methods, but also to compare particle bed breakage, as well as comminution in a number of mills [1,10,103]. Recognizing that most often product speciﬁcation is determined on the basis of a ﬁnal passing size of the product, not surface area, Scho¨nert [104] used a procedure that allows calculating the energy required to reach a given product size. While Scho¨nert [104] used single-particle breakage data by slow compression, Tavares [87] recently used single-particle breakage data from the ILC. These studies have shown that optimal breakage conditions are reached by approximately matching the input energy and the fracture energy of the particle. If the input energy is lower than the particle fracture energy, breakage does not occur and the particle may only be damaged [88]. If the stressing energy is larger than the particle fracture energy, then the excess energy

62

L.M. Tavares

Table 6. Deformation rates in different comminution machines

Comminution machine

Deformation velocity (m s–1)

Fluid energy mills Impact crushers Ball, rod, autogenous and semi-autogenous mills Gyratory and cone crusher Jaw crusher High pressure grinding rolls

Up to 400 20 to 200 Up to 15 0.1 to 2 0.1 to 1 Up to 0.1

can only be used for secondary breakage events of the primary fragments, however, with a lower efﬁciency [12]. In light of this, the comminution of particles between the working surfaces of jaw, cone and roll crushers as well as the comminution at the impact elements of impact crushers or at the hammers of hammer mills are advantageous variants if in the last two cases the impact velocity is adjusted to the strength of the particles. At the working surfaces of these machines each particle absorbs only as much energy as it needs for breakage, i.e., the particle fracture energy. In contrast to this, by the tumbling bodies in media mills, energy is impressed on the particles, which is commonly higher or lower than the particle fracture energy. In this case, it is important to reconcile the stress energy distribution as well as possible to the breakage probability distribution of the particles to be comminuted by choice of appropriate process parameters [88]. In principle, a successful application of single-particle breakage to modelling of an industrial comminution process requires that the predominant stressing conditions in the mill are matched by the method used in testing single particles in the laboratory. Given the importance of deformation rate in the fracture strength of some materials, the deformation velocity in the single-particle breakage test should ideally match that of the comminution process to be described (Table 6). Further, given the importance of the type of stressing, the appropriate type of single-particle testing method should be chosen, and, considering the important effect of particle size on a number of particle breakage characteristics, the range of particle sizes that are tested individually should essentially match the one that is actually comminuted in the industrial equipment. Use of data from breakage of coarse particles to modelling comminution processes that deal with ﬁner particles can lead, for example, to an overestimation of capacity or underestimation of energy consumption.

ACKNOWLEDGEMENTS The author would like to express his gratitude to Professor R.P. King and a number of fellows from the University of Utah and the Universidade Federal do

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Rio de Janeiro, who co-authored some of the publications reported throughout the chapter. The author is also grateful to the Brazilian agencies CNPq, FINEP and FAPERJ, which supported most of the original work presented here.

APPENDIX: DEFINITION OF TERMS Given the importance of energy in single-particle breakage, a deﬁnition of terms is important. The term particle fracture energy has been used in the present chapter to represent the strain energy that is stored in the particle from the instant of contact until failure of the particle, and should not be confused with similar terms used in fracture mechanics literature [28,29]. It is the minimum amount of energy required to fracture the particle under the particular orientation and loading conditions of the test. Since it depends on orientation and loading rate, and on the location, orientation and size of internal microﬂaws, any one particle does not have a unique particle fracture energy. This quantity is called (mass) speciﬁc fracture energy by Yashima et al. [16], comminution energy by Baumgardt et al. [38] and breakage energy by Scho¨nert [1]. The term speciﬁc breakage energy is reserved for the energy consumed during the entire single-particle breakage event including both primary fracture and all subsidiary fracture events that occur during a loading cycle in a particle, such as an impact. This quantity is called comminution energy by Narayanan and Whiten [22], a designation that has also been used as an alternative for breakage energy in the present chapter, in spite of the fact that it would be more appropriately used to represent the total energy that is required to reduce a given material from a feed size distribution to the ﬁnal ground product [20]. The term input energy is the amount of energy that is introduced in the system, such as the kinetic energy of the striker. This quantity is also called load energy by Baumgardt et al. [38] or stress energy by Schubert [12].

Nomenclature

A Ar aT b b0 bT C

parameter of equation relating t10 and Ecs – equation (54) (%) cross-sectional area of the ILC rod (m2) constant of the equation of the period of the pendulum (equation (12)) (–) parameter of equation relating t10 and Ecs – equation (54) (kWh1 t) parameter of equation (56) (–) constant of the equation of the period of the pendulum (equation (12)) (–) wave propagation velocity in the ILC rod (m s1)

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CB

impact energy per unit thickness, also called crushing resistance (J mm1) distance between loading points (m) representative particle size (m) characteristic size of the microstructure in equation (55) (m) particle size below which deformations become predominantly plastic (m) distance from the impact face to the strain gauge position (m) outer diameter of the rotor in the rotary impact tester (m) particle fracture energy (J) coefﬁcient of restitution (–) fracture energy associated with the deformation only on the particle (J) comminution or breakage energy (J) speciﬁc comminution energy (J kg1; kWh t1) input energy (J) speciﬁc input energy (J kg1) speciﬁc particle fracture energy (J kg1) residual particle fracture energy (J kg1) energy absorbed in bulk deformation of the rod (J) residual energies of the rebound and the input pendulums, respectively (J) residual energy of the drop weight (J) load on the particle (N) critical force at primary fracture (N) force on the drop weight (N) force on the rod (N) acceleration due to gravity (m s2) gauge factor (–) height of the pendulums after collision (m) net drop height (m) maximum height of the drop weight after impact (m) local deformation coefﬁcient of the Hertzian contact (GPa) stiffness of the tools (platens of rod and drop weight) (GPa) particle stiffness (GPa) mode I fracture toughness (MN m1.5) distance from the axis of rotation to the centre of gravity of the hammer (m) length of the ILC rod (m) mass of the impact and rebound pendulums, respectively (kg) weight of the striker in the drop weight tester or the ILC (kg) particle weight (kg) number of valid traces in an ILC test (–) frequency of rotation of the impact tester (rad s1) resistances in Wheatstone bridge circuit (O)

D d do dp dr Dr E e E0 Ec Ecs Ei Eis Em Em,N Er ER1, ER2 E1 F Fc Fb Fr g GF h ho h1 K k kp KIc L l m1, m2 mb mp N n R

Breakage of Single Particles: Quasi-Static

r s T t t10 tc tf u1, u2 ub, ur V Vm, Vin vo v1, v2 w Wi Y Yp

65

parameters in the deconvolution of the output signal from the ILC (–) parameters in the deconvolution of the output signal from the ILC (–) corrected period of the pendulum (s1) time (s) proportion of material passing 1/10th of the parent particle size (%) time at primary fracture (s) ﬁnal time of contact (s) velocities of the impact and rebound pendulum at the instant of contact (m s1) position of the centre of gravity of the drop weight and the ILC rod (m) deconvoluted output voltage (V) voltage output and input from the bridge circuit, respectively (V) impact velocity (m s1) velocities of the impact and rebound pendulums after impact, respectively (m s1) particle thickness (m) Bond work index (kWh t1) modulus of elasticity of the rod (N m2) modulus of elasticity of the particle (N m2)

Greek letters a ac b D Dc Df e f l m mp y r rp sp sPLT

local deformation of the system (m) local deformation of the system at primary fracture (m) particle shape factor ½¼ mp =ðd 3 rp Þ (–) particle deformation (m) particle deformation at primary fracture (m) ﬁnal particle deformation (m) unit deformation (–) parameter of the particle fracture energy versus size model (equation (55)) (–) proportionality constant of the bridge circuit (–) Poisson’s ratio of the rod (–) Poisson’s ratio of the particle (–) angle swept by the pendulum (degrees) rod density (kg m3) particle density (kg m3) particle strength (N m2) point-load test strength (N m2)

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REFERENCES [1] K. Scho¨nert, Aufbereit.-Tech. 32 (1991) 487. [2] L.M. Tavares, R.P. King, ZKG Int. 58 (2005) 49. [3] D.D. Chabtree, R.S. Kinasevich, A.L. Mular, T.P. Meloy, D.W. Fuerstenau, Trans. SME/AIME 229 (1964) 201. [4] J.A.S. Cleaver, M. Ghadiri, N. Rolfe, Powder Technol. 76 (1993) 15. [5] L.G. Austin, J.M. Menacho, F. Pearcy, A general model for autogenous and semiautogenous milling, Proc. APCOM, SAIMM, 1987, vol. 2, p. 107. [6] J.J. Gilvarry, B.H. Bergstrom, J. Appl. Phys. 32 (1961) 400. [7] B.H. Bergstrom, C.L. Sollenberger, Trans. SME/AIME 220 (1961) 373–379. [8] B.H. Bergstrom, C.L. Sollenerger, W. Mitchel Jr., Trans. SME/AIME 220 (1961) 367. [9] N. Arbiter, C.C. Harris, G.A. Stambolzis, Trans. SME/AIME 244 (1969) 118. [10] H. Rumpf, Powder Technol. 7 (1973) 145. [11] K. Scho¨nert, Trans. SME/AIME 252 (1972) 21. [12] H. Schubert, Aufbereit.-Tech. 5 (1987) 237. [13] S. Yashima, F. Saito, T. Sagawa, H. Suzuki, S. Sano, Kagaku Kogaku Ronbunshu 1 (1975) 344; (in Japanese). [14] S. Yashima, S. Morohashi, F. Saito, Science Reports of Research Institutes, Tohoku University, 28 (1979) 116. [15] S. Yashima, F. Saito, T. Mikuni, Kagaku Kogaku Ronbunshu 2 (1976) 150; (in Japanese). [16] S. Yashima, Y. Kanda, S. Sano, Powder Technol. 5 (1987) 277. [17] C.C. Dan, H. Schubert, Aufbereit.-Tech. 31 (1990) 241. [18] S.E.A. Awachie, Development of crusher models using laboratory particle breakage data, Ph.D. thesis, University of Queensland, Brisbane, 1983. [19] L.M. Tavares, Miner. Eng. 12 (1999) 43. [20] L.M. Tavares, R.P. King, Int. J. Miner. Process. 54 (1998) 1. [21] E.L. Piret, Chem. Eng. Prog. 49 (1953) 56. [22] S.S. Narayanan, W.J. Whiten, Trans. Inst. Min. Metall. 97 (1988) C115. [23] S.R. Krogh, Powder Technol. 27 (1980) 171. [24] T.J. Napier-Munn, S. Morrell, R.D. Morrison, T. Kojovic, Mineral Comminution Circuits: Their Operation and Optimization, University of Queensland, Brisbane, 1996. [25] W.J. Barnard, F.A. Bull, Primary breakage of brittle particles, Fourth Tewksbury Symposium University of Melbourne, Melbourne, 1979, p. 20.1. [26] L. Sikong, H. Hashimoto, S. Yashima, Powder Technol. 61 (1990) 51. [27] T.L. Anderson, Fracture Mechanics: Fundamentals and Applications, 2nd edition, CRC Press, Boca Raton, 1995. [28] B. Cottrell, Y.W. Mai, Fracture Mechanics of Cementitious Materials, Blackie Academic and Professional, 1996. [29] ASTM, Standard drop shatter test for coal. Annual Book of ASTM Standards, 1991, vol. 5.05, p. 214. [30] L.M. Tavares, R.R. Diniz, Analysis of self-breakage for modelling media competence in autogenous mills, Proc. VI Southern Hemisphere Meeting on Mineral Technol., Rio de Janeiro, 2001, vol. 1, p. 47. [31] A. Jowett, K.R. Weller, A critical assessment of comminution test methods, Fourth Tewksbury Symp., University of Melbourne, Melbourne, 1979, p. 18.1. [32] S.S. Narayanan, Bull. Proc. Australas. Inst. Min. Metall. 291 (1986) 49. [33] P.M.M. Vervoorn, B. Scarlett, Particle impact testing, Proc. 4th Eur. Symp. Commin., Ljubljana, 1990, p. 195. [34] K. Scho¨nert, M. Marktscheffel, Liberation of composite particles by single particle compression, shear and impact loading, Proc. 6th Eur. Symp. Commin., Nu¨rnberg, 1986, p. 29. [35] L. Vogel, W. Peukert, Powder Technol. 129 (2003) 101.

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[36] J.M. Karpinski, R.M. Tervo, Trans. SME/AIME 229 (1964) 126. [37] S.M. Hendersen, R.C. Hansen, Trans. ASAE 17 (1974) 315. [38] S. Baumgardt, B. Buss, P. May, H. Schubert, On the comparison of results in single grain crushing under different kinds of load, Proc. 11th Int. Miner. Process. Congr., Cagliari, 1975, p. 3. [39] J. Gross, S.R. Zimerley, Trans. SME/AIME 87 (1930) 7–26, 27–34, 35. [40] J. Gross, US Bureau Mines Bull. 402 (1938) 148. [41] R.J. Charles, Trans. SME-AIME 208 (1957) 80. [42] R.A. Bearman, C.A. Briggs, T. Kojovic, Miner. Eng. 10 (1997) 255. [43] R.G. Frandrich, J.M.F. Clout, F.S. Bourgeois, Miner. Eng. 11 (1998) 861. [44] V.F. Pereira, C.M. Vasques, P.B. Neves, L.M. Tavares, Breakage of coarse particles by impact, Proc. 21st Encontro Nacional de Tratamento de Mine´rios e Metalurgia Extrativa, Floriano´polis, 2005, vol. 1, p. 33 (in Portuguese). [45] F.S. Bourgeois, G.A. Banini, Int. J. Miner. Process. 65 (2002) 31. [46] S.S. Narayanan, W.J. Whiten, Proc. Australas. Inst. Min. Metall. 286 (1983), 31. [47] L.M. Tavares, Micro scale investigation of particle breakage applied to the study of thermal and mechanical predamage, Ph.D. thesis, University of Utah, Salt Lake City, 1997. [48] R. Sahoo, Powder Technol. 161 (2006) 158. [49] A.W. Fahrernwald, J. Newton, E. Herkernhoff, Eng. Min. J. 139 (1938) 43. [50] F.C. Bond, Mining Technology, Technical Preprint No. 1895 169 (1946) 58. [51] A.M. Gaudin, R.T. Hukki, Trans. SME/AIME 169 (1946) 67. [52] B.H. Bergstrom, in: N.L. Weiss, (Ed.), SME Mineral Processing Handbook, SME, Littleton, 1985, p. 30. [53] S.S. Narayanan, Development of a laboratory single particle breakage technique and its application to ball mill modelling and scale-up, Ph.D. thesis, The University of Queensland, Brisbane, 1985. [54] P. Szczelina, V. Raaz, Aufbereit.-Tech. 43 (2002) 28. [55] K. Steier, K. Scho¨nert, Dechema-Monographien 69 (1972) 167. [56] G. Unland, P. Szczelina, Int. J. Miner. Process. 74S (2004) S209. [57] C.H. Dowding, G. Lytwynyshyn, Powder Technol. 31 (1982) 277. [58] Y. Oka, W. Majima, Can. Metall. Q. 9 (1970) 429. [59] L.J. Taylor, D.G. Papadopoulos, P.J. Dunn, A.C. Bentham, J.C. Mitchell, M.J. Snowden, Powder Technol. 143–144 (2004) 179. [60] R. Pitchumani, O. Zhupanska, G.M.H. Meesters, B. Scarlett, Powder Technol. 143–144 (2004) 56. [61] D.W. Fuerstenau, P.C. Kapur, O. Gutsche, Powder Technol. 76 (1993) 253. [62] O. Gutsche, P.C. Kapur, D.W. Fuerstenau, Powder Technol. 76 (1993) 263. [63] P.C. Kapur, O. Gutsche, D.W. Fuerstenau, Powder Technol. 76 (1993) 271. [64] B. Hopkinson, Phil. Trans. Royal Soc. A213 (1914) 375. [65] M.K. McCarter, D.S. Kim, in: R. Rossmanith, (Ed.), Rock Fragmentation by Blasting, Balkema, Rotherdam, 1993, p. 63. [66] B. Lundberg, Int. J. Rock Mech. Min. Sci. 13 (1976) 187. [67] H. Kolsky, Stress Waves in Solids, Dover, New York, 1963. [68] R. Weichert, J.A. Herbst, An ultra fast load cell for measuring particle breakage, Proc. 6th Eur. Symp. Commin., Nurnberg, 1986, p. 3. [69] R.P. King, F.S. Bourgeois, Miner. Eng. 6 (1993) 353. [70] A. Ho¨fﬂer, Fundamental breakage studies of mineral particles in an ultra fast load cell device, Ph.D. thesis, University of Utah, Salt Lake City, 1990. [71] F.S. Bourgeois, Single-particle fracture as a basis for micro scale modelling of comminution processes, Ph.D. thesis, University of Utah, Salt Lake City, 1993. [72] L.M. Tavares, R.P. King, Int. J. Miner. Process. 74S (2004) S267. [73] C.A. Briggs, A fundamental model for cone crusher, Ph.D. thesis, The University of Queensland, Brisbane, 1997.

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L.M. Tavares L.M. Tavares, A.S. Lima, Rem: R. Esc. Minas 59 (2006) 61. P. Hildinger, Chemie-Ing.-Techn. 41 (1969) 278. R. Weichert, ZKG Int. 45 (1992) 51. L.M. Tavares, M.C. Cerqueira, Cem. Concr. Res. 36 (2006) 409. L.M. Tavares, L.G. Austin, R.P. King, in: S.K. Kawatra, (Ed.), Advances in Comminution, SME, Littleton, 2006, pp. 205–222. Y. Hiramatsu, Y. Oka, Int. J. Rock Mech. Min. Sci. 3 (1966) 89. R.A. Bearman, P.J. Pine, B.A. Wills, Use of fracture toughness testing in characterizing the comminution potential of rock, Proc. MMIJ/IMM Joint Symp., Kyoto, 1989, p. 161. R.A. Bearman, Int. J. Rock Mech. Min. Sci. 36 (1996) 257. W. Goldsmith, Impact, Edward Arnold, London, 1960. R.P. King, Modelling and simulation of mineral processing systems, ButterworthHeinemann, Oxford, 2001. P.C. Kapur, D. Pande, D.W. Fuerstenau, Int. J. Miner. Process. 49 (1997) 223. K. Klotz, H. Schubert, Powder Technol. 32 (1982) 129. L. Milin, Incomplete beta function modelling of the t10 procedure, Technical Report, Comminution Centre, University of Utah, 1994. L.M. Tavares, Powder Technol. 142 (2004) 81. L.M. Tavares, R.P. King, Powder Technol. 123 (2002) 138. D.E. Grady, M.E. Kipp, in: B.K. Atkinson, (Ed.), Fracture Mechanics of Rock, Academic Press, London, 1987, p. 429. H.H. Gildemeister, K. Scho¨nert, Dechema Monographien 69 (1972) 233. S. Yashima, Y. Kanda, F. Saito, T. Sasaki, M. Iijima, Kagaku Kogaku 37 (1973) 1218; (in Japanese). W. Weibull, J. Appl. Mech. 9 (1951) 293. L.M. Tavares, Role of microstructure in comminution, Proc. 23rd Int. Miner. Process. Congr., Rome, 2000, vol. C, pp. 4–99. G.E. Dieter, Mechanical Metallurgy, 3rd edition, McGraw-Hill, New York, 1986. W.J. Kenny, E.L. Piret, Trans. AICHE 7 (1961) 199. K. Scho¨nert, H. Umhauer, H. Rumpf, Glastech. Ber. 35 (1962) 272. S. Yashima, F. Saito, T. Mikuni, Kagaku Kogaku Ronbunshu 2 (1976) 150; (in Japanese). R.P. King, F.S. Bourgeois, A new conceptual model for ball milling, Proc. 18th Int. Miner. Proc. Congr., Sydney, 1993, vol. 1, p. 81. B.K. Mishra, R.K. Rajamani, Appl. Math. Model. 16 (1992) 598. D.M. Weedon, F. Wilson, Int. J. Miner. Process. 59 (2000) 195. JKTech, JKSimMet Users Manual, Brisbane, 1999. S. Morrell, W.I.L. Lim, L.A. Tondo, D. David, Modelling the high pressure grinding rolls, Mining Technology Conference, 1996, p. 169. D.W. Fuestenau, A.-Z.M. Abouzeid, Int. J. Miner. Process. 67 (2002) 161. K. Scho¨nert, Comminution from theory to practice, Proc. 19th Int. Miner. Process. Congr., San Francisco, 1995, vol. 1, p. 7.

CHAPTER 2

Impact Breakage of Single Particles: Double Impact Test Kam Tim Chaua, and Shengzhi Wub a

Department of Civil and Structural Engineering,The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong,China b Department of Mechanics, Lanzhou University, Lanzhou,Gansu 730000,China

Contents 1. Introduction 2. Background and literature review 3. Double impact tests 4. Key issues 5. Summaries of recent development 5.1. Theoretical solution 5.2. Experimental study 5.3. Numerical simulations 5.3.1. Development of DIFAR 5.3.2. Numerical simulation of fragmentation 6. Forward look Acknowledgments References

69 70 71 73 75 76 79 82 82 83 83 83 84

1. INTRODUCTION Fragmentation, the process of breaking brittle solids into smaller pieces, is caused by the propagation of multiple fractures and crushing at different scales. Such fractures can be induced by either quasi-static loading (either compression or tension) or wave interference under dynamic loading [1]. The scale of fragmentation phenomena ranges from asteroid impact resulting in craters of hundreds of kilometers to the artiﬁcial dynamic crushing of the Ganoderma lucidum spores (lingzhi) to extract its constituents in 10 mm or so. For example, the egg-shaped shell of G. lucidum spores is formed by a composite layer consisting of a very hard outer shell connecting to a tough and ductile inner shell by a shockabsorbent material. This composite structure of the shell makes the G. lucidum Corresponding author. Tel.: +852 2766 6015; Fax: +852 2334 6389; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12005-4

r 2007 Elsevier B.V. All rights reserved.

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Fig. 1. The meteor crater at Arizona (USA) caused by dynamic impact and particles of Ganoderma lucidum spores.

spores one of the toughest materials to break on earth. Dynamic impact on these spores was found to be an efﬁcient way to break their shell. Therefore, dynamic impact of particles can manifest as either a natural phenomenon or as an artiﬁcial process in manufacturing and material processing. Figure 1 shows the meteor crater at Arizona, USA, which was formed by huge meteor impact, and the microscopic photographs of the particles of G. lucidum spores, which have to be broken by dynamic impacts. Today there are very few industrial, agricultural, or domestic processes that do not involve size reduction of solid materials in some form. The industries for mining, iron and steel, ceramics, concrete, plastics and ﬁbers, chemicals, food, and pharmaceuticals are just a few examples in which fracturing and crushing of particles play an important part [2]. In civil engineering applications, impactinduced fragmentation relates to crushing of rock mass during mining process, blasting of rock material in tunnelling, and aggregate production for roadbeds and concrete mixing [3]. A detailed understanding of the fragmentation process is paramount in many disciplines including geology, applied rock mechanics, and soil science. A reliable way to study the process of impact breakage of particles can be made in well-controlled laboratory environments, and dynamic tests in laboratories in understanding the process of impact breakage are of fundamental importance in investigating breakage mechanisms. The purpose of this chapter is to review one of these test methods – the impact breakage of particles in double impact tests.

2. BACKGROUND AND LITERATURE REVIEW Since the 1960s, single particle fracture studies have provided the basis for particle fragmentation researchers. There are at least three different tests

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commonly used in single sphere or particle breakage, namely slow compression test, free-fall impact test, and double impact test [4]. Slow compression testing involves the compression of spheres or particles between two ﬂat rigid platens, and has been applied to the testing of concrete [4], glass [5–10], and soda-lime glass [5–10]. Previous theoretical models have involved analysis of the state of stress inside spherical grains loaded diametrically, the most popular one proposed by Hiramatsu and Oka [11], which has been applied by various authors [9,12]. It should be noted that similar solutions have also been obtained by Dean et al. [13], and Sternberg and Rosenthal [14]. The drawback of the solution by Hiramatsu and Oka [11] is that contact stress is assumed as uniform radial stress. The contact between a sphere and two rigid platens has been considered analytically by Chau and Wei [15] and Meyers and Meyers [16]. Tatara obtained a solution for an elastic sphere under large deformation [17,18]. Free-fall impact test normally involves the dynamic impact of spheres on a ﬂat surface, and has been carried out on spheres of steel [19], aggregate [20], ceramic [21], glass [22], sapphire [10], and sand-cement [4]. Theoretical analyses for failure of spheres under such dynamic impacts include works by Arbiter et al. [4], Dean et al. [13], Shipway and Hutchings [9], Thornton et al. [23], and Andrews and Kim [21,22]. Double impact testing, a term coined by Arbiter et al. [4], is the dynamic compression of spheres between two rigid platens (i.e. the dynamic counterpart of the slow compression of spheres). This test is normally done by putting a sphere on a ﬂat rigid surface, then another weight with a ﬂat rigid bottom was dropped on the spherical specimen following a guiding system. The weight as well as the vertical height of the dropping mass can be adjusted. Therefore, this test is also sometimes referred to as the drop weight test. Such experimental tests have been conducted by Arbiter et al. [4], Chau et al. [3], Wu et al. [24], and Shipway and Hutchings [9].

3. DOUBLE IMPACT TESTS Figure 2 shows the apparatus of the double impact tests used by Arbiter et al. [4] (Fig. 2(a)), and those used by Chau et al. [3] and Wu et al. [24] at the Hong Kong Polytechnic University (Fig. 2(b)) and the Hong Kong University of Science and Technology (Fig. 2(c)) respectively. A typical result for the double impact test compared to the result for a slow compression test obtained by Arbiter et al. [4] is shown in Fig. 3. The cumulative percentage of mass is plotted against the logarithm of fragment sizes (similar to the particle size distribution curve used in soil mechanics). For fragments

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Fig. 2. Double impact testers: (a) used by Arbiter et al. [4], (b) at PolyU and (c) at HKUST.

less than 0.4 in., the distribution falls onto a straight line for both slow compression and double impact tests. For fragments larger than 0.4 in. (or so-called residue part in the graph), the fragment size distribution is no longer a straight line. Clearly, there are two kinds of fragmentation mechanisms existing for ﬁne fragmentation and coarse fragmentation processes. This phenomenon is still today not well-understood. Later in this chapter this phenomenon will be discussed again in the summary of the latest experimental work by Wu et al. [24]. In addition, as shown by Wu et al. [24], the slope of the plot also depends on the impact energy subjected to the sphere and the type of material being tested. In general, there are two guide bars that guide the free fall of the weight. The impact tester at HKUST is a Dynatup 8250 impactor and can record the impact energy, impact velocity, and impact force as a function of time every 0.01 ms. The typical fracturing pattern of spheres under double impact testing is summarized in Fig. 4. The symbols A-50 and B-75 used in Fig. 4 represent spheres of material A (uniaxial compressive strength (UCS) ¼ 37 MPa) with diameter 50 mm, and of material B (UCS ¼ 59 MPa) with diameter of 75 mm. The failure patterns were named II, IIIa, IIIb, IIIc, IVa, IVb, IVc, Va, Vb, VI, and X. In this symbology, II indicates two main fragments in the shape of slice, III for three main fragments in the shape slice, and so on for roman numerals; the second roman letter indicates the type of ﬁne powder crushing. The full details are described by Wu et al. [24]. Nevertheless, the occurrence of this pattern depends on the applied impact energy level, the UCS of the spheres, and the size of the sphere.

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Fig. 3. Cumulative weight distribution curve of fragments resulting from double impact testers [4].

However, due to mathematical complexity, there has been little attempt to apply theoretical solution to predict the full range of possible particle failure modes [3,24–26].

4. KEY ISSUES As shown in the previous section the particle distribution after dynamic crushing is very complicated. In attempting to resolve the problem, we have to look at two

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Fig. 4. Twelve breakage patterns of spheres observed in double impact testing. The ﬁrst letter A is for material of UCS of 37 MPa and B is for 59 MPa. The digit after the letter indicates the diameter of the tested spheres [24].

main issues. The ﬁrst is the stress evolution with time in the sphere and the second the inherent distribution and size of pre-existing ﬂaws within the sphere. For the ﬁrst issue, a closed-form analytical solution of a sphere subject to double impact testing was recently derived by Wu [25], which is a big improvement over numerical stress analysis [27]. For the second issue, the inherent distribution of defects are likely to be the points of fracture initiation under local tensile stress created during the stress wave propagation which is induced by the dynamic impact of the dropping mass. These defects may control the main fracturing as well as the ﬁnal breakage patterns of the sphere. It is likely that more dispersed local defects might also lead to ﬁner fragment distribution. Micro-structures (such as pre-existing ﬂaws) and their relevance to the dynamic fracture patterns have been performed by various authors [26,28,29]. The contact stress at impact is modelled by using classical Hertz analysis [30] for the normal stress distribution over a circular contact area. In the limiting case, when the contact area approaches a point, the Hertz solution becomes Boussinesq’s solution for a stress ﬁeld due to a point load applied to the surface of an elastic half space [31]. Therefore, the static solution to the stress distribution problem of a spherical grain under diametric compression may be approximated

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by superposition of the near-ﬁeld Hertzian contact stresses onto the far-ﬁeld stresses obtained for diametric loading of a sphere either point loading or by uniform normal stresses [11,31,32]. This superposition approach is also supported by Saint-Venant’s principle which states that a system of forces applied to a small element on the surface of the body will result in only localized stresses [30]. The quasi-static solution of the problem is also discussed by Tavares in this handbook.

5. SUMMARIES OF RECENT DEVELOPMENT Although a vast body of practical knowledge has been accumulated in industries using dynamic impact, there is a lack of detailed knowledge about the failure mechanisms in fracturing process under dynamic impact. The theory of dynamic fragmentation is relatively less developed compared with the static counterpart, and dynamic fragmentation mechanisms remain unknown. Because of the difﬁculty in monitoring the fragmentation sequence inside a solid under an impact load of an extremely short duration, our understanding of dynamic fragmentation processes is quite limited. Thereby, fragmentation processes in industry are still modelled using an empirical approach. As a ﬁrst step, an understanding of single particle failure mechanisms and their role in fragmentation processes is essential. In this section, our comprehensive approach is outlined using analytical, experimental, and numerical analyses to investigate the fragmentation of solids. First, an analytical solution of an elastic sphere subject to a pair of suddenly applied patch loads along a diameter is obtained. For the special case that the patch loads converge to a pair of point loads, our solution is comparable to those obtained by Jingu and Nezu [33]; when transmission of waves through the two rigid platens is allowed, the long-term solutions converge to the static solutions given by Hiramatsu and Oka [11] and Chau et al. [15] for the cases of uniform and Hertz contact loads, respectively. Contour plots provide the time evolution of dynamic stress patterns and can be used to interpret the position of fracture initiation and patterns of fragmentation [25]. In experimental tests, brittle spheres made of plaster of two different strengths and three different sizes were compressed dynamically between two rigid platens at various impact energy levels (i.e. under double impact test). Both impact velocity and contact force at the impactor can be measured accurately as a function of time. Third, a newly developed computer program, Dynamic Incremental Failure Analysis for Rocks (DIFAR) is used to simulate the dynamic failure and fragmentation of a sphere subject to double impacts [34]. The computer program is based on an elastic ﬁnite element analysis of solids incorporated with loading-rate-sensitive Mohr–Coulomb criterion and a tensile cut-off for damage

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checking. Both the elastic modulus and strength of all elements follow a Weibull distribution spatially; thus, the random nature of the initiation of fragmentation can be modelled. The numerical simulations agree well with the general pattern of observations in experiments. The full details can be found in Wu [25]. The results of this work should provide some insight on dynamic fragmentation for spheres or non-spherical particles and a benchmark study for further research in the area.

5.1. Theoretical solution Consider a spherical polar coordinate system (r, y, j) with the origin located at the center of the sphere, as shown in Fig. 5. The spherical specimen is assumed to be linear elastic, homogeneous, and isotropic; the stress and strain components are related by the generalized Hooke’s law. For the present problem of the spheres under diametral compression, body forces can be neglected. Hence, the motion equations can be simpliﬁed to @srr 1 @srj 1 @sry 2srr syy sjj þ sry cot y @2 ur ¼r 2 þ þ þ r sin y @j r @y r @r @t

ð1Þ

@srj @2 uj 1 @sjj 1 @syj 3srj þ 2syj cot y ¼r 2 þ þ þ r sin y @j r @y r @r @t

ð2Þ

@sry 1 @syj 1 @syy 3sry þ ðsyy sjj Þ coty @2 uy ¼r 2 þ þ þ r siny @j r @y r @r @t

ð3Þ

For the uniform contact stress boundary conditions, the impact load is modelled by uniform radial stress pH(t) applied over two opposite spherical areas on r ¼ a which subtend an angle 2y0 from the origin symmetrically with respect to the z pH(t) A

p()H(t) A 20

20

ϕ

ϕ a

a B

B (a)

(b)

Fig. 5. Sketch for a sphere under double impact loading (Heaviside step function in time): (a) uniform contact stress; and (b) Hertz contact stress.

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axis; all other tractions are zero on r ¼ a. Mathematically, this boundary condition can be expressed as 0; y0 oyop y0 ; srr ¼ ð4Þ sry ¼ srj ¼ 0 on r ¼ a pHðtÞ; 0 y y0 ; p y0 y p; For the case of Hertzian contact, the interaction between the sphere and the two rigid ﬂat platens can be modelled as contact stress by considering the Hertz contact theory 0; y0 oyop y0 ; srr ¼ sry ¼ srj ¼ 0 on r ¼ a ð5Þ pðyÞHðtÞ; 0 y y0 ; p y0 y p; where the contact pressure p(y) is given by Timoshenko and Goodier [30]. The analytical solution for this problem can be decomposed into two parts. One is a corresponding quasi-static solution, the other is a free vibration solution [35]. The free vibration problems are subject to an initial displacement and velocity of the sphere, which is obtained from the corresponding static problem [24]. Regarding the initial condition, all tractions are initially zero. The full details are given in [36] and will not be repeated here. To fully visualize the prediction of this solution, Fig. 6 shows the maximum principal stress (or the most compressive stress) in the sphere. The time of the plot at a/c1 where a is the radius of the sphere and c1 is the compressional wave speed of the solid sphere. That is, the 20 10 9 8 7 6 5 4 3 2 1 -1 -2.5 -3.5 -4.5 -7.5 -200

Fig. 6. Theoretical contour plot for the maximum compressive stress at time a/c1 induced by double impact test.

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20 10 9 8 7 6 5 4 3 2 1 -1 -2.5 -3.5 -4.5 -7.5 -200

Fig. 7. Theoretical contour plot for the most tensile stress at time a/c1 induced by double impact test.

20 10 9 8 7 6 5 4 3 2 1 -1 -2.5 -3.5 -4.5 -7.5 -200

Fig. 8. Theoretical contour plot for the maximum shear stress at time a/c1 induced by double impact test.

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plot is for the time when the compression wave reaches the centre of the sphere. Similarly, Figs. 7 and 8 plot the minimum principal stress (or the most tensile stress) and the shear stress in the sphere. It is clear from Figs. 6–8 that the locations of the maximum compressive, tensile, and shear zones coincide in these diagrams. Thus, these plots clearly indicate that certain regions are more conducive to fracture initiation than others at a certain time. Wu and Chau [36] have more thoroughly analyzed the region of highest stress concentrations at different time [25,36], but the details will not be given here.

5.2. Experimental study As part of our comprehensive studies on the fragmentation of a single particle, some double dynamic impact tests similar to those considered in previous section have been conducted at HKUST using the Dynatup 8250 impactor (shown in Fig. 9), using which the impact energy, impact velocity, and impact force can be accurately measured in the order of 0.01 ms. A total of 151 plaster spheres with different UCS (37 and 59 MPa) and three different sizes (50, 60, and 75 mm) were cast; 16 of them were tested by compressing to failure under static loads while 135 were subjected to double impacts. The impact energies varied from 8 J to over 310 J with impact velocities ranging from 1 to 9 m s1 while for the static test the applied energy is typically from 3 to 9 J.

(b)

(a)

(c)

Fig. 9. (a)The Dynatup 8250 impactor at HKUST. (b) High-speed cameras for capturing digital images of dynamic failure of sphere. (c) Three spheres with different diameters.

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K.T. Chau and S. Wu (a)

(b)

Fig. 10. A typical fractured sphere under double dynamic impact testing: (a) a specimen with diameter of 60 mm and UCS of 37 MPa subject to an impact of 132 J; (b) a sketch showing internal fractured sections of a typical specimen.

Fig. 11. A typical distribution of fragments resulted from the impacted sphere.

It was discovered that there can be up to 12 different failure or fracture patterns of the spheres, depending on the applied impact energy level, the UCS of the spheres, and the size of the sphere. A typical fractured sphere under double dynamic impact test is shown in Fig. 10(a) for a specimen with diameter of 60 mm and UCS of 37 MPa subject to an impact of 132 J; and a sketch showing the internal fractured sections of a typical specimen is shown in Fig. 10(b). The calculated speciﬁc surface energy is from about 0.01 to 0.027 J mm2. The full details of these experiments are available in [24,25].

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It was also discovered that the fragments (Fig. 11) of the fractured spheres can be described by a Gates–Gaudin–Schuhmann (GGS) size distribution [37]: RðdÞ ¼

d d

n ð6Þ

where R(d) is cumulative weight percentage passing diameter d, and d* is the equivalent maximum size of fragments. We found that typically for the plaster spheres under double impact, the n value ranges from about 0.89 to 1.75, which is slightly smaller than the results for real rocks [38]. It is interesting that there are two distinct slopes of the double-log scale plot of R(d) versus d. This is because fragments resulting from the double dynamic impact are of two different sizes, larger fragments resulting from primary fractures (mainly passing through a diameter of the sphere) and smaller fragments resulting from secondary fractures (do not pass through a diameter of the sphere). The power index n of the larger-fragment-ﬁt is, in general, larger than that of the smaller-fragment-ﬁt. Typically, n ranges from 0.8 to 1.6 and decreases with the impact energy and strength of the sphere. The diameter of the end crushing zone is found approximately proportional to m1=4 r 1=4 v 1=2 as shown in Fig. 12, where m is the mass of the impactor, r the radius of the sphere and v the impact velocity. In addition, taking into account the kinetic energy loss, crushing modulus as well as speciﬁc fracture energy were obtained for the spheres.

18

L(mm)

16

14

12

10 0.5

1

1.5 m1/4r1/4v1/2

2.0

2.5

Fig. 12. The crushed size L (mm) vs. combination parameter m1=4 r 1=4 v 1=2 for a spherical specimen with diameter of 60 mm and UCS of 59.3 MPa.

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5.3. Numerical simulations In view of the limitation of the theoretical solution in predicting the progressive sequences of fragmentation, a new computer code has been developed as a result of joint collaboration between the Hong Kong Polytechnic University and the Northeastern University (China). This model is called DIFAR, which is the abbreviation for ‘‘Dynamic Incremental Failure Analysis for Rocks’’ and is the dynamic extension of the computer model Rock Failure Process Analysis (RFPA) [34].

5.3.1. Development of DIFAR In order to reﬂect the heterogeneity of rock material, the rock is assumed to be composed of many elements with the same size, and the mechanical properties of these elements are assumed to conform to a given Weibull distribution: m u m1 u m f ðuÞ ¼ exp ð7Þ u0 u0 u0 where u is the considered parameter of each element (such as strength or elastic modulus); and the average of it among all elements is denoted by u0. A key parameter in this Weibull distribution is m, which deﬁnes the shape of the distribution function such that a larger m implies a more homogeneous material while a smaller m indicates a more heterogeneous material. Once the Mohr–Coulomb failure criterion is satisﬁed at the element level, the elastic modulus of the element will be reduced to a damage level as E ¼ ð1 oÞE 0

t=55.10μs

74.45μs

78.0μs

ð8Þ

84.0μs

90μs

Fig. 13. The mechanism of progressive failure in a solid sphere of 60 mm diameter subject to 20 J of impact energy.

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Considering the dynamic UCS is related to the loading rate, the following assumption is adopted [39]: scd ¼ A log s_ cd =s_ c þ sc ð9Þ

5.3.2. Numerical simulation of fragmentation To illustrate the progressive failure, Fig. 13 shows the progressive failure of a solid sphere of diameter 60 mm and UCS of 153 MPa subject to a double impact test of energy of 20 J. The process of fragmentation can clearly be seen in Fig. 13. More numerical simulations for other conditions are referred to [25].

6. FORWARD LOOK Although the theoretical investigations reported in this chapter concentrate on much simpler, idealized conditions than actual impact crushing, the results do appear to offer some insights that may be of practical value. The high-speed camera available at HKUST for our experiments is not capable of capturing the exact instance of the fracture initiation. Thus, a high-speed camera of higher capability (upto million frames per second) would prove invaluable for future experiments. Fragmentation models offer both descriptive and prediction capabilities. Previous studies have tended to focus on the use of statistics as a descriptive tool for characterizing fragment size distributions. There is a need to move beyond this empirical approach, toward using statistics for prediction, based on an improved physical understanding of the fragmentation mechanism. In this regard, it is important that the material to be fragmented is well characterized before energy input, that the fragmentation energy is quantiﬁed, that probabilities of failure can be estimated and that the resulting number size distribution of fragments is determined. Some of the studies described in this chapter have already made a start in this direction, but many more are needed.

ACKNOWLEDGMENTS This research was supported by the Research Grants Council (RGC) of the Hong Kong SAR Government under the Competitive Earmarked Research Grants (CERG) PolyU 70/96E, PolyU 5079/97E, PolyU 5044/99E and PolyU fund 1-BBZF through KTC as the principal investigator.

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Symbols

a A d d* E0 E f(u) H(t) m n o p r, j, y R(d) r sc sdc s_ dc u u u0

radius of the sphere material constant for strain rate effect size of fragment equivalent size of maximum fragment initial Young’s modulus updated Young’s modulus Weibull distribution Heaviside step function parameter for Weibull distribution power index of size distribution damage parameter applied pressure on surface of sphere polar coordinates size distribution of fragments Cauchy stress tensor compressive strength of solid dynamic compressive strength of solid dynamic compressive stress rate applied to solid displacement vector material parameter in Weibull distribution average of material parameter

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

E. Perfect, Eng. Geol. 48 (1997) 185–198. G.C. Lowrison, Crushing and Grinding, CRC Press, INC., OH, 1974. K.T. Chau, X.X. Wei, R.H.C. Wong, T.X. Yu, Mech. Mater. 32 (9) (2000) 543–554. N. Arbiter, C.C. Harris, G.A. Stamboltzis, Trans. AIME 244 (1969) 118–133. B.H. Bergstorm, C.L. Sollenberger, Trans. AIME 220 (1961) 373–379. B.H. Bergstorm, C.L. Sollenberger, W. Mitchell Jr., Trans. AIME 223 (1962) 362–372. J.J. Gilvarry, B.H. Bergstorm, Trans. AIME 220 (1961) 380–389. J.J. Gilvarry, B.H. Bergstorm, J. Appl. Phys. 32 (1961) 400–410. P.H. Shipway, I.M. Hutchings, Philos. Mag., A 67 (6) (1993) 1389–1404. P.H. Shipway, I.M. Hutchings, Philos. Mag., A 67 (6) (1993) 1405–1421. Y. Hiramatsu, Y. Oka, Int. J. Rock Mech. Mining Sci. 3 (1966) 89–99. Y. Oka, W. Majima, Can. Metall. Quart. 9 (2) (1970) 429–439. W.R. Dean, I.M. Sneddon, H.W. Parsons, Selected Government Research Reports: Strength and Testing of Materials: Part II: Testing Methods and Test Results, HMSO, London, 1952, pp. 212–234. [14] E. Sternberg, F. Rosenthal, J. Appl. Mech. 19 (1952) 413–421. [15] K.T. Chau, X.X. Wei, Int. J. Solids Struct. 36 (29) (1999) 4473–4496. [16] M.A. Meyers, P.P. Meyers, Trans. Soc. Mining Eng. – AIME. 274 (1983) 1875–1884.

Impact Breakage of Single Particles: Double Impact Test [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39]

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Y. Tatara, J. Eng. Mater. Technol. – ASME 111 (1989) 163–168. Y. Tatara, JSME Int. J. A 36 (2) (1993) 190–196. G.S. Knight, M.V. Swain, M.M. Chaudhri, J. Mater. Sci. 12 (1977) 1573–1586. A. Hadas, D. Wolf, Soil Sci. Soc. Am. J. 48 (1984) 1157–1164. E.W. Andrews, K.S. Kim, Mech. Mater. 29 (1998) 161–180. E.W. Andrews, K.S. Kim, Mech. Mater. 31 (1999) 689–703. C. Thornton, K.K. Yin, M.J. Adams, J. Phys. D 29 (1996) 424–435. S.Z. Wu, K.T. Chau, T.X. Yu, Powder Technol. 143–144 (2004) 41–55. S.Z. Wu, Theoretical and experimental studies on dynamic impact on brittle solids, PhD Thesis, The Hong Kong Polytechnic University, 2003. D.R. Curran, L. Seaman, D.A. Shockey, Phys. Today 30 (1977) 46–55. R. Kienzler, W. Schmitt, Powder Technol. 61 (1990) 29–38. D.E. Grady, M.E. Kipp, J. Appl. Phys. 58 (3) (1985) 1210–1222. C.A. Tang, H. Liu, P.K.K. Lee, Y. Tsui, L.G. Tham, Int. J. Rock Mech. Mining Sci. 37 (2000) 555–569. S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, 3rd edition, McGraw-Hill, 1973. J. Boussinesq, Application des Potentials a l’etude de l’equilibre et du mouvement des solides elastiques. Gauthier-Villars, Paris, 1885. R.H. Brzesowsky, Micromechanics of sand grain failure and sand compaction, PhD Thesis, University of Utrecht, The Netherlands, 1995. T. Jingu, K. Nezu, Bull. JSME 28 (245) (1985) 2553–2561. K.T. Chau, W.C. Zhu, C.A. Tang, S.Z. Wu, Key Eng. Mater. 261–263 (2004) 239–244. A.C. Eringen, E.S. Suhubi, Elastodynamics, vols. I and II, Academic Press, 1975. S.Z. Wu, K.T. Chau, Mech. Mater. 38 (2006) 1039–1060. E.G. Kelly, D.J. Spottiswood, Introduction to Mineral Processing, Wiley, New York, 1982. D.A. Shockey, D.R. Curran, L. Seaman, J.T. Rosenberg, C.F. Petersen, Int. J. Rock Mech. Mining Sci. 11 (1974) 303–317. J.W. Tedesco, C.A. Ross, P.B. McGill, B.P. O’Neil, Comput. Struct. 40 (2) (1991) 313–327.

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CHAPTER 3

Particle Breakage due to Bulk Shear John Bridgwater Department of Chemical Engineering, University of Cambridge, Pembroke Street, Cambridge CB2 3RA, UK Contents 1. Breakage in the bulk 1.1. Industrial importance 1.2. Mechanisms: bulk solids ﬂow and attrition 2. Testing methods 2.1. Shear cell test 2.2. Other tests 3. Understanding and application 3.1. Comparison of materials 3.2. Product size distribution 3.3. Extent of attrition 3.4. Equipment type 3.5. Stress and strain 4. Subject development 5. Recent work 5.1. Inﬂuence of particle strength and shape, extensive stress and shear strain 5.2. Population balance modelling 5.3. Breakage in narrow clearances 6. What next? References

87 87 89 92 92 94 95 95 98 99 100 101 102 104 105 108 110 113 115

1. BREAKAGE IN THE BULK 1.1. Industrial importance The breakage of particles during processing can be required or, conversely, it can be the very behaviour that is not wanted. Breakage can arise from a number of interactions including particle impact onto walls and internal surfaces, collisions between particles in free space and the motion of solid objects such as hammers or blades into and through bulk materials. Some of these processes, for example the ones using hammers, are aimed at causing damage to the particles. In other processes, the mechanical collision is a consequence of an interaction needed Corresponding author. Tel.: (44) 1954 267235; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12006-6

r 2007 Elsevier B.V. All rights reserved.

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for other good reasons and damage to the particles is not required. While damage is sometimes trivial, it can have the most serious consequences either to the operation of the process or to the properties and quality of the product. Bulk shear of beds of particles is widespread in the processing industries. For example, particles might be held in a storage vessel; in the very act of emptying the vessel, much of the material will move as a coherent block in the upper part of the container but there will be a narrow region of material, just a few particle diameters in width close to the wall, where there is signiﬁcant shear strain. Furthermore, as the particles converge in order to escape from the outlet, an initial elemental cube of the bulk material will be extended to form a long thin pencil of material of square cross-section. This is necessarily accompanied by substantial shear strain within the element. Additionally, the solids discharged from the hopper might be regulated by a rotary valve which has a number of vanes mounted on a rotating shaft. The particles fall into the space between the vanes and are then conveyed to an outlet in the lower part of the housing. Considerable shear will arise as the particles ﬂow into the space between the vanes with further effects due to the closing gap between the blades and the containing cylindrical shell of the valve. Thus the simple process of withdrawing a material from storage and moving it from one place to another can cause shear strain to occur with breakage occurring as a result. For the mining and minerals businesses, breakage is often a key step in the release of a mineral from an ore. In these industries, high breakage is sought that exposes the mineral that is needed. In the process industries one may surmise that the proportion of material subject to degradation is not great in many instances. If processes such as drying or chemical reaction are taking place, these may well be done in equipment in which the solids move as a block with a wall shear layer or in a bed subject to stirring by using an impeller. Alternatively, there may be rotation of the equipment itself. Examples here include rotating horizontal drums and solids mixers having the form of a hollow letter V or Y. So why is particle degradation a matter of great practical and, as a consequence, of great theoretical interest too? One result of particle damage is that the particle size distribution and the shape distribution are each altered. Large pieces of angular product can be created by fragmentation as can ﬁne particles formed by abrasion at corners, edges or ﬂat surfaces. This shows up as differences in the properties of a product some of which may be sought, some of which may be unwanted. In practice, the disadvantages dominate. The material becomes more difﬁcult to handle, less permeable to the passage of ﬂuid, has altered compression characteristics, is more easily aerated and has a different feel. If there is a ﬁne dust created, it can become entrained in a ﬂuid ﬂow; a ﬁltration system will become a necessary additional step in the process. The inability to assess the amount of ﬁner material that will be made on an industrial scale plant from tests on a smaller scale, let

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alone by a more fundamental means of evaluation, renders this a difﬁcult design proposition. Thus a process may need to be substantially modiﬁed to cater for the creation of ﬁne material and to meet the necessary requirements of health and safety. There is loss of useful material. Thus, in a system that employs recycle, as in the movement of catalyst particles through the standard cycle of use and regeneration, the loss to material due to abrasion of the surface or to cracking of the particles reduces the usable lifetime of the particles within a plant. Many instances arise when this is more signiﬁcant economically than the loss of catalytic activity with age. It is evident that breakage in all sorts of environments needs to be understood. Sometimes this is to secure breakage, in others it is to minimise or avoid breakage. Since most of the literature speciﬁcally studying breakage due to shear has been motivated by the undesired consequences, the term mainly used in the relevant literature is attrition. This usage will now be employed.

1.2. Mechanisms: bulk solids ﬂow and attrition The attrition of particles arises from stresses that are put onto the particles during ﬂow. We thus need to ﬁrst discuss some of the underlying physics of the ﬂow of bulk solids before considering how the breakage of particles within a ﬂow occurs. Consider a rectangular block of material held between two horizontal plates (Fig. 1, top part). If the material is a viscous liquid, when the top plate is moved the velocity distribution will be linear in the steady state, as depicted in the middle sketch. However, if the material were to be a packing of particulate solids, the pattern is then generally as shown in the bottom sketch. The motion is now taken up in quite a narrow region termed a failure zone, in which the rate of strain is high, typically 10 particle diameters in width. There are also two blocks of materials, one above and one below the failure zone, in which there is little internal displacement. Suppose that the shear stress to cause the displacement (a proxy for strain) for the solids is measured at a given and constant normal stress (Fig. 2). It is found that increasing strain leads to a rapid rise in the shear stress; it attains a maximum and then drops to a steady state value. Such behaviour is commonly labelled ‘‘Densely packed’’. The initial increase in shear stress is due to a deformation of structure of the particle bed with the nearest neighbours of particles not being altered. When the maximum stress is attained, internal rearrangement of the structure is precipitated and ﬁnally, in the steady state, there is a continual change of the pattern of contact between neighbouring particles. In regions away from high rate of strain, the particle contacts are either constant or undergoing little adjustment. If the packing is more consolidated, then the peak stress is

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Fig. 1. Comparison of the pattern of deformation of a viscous liquid and a bulk particulate solid. Top: initial shape. Middle: viscous Newtonian liquid. Bottom: bulk particulate solid.

Fig. 2. Shear stress as a function of displacement for a powder tested at constant normal stress.

increased. However, the stress attained at high strain remains unchanged. The ‘‘Densely packed’’ behaviour is that encountered in the vast majority of practical cases. It is the state arising if a set of particles that have been allowed to settle normally to form a packing. Only if very great care is taken to form an open packing, say with gentle settling in a vibration-free environment, is the behaviour different with the stress rising steadily to a limiting value, the ‘‘Loosely packed’’ case. For a given normal stress, the limiting value of stress is the same as that

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encountered previously for the densely packed case. If the normal stress is varied, then the limiting stress varies together with the local bulk density of the particles in the failure zone. Such notions are maintained when the ideas are transposed to three dimensions and are embodied in the concepts of the critical state, the formal name for asymptotic stress state, as it is described in texts on soil mechanics. Thus, during the bulk processing of particles, it can be seen that the breakage in the bulk will occur principally in the failure zones. Other important remarks are necessary at this point. Firstly, despite the enormous importance of understanding the mechanics of bulk solids subject to shear, there is no comprehensive theory available to us from physics to predict the stress and velocity distributions that arise during ﬂow. We are not able to take one equation to describe the material and then apply it across the whole range of processing equipment and processing conditions. This is a most intractable problem that has deﬁed the attention of some of the world’s leading physicists, with there still being no sign of a solution. Some progress has been made using computer-based simulation of particle interactions but the use of such methods in an integrated way for practical problems seems a long way off. Although we know that shear is important and causes particle breakage to form both large fragments and ﬁne dust, there is necessarily a major problem in using knowledge in a quantitative predictive manner. Furthermore, in most cases we have very little direct experimental knowledge of internal structure of the ﬂows, the media usually being opaque to probing radiation. This is even more true on the industrial scale. Secondly, one signiﬁcant ﬁnding from experiments in two dimensions on shearing photoelastic discs is that a normal stress is not borne by particles equally at all. This is conﬁrmed by simulation. It is found that, at an instant of time, a number of chains of particles carry most of the load. An example to illustrate this behaviour in three dimensions obtained by numerical simulation is given (Fig. 3). The overall force is carried by the network of particle contacts with many of the contacts carrying a low force than the few that have a large force. Furthermore, the particles carrying a large force are linked to one another in a sequence, developing a large-scale fabric of force paths known as a force chain network. While the length scales in the fabric remain unchanged with strain, the structure switches rapidly and frequently with strain. The probability of a particle being damaged during shear is thus determined by its probability of being part of a force chain that carries a high force. One can deduce that the breakage of particles occurs in failure zones in which the rate of strain is high. Within such zones, the mechanism of force transmission is through a network of force chains, the conﬁguration of which is varying as motion occurs. The high stresses occurring around points of contact in those chains carrying a great load are those most responsible for breakage. This is a recent view of the subject.

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Fig. 3. Distribution of contact forces in a sheared bed of mono-sized material at a shear strain of 0.11. The magnitude of the force carried between particles is proportional to the line thickness. The bed is approximately 10 particle diameters deep and the top surface has been moved to the left [1].

Thirdly, if the ﬂow is sufﬁciently fast, say due to the action of a high-speed blade, material can then become sufﬁciently dispersed such that the number of particle contacts is reduced and the usual packing state no longer exists. The material then behaves rather like a dense gas with the important distinction that the ﬂow continually loses kinetic energy due to the particle-particle collisions. If the input of energy ceases, the bed collapses immediately to yield a static packed bed. The behaviour is a strong function both of time and position and the notion of the failure zone is not helpful then. The attrition will also be varying time and position. This behaviour arises in equipment for granulation using blades operating at high speed; it does form part of the present discussion.

2. TESTING METHODS The observation that strain occurs in conﬁned regions termed failure zones, these being some 10 particle diameters in width, leads directly to a shear cell test that is described below. Other possible tests are then examined.

2.1. Shear cell test An attrition cell has been developed by Paramanathan and Bridgwater [2] using the principle of the annular design created by Hvorslev [3] for evaluating soils. Tests for measuring the strength and ﬂow properties of bulk particles and powders that have become widespread in the last decade also rely on his approach. The Hvorslev design was developed for studies of particle mixing by Stephens and Bridgwater [4] and was then further developed to yield a design for the evaluation of attrition.

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Pulleys Top ring (Perspex)

Grooved ring

Sample

Balancing weight Cell base

Bottom ring

Fig. 4. Annular shear cell shown in a form for studying attrition at low normal stresses. The outer diameter of each grooved ring is 160 mm and the inner diameter is 120 mm.

The principle of the cell is shown in Fig. 4. It has an annular region of inner diameter of 120 mm and an outer diameter of 160 mm, giving an annular width of 20 mm. Assuming that the region of high strain has a thickness that is not dependent on the radial position, the strain imposed on the material thereby varies by a ratio of four to three across the annulus. The annular width gives a sample that is 10 particle diameters across, a size around 10 times that of many granulated products of around 2 mm, but much smaller particle sizes that can, of course, be studied. Early studies showed that a gripping ring was needed at both the top and the bottom of the annulus to ensure that the material under test developed a velocity proﬁle, the rings being necessary to avoid material slipping against either the upper or lower annular boundaries. Gripping rings with a groove size appropriate to the particles to be tested are ﬁtted to the cell. The material is then placed in the cell and gently levelled prior to ﬁtting the top grooved ring. This ring is free to move up and down; the cell thus provides a test at constant stress, not constant volume. If tests are to be performed in a precisely controlled velocity ﬁeld, the mass of sample is such as to give a bed depth of about ﬁve particle diameters. Deeper beds can be used for comparison of materials but the deduction of the shear strain imposed is then no longer possible. For the higher range of stress to be achieved, pressure is exerted on the top ring plate via a pneumatic or hydraulic cylinder. The use of

94

J. Bridgwater

weights is another possibility. The ﬁgure shows the cell being used at low stresses with a counterweight to reduce the load arising from the lid and the upper annular gripping ring. After rotating the base of the cell through the required angle at the required speed of rotation, the top-gripping ring is removed and the contents of the cell are emptied out. The product size is then analysed using any suitable technique such as sieving, or electrical methods for particle volume. Within the constraints imposed by the strength of the equipment and the power of the motor, the test can be conducted at any normal stress that is desired and the shear strain imposed can likewise be set by the number of revolutions of the cell. The speed of rotation of the cell can also be adjusted to the desired value. The strain imposed on the material is the average circumferential displacement, very closely the displacement evaluated at the centre of the annulus divided by the height of the sample. The average rate of strain is approximately the velocity at the centre of the annulus divided by the height of the sample. The breakage of the material can thus be evaluated in terms of the physical variables strain, normal stress and rate of strain. The selection of the correct gripping rings with grooves of appropriate design is important. If the grooves are too ﬁne, the particles are not gripped properly whereas if the grooves are too coarse, then there is inactive material collected in the grooves. Grooves with a 901 at the base have been shown to be effective. Ghadiri et al. [5] indicate that two criteria should be satisﬁed. One states that a gripping ring parameter Z may be given Z ¼ 2Dz=d p

ð1Þ

where Dz denotes the distance of the centre of the particle of diameter dp above the top of the groove. It is recommended that Z lies in the range of 0.25–0.75. The other states that the groove width should be greater than dp. The selection of the sample size is important if the strain applied is to be known. It is found that the thickness of the region of the cell in which the shear behaviour of the material is uniform is smaller than the width of a failure zone, around ﬁve particle diameters being ideal. Experiments in which the displacement of a carefully inserted column of tracer is followed by stopping the cell and dissecting it to discover the displacement have been used to develop this proposal. Thus the cell seeks to study the central part of a failure zone in which the strain rate does not vary with vertical position.

2.2. Other tests A wide variety of other procedures can be employed to assess breakage. A listing of methods employed is to be found in a survey [6] and there has been little change since then.

Particle Breakage due to Bulk Shear

95

For instance, tests can be carried out on standard pieces of equipment. Thus, the rotating drum provides the basis of various standard tests for evaluating the tendency of particles to lose dust by surface abrasion. The damage is likely to arise principally by the tumbling action on the surface and so is not relevant to this discussion. In any event, the information gained is hard to transfer from one operating condition to another or from one type of equipment to another. Likewise, tests on small stirred mixers cannot be understood as the velocities and positions where the actual damage is occurring are not known. The deduction that within zones breakage occurs principally in force chains leads to the suggestions of other types of tests that could be useful. In one important group of tests, particles are crushed one by one in a compression machine. Unless there is to be sophisticated robotic control, this is a labour intensive and tedious procedure. However, the earlier observation that breakage during shear occurs in force chains gives one reason to believe that the results might be able to provide guidance about breakage due to bulk shear. Tests in which the single particle is surrounded by others in order to better describe behaviour in the bulk are described by Couroyer et al. [7]. Perhaps, such tests will give insight into the process of bodily fracture in force chains but will be less helpful for the processes of the smoothing out of edges and corners due to the rolling and sliding of the external surfaces of particles during shear. While the single particle testing methods offer a greater experimental simplicity than the shear cell method, these are not, at least yet, able to advance understanding in the same way as the shear cell.

3. UNDERSTANDING AND APPLICATION 3.1. Comparison of materials The shear test can be used without concern about its linkage to a velocity ﬁeld in certain circumstances. This application can arise in comparing the attrition properties caused by the method of particle manufacture. It can often happen that particles of a certain size are needed for an application but the product may be formed by a number of methods. For instance, there may be a need to assess a granulated product that can be made using a number of different binders. A sieve cut of each of the products can then be taken and used in the annular cell. Ideally, one then makes an estimate of the stress that is likely to arise in the process and use this in the test. The size of sample taken can be one that exceeds the ﬁve particle diameters criterion; there is then a velocity gradient of zero at the lower gripping ring. The ﬁrst work was carried out on the cell by Paramanathan and Bridgwater [8], (Fig. 5). The materials were three in number, all being forms of sodium chloride.

96

J. Bridgwater

Fig. 5. The effect of strain on attrition for various types of sodium chloride. Normal stress 41 kPa, cell speed 5 rpm, sample thickness 10.0 mm. W denotes mass fraction.

Two were granular salts taken from a process, one before and the other after a centrifuge; each of the materials had an initial size lying between the 1.7 and 2.0 mm sieve cuts. Before the centrifuge, the salt was spherical and after the centrifuge it was a mixture of spheres and half-spheres. The third salt was again spherical, now being made in the laboratory with an initial of sieve size of 355–500 mm. The material broken is determined by sieving the material after shear in the cell, the mass of material passing the coarsest sieve W, expressed here as mass fraction, being taken as the measure of attrition. The plot uses a logarithmic form for each axis. In this format, a linear relationship is found over a wide range of attrition time or shear strain, the latter expressing the deformation more fundamentally. The two lines for granular salt are of the same gradient and of similar intercept. The material coming from the ﬁrst piece of equipment, the crystallizer, is slightly more prone to attrition. This deﬁed the expectations of plant personnel who thought that the action of the centrifuge that followed the crystallizer in the process would be to create very much weaker particles through the introduction of defects. Rather, it seems that the centrifuge serves to break those particles that are weak and leave strong ones that are little changed.

Particle Breakage due to Bulk Shear

97

When tested under the same conditions, the laboratory salt showed a lower extent of attrition and a much lower rate of attrition. In view of the different initial particle size of the material, deductions on material behaviour cannot then be made on the basis of this evidence alone. Figure 6 shows results for another material, soda ash of initial size 355–500 mm, where the size of sample is varied from 6.3 to 18.7 particle diameters in thickness, all of which lie above the recommendation of ﬁve particle diameters needed for more fundamental work. There is a change in the extent of attrition with sample size, but the gradient of the plot changes little with the mass of sample. This points to the material in the failure zone behaving in a consistent manner, with an increasing amount of inactive material lying outside the failure zone as the sample size increases. In instances where the sample thickness much exceeds ﬁve particle diameters, it is then best to consider W in terms of the mass broken. Tests in the annular cell allow the physical form of the attrition product to be studied. For each of these three materials studied in Fig. 5, there is a smoothing of the surface but there is also fracturing, generally on a radial plane for the granular salt. There is thus seen the two important processes, (i) bodily fragmentation and (ii) the smoothing of the exterior by abrasion to create a ﬁne dust. For the soda ash, there is abrasion of edges.

Fig. 6. The effect of strain on attrition at different sample weights for soda ash. Normal stress 41 kPa, cell speed 5 rpm. Sample thicknesses: 2.7, 5.3, 8.0 mm.

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J. Bridgwater

3.2. Product size distribution Sample results for the product size distribution are given in Fig. 7 obtained for molecular sieve beads at one stress at various times of attrition. Both axes are in logarithmic form. It is seen that a linear relationship exists over the size range below the lower size limit of fresh material. A relationship similar to that seen here is common to all materials tested to date, ones that have been made by a wide variety of processes and ones which break down by a variety of ways. These have various modes of manufacture and prior processing steps. It is a remarkable experimental observation that, whatever material is considered, the size distribution is effectively described by a relationship, the Gaudin–Schuhmann distribution [9], which states G FðD; tÞ ¼ D=D0

ð2Þ

Here, F(D,t) represents the cumulative size distribution of product particles of diameter D at a time t which are ﬁner than a top sieve size, D0. The parameter G is known as the cumulative size distribution modulus.

Fig. 7. Particle size distributions obtained at various times for molecular sieve beads of 1.7–2.0 mm, sample mass 80 g, normal stress 23 kPa [10].

Particle Breakage due to Bulk Shear

99

2.0 Granular salt

1.5 PDV salt Urea Size distribution modulus G

1.0 Alumina Rock salt

0.5 MSB (i) 22.7 kN m-2 0.0 0.0

0.1

MSB (ii) 84.0 kN m-2

0.2 0.3 Fractional mass attrited, 1-R (xo, T )

MSB (iii) 168 kN m-2

0.4

Fig. 8. Experimental data from Neil and Bridgwater [10] showing the size distribution modulus G as a function of fractional mass that has undergone attrition, 1-R(xo,T), for various materials. Solid lines represent best ﬁt trend lines through data points. Illustrative data points are shown for (m) urea prills and (J) molecular sieve beads (MSB). MSB evaluated at three normal stresses. Alumina ¼ alumina extrudate, rock salt ¼ mined NaCl, PDV salt ¼ vacuum dried NaCl.

Observation of the form of the plots found in Fig. 7 also suggests an improved way of evaluation of the breakage as the amount of material falling below some size [10]. Let the linear form of the cumulative size distribution be extrapolated to the size deemed to determine attrition, here denoted by dT Then a revised estimate of the breakage WT can then be found. Whether one should choose to analyse the amount of an attrition product in this way is a matter of judgment and may not always be necessary. It perhaps serves to separate out the effects of attrition caused by fragmentation as opposed to abrasion. As the majority of data is now obtained in digital form, the calculation is not difﬁcult. The evolution of the products of attrition for several materials can then be given by graphs of the form shown (Fig. 8). G is found to decrease as attrition proceeds; this behaviour is discussed later.

3.3. Extent of attrition Figure 5 showed that if the amount of material broken is plotted versus shear strain, each expressed in logarithmic form, then a linear relationship is found.

100

J. Bridgwater

Whatever the normal stress, the relationship is linear, with the slope and intercept being a function of normal stress. Thus, W T ¼ Kgm

ð3Þ

The parameters used in this equation can take a variety of forms. If extrapolation of the cumulative size distribution is not employed to ﬁnd WT, then WT is replaced by W, the total experimentally observed mass falling below dT. Care is also necessary as it is best to work in terms of mass taken as attrition product especially if the sample size is larger than ﬁve particle diameters, rather than the fraction of the total mass present. The relationship also is used with the shear strain g replaced by time t or by the number of cell revolutions. The parameter K can be thought of as a measure of the attrition at a shear strain of unity. m determines the inﬂuence of the amount of shear strain on attrition, thus combining together the changing properties of particles in the system and the changing local packing in which attrition is occurring. m has provided a valuable basis to analyse data and is often known as the Gwyn parameter. The packing will be changed by the changing size and shape distribution and there are likely to also be effects coming from the occurrence of segregation due to the shear strain; m absorbs all these effects. This form of equation was ﬁrst proposed by Gwyn [11] to describe the attrition that occurs in jetting ﬂuidised beds; in the attrition literature, it has become known as the Gwyn equation. The success of this equation in the ﬁtting of data is generally good but, lacking quantitative fundamental physical models of the dynamics of attrition, no explanation has been offered for its success. The only theoretical attempt [8] develops a model for surface abrasion. It is partially successful, though being found to be rather better for fracture than abrasion. Equation (3) is widely accepted for use. However, the limit arising from the form of equation (3) is inﬁnity. It has to be recognised that the equation fails to meet the requirement that at a large shear strain, the value of WT tends to limit.

3.4. Equipment type Although there is a wealth of industrial knowledge on attrition, the literature contains little systematic work on behaviour in various forms of mechanical device. There is information on attrition in a double Z blade mixer including data on particle lubrication [12]. One reference [13] compares attrition in various pieces of equipment. Three sets of particles, these being two sieve cuts of a tetra-acetylethylene-diamine (TAED) agglomerate, the ﬁnal material being a heavy soda ash of 0.36–0.50 mm. Each of these was studied in the annular shear cell at three normal stresses, in a ﬂuidised bed and in a batch screw pugmill. Data for the pugmill are reproduced in Fig. 9 which plots, in the manner of Gwyn equation, the fraction of material x broken as a function of time t in the

Particle Breakage due to Bulk Shear

101 Screw Pugmill

0

ln x

-1 Heavy Soda Ash (0.360.50mm) 10 rpm TAED (0.50-0.71mm) 10 rpm TAED (0.50-0.71mm) 14 rpm TAED (1.0-1.6mm) 10 rpm

-2

-3

-4 0

2

4

6

8

10

ln t

Fig. 9. Attrition of materials in a pugmill as a function of time [13].

pugmill. The shear strain is unknown. The lines for the two cuts of TAED are parallel indicating that the value of m is the same for the two initial sizes. For the heavy soda ash, m ranges from 0.72 to 0.90 for the three different sorts of equipment, though having quite some variability in the annular shear cell. For the larger TAED, m lies between 0.46 and 0.52 for the three pieces of equipment; for the smaller TAED it lies between 0.43 and 0.51. There is thus reason to believe that the value of m is constant but the evidential basis for this is far from ideal. More studies are needed. However, facing a lack of other guidance, when providing an estimate of behaviour the treatment of m as constant is the best that can be suggested.

3.5. Stress and strain The ability to interpret results for a material at various stresses and strains has been studied using data from the shear cell. The ﬁrst approach suggested that the mass of material formed by attrition can be described by the product of normal stress and shear strain, a measure of the work that has been done on the system. For early studies on alumina pellets this seemed to be satisfactory but this was not found to be adequate when other materials were studied. To proceed, the Gwyn equation was then written as [10] f b sg W T ¼ KN ð4Þ sscs Here KN is a Gwyn constant, the equivalent to K, and the product b. f is the equivalent of m. s is the normal stress applied. sscs is the side crushing stress,

102

J. Bridgwater

this being determined by separate testing. In this test, particles are crushed one by one between two platens and the average peak load F0 is calculated. Following Hiramatsu and Oka [14], the failure stress sscs is given by 0.9 F0 /s2 where s is the distance between the loading planes, a result modiﬁed from that for the crushing of spheres. The basis for equation (4) is empirical. On intuitive grounds one may argue that the parameter b is linked to the change in the force chain structure associated with the stress and the term f to the degradation behaviour of the particles when subject to strain. There is, however, as yet nothing to substantiate such assertions. Figure 10 provides the data for the materials studied by Neil and Bridgwater [10], when analysed in this way. The range of materials is wide with urea of two different internal structures, a molecular sieve bead, three forms of sodium chloride and alumina pellets of three strengths. Values of f range from 0.26 (urea B) to 1.0(0) (alumina E) and b from 0.71 (aluminas D and E) to 4.8 (urea B). This approach has been scrutinised by Ghadiri et al. [5] using data they had obtained on the attrition of silica beads in an annular cell. They recommend the use of a rigorous statistical procedure to determine f and b. They consider a sequence of values of f and, for each, determine the associated value of b that gives the best correlation. They then determine the values of f and b that gives the best ﬁt overall. It is found, however, that the optimum is not very sharply deﬁned.

4. SUBJECT DEVELOPMENT The direct knowledge as revealed by the published literature on the attrition of bulk powders and granular material is restricted. The extent can be judged by the list of references here which, while certainly not comprehensive, is certainly not great in amount. There is much in-house knowledge in those companies that have chosen to buy the annular attrition equipment but the extent and value of the knowledge obtained has not been revealed publicly. There is a greater wealth of information in industry using other tests and certainly there is a great deal known about the rates of attrition to form gas-borne dust from the very need to size gas cleaning equipment. However, there is no integration of the knowledge that has arisen. There are important pieces of work. Ouwerkerk [15] carried out studies to understand the relationship between the deformation of individual particles and breakage in tests conducted in the annular shear cell. He points out that the shear strain g is a more appropriate physical parameter to time. He examined the attrition of amorphous vitreous silica spheres of diameter 2.2 mm. For present purposes, the most important observation is that the attrition of their material is best correlated by devising a normalised shear strain given by g(s/sref)2, where s is the normal stress and sref is a reference stress. sref is not linked to a speciﬁc physical measurement. A good correlation was conﬁrmed [5] for this same material but it does not seem to be successful for other materials [16].

Particle Breakage due to Bulk Shear

103

Fig. 10. The results of the ﬁtting exercise using equation (4) as reported by Neil and Bridgwater [10] for nine materials.

Numerical simulations following the process of attrition in an annular cell have been reported by Potapov and Campbell [17]. Their simulation is two-dimensional and they ﬁnd that the amount of breakage of agglomerates made of glued assemblages of smaller particles is proportional to the work done, i.e. is proportional to s g. Equation (4) indicates a more complex behaviour for real materials. The works by Ghadiri and his collaborators have been extensive and signiﬁcant [e.g. 18–21]. Couroyer et al. [7] provide a preliminary account of applying the methods of distinct element method (DEM) to understanding attrition. However, there is now a full account of such work from Ning and Ghadiri [1]. Their object of study is silica particles attrition in an annular shear cell. They proceed by the DEM allowing the forces between particles to be governed by Newton’s laws and

104

J. Bridgwater

Weight % broken material

50 45

Simulation: fragmentation + wear

40

Simulation: fragmentation only Experimental data

35 30 25 20 15 10 5 0 0

2

4 Shear strain Γ

6

8

Fig. 11. Attrition of silica beads under a normal stress of 200 kPa and a shear strain rate of 7.0 s-1 [1].

calculating positions at time increments for their material of about 109 s. This is related to the time for a pressure wave to pass through a particle. They allow surface abrasion to occur from the modelling of sub-surface cracks and they consider fragmentation too, the latter determined by the side crushing strength. The analysis is conﬁned to small shear strains as there is no way to allow for the creation of fragments and for modelling of subsequent fragment behaviour into the code at the present time. The extent of attrition is taken by removing those particles that reach the fragmentation criterion from the system and adding to these the material made by abrasion. This is a demanding computational problem which, for further development, will need a supercomputer. They ﬁnd that the particle properties of Young’s modulus and coefﬁcient of friction have a marked effect upon ﬁndings. This emphasises the importance of internal particle properties. The agreement with work on the attrition of silica spheres has been over quite a narrow range of conditions and in most cases the agreement is good (Fig. 11) though this is less so at low normal stresses (50 kPa). They ﬁnd that their results are insensitive to strain rates up to 1 s1, the range of experimental work in the annular cell, and are thus consistent with experimental ﬁndings.

5. RECENT WORK Here three pieces of work are considered, all being pieces of work with which the author has been associated. These are as follows: The ﬁrst is concerned with the inﬂuence of shape on the attrition of an extrudate; it also follows on the work of Ghadiri et al. [5] being further concerned with integrating the roles of stress and strain over wide ranges of these parameters.

Particle Breakage due to Bulk Shear

105

The next is concerned with the taking of the concepts of selection function and breakage function to seek an explanation for the observations on the effect of breakage on the particle size distribution in the annular shear cell. The third looks at another form of breakage that arises in the bulk, namely that in narrow clearances. Though the work is not so recent, it raises further issues that have not been pursued so far.

5.1. Inﬂuence of particle strength and shape, extensive stress and shear strain This work is fully reported [16]. Tests were conducted on alumina pellets made by extruding and ﬁring a paste made from alumina powder of known and controlled powder size distribution, starch, clay and water. By varying the detailed method of preparation, it was possible to obtain materials of different strengths and, by use of various dies and further processing steps, to make particles of a number of shapes. The two types of materials were designated C and E. Tensile strengths deduced from side crushing tests were 9.0 and 37 MPa respectively. The geometric measurements on extrudate C were 3.13 mm diameter, 3.68 mm long, and on extrudate E were 2.97 mm diameter, 2.94 mm long. These were used to explore the roles of normal stress and strain. For work on particle shape, this was carried out with a variety of shapes all made with extrudate C. The geometries created were (i) cylinders of all of length 3.2 mm and with diameters of 3.1, 6.2, and 9.4 mm, (ii) 3 mm cubes, (iii) triangular pyramids of 3 mm side and (iv) spheres, made by rolling of paste between two plates, of diameter 3.1 mm and of 6.5 mm. When the attrition tests were performed, the values for the ﬁtting of the parameters KN , b and f in equation (4) were obtained. This was done selecting a number of different size cuts to deﬁne the boundary of what one chooses to call attrition, these ranging from 106 to 2800 mm. Some of the results of the analysis, which follows the statistical protocol advocated above, are listed in Table 1 for experiments in which the normal stress varied from 0.28 to 290 kPa. Figure 12 shows the result of the analysis with the sieve sizes set to describe attrition being 106 and 1000 mm. The following points are made with reference to Table 1 and Fig. 12: The data are described by the Gwyn form of relationship modiﬁed as given in

equation (4). There is considerable scatter about the mean trend line. The maximum values

of R2, the correlation coefﬁcient used to select values of parameters on statistical grounds, yields values that are not close to unity in any of the cases. There are thus other parameters that have an inﬂuence on the ﬁndings. For instance the data at the highest normal stress are grouped above the correlation lines shown in the ﬁgure.

106

J. Bridgwater

Table 1. Attrition parameters for experiments on extrudates C and E over the stress range 0.28–290 kPa. R2 denotes the correlation coefﬁcient

Extrudate C Attrition boundary (mm) KN 106 180 425 1000 2,800

4.3 5.6 10 15 40

Attrited weight %

100

Extrudate E

b

f

bf

R2

KN

b

f

bf

R2

0.34 0.37 0.43 0.46 0.53

1.30 1.30 1.15 1.10 0.90

0.44 0.48 0.49 0.51 0.48

0.79 0.85 0.86 0.88 0.92

0.89 1.12 1.62 3.16 9.87

0.48 0.47 0.47 0.46 0.55

1.10 1.10 1.10 1.05 0.90

0.53 0.52 0.52 0.48 0.50

0.87 0.88 0.89 0.90 0.90

Particle size < 106 μm

10

Stress (kPa) 1

0.15

0.28 0.55

1.22

2.53 5.32

10.3

24.8 150

290 0.1 0.0001

0.001

0.01

0.1

1

10

100

(/scs)1.30 100

Attrited weight %

Particle size < 1000 μm

10

Stress (kPa) 0.15 0.28 0.55 1.22 2.53 5.32 10.3 24.8 150 290

1

0.1 0.0001

0.001

0.01

0.1

1

10

100

(/scs)1.10

Fig. 12. Normalisation of the shear cell data, selecting sizes to describe attrition as 106 mm, 1000 mm. Extrudate C.

Particle Breakage due to Bulk Shear

107

The values of KN decrease as the product size being deemed to deﬁne attrition

is reduced. This is necessarily so because the proportion of material being considered is reduced, the lower the size selected. Values of f rise as the size chosen to deﬁne attrition products reduces; this occurs for both extrudates but is more pronounced for C. Values of b fall as the size chosen to deﬁne attrition products reduces; this occurs for both extrudates but is more pronounced for C. For the experiments concerning the shape of the initial material (Table 2) made from material C, conducted over a modest range of normal stresses, 150, 290, 390 and 490 kPa., the following points may be noted: For all particle shapes taken together, it is evident that the ﬁndings were gen-

erally of a very similar character, except when analysed for the largest sieve cut of 2800 mm. However, a number of initial materials have a dimension that differs signiﬁcantly from 2800 mm. Values of R2 are 0.95 to 0.98. The linked values of f, KN and b are given in Table 2. A value of f of around 0.9 is found except for the larger spheres of diameter 6.5 mm. Values of b range from 0.84 to 1.16. For the spherical materials, the values of KN are high. For these, the ﬁnal stage of manufacture relied upon rolling between two plates. The material also offers no edges or corners. A different mechanism may be at work. The values of KN for the three cylindrical materials show that it is smallest for the longest cylinders and largest for the intermediate cylinders; the origins of this behaviour require further work. The product bf depended little on the shape and particle size selected for attrition.

Table 2. Attrition parameters for experiments on extrudate C for a variety of initial shapes of particle. R2 denotes the correlation coefﬁcient

Extrudate shape

KN

b

f

bf

R2

Cylinder 3.2 mm long Cylinder 6.2 mm long Cylinder 9.4 mm long Cube 3 mm Pyramid 3 mm Sphere 3.1 mm Sphere 6.5 mm

27 45 18 50 35 71 130

1.08 1.16 0.91 0.99 0.84 1.30 1.68

0.9 0.8 1.0 0.8 0.9 0.8 0.7

0.97 0.93 0.91 0.79 0.76 1.04 1.18

0.98 0.96(5) 0.95 0.96 0.95 0.98 0.96

108

J. Bridgwater

The work shows that equation (4) draws together the data well for a stress range covering three orders of magnitude. There is, however, some noticeable scatter. The work also shows that, although there are some minor differences, the initial shape of the material does not have a great effect on the parameters describing attrition of these extrudates.

5.2. Population balance modelling This work by Ouchiyama et al. [22] seeks to develop a method of describing the evolution of the size distribution as attrition proceeds using the methods of population balance modelling. It takes into account the processes of both abrasion and fracture. The frequency size distribution of the particles, on a weight basis, is considered to be a function of particle size D and time t, and is denoted by g(D,t). The fraction of particles existing at time t within an arbitrarily speciﬁed size fraction between D and (D+dD) is g(D,t) dD. The fractional mass of the particles of size D selected for breakage in unit time at time t is given by the selection function S(D,t). Hence the mass of particles leaving this speciﬁed size fraction within a small time interval dt can be calculated. Now B(D,d) is the breakage function, which represents the cumulative size distribution of the ground particles ﬁner than D that result from breakage of a particle of size d. This enables a mass balance to be written for an interval of the size distribution. It is now argued that two different mechanisms can exist in attrition, namely abrasion and fracture. The selection and breakage functions for each mechanism of degradation are denoted respectively as Sa(d,t) and Ba(D,d) for abrasion, and Sf(d,t) and Bf(D,d) for fracture. The overall selection function, S(d,t), of degradation due to the combined mechanisms is then given by the sum of the two individual selection functions, Sðd; t Þ ¼ Sa ðd; t Þ þ Sf ðd; t Þ

ð5Þ

The probabilities of selection leading to abrasion, pa, and fracture, pf, are deﬁned such that pa ¼

Sa ðd; t Þ Sðd; t Þ

ð6Þ

pf ¼

Sf ðd; t Þ Sðd; t Þ

ð7Þ

and

where pa þ pf ¼ 1. Then the overall breakage function, B(D,d), can be described by BðD; d Þ ¼ pa Ba ðD; d Þ þ pf Bf ðD; d Þ

ð8Þ

Particle Breakage due to Bulk Shear

109

The process of attrition comprising both these mechanisms gives a mass balance as follows: Z Dmax @RðD; t Þ Sa ðd; t ÞBa ðD; d Þgðd; t Þdd ¼ @t D Z Dmax Sf ðd; t ÞBf ðD; d Þgðd; t Þdd ð9Þ D

Here R(D,t) is the residual size distribution and Dmax is the largest particle size in the system. In this approach, the joint mechanisms are analysed by considering the effects of individual mechanisms separately. The work shows how surface abrasion may be treated by taking dL=dt Sa ðd; t Þ ¼ ð10Þ D where L is a linear dimension of a particle and D is a small change in particle size. Since the breakage function is deﬁned as the cumulative size distribution of ground particles ﬁner than D that result in breakage of a particle of size, the breakage function for abrasion becomes Ba ðD; d Þ ¼ 1

for DodoD þ D

ð11Þ

Ba(D,d) becomes 3 D/d for d4D+ D for particles of spherical shape. A simple mathematical model was assumed for particle fracture following Nakajima and Tanaka [23]: Sf ðd; t Þ ¼ K f d n

ð12Þ

and Bf ðD; d Þ ¼

m D d

ð13Þ

where n and m are taken as constants independent of both size and time. Kf can be a function of time, but for simplicity it is taken as constant. A degradation parameter a is deﬁned so that for surface abrasion alone, a ¼ 1 and for fracture alone it is zero. It is thus possible to combine the two mechanisms and to solve equation (9) to obtain a model for attrition with combined mechanisms. The following points can be made [22]: The product size distribution arising from the simulation is consistent with that

found empirically, namely the Gaudin–Schuhmann distribution (Fig. 7). This arises for a range of values of a, m and n. The size distribution modulus given by the simulation decreases as the fractional mass damaged increases, which is consistent with experiment. This is seen in the example from the results given in Fig. 13 and which should be compared to Fig. 8.

110

J. Bridgwater 2.0 =0, n =1 n =0 1.5 n =-1 Size distribution 1.0 modulus

n =-2 =0.1, n =-2

n =1

n =3

0.5 m =2

=0.5, n=1 0.0 0.0

0.1

0.2

0.3

0.4

Fractional mass attrited, 1-R (xo ,T)

Fig. 13. Simulation data showing the size distribution modulus as a function of fractional mass suffering attrition, for m ¼ 2, various values of a and n.

The pattern of breakage for different materials is consistent with the values of m

obtained. Thus a material giving predominantly coarse particles such as granular salt has a high value of m. Rock salt and alumina extrudates each generate signiﬁcant ﬁnes and have a low value of m. The simulation shows that the breakdown of the initial narrow-sized material follows ﬁrst order kinetics, and indeed this can be seen in experiment. However, the origin of this behaviour is linked here to the narrow initial size distribution whatever value is selected for n. Examination of the data suggests that as attrition proceeds n decreases, there being no reason why it should not even be negative. This would be consistent with the changing environment of particles due to breakage, with a changing force chain pattern and a cushioning of large particles by small ones. More sophisticated approaches might enable us to understand the origins of the continued success of the Gwyn formulation in describing attrition over a broad range of conditions. The selection function is then likely to depend on the size distribution in existence at a moment of time and this would signiﬁcantly complicate matters. As it stands, the model is straight forward, but it has properties consistent with experiment and imparts a coherence.

5.3. Breakage in narrow clearances The work sets out to examine what happens when the external boundaries dictate the behaviour rather than it being determined by what happens in failure zones and was reported in 1997 [24]. The particular goal was to carry out experiments in which particles are drawn systematically into a narrow gap.

Particle Breakage due to Bulk Shear

111

Fig. 14. Apparatus to study attrition in narrow gaps. The gap between the rotating blade and the wall is adjusted by raising or lowering the blade assembly.

In the equipment (Fig. 14), particles are fed into an inverted conical hopper and withdrawn at a controlled rate by control of the ﬂow out of a pipe at the bottom. In the conical section there is a rotating blade having its ends cut to be parallel to the sloping sides of the surface of the cone. As particles move down the hopper, some get caught close to the blade and then may undergo attrition. By adjustment of the vertical position of the blade, the width of the gap can be varied. One material was urea spheres available in a number of sieve cuts. Figure 15 shows the percentage of particles being broken varies with gap width. For each size, there is no attrition when the particle size is small. Increasing the gap width, attrition suddenly increases, reaches a maximum and then declines with an equal rapidity to a minimum that is near zero. However, the extent of attrition then rises rapidly once more, attaining a maximum and then falls rapidly to zero, with no particle damage occurring for larger gap widths. The second maximum has a value slightly lower than the ﬁrst. A third, perhaps rather reduced, maximum might be anticipated but none was found. This behaviour is seen whatever the

112

J. Bridgwater

Fig. 15. Percentage attrition as a function of gap size for urea particles.

rate of rotation of the blade and whatever the ﬂow rate of material through the cone. It is also found for the other materials tested, namely molecular sieve beads of three diameters and a catalyst base of two diameters. All materials were spherical. There is an extreme sensitivity to attrition stemming from the exact value of the gap size. When the gap width is plotted in a dimensionless form, the results then show (Fig. 16) that the maxima and minima in the extent of attrition for the three particle sizes of the urea occur at the same values of dimensionless gap width. Indeed, the data for all the materials conform to this pattern. It was also found that the size distribution was affected by the gap size and the speed of rotation of the blade. This calls into question the use of the notions of selection function and breakage function as is commonplace in the literature for related systems. However, the most important ﬁnding is that the particle size to gap size ratio is of dominant importance and it is deduced that is the packing created ahead of the face of the blade that is advancing into the material that is signiﬁcant for the occurrence or absence of attrition. The shifting patterns of the product size distribution are logically linked to the internal structure and properties of the particles. There are practical instances where attrition occurs due to the existence of small clearances. However, no work has been found that follows this paper.

Particle Breakage due to Bulk Shear

113

Fig. 16. Percentage attrition expressed as a function of dimensionless gap size for urea particles.

6. WHAT NEXT? The damage to particles during processing often arises due to shear; it is a matter of very great importance which has signiﬁcant consequences for processing. The work described here seeks to extract information about the breakage that occurs in a failure zone, taken on its own. Carrying out and analysing these tests is demanding and many issues arise. Why is the Gwyn formulation so often successful, for instance? It leaves open the difﬁcult question as to how the ﬂow pattern developed in processing equipment is to be predicted so as to make full use of the data. We usually lack a precise understanding of the distribution of velocities in the ﬂow, this stemming from the difﬁculty of measuring these experimentally. Methods are available such as the use of a positron camera but these are not easy to use on the industrial scale. Strain rates are important which rely on the derivative of velocity. However, it is certain that we shall be gaining a gradually increased insight into the ﬂow in equipment both from experiment and from the application of modelling DEM code. Understanding behaviour in narrow gaps is more accessible both experimentally and theoretically using DEM and this may be a good target for immediate work. Use of further code in which the movement of

114

J. Bridgwater

stress waves through individual particles has been demonstrated and offers a link between ﬂow and attrition but is going to be most demanding of computer resources. Dealing with changing particle shape and particle segregation adds to the complexity. The indications from the work on population balance modelling suggest some problems in the formulation relying on a selection function for breakage as well as a breakage function itself. The event of selection and the form of breakage are each dependent on the local packing structure. Is there scope for novel imaginative physical thinking? Can we develop a model of force chains that deals with the frequency of a particle lying in a chain, its critical positions therein, and the consequential particle damage, for example? The use of the annular cell to control the design of particles through optimisation of the mode of manufacture is strongly advocated. Also, a great deal could be gained from the accumulation and reporting of attrition rates as a function of equipment type, material and operating conditions. It may be possible to use the attrition caused in equipment to be turned into an equivalent behaviour in the annular shear cell. The annular cell could be improved by the simultaneous measurement of torque so that there is a direct measurement of shear stress at the imposed normal stress and additionally as a function of the extent of attrition. The recording of cell height to provide information for ﬁnding the change in volume is also desirable. Methods of viewing both ﬂow and the detailed occurrence of attrition during shear, say using MRI, would be very welcome. It is not an easy area but it should be one that commands focus from grant-giving organisations but sadly would not beneﬁt the author writing from an Emeritus position. It needs more people to be engaged in the ﬁeld. This is an area of great importance.

Nomenclature

dp dT D Do F G K KN m R2 t W

initial diameter of particle in cell (m) particle size denoting attrition (m) particle size in size distribution function (m) smallest particle size in feed (m) cumulative size distribution function (–) cumulative size distribution modulus equation (1) (–) constant determining attrition equation (3) (–) constant determining attrition equation (4) (–) index determining rate of attrition equation (3) (–) regression coefﬁcient (–) time (s) mass fraction of particles broken (–)

Particle Breakage due to Bulk Shear

WT x b Dz e Z f s sscs

115

mass fraction of particles broken extrapolated to dT (m) fraction of material broken in equipment tests (–) parameter describing the effect of stress on attrition shear strain in annular cell (–) distance of centre of particle of above top of groove (m) stf/sscs (sf) gripping ring parameter equation (1) (–) parameter describing the effect of strain on attrition (–) normal stress applied (MPa) side crushing stress of particles (MPa)

Population balance B d D Dmax Ds g Ka Kf L m n p R S t T xo a G D

breakage function (–) particle size chosen for breakage (m) particle size (m) largest particle size in feed (m) smallest particle size to be measured (m) mass frequency size distribution function (m1) abrasion rate coefﬁcient (m(1r) s1) fracture rate coefﬁcient (mn s1) linear dimension (m) fracture breakage index in equation (18) (–) fracture selection index in equation (17) (–) probability function (–) residual size distribution (–) selection function (s1) time (s) dimensionless time [22] (–) ratio of smallest to largest particle sizes in feed (–) degradation mechanism parameter; a ¼ 1 for fracture alone, a ¼ 0 for surface abrasion alone (–) size distribution modulus (equivalent to G in experiments) (–) small change in particle size (m)

Subscripts a f

abrasion fracture

REFERENCES [1] Z. Ning, M. Ghadiri, Chem. Eng. Sci. 61 (2006) 5991–6001. [2] B.K. Paramanathan, J. Bridgwater, Chem. Eng. Sci. 38 (1983) 197–206. [3] M.J. Hvorslev, Proc. Am. Soc. Test. Mater. 39 (1939) 999.

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[4] [5] [6] [7]

D.J. Stephens, J. Bridgwater, Powder Technol. 21 (1978) 17–28. M. Ghadiri, Z. Ning, S.J. Kenter, E. Puik, Chem. Eng. Sci. 55 (2000) 5445–5456. C.R. Bemrose, J. Bridgwater, Powder Technol. 49 (1987) 97–126. C. Couroyer, M. Ghadiri, P. Laval, N. Brunard, F. Kolenda, Oil Gas Sci. Technol. 55 (1) (2000) 67–85. B.K. Paramanathan, J. Bridgwater, Chem. Eng. Sci. 38 (1983) 207–224. A.M. Gaudin, T.P. Melloy, Trans. AIME 223 (1962) 40–43. A.U. Neil, J. Bridgwater, Powder Technol. 80 (1994) 207–219. J.E. Gwyn, AIChE J. 15 (1969) 35–39. M.L. Wyszynski, J. Bridgwater, Tribol. Int. 26 (1993) 311–317. A.U. Neil, J. Bridgwater, Powder Technol. 106 (1999) 37–44. Y. Hiramitsu, Y. Oka, Int. J. Rock Mech. and Miner. Sci. 3 (1985) 89. C.E.D. Ouwerkerk, Powder Technol. 65 (1991) 125–138. J. Bridgwater, R. Utsumi, Z. Zhang, T. Tuladhar, Chem. Eng. Sci. 58 (2003) 4649–4665. A.V. Potapov, C.S. Campbell, Powder Technol. 94 (1997) 109–122. S.J. Antony, M. Ghadiri, First MIT Conf. on Comput. Fluid and Solid Mech., MIT USA, 2001, pp. 36–38. S.J. Antony, M. Ghadiri, J. of Appl. Mech. 68 (2001) 772–775. S.J. Antony, M. Ghadiri, Proc. of the ASME World Congress, New Orleans, USA, 2002, pp. 1–4. D.J. Golchert, R. Moreno, M. Ghadiri, J.D. Litster, Powder Technol. 143–144 (2004) 84–96. N. Ouchiyama, S.L. Rough, J. Bridgwater, Chem. Eng. Sci. 60 (2005) 1429–1440. Y. Nakajima, T. Tanaka, Funsai 19 (1974) 2–11. C.C. Crutchley, J. Bridgwater, Kona 15 (1997) 21–31.

[8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

CHAPTER 4

The Principles of Single-Particle Crushing Georg Unland Technische Universitaºt Bergakademie Freiberg, Institut fuºr Aufbereitungsmaschinen, Germany Contents 1. Introduction 2. Terminology 3. Deﬁnition 3.1. Deﬁnition based on physical phenomena 3.1.1. Comminution effects 3.1.2. Comminution phases 3.2. Deﬁnition based on grain sizes 3.3. Deﬁnition of the term crushing 4. Concepts of investigation 4.1. Physical models 4.1.1. Physical formulation and mathematical methods 4.1.2. Loading conditions 4.1.3. Material model 4.2. Empirical models 4.3. Assessment of concepts 5. Crusher as a system 6. Crushing parameters 6.1. Deﬁnition of related and equivalent features 6.2. Crushing resistance 6.2.1. Breakage probability 6.2.2. Crushing force 6.2.3. Loading time 6.2.4. Energy consumption 6.3. Crushing product 6.3.1. Particle size distribution 6.3.2. Particle shape distribution 6.3.3. Energy utilization 7. Applications 7.1. Mechanical design 7.2. Process design

118 118 119 120 120 122 123 125 126 126 126 131 132 142 143 145 148 150 153 154 164 176 179 184 185 202 207 214 214 216

Corresponding author. Tel.: +49 3731 392558; Fax: +49 3731 393500; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12007-8

r 2007 Elsevier B.V. All rights reserved.

118 7.3. Properties of particulate materials 7.4. Energetic assessment of crushers Acknowledgements References

G. Unland 217 218 218 222

1. INTRODUCTION Crushing is a mechanical process where larger solid particles of brittle material are subjected to sufﬁciently high forces and energy with the consequence that the material of the particles fails and fragments are generated. This process happens predominantly in crushers. Besides screens, crushers treat by far the highest amount of solids among all mineral processing machines. In the world more than 25 billion tons of rocks, ores, and coals are crushed annually. Crushers are installed in processing plants for various reasons. The most common application is to produce a certain product size distribution or upper size for requirements of a special market or subsequent machines. Further applications are related to the production of special particle shapes, the selective crushing and liberation of desired and/or undesired minerals or materials, and ﬁnally the generation of larger or special particle surfaces. Therefore, crushers are installed as ﬁrst and intermediate machines within a processing plant or as a machine to produce the ﬁnal product. The conﬁguration can be in open or closed circuit. For those purposes a variety of different crushers can be considered; machines, which mainly apply the effect of compression, such as jaw, gyratory, cone (Hydrocone type) and roll crushers and others, which use predominantly the effects of impact (impact crushers and rotary breakers) and percussion (special cone (Simons type) and hammer crushers). Each group of machines can be subdivided in numerous different designs. Inside the process zone of such crushers particles are loaded and comminuted as single particles and/or in a particle bed, whereas the amount of single-particle comminution depends on the type, design and operation of the machine. Figure 1 gives a rough estimate. This section deals with ‘‘single-particle crushing’’ as an important microprocess in the ﬁeld of comminution. It is a part of the macroprocess ‘‘crushing’’ which happens inside the process zone of a crusher.

2. TERMINOLOGY The term ‘‘crushing’’ is related to a certain kind of comminution. It depends on the material properties and the grain size distribution of the feed and/or the product.

The Principles of Single-Particle Crushing type of crusher

119

share of single-particle comminution[%]

jaw crusher

minor

predominantly

gyratory crusher 0%

100%

cone crusher Simons type Hydrocone type

roll crusher smooth surface contoured surface

rotary breaker

impact crusher HSI (horizontal shaft) VSI (vertical shaft)

hammer crusher

Fig. 1. Estimated share of single-particle comminution.

Usually the word ‘‘crushing’’ is used if brittle material, such as hard rock or hard coal, is fractured by a tool. However, in certain industries this word is applied for other materials as well. Lignite, salt or clay are not brittle, but the corresponding industries use crushers for their comminution. The disintegration of moist clay by crushers could be for instance better described by the term ‘‘cutting’’. Similar problems occur with the use of the word ‘‘crushing’’, if the size of the feed and/or the product is considered. Generally, crushers are installed for coarser lumps and mills for smaller particles ([1], p. 3A–5f, [2], p. 110ff). Are SAGmills then mills although their feed size exceeds occasionally 500 mm and VSIcrushers then crushers although their feed size can be as small as 5 mm? There is no consistency with the application of the term ‘‘crushing’’. Therefore, it is necessary to deﬁne the related words.

3. DEFINITION In process engineering comminution comprises a major group of unit operations. In principle it can be divided by physical phenomena, mechanisms, effects, grain

120

G. Unland

sizes and materials to be crushed, whereas the borderline is not well deﬁned. Generally there are two options to deﬁne ‘‘crushing’’. The ﬁrst option relates to phenomena during the process of comminution, the second one to the grain size. The deﬁnition needs to consider both options in order to be applied in science and engineering.

3.1. Deﬁnition based on physical phenomena The process of comminution is characterized by different effects and by various phases on a macroscopic scale. They depend on the location, kind and magnitude of the energy respectively the force applied to a particle.

3.1.1. Comminution e¡ects With the transfer of forces and energy to a particle various effects can be observed. The surface and/or the interior of the particle change their cohesion. The following effects can be distinguished (Fig. 2): a. Weakening: A particle subjected to repeated loading events below the level where cracking occurs experiences a weakening of the material compound. No.

effect

a

weakening

b

cracking

c

breaking

d

crumbling

e

chipping

f

splitting

g

disintegrating

Fig. 2. Comminution effects.

feed

loaded particle

product

...

...

The Principles of Single-Particle Crushing

b.

c. d.

e. f.

g.

121

The strength of the material decreases with every additional loading event. This effect is termed as fatigue too. The particle retains its shape. Cracking: Cracking occurs, if the material fails due to the loading of the particle. The energy is not sufﬁcient in order to penetrate the crack through the whole particle. The particle keeps its shape. Breaking: Breaking happens, if the energy is sufﬁciently high to move the crack through the whole particle. Two fragments are produced. Crumbling: In most cases the loads are introduced into the particle via the surface. Crumbling occurs, if the energy is only high enough to disintegrate the surface. Many ﬁne particles are generated besides a bigger one only slightly smaller than the feed particle. Crumbling can be termed as attrition or abrasion as well. The direction determines the loading. Thus, it may occur either perpendicularly (attrition) or tangentially (abrasion) to the surface of the particle. Chipping: Particles often show edges and corners. If they brake apart, chipping happens. The product consists of one large particle and a few smaller fragments. Splitting: With higher input of energy several major cracks can move through a particle. It is fractured into a few bigger fragments. This process can be described as splitting or cleavage ([3], p. 117). Disintegrating: The whole compound is disintegrated, if the offered energy to the particle exceeds by far the required energy to break. The disintegrating yields a large number of smaller fragments with a wide size distribution.

As a consequence the various effects produce different kinds of fragment size distributions (Fig. 3). Typical are bimodal distributions with crumbling and a

b

f 100%

feed

d e c g f

e

c

d 0%

dF

d

Fig. 3. Quantity f of fragments with different comminution effects (comminution of only one particle).

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G. Unland

chipping. In reality these effects do not occur isolated. They happen in series or in parallel during the various phases of comminution.

3.1.2. Comminution phases The comminution of particles inside machines uses various physical principles, such as compression or impact. The generation of cracks and fractures requires a certain amount of load and energy. During compression a particle is exposed to a certain displacement, during impact to a certain impact velocity. The loading is accompanied by an induction of stresses and deformations within the particle and an energy transfer from the machine to the particle, whereas not the total transferred amount of energy is used for comminution. During the process of comminution various phases can be observed on a macroscopic scale. As an example a typical compression diagram can be used (Fig. 4). In a ﬁrst phase (preparatory phase) the particle is relocated and adjusted. The irregularities of the particle surface are smoothened at the contact planes with the machine (effects d and e) until sufﬁciently large contact planes exist and the particle remains in a stable position in order to sustain the increasing displacement and loading of the machine. In this phase only smaller fragments separate from the major particle. In a second phase (initial breakage phase) a major portion of the particle volume can be deformed and stressed until a certain limit is reached. Cracks develop and grow with the consequence that ﬁnally the particle breaks, followed by a release of the contact forces. This second phase (effects b, c, f) usually 120 preparatory phase

100

F [kN]

80 phase 1 preparatory phase

60

grinding phase

crushing phase phase 2 initial breakage phase

phase 3 multiple breakage phase

phase 4 high pressure grinding phase

40 breakage point

maximum crushing force

20

0 0

10

20

30

40

50

h [mm]

Fig. 4. Compression diagram (granodiorite, Kindisch/Sachsen, dF ¼ 75 mm).

60

The Principles of Single-Particle Crushing

123

generates some bigger pieces. With further displacement the contact forces rise again after the contact planes are smoothened. Depending on the particle they may either exceed or not exceed the initial breakage force, and cause additional fractures until a particle bed is formed. During this phase the fragments do not act in many cases as isolated individual particles although they are separated. They interact on the contact planes of the separated fragments by friction, interlocking, etc. They form a compound of fragments. Only in special cases (i.e. spheres of brittle material) the fragments ﬂy away and each fragment is then comminuted as an individual particle. In this third phase (multiple breakage phase) more ﬁne particles are produced (effects b, c, d, e, f, g). The crushing phase comprises the second and third phase. With an additional increase of the displacement the particle bed is deformed and the contact forces rise steeply to extremely high forces as the porosity of the particle bed approaches zero. In this phase (high pressure grinding phase) the total structure of the material is destroyed (effect g). The initial particle is ground to ﬁne particles in a high pressure particle bed. Particles subjected to impact or percussion show often an additional effect. If the initial loading of the particle is not sufﬁcient, no cracks occur. However, the repeated loading of a particle below the level of cracking weakens the material. The strength of the material decreases because of fatigue (effect a). The crushing happens then on a lower level of loading. The description of the comminution process makes obvious that it is not sufﬁcient to categorize the process by just mentioning the physical phenomena. There is an uninterrupted shift from one phenomenon to the other one by the application of one physical principle during one process. The preparatory phase generates only small fragments and maintains the principal dimensions of the initial particle. The crushing phase is characterized by the destruction of the macroshape of the initial particle, the generation of bigger fragments besides smaller ones, the occurrence of one or more peak forces, the following force releases and not necessarily increasing forces during the progression of the process. The breakage point is deﬁned as located at the ﬁrst maximum force and lies within the crushing phase. It separates the initial breakage phase from the multiple breakage phase. In the grinding phase the total structure of the initial particle is dissolved and only smaller fragments are generated.

3.2. Deﬁnition based on grain sizes Many authors deﬁned the borderline between crushing and grinding by a certain grain size. They use either a deﬁned feed or product size. Figure 5 shows the various deﬁnitions, which are based on feed sizes, whereas Fig. 6 depicts the ones related to product sizes.

124

G. Unland 10 0

10

1

10 2

103

author

Kelly et al [3], p. 127

crushing

grinding

dF

Hukki [4], p. 404

grinding

Schubert [5], p. 110

grinding

grinding

Höffl [6]

104 [mm]

secondary crushing

primary

explosive

crushing

shattering

intermediate crushing

coarse crushing

intermediate

coarse

crushing

crushing

dF

dF

dF

Fig. 5. Deﬁnition of crushing based on feed sizes. 10-1

100

101

102

author

Gaudin [7], p. 25

grinding

Pahl [8], p. 45

grinding

crushing

fine crushing

103 [mm]

dP

coarse cru shing dP

Fig. 6. Deﬁnition of crushing based on product sizes.

Another easy approach to deﬁne ‘‘crushing’’ and to distinguish between ‘‘grinding’’ and ‘‘crushing’’ is the investigation of the feed or the product sizes in front of or after a crusher or a mill. The analysis of the feed to crushers yields maximum dimensions of rocks up to 2 m for the case of gyratory or roll crushers as a ﬁrst stage in the mining industry and minimum dimensions of approximately 5 mm in front of a VSI-crusher as a ﬁnal stage to produce chippings in the aggregate industry. The product sizes are maximum 500 and 0–2 mm, respectively. The analysis of the transfer product between the crushing and grinding plant gives different results as well. A typical ball mill feed in ore grinding is up to 15–25 mm, a roller mill feed in limestone grinding up to 50–100 mm, an impact mill feed in lignite grinding up to 40–60 mm. The compilation indicates clearly, what Taggert already stated in 1956: There is a twilight zone in which the product is 6- to 10- or 14-mesh limiting size,

The Principles of Single-Particle Crushing

125

which is either crushing or grinding according to the type of machine used ([9], p. 4-01). As a consequence the examples show that it is impossible to derive a scientiﬁcally correct deﬁnition, i.e. in this case any deﬁnition is somehow arbitrary.

3.3. Deﬁnition of the term crushing Crushing is a size reduction process for particles of brittle materials. It is characterized by particle diameters, volumetric equivalent diameters and by force releases on the working surfaces of the machines after the crushing events happen. Furthermore, crushing is a technical process with numerous particles. In industry particle size distributions are often assessed by one characteristic diameter. This diameter relates usually to a certain cumulative percentage (e.g. 80%) passing or retaining a size, i.e. the particle size distribution is characterized by just one point on the cumulative undersize or oversize curve (which is not correct and sufﬁcient). Crushing produces fragments. The borderline between small and large fragments is set by their volumetric equivalent diameter and by a tenth of the original feed diameter. Fines are all fragments with an equivalent diameter smaller than 1 mm. Crushing is then deﬁned, if an assembly of particles of a brittle material with diameters of up to 2 m is reduced in size and yields a product size distribution, where minimum 80%Vol of the product consists of particles larger than 1 mm (ﬁnes), i.e.: ‘‘Crushing’’ as macroprocess occurs, if feed is def

d F 100% P 2 m

ð1Þ

and product is def

80% R1 mm d p

ð2Þ

For single particles there are several additional deﬁnitions possible to deﬁne crushing. It can be related to a certain loss of mass or volume of the initial particle or to a certain decrease of crushing forces. In literature breakage as the ﬁrst crushing step is mostly deﬁned, if the initial particle looses 10% of its original mass [5]. Since crushing is a size reduction process it is better to relate the loss to the volume. The consequences are often the same, because many particles show a homogeneous distribution of the density on a macroscopic level. Other authors deﬁne breakage, if the comminution force drops steeply by more than 50% ([10], p. 120). It is very simple to indicate breakage with spheres of brittle material. They break instantaneously so that the sudden release of the force to zero indicates the end of the breakage. Usually the fragments ﬂy away in this case.

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G. Unland

Therefore, it is recommendable to extend for the microprocess ‘‘single-particle crushing’’ the conditions (1) and (2) by product is ﬃﬃﬃﬃﬃﬃﬃ def p 3 d P 0:9d F ð3Þ

4. CONCEPTS OF INVESTIGATION The process of crushing can be described and explained by different concepts. They are related to various kinds of models, which are formulated by mathematical methods. Basically two kinds of models can be used, physical and empirical models.

4.1. Physical models The physical models are based in general on mathematical formulations of relations between forces (stresses), displacements (strains), time and temperature. They use laws to describe an equilibrium or a motion of a body of a material. The body itself can be considered as continuum or as discontinuum, i.e. it consists of sub-bodies. Those models were developed to formulate the behaviour of a body on an atomistic, a microstructural and a macroscopic level. It must be emphasized that in principle all laws rely and base on constitutive assumptions and need to reﬂect ﬁnally phenomenological effects, even if a model is theoretically derived at ﬁrst. The physical models follow a certain pattern (Fig. 7). Crushing happens, when a body, a particle is subjected to external and internal loads (loading conditions). The particle itself can be characterized by its geometry and material (geometry and material model). Inside the particle the loading results in a certain state, which depends on the location and time as well as the boundary and initial conditions (formulation of the physical model). There are several mathematical options to determine the internal state of the particle (mathematical methods). If the state conditions exceed a certain limit, the material fails (failure criteria of the material model) and the particle breaks into fragments.

4.1.1. Physical formulation and mathematical methods The crushing of a particle is a physical process, which needs to be formulated mathematically. There are several physical formulations possible and mathematical methods available. They depend on the complexity of the models (Fig. 8).

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127

loading boundary initial conditions mathematical methods

geometry model

physical model

result

material model stress-strain model

failure criteria

Fig. 7. Physical model.

Particle as continuum

On a microstructural and macroscopic scale a body can be seen as a continuum. Its behaviour can be assessed by applying the classical ﬁeld theories. With the boundary and initial conditions the state of the ﬁeld can be formulated, such as the stress or strain distribution inside the body. The disadvantage of classical continuum models is their difﬁculty with or their inability to cope with small scale resolution (atomistic level). There are compromises to overcome those problems by incorporating non-linear terms. Another option is the application of non-local continuum mechanics [11]. In engineering shape and notch factors are widely used to determine the inﬂuence of certain specimen geometries on the stress ﬁeld [12,13]. These factors are derived either from theoretical calculations or from experiments. Concerning comminution the theories of damage and fracture mechanics are interesting as well. The ideas of damage mechanics explain the phenomena which happen in a material free of any damage at the microscopic scale and lead to macroscopic crack initiation [14] ([15], 346ff). The ﬁndings of fracture mechanics describe the phenomena which deal with the evolution of cracks up to the complete break of the body. The concept of fracture mechanics basically introduces an additional structural variable, the ﬂaws and their sizes, and considers a certain part of a body with a crack as a thermodynamic system. With brittle materials the system energy associated with crack formation is comprised of the

128

G. Unland physical models macroscopic level

microstructural atomistic

physical formulation

continuum

infinitesimal elements

discrete elements

discrete elements

body

discretization

finite

infinite

degrees of freedom of internal elements

defined

identified

location of element coupling

complex models

numerical methods solution

discontinuum

differential methods (FEM, FDM)

integral methods (BEM)

simple models

analytical / numerical methods

complex models

complexity of physical model

numerical methods Distinct Element Method (DEM)

mathematical methods

Fig. 8. Formulation of physical models and mathematical methods (selection).

energy offered by the system and the free energy of the new crack surfaces. Depending on the energy balance a crack extends or retracts. Besides stresses and strains the approach of fracture mechanics uses stress intensity factors K, contour integrals J and energy release rates G to characterize the singularity of the stress ﬁeld in the vicinity of the crack tip and to describe the effects of fracture. The stress intensity factors K determine the stresses, strains and displacements near the crack tip, while the energy release rates G quantify the net change in potential energy that accompanies an increment of crack extension. G describes the global behaviour, while K is a local parameter ([16], p. 69f). The contour integrals J are line integrals related to energy in the vicinity of a crack. For perfectly brittle, linear elastic materials the energy release rates G are related to the stress intensity factors K(GK2) and are equal to the contour

The Principles of Single-Particle Crushing

I opening mode

II sliding mode

129

III tearing mode

Fig. 9. Basic modes of fracture ([17], p. 24).

integrals J (G ¼ J). In general only stress intensity factors K are used for brittle materials, since they are easier to determine. The stress intensity factors K depend on the applied loads and the body geometry. They determine the intensity of the local stress and strain ﬁeld. The distribution of the ﬁeld is given by additional terms, which are functions of the spatial coordinates. Three different modes of crack-surface loading and displacements are to be distinguished (Fig. 9). Mode I (opening mode) is related to normal separation of the crack planes due to tensile stresses, mode II (sliding or in-plane shear mode) to longitudinal shearing of the planes in a direction perpendicular to the crack tip and mode III (tearing or out-of-plane shear mode) to lateral shearing parallel to the fracture front. However, mode I is the most important one for brittle materials, since cracks in those materials tend to orientate in a direction that the shearing stresses are minimized ([17], p. 23f). Particle as discontinuum

Alternatively, a body can be considered as discontinuum. In this case a body consists of real or virtual sub-bodies which interact by forces. It depends on the scale whether those sub-bodies are, for example atoms, molecules (atomic scale), crystals, discontinuities (microstructural scale) or grains, pores (macroscopic scale). On an atomic scale especially the ﬁndings of quantum mechanics can be applied. A particle of certain materials and with a deﬁned fabric (e.g. a granite rock) can be represented by a dense packing of polydisperse bodies that are bonded together at their contact points by cement, forming a bonded-body model, a bonded-particle model (BPM) [18]. Depending on the fabric and the material the cement can be true or notional. The bonded contacts can transmit compressive,

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tensile and shear loading as well as bending moments, whereas the loose contacts experience only compressive and shear loading. Mathematical methods

For very simple and ideal geometries (e.g. cube), material laws (linear elastic) and loading conditions (tension at two opposite planes) as well as ﬁeld and failure assumption (homogeneous ﬁeld, failure at tensile strength), the mathematical formulation comprises only simple equations which can be solved analytically. If those conditions and assumptions become more complex, the particle needs to be described by a sum of inﬁnitesimal elements (particle as continuum) or by a sum of simple ﬁnite (discrete) elements (particle as discontinuum). The behaviour can then be assessed by solving one or a series of differential equations. Only in a few cases can exact solutions be obtained. If this is not possible, numerical solving methods are available to obtain approximate solutions, such as the Euler or the Runge-Kutta methods. Only numerical methods are applicable, if complex conditions and assumptions are taken. Several methods were developed. They can be classiﬁed according to the kind of the physical formulation (continuum or discontinuum methods), the kind of domain discretization (differential or integral methods) and the kind of time discretization (implicit or explicit methods) ([19], p. 9). In case a particle is considered as a continuum it is subdivided into a ﬁnite number of elements whose behaviour is approximated by physical and geometric relations with ﬁnite degrees of freedom. The elements must satisfy the differential equations of the problem (same as with analytical methods) and the continuity conditions at their interfaces with adjacent elements. The continuum method is an approximation of a continuum with inﬁnite degrees of freedom and variable geometry by discrete elements with ﬁnite degrees of freedom and deﬁned geometry. The displacement compatibility has to be enforced between these internal elements ([20], p. 288). The differential methods (e.g. ﬁnite element method (FEM); ﬁnite difference method (FDM)) use the interior discretization and necessitate the complete discretization of the body (particle). The integral methods (e.g. boundary element method (BEM)) require only the discretization of the surface of the body or regions inside a body in order to apply the boundary conditions. With the application of discontinuum mechanics (e.g. distinct element method (DEM)) a body (particle) is subdivided into blocks, which interact by contacts. Complete decoupling and individual motions of the blocks are possible. Therefore, displacement compatibility is not required between the blocks. However, the contacts between the blocks need to be identiﬁed and classiﬁed during the entire computing process. In case of contact it has to be described by constitutive models. Figure 10 illustrates the different discretization concepts of a particle with joints and faults.

The Principles of Single-Particle Crushing joints

131

faults

joint element rock mass

FDM, FEM

region 1 block region 4

region 2

region 3

block

element of displacement discontinuity

BEM

regularized discontinuity DEM

Fig. 10. Discretization concepts ([20], p. 289).

The numerical methods comprise hybrid models (e.g. FEM/BEM, DEM/DEM, DEM/FEM) as well. A particle which is modelled for instance by DEM (external behaviour of the blocks) can incorporate deformable blocks with FEM discretization (internal behaviour of the blocks).

4.1.2. Loading conditions In crushers particles are subjected to different kinds of loads. Thereby various physical principles can be applied [21]. However, in crushers only those principles are used, which introduce the load by the working surfaces (Fig. 11). The application of a certain loading case depends on the material and its behaviour. Since predominantly brittle materials are comminuted in crushers mainly three loading cases are only used. In jaw, cone, gyratory and roll crushers the particles are loaded by two working surfaces causing compression or percussion and to a lesser degree bending and shearing. Impact and hammer crushers as well as rotary breakers load a particle mainly with impact by one working surface. On purpose, several machines are designed in a way that the offered energy is in excess to the energy necessary to crush the particle. In case of compression (e.g. jaw crusher) and percussion (e.g. cone crusher, type Simons) the machine imposes a certain stroke and in case of impact (e.g. impact crusher) a certain velocity on a particle no matter how much resistance it shows. The energy of the machine is so big that hardly any reaction can be noticed. There are two

132

G. Unland v

F

F

F F compression

F

v

v

F

friction

percussion

bending

F

F

F

F cuttings

shearing

F

F splitting

v v

v

impact

v

impact

impact

Fig. 11. Loading cases ([22], p. 4).

exceptions where the magnitude of the offered energy is limited. In hammer crushers the hammers are suspended in bearings. In case of impact and percussion the hammers swing, occasionally rotate. The offered energy of a hammer is purposely limited due to the design. A similar limitation occurs in rotary breakers because of the deﬁned falling conditions of particles inside the drum.

4.1.3. Material model The material model comprises the relationship between loading and deformation as well as the failure criteria.

4.1.3.1. Stress– strain behaviour The behaviour of a material under load can be best described by the stress–strain curve. The simplest form is generated by an uniaxial compression test on a cubical or cylindrical specimen (Fig. 12).

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133

σ σmax

ε

Fig. 12. Stress–strain curve ([23], p. 87).

σ

σ'

tangent modulus dσ M t = dε

dσ dε

secant modulus σ' M s = ε'

ε'

ε

Fig. 13. Deﬁnition of tangent and secant modulus.

The curve is characterized by its maximum, the peak strength smax and its slopes. The peak strength indicates a failure locus and is termed also as the c . The slope can be determined in two ways. The uniaxial compressive strength R slope of the stress–strain curve at a given point is named tangent modulus Mt and the slope of a line connecting a point of the curve with the zero point secant modulus Ms (Fig. 13). Therefore, the moduli of a given stress–strain curve look differently (Fig. 14). Usually, the tangent modulus is given at 50% of the peak strength, except when otherwise mentioned. For ideal elastic materials the tangent and secant

134

G. Unland Ms

Mt

ε

ε

Fig. 14. Tangent and secant moduli of stress–strain curve (Fig. 12) ([23], p. 87). σ

B

A C

ε

Fig. 15. Slopes of stress–strain curves.

moduli are identical and constant up to the peak strength and are then called The secant modulus is often referred as the deformation Young’s modulus E. modulus of the material. The tangent modulus at the peak strength is an indication of the brittleness. The higher this modulus the more brittle is the material ([24], p. 60). The ﬁrst phase of the stress–strain curve can depict three different kinds of slopes (Fig. 15). The type A material shows a linear elastic behaviour up to the point of failure, i.e. E is constant. Many hard and strong materials exhibit this type of curve, such as granite, gabbro, basalt, quartzite, very strong and dense sandstones and limestones. The type B material depicts a lower stress increase with every additional increment in strain. The tangent modulus is highest at the beginning of loading and continuously decreases till the failure occurs. This effect is usually termed as strain-softening behaviour. Many softer materials, such as shales, tufts, softer limestones, and stratiﬁed coals, loaded parallel to the bedding planes show this shape of curve. The type C material is characterized by an additional increase of stress with every further increment of strain. The tangent modulus increases continuously with the loading of the material. This effect is named strain-hardening behaviour as well. Rock salt and stratiﬁed rocks, such as softer sandstone or coal, loaded perpendicularly to the bedding planes exhibit this behaviour.

The Principles of Single-Particle Crushing

135

σ

class I class II

ε

Fig. 16. Stress–strain curves with different post-failure behaviour ([25], p. 569).

Beyond the failure locus the stress–strain curve characterizes the post-failure behaviour of the material (Fig. 16). The behaviour varies to a great extent and can be assessed by its tangent modulus. This modulus is often referred as postfailure modulus or stiffness as well. Two fundamental types of material behaviour can be distinguished [25]. Class I materials exhibit a negative tangent modulus and stable fracture propagation. Energy is still needed to be transferred to the particle to cause further load reduction. Even after the failure of the specimen at the peak strength the material remains some strength although it is fractured. Class II material has a positive tangent modulus. The fracture process is unstable and self-sustaining. Whenever the stress equals the strength of the material the elastic strain energy stored in the particle is sufﬁcient to maintain the fracture propagation until the material has lost virtually all strength. The energy stored is high enough for the total breakdown of the specimen. Many particles exhibit an ‘‘explosive’’ failure, fragments ﬂy away. Materials with a positive high tangent modulus at and beyond the peak strength (type A or C/Class II materials) are considered as perfectly brittle. The vertical dashed line separates materials with class I post-failure behaviour from those with class II behaviour. It has to be mentioned that the post-failure behaviour of the particle – like the pre-failure behaviour – does not depend solely on the material, but also on the shape of the particle, the strain rate, the stiffness and the operation mode of the machine [23].

4.1.3.2. Failure criteria There are several theories available to explain the failure of a material, but as yet there is no comprehensive understanding. Therefore, numerous criteria were developed. In engineering they are based on forces and displacements (or stresses and strains), and their combinations, such as stress, strain or energy

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G. Unland

criteria: ¼ f ðsi ; j Þ R

ð4Þ

For simple loading conditions the strengths are related to the four basic loading cases: Tension/compression: Bending: Shear: Torsion:

t =R c uniaxial tensile/compressive strength R bending strength Rb s shear strength R ts torsional strength R

A compilation of uniaxial compressive strengths for some geomaterials is summarized in Table 1 and Table 2 gives an indication of approximated relations between different kinds of strengths for hard rocks. Usually the loading conditions are more complex. They consist of several basic loading cases. There are two options to formulate the limits. The ﬁrst option is based on different theories. They determine how the various stresses can be converted into an equivalent normal stress. This stress is then

Table 1. Uniaxial compressive strength (UCS) ([22], p. 5)

material Clay

UCS (MPa) 0.2C6

Lignite Hard coal Coke

1.5C4 10.0C50 10.0C20

Brick Concrete

10.0C20 20.0C50

Rock salt Limestone Argillite Sandstone

25.0C55 4.0C200 25.0C170 10.0C320

Granite Gabbro Diabase Basalt

80.0C300 100.0C280 120.0C300 80.0C580

Gneiss Greywacke

60.0C250 180.0C360

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137

Table 2. Approximated relations between strengths for hard rocks (data from [26], p. D21, [27], p. 78, [28], p. 2, [29], p. 205, [30], p. 101ff)

Load

Strength

Approximate share of compressive strength

Tension Bending

t R b R

t ð 1 C 1 ÞR c R 25 10 1 1 b ð C ÞR c R

Shear

s R

s ð 1 C 1 ÞR c R 15 10

20

5

σ1

σn σ3

σ3

τ

β

σ1

Fig. 17. Stresses on failure plane.

assessed by a certain strength, usually the uniaxial tensile or compressive strength. A typical example is the theory according to von Mises–Hencky, a concept of failure related to the constant energy of distortion. A material fails, if a certain amount of shear energy is exceeded. The second option uses different relations between stresses at the point of failure. A widely applied theory for geomaterials is the Mohr–Coulomb criterion. A rock fails at a critical combination of normal and shear stresses (Fig. 17). The transmitted shear stress on an inclined plane is comprised of two components, cohesion and friction ([23], p. 108): jtj ¼ to þ msn

ð5Þ

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Mohr envelope

τ0

tensile cut-off Rt

2β

σ3

Rt

σ1

Rc

σ

Fig. 18. Mohr–Coulomb failure criterion with tensile cut-off.

with to cohesion, m coefﬁcient of friction and 1 ðs1 s3 Þ sin 2b 2

ð6Þ

1 1 ðs1 þ s3 Þ þ ðs1 s3 Þ cos 2b 2 2

ð7Þ

jtj ¼ sn ¼

The equations for jtj and sn are the equations of circles in the (st) space, the Mohr’s circles (Fig. 18). The limit of the Mohr’s circles is represented by a Mohr envelope, derived from the cohesion to and the coefﬁcient of friction: m ¼ tan j. t the limit of sustained stresses Because of the limited uniaxial tensile strength R is set by a tensile cut-off. For any st combination below the envelope no failure occurs. The failure of the material is represented by the envelope. The plane of failure is oriented at j b ¼ 45 þ ð8Þ 2 The investigation of several geomaterials has shown that the envelope is not a straight line. Therefore, several empirical strength criteria were developed in order to match the experimentally found strength data (compilation see [23], p. 112). A wide range of geomaterials can be reasonably described by the Hoek–Brown failure criterion: c s3 þ bR 2 Þ0:5 s1 ¼ s3 þ ðmR c

ð9Þ

or normalized by s3 c R

ð10Þ

s1N ¼ s3N þ ðms3N þ bÞ0:5

ð11Þ

s1N ¼

s1 c R

and

s3N ¼

The strength can be illustrated by an envelope ([31], p. 191) as well (Fig. 19), ci represents the unconﬁned uniaxial compressive strength of the intact where R

The Principles of Single-Particle Crushing

139

σ 1N

σ1

σ3

σ1 β

uniaxial compression Rc = ( bRci2)0.5

σ3 σ1

triaxial compression σ1N = σ3N + (m σ3N + b)0.5

σ1 σ3

σ3

uniaxial tension R t = ½ R ci [m - (m2 + 4b)0.5] σ3N

Fig. 19. Hoek–Brown empirical failure criterion.

material, a factor m describes the disturbance of the material and a constant b assesses the jointing. The value of m varies between 0.001 for highly disturbed geomaterials and approximately 25 for hard intact materials. Rocks can be grouped as follows ([32], p. 77f): Group a

m E7

Group b

m E10

Group c

m E15

Group d

m E17

Group e

m E25

Carbonate rocks with well developed crystal cleavage (dolomite, limestone, marble) Lithiﬁed argillaceous rocks (mudstone, siltstone, shale, slate) Arenaceous rocks with strong crystals and poorly developed crystal cleavage (sandstone, quartzite) Fine-grained polyminerallic igneous crystalline rocks (andesite, dolerite, diabase, rhyolite) Coarse-grained polyminerallic igneous and metamorphic rocks (amphibolite, gabbro, granite, quartz-diorite, norite, gneiss)

The value of b can range from 0 for jointed and broken to 1.0 for intact rocks. The corresponding failure envelopes are shown in Fig. 20. Consequently, the pre- and post-failure behaviour of a rock can then be described by the Hoek–Brown criterion (see also [33]) and characterized by a series of failure envelopes (Fig. 21).

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G. Unland τN m =15

1.0

m=7 0.8

m=3

0.6 0.4

b=1

σN

1.0 τN 1.0

b=1

b = 0.5

0.8

b=0

0.6 0.4

m = 10

1.0

σN

Fig. 20. Failure envelopes according to Hoek–Brown failure criterion ([31], p. 193f).

60

Rc = 178 MPa m = 15 b=1

50

Peak strength m = 12 b=0

τ [MPa]

40

Residual strength m=2 b=0

30

intact rock fractured rock

10

-10

10

20

30

40

50

σ [MPa]

Fig. 21. Failure envelopes with corresponding data of an intact and fractured rock (greywacke sandstone) ([31], p. 210).

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141

Table 3. Fracture troughness KIc of different geomaterials (compiled from [35], p. 115, [36], p. 20, [37], p. 227f)

material

pﬃﬃﬃﬃﬃ KIc MPa m

Coal Oil shale Limestone Sandstone

0.01C0,3 0.3C0,6 0.7C1,7 1.0C2,1

Granite Diorite Gabbro

0.7C2,4 2.2C2,8 2.2C2,9

Marble Greywacke

1.3C2,3 2.4C3,1

A different kind of failure criteria is introduced by fracture mechanics. The resistance of a material against failure, i.e. crack propagation, is assessed by the fracture toughness Kc or the crack resistance energy Gc. For brittle materials Kc is directly related to Gc for mode I ([34], p. 2): Ic Þ1=2 K Ic K Ic ð1 n2 Þ1=2 ¼ ðEG

ð12Þ

Therefore, it is sufﬁcient to compile just the fracture toughness Kc (Table 3). However, the determination of the material fracture parameters is complicated and expensive. There were several investigations done to ﬁnd an easier and less expensive way to determine the fracture toughness of brittle material. Bearman et al. [38] found that the point load test can provide a highly accurate estimate of the fracture toughness. For round, diametrically loaded, core based samples the relationship is K Ic ¼

35:97 F PL D1:55

ð13Þ

and for irregular lumps and axially loaded core samples K Ic ¼

29:8 F PL ðW DÞ0:775

ð14Þ

with D and W in (mm) and FPL in (kN). With BPMs the microscopic behaviour of the particles is determined by microproperties such as stiffness and strength parameters of the bodies and bonds and the dimensions of the bodies, thus affecting the simulated fracture toughness of the material. An applied load to the particle is carried by the body and bond skeleton in the form of force chains. The forces propagate from one body to the next across the

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cement. If the strength of the cement is locally reached, the bond brakes. If the strength of the neighbouring bonds is reached as well, the broken bonds can form and coalesce into macroscopic fractures [18].

4.2. Empirical models Empirical models follow a different strategy (Fig. 22). They are applied, if no physical model is available which can interpret the phenomena. These models are based on experiments. Usually the different inﬂuencing parameters are varied within a limited framework of deﬁned steps and combinations in order to investigate the inﬂuence on the resulting parameters. In case of crushing these are for instance the kinds of materials and geometries of the grains, the loading conditions and their inﬂuences on the crushing forces. The results of the experiments are then processed by different mathematical methods: Ratings: The experimental results are categorized by deﬁned limits and as-

signed to a certain rating system. There are no equations describing the relationship between parameters (see [39]). Logical descriptions: These methods are based on rules (e.g. IF y THEN y) concerning causes and results to describe the behaviour of a system; in case of crushing the impact of the inﬂuencing parameters on the resulting parameters. This can be done verbally or mathematically. But there is no relationship in terms of equations between the parameters. A typical method is the fuzzy logic. Regression and correlation analysis: This group of methods is based on statistical means to determine a relationship between parameters. It can be expressed by equations and/or graphs. mathematical methods

model

load

geometry

material

result

Fig. 22. Empirical model.

experiment

The Principles of Single-Particle Crushing

143

Neural networks: This approach tries to mirror the cognizable behaviour of a

system by a network of neurons. The neurons operate as processors which perform mathematical operations. The interaction of the results and parameters of the experiments are approximated by a model, which is a priori not given. The model needs to be learnt on the basis of these interactions.

4.3. Assessment of concepts The process of crushing can be explained and described by different concepts. They are based on physical as well as empirical models and allocated at different levels of the dimension scale. The models on the atomic scale can assess the different bonds (e.g. ionic, metallic, covalent bonding), the various defects (e.g. vacant, interstitial and substitutional atoms) and the theoretical necessary stresses/forces to dissolve the lattice, to initiate and extend cracks. The theoretical bonding stresses for matters without defects exceed the real ones by approximately a magnitude of one to three decimal powers ([8], p. 7). The same magnitude of breaking stresses can only be observed with ‘‘perfect’’ materials such as whiskers ([15], p. 13). However, the atomic concepts are valuable to explain certain effects of crack initiation and development, but they cannot be used to size a crushing process or machine. The models on the microstructural scale consider different types of defects (volume defects: voids, inclusions, different phase grains; surface defects: grain, twin and phase boundaries, secondary crack surfaces, stacking faults; line defects: edge and screw dislocations) ([17], p. 194, [40], p. 164f). They explain for instance the effects of twinning, slip and the nucleation of cracks due to elastic incompatibility or boundary sliding of neighbouring grains ([41], p. 18). If they are incorporated in concepts on the macroscopic level, they contribute to the understanding of the process. On a macroscopic level several concepts are applicable. With the concept of classical continuum mechanics a body is subjected to contact and ﬁeld forces and/or externally implied deformations causing stresses and strains inside the body. The material behaviour is described by an experimentally derived relationship between stress and strain (elastic, plastic or viscous). If an analytic solution of the stress or strain calculations is not possible within the body, experimentally determined factors (e.g. notch factors) or numerical approaches (e.g. FEM) are applied. Several hypotheses were developed to predict the failure of the material at given stress/strain conﬁgurations. For geomaterials very often the failure criteria of the Mohr–Coulomb hypothesis with a tensile cut-off or the Hoek–Brown hypothesis are used. With this concept it is not possible to explain why a particle fails, either in terms of the initiation and propagation of cracks or in terms of the total breakdown of the matrix as cracks propagate,

144

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bifurcate and coalesce. Additionally no information of the crushing product can be given. Therefore, it is not possible to use this concept for sizing a process. On the other side this concept enables a basic understanding of many reactions of and effects inside a loaded or deformed particle as well as gives a simple method to characterize a material. As a consequence this concept can be used as a preliminary assessment of the material to be crushed and a basis to size a crusher in respect to its structural design. The classical concept of continuum mechanics does not enable the investigation of the stress/strain ﬁeld and the material behaviour around a crack tip. With the introduction and development of fracture mechanics the initiation and extension of cracks, their stable or unstable propagation and their possible bifurcation can be predicted. The material is characterized by the distribution of its defects (defect statistics) as well as by the fracture toughness Kc or the fracture resistance Rc at three different crack loading cases. The crack propagation is determined by the equilibrium of the energy available at the crack tip (characterized by the stress intensity factor K or the energy release rate G) and the energy consumed by the crack during its propagation (characterized by the fracture toughness Kc or the crack resistance energy Gc). As a result the breakage of a particle and the resulting fragment size distribution with the exception of large fragments can be theoretically calculated ([42], p. 1ff). Additionally, several tests were successfully performed to ﬁnd a correlation between the fracture toughness, machine parameters (e.g. css in jaw and cone crushers) and the fragment size distribution as well as the power consumption of a crusher [43,38]. The application of fracture mechanics enables a basic understanding of the process of particle breakage. It bridges the gap of understanding of effects and results which are derived from concepts on the atomic and microstructural level and classical concepts of continuum mechanics. Additionally, models of fracture mechanics are incorporated in models of continuum mechanics (e.g. in FEM codes). These concepts can be partially used to size the process and the crusher. The concept of discontinuum mechanics is based on single, discrete bodies, which can translate and rotate. They interact at contacts. BPMs can describe the microscopic and macroscopic behaviour of particles during crushing. Therefore, this concept can be used not only to analyse the different effects but also to be potentially applied to size the crushing process and the machine. However, considerable research work is still necessary to determine the microproperties of bodies and bonds as well as to provide sufﬁcient computing power due to the necessary resolution of the structure. The concept of discontinuum mechanics has the potential to be applied on all scales. Owing to the stochastic nature of fracture occurrences within a loaded and irregularly shaped particle with inhomogeneous fabric all attempts to develop an all explaining theory which covers the atomic, microstructural or particle level by

The Principles of Single-Particle Crushing

145

means of atomic physics, classical continuum, fracture or discontinuum mechanics remain unsatisfactory. In order to reliably size crushing processes and crushers phenomenological concepts are developed which investigate and determine the particle behaviour by statistical means. These concepts cannot principally fulﬁl any requirements to explain the nature of fracture, but they yield all necessary information concerning the phenomenological reaction of a particle during loading and of the fragments after crushing. Reliable information of logical descriptions and neural networks are restricted within the limits set by the number and variety of previously executed experiments. Ratings are very often helpful to give a rough estimate and assessment of a crushing problem. Widely used are concepts based on regression and correlation analysis. Properly done these concepts are reliable within certain limits and allow extrapolations to a certain extent as well. Summarizing the assessment of the different concepts one can say that the concepts based on physical models contribute to the understanding of the microprocess ‘‘single-particle crushing’’ but those related to the empirical models based on statistics are the only ones which can describe sufﬁciently the features of this microprocess. Therefore, the empirical concepts on the basis of regression and correlation analysis are mainly applied as a tool to size the process and the machine. Several results are compiled in the following subsections. At ﬁrst it is necessary to structure systematically the microprocess by parameters. This can be done on the basis of an analysis of the system ‘‘crusher’’ from which the different parameters of the microprocess ‘‘single-particle crushing’’ can be derived.

5. CRUSHER AS A SYSTEM A crusher is a machine, where inside the process zone the macroprocess ‘‘crushing’’ occurs. The comminution behaviour of the particles inside the process zone depends on their features, such as the kind of material, their shapes and dimensions. Furthermore, it depends on the boundary conditions. These conditions comprise the atmosphere with its chemical components and its temperature in which the particles break as well as the way and kind, how the loads are applied to the particles. The contact conditions with the working surfaces of the machine determine the reaction of the particles. Finally, the operation of the crusher inﬂuences the comminution too, such as the feeding of the process zone (choked or controlled feed). Therefore, these features are not properties of particles, they are properties of a system. However, the processes (macroprocesses) and machines for crushing need to be sized. As a basis the macroprocess can be divided into the different microprocesses. If the microprocesses are then well understood, the macroprocess

146

G. Unland system

material

material’ crusher

energy

Fig. 23. Crusher as a system.

can be assessed and sized. But since there is a strong interrelation between the macroprocess and the microprocesses, it is necessary to develop at ﬁrst the features of the macroprocess in order to ﬁnd the right ones for the microprocesses. That can be done on the basis of the analysis of the crushing system. The crushing system is a system of particulate material conversion, where the dispersion of a solid is changed by the transfer of energy (Fig. 23). Inside a crusher, the macroprocess of crushing happens. The in- and outgoing material and energy ﬂow as well as the machine can then be characterized by different features (Fig. 24). It is necessary to mention that besides crushing other processes happen within the process zone, such as transportation and sizing. This whole section is, however, only dedicated to comminution. The system ‘‘crusher’’ can be structured by three different sets of features, which are related to the particulate material ﬂow, the machine and the system itself. The ingoing material ﬂow, the feed, can be characterized by the kind of the material, the diameter, the shape and the feed rate of the particles. The machine changes the dispersion of the particulate material ﬂow. Therefore, the product shows different features, such as diameter and shape distributions of the outgoing particles. The machine transfers energy and applies forces to the particles by one or two working surfaces, which can be described by their geometry and material. Additionally, it is possible to operate the machine differently. The working surfaces impose various displacements to the particles and their velocities vary as well. Furthermore, the operation conditions are determined by the conditions of the working surfaces (e.g. dry, wet, dirty), under which the particles are loaded. During operation the particles can be in different conditions. They can show various temperatures and moisture contents. Finally, the contact locus of the particles with the working surface has a major impact on the crushing result. The assessment of the macroprocess and the machine is based on typical system parameters, which describe the interaction of the particles with the machine, such as the probability of breakage of the particles, the contact times and

particle

feed

particle/machine

machine

product

design

operation

interaction

features

particle: - material - diameter - shape - rate

particle: - diameter - shape

surface: - number - material - geometry

surface: - displacement - velocity - condition particle: - moisture - temperature particle/surface: - contact locus

The Principles of Single-Particle Crushing

system

particle: - probability of breakage surface: - contact force - contact time - energy consumption - wear

Fig. 24. Features of the system ‘‘crusher’’ (only macroprocess ‘‘crushing’’).

147

148

G. Unland

forces with the working surfaces, the power consumption of the machine and ﬁnally the wear. There are complex interdependencies of these parameters. Some are the inﬂuencing parameters, such as the feed and machine parameters and others are the resulting features, such as the product and interaction parameters. It is necessary to know these interdependencies. In the design stage of a crusher, for instance, certain feed parameters have to be considered as well as requirements on the product and on certain limits of interaction parameters such as energy consumption or wear. In this stage the design and potential operation parameters are set and sized according to the interaction and product parameters. During the operation of a crusher the feed and design parameters are mostly given and the product and interaction parameters are optimized by adjusting the operation parameters.

6. CRUSHING PARAMETERS The features of the system ‘‘crusher’’ characterize the macroprocess and the machine in total. The different microprocesses, including single-particle or particle bed comminution, are the basis of the macroprocess. In this subsection the microprocess ‘‘single-particle crushing’’ is structured by crushing parameters. The determination of these parameters faces two principal challenges. The crushing occurrences of loaded and irregularly shaped particles with inhomogeneous fabric do not ﬁrstly depend solely on themselves and are secondly of stochastic nature. The crushing parameters of one single particle do not depend only on the features of the particle but also on the design and operation parameters of the test apparatus. Therefore, these parameters are not properties of a particle, they are per se system parameters too. There is no intrinsic property of a particle or a material related to crushing! An intrinsic property would not depend on the particle geometry or the loading conditions. Since those properties are not available the crushing parameters are derived from the features of the system ‘‘crusher’’ (see Fig. 24), whereas the term test apparatus can stand for the word crusher. In a test apparatus a single particle can be investigated under deﬁned conditions. For the most common loading cases, compression, percussion and impact, several test apparatuses were developed and used, e.g. hydraulic press [44], drop weight test [45], ultra fast load cell [46], Hopkinson bar [47], pneumatic cannon [48] and high resolution impact analyser [49]. In those test apparatuses parameters can be systematically varied (inﬂuencing parameters) and test results are gained (crushing parameters). The tests and their results need to be grouped (Fig. 25) in order to achieve a systematic and

The Principles of Single-Particle Crushing

149

influencing parameters

feed parameters particle: - material - diameter - shape

design parameters surface: - number - material - shape

operation parameters surface: - displacement - velocity - condition particle: - moisture - temperature particle/surface: - contact locus

crushing parameters

product parameters

interaction parameters

particle: particle: - fragment size distribution - probability of breakage - fragment shape distribution surface: - contact force - contact time - energy consumption - wear

Fig. 25. Inﬂuencing and crushing parameters.

clearly structured compilation of parameters for requirements of the characterization of the particles, the design, sizing and operation of the machine as well as the modelling and simulation of the macroprocess. In the literature several parameters are used to characterize the comminution of a particulate material. The breakage point (see Fig. 4), for instance, is taken to deﬁne three fundamental properties, the particle strength, the mass speciﬁc breakage energy and the breakage fragment size distribution ([50], p. 99) (see Section 7.3). Because of the tremendous impact of other inﬂuencing parameters, such as the condition and geometry of the working surfaces, the difﬁculties

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concerning the determination of e.g. the breakage point (in several cases it is impossible to determine reliably this point) and the usage of operation parameters in technical applications (imposed displacement e.g. in jaw crushers) a different approach is used. The inﬂuencing and crushing parameters are features of a system ‘‘singleparticle crushing’’, which can describe and characterize the system and its behaviour. They are system parameters and depend on each other. There are very complex interrelations between those parameters, where many of them are not well understood. In the following subsections the crushing parameters and their inﬂuencing parameters with reference to the particle comminution are presented. The wear and surface material parameters can be omitted, since they cannot be measured or do not show a signiﬁcant effect within the usual limits. The crushing parameters are grouped as crushing resistance, crushing product and energy utilization ([51], p. 213). Since it is impossible to show all interrelations between the crushing and inﬂuencing parameters the subsections concentrate on the most common interrelations. As an extended example the subsection on contact forces, however, shows a wide variety of interrelations. Furthermore, the crushing occurrences are of stochastic nature. The physically based theories are insufﬁcient to explain the interrelations. Even though the following subsections on crushing parameters attempt to formulate the phenomenological relations by physical and mathematical means. The added exemplary results of experiments are referred to more or less spherical bodies, such as glass spheres and cement clinker, as well as to irregularly shaped particles of broken glass, ores, coals, and hard rocks. Some applications of these interrelations in science and engineering are then summarized in Section 7. Since many of the crushing and inﬂuencing parameters are related or equivalent features and many of the results are referred to related or equivalent parameters, they need to be deﬁned before.

6.1. Deﬁnition of related and equivalent features For better comparison and assessment many parameters and features are related to the geometry of the particle and the intensity of loading. Geometry of particle: It is easy to ﬁnd a characteristic size to assess a regularly

shaped body. It is the diameter for instance for a sphere. Since the particles are usually irregularly shaped it is necessary to deﬁne equivalent sizes. Such a particle is transformed into a volume or mass equivalent sphere. The characteristic features of this sphere can be used as equivalent sizes. The volume

The Principles of Single-Particle Crushing

151

equivalent diameter dV of an irregularly shaped particle with a volume VP is then rﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 6V P dV ¼ ð15Þ p Another option is related to the area of projection APr of a particle. This area is converted into a circle with the same area. The projection area equivalent diameter dPr of an irregularly shaped particle with a projection area APr is correspondingly rﬃﬃﬃﬃﬃﬃﬃ APr d Pr ¼ 2 ð16Þ p Besides other equivalent diameters they can be used to calculate crosssections, projection areas, surfaces, volumes, etc. Intensity of loading: Particles are subjected to different kinds of loads. The process of loading can be characterized among others by its intensity. It depends on the loading case which feature is applicable as intensity factor. Scho¨nert [21] for instance introduced dynamic and energetic intensity factors. Since the three most common loading cases use different ways to impose loads on particles the following intensity factors can be distinguished: a. Kinematic intensity factors: In several crushers the effects of compression and percussion are applied. The machine imposes a certain displacement or stroke h on a particle of a diameter dF no matter how much energy is necessary and how high the forces are. The relative displacement hr hr ¼

h dF

ð17Þ

can be considered as a kinematic intensity factor. b. Dynamic intensity factors: The loading of a particle is accompanied by the occurrence of forces no matter whether a particle is subjected to compression, percussion or impact. The force at the breakage point FBP or the maximum force Fmax during loading can be used as a dynamic intensity factor. Those forces are mostly related either to the projection area APr, F APr , or to the cross-section of a volume equivalent sphere AV, F A V Occasionally, the forces are referred to the particle mass as well, Fm. c. Energetic intensity factors: During the process of loading energy is transferred to the particle. This happens with all load cases, i.e. compression, percussion and impact. There are different kinds of energy to be considered depending on the phases or effects of the comminution process and the mechanical work done. The following kinds of energy have to be distinguished O¡ered energy EO: A comminution system offers energy to a particle. In case of impact a particle ﬂies for instance with a certain velocity. The

152

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kinetic energy of the particle E kin; P ¼ 12mP v 2P just before the impact is the offered energy (EO ¼ Ekin, P). Transferred energy ET: Between the particle and the working surface energy is exchanged. The energy transferred to the particle can be used for various processes, such as rotation, acceleration, deformation, comminution, etc. of the particle and the fragments. Breakage energy EBP: The ﬁrst crushing event happens at the breakage point. The energy used up to the breakage point is termed the breakage energy. Comminution energy EC: The process of comminution usually exceeds the stage of breaking the particle at the breakage point. The particle is then further comminuted. However, the comminution process can stop before the breakage point is reached. This happens for instance in case of crumbling or chipping. The total energy absorbed for the whole process of comminution is the comminution energy no matter which effect is achieved and how far the process goes. Fracture energy EF: The sole energy which is necessary to develop and run the cracks through the particle is considered as fracture energy. This energy cannot be used as intensity factor. The offered energy is bigger than the transferred energy, whereas the transferred energy does not necessarily have to be larger than the breakage or comminution energy. Only if the transferred energy is larger then comminution happens. The transferred energy comprises also all portions of energy which are not directly used to comminute the particle, such as the energy to orientate the particle. The comminution energy can be larger, equal to or smaller than the breakage energy. It depends which comminution effect is reached. If the loading exceeds the breakage point the comminution energy is bigger. The comminution energy always exceeds the fracture energy, since the comminution energy comprises for instance the dissipated energy, such as the unused amount of strain energy. The comminution energy as well as the breakage energy is the work necessary for comminution, it is the integral of the force–displacement diagram. In practice it is sometimes difﬁcult to measure the different kinds of energy. In case of compression the amounts of energy which are not used for comminution are usually small and can be neglected. For impact or percussion the measurement of the transferred energy is very often complicated. Therefore, the offered energy is frequently used in those cases. For reasons of better comparison or assessment the energy is related to different ﬁgures. If the energy is related to the mass or the volume of the particle these factors are termed mass or volume speciﬁc energy, i.e. Em

The Principles of Single-Particle Crushing

153

and EV, respectively: Em ¼

E E resp: E V ¼ mP VP

ð18Þ

From a physical point of view it is more suitable to use the volume speciﬁc energy EV as an intensity factor since the volume speciﬁc energy EV is directly related to the stresses inside the particle due to elastic deformations. Since E mP 1 ¼ v2 2

Em ¼

ð19Þ

the impact velocity v is often used as well as an intensity factor in case of percussion or impact. Occasionally, it is necessary to distinguish two deﬁnitions of mass related energy. If the mass m* of the counter body is used, the mass related energy Em* needs to be clearly indicated. Another option is to relate the energy to the newly produced surface DS, the surface speciﬁc energy EDS. However, this factor combines an intensity factor with a feature of the crushing result. Therefore, the surface speciﬁc energy cannot be used to describe the loading features, it can be applied to assess the efﬁciency of the process. Usually, this is done by the reciprocal value, the energy utilization eSE: SE ¼

1 E DS

ð20Þ

This factor is discussed in a separate Section 3.3 In the following subsections the crushing parameters are now presented by using the above mentioned and deﬁned parameters.

6.2. Crushing resistance Particles subjected to loads show a resistance against fracture. Only if the loads exceed a certain limit, the material of the particle fails. In technical applications a particle is subjected to a deﬁned intensity of loading, to a displacement in case of compression, to a velocity in case of impact and to a sudden displacement or energy input in case of percussion. The resistance comprises the breakage probability, the maximum crushing force, the loading time and the energy consumption.

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6.2.1. Breakage probability The crushing phase starts with the initial breakage phase, where the ﬁrst fragments are generated. Breakage does not occur always at the same level of loading, i.e. at the same intensity of loading with all particles of the same diameter, shape and material. The magnitude of loads at the breakage point shows a distribution. The breakage probability, also termed the likelihood of breakage, can be deﬁned as the share of particles of the same diameter, shape and material, which breaks at a deﬁned magnitude of loading.

6.2.1.1. Physical and mathematical formulation The strengths of particles are random ﬁgures. Therefore, the graphical representation of the relationship between the breakage probability P and the loading parameters x can often be approximated in certain limits by a straight line in a logarithmic probability net. From a statistical and physical point of view, only the forces and the absorbed energy at the breakage point or the maximum forces and the energy absorbed during the process of comminution can be considered as pure random variables. However, the other variables (e.g. impact velocity, offered kinetic energy) can be and often are assessed the same way although it is physically not correct. pﬃﬃﬃﬃﬃ The probability density function p(ln MPa mx) of the normal distribution of ln x is described by two parameters. The median value m of ln x, the variance s2 and the standard deviation s of ln x are m ¼ ln x 50

ð21Þ

1 x 84 s ¼ ln 2 x 16

ð22Þ

and

¼ ln

x 84 x 50

ð23Þ

¼ ln

x 50 x 16

ð24Þ

wherein x16, x50 and x84 are the values on the abscissa which correspond to the values of the ordinate P(ln x) ¼ 15.87%, 50% and 84.13% (see also Section 6.3.1.1). The loading parameter x is considered a random variable. Then it is 1 1 ln xm 2 pðln xÞ ¼ pﬃﬃﬃﬃﬃﬃ e2ð s Þ s 2p

ð25Þ

The Principles of Single-Particle Crushing

155

for the distribution density function and 1 Pðln xÞ ¼ pﬃﬃﬃﬃﬃﬃ s 2p

Z

ln x

2

1 ym e2ð s Þ dy

ð26Þ

0

for the distribution function. Since there are lower and/or upper limits and no negative values of the loading parameters the distribution function can better be described by one or two limits in addition to the median value and the variance. The 2-parameter logarithmic normal distribution can be transformed by xT into a 4-parameter logarithmic distribution function (see also [52], p. 53) Z ln x T 1 ym T 2 2 s 1 TF pﬃﬃﬃﬃﬃﬃ Pðln x T Þ ¼ e dy ð27Þ sTF 2p 0 with xT ¼ xO

sTF ¼

x xU xO x

ð28Þ

m T ¼ ln x T50

ð29Þ

1 ðln x T84 ln x T16 Þ 2

ð30Þ

and Pðx T16 Þ ¼ 0:1587

ð31Þ

Pðx T50 Þ ¼ 0:5

ð32Þ

Pðx T84 Þ ¼ 0:8413

ð33Þ

0 xT 1

ð34Þ

0 xU x xO 1

ð35Þ

as well as

The 4-parameter function can be transformed into a 3-parameter function by setting xU ¼ 0, if there is no lower limit. An example is given with Figs. 26 and 27. The shape of the curves in Fig. 26 indicates a lower and upper limit of the percussion energy EO,m related to the particle mass. The transformation of the curves conﬁrms the assumption that the function of the breakage probability can be described by a 4-parameter logarithmic distribution function.

156

G. Unland 98 96

3

2

1

90

P [%]

80

60 40 1: dF = 5 mm 2: dF = 10 mm 3: dF = 16 mm

20 10 5 2 6

8

10-1

2

4

6

10 0

8

2

EO,m [J/g]

Fig. 26. Breakage probability P as function of mass speciﬁc offered energy EO,m and particle feed diameter dF (percussion; glass spheres; v ¼ 3.8 m s–1) ([52], p. 55).

98 96

1: dF = 5 mm 2: dF = 10 mm 3: dF = 16 mm

90

P [%]

80

60 40

20 10 5 2

3

2

6 810 -2

1 2

4

6 810 -1

2 EO,mT [J/g]

Fig. 27. Transformed curves of Fig. 26 ([52], p. 56).

4

6 810 0

2

4

6 8 10 1

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157

6.2.1.2. Results 6.2.1.2.1. Compression. The distributions of the breakage probability are investigated for several materials. Figure 28 (glass) and Fig. 29 (cement clinker) show the distributions for more spherical bodies, whereas Fig. 30 (limestone) and Fig. 31 (quartzite) are related to irregularly shaped particles. In those ﬁgures the breakage probability is depicted as a function of the mass speciﬁc breakage force Fm, whereas Fig. 32 shows as an example the breakage probability as a function of the comminution energy EC for quartzite. It is obvious that there is a strong inﬂuence of the particle diameter and material as well as the magnitude of load. Figure 33 elucidates especially the inﬂuence of the imposed relative displacement hr on the breakage probability of diorite. In this case the inﬂuence of the particle diameter vanishes because it is incorporated in the relative displacement. 6.2.1.2.2. Impact. In the case of impact the breakage probability P can be evaluated as a function of the impact velocity v. It is either the velocity of the particle or the velocity of the working surface. The impact velocity v is a criterion for the offered energy as well, since E O ¼ 12mP v 2 . Figures 34 and 35 show the inﬂuence of the impact velocity on spherical and irregularly shaped particles, whereas Fig. 36 uses the mass speciﬁc offered energy E O;m ¼ 12v 2 as the abscissa.

99.5 99 98 97 95

P [%]

90

2

1

1: dF =16 mm 2: dF = 9 mm 3: dF = 5 mm 4: dF = 3 mm

3

4

80 70 60 50 40 30 20 10 5 3 2 1 103

2

3

5

7

104

2

3

5

Fm [N/g]

Fig. 28. Breakage probability P as function of mass speciﬁc breakage force Fm and particle feed diameter dF (compression; glass spheres) ([10], p. 119).

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99.9 99.8 99.5 99 98 95 90

1: dF = 20 mm 2: dF = 16 mm 3: dF = 10 mm

1

2

3

4

4: dF = 4 mm

P [%]

80 70 60 50 40 30 20 10 5 2 1 0.5 0.1 2

3

5

7

10 2

2

3 5 Fm [N/g]

7 103

2

3

5

10 4

7

P [%]

Fig. 29. Breakage probability P as function of mass speciﬁc breakage force Fm and particle feed diameter dF (compression; cement clinker, Bernburg/Sachsen-Anhalt) ([10], p. 121). 99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20

1: dF = 15 mm 2: dF = 8 mm 3: dF = 5 mm

10 5 2 1 0.5

1

2

3

0.1 2

3

5 Fm [N/g]

7

10 3

2

Fig. 30. Breakage probability P as function of mass speciﬁc breakage force Fm and particle feed diameter dF (compression; limestone, Haldensleben/Sachsen-Anhalt) ([10], p. 120).

The Principles of Single-Particle Crushing

99.9 99.8 99.5 99 98

P [%]

95 90

159 1

2

3

4

1: dF = 32 ÷ 40.0 mm 2: dF = 16 ÷ 18.0 mm 3: dF = 10 ÷ 12.5 mm 4: dF = 5 ÷ 6.3 mm

80 70 60 50 40 30 20 10 5 3 2 1 0.5 0.2 0.1

2

3

5

7

102

2

3 5 Fm [N/g]

7

103

2

3

5

7

10 4

Fig. 31. Breakage probability P as function of mass speciﬁc breakage force Fm and particle feed diameter dF (compression; quartzite, Sproitz/Sachsen) ([53], p. 33).

1

99.9 99.5 99 98

P [%]

95 90

2

3

4

1: dF = 5 ÷ 6.3 mm 2: dF = 10 ÷ 12.5 mm 3: dF = 16 ÷ 18.0 mm 4: dF = 32 ÷ 40.0 mm

80 70 60 50 40 30 20 10 5 2 1 0.5 0.1 2 3

5 7 10 -2

2

3

5 7 10 -1

2 3 EC [J]

5 7 10 0

2 3

5 7 101

2 3

Fig. 32. Breakage probability P as function of comminution energy EC and particle feed diameter dF (compression; quartzite, Sproitz/Sachsen) ([53], p. 39).

160

G. Unland 100

P [%]

80 60 40 20

5

10

15

20

25

30

hr [%]

Fig. 33. Breakage probability P as function of relative displacement hr (compression; diorite, Hohwald/Sachsen) ([44], p. 88).

8 mm

P [%]

99.9 99.8 99.5 99 98 95 90

4 mm

glas spheres

4 mm

cement clinker

80 70 60 50 40 30 20 10 5 3 2 1 0.5 0.2 0.1 5

7

10

2

3

5

7

10 2

v [m/s]

Fig. 34. Breakage probability P as function of impact velocity v and particle feed diameter dF (impact; glass spheres; cement clinker, Mainz-Weisenau/Rheinland-Pfalz) ([54], p. 46f).

The results indicate a size effect as well as the inﬂuence of the material and the intensity of the loading ðv; E O;m Þ. It is interesting to see similar inﬂuences, if the absorbed energy EC is used as the intensity factor (Fig. 37). 6.2.1.2.3. Percussion. Particles subjected to percussion are positioned on an anvil and are loaded by a moving weight or body. The loading can be

The Principles of Single-Particle Crushing

99.9 99.8 99.5 99 98

161

limestone

3

2

1

quartz

4

6

diabase

95

5

P [%]

90 80 70 60 50 40 30 20

1 + 3: dF = 6 ÷ 7 mm 2 + 4: dF = 3 ÷ 4 mm 5: dF = 113 mm 6: dF = 90 mm

7

10

7: dF =

5 3 2 1 0.5 0.2 0.1 2

3

5 v [m/s]

60 mm

10 2

7

2

Fig. 35. Breakage probability P as function of impact velocity v and particle feed diameter dF (impact; limestone, Lauingen/Bayern; quartz, Frechen/Nordrhein-Westfalen; diabase, Hausdorf/Sachsen) ([54], p. 46f, [55], p. 65).

99.9 99.8 99.5 99 98

limestone quartzite

P [%]

95 90 80 70 60 50 40 30 20 10 5

3

1: dF = 8.0 ÷ 10.0 mm 1

2

2: dF = 6.3 ÷ 8.0 mm

4

3: dF = 8.0 ÷ 10.0 mm 4: dF = 5.0 ÷ 6.3 mm

2 1 0.5 0.1 2

3

5

7

100

2

3

EO.m [J/g]

Fig. 36. Breakage probability P as function of mass speciﬁc offered energy EO,m and particle feed diameter dF (impact; limestone, Ru¨beland/Sachsen-Anhalt; quartzite, Sproitz/ Sachsen) ([56], p. 36f).

162

G. Unland

99.9 99.8 99.5 99 98

1: dF = 113 mm 2: dF = 90 mm 3: dF = 60 mm

1

P [%]

95 90

2 3

80 70 60 50 40 30 20 10 5 2 1 0.5 0.1 2

3

5

7

100 EC,m [J/g]

2

3

5

7

101

Fig. 37. Breakage probability P as function of mass speciﬁc comminution energy EC,m and particle feed diameter dF (impact; diabase, Hausdorf/Sachsen) ([55], p. 66).

99.9 99.8 99.5 99 98

3.4 mm

9 mm

glas spheres cement clinker

P [%]

95 90 80 70 60 50 40 30 20 10 5

10 mm 5 mm

2 1 0.5 0.1 2

3

5 7 10 -1

2

3

5 7 100 2 EO,m [J/g]

3

5 7 101

2

3

5 7 10 2

Fig. 38. Breakage probability P as function of mass speciﬁc offered energy EO,m and particle feed diameter dF (percussion; glass spheres; cement clinker, Bernburg/SachsenAnhalt) ([52], p. 61, [45], p. 27).

The Principles of Single-Particle Crushing

P [%]

99.9 99.8 99.5 99 98 95 90

163

limestone quartz halite

9.8 mm 4.8 mm

80 70 60 50 40 30 20 10 5 2 1 0.5

7.7 mm 8 mm

19.1 mm

0.1 2

3

5

7 10-1

2

3 5 EO,m [J/g]

7 100

2

3

5

101

7

Fig. 39. Breakage probability P as function of mass speciﬁc offered energy EO,m and particle feed diameter dF (percussion; limestone, Haldensleben/Sachsen-Anhalt; quartz, Frechen/Nordrhein-Westfalen; halite, Bernburg/Sachsen-Anhalt) ([52], pp. 58, 64, [45], p. 27).

P [%]

99.9 99.8 99.5 99 98 95 90

limestone iron ore

4

1

5

2

80 70 60 50 40 30 20

1: dF = 75.0 ÷ 90.0 mm 2: dF = 26.0 ÷ 37.0 mm

10 5 2 1 0.5

3: dF = 2.4 ÷ 3.4 mm 4: dF = 5.6 ÷ 6.7 mm

3

5: dF = 3.4 ÷ 4.0 mm

0.1 2

3

5

7 10-2

2

3 5 EC,m [J/g]

7 10-1

2

3

5

7

100

Fig. 40. Breakage probability P as function of mass speciﬁc comminution energy EC,m and particle feed diameter dF (percussion; iron ore/Australia; limestone/Utah) ([50], p. 105, [47], p. 867).

164

G. Unland

characterized by the kinetic energy EO of the weight/body or the transmitted force F. Figures 38 and 39 indicate the breakage probabilities of various materials as functions of the mass speciﬁc offered energy EO,m and the particle diameter d. With special devices (e.g. Hopkinson bar) it is possible to measure the absorbed energy till the breakage point (Fig. 40). All the investigations for the load case percussion conﬁrm the ﬁndings of the other load cases. There is a size, material and intensity effect on the breakage probability.

6.2.1.3. Conclusions The breakage probability can be described as a function of characteristic features of the loading, such as the comminution energy. The best approximation shows the function of the logarithmic normal distribution with three or four parameters, since there are upper and usually lower limits of the loading parameters. The breakage probability shows a size effect. The smaller the diameter of the particle the less likely is the breakage at a certain magnitude of loading (speciﬁc loading). Furthermore, the higher the intensity of the loading the more likely is the breakage of a particle of a deﬁned diameter. Finally, there is a major inﬂuence of the material on the probability of breakage.

6.2.2. Crushing force During the whole process of comminution the crushing forces change to a very large extent. Usually two different characteristic forces assess the range of occurring contact forces. The ﬁrst one coincides with the force FBP at the breakage point, whereas the breakage point is deﬁned according to Sections 3.1.2 and 3.3. The second force characterizes the maximum force during the whole process of loading, FCmax. Each of the two characteristic forces can be equal, larger or smaller than the other one, i.e. F BP F C max or F BP F C max . For requirements of comparison these forces are usually related to the mass m of the particle, to the cross-section A or to the projection area APr of the real or equivalent particle. The volume equivalent sphere is very often used as equivalent particle. Although the dimensions of the area or cross-section related forces are the same as the ones of a stress the related forces cannot be compared with stresses on or inside a particle of irregular shape. The contact forces depend heavily on several parameters as the other crushing parameters do as well. Therefore, as an example a wide investigation of those dependencies are summarized in Subsection 6.2.2.2.

The Principles of Single-Particle Crushing

165

6.2.2.1. Physical and mathematical formulation The maximum contact forces between a particle and the working surface(s) are random values. They depend on the material and the geometry of the particles, the loading and contact conditions as well as the geometry of the working surface(s). For sizing and operation of a crusher the inﬂuence of the particle diameter is important to know. This so-called size effect can be mathematically characterized (see [57]). The results of measurements yield graphically a straight line in a logarithmic net (Fig. 41), i.e. the relationship is d log F A max ¼ log F A maxðd c Þ þ r F log ð36Þ dc with the approximate solutions of the equations of the linear regression n rF ¼

n P

n n

P P log F A maxðd i Þ log d i log F A maxðd i Þ log d i

i¼1

i¼1

n

n P

2

ðlog d i Þ

i¼1

n P

2

i¼1

ð37Þ

log d i

i¼1

and F A maxðd c Þ ¼ 10

1 n

n P

log F A maxðd i Þ r F

i¼1

1 n

n P

log d i log d c

ð38Þ

i¼1

The deviation of the measurement results in relation to the regression curve is vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u n u 1 X F A maxðd c Þ di 2 t slog F A max ¼ log þ r F log ð39Þ n 2 i¼1 F A maxðd i Þ dc

log FA max

F • log FA max log FA max (dc)

log dc

log d

Fig. 41. Cross-section related maximum crushing force FA max with characteristic parameters lF , slog F A max as function of particle diameter d.

166

G. Unland

The conﬁdence interval lF slog F A max for the crushing forces can be assessed by the conﬁdence factor lF. For lF ¼ 1 the conﬁdence interval is 68.26%, for lF ¼ 1.96, 2.58 and 3.29 they are 95%, 99% and 99.9%. The crushing forces or the strength of a material can be described by three parameters: 1. Strength parameter FA max(50): F A maxðd c Þ characterizes the basic strength of the material. Since a typical geological specimen has a diameter of approximately 50 mm, this strength parameter should be related to dc ¼ 50 mm. 2. Size parameter rF: This parameter describes the inclination of the curve. It characterizes the size effect. 3. Anisotropy parameter slog F A max : The deviation of the crushing forces are indicated by this parameter: It is a criterion to characterize the anisotropy of the material. The investigation of several rocks yields that the anisotropy parameter is comprised of two components. The ﬁrst is caused by the anisotropic fabric of a rock due to orientated planes of weakness (e.g. with stratiﬁed rocks). The other occurs with rocks despite their isotropic fabrics. The material of the rock itself breaks at a wide variety of forces. In practice it is hardly possible to distinguish the two effects.

6.2.2.2. Results The magnitude of the crushing forces depends (as with other parameters) on several inﬂuencing parameters. As an example these interrelations are presented for the loading case compression in detail. 6.2.2.2.1. Compression. The inﬂuencing parameters can be grouped into three sections, feed, design and operation parameters. 1. Feed parameters Material, diameter: The inﬂuence of the material and diameter must be described by three parameters although all other parameters are kept constant. Figures 42 and 43 show the inﬂuence of the particle diameter on the projection area related maximum force F A max for two typical hard rocks, 20% relative displacement hr and a conﬁdence interval lF slog F A of 99.9%. The compilation of the crushing force parameters (Table 4) clearly indicates that the size and deviation parameters are not constant. They are characteristic features of the material. Both rocks are of volcanic origin but the basalt necessitates signiﬁcantly higher crushing forces at 50 mm and smaller forces at 500 mm than the quartz porphyry. Furthermore, the crushing forces of the quartz porphyry vary signiﬁcantly more than those of the basalt although the former is very dense and of isotropic fabric.

The Principles of Single-Particle Crushing

167

102

FA max [MPa]

101

100

2

3

5

7

102 dF [mm]

2

3

5

7

103

Fig. 42. Projection area related maximum crushing force FA max as function of particle diameter dF (compression; basalt, Kulmain/Bayern; hr ¼ 20%; lF slog F A max ¼ 99:9%).

102

FA max [MPa]

101

100

2

3

5

7

102 dF [mm]

2

3

5

7

103

Fig. 43. Projection area related maximum crushing force FA max as function of particle diameter dF (compression; quartz porphyry, Lo¨beju¨n/Sachsen-Anhalt; hr ¼ 20%; lF slog F A max ¼ 99:9%).

168

G. Unland

Table 4. Crushing force parameters

Basalt/Kulmain

Quartz porphyry/Lo¨beju¨n

13.90 MPa 0.57 0.12

8.30 MPa 0.18 0.20

FA max(50) rF slog F A max

103

7 6

FAmax [MPa]

102

101

4 8

1: salt (c, cy) 2: coal (c) 3: coal (c) 4: coal (c) 5: quartz diorite (p, cy) 6: granite (cy) 7: granite (p) 8: marble (p) 2

5

100

2

1

3 5 2

5 101

2 5 102 dF [mm]

2

5

103

2

5 104

Fig. 44. Area related maximum crushing force FA max as function of particle diameter dF, various materials and shapes (compression; c: cube; cy: cylinder; p: prism) (data from [58], p. 330, [59], p. 523, [44], p. 74).

The slope of the related force–diameter curve is negative for brittle materials, i.e. the strength parameter decreases with rising dimension (Fig. 44). However, for materials with plastic/viscous behaviour the inclination can be positive, such as for salt. An investigation on rocks with dimensions of 1 m and bigger show that the area related maximum crushing force F A max approaches asymptotically a constant value. Pratt et al. [59] for instance tested a quartz diorite and found a constant related force for specimens bigger than 1 m, whereas Bieniawski [60] found for hard coal constant values for pieces larger than 1.2 m (see also [61]).

The Principles of Single-Particle Crushing

169

Shape: The shape inﬂuences the stress distribution inside the body of the

particle. Thereby, two geometric features are important to consider. The introduction of the loads into the particle are determined by the contact geometry between the particle and the working surface. If the working surface is a ﬂat plate, the mesoshape of the particle (the curvature at the contact point) is important. The curvature can be characterized by a shape anglea ([44], appendix 17). With larger angles (a ¼ 1801: contact plane) the contact forces and their deviations increase (Fig. 45). The other important geometric feature is the macroshape of the particle. Compact specimens show higher crushing forces than taller ones (Fig. 46). 2. Design parameters Geometry: The geometry of the contact surface inﬂuences the introduction of the load into the particle as well as the stress distribution inside the body of the particle. With a curved shape of the contact surface the related crushing force can be signiﬁcantly reduced. Figure 47 depicts the related force–diameter curves of diorite. One curve represents the relationship for a contact of irregularly shaped particles with a ﬂat working surface (hr ¼ 20%), whereas the other curve refers to a contact of the same shaped particles with a spherically shaped working surface (r ¼ 5 mm). The

50 1: fraction: dF = 25 ÷ 40 mm, hr: 20 % 2: fraction: dF = 56 ÷ 80 mm, hr: 22 % 3: fraction: dF = 80 ÷ 140 mm, hr: 22 %

40

4: fraction: dF = 140 ÷ 200 mm, hr: 18 % 5: fraction: dF = 200 ÷ 250 mm, hr: 22 %

FAmax [MPa]

6: fraction: dF = 250 ÷ 320 mm, hr: 20 %

30

1 2

20

3 4 10 5 6 120

130

140

150

160

170

α [°]

Fig. 45. Area related maximum crushing force FA max with standard deviation as function of shape angel a, particle diameter fraction and relative displacement hr (compression; diorite, Hohwald/Sachsen) ([44], p. 92).

170

G. Unland

300

FAmax [MPa]

250

2 1

200

150 3 100 1: dolomite / Dunham 2: granite / Westerley 3: trachyte / Mizuho

50

1

2

3

4

5

l/d [-]

Fig. 46. Area related maximum crushing force FA max as function of length–diameter ratio l/d of clamped cylindrical specimens (compression) [62].

102 7 5

flat surface (hr = 20%) spherical surface

3 FAmax [MPa]

2 101 7 5 3 2

101

2

3

5

7

102 dF [mm]

2

3

5

7

103

Fig. 47. Area related maximum crushing force FA max as function of particle diameter dF and geometry of working surface (compression; diorite, Hohwald/Sachsen).

The Principles of Single-Particle Crushing

171

necessary force for breaking the particle is reduced by a factor of approximately 2. The investigation of several hard rocks shows that the factor is between 1.0 and 4.1 and usually increases with larger diameters [63]. It is necessary to mention that the inclinations of the related force–diameter curves for different contact geometry conﬁgurations are in general not equal. 3. Operational parameters Displacement (surface): During compression the particles are subjected to different magnitudes of displacements. The displacement h is usually related to the particle diameter d, giving the relative displacement hr . The curves for the related forces F A max show the typical size effect as well as an increase with larger relative displacements (Fig. 48). This increase can be marginal or not existing, if the maximum force happens already at lower displacements. This is especially the case with very brittle materials and compact particle shapes. Velocity (surface): The velocity of the working surface during compression of a particle inﬂuences the related crushing forces. If the velocity increases, the state of compression changes to percussion. Many rocks show a signiﬁcant increase of related crushing forces at strain rates of 102 to 103 s1 (Fig. 49). Typical strain rates of crushers, which apply the effect of compression, are between approximately 0.1 and 10 s1 (e.g. jaw crushers).

40 1: fraction: dF = 450 ÷ 560 mm 2: fraction: dF = 250 ÷ 320 mm 3: fraction: dF = 200 ÷ 250 mm 30

4: fraction: dF = 140 ÷ 200 mm

FAmax [MPa]

5: fraction: dF = 80 ÷ 140 mm

7

6: fraction: dF = 56 ÷ 80 mm 7: fraction: dF = 25 ÷ 40 mm 20

6 5 4

10

2

5

10

15

3 1

20

25

hr [%]

Fig. 48. Area related maximum crushing force FA max as function of relative displacement hr and particle diameter dF (compression; diorite, Hohwald/Sachsen) ([44], p. 91).

172

G. Unland 700

600

FAmax [MPa]

500

1: porphyritic tonalite (cylinder) 2: andesite / Ishikoshi (prism) 3: granite / Inada (prism) 4: sandstone / Iwaki (prism) 5: andesite / Emochi (prism) 6: marble /Tohoku (prism) 7: glass (sphere)

2

3

400

300

1 6

200 5

100 7

4 10-6

10-5

10-4

10-3

10-2 10-1 ε [1/s]

100

101

102

103

104

Fig. 49. Area related maximum crushing force FA max as function of strain rate _ for various materials (compression) (data from [64], p. 57, [10], p. 126, [65], p. 533).

Some investigations, however, yield opposite results. May ([10], p. 126) found with spherical bodies of glass and cement clinker reduced related forces at a breakage probability of 50% with increased loading velocities. Condition (surface): The conditions of the working and the particle surfaces inﬂuence the radial and tangential relative movement of the two contact planes during the process of loading. This happens especially, if there is an elastic mismatch, i.e. different Young’s moduli E and Poisson’s ratios u between the materials of the two planes at the area of contact. The surface conditions can be characterized by the coefﬁcient of friction m . The lower the coefﬁcient the lower the inﬂuence on the different dilatations of the two contact faces. Therefore, the related forces increase with larger coefﬁcients of friction (Fig. 50). Moisture (particle): The moisture content of the particle inﬂuences the coefﬁcient of friction at the contact plane with the working surface as well as the strength of the material of the particle. In particular, porous materials or minerals which can swell reduce the strength with increasing moisture content, such as with certain sandstones, mudstones or shales (Fig. 51). However, some authors report on a few exceptions where the strength increases by up to 15% with higher moisture content (e.g. [67]). Temperature (particle): Only little work has been done on the inﬂuence of temperature on the strength of various materials. The ﬁndings are different;

The Principles of Single-Particle Crushing

173

103 7

1: μ = 0.30 2: μ = 0.15 3: μ = 0.04

5 3

FAmax [MPa]

2

1 102

2

7 5

3

3 2

0.2

0.3

0.4

l/d [-]

Fig. 50. Area related maximum crushing force FA max as function of coefﬁcient of friction m and length–diameter ratio l/d (compression; coal, Kentucky; ﬂat surfaces) (data from [66]). 200

quartzitic shale / Sallies Gold Mine / South Africa quartzitic sandstone / Sigma Colliery / South Africa

FA max [MPa]

150

100

50

10

20

30

40

50

60

70

80

90

100

sH2O [%]

Fig. 51. Area related maximum crushing force FA max as function of water saturation sH2 O (compression; ﬂat surfaces; sH2 O ¼ mH2 O =mH2 Omax ) (data from [68]).

174

G. Unland

some authors found increasing strengths with decreasing temperatures [65,69] whereas others found the opposite [70].

6.2.2.2.2. Impact. In case of impact the crushing forces are inﬂuenced not only by the strength of the particle but additionally by its inertia. Figures 52 and 53 show the related crushing forces F A max as functions of the particle diameters and the impact velocities, whereas Fig. 54 depicts the inﬂuences of the different locations of impact (central and eccentric impact). The related crushing forces increase with smaller particles and higher impact velocities. They are highest with central impact. It is noteworthy to imagine the magnitudes of forces which are provoked by impact. A limestone particle, e.g. of 0.3 kg (dF ¼ 60 mm) causes on average a crushing force of 236 kN (central impact, v ¼ 50 m s–1), which corresponds to a mass of about 24,000 kg, i.e. the load is multiplied in this case by a factor of 80,000! 6.2.2.2.3. Percussion. For the load case percussion there are hardly any systematic investigations available concerning the crushing forces, since the target of those investigations was to determine the particle strength. With the assumption that the maximum force coincides with the force at the breakage point the results of the particle strength (e.g. [47,71]) can be used.

102 1: v = 30 m/s 2: v = 40 m/s 3: v = 50 m/s 5 FAmax [MPa]

3 2

1 2

5

7

102 dF [mm]

2

Fig. 52. Area related crushing force FA max as function of particle diameter dF and impact velocity v (impact; glass fragments) ([55], p. 73).

The Principles of Single-Particle Crushing

175

102 1: v = 30 m/s 2: v = 40 m/s 3: v = 50 m/s 5 FAmax [MPa]

3 2 1 2

5

7

102

2

dF [mm]

Fig. 53. Area related crushing force FA max as function of particle diameter dF and impact velocity v (impact; limestone, Bernburg/Sachsen-Anhalt) ([55], p. 73).

102 central impact eccentric impact

5

3

FA max [MPa]

2 6 1 5 4

2

1 and 4: v = 30 m/s 2 and 5: v = 40 m/s 3 and 6: v = 50 m/s 5

7

102 dF [mm]

2

Fig. 54. Area related crushing force FA max as function of particle diameter dF, impact velocity v and contact locus (impact; diabase, Hausdorf/Sachsen) ([55], p. 73).

176

G. Unland

6.2.2.3. Conclusions Apart from the size effect the crushing forces vary to a very large extent with particles of the same material, diameter and shape at the same conditions. Therefore, the logarithmic relationship between the crushing forces and the particle diameters, which is deﬁned by a strength and a size parameter, considers an anisotropy parameter in order to characterize the whole spectrum of forces. The crushing forces depend on the various feed, design and operation parameters, whereas the inﬂuence is not marginal with the intensity of loading, the material and diameter of the particle as well as the shape of the contact planes. Particularly noticeable is the tremendous increase of forces by the rise of the loading velocity.

6.2.3. Loading time For the load cases impact and percussion it is quite easy to deﬁne the loading time Dt of the particle. Usually there is a more or less sinusoidal shape of the load development during contact with deﬁned terminations (Fig. 55). In case of compression it is difﬁcult to deﬁne the loading time. During loading the forces increase and very often rapidly decrease down to zero after a breakage event has occurred. 330 300 270 240

F [kN]

210 180 150 120 90 Δt

60 30

0.1

0.2

0.3

t [ms]

Fig. 55. Development of loading F during contact time Dt (central impact; limestone, Bernburg/Sachsen-Anhalt; dF ¼ 90 mm, v ¼ 40 m s–1) ([55], p. 53).

The Principles of Single-Particle Crushing

177

However, it is important to know the loading time development even in this case because the intensity of loading is comprised of the magnitude of the load, the time interval, in which it occurs and the partial loading time intervals, where load changes occur. This information is of prime interest for the mechanical sizing of the machine. Hardly any research work has been done on this subject so far.

6.2.3.1. Mathematical formulation During the time of loading the magnitude of the loading and its frequency vary. Appropriate tools to assess the dynamics of the loading process include the Fourier’s analysis ([50], p. 106).

6.2.3.2. Results 6.2.3.2.1. Compression. The total time of loading and the time when the load is built up are usually not critical with regard to the dynamic reaction of the machine. However, the release of the load while the breakage occurs can be very abrupt with several materials. This happens especially with class II materials. The sudden release of the load acts like a sharp impulse. 6.2.3.2.2. Impact. In case of impact the loading time is clearly terminated. In Figs. 56 and 57 this time is presented as a function of two different types of materials (limestone and diabase), the particle diameter and the impact velocity. The biggest impact on the loading time has the particle diameter. There is hardly any inﬂuence of the impact velocity. 0.45 2

0.4

1

3

Δt [ms]

0.3

0.2

1: v = 30 m/s 2: v = 40 m/s 3: v = 50 m/s

0.1

60

80

100

120

dF [mm]

Fig. 56. Loading time Dt as function of particle diameter dF and impact velocity v (central impact; limestone, Bernburg/Sachsen-Anhalt) ([55], p. 75).

178

G. Unland 0.45 3

0.4

1

Δt [ms]

0.3

0.2 2 1: v = 30 m/s 2: v = 40 m/s 3: v = 50 m/s

0.1

60

80

100

120

dF [mm]

Fig. 57. Loading time Dt as function of particle diameter dF and impact velocity v (central impact; diabase, Hausdorf/Sachsen) ([55], p. 75). 1

0.8 1

Δt [ms]

0.6

0.4

2

0.2

1: dF = 9.0 mm 2: dF = 3.4 mm

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

v [m/s]

Fig. 58. Loading time Dt as function of percussion velocity v and particle diameter dF (percussion; glass spheres; mass m* of moving weight 0.18 and 0.38 kg for dF ¼ 3.4 and 9 mm, respectively) ([45], p. 37).

6.2.3.2.3. Percussion. Only a few results are published for the load case percussion, e.g. a time of 0.11 ms till the breakage of an iron ore particle with a diameter of 75–90 mm and at a velocity of about 3.4 m s–1 ([47], p. 865) (see also [71,72]). Figure 58 shows some results for glass spheres at different velocities.

The Principles of Single-Particle Crushing

179

6.2.3.3. Conclusions Crushing is a very dynamic process and is characterized by sudden load changes. They happen within a few microseconds. But still there are not many results available. The dynamic responses of the crushers can be more violent with the load case compression than with the other cases although the total loading time is longer. The sudden release of the applied crushing forces and the stored elastic energy of the pre-stressed machine parts with brittle class II materials can provoke a more intensive impulse on the crusher than the one caused by impulse or percussion.

6.2.4. Energy consumption The energy is the most suitable parameter to characterize the intensity of loading. Besides the necessary forces it is required to transfer sufﬁcient energy into the particle in order to crush it. The energy is transferred from the working surface into the particle and vice versa.

6.2.4.1. Physical and mathematical formulation The energetic assessment of the crushing process is based on different kinds of energy (see Section 6.1). Therefore, it is very often difﬁcult or even impossible to compare results of different measurements. One reason is that various authors use different phases or effects to characterize the comminution of single particles and the mechanical work done during the process. Another reason is associated with the difﬁculties in measuring the mechanical work. The energy consumed or transferred is a random feature, whereas the energy offered is given. For a given material, loading and contact case the assessment of the measuring results has shown that the consumed or transferred energy with regard to its dependence on the particle diameter can be best described by a logarithmic function similar to Section 6.2.2.1. In case of the comminution energy EC it is d log E C ¼ log E Cðd c Þ þ r E log ð40Þ dc with n rE ¼

n n

P P P log E Cðd i Þ log d i log E Cðd i Þ log d i i¼1

i¼1

n

n P i¼1

2

ðlog d i Þ

n P

i¼1

2 log d i

ð41Þ

180

G. Unland

E Cðd c Þ ¼ 10 and slog E C

1 n

n P

log E Cðd i Þ r E

i¼1

1 n

n P

log d i log d c

i¼1

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u n u 1 X E Cðd c Þ di 2 t ¼ log þ r E log n 2 i¼1 E Cðd i Þ dc

ð42Þ

ð43Þ

The variation of the energy consumed, the work, can be assessed by a conﬁdence factor lE as well. For lE ¼ 1, 1.96, 2.58 and 3.29 the conﬁdence interval lE slog E is then 68.26%, 95%, 99% and 99.9%. Therefore, the energy used can be described by three parameters as well: Work parameter EC(50): Since the work parameter is related to a typical geo-

logical specimen, it is based on a particle of a diameter dc ¼ 50 mm. Size parameter rE: This parameter characterizes the inclination of the curve. Deviation parameter slog EC : The deviation of the energy used for a given dia-

meter is described by this parameter. It is necessary to mention that the above relations are valid for the transferred and breakage energies as well as for their related ﬁgures too, i.e. the mass, volume and surface speciﬁc energies. Additionally, the loading parameters have to be speciﬁed, such as the relative displacement in case of compression.

6.2.4.2. Results 6.2.4.2.1. Compression. During compression a particle is subjected to a deﬁned displacement h. Along the way the crushing forces have to be overcome and energy is transferred from the working surface into the particle. The integral of the force–displacement curve is the work done by the working surface. It is the transferred energy. The transferred energy as a function of the particle diameter is shown in Fig. 59. This curve for granodiorite can be described by a size parameter rE,20% ¼ 0.37 and a deviation parameter slog E T;m;20% ¼ 0:21. The mass speciﬁc energy ET,m for a diameter of dc ¼ 50 mm and at a relative displacement hr ¼ 20% is 0.8 J g–1. The mass speciﬁc transferred energy ET,m depends on the relative displacement hr (Fig. 60). It rises with increasing displacements hr and decreasing particle diameters d. There is a signiﬁcant size effect to be considered (Fig. 61). For this particular type of diorite the size effect is equal with various relative displacements. However, it is noteworthy that for other materials the size parameter varies with different relative displacements. 6.2.4.2.2. Impact. Most investigations are related to the offered energy because of difﬁculties in measuring the transferred energy. In those cases an

The Principles of Single-Particle Crushing

181

101 7 5 3

ET,m [J/g]

2

100 7 5 3 2

101

2

3

5

7

102 dF [mm]

2

3

5

7

103

Fig. 59. Mass speciﬁc transferred energy ET,m as function of particle diameter dF (compression; granodiorite, Kindisch/Sachsen; hr ¼ 20%).

3 1: fraction: dF = 450 ÷ 560 mm 2.5

2: fraction: dF = 250 ÷ 320 mm 3: fraction: dF = 200 ÷ 250 mm

ET,m [J/g]

2

4: fraction: dF = 140 ÷ 200 mm

6

5: fraction: dF = 80 ÷ 140 mm 6: fraction: dF = 56 ÷ 80 mm 1,5

5 1 3

4 0,5

2

5

10

15

1

20

25

hr [-]

Fig. 60. Mass speciﬁc transferred energy ET,m as function of relative displacement hr and particle feed diameter dF (compression; diorite, Hohwald/Sachsen) ([44], p. 98).

182

G. Unland 101 7 1: hr = 25 % 2: hr = 20 % 3: hr = 15 % 4: hr = 10 %

5 3

ET,m [J/g]

2

1 2 3

100

4

7 5 3 2

5

102

7

2 dF [mm]

3

5

7

103

Fig. 61. Mass speciﬁc transferred energy ET,m as function of particle diameter dF and relative displacement hr (compression; diorite, Hohwald/Sachsen) ([44], p. 100). 4 3

2 ET,m [J/g]

2

1 100 1: v = 30 m/s 2: v = 40 m/s 3: v = 50 m/s

7

5

7

102 dF [mm]

2

Fig. 62. Mass speciﬁc transferred energy ET,m as function of particle diameter dF and impact velocity v (impact; glass fragments) ([55], p. 78).

energetic assessment of the crushing process is not very meaningful. There are interesting ﬁndings only in conjunction with the crushing result (see Section 6.3.3). The transferred amount of energy is determined only by a few authors. Figures 62–64 show the inﬂuence of the particle diameter and the impact velocity on the

The Principles of Single-Particle Crushing

183

5 3

ET,m [J/g]

2 2 1

100 1: v = 30 m/s 2: v = 40 m/s 3: v = 50 m/s

7

5

102

7

2

dF [mm]

Fig. 63. Mass speciﬁc transferred energy ET,m as function of particle diameter dF and impact velocity v (impact; limestone, Bernburg/Sachsen-Anhalt) ([55], p. 78).

5 3

ET,m [J/g]

2

2 1

100 1: v = 30 m/s 2: v = 40 m/s 3: v = 50 m/s

7

5

102

7

2

dF [mm]

Fig. 64. Mass speciﬁc transferred energy ET,m as function of particle diameter dF and impact velocity v (impact; diabase, Hausdorf/Sachsen) ([55], p. 79).

184

G. Unland 101 apatite quartz marble copper ore taconite

5 2

EBP,V [J/cm3]

100 5 2 10-1 5 2

3

5

7

100

2 3 dF [mm]

5

7

101

Fig. 65. Volume speciﬁc breakage energy EBP,V as function of particle diameter dF and various materials (percussion; apatite; quartz; marble; copper ore, Bingham Canyon Mine/ Utah; taconite, Eveleth Mines/Minnesota) ([71], p. 17).

mass speciﬁc transferred energy ET,m for various materials. With decreasing diameters and increasing velocities the amount of speciﬁc energy rises, i.e. the size and intensity effect is similar to the results of comminution by compression. There is hardly any inﬂuence of the material. 6.2.4.2.3. Percussion. For this load case only a few results of investigations are accessible (Fig. 65). Most of the experiments determine the energy necessary to initiate the breakage of the particle ([50], p. 105).

6.2.4.3. Conclusions The work necessary and used to crush a single particle can be characterized by three parameters of a logarithmic energy-diameter function, a work parameter at a particle diameter of 50 mm, a size parameter and a deviation parameter. The mass speciﬁc transferred energy increases with rising loading intensities and decreasing particle diameters. The inﬂuence of materials is highly different.

6.3. Crushing product After comminution the resulting fragments can be assessed by their size and shape. Since the fragments are a polydisperse product, they need to be described as distributions.

The Principles of Single-Particle Crushing

185

6.3.1. Particle size distribution The particle size distribution is usually the most important feature of the product, because several industries demand a certain distribution. Furthermore, the maximum particle size in a product can be of paramount importance, if subsequent machines are restricted to a maximum feed size. The particle size distribution can be presented as a function of the different variables. If the cumulative mass per cent passing is shown as a function of the fragment size and the comminution energy, this relationship is also termed the breakage function. The breakage function gives the size distribution of fragments resulting from a single breakage event of a particle of a certain diameter and material at a deﬁned loading mode and energy input.

6.3.1.1. Physical and mathematical formulation Several types of distribution functions were developed in order to describe mathematically the particle size distribution of a crusher product, such as the Gates–Gaudin–Schuhmann (GGS) distribution or the Rosin–Rammler–Sperling–Bennett (RRSB) distribution. A very good approximation can be achieved by multiple parametric logarithmic normal distributions (see also Section 6.2.1.1) or multiple parametric RRSB distributions, which consider lower and upper limits of the particle size distribution. For the general case, i.e. there are lower and upper limits, the logarithmic normal distribution Z u 1 2 QðdÞ ¼ pﬃﬃﬃﬃﬃﬃ ez =2 dz ð44Þ 2p 1 is transferred by the substitution u¼

ln d ln d50 sTQ

ð45Þ

to the 4-parametric logarithmic normal distribution ([73], p. 399) with d ¼ dO

d dU dO d

ð46Þ

1 ðln d84 ln d16 Þ 2 ¼ ln d84 ln d50

sTQ ¼

¼ ln d50 ln d16

ð47Þ

Qðd16 Þ ¼ 0:1587

ð48Þ

Qðd50 Þ ¼ 0:5

ð49Þ

Qðd84 Þ ¼ 0:8413

ð50Þ

and

186

G. Unland

where 0od U d d O o1

ð51Þ

In general the lower limit dU and the upper limit dO are unknown. In those cases the limits need to be estimated ([74], p. 481). If there is only one limit, then dU or dO is set to zero. In most technical applications the comminution of irregularly shaped particles is characterized by friction between the fragments. Furthermore, there are crushers which offer more energy EO than the particle is able to absorb during the process of comminution, e.g. VSI crushers. In many of those cases the size distribution of the fragments can better be approximated by mixed distributions ([75], p. 467 [76]). The size distribution q(d) (density function) of fragments formed by crushing of single particles is then approximated by a superposition of several (N) statistical particle assemblies, whereas every assembly is weighted by the comminution energy wk (EC): qðd; E C Þ ¼

N X

w k ðE C Þqk ðd; d O;k ; d 50;k ; sln;k Þ

ð52Þ

k¼1

with q ðdÞ ¼

dQ ðdÞ d ðdÞ

ð53Þ

Each assembly is described by a truncated 3-parameter logarithmic normal distribution qk with an upper limit dO (dU ¼ 0) for each comminution energy range EC.

6.3.1.2. Results 6.3.1.2.1. Compression. For several materials and different particle shapes the breakage functions Q(d) are determined. As an example of regularly shaped particles Fig. 66 shows the inﬂuence of the mass speciﬁc energy Em and the feed diameter dF on the breakage function of glass spheres. With the transformation into a 3-parameter logarithmic normal distribution (see Section 6.3.1.1) those curves can be transformed into straight lines (see [75], p. 470, [10], p. 135). There are several approaches to formulate the fragment size distribution Q(d) of irregularly shaped feed particles. Two examples are given. For diorite the fragment size distributions are approximated by 3-parametric logarithmic normal distributions with the parameters dO, d50 and sTQ. In the case of compression it is easier to determine the size distribution, if the relative displacement hr is used as a load parameter. If the parameters dO and d50 are then related to the feed diameter dF, these ratios are only functions of the relative

The Principles of Single-Particle Crushing

Q(d) [%]

99.9 99.8 99.5 99

187

6 5

1: EC,m = 7.1.10-2 J/g 2: EC,m = 22.4.10-2 J/g 3: EC,m = 44.0.10-2 J/g 4: EC,m = 12.1.10-1 J/g 5: EC,m = 38.2.10-2 J/g 6: EC,m = 12.1.10-1 J/g 7: EC,m = 30.9.10-1 J/g

95 90 80 70 60 50 40 30 20 10 5 2 1

3 2

7

1

dF = 16 mm dF = 5 mm

4

2

3

5 7 10-1

2

3

5 7 100 2 d [mm]

3

5 7 101

2

3

5 7 102

Fig. 66. Fragment size distribution Q(d) as function of mass speciﬁc comminution energy EC,m and particle feed diameter dF (glass spheres) (data from [75], p. 470, [10], p. 131).

dO [-] dF

1

0 5

10

15

20

25

hr [%]

Fig. 67. Upper fragment size ratio dO/dF as function of relative displacement hr and various particle feed fractions (compression; diorite, Hohwald/Sachsen; dF ¼ 56–80, 80–140, 140–200, 200–250, 250–320, 450–560 mm) ([44], Appendix 27).

displacement hr (Figs. 67 and 68), whereas the deviation sTQ tends to increase with the displacement hr (Fig. 69). The second example relates to the approximation of the measured fragment size distribution by a size density function q(d) with energy weighted terms.

188

G. Unland

d50 [-] dF

1

0 5

10

15

20

25

hr [%]

Fig. 68. Median fragment size ratio d50/dF as function of relative displacement hr and various particle feed fractions (compression; diorite, Hohwald/Sachsen; dF ¼ 56–80, 80–140, 140–200, 200–250, 250–320, 450–560 mm) ([44], p. 104). 2.8 2.4

sTQ [-]

2.0 1.6 1.2 0.8 0.4

5

10

15

20

25

hr [%]

Fig. 69. Deviation sTQ as function of relative displacement hr and various particle feed fractions (compression; diorite, Hohwald/Sachsen; dF ¼ 56–80, 80–140, 140–200, 200–250, 250–320, 450–560 mm) ([44], p. 106).

Figure 70 shows the graph of the fragment size distribution, the approximation by four fragment size density functions and their parameters. The investigation of the functions qk(d) in relationship to the comminution energy EC yields the following results: The maximum fragment size dO,k is independent of the comminution energy EC

and raises from the ﬁne to the coarse particle assembly (Fig. 71). The

The Principles of Single-Particle Crushing

189

q(d) [%/mm]

99.9 99.8 99.5 99 98 95 90 80 70 60 50 40 30 20

d0.1 = 0.4 mm d50.1 = 0.11 mm σln,1 = 1.1 w1 = 2.4 %

d0.2 d50.2 σln,2 w2

= 1.0 mm = 0.5 mm = 1.1 = 0.4 %

d0.3 = 4.5 mm d50.3 = 2.85 mm σln,3 = 1.0 w3 = 2.4 %

d0.4 = 18.0 mm d50.4 = 13.0 mm σln,4 = 0.9 w4 = 94.8 %

10 5 2 1 0.5 0.1 2

3

5 7 10-1

2

3

5 7 100 d [mm]

2

3

5 7 101

2

3

Fig. 70. Fragment size density function q(d) as superposition of particle assemblies with characteristic parameters (compression; irregularly shaped particle; quartzite, Sproitz/ Sachsen; dF ¼ 16–18 mm; EC ¼ 1.63–2.04 J) ([53], p. 50). 20 d0.4

18 16

dO,k [mm]

14 12 10 8 6 d0.3

4

d0.2

2

d0.1 0

2

4 EC [J]

6

8

Fig. 71. Maximum fragment size dO,k as function of comminution energy EC (compression; quartzite, Sproitz/Sachsen; dF ¼ 16–18 mm) ([53], p. 56).

190

G. Unland

maximum fragment sizes dO,1 and dO,2 of the two ﬁner fragment assemblies are not dependent on the initial particle feed size dF ([76], p. 133). The median fragment size d50,k is constant with the comminution energy except for the most coarse assembly N. This diameter decreases with the energy EC (Fig. 72). The median fragment sizes d50,1 and d50,2 do not depend on the particle feed size dF ([76], p. 133). The deviation sln,k does not depend on the comminution energy or the ﬁneness of the assembly (Fig. 73). According to Klotz et al. ([76], p. 133f) the deviation sln,k is not dependent on the feed diameter either. The mass fractions wk depend on the comminution energy EC. The share wN of the coarsest assembly N falls, whereas all the other shares increase with rising comminution energy EC (Fig. 74). The number N of fragment assemblies increases with rising feed diameter dF and constant comminution energy EC (Fig. 75).

6.3.1.2.2. Impact. For the loading case ‘‘impact’’ the most common used intensity factor is the impact velocity. The fragment size distributions from more spherical (Figs. 76 and 77) and irregularly shaped particles (Figs. 78 and 79) make it obvious that there are still feed particles in the product with lower impact

20 18 16

d50,k [mm]

14 d50.4

12 10 8 6

d50.3

4 2

d50.2 0

2

4

6

d50.1 8

EC [J]

Fig. 72. Median fragment size d50,k as function of comminution energy EC (compression; quartzite, Sproitz/Sachsen; dF ¼ 16–18 mm) ([53], p. 57).

The Principles of Single-Particle Crushing

191

σln,1

1.2

σln,2 σln,3

1.0 σln,k

σln,4 0.8

0.6 0

2

4

6

8

EC [J]

Fig. 73. Deviation sln, k as function of comminution energy EC (compression; quartzite, Sproitz/Sachsen; dF ¼ 16–18 mm) ([53], p. 57).

1 w4 0.8

wk [-]

0.6

0.4

0.2 w1

w2 w3

2

4 EC [J]

6

8

Fig. 74. Mass fraction wk as function of comminution energy EC (compression; quartzite, Sproitz/Sachsen; dF ¼ 16–18 mm) ([53], p. 58).

velocities and that the shape of the size distribution curve varies signiﬁcantly with the impact velocity and the material to be crushed. Therefore, different functions are applied to formulate mathematically the distribution curves. For three materials the regression curves are depicted in Fig. 80. Those curves are based on

192

G. Unland

50 20 10 5

dF = 32 - 40 mm EC = 0.315 J/g

1

q(d) [%/mm]

50 20 10 5

dF = 22 - 25 mm EC = 0.299 J/g

1 50 20 10 5

dF = 12.5 - 16 mm EC = 0.312 J/g

1 50 20 10 5

dF = 6.3 - 8 mm EC = 0.310 J/g

1 2

3

5 7 10-1

2 3

5 7 100 2 d [mm]

3

5 7 101

2 3

5 7 102

Fig. 75. Fragment size density function q(d) as function of particle feed diameter dF (compression; quartzite, Sproitz/Sachsen; ECEconst.) ([53], p. 60).

the transformed RRSB distribution with an upper limit dO. The evaluation of the experiments yields an upper limit dO, which corresponds to the feed diameter dF, i.e. dO ¼ dF. There are still unbroken feed particles in the product. The ratio of the characteristic diameter dRRSB63,2 and the feed diameter dF tends to be constant with the same impact velocity v and various particle feed diameters dF but decreases with larger impact velocities v (Fig. 81). The exponent nRRSB T of the transformed RRSB distribution decreases slightly with larger particle feed diameters dF, but it is independent of the impact velocity v (Fig. 82). There is hardly any noticeable inﬂuence of the material type. For impact loading it is possible to enhance the approximation of the fragment size distribution by a superposition of partial size distributions as well. Figures 83 and 84 show density functions q(d, EO) for limestone and quartzite. It is to consider that the energy denoted is the offered energy EO. The following results are derived from the assessment of the functions qk(d): The maximum fragment size dO,k is independent of the offered energy EO,m

(Fig. 85).

The Principles of Single-Particle Crushing 99.9 99 90

Q(d) [%]

50 40 30 20 10

1: v = 200 m/s 2: v = 200 m/s 3: v = 140 m/s 4: v = 50 m/s 5: v = 30 m/s 6: v = 30 m/s 7: v = 30 m/s 8: v = 20 m/s 9: v = 20 m/s 10: v = 15 m/s

193

dF = 2 ÷ 2.5 mm dF = 4 ÷ 4.5 mm dF = 7 ÷ 8.0 mm

5

4

1

1

6 0.5

2

3

5

8

9

10

7 0.1 2

5

10-2

2

5

10-1

2

5

100

2

5

101

d [mm]

Fig. 76. Fragment size distribution Q(d) as function of impact velocity v and particle feed diameter dF (impact; glass spheres) (data from [54], p. 52). 99.9 99

1: v = 200 m/s 2: v = 150 m/s 3: v = 105 m/s 4: v = 50 m/s 5: v = 30 m/s 6: v = 15 m/s

90

Q(d) [%]

50 40 30 20

2

10 5

1 0.5

5 1

3

4

0.1 2

5

10-2

6

2

5

10-1 2 d [mm]

5

100

2

5

101

Fig. 77. Fragment size distribution Q(d) as function of impact velocity v (impact; cement clinker, Weisenau/Rheinland-Pfalz; dF ¼ 5–6 mm) (data from [54], p. 53).

194

G. Unland 99.9 99

1: v = 200 m/s 2: v = 200 m/s 3: v = 110 m/s 4: v = 100 m/s 5: v = 50 m/s 6: v = 50 m/s 7: v = 35 m/s 8: v = 30 m/s

90

Q(d) [%]

50 40 30 20 10 5 1

2 6

1 0.5

3 4

5

7

dF = 3 ÷ 4 mm

8

dF = 6 ÷ 7 mm 0.1 2

10-2

5

2

10-1 2 d [mm]

5

100

5

2

101

5

Fig. 78. Fragment size distribution Q(d) as function of impact velocity v and particle feed diameter dF (impact; limestone, Lauingen/Bayern) (data from [54], p. 52). 102 5

dF = 3 ÷ 4 mm dF = 6 ÷ 7 mm

2 101

Q(d) [%]

5 2

1

2

100 3

5

5

4

2

1: v = 200 m/s 2: v = 190 m/s 3: v = 102 m/s 4: v = 100 m/s 5: v = 50 m/s 6: v = 31 m/s 7: v = 20 m/s 8: v = 20 m/s 9: v = 15 m/s

6

9

7

10-1

8

2

5

10-2

2

5

10-1 d [mm]

2

5

100

2

5

101

Fig. 79. Fragment size distribution Q(d) as function of impact velocity v and particle feed diameter dF (impact; quartz, Frechen/Nordrhein-Westfalen) (data from [54], p. 53).

The Principles of Single-Particle Crushing 99.9 99

195

glass fragments limestone diabase

90

v = 50 m/s

Q(δ) [%]

50 40 30 20 10 5

1 0.5

v = 30 m/s

0.1 2

5

100

2

5

101

2

5

102

2

5

103

δ [mm]

Fig. 80. Transformed fragment size distribution Q(d) as function of impact velocity v and different materials (impact; glass fragments; limestone, Bernburg/Sachsen-Anhalt; diabase, Hausdorf/Sachsen; dF ¼ 113 mm) (data from [55], 81f). 2 v = 50 m/s v = 40 m/s v = 30 m/s

dRRSB63.2 /dF [-]

diabase 1 0.9 0.8 0.7 0.6 0.5 0.4 limestone 0.3

glass fragments 62.5

75 dF [mm]

87.5

100

125

Fig. 81. Ratio dRRSB63,2/dF as function of particle feed diameter dF and impact velocity v (impact; glass fragments; limestone, Bernburg/Sachsen-Anhalt; diabase, Hausdorf/Sachsen) (data from [55], p. 83).

196

G. Unland 1 0.96 0.92

limestone

0.88 0.84 0.80 glass fragments

nRRSB T [-]

0.76 0.72 0.68

diabase 0.64 0.60 v = 50 m/s v = 40 m/s v = 30 m/s 70

80

90

100

dF [mm]

Fig. 82. Exponent nRRSSB T as function of particle feed diameter dF and impact velocity v (impact; glass fragments; limestone, Bernburg/Sachsen-Anhalt; diabase, Hausdorf/Sachsen) (data from [55], p. 86). 50 d0.1 = 0.35 mm d50.1= 0.13 mm σln,1 = 1 20 w = 0.77 % 1

d0.2 = 1.7 mm d50.2 = 0.78 mm σln,2 = 0.8 w2 = 3.89 %

d0.3 = 3.5 mm d50.3 = 2.2 mm σln,3 = 0.76 w3 = 12.6 %

d0.4 = 8.5 mm d50.4 = 5.3 mm σln,4 = 0.53 w4 = 82.33 %

q(d) [%/mm]

10

5

2

0.2 10-2

2

5

10-1

2

5

100

2

5

101

d [mm]

Fig. 83. Fragment size density function q(d) as superposition of partial particle assemblies with characteristic parameters (impact; limestone, Ru¨beland/Sachsen-Anhalt; dF ¼ 6.3–8 mm; EO,m ¼ 0.456 J g–1) ([56], p. 52).

The Principles of Single-Particle Crushing

197

50 d0.1 = 0.47 mm d50.1 = 0.16 mm σln,1 = 0.92 w1 = 3.93% 20

d0.2 = 1.7 mm d50.2 = 0.85 mm σln,2 = 0.85 w2 = 4.47 %

d0.3 = 3.5 mm d50.3 = 2.35 mm σln,3 = 0.76 w3 = 17.28 %

d0.4 = 8.5 mm d50.4 = 5.3 mm σln,4 = 0.6 w4 = 74.62 %

q(d) [%/mm]

10

5

2

0.2 10-2

2

10-1

5

2

5

100

2

101

5

d [mm]

Fig. 84. Fragment size density function q(d) as superposition of partial particle assemblies with characteristic parameters (impact; quartzite, Sproitz/Sachsen; dF ¼ 6.3–8 mm; EO,m ¼ 0.638 J g–1) ([56], p. 52). 11 d0.4

10 9 limestone

8

quartzite

d0,k [mm]

7 6 5 4

d0.3

3 2

d0.2

1

d0.1 0.2

0.4

0.6

0.8

1.0

1.2

1.4

EO,m [J/g]

Fig. 85. Maximum fragment size dO,k as function of mass speciﬁc offered energy EO,m (impact; limestone, Ru¨beland/Sachsen-Anhalt; quartzite, Sproitz/Sachsen; dF ¼ 8–10 mm) ([56], p. 57).

198

G. Unland

The median fragment size d50,k decreases with the offered energy EO,m for the

most coarse assembly N and is constant for the ﬁner assemblies (krN1) (Fig. 86). The deviation sln,k slightly increases with the energy EO,m (Fig. 87). The mass fractions wk depend on the energy EO,m, whereas the fraction wN of the coarse assembly N decreases and the other ones (koN) increase with the offered energy EO,m (Fig. 88).

6.3.1.2.3. Percussion. For the loading case ‘‘percussion’’ the passing/breakage functions from more regularly (Figs. 89 and 90) and irregularly shaped particles (Figs. 91 and 92) are presented. The different curves reﬂect the crushing behaviour of the particles. The graphs related to glass spheres clearly show that the size distribution remains constant for offered energies EO,m bigger than Ea,m, apart from secondary comminution events of already crushed fragments. Every glass sphere can only absorb energy up to a certain limit, the maximum absorbed energy Ea,m. For brittle spherical particles, such as glass spheres, this energy coincides with the energy absorbed EBP,m at the breakage point, since the fragments ﬂy away, when the sphere breaks. The breakage event is clearly deﬁned.

9 limestone

8

quartzite 7 d50.4

d50,k [mm]

6 5 4 3

d50.3 2 d50.2

1

d50.1 0.2

0.4

0.6 0.8 EO,m [J/g]

1.0

1.2

1.4

Fig. 86. Median fragment size d50,k as function of mass speciﬁc offered energy EO,m (impact; limestone, Ru¨beland/Sachsen-Anhalt; quartzite, Sproitz/Sachsen; dF ¼ 8–10 mm) ([56], p. 58).

The Principles of Single-Particle Crushing

199

1.2 σln,1

σln,k [-]

1.0

σln,2 0.8 σln,3

0.6 σln,4

limestone quartzite 0.2

0.4

0.6

0.8

1.0

1.2

1.4

EO,m [J/g]

Fig. 87. Deviation sln,k as function of mass speciﬁc offered energy EO,m (impact; limestone, Ru¨beland/Sachsen-Anhalt; quartzite, Sproitz/Sachsen; dF ¼ 8–10 mm) ([56], p. 56). 1 limestone quartzite 0.8

w4

wk [-]

0.6

0.4

w3

0.2

w2 w1 0.4

0.8 EO,m [J/g]

1.2

1.6

Fig. 88. Mass fraction wk as function of mass speciﬁc offered energy EO,m (impact; limestone, Ru¨beland/Sachsen-Anhalt; quartzite, Sproitz/Sachsen; dF ¼ 8–10 mm) ([56], p. 59).

200

G. Unland 99.9 99.8 99

Q*(dp,EO,m) [%]

96 90 80 60

7

1: dp = 0.1 mm 2: dp = 0.5 mm 3: dp = 1.0 mm 4: dp = 2.0 mm 5: dp = 5.0 mm 6: dp = 8.0 mm 7: dp = 10.0 mm

6

5

40

4

20 10 5 2 1

3 2 1

0.1 0.02 7 8

10-1

2

4

6

8

100

2

Ea,m EO,m [J/g]

Fig. 89. Cumulative percent Q* passing dP as function of mass speciﬁc offered energy EO,m (percussion; glass spheres; dF ¼ 16 mm; v ¼ 3.8 m s–1) ([52], p. 69).

102 5 2

Q(d) [%]

101 5

1: EO,m = 8.0 J/g 2: EO,m = 6.33 J/g 3: EO,m = 2.7 J/g 4: EO,m = 2.29 J/g 5: EO,m = 1.0 J/g 6: EO,m = 0.864 J/g 7: EO,m = 0.3 J/g 8: EO,m = 0.204 J/g 9: EO,m = 0.09 J/g

2 5

100 5

2

7

dF = 6.9 mm dF = 2.5 mm

1 4 2 3 2

8

6 5

-2

10

9 2

5

10-1 2 d [mm]

5

100

2

5

101

Fig. 90. Fragment size distribution Q(d) as function of mass speciﬁc offered energy EO,m and particle feed diameter dF (percussion; cement clinker, Weisenau/Rheinland-Pfalz) ([45], p. 48).

The Principles of Single-Particle Crushing

201

99.5

1: EO,m = 4.18 2: EO,m = 1.71 3: EO,m = 1.01 4: EO,m = 0.38 5: EO,m = 0.18 6: EO,m = 0.12

96 90 80

Q(d) [%]

60

J/g J/g J/g J/g J/g J/g

40 20 10 1

5 2 1

2 3 6

4 0.1

5

0.02 2

10-1

5

2

100

5

2

101

5

d [mm]

Fig. 91. Fragment size distribution Q(d) as function of mass speciﬁc offered energy EO,m (percussion; limestone, Bernburg/ Sachsen-Anhalt; dF ¼ 11.1 mm; v ¼ 2 m s–1) ([52], p. 86). 102 dF = 9.8 mm dF = 4.8 mm dF = 2.0 mm

1: EO,m = 40.00 J/g

50 2: EO,m = 16.00 J/g

Q(d) [%]

3: EO,m = 9.00 J/g 4: EO,m = 6.70 J/g 20 5: EO,m = 10.00 J/g 6: EO,m = 2.75 J/g 101 7: EO,m = 1.15 J/g

1

2

5 2 6 100 0.5

3 4

5

9 11

0.2 7

10-1 2

8: EO,m = 1.00 J/g 9: EO,m = 0.80 J/g 10: EO,m = 0.42 J/g 11: EO,m = 0.38 J/g 12: EO,m = 0.09 J/g 13: EO,m = 0.10 J/g

8 5

10-2

10

12

2

5

13 10-1

2

5

100

2

5

101

d [mm]

Fig. 92. Fragment size distribution Q (d) as function of mass speciﬁc offered energy EO,m and particle feed diameter dF (percussion; quartz, Frechen/Nordrhein-Westfalen) ([52], p. 86).

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With other materials and particle shapes the fragments can remain in the process area and are subjected to additional comminution and occasionally to agglomeration, if the energy offered is big enough. In technical applications these effects cannot be avoided. Baumgardt ([52], 66ff) developed a method to separate in theoretical calculations the additional effects from the initial breakage event by assessing the fragment size distribution at the offered energy with the function of the breakage probability. In general, it can be postulated that the size distribution does not depend on the offered energy EO,m. It depends on the absorbed comminution energy EC,m.

6.3.1.3. Conclusions The fragment size distributions can be mathematically described by 3-parametric logarithmic normal distributions or occasionally better by 3-parametric RRSB distributions. These approximations yield especially good results with spherical particles of brittle materials. The size distributions of fragments from crushed irregularly shaped particles can be very often best approximated by a superposition of distributions of partial fragment assemblies. The fragment size distribution depends on the intensity of loading, especially on the absorbed comminution energy, but not on the offered energy.

6.3.2. Particle shape distribution The shape of particles is of great importance in many industries. The macroshape, i.e. the relation of the overall dimensions, inﬂuences the load capacity of buildings, railroads or roads. The mesoshape, i.e. the angularity of corners and edges, is one of the most important features in the production of abrasives, whereas the microshape, i.e. the roughness of the surface, controls the consumption of binding material, such as cement or asphalt, in the construction industry. Furthermore, the shape inﬂuences the behaviour of the particles in a processing plant, such as the ﬂowability in chutes and bins or the pressure drop of gas vented particle layers due to different porosities. Despite the importance of the particle shape only limited research work has been done and hardly any results are published.

6.3.2.1. Physical and mathematical formulation Because of the importance of the particle shape many countries and industries have developed their own standards. Basically, the characterization of the particle shape uses three principal methods (compilation of methods [77]): Verbal descriptions, Shape factors (mathematical and physical) and Shape functions.

The Principles of Single-Particle Crushing

203

Since the particles exhibit different shapes it is important that a collective of particles is characterized by a shape distribution. Besides the few publications on the systematic investigation of the inﬂuence on the particle shape it is the biggest problem that there is no common method which is internationally agreed upon or applied.

6.3.2.2. Results There are only a few systematic investigations accessible for the load case compression and impact. 6.3.2.2.1. Compression. For an isotropic and homogeneous material (diorite) systematic experiments were carried out in order to investigate the inﬂuence of the particle size and the relative displacement on the shape of the fragments. The shape is assessed by the macroshape. The longest dimension ll together with the perpendicular shortest dimension ls dictates the orientation of the particle. The medium dimension lm is then perpendicular to the longest and shortest dimension. The elongation ie is deﬁned as the ratio of the longest to the medium dimension and the ﬂatness if as the ratio of the medium to the shortest dimension. Figure 93 shows the inﬂuence of the particle sizes and relative displacements on the magnitude of ﬂatness if of the fragments, whereas Fig. 94 depicts the inﬂuence on the elongation ie. A detailed analysis of the fragment shapes within the size fractions is presented in Figs. 95 and 96. It is obvious that there is hardly any inﬂuence of the feed diameter and the relative displacements on the fragment shape. However, there tends to be an enrichment of elongated and ﬂat fragments within the smaller size fractions. 6.3.2.2.2. Impact. The particle shape of fragments from diabase were investigated (Fig. 97) according to the German standard DIN 52114. Particles are deﬁned as misshapen if their ratio between the longest and shortest dimension exceeds 3. The assessment of fragments from impacted cubical rocks comes to the conclusion that misshapen particles are enriched within the ﬁner product fractions and that an increase in impact velocity ameliorates the fragment shape. 6.3.2.2.3. Percussion. Several crushers which apply the load case percussion were developed in order to produce cubical products, such as percussion jaw crushers (Krupp type) or cone crushers (Simons type). Besides percussion there is a turning of the particles within the process zone of the crusher. Although those crushers were introduced to improve the product shape there are no investigations published on the generation of particle shapes for the percussion of single particles.

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50 dF = 140 - 200 mm

30

hi [%]

40

20 10 1- 2

2- 3

18

3- 4

4- 5 if [-]

5- 6

6- 7

9 7- 8

hr [%]

6

8- 9

60

dF = 80 - 140 mm

hi [%]

40

20

1- 2 2- 3 3- 4 4- 5 5- 6 6- 7 if [-] 7- 8

22 14 9 5

8- 9

hr [%]

60

40

hi [%]

dF = 56 - 80 mm

20

1- 2

2- 3

22

3- 4

4- 5 5- 6 6- 7 if [-] 7- 8

15 8- 9

7

hr [%]

Fig. 93. Share hi of fragments as function of relative displacement hr and ﬂatness if (compression; diorite, Hohwald/Sachsen) ([44], Appendix 29).

The Principles of Single-Particle Crushing

205

90 80 60 40

hi [%]

dF = 140 - 200 mm

20

1- 2

2- 3

18

3- 4

4- 5 5- 6 ie [-] 6- 7

9 7- 8

6

8- 9

hr [%]

90 80 dF = 80 - 140 mm

40

hi [%]

60

20

22 1- 2

2- 3

14 3- 4

4- 5

ie [-]

5- 6

9

hr [%]

5

6- 7

100 80 60 40

hi [%]

dF = 56 - 80 mm

20 1- 2

2- 3

3- 4

22 4- 5

ie [-]

5- 6

6- 7

7- 8

8- 9

7

15 hr [%]

Fig. 94. Share hi of fragments as function of relative displacement hr and elongation ie (compression; diorite, Hohwald/Sachsen) ([44], Appendix 30).

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60 50

30

hi [%]

40

20 10

125 + 90 -125 1-2 2-3 3-4

63 - 90 45 - 63 4-5 5-6 if [-]

6-7

31.5 - 45 7-8

8-9

dp [mm]

22.4 - 31.5

Fig. 95. Share hi of ﬂat fragments (if) within fragment size fraction dP (compression; diorite, Hohwald/Sachsen; dF ¼ 250–320 mm; hr ¼ 20%) ([44], p. 110).

100 80

40

hi [%]

60

20

125 + 90 -125 63 - 90 45 - 63 1-2

31.5 - 45

2-3

3-4 4-5 ie [-] 5-6

dp [mm]

22.4 - 31.5

Fig. 96. Share hi of elongated fragments (ie) within fragment size fraction dP (compression; diorite, Hohwald/Sachsen; dF ¼ 250–320 mm; hr ¼ 20%) ([44], p. 111).

6.3.2.3. Conclusions Despite the problem of not ubiquitously accepted and applied shape determinations and apart from too few scientiﬁc investigations it is known from practical experience that very often the more misshapen fragments are more abundant in the ﬁner size fractions independent of the load case.

The Principles of Single-Particle Crushing

207

50

20

pms [%]

40

30 40 2-8

v [m/s]

8-22 dp [mm]

22-50

50

Fig. 97. Percentage pms of misshapen particles as function of fragment fraction dP and impact velocity v (impact; diabase, Mu¨hlbach/Sachsen; dF ¼ 113 mm; cubical feed shape) ([78], p. 220).

6.3.3. Energy utilization The comminution of particles consumes energy of various amounts. It depends on the applied process parameters, the material and the diameter of the particles among others. For reasons of comparison and assessment it is appropriate to deﬁne an energetic efﬁciency factor of the comminution process.

6.3.3.1. Physical and mathematical formulation The energetic assessment necessitates a relation between the intensity of the loading and the result of the comminution process. One frequently used option refers to the newly produced surface DS of the fragments and the amount of energy E involved in the process of comminution, i.e. SE ¼

DS E

ð54Þ

This factor is termed energy utilization [79]. However, the application of the energy utilization faces two major problems. The ﬁrst one relates to the determination of the surface (e.g. calculations from size distributions, permeability or gas adsorption data) and the second one to the kind of energy, which is taken for the factor eSE. Since crushing deals with bigger particles the surface is most often calculated from data of measured particle size distributions. For a fair comparison it is necessary to use the same kind of energy in equation (54). It depends on the

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target, whether the transferred energy ET, the breakage energy EBR or the comminution energy EC is most suitable. Owing to measuring difﬁculties the offered energy EO is most frequently taken in equation (54) with comminution by impact and percussion, although such a factor has hardly anything to do with the energy utilization from a physical point of view. In case the comminution energy EC is used an efﬁciency ZSP of the singleparticle comminution can be deﬁned by multiplying the energy utilization eSE with the surface energy g per unit area ([10], p. 157, [80]): ZSP ¼ gSE

ð55Þ

It is necessary to mention that this efﬁciency is not helpful to assess the singleparticle comminution, since all unavoidable energy losses, such as for deformation, are not considered with the speciﬁc surface energy g. Furthermore, this efﬁciency is not applicable for the assessment of the technical process at all. In those processes several additional unavoidable losses occur, such as friction.

6.3.3.2. Results 6.3.3.2.1. Compression. The investigations on glass spheres do not show a size effect with the energy utilization (Fig. 98). As described in Section 6.3.2 glass spheres absorb only energy up to the breakage point. With the speciﬁc surface energy g of glass of 0.92 J m–2 ([10], p. 157) the efﬁciency of single-particle comminution for glass spheres is then 0.42% and independent of the mass speciﬁc energy. The energy utilization of the comminution of irregularly shaped particles of diabase and quartzite, however, shows a size effect (Figs. 99 and 100) and decreases with higher relative displacements resp. comminution energy. These ﬁndings are supported by Fig. 101, where the energy utilization is independent of the diameter of the glass spheres and the comminution energy. A decrease of the energy utilization is indicated with a decrease of the diameter of the broken glass and an increase of the comminution energy. 6.3.3.2.2. Impact. The assessment of the energy utilization for the load case impact needs to be grouped because different kinds of energy are applied. Figures 102–104 relate to the offered energy EO. The graphs show a size effect and an optimal energy utilization for a deﬁned impact velocity. The relations shown in Fig. 105 are based on the comminution energy EC. They conﬁrm an existing size effect, but no inﬂuence of the impact velocity can be found. There is a limit how much energy a particle can absorb during the process of comminution. 6.3.3.2.3. Percussion. The energy utilization eSE,O in case of percussion indicates a dependency on the particle size and the offered energy EO. For every material and particle diameter there is a deﬁned optimum (Figs. 106 and 107).

The Principles of Single-Particle Crushing

209

2

10-1 εSE [m²/J]

8 6

4

2 2

3

4

5 6 7 dF [mm]

10

15

20

Fig. 98. Energy utilization eSE as function of particle feed diameter dF (compression; glass spheres) ([10], p. 156).

10-3 8 6 4

εSE [m²/J]

2

10-4 8 6 hr = 5 % hr = 10 % hr = 20 %

4

60

70

80

90 100

200

300

dF [mm]

Fig. 99. Energy utilization eSE as function of particle feed diameter dF and relative displacement hr (compression; diabase, Mu¨hlbach/Sachsen) ([51], p. 55).

210

G. Unland 2 100 5

εSE [m²/J]

2 10-1

5 2 1: dF = 5.0 ÷ 6.3 mm 2: dF = 10.0 ÷ 12.5 mm 3: dF = 22.0 ÷ 25.0 mm

10-2

2

5

3

1

2 2

5

10-2

2

5

10-1 2 Ec [J]

100

5

2

101

5

2

Fig. 100. Energy utilization eSE as function of comminution energy EC and particle feed diameter dF (compression; quartzite, Sproitz/Sachsen) ([53], p. 93). 10-1 5

dF = 32 ÷ 40 mm

2

dF = 22 ÷ 25 mm dF = 16 ÷ 18 mm glass spheres dF = 5 ÷ 16 mm

10-2 εSE [m²/J]

broken glass

3

x

5

x

x

3 2

x

x

x x x

10-3

x

5 3 2 2

3

5

100

2

3

5

101

2

3

EC [J]

Fig. 101. Energy utilization eSE as function of comminution energy EC and particle feed diameter dF (compression; broken glass; glass spheres) ([79], p. 500).

The Principles of Single-Particle Crushing

211

6

εSE,O [10-3 m2/J]

5

4

3

2

glass spheres dF = 4 ÷ 8 mm dF = 2.3 mm dF = 1.2 mm cement clinker dF = 5÷ 6 mm

1

50

100 v [m/s]

150

200

Fig. 102. Energy utilization eSE,O as function of impact velocity v and particle feed diameter dF (impact; glass spheres; cement clinker, Weisenau/Rheinland-Pfalz) (data from [54], pp. 67, 69).

8 7

εSE,O [10-3 m2 /J]

6

1

5

2 4

4

3

5 3

6

2 limestone quartz

1

50

100

150

1, 4:dF = 1÷2 mm 2, 5:dF = 3÷4 mm 3, 6:dF = 6÷7 mm 200

250

v [m/s]

Fig. 103. Energy utilization eSE,O as function of impact velocity v and particle feed diameter dF (impact; quartz, Frechen/Nordrhein-Westfalen; limestone, Lauingen/Bayern) (data from [54], p. 67).

212

G. Unland 0.12 5

1: dF = 1 mm 2: dF = 5 mm 3: dF = 20 mm 4: dF = 50 mm 5: dF = 200 mm

4 0.1 3 εSE,O [m2/J]

0.08 2

0.06

0.04

1

0.02

0 10

20

30

40 50 v [m/s]

60

70

80

90

Fig. 104. Energy utilization eSE,O as function of impact velocity v and particle feed diameter dF (impact; halite, Bernburg/Sachsen-Anhalt) ([81], p. 61).

7.5 5

εSE [10-4 m2/J]

2.5

1 0.75 0.5

broken glass limestone diabase 62.5

75

87.5

100

125

dF [mm]

Fig. 105. Energy utilization eSE as function of particle feed diameter dF and impact velocity v (impact; broken glass; limestone, Bernburg/Sachsen-Anhalt; diabase, Hausdorf/Sachsen; v ¼ 30, 40, 50 m s–1) ([55], p. 88).

The Principles of Single-Particle Crushing

213

16 14

glass spheres cement clinker

12

1: dF = 9.0 mm 2: dF = 3.4 mm 3: dF = 6.9 mm 4: dF = 2.5 mm

εSE,O [10-3 m2/J]

1 10

3

8 2 6 4 4 2

10-2

2

10-1

5

2

5

100 2 EO,m [J/g]

5

101

2

102

5

Fig. 106. Energy utilization eSE,O as function of mass speciﬁc offered energy EO,m and particle feed diameter dF (percussion; glass spheres; cement clinker, Weisenau/Rheinland-Pfalz) ([45], 62f). 16 14

quartz limestone

εSE,O [10-3 m2/J]

12

1: dF = 9.8 mm 2: dF = 4.8 mm 3: dF = 11.3 mm 4: dF = 6.3 mm

10 8 3

6

4

1

4 2 2

10-2

2

5

10-1

2

5

100 2 EO,m [J/g]

5

101

2

5

102

Fig. 107. Energy utilization eSE,O as function of mass speciﬁc offered energy EO,m and particle feed diameter dF (percussion; limestone, Tel/Su¨dtirol; quartz, Frechen/NordrheinWestfalen) ([45], 62f).

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G. Unland

6.3.3.3. Conclusions The energy utilization can be used as an integral ﬁgure to assess a comminution process of a single particle. Since different kinds of energy are applied to formulate the energy utilization it is in many cases impossible to compare the crushing results. If the used comminution energy is applied, the comparison shows that the energy utilization is better with compression than with impact. Furthermore, there is a size effect with irregularly shaped particles. In case of impact and percussion there is an optimum energy utilization at a deﬁned offered intensity (EO,v).

7. APPLICATIONS The crushing parameters which are compiled in this chapter can be applied in the various phases of the design and operation of crushers. The parameters determine the mechanical sizing of the machine and its parts as well as being a basis with which to select and size the process and to optimize the operation. Furthermore, the crushing parameters can be used to characterize the particulate material. For this it is necessary to deﬁne characteristic crushing events. The features of these events determine the crushing properties of the particulate material.

7.1. Mechanical design The mechanical sizing of machines and structures is principally based on applied loads and deﬂections. Together with the support and fastening situation they determine the boundary conditions under which the machine and structure operates and reacts. There is a strong interrelation between the machine and the boundary, i.e. the boundary conditions inﬂuence the behaviour of the machine and vice versa. The loading of the machine and the structure can be classiﬁed by the following features: Magnitude of loading: The amount of loading describes the magnitude of the

forces and deﬂections. Since the crushing forces are not constant it is necessary to evaluate the spectrum of forces. Rate of loading: The rate of loading refers to the load–time curve. The inclination of the curve describes the rate. This timely load change can be positive if the loading rises or negative if the loading decreases. Time of loading: The time of loading is the time interval from zero load via distinct loads to zero load again. This interval can be subdivided by interval sections, where the rate of loading is positive, negative or zero. Frequency of loading: The process of crushing is associated with many rapidly changing loads. The various amounts of load occur differently often, i.e. there is a spectrum of frequencies for the different loads and load rates.

The Principles of Single-Particle Crushing

215

The knowledge of the crushing loads is the basis of the design and sizing of the machine and process. The crusher has to apply the necessary loads to the particle in order to produce fragments and on the other side all parts of the crusher have to sustain the loads due to the crushing process. The main problem is that it is not sufﬁcient to analyse the loads as they are given by the crushing parameters. The machine and structure reacts in conjunction with the particle, they behave as a system. Different designs of a machine respond differently even with the same particles and at the same operation conditions. One of the main inﬂuencing parameters is the sustainability to dynamic loads, characterized among others by the eigenfrequencies, i.e. the natural frequencies of the machine and structure. They react differently with various ratios between the periods of eigenfrequencies and the time intervals of loading. Consequently, the loading features do not only depend on the crushing parameters but also on the dynamic responses of the machine and structure. The following principal types of loading and design strategies can then be distinguished: Static loading: In this case the machine runs smoothly and all loads are bal-

anced. The accelerations of all mass elements are assumed to be zero. Additionally, a static loading case can be considered in many applications, if the time interval section where the rate of loading is not zero exceeds minimum three times the natural period. In the static loading case where the load rates are low the maximum load is of importance for sizing the machine and structure. Especially the peak forces need to be known in order to prevent the crusher from catastrophic failures. Therefore, it is advisable to use a conﬁdence factor l of about 3 in the design stage of the crusher (see Section 6.2.2.). Additionally certain overload elements have to be considered as protection devices. The materials for the crusher and structure are chosen according to their yield strength and ultimate strength, respectively. Cyclic loading: Smoothly running machines at constant speed and unbalanced loads need to be sized according to the cyclic loading case. The cyclic loading of the structural and machine parts can lead to the problem of fatigue. Therefore, the range (constant with cyclic loading) and the frequency of the loads as well as the material properties fatigue or endurance strengths are used for sizing. Impulsive loading: The kind of loading in addition to the magnitudes and frequencies of the loads are the basis for design of all machines and structures which are subjected to impulsive loading. Impulsive loading does not necessarily mean impact or percussion, but simply sudden changes of loads on the machine (see Section 4.1.3.1.). The deﬁnition of sudden load changes is that the time interval section with load changes is shorter than or in the order of the period of the eigenfrequencies (period of natural frequencies) – in general the lowest – of the machine or structure. If this is the case, impulsive loading

216

G. Unland

can be assumed. Impulsive loading deﬁnitely happens, if the time interval section with load change is shorter than or equal to the half of the natural period. The dynamic response of the machine and structure is determined in this case by the impulse, i.e. the area under the load–time curve. Z Dt I¼ Fdt 0

During the process of crushing various amounts of loads occur differently often. If the variations of the loads are small, i.e. fairly constant load ranges, fatigue is the most damaging effect. Then the endurance limit of the material has to be considered, i.e. the fatigue strength. In many applications of crushers the loads vary to a high extent, especially with several impulsive loading cases. In such a case the yield strength and the fatigue strength or better the damage lines and the endurance limits of the Wo¨hler diagrams, i.e. the S–N diagrams, of the materials have to be applied.

7.2. Process design The purpose of a crusher installation is the production of fragments. The sizing of the crushing process needs to consider the minimum level of loading intensity in order to generate fragments. The breakage point represents this level. Therefore, a crusher needs to apply or exceed the minimum level of loading in all intermediate stages of comminution inside the process zone in order to fulﬁl its duty. For sizing and optimizing a comminution process it is advantageous to simulate the operation of a crusher. The most widely applied models are based on a mechanistic approach ([82], pp. 3A–31, [6,83]). It depends on four functions: Probability of breakage, also called selection or breakage rate function. This

function yields the fractional rate of breakage at a given intensity. The distribution functions of the breakage probability for different materials, particle diameters and intensities can be applied (see Section 6.2.1.). Fragment size distribution, also termed breakage, distribution, or appearance function. The breakage function gives the size distribution of the fragments due to the comminution event of particles of a deﬁned diameter, material and at a deﬁned intensity of a certain loading case. Here, the fragment size distribution functions can be used (see Section 6.3.1.). Selection of particles, which are subjected to loading or transportation through or out of a crusher, also named classiﬁcation, discharge rate function, or sizedependent diffusion coefﬁcient. The decision whether a particle is held and subjected to loading or can move through or out of a crusher depends on its dimensions, the arrangement of the particles inside the process zone and the kinematic boundary conditions of the crusher.

The Principles of Single-Particle Crushing

217

Transportation of particles due to ﬁeld and contact forces, also called trans-

portation or movement function. The movement of the particles, i.e. the direction and velocity of the particles, depends on the ﬁeld forces and the contact forces between the particles and the boundaries of the process zone given by the working surfaces of the crusher. These functions are used in different types of models: Black-box models: These models are based on the integral behaviour of the

crusher. Crushing is considered a continuous process. Whenever a particle enters a crusher the equivalent fragments are discharged as product after a certain time. Additionally, different retention times of fragments can be considered by integral mixing and diffusion models. Stepped zones models: The process zone of a crusher is divided into stepped sub-zones. Within every sub-zone a generalized crushing event happens. These models describe a succession of breakage events, where the feed of the next sub-zone is the product of the proceeding one. Additional mixing and diffusion models enhance the adaptation of the models. Single-particle models: In those models every particle is traced through the crusher. It is subjected to certain loads and movements. In every position it is decided whether and in which direction a particle moves or whether and how a particle is crushed. The best basis for design and operation investigations is the singleparticle model, especially considering that ever-increasing computational power enables a very detailed assessment of the interaction between particles and crusher.

7.3. Properties of particulate materials Crushing properties can be determined at different comminution events and intensities. If the particulate material has to be characterized, it is necessary to ﬁnd and deﬁne a unique event. This is possible at the breakage point. The process features at the breakage point for a given load case characterize then the breakage properties of a particulate material. They can be considered as fundamental fracture properties as well [50]. At the breakage point it is possible to derive and to distinguish three sets of properties: Particle strength FA

max:

The strength is again described by three parameters (see Subsection 6.2.2.), a strength parameter FA max(50) at a particle diameter of 50 mm, a size parameter rF and an anisotropy parameter slog F A . Breakage energy EBP,m: The breakage energy is usually related to the mass of the particle. This mass speciﬁc breakage energy is also characterized by three

218

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parameters (see Section 6.2.4.), a work parameter EBP,m(50), a size parameter rE,BP and a deviation parameter slog E BP;m . Breakage fragment size distribution QBP(d): The breakage fragment size distributions QBP(d) vary with the particle diameter (see Section 6.3.1.). The size distribution itself can be characterized by an upper size dO, a transformed size parameter d50 and a transformed deviation sTQ. These sets of properties are typical for a certain particulate material; together they characterize the breakage behaviour. In addition to these elaborate properties it is possible to deﬁne features which are easier to determine. The particle strength for instance can be based on a certain value of the breakage probability, such as 50% or the resistance against crushing can be characterized by the fracture toughness, which is derived from the point load test (see Section 4.1.3.2.).

7.4. Energetic assessment of crushers The energy used by a crusher and during the comminution of a single particle can be taken as a basis to assess technically the efﬁciency of a crusher (see also [84]). The energy consumed by single-particle comminution represents the lowest possible work to produce certain fragment sizes. If the conditions are the same, i.e. for instance the same material, particle feed diameter, loading conditions, the comparison of the ﬁgures from the crusher and the single-particle comminution test yields the technical efﬁciency of the crusher operation.

ACKNOWLEDGEMENTS The author wishes to express his gratitude to Mrs. Richter and Mr. Peukert for the preparation of the manuscript and the diagrams. Nomenclature

A APr AV b d D dc

area (mm2) projection area (mm2) cross-section of volume equivalent sphere (mm2) factor (characterizes jointing of material) (–) diameter (mm) distance between load application points (mm) characteristic diameter (mm)

The Principles of Single-Particle Crushing

dF dO dP d P dPr dU dV d0,1, d0,2, d0,3, d0,4, dO,k d50 d50,1, d50,2, d50,3, d50,4, d50,k E E Ea,m EBP EBP,m EBP,m(50) EBP,v EC E Cðd c Þ EC,m EC(50) EF Ekin,P Em Em* EO EO,m EO,mT ET ET,m ET,m,20% EV EDS f F FA max F A maxðd c Þ F A maxð50Þ F APr

219

feed diameter (mm) upper diameter (mm) product diameter (mm) mean product diameter (mm) equivalent diameter related to projection area (mm) lower diameter (mm) equivalent diameter related to volume (mm) upper diameter of 1st, 2nd, 3rd, 4th, k-th assembly (mm) median size (mm) median size of 1st, 2nd, 3rd, 4th, k-th assembly (mm) energy (J) Young’s modulus (N mm2) mass speciﬁc absorbed energy (J kg1) breakage energy (J) mass speciﬁc breakage energy (J kg1) mass speciﬁc breakage energy at dc ¼ 50 mm (J kg1) volume speciﬁc breakage energy (J cm3) comminution energy (J) comminution energy at dc (J) mass speciﬁc comminution energy (J kg1) work parameter at dc ¼ 50 mm (J) fracture energy (J) kinetic energy of particle (J) mass speciﬁc energy (J kg1) energy related to counter body mass (J kg1) offered energy (J) mass speciﬁc offered energy (J kg1) transformed mass speciﬁc offered energy (J kg1) transferred energy (J) mass speciﬁc transferred energy (J kg1) mass speciﬁc transferred energy at hr ¼ 20% (J kg1) volume speciﬁc energy (J m3) surface speciﬁc energy (J m2) quantity of fragments (%) force (N) area speciﬁc maximum force (MPa) area speciﬁc maximum force at dc (MPa) strength parameter at dc ¼ 50 mm (MPa) force related to projection area (MPa)

220

G. Unland

F AV

force related to cross-section of volume equivalent sphere (MPa) force at breakage point (N) mass speciﬁc force (N g1) maximum force (N) force (point load) at failure (kN) energy release rate (J mm2) crack resistance energy (J mm2) crack resistance energy, mode I (J mm2) displacement (mm) relative displacement (–) impulse (kg m s1) parameter (–) elongation (–) ﬂatness (–) contour integral (J mm2) stress intensity factor ( MPa m1=2 ) fracture toughness ( MPa m1=2 ) fracture toughness, mode I ( MPa m1=2 ) length (mm) longest dimension (mm) medium dimension (mm) shortest dimension (mm) factor (characterizes disturbance of material) (–) counter body mass (kg) mass of water (kg) maximum mass of water (kg) particle mass (kg) secant modulus (MPa) tangent modulus (MPa) number (–) exponent of transformed RRSB distrubution (–) number (–) breakage probability (distribution function) (–) passing (%) probability density function (% mm1) probability function (%) percentage of misshapen particles (%) size density function (% mm1) size distribution function, cumulative mass percentage undersize curve (%) cumulative mass percent ﬁner at diameter dP (%)

FBP Fm Fmax FPL G Gc GIc h hr I i ie if J K Kc KIc l ll lm ls m m* mH2 O mH2 Omax mP Ms Mt n nRRSB T N P P p(d) P(d) pms q(d) Q(d) Q*(dP)

The Principles of Single-Particle Crushing

QBP(d) qk r R R b R c R ci R Rc rE rE,BP rE,20% rF i R s R t R ts R s sH 2 O DS sTF sTQ t Dt u v vP VP w W w1, w2, w3, w4, wk x xO xT xU y z a b g d e

221

fragment size distribution at breakage point (%) size density function of kth assembly (% mm1) radius (mm) retaining (%) strength (MPa) bending strength (MPa) uniaxial compressive strength (MPa) compressive strength of intact material (MPa) fracture resistance (J mm2) energy related size parameter (–) energy related size factor at breakage point (–) energy related size factor at hr ¼ 20% (–) force related size parameter (–) strength of intact material (MPa) shear strength (MPa) uniaxial tensile strength (MPa) torsional strength (MPa) deviation (–) saturation (%) newly produced surface (m2) transformed deviation (–) transformed deviation (size) (–) time (s) loading time (ms) parameter (–) velocity (m s1) particle velocity (m s1) particle volume (mm3) share (%) width of specimen (mm) share of 1st, 2nd, 3rd, 4th, kth assembly (%) parameter (–) upper parameter (–) transformed parameter (–) lower parameter (–) variable (–) variable (–) shape angle (1) angle (1) surface energy per unit area (J m2) transformed diameter (mm) strain (–)

222

0 _ eSE Zse lE lF m m m T n s s0 sln;1 ; sln;2 ; sln;3 ; sln;4 ; sln;k slog E slog E C slog F A max smax sn s1, s2, s3 s1N, s3N t to j

G. Unland

strain (–) strain rate (s1) energy utilization (m2 J1) efﬁciency of single-particle comminution (%) energy related conﬁdence factor (–) force related conﬁdence factor (–) coefﬁcient of friction (–) median value (–) transformed median value (–) Poisson’s ratio (–) stress (MPa) stress (MPa) deviation of 1st, 2nd, 3rd, 4th, kth assembly (–) energy related deviation (–) deviation parameter (–) force related deviation, anisotropy parameter (–) maximum stress (MPa) normal stress (MPa) stress (MPa) normalized stress (–) shear stress (MPa) cohesion (MPa) angle of friction (1)

REFERENCES [1] W. Horst, Purpose and terminology, in: N. Weise, (Ed.), SME Mineral Processing Handbook, Vol. 1, American Institute of Mining, Metallurgical and Petroleum Engineers, Inc., New York, 1985, pp. 3A–5f. [2] B. Wills, Mineral Processing Technology, Butterworth-Heinemann, Oxford, 1997, 110ff. [3] E. Kelly, D. Spottiswood, Introduction to Mineral Processing, Wiley, New York, 1982, p. 117. [4] R.T. Hukki, Proposal for a Solomonic settlement between the theories of von Rittinger, Kick and Bond, Trans. AIME 220 (1961) 403–408. [5] H. Schubert, Aufbereitung fester mineralischer Rohstoffe, Bd. 1, VEB Deutscher Verlag fu¨r Grundstofﬁndustrie, Leipzig, 1989. [6] K. Ho¨fﬂ, Zerkleinerungs- und Klassiermaschinen, Schlu¨tersche Verlagsanstalt und Druckerei GmbH & Co., Hannover, 1994. [7] A. Gaudin, Principles of Mineral Dressing, McGraw-Hill Book Company, Inc., New York, 1939. [8] M. Pahl, Einteilung der Zerkleinerungsmaschinen, in: M. Pahl, (Ed.), Zerkleinerungstechnik, Verlag TU¨V Rheinland GmbH, Ko¨ln, 1993, p. 7. [9] A. Taggert, Handbook of Mineral Processing, Wiley, New York, 1956, p. 4-01. [10] P. May, Einzelkorndruckzerkleinerung von spro¨den Stoffen, Freiberger Forschungsheft A 550, Bergakademie Freiberg, 1974, pp. 86–179.

The Principles of Single-Particle Crushing

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[11] C. Eringen, Continuum mechanics at the atomic scale, Crystal Lattice Defects 7 (1977) 109–130. [12] H. Neuber, Kerbspannungslehre, 2. Auﬂage, Springer-Verlag, Berlin, 1958. [13] A. Burr, K.-H. Habig, G. Harsch, K. Kloos, Werkstofftechnik, in: W. Beitz, K.-H. Ku¨ttner (Eds.), Dubbel-Taschenbuch fu¨r den Maschinenbau, Springer-Verlag, 18. Auﬂage. Berlin, Heidelberg, 1995, pp. E1–E120. [14] D. Krajcinovic, Damage mechanics, Mechanics of Materials 8 (1989) 117–197. [15] J. Lemaitre, J.-L. Chaboche, Mechanics of Solid Materials, University Press, Cambridge, 1994. [16] T. Anderson, Fracture Mechanics, 2nd edition, CRC Press, Boca Raton, 1995, 69f. [17] B. Lawn, Fracture of Brittle Solids, University Press, Cambridge, 1993. [18] C. Potyondy, P. Cundall, A bonded-particle model for rock, Int. J. Rock Mech. Mining Sci. 41 (2004) 1329–1364. [19] H. Konietzky, Numerische Simulation in der Geomechanik mittels expliziter Verfahren, Habilitationsschrift, Technische Universita¨t Bergakademie Freiberg, 2001, p. 9. [20] L. Jing, A review of techniques, advances and outstanding issues in numerical modelling for rock mechanics and rock engineering, Int. J. Rock Mech. Mining Sci. 40 (2003) 283–353. [21] K. Scho¨nert, Bruchvorga¨nge und Mikroprozesse des Zerkleinerns, in: H. Schubert, (Ed.), Handbuch der mechanischen Verfahrenstechnik, Wiley-VCH Verlag, Wein heim, 2003. [22] G. Unland, Brecher – Eine U¨bersicht, in Markt Focus, Bauverlag GmbH, Walluff, 2000. [23] J.A. Hudson, J.P. Harrison, Engineering Rock Mechanics, Pergamon, Oxford, 1997. [24] R.D. Lama, V.S. Vutukuri, Handbook on Mechanical Properties of Rocks Vol. II, Trans Tech Publications, Clausthal, 1978, p. 24. [25] W. Wawersik, C. Fairhurst, A study of brittle rock fracture in laboratory compression experiments, Int. J. Rock Mech. Mining Sci. 7 (1970) 561–575. [26] H. Czichos, Werkstoffe, in: H. Czichos, (Ed.), HU¨TTE, Springer-Verlag, Berlin, Heidelberg, 1996, pp. D1–D90. [27] J.P. Harrison, J.A. Hudson, Engineering Rock Mechanics, Part 2, Pergamon, Oxford, 2000, p. 78. [28] K. Kegel, Bergma¨nnische Gebirgsmechanik, Verlag Wilhelm Knapp, Halle, 1950, p. 2. [29] H. Koppers AG, Handbuch der Brennstofftechnik, Verlag W. Girardet, Essen, 1928, p. 205. [30] A. Peschel, Natursteine, VEB Deutscher Verlag fu¨r Grundstofﬁndustrie, Leipzig, 1983, 101ff. [31] E. Hoek, Strength of jointed rock masses, Ge´otechnique 33 (3) (1983) 187–223. [32] R.H.G. Parry, Mohr Circles, Stress Paths and Geotechnics, E & FN Spon, London, 1995, 77f. [33] T. Joseph, K. Barron, The post-failure characteristics of rock, CIM Bull. 96 (1070) (2003) 66–74. [34] D. Tromans, J. Meech, Fracture toughness and surface energies of covalent minerals: theoretical estimates, Miner. Eng. 17 (2004) 1–15. [35] R.A. Bearman, The application of rock mechanics parameters to the prediction of crusher performance, Doctoral dissertation, Camborne School of Mines, Redruth, 1991, p. 115. [36] H. Herrmann, Introduction to basic notions and facts, in: H. Herrmann, S. Roux (Eds.), Statistical Models for the Fracture of Disordered Media, North-Holland, Amsterdam, 1990, pp. 1–31.

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[37] S. Shah, S. Swartz, C. Ouyang, Fracture Mechanics of Concrete, Wiley, New York, 1995, 227f. [38] R.A. Bearman, C.A. Briggs, T. Kojovic, The application of rock mechanics parameters to the prediction of comminution behaviour, Miner. Eng. 3 (1997) 255–264. [39] B. Singh, R. Goel, Rock Mass Classiﬁcation, Elsevier Science Ltd, Oxford, 1999. [40] D. Green, An introduction to the mechanical properties of ceramics, University Press, Cambridge, 1998, 164f. [41] D. Hull, Fractography, University Press, Cambridge, 1999, p. 18. [42] R. Weichert, Anwendung von Fehlstellenstatistik und Bruchmechanik zur Beschreibung von Zerkleinerungsvorga¨ngen, Zement-Kalk-Gips. Nr. 1 (1992) 1–8. [43] R.A. Bearman, R.W. Barley, A. Hitchcock, Prediction of power consumption and product size in cone crushing, Miner. Eng. 12 (1991) 1243–1256. [44] P. Szczelina, Auslegung von Backenbrechern durch Modellierung des Ko¨rnerverhaltens, Doctoral dissertation, Technische Universita¨t Bergakademie Freiberg, 2000. [45] P. Hildinger, Einzelkornzerkleinerung durch Fallko¨rper, Doctoral dissertation, Universita¨t Karlsruhe (TH), 1968. [46] R. Weichert, J.A. Herbst, An ultra fast load cell device for measuring particle breakage. Proceedings 1. World Congress on Particle Technology, Part 2 (pp. 3–14), Nu¨rnberg, 1986. [47] R.G. Fandrich, J.M.F. Clout, F.S. Bourgeois, The CSIRO Hopkinson bar facility for large diameter particle breakage, Miner. Eng. 9 (1998) 861–869. [48] J. Tomas, M. Schreier, T. Gro¨ger, S. Ehlers, Impact crushing of concrete for liberation and recycling 105 (1999) 39–51. [49] G. Unland, T. Wegner, V. Raaz, J. Holla¨nder, Ein Hochauﬂo¨sender StoXanalysator (HASA) zur Untersuchung der Prallzerkleinerung, Aufbereitungstechnik 10 (2002) 31–37. [50] F. Bourgeois, R.P. King, J.A. Herbst, Low-impact-energy single-particle fracture, in: S. Komar Kawatra, (Ed.), Comminution Theory and Practice, Society for Mining, Metallurgy and Exploration, Inc., Littleton, CO, 1992. [51] G. Unland, P. Szczelina, Coarse crushing of brittle rocks by compression, Int. J. Miner. Process. 74 (2004) 209–217. [52] S. Baumgardt, Beitrag zur Einzelkornschlagzerkleinerung spro¨der Stoffe, Freiberger Forschungsheft A 560, Bergakademie Freiberg, 1976, pp. 29–106. [53] M. Wedekind, Einzelkorndruckzerkleinerung unregelma¨Xig geformter Teilchen, Doctoral dissertation, Bergakademie Freiberg, 1985. [54] J. Priemer, Untersuchungen zur Prallzerkleinerung von Einzelteilchen, VDI-Zeitschift Fortschritt-Berichte Reihe 3, Nr. 8, VDI-Verlag, Du¨sseldorf, 1965. [55] T. Wegner, Prallzerkleinerung grober Einzelpartikel als Auslegungsgrundlage fu¨r Rotorprallbrecher, Doctoral dissertation, Technische Universita¨t Bergakademie Freiberg, 2005. [56] C. Dan, Zur Prallzerkleinerung unregelma¨Xig geformter Teilchen, Doctoral dissertation, Bergakademie Freiberg, 1989. [57] V. Raaz, Charakterisierung der Gesteinsfestigkeit mit Hilfe eines modiﬁzierten Punktlastversuches, Zeitschrift fu¨r geologische Wissenschaften 30 (3) (2002) 213–226. [58] Z. Bieniawski, In situ strength and deformation characteristics of coal, Eng. Geol. 2 (5) (1968) 325–340. [59] H. Pratt, A. Black, W. Brown, W. Brace, The effect of specimen size on the mechanical properties of unjointed diorite, Int. J. Rock Mech. Mining Sci. 9 (1972) 513–529. [60] Z. Bieniawski, The effect of specimen size on compressive strength of coal, Int. J. Rock Mech. Mining Sci. 5 (1968) 325–335. [61] K. Barron, Y. Tao, Inﬂuence of specimen size and shape on the strength of coal, Proceedings Workshop on Coal Pillar Mechanics and Design, U.S. Bureau of Mines, 1992, pp. 5–24.

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[62] K. Mogi, Some precise measurements of fracture strength of rocks under uniform compressive stress, Rock Mech. Eng. Geol. 4 (1966) 41–55. [63] R. Sobol, Personal communication, 2006. [64] R. Kobayashi, On mechanical behaviours of rocks under various loading-rates, Rock Mech. Japan 1 (1970) 56–58. [65] R. Perkins, S. Green, M. Friedman, Uniaxial stress behaviour of porphyritic tonalite at strain rates to 103/second, Int. J. Rock Mech. Mining Sci. 7 (1970) 527–535. [66] P. Meikle, C. Holland, The effect of friction on the strength of model coal pillars, Trans. Soc. Mining Eng. 232 (2) (1965) 322–327. [67] M. Ruiz, Some technological characteristics of twenty-six Brazilian rock types, Proceedings 1st Congress of International Society of Rock Mechanics, Lisboa, 1966, Vol. 1, pp. 115–119. [68] P. Colback, B. Wiid, The inﬂuence of moisture content on the compressive strength of rocks, Proceedings 3rd Canadian Rock Mechanics Symposium, Toronto, 1965, pp. 65–83. [69] G. Brighenti, Inﬂuence of cryogenic temperatures on the mechanical characteristics of rocks, Proceedings 2nd Congress of International Society of Rock Mechanics, Beograd, 1970, Vol. 1, pp. 473–477. [70] D. Simpson, J. Fergus, The effect of water on the compressive strength of diabase, J. Geophys. Res. 73 (20) (1968) 6591–6594. [71] L. Tavares, R. King, Single-particle fracture under impact loading, Int. J. Miner. Process. 54 (1998) 1–28. [72] L. Tavares, R. King, Measurement of the load–deformation response from impactbreakage of particles, Int. J. Miner. Process. 74S (2004) S267–S277. [73] S. Baumgardt, B. Buss, P. May, H. Schubert, Zum Vergleich der Zerkleinerungsergebnisse bei der Einzelkornzerkleinerung mit verschiedenen Beanspruchungsarten, Teil 1, Aufbereitungstechnik, 8 (1975) 397–400. [74] F. Binder, Die einseitig und beiderseitig begrenzte lognormale Ha¨uﬁgkeitsverteilung, Radex-Rundschau, Heft, 3 (1963) 471–485. [75] S. Baumgardt, B. Buss, P. May, H. Schubert, Zum Vergleich der Zerkleinerungsergebnisse bei der Einzelkornzerkleinerung mit verschiedenen Beanspruchungsarten, Teil 2, Aufbereitungstechnik, 9 (1975) 467–476. [76] K. Klotz, H. Schubert, Crushing of single irregularly shaped particles by compression: size distribution of progeny particles, Powder Technol. 32 (1982) 129–137. [77] M. Zlatev, Beitrag zur quantitativen Kornformcharakterisierung unter besonderer Beru¨cksichtigung der digitalen Bildaufnahmetechnik, Doctoral dissertation, Technische Universita¨t Bergakademie Freiberg, 2005. [78] G. Unland, T. Wegner, Coarse crushing of rocks by impact, Proceedings XXII, International Mineral Processing Congress, Cape Town, 2003, Vol. 1, pp. 214–221. [79] H. Schubert, Zur Energieausnutzung bei Zerkleinerungsprozessen, Aufbereitungs technik, 34(10) (1993) 495–505. [80] H. Rumpf, U¨ber grundlegende physikalische Probleme bei der Zerkleinerung, Proceedings 1, Europa¨isches Symposium Zerkleinern, Frankfurt/Main, 1962, pp. 1–30. [81] B. Buss, Untersuchungen zur Einzelkornprallzerkleinerung spro¨der Stoffe, Doctoral dissertation, Bergakademie Freiberg, 1972, p. 61. [82] A. Lynch, M. Less, Simulation and modeling, in: N.L. Weiss, (Ed.), SME Mineral Processing Handbook, American Institute of Mining, Metallurgical and Petroleum Engineers, Inc., New York, 1985. [83] F. Silbermann, Beitrag zur Modellierung der Arbeitsweise von Kegelbrechern in Bezug auf den Materialstrom, Doctoral dissertation, Technische Universita¨t Bergakademie Freiberg, 2004. [84] C. Stairmand, The Energy efﬁciency of milling process, Proceedings 4. Europa¨isches Symposium Zerkleinern, Nu¨rnberg, 1976, pp. 1–18.

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Part II: Milling

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CHAPTER 5

Rotor Impact Mills Roland Nied Dr. Nied Consulting, Rai¡eisenstraX e 10, 86486 Bonstetten,Germany Contents 1. Introduction 2. Model of the milling process in rotor impact mills 2.1. Impact processes in the rotor 2.2. Impact processes in the milling gap 3. Control of ﬁneness 3.1. Impact type 3.2. Stress speed 3.3. Inﬂuence of impact frequency (dwell time) 3.3.1. Non-classifying processes 3.3.2. Classifying processes 4. Scale-up 5. Designs References

230 230 232 235 237 237 237 238 239 240 242 244 249

Abstract The stress in rotor impact mills is characterised by impact processes in the rotor and in the milling gap. A distinction can be made between particle impacts with the impact beaters, particle–particle impacts in the milling gap and particle–stator impacts. The main inﬂuences on the milling result are produced by the rotor circumferential speed, the particle acceleration in the rotor and the conditions in the milling gap. The ﬁneness of the milled material can be signiﬁcantly inﬂuenced amongst other things by the impact frequency. Integrated classiﬁcation in particular is a suitable means of selectively increasing the impact frequency for the large particles due to the longer dwell time. By means of the model presentation and empirical values, rules will be developed for the scaling-up of rotor impact mills. Finally, the most important designs will be described, such as universal mills, pin disc mills and classiﬁer mills.

Corresponding author. Tel.: +49-8293-6756; Fax: +49-8293-7136; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12008-X

r 2007 Elsevier B.V. All rights reserved.

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R. Nied

1. INTRODUCTION The term rotor impact mills refers to crushing machines, in which the crushing is carried out by impact stresses, and the kinetic energy (impact energy) required to produce the impact stress is applied by the rotary movement of rotors. Rotor impact mills cover a wide ﬁneness range, which extends from a ﬁnal particle size in the area of a few millimetres, down to less than about 10 mm. In the coarser range, rotor impact mills are frequently referred to as crushers (hammercrushers) and in the ﬁner range as blast rotor mills, wing beater mills, pin disc mills etc. The common feature of all rotor impact mills is the usually centrally arranged rotor, which is equipped with tools of different shapes. Static milling elements can be arranged concentrically around the rotor. Classiﬁcation of the milled material can be achieved by means of grills, screens or integrated classiﬁers. The main area of application of rotor impact mills is the crushing of brittle materials with a Mohs-hardness of up to 3. Cold milling processes are available for special purposes (milling of plastics, spices etc.). The air is usually warmed for drying during the milling process. Further special versions include the pressure shock resistant, gas-tight (for circulating gas operation) or sterilisable design. Rotor impact mills represent a universally used type of mill. The following systematic description of rotor impact mills cannot claim to be comprehensive, in view of the wide existing variety, so that only little space is devoted, for example, to pin disc mills. The article deals essentially with rotor impact mills using impact beaters as the rotating tools, for which the most comprehensive research results are also available.

2. MODEL OF THE MILLING PROCESS IN ROTOR IMPACT MILLS The most common form of rotor impact mills makes use of plate-shaped milling tools on the rotor, with a central milling material feed into the interior of the rotor (Fig. 1). The milling material, together with the air, which is carried along by the centrifugal force in the rotor, is fed through the inlet pipe (1) arranged centrally with the rotor (2) and is carried by the air into the area of the impact beaters (3). The crushing is carried out by impacts with the beaters and other particles in the milling gap (4) or with the stator (5). For the impact of one particle against another or against a surface, the following types of impacts can be distinguished ([1], Fig. 2): Direct impact, the angle between the direction of impact and the direction of

movement is zero.

Rotor Impact Mills

231 5

6

3

2 1

3 4 7

Fig. 1. Section through a rotor impact mill with plate-shaped milling elements: 1, milling material feed; 2, rotor; 3, plate-shaped milling element (impact beater); 4, milling gap; 5, stator (grinding track) with outlet gap 6; 7, alternative stator (screen) with outlet through the screen perforation.

Fig. 2. Impact types to [1]: (a): direct impact; (b) angular impact; (c) edge impact.

Angular impact, the angle between the direction of impact and the direction of

movement differs from zero; this type also includes direct eccentric impact. Edging impact, in which the centre of the particle lies outside the surface vol-

ume.

The highest level of energy transfer usable in the crushing process is achieved in the case of direct impact; in the case of angular or edge impact, rotation or slippage also occurs. In addition to the type of impact, the point of application of the stress also largely determines the milling result. In rotor impact mills, there are two main

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stress points: Impact processes on the broad side and the edges of the impact beaters (in the

rotor) and Impact processes in the milling gap and at the stator.

2.1. Impact processes in the rotor As already described, the milling material enters the interior of the rotor together with the milling gas transported by the mill. The radial acceleration of the particles, apart from wall inﬂuences, is provided by the drag force of the milling gas; the particle movement is essentially radial [2,3]. Provided that the particles in the interior of the rotor are evenly distributed, and that their radial speed corresponds to the radial speed of the milling gas, their penetration depth h between the impact beaters can be estimated (Fig. 3). A group of particles of approximately hmax si moves radially outward at the speed vP. At the same time, the impact beaters ‘‘1’’ and ‘‘2’’ move at the circumferential speed wi. The particles of the group which ﬁrst passes the side opposed to the movement direction of impact rail ‘‘1’’ achieves the greatest penetration depth hmax. The particle which is simultaneously at the position Ri – hmax, achieves the lowest penetration depth h ¼ 0. If the time window needed by the impact rail ‘‘2’’ for traversing the distance si and the radial speed of the group of particles are known, the penetration depth h can be calculated. We ﬁrst calculate the time required by the impact beater ‘‘2’’ to traverse the distance si: t ¼ si =w i

ð1Þ

1

2

Wa hmax

Vp si

R

i

R

a

Wi

Fig. 3. Model of the penetration depth h between the impact beaters of a rotor impact mill.

Rotor Impact Mills

233

100.00 Cumulative particle proportion [%]

90.00 80.00 70.00 60.00 50.00 40.00 30.00 20.00 10.00 0.00 0

0.5

1 Penetration depth [mm]

1.5

2

Fig. 4. Cumulative proportion of particles which reach a particular penetration depth.

During this time, the particle h ¼ hmax moves by the distance hmax ¼ tv p

ð2Þ

into the impact circle. All other particles of the group have a shorter time available for their radial penetration into the impact circle. If we calculate the penetration depth for different points in time t, we obtain a cumulative proportion of particles which achieve a particular penetration depth up to h ¼ hmax. Figure 4 graphically interprets the result of such an estimate for a rotor impact mill with a rotor external diameter of 300 mm (vP ¼ 4.3 m/s, wi ¼ 73 m/s, si ¼ 28.4 mm, free surface area between the impact rails 90.5%). A particle which penetrates into the impact circle immediately after passing the impact beater ‘‘1’’ therefore achieves a maximum penetration depth of 1.85 mm. The later the particle enters, the lower the penetration depth. To this is added a proportion of particles which do not penetrate into the impact circle: this corresponds to the total of the impact rail surface area facing the interior of the rotor, divided by the total surface area (in this example 9.5%). The typical particle diameter for the feed material in rotor impact mills lies in the range from 1 mm up to several millimetres. In order to estimate the type of impact to which the particles are subjected, we will assume a particle radius of 1 mm. A surface impact only occurs in the case of particles which achieve a penetration depth greater than 1 mm (in this example about 40%). If the penetration depth is less than 1 mm, the particles undergo an edge impact. Pieces and uncrushed particles are transported back into the interior of the rotor at a speed increased by a factor of about 10 ([2], Fig. 5). This increased speed now enables them to

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R. Nied

C2=24m/s

C1=1.9m/s

Fig. 5. Rotor impact mill with screen and milling jaws showing the particle movement in the inlet, together with the speed c1 prior to the 1st impact and c2 after the 1st impact (from [2]). Rotor circumferential speed 40 m/s.

penetrate between the impact beaters, where they are also subjected to a surface impact, before being accelerated along the broad side of the impact rail [2]. Using the following assumptions: friction-free particle movement acceleration path ¼ width of impact rails ¼ Ra Ri radial speed of particles prior to impact vP,i ¼ 0

we can derive the movement equation for the acceleration process as follows: br ¼ dv P =dt ¼ dv P =dR dR=dt ¼ dv P =dR v P

ð3Þ

br ¼ o2 R

ð4Þ

) o2 R dR ¼ v P dv P

ð5Þ

By integration within the limits Ri and Ra and with o ¼ w R;a =Ra we obtain v P;a ¼ w R;a ð1 R2i =R2a Þ0:5

ð6Þ

Under the further assumption that the particle circumferential speed wP,a equals the impact beaters circumferential speed wR,a, the particle ejection speed cP can be expressed as cP ¼ w R;a ð2 R2i =R2a Þ0:5

ð7Þ

Rotor Impact Mills

235

It can easily be seen that for Ri ) 0 (i.e. impact beater width B ¼ Ra) and friction-free consideration the particle p ejection speed cp lies above the rotor cirﬃﬃﬃ cumferential speed wR,a by the factor 2 [3]. The particle ejection angle can ﬁnally be calculated as b ¼ arctanðv P;a =w P;a Þ

ð8Þ

2.2. Impact processes in the milling gap Following the acceleration process, the particles enter the ring-shaped area between the impact beaters and the stator. In this milling gap, the milling material rotates in a cloud. The average free path length l and the particle braking path s0 in the milling gap can be estimated ([4], Fig. 6). Depending on the volume concentration 1 – e and the particle speed cP, which is correlated with the rotor circumferential speed wR,a in accordance with the equation (7), the following areas can be distinguished for a typical milling gap s of from 2 to 8 mm: For particles 41 mm is l4s, s0cs. Mutual particle impacts in the milling gap

are unlikely. The impact on the stator takes place at almost unchanged speed.

1-

ε= 0

.0

01

102

s0

100

s m/ 10

.1

=0

ε 1-

C=

10

0m

/s

101

C=

Effective braking path of particles so / mm Average free path length of particles λ / mm

103

λ

10-1

10-2 -1 10

100

101 102 Partical size x /μm

103

104

Fig. 6. Free path length and braking path so of spherical particles (r ¼ 103 kg/m3) in stationary air from [4]. Starting speed of particles 10 – 100 m/s; Volume concentration of the milling material in the milling gap 1e ¼ 0.1 – 0.001.

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R. Nied

In the size range from 100 mm to 1 mm is lES, s0Zs. Both particle–particle

impacts and impacts on the stator occur. The inﬂuence of the particle braking as a result of the ﬂight path is negligible. For particle sizes between 10 mm and 100 mm is lEs, s0 s. The particle–particle impacts in the milling gap predominate, the braking path s0 can fall to the order of size of the milling gap s. In this range, the parameters volume concentration 1–e , size of the milling gap s and the particle speed cP are very important. For particles below 5 mm to 10 mm the stress limits are reached in rotor impact mills. The particle braking path s0 lies in the range of the milling gap s, and even particle–particle impacts often take place at insufﬁcient speed. Finally, for even smaller particles, the braking path s0 reaches the order of size of the average free path length l; impact events are no longer probable. In addition to particle–particle impacts and the impact with the stator, impacts with the outer edges of the impact beaters can also be observed. Particles from the rotating material ring re-enter the area of the impact beaters due to momentum transfer with other particles or with the stator, although in this case, the intensity of the stress is usually lower, since only speed difference between the surrounding ring and the rotor circumferential speed is effective. From [2], the ratio between the particle speed in the surrounding cloud of material in the area of the impact circle and the circumferential speed of the rotor is approximately 0.44 (stator: screen without ﬁttings) to approximately 0.1 (stator: screen with milling jaws; rotor circumferential speed: approximately 43 m/s). Depending on the ﬁneness range and the selected operating and geometrical parameters, the main impact events taking place are particle–particle or particle–plate impacts. For both cases, the maximum stress force for the central, elastic impact can be estimated according to the Hertz–Huber theory [2,5]: m1 m2 3=5 r 1 r 2 1=5 1 v 1 1 v 2 2=5 6=5 F max / þ crel ð9Þ m1 þ m2 r1 þ r2 E1 E2 where m is the mass; r the radius; E the elasticity module; and v the contraction number. The following applies for the impact of a spherical particle with a plate: m1 ; r 1 ; E 1 m2 ; r 2 ; E 2 F max /

3=5 m1

1=5 r1

1 v1 E1

2=5

6=5

crel

For the impact of two spherical particles of the same size, this gives: m1 ; r 1 ; E 1 ¼ m2 ; r 2 ; E 2

ð10Þ

Rotor Impact Mills

237

F max /

3=5 m1

1=5 r1

1 v1 E1

2=5

crel 6=5 2

ð11Þ

If one compares equations (10) and (11), it can be seen that for the same stress force, the required relative speed in the case of a particle–particle impact must be twice as high as that of a particle–plate impact.

3. CONTROL OF FINENESS There are three possible procedures (or combinations of these) for the control of ﬁneness in rotor impact mills: Inﬂuence of the impact type, Selection of the stress speed, Inﬂuence of the impact frequency (dwell time).

3.1. Impact type The inﬂuence of impact type can largely be concluded from the preceding chapter, according to which particle–plate impacts are preferable to particle–particle impacts. Over and above all other considerations therefore, it will have a favourable effect on the achievable ﬁneness, if the volume concentration of the milling material and the milling gap are low.

3.2. Stress speed In order to initiate rupture, a particle must be supplied with a certain minimum stress energy [2,6]. For the individual impact, the following areas can be distinguished ([1], Fig. 7) for the stress speed, and thus the stress energy: a. If the stress energy lies below a critical limit, no rupture or crushing will take place, and the energy utilisation is therefore 0. b. With increasing stress energy, the energy utilisation rises to a maximum value, which corresponds largely to a rupture probability of 100%. c. After exceeding the optimum stress energy, the energy utilisation declines again. It will have a favourable inﬂuence, both with regard to the energy requirement and the ﬁneness achieved, if the stress speed, and thus also the stress energy are selected so that the energy utilisation is in the area of the optimum.

238

R. Nied 1000 d/mm Energy utilisation, cm2/J

0.1 1.0

Limestone pressure impact

5.0 100

0.1 1.0 5.0 d/mm

10 0.4

1 10 Specific stress energy, J/g

40

Fig. 7. Energy utilisation of individual particles (limestone) under compression and impact stress [1].

3.3. Inﬂuence of impact frequency (dwell time) If a ﬁner milling result is required, increasing the impact frequency [3,6] can also be considered, in addition to increasing the stress speed. This method can however only be used if the level of the stress speed permits initiation of rupture, i.e. for cases (b) and (c). For the central, direct, elastic impact, the kinetic energy, in the case of multiple impacts multiplied by the impact frequency, can be expressed as E kin ¼

m 2 c 2 rel

ð12Þ

It can easily be shown that the balanced energy of one individual impact corresponds to that of several impacts at reduced relative speed crel: E kin;1 ¼

m 2 c 2 rel;1

n E kin;2 ¼ n

m 2 c 2 rel;2

ð13Þ ð14Þ

E kin;1 ¼ n E kin;2

ð15Þ

crel;1 2 crel;2

ð16Þ

)n¼ (n ¼ number of individual impacts).

Rotor Impact Mills

239

100

PB /%

75

50 1.ZK 2.ZK 3.ZK Modell Gl.4-44

25

0 0.0

0.5

1.0 1.5 K-(Wm,kim-Wm,min)/ kJ/kg

2.0

Fig. 8. Probability of rupture of potassium–alum (2 to 2.5 mm, one to three stresses) in relation to the speciﬁc impact energy [6].

Figure 8 shows the likelihood of rupture of potassium-alum crystals in relation to the speciﬁc impact energy multiplied by the impact frequency [6]. As can be seen, the measured values for one to three stresses all fall along a common curve, representing an experimental conﬁrmation of the relationship derived above. In order to inﬂuence the impact frequency, the dwell time of the milling material in the mill is usually changed, using the following methods:

3.3.1. Non-classifying processes In rotor impact mills with a proﬁled grinding track (stator) and an outlet gap (see Fig. 1), the proﬁling of the grinding track and/or the impact beaters can for example be inclined, in order to produce a conveyor effect away from the outlet gap (Fig. 9). A further possibility is the use of so-called closed rotors. If one assumes that the milling material is evenly distributed over the length l in open blast rotors, the particles that pass the rotor in the vicinity of the outlet gap have only a short dwell time in the milling gap. On the other hand, the particles that pass the rotor at the same height as the inlet are stressed along the whole milling gap length l (Fig. 10). This results in an average effective milling gap length of l/2 (and a wide dwell time range). When using a closed rotor, all particles must pass the milling gap along its complete length.

240

R. Nied

Outlet gap

Stator Rotation direction

Fig. 9. Schematic representation of a proﬁled grinding track without classifying effect (plan view). 4 3

2

1

Fig. 10. Section through a rotor impact mill with closed rotor: 1, milling material inlet; 2, closed rotor; 3, grinding track; 4, outlet gap.

The disadvantage with all these procedures is its non-speciﬁc effect; even particles which have already attained the required ﬁnal particle size will be subjected to further impacts, and therefore use up energy unnecessarily. Finally, there is no complete probability of rupture even for coarser particles; instead, quite a wide particle size distribution is generally observed.

3.3.2. Classifying processes Classifying processes are generally characterised by the fact that only the coarse proportions are subjected to further stress. The ﬁne proportions can leave the mill by suitable means and no longer affect the energy balance. The particle distribution curves of the milled products are therefore generally narrower than in the case of non-classifying processes (no large particles, lower ﬁne proportion). One

Rotor Impact Mills

241

possibility of classifying the output consists in the equipment of rotor impact mills with grills or screens. Depending on the angle of incidence of the particles on the screen, the available perforated surface, the perforation dimensions and the thickness (plate thickness) of the screen, the likelihood of penetration can be expressed as a function of the particle size (Fig. 11). Depending on the rotor speed (see Chapter 2, Ejection angle of the accelerated particles from the impact beaters), the volume concentration of the milling material in the milling gap (momentum transfer with other particles) and the air volume ﬂow (air speed at the screen), the angle of incidence will assume different values. In general however, it can be assumed that the resulting particle size will be signiﬁcantly smaller than the screen perforation used. The disadvantage of grills or screens as a stator is that proﬁling of the stator, such as would favour the maximum possible efﬁciency of the milling, is only possible to a limited extent (see ‘‘Milling jaws’’ in Fig. 5). They are also limited in the end ﬁneness that can be achieved: ﬁne screens (approximately o500 mm) wear quickly and tend to cause blockage of the screen perforation. Finer milling results (down to approximately o100 mm) can however be achieved by special screen designs (e.g. corrugated trapezoid sieves, whose perforations are inclined toward the sieve circumference, and which thus result in a further reduction of the angle of incidence).

Normalized penetration probability Wo/cf

1

Angle of incidence 90°

Free screen surface proportion cf d

5

γ

γ =30°

AP γ =15°

0 0

0.5 Normalized particle size xp/d

1

Fig. 11. Penetration probability WD over the normalised particle size for different angles of incidence from [2]. Thickness of screen not taken into account.

242

R. Nied

These disadvantages are avoided by rotor impact mills when combined with a classiﬁer1 to form a so-called classiﬁer mill (see also Fig. 16). The milling material in the mill is transported with the milling air in an inner circuit to the classiﬁer, where the coarse particles are separated from the ﬁne material produced. The ﬁne material leaves the classiﬁer mill together with the milling air. The coarse material, together with the feed material, undergoes further stress. The separation between milling and classifying enables the rotor and stator to be optimised solely for the task of milling. The ﬁneness of separator mills is limited in the coarse range by the classiﬁer (to approximately 200 mm), and in the ﬁne range by the limits of stress possible in rotor impact mills (to approximately. 10 mm).

4. SCALE-UP It will only rarely be possible to perform a scale-up without being forced to compromise with regard to individual criteria. In the scaling-up of rotor impact mills, two parameters in particular present difﬁculties: 1. The working length of the rotor l If a screen is used as the stator, care must be taken to ensure that the product is distributed evenly along the working length l. If the material load becomes too great, this will result in zones of higher stress, particularly on the rear side of the rotor, while the stress will be relatively low in the vicinity of the product inlet. When using a grinding track with outlet gap as the stator, the dwell time of the milling material in the milling gap must also be taken into account, which under otherwise similar conditions will depend on the working length l. Special importance must be attached to this when using closed rotors. 2. The size of the milling gap s The milling gap s should basically be regarded as a constant in the scaling-up process. Under otherwise similar conditions, the particle-braking path is also constant. If allowed by the production tolerances, this requirement can be fulﬁlled by using a screen as the stator (in which the through-ﬂow takes place radially). When using a grinding track as the stator, the through-ﬂow and the product transport take place axially. For the same axial speed in the milling gap, the ring surface [(Ra+s)2R2a] must be scaled-up accordingly. The milling gap will then also increase in size in the case of larger mills. 1 For further information on classiﬁcation see: R. Nied, Fine classiﬁcation with vaned rotors, Int. J. Mineral. Process 74S (2004) 137–145.

Rotor Impact Mills

243

This needs only to be noted however for very ﬁne milling (xo20–50 mm). For xZ50 mm the braking path so is already about 10 times that of the milling gap s to be expected for larger mills. The scaling-up model described below has been derived partly empirically, and partly from the models and observations described previously in Section 2. This model assumes the following requirements:

equal equal equal equal

product ﬁneness, energy utilisation,2 ratio of milling material mass ﬂow to milling air volume ﬂow, circumferential speed at the impact beaters outer edge.

From the point of view of the mill designer, the scale-up factor f is determined as the ratio of the mechanical drive of the mills: f ¼

P M;2 P M;1

ð17Þ

These requirements give :

m2 : ¼ f m1

ð18Þ

:

V2 :

¼f

ð19Þ

V1 The further scaling-up is performed by means of the following steps: a. From experience, one approach has proven itself, which links the radius ratio at the impact beater outer edge with the scale-up factor: Ra;2 ¼ f 0;6 Ra;1

ð20Þ

The exponent of this equation was determined empirically. Manufacturer’s data published in the literature [7] demonstrate good conformity with this approach. Starting from the basic mill, the impact circle radius Ra,2 can therefore be deﬁned. b. The impact beaters width Ra Ri is determined from considerations of the ejection speed and ejection angle (see equation (7) and (8)), according to 2 Wolf and Pahl [7] found in their investigations of ‘‘turbo-mills’’ with sieve inserts that the energy utilisation was even initially improved following scaling-up, based on a laboratory mill. Constant energy utilisation was then found in the case of larger mills.

244

R. Nied

which for wR ¼ const. the ratio Ri/Ra must be kept constant: Ri;1 Ri;2 ¼ Ra;1 Ra;2

ð21Þ

) Ra;2 Ri;2 ¼ Ra;2

1

Ri;1 Ra;1

ð22Þ

c. The impact beaters number n results from the requirement for the same impact beaters distance a at radius Ri (so that the impact and penetration conditions of the particles remain approximately equivalent): a¼

2 Ri p ¼ const. n

ð23Þ

2 Ri p a

ð24Þ

or n¼

(a ¼ impact beaters distance, n ¼ impact beaters number). d. In order to achieve similar impact conditions, the total of the impact beaters _ should be constant (aslengths n l, divided by the material mass ﬂow m, suming the similarity of the hold-up in the mill): :

:

m2 m1 ¼ n2 l 2 n1 l 1

) l2 ¼ f l1

n1 n2

ð25Þ

ð26Þ

e. The milling gap can be kept approximately constant for rotor impact mills using a screen as the stator (radial through-ﬂow of the milling gap). If a grinding track is used as the stator, the gap width must be determined in accordance with :

h

V ðRa þ sÞ2 R2a

i ¼ const.

ð27Þ

(equal axial ﬂow speed in the milling gap).

5. DESIGNS The classiﬁcation of rotor impact mills according to only one feature (e.g. central/ tangential milling material feed, with/without classiﬁcation) is almost impossible:

Rotor Impact Mills

245

the known types and designs are too varied. The distinction is therefore often made according to the achievable ﬁneness. Figure 12 shows a possible classiﬁcation. The basic features of hammer mills (Fig. 13) are the tangential milling material feed and the pendulum-type suspended hammers: These are generally equipped with the grill and/or screens, and can produce ﬁnenesses of approximately. o1 mm. Wing beater mills, blast rotor mills and pin disc mills with a rotating disc are today generally classiﬁed together as so-called universal mills. These allow the use of different milling tools (rotors and stators) in the same housing (Fig. 14). The milling material feed for universal mills generally takes place centrally. The rotors are equipped with wing beaters, rigid impact beaters with open (Fig. 14c), semi-closed (Fig. 14b) or closed rotors (see Fig. 10) or also with pins (Fig. 14a). Screens, short grinding track with screen (Fig. 14b), grinding track with outlet gap (Fig. 14c) or again pinned discs are used as the stator. The achievable ﬁneness ranges to a median value of approximately 10 mm.

superfine

fine

medium

coarse

Pin Mill with one rotating disc

Hammer Mill With screen

Pin Mill with two rotating discs

Beater Mill with screen

Spiral Jet Mill

without screen

without screen

Blast Rotor with screen

without screen

Classifier Mill

Counter Jet Mill with classifier

10-3

10-2

10-1 x50 [mm]

Fig. 12. Fineness range of different impact mills.

100

101

246

R. Nied

Fig. 13. Schematic representation of a hammer mill. Reproduction with approval of Netzsch–Condux Mahltechnik GmbH, Hanau, Germany.

Fig. 14. Schematic representation of a universal mill: (a) rotor: pin disc, stator: pin disc; (b) rotor: wing beaters, stator: short grinding track with screen; (c) rotor: blast rotor, stator: long grinding track with outlet gap. Reproduction with approval of Netzsch–Condux Mahltechnik GmbH, Hanau, Germany.

Pinned disc mills with counter-rotating discs (Fig. 15) achieve the maximum ﬁneness (up to x50 5 mm) amongst rotor impact mills without integrated classiﬁers. Thanks to the different rotation directions of the rotors, relative speeds of up to 250 m/s can be achieved at the outer pin rows. Rotor impact mills with integrated classiﬁers (see also Section 3.3.2) are frequently referred to as classi¢er mills (Fig. 16). In the group of rotor impact mills, the maximum ﬁnenesses (x50 3 mm) can be achieved by such mills. Another design worthy of peripheral mention are rotor impact mills with socalled corrugated-milling discs (Fig. 17).

Rotor Impact Mills

247

Fig. 15. Section through a counter-rotating pinned disc mill. Reproduction with approval of Netzsch–Condux Mahltechnik GmbH, Hanau, Germany.

Fig. 16. Section through a classiﬁer mill: 1, 2, milling material feed; 2, 3, Guide vanes; 4, classiﬁer wheel; 5, rotor; 6, impact beaters 7; stator; 8, air inlet; 9, ﬁne material outlet. Reproduction with approval of Netzsch-Condux Mahltechnik GmbH, Hanau, Germany.

248

R. Nied

Fig. 17. Schematic representation of a rotor impact mill with ripple-milling discs. Reproduction with approval of Netzsch-Condux Mahltechnik GmbH, Hanau, Germany.

The milling gap can be set very narrow (so1 mm) with such mills. This favours the additional shear stressing of the milling material, making them ideally suitable for the milling of ﬁbrous, tough elastic materials.

Nomenclature

br B c crel EA Ekin f F h l m _ m P r R

radial acceleration (m/s2) impact rail width (m) speed (m/s) relative speed (m/s) energy utilisation (m2/kJ) kinetic energy, impact energy (kJ) scaling-up factor (–) stress force (N) penetration depth (m) length of milling gap (m) mass (kg) mass ﬂow (kg/s) mechanical drive performance (kW) contact radius (m) radius (m)

Rotor Impact Mills

so s t v V_ w x b g e l o

249

particle braking path (mm) path, milling gap width (mm) time (s) radial speed (m/s) volume ﬂow (m3/s) circumferential speed (m/s) particle size (mm) particle ejection angle (1) penetration angle (1) porosity (–) free path length (mm) angular speed (1/s)

Frequently used indices: a external i internal P particle R rotor

REFERENCES [1] K. Scho¨nert, Prallmu¨hlen. Handbuch der Mechanischen Verfahrenstechnik, H. Schubert (Ed.), Wiley-VCH, 2003, pp. 207ff, 355ff. [2] D. Landwehr, Kaltzerkleinerung in Turbomu¨hlen am Beispiel von Gewu¨rzen, Fortschr.Ber. VDI Reihe 3, Nr.141, VDI-Verlag, Du¨sseldorf, 1987. [3] K. Leschonski, R. Dro¨gemeier, Ultra ﬁne grinding in an impact grinding machine and its limits of application, Australas. Inst. Mining Metall., Publication Series No 3/93, 1 (1993) 227–236. [4] H. Rumpf, Prinzipien der Prallzerkleinerung und ihre Anwendung bei der Strahlmahlung, CIT 32 (3) (1960) 129–135. [5] I. Scabo, Ho¨here Technische Mechanik, Springer Verlag, Berlin, 1977. [6] L. Vogel, Zur Bruchwahrscheinlichkeit prallbeanspruchter Partikeln, Diss, TU Mu¨nchen, 2003, pp. 55ff, 146ff. [7] Th. Wolf, M.H. Pahl, Scale-Up-Kriterien fu¨r die Prallzerkleinerung, Aufbereitungstechnik 33 (10) (1992) 552–561.

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CHAPTER 6

Wet Grinding in Stirred Media Mills Arno Kwade and Jo¨rg Schwedes Institute for ParticleTechnology,Technical University of Braunschweig,Volkmaroder Strasse 5, 38104 Braunschweig,Germany Contents 1. Introduction: design and principal operation of stirred media mills 1.1. Development 1.2. Development of stirred media mills 1.3. Principal arrangement 1.4. Movement of the grinding media 2. Fundamental considerations 2.1. Introduction 2.1.1. Description of production rate 2.1.2. Determination of product quality as function of the grinding time and speciﬁc energy 2.2. Stress models 2.2.1. Mill related stress model 2.2.2. Product related stress model 2.2.3. Relation between the model parameters and speciﬁc energy, power input and production capacity 2.3. Application of the stress models on stirred media mills 2.3.1. Estimation of number of stress events and stress frequency 2.3.2. Estimation of stress energy and stress intensity 2.3.3. Speciﬁc energy and energy transfer factor 3. Inﬂuence of important operating parameters on the grinding and dispersing result 3.1. Tip speed of the stirrer as well as size and density of the grinding media 3.1.1. Grinding of weak to medium-hard crystalline materials 3.1.2. Grinding of crystalline materials with high modulus of elasticity 3.1.3. Deagglomeration and cell disintegration 3.1.4. Conclusions from the inﬂuence of stress number and stress energy 3.1.5. Determination of optimum operating parameters 3.2. Filling ratio of grinding media 3.3. Solids concentration and ﬂow behaviour of the suspension 3.4. Construction and size of the stirred media mill 3.5. Formation of nano-particles by wet grinding in stirred media mill 3.5.1. Conditions of producing nano-particles with stirred media mills 3.5.2. Grinding of alumina down to sizes in the nanometre range

253 253 254 255 258 262 262 263 265 269 269 270 274 274 276 278 281 283 284 284 294 296 301 303 304 307 311 314 314 316

Corresponding author. Tel.: +49 531/3919610; Fax: +49 531/3919633; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12009-1

r 2007 Elsevier B.V. All rights reserved.

252

A. Kwade and J. Schwedes

4. Transport behaviour and operation mode 4.1. Basic considerations 4.2. Modelling the axial transport in stirred media mills 4.3. Effect of the operation mode on the residence time distribution 4.4. Effect of residence time distribution on the particle size distribution 5. Operation of stirred media mills 5.1. Power draw 5.1.1. Power-number diagram without grinding media 5.1.2. Power-number diagram with grinding media 5.1.3. Inﬂuence of important operating parameters 5.1.4. Inﬂuence of mill geometry 5.1.5. Summary on power draw 5.2. Pressure and packing of grinding media 5.2.1. Experimental results on media packing 5.2.2. Grinding media distribution model 5.3. Wear 5.3.1. Wear of mills 5.3.2. Wear of grinding media 5.4. Autogenous grinding 6. Scale-up 6.1. Practical methods – consideration of cooling area of grinding chamber 6.1.1. Stirred media mills with disc stirrer 6.1.2. Stirred media mills with an annular gap 6.2. Exact method based on stress model 6.2.1. Grinding behaviour of different mill sizes 6.2.2. Calculation of stress energy distribution and mean stress energy 6.2.3. Calculation of energy transfer factor 6.2.4. Scale-up with Newton–Reynolds diagrams References

321 322 323 327 332 335 335 336 337 339 341 341 342 342 344 347 347 348 358 362 363 363 364 366 366 370 372 375 380

Grinding and dispersing in stirred media mills is a process, which is widely used in different industries such as chemical, ceramic, pharmaceutical, ﬁller, ink and cosmetic industry. While usually stirred media mills are employed to produce products in the micro and submicron particle size range, today more and more applications to produce stable suspensions with nano-particles are developed. Details on nanogrinding in stirred media mills are given in Section 3. This section discusses the design, physical fundamentals, grinding behaviour, operating behaviour and scale-up of stirred media mills. Since stirred media mills are operated usually wet, this section focuses on wet grinding. It will show possibilities to develop new and optimise existing grinding processes with stirred media mills. The section is a compendium of the short course on grinding and dispersing in stirred media mills which is held every year by the Institute for Particle Technology, the former Institute of Mechanical Process Engineering, at the Technical University of Braunschweig [1].

Wet Grinding in Stirred Media Mills

253

1. INTRODUCTION: DESIGN AND PRINCIPAL OPERATION OF STIRRED MEDIA MILLS 1.1. Development Stirred media mills belong to the group of mills with free movable grinding media [2,3]. Classical representative is the tumbling mill, consisting of a horizontally oriented rotating cylinder being ﬁlled with up to 40% by volume with grinding media. Owing to the rotation the grinding media are lifted and get potential energy which will be transferred into kinetic energy in a cascading and/or cataracting manner. The feed material to be ground is dispersed within the grinding media and is stressed by pressure and friction between layers of media or by impact of falling beads. The power input of a tumbling mill is limited by the speed of rotation. Typical speeds of rotation are in the range of 2/3 to 3/4 of the critical speed of rotation, where the centrifugal acceleration equals the acceleration due to gravity. This limit of the power input leads to relatively small energy densities. An alternative approach to provide the grinding media with the necessary energy is the use of a mixer in the form of a stationary grinding chamber equipped with a grinding agitator. The chamber is ﬁlled with grinding media which are put into relative motion by the rotating agitator. An equivalent device was ﬁrst introduced by Klein and Szegvary in 1928 [4]. In Fig. 1 a latter prototype is shown, also proposed by Szegvary. An agitator equipped with bars rotates in a grinding chamber oriented vertically. The grinding chamber is ﬁlled with grinding media (balls of 6–8 mm size). The material to be ground is suspended in a liquid. With help of a pump the suspension is removed from the chamber at the bottom and is recirculated at the top. The circumferential speeds at the agitator tip are in the order of 4 m s–1 and smaller. Those mills are low speed mills and are called ‘‘attritors’’. Parallel to their development high speed mills were investigated, ﬁrst invented by the DuPont Company. Since a ﬁne grained sand was used as a grinding media today they are still called ‘‘sandmills’’. The agitator shaft was equipped with discs. Figure 2 shows a sandmill from 1950 [5] for continuous operation. The suspension (carrier ﬂuid plus material to be ground) enters the mill through the bottom and leaves the mill at the top. To keep the grinding media in the mill a cylindrical sieve is placed in this region. With increasing speed of rotation a vortex will be formed at the top. This can lead to an entrainment of gas into the suspension and thus will reduce the energy input. To avoid this effect closed stirred media mills were developed which exclude a gas contact within the mill. Thus, an operation without embedded gas and under pressure is possible. These mills have a lot of advantages. Therefore, today open mills are used very seldom.

254

A. Kwade and J. Schwedes

Fig. 1. Attritor.

1.2. Development of stirred media mills Wet grinding and therefore the use of stirred media mills have many advantages [6]: reduced agglomeration tendency compared to dry grinding; material losses are avoided; no dust explosions and oxidations; easier handling of toxic materials; no devices for air cleaning necessary; improvement of heat transfer and others. Stirred media mills are used for dispersion and deagglomeration processes as well as for true grinding of crystalline substances or the disintegration of microorganisms. The attritor was developed for true grinding processes, whereas the sandmill was ﬁrst used for deagglomeration of pigments. But due to the many advantages of closed high speed stirred media mills these mills have increased their importance tremendously in the last few decades. For many applications they could displace other grinding systems. Stirred media mills are used for many applications in different industries and can be found especially where a high product ﬁneness is demanded. The

Wet Grinding in Stirred Media Mills

255

Fig. 2. Sandmill DuPont.

following industries can be mentioned [7]: paint and lacquer, pigments, chemical and pharmaceutical industry, agrochemistry, food industry, ceramics, papers and plastics, bioengineering, the rubber industry and nanotechnology.

1.3. Principal arrangement Figure 3 shows a schematic drawing of a stirred media mill with horizontal axis. In continuous operation the suspension is pumped into the grinding chamber through the front face. The driven shaft with the agitator elements on it projects into the grinding cylinder and brings the contents into motion necessary for the grinding process. In the shown version the discs are perforated. Circumferential velocities of the disc-tips of up to 20 m s–1 are used leading to centrifugal accelerations of more than 50 times the acceleration due to gravity. Up to 85% of the grinding chamber volume is ﬁlled with a bulk of grinding media. At the end of the grinding chamber the suspension leaves the mill. The grinding

256

A. Kwade and J. Schwedes

Fig. 3. Grinding chamber of a stirred media mill [2].

media are kept inside the mill, either by a rotating separation gap, as shown in Fig. 3, or by a sieve or a special centrifugal separation. Many different constructions with respect to grinding chamber and stirrer geometries and devices for the separation of the beads exist. In general it can be said that the width of a separation gap and the mesh size of a sieve should be smaller than half of the diameter of the grinding media. As can be seen in Fig. 3 grinding chambers can be built double-walled for cooling purposes. Also the rotor can be cooled. The cooling is necessary if temperature-sensitive products, as for instance microorganisms, are stressed, since almost 100% of the energy input is dissipated into heat. Extremely high energy inputs are a characteristic advantage of stirred media mills. With respect to the geometries of grinding chamber and agitator three different types of stirred media mills can be distinguished [8]: disc agitator pin-counter pin agitator annular gap geometry

The three different types are shown in Fig. 4. The simplest agitator geometry is the disc agitator. The energy transfer from the agitator to the mixture of grinding media and the material to be ground mainly takes place due to adhesion forces. The discs are provided with circular or elliptical holes or slits and are sometimes placed eccentrically. Thus, displacement forces are produced which increase the energy input. The parts of the mill in contact with the suspension are made from metallic or ceramic materials, as well as out of rubber or polymers. The movement of the grinding media in the pin-counter pin geometry is mainly determined by displacement forces. Especially with counter pins a higher energy

Wet Grinding in Stirred Media Mills

257 A Rotating Gap

A-A

A (a) A

A-A

A (b)

A-A

Annular Gap

A

A (c)

Fig. 4. Different types of stirred media mills. (a) Disc – geometry; (b) pin-counterpin – geometry; (c) annular gap – geometry.

input can be realised at identical operating parameters compared to the disc agitator. Compared to the disc agitator it is more difﬁcult to manufacture the pin agitator from ceramic materials. The highest energy densities can be realised in an annular gap geometry. Often only smooth surfaces of rotor and stator are used. Therefore, an energy transfer takes place only due to adhesion forces. However, the rotor and/or stator can also be equipped with pins which produce displacement forces. Generally the width of the gaps equals four to ﬁve times the diameter of the grinding media. With smaller values the blocking tendency of the beads is increased. At higher values the energy transfer from the rotor to the beads becomes less effective. Many different special annular gap mills are offered, which can be characterised by very high energy densities, large surface areas for heat transfer or extremely narrow residence time distributions. The axis of the grinding chamber can be horizontal or vertical. As long as the grinding media are equally distributed in the grinding chamber its orientation is of

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no inﬂuence with respect to energy input and the product ﬁneness. Having very large units the horizontal orientation has an advantage, because less increased torque at the start due to the sedimentation of the grinding media is necessary. At high throughputs a compression of the grinding media ﬁlling at the exit is possible. In this respect a vertical orientation with an upward ﬂow of the suspension is advantageous. Stirred media mills can be operated continuously or batchwise. Mill sizes from 0.1 l to several cubic metres exist. Grinding media made from metals, glass and many ceramics with densities between 2500 and 7800 kg m–3 and sizes between 0.1 and 410 mm are used [9]. For special applications also polymer beads are used. Based on the three principal mill geometries today a lot of different mill constructions exist. During the last two decades in particular, different kinds of annular gap mills with or without additional pins were developed [1]. Usually the mills are designed in a way that the mills can be run at relatively high throughputs without grinding media packing and that an effective operation in circuit mode is possible. Moreover, some annular gap mills are very effective regarding cooling of the product.

1.4. Movement of the grinding media Bosse [5], who was responsible for the development of the sandmill at DuPont, was the ﬁrst publishing ideas of the movement or the grinding media within a stirred media mill with a vertically oriented grinding chamber (Fig. 5 from [5]): The grinding media are accelerated close to the stirrer discs towards the grinding chamber wall and due to continuity they have to move back to the agitator shaft near the symmetry plane of two discs. Thus, a cyclic movement of the grinding media is originated. An inﬂuence of the acceleration due to gravity can only be noticed close to the agitator shaft, where the centrifugal acceleration is relatively small. The result is an asymmetric ﬂow pattern of the beads with respect to the symmetry plane between two discs. The zone of highest energy density, called milling zone due to Bosse [5], is located close to the disc at their outer tip and has an assumed thickness of about one inch. Following Bosse’s arguments further ideas of the movement of the grinding media were published ([4,10,11] and others). Blecher [12–14] was the ﬁrst investigating theoretically the velocity and pressure ﬁeld, the distribution of the speciﬁc energy as well as the motion of single grinding media in the grinding chamber of a stirred media mill. With his results quantitative estimates are possible even though simpliﬁed assumptions were made. Blecher assumed Newtonian behaviour of the ﬂuid and started his numerical calculations without any grinding media. He solved the conservation laws regarding momentum and mass in Eulerian coordinates with help of a ﬁnite volume method with staggered grids. The parameters varied in his calculations are

Wet Grinding in Stirred Media Mills

259 TOP

MILLING ZONE

VERTICAL CIRCULATION PATTERN

CENTRIFUGAL FORCE

NOTE : MAJOR DIFFERENTIAL VELOCITY IN HORIZONTAL PLANE

GRAVITY

Fig. 5. Circulation of the grinding media due to Bosse [5].

Re = 2000 0.35 0.30 0.25 z ->

0.20 0.15

0.20E 00

0.10 0.05 0.00 -0.05 0.4

0.5

0.6

0.7

0.8 r ->

0.9

1.0

1.1

1.2

Fig. 6. Dimensionless radial and axial velocities, Re ¼ 2.000.

combined in a Reynolds number in a way used in stirring processes to characterise the power consumption (Power-number vs. Reynolds-number diagram). The laminar range was investigated up to Reynolds numbers of 8.000. Velocity ﬁelds are calculated in tangential, radial and axial directions. Only one example, the radial-axial velocity proﬁle for Re ¼ 2.000, is shown in Fig. 6. The operation domain is bound by the agitator shaft, the cylinder wall and the symmetry planes existing in the middle of the agitator disc and between the

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two discs, respectively. In the ﬁgure dimensionless r,z coordinates are used – related to the outer radius of the disc. The lengths of the arrows are related to the tip speed of the discs with a scaling factor of 0.2 (see scale on the right). The proﬁle conﬁrms the ideas of Bosse. High velocities and high velocity gradients occur in the periphery of the cyclic ﬂow (vicinity of the disc surface, symmetry ﬁeld between the discs and vicinity of the grinding chamber wall). In these areas the velocity vectors get maximum values of 20% of the circumferential velocity of the agitator disc. If the local velocity gradients are known local energy dissipations can be calculated. With respect to local volumes local speciﬁc energies arise. If these are related to the mean speciﬁc energy (total amount of energy divided by the net volume of the grinding chamber) energy densities are deﬁned. In Fig. 7 isolines of the energy density are plotted for Re ¼ 2.000. The isoline with the value of 1 is of great importance. It separates the grinding chamber into different characteristic zones. In these zones the local speciﬁc energy is larger or smaller than the mean speciﬁc energy. At Re ¼ 2.000 two zones characterised by a high energy exist, one close to the disc surface and the other close to the wall. For Re ¼ 10 (not shown here) the total area with high energy distributions is much smaller. Hence, the speciﬁc energy distribution at low Reynolds numbers is much more homogeneous than at high Reynolds numbers. Figure 8 quantiﬁes this statement. It shows the volume percent of the entire grinding chamber with a high energy density (higher than the mean speciﬁc energy) as a function of the respective Reynolds number. The second curve indicates the portion of the entire energy consumption which is dissipated within this volume part. At Re ¼ 8.000 90% of the energy is dissipated in just 10% of the volume. The results shown only characterise the ﬂuid ﬂow, i.e. without any beads. Blecher also calculated individual bead trajectories. Independent of the starting

Re = 2000 0.35

0.1

0.30 0.1

0.25

0.5

0.1

1.0

z ->

0.20 0.15 0.10

5.0

0.05 15.0

0.00

1.0 5.0

-0.05 0.4

0.5

0.6

1.0 0.5

15.0 0.7

0.8 r ->

Fig. 7. Isolines of the energy density, Re ¼ 2.000.

0.9

1.0

1.1

1.2

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261

100

80 percentage [%]

part of energy 60

40 part of volume 20

0 10

2000

4000 6000 Re = VuRs/

8000

Fig. 8. Part of the grinding chamber volume with high energy density and part of the entire energy input dissipated there.

point stationary trajectories are obtained [12–14]. In extension to the calculations of Blecher, Theuerkauf [15] investigated the turbulent range and a bulk of grinding media. He also assumed Newtonian behaviour and treats the ﬂow incompressible, rotationally symmetrical and stationary. For the turbulent ﬂow the time-averaged Reynolds equations are solved using the standard k,e-model. The calculation of the turbulent ﬂow is an extension of the laminar calculation, because the system of equations is extended by two equations, one for the turbulent ﬂuctuating energy k and one for the energy dissipation e. The resulting velocity ﬁelds follow the ones for the laminar range. The areas with a high energy density are decreasing further, whereas the energy dissipation in these areas is increasing. The calculation of the continuous phase – the grinding media ﬁlling – is based on a one way coupling to the continuous phase. The motion of the grinding media within the calculated velocity ﬁeld of the ﬂuid is simulated using a Lagrangian approach. The equations for the beads movement are solved with respect to translational and rotational movement. Contacts of beads with borderlines of the calculated domain are regarded as central collisions. The simulation leads to zones with high collision frequencies. The kinetic energy dissipated at those collisions is calculated. The results deﬁne areas with high energy densities, being of importance for grinding processes. Currently a discrete-element method (DEM) based approach is developed at the Institute for Particle Technology, TU Braunschweig, to realise a two way coupling between the grinding media ﬁlling and the continuous phase. For this at ﬁrst a dry operated stirred media mill with different geometry and operating parameters was modelled. At this stage with the existing model it is possible to

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Fig. 9. With discrete elements ﬁlled grinding chamber of a stirred media mill.

investigate the movement of grinding media and energy distribution inside dry operated stirred media mills (Fig. 9). Based on the dry model different possibilities to simulate the motion of beads in a wet operated mill will be investigated. The different methods range from adding adhesive forces at the stirrer to the implementation of a two-way coupling by using a combined DEM and computational ﬂuid dynamic (CFD) approach.

2. FUNDAMENTAL CONSIDERATIONS 2.1. Introduction By trying to describe the processes in stirred media mills very soon it becomes obvious that many inﬂuencing parameters exist and affect the grinding and dispersing result. Mo¨lls and Ho¨rnle [16] listed 44 inﬂuencing parameters. Certainly, not all these are of major importance, but the large number demonstrates the complexity of the problem. The most important parameters, which can be divided into four groups, are (a) Operating parameters of the mill Grinding or dispersing time Throughput Stirrer tip speed Grinding media size Grinding media material (density, elasticity and hardness) Filling ratio of the grinding media

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(b) Operation mode of the mill (one or multiple passage mode, pendulum or circuit operation) (c) Formulation (composition of the suspension) Solids concentration of the particles Fluid (water, solvents, resins and so on) Additives or dispersing agents (Reduction of the viscosity and/or avoidance of reagglomeration or ﬂocculation) (d) Mill geometry Type of the mill Size and dimensions of the mill In order to ﬁnd the parameter set, which is most favourable for a certain grinding or dispersing process, usually experiments with a laboratory or other small scale mill are carried out. In some cases the grinding or dispersing process is directly tested in production scale. During the experiments usually the product quality is measured as function of grinding time (charge operation) or throughput (continuous operation). By comparing the grinding times or throughputs of the different parameter sets at which the demanded product quality is obtained hopefully the optimum set of parameters can be found. Finding the optimum parameter set the question arises which is the exact aim and which means ‘‘optimum’’ for the grinding and dispersing process looked at. In this connection two main aims or claims can be distinguished: (a) Product quality, which is deﬁned among others by Particle size distribution, gloss, intensity of colour, transparency Product purity (no contamination by wear of mill and grinding media) No product degradation (e.g. by too high temperatures) Stability (against reagglomeration, ﬂocculation, sedimentation and so on) (b) Economy, which is determined above all by Investment costs Operating costs (energy, cooling water, maintenance and so on) Production capacity Cleaning expenditure

2.1.1. Description of production rate Besides the investment and operation costs the production capacity established at the demanded product quality mainly determines the economy of a process. At continuous operation the production capacity is determined by the product mass ﬂow rate, mp ﬂowing through the mill and in a charge process by the ratio of Å

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product mass mP of the charge and grinding time tgrind of the charge. The dependency of the production capacity on the operating and geometry parameters is described by the following equation: _P ¼ m

mP t grind

¼

P P 0 Em

ð1Þ

where P is the average power draw of the motor, P0 is the no-load power which is consumed by the friction inside the bearings and the seal without a ﬁlling of the grinding chamber (i.e. also without grinding media) and Em is the speciﬁc energy required to produce the demanded product quality. The speciﬁc energy is deﬁned as the energy transferred into the grinding chamber related to the mass of the product inside the process. Therefore, the production capacity is proportional to the power input into the grinding chamber, P P 0 and inversely proportional to the speciﬁc energy required for the demanded product quality. From equation (1) it follows directly that the maximum production capacity is reached if the power input into the grinding chamber is the highest the speciﬁc energy requirement for the production of the demanded product

quality is the lowest. The problem regarding an increase in production capacity is that several operating and geometric parameters inﬂuence the power input as well as the speciﬁc energy requirement. For example the power input and thus the production capacity increases with increasing stirrer tip speed. Since, as a rule, also the speciﬁc energy requirement increases with increasing stirrer tip speed, the increase of the production capacity is lower than it could be theoretically due to the increase in power input. Therefore, power input and speciﬁc energy requirement cannot be optimised separately. The problem is to ﬁnd a parameter set which results in a power input being as high as possible and simultaneously a speciﬁc energy requirement being as low as possible. Independently on the mill type and construction the maximum possible power input into the grinding chamber is limited by the installed motor power and the installed cooling capacity. At sufﬁcient cooling capacity the operation parameters should be chosen in a way that the installed motor power decreased by the noload power is fully consumed by the stirrer or inside the grinding chamber, respectively. If the cooling capacity is not sufﬁcient at maximum power input, the operating parameters have to be chosen in a way that the power transferred to the product suspension can just be removed by the cooling jacket without an overheating of the product. The speciﬁc energy requirement for the production of the demanded product quality depends on many parameters. For the reduction of the number of

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265

inﬂuencing parameters it is useful to combine the parameters in characteristic numbers or parameters which describe the grinding or dispersing process. For the derivation of these characteristic numbers ﬁrst of all an idea of the physical processes acting in stirred media mills must be developed. These physical processes can be described by so-called stress models, which are discussed in Section 2.2 in more detail.

2.1.2. Determination of product quality as function of the grinding time and speci¢c energy In order to ﬁnd out if a certain grinding or dispersing task can be solved by using a stirred media mill, which parameter values are most favourable for the process under investigation and how high the ﬁnancial and technical expenditure is in production scale, usually tests with a laboratory or small scale mill are carried out. Depending on the grinding or dispersing task, the available quantity of feed material, the available test equipment and the demanded quantity of product the tests can be carried out differently (see also Fig. 10): 1. 2. 3. 4.

Batch operation (demanded quantity of product should be small) One passage mode (continuous operation) Multiple passage mode (e.g. pendulum operation) Circuit mode (circuit with stirred vessel)

The determination of grinding time and speciﬁc energy depends on the type of grinding test: 1. Batch operation (discontinuous test) The mean residence time is equal to the test time, at which the product sample is taken from the grinding chamber The speciﬁc energy input after the grinding time t results from the following equation (without consideration of wear of grinding media): Z t EðtÞ ðPðtÞ P 0 Þdt E m ðtÞ ¼ ¼ ð2Þ mP mP 0 2. One passage (continuous grinding) The grinding time is equal to the mean residence time of the product inside the grinding chamber, which can be determined strictly speaking only from the residence time distribution of the particles in the mill. The mean residence time t inversely proportional to the volume ﬂow rate of the product suspension and agrees approximately with the so-called ideal ﬁlling time tf. The ideal ﬁlling time describes the time which is necessary to ﬁll the free volume of the grinding chamber (grinding chamber ﬁlled with grinding media)

266

A. Kwade and J. Schwedes 1. Batch operation

Sample

2. One Passage mode (continuous operation)

3. Multiple passage (pendulum operation) 2,4,...passage

1,3,... passage

1,3,... passage

2,4,...passage

4. Circuit operation (with stirred vessel)

Fig. 10. Different possibilities of running a grinding or dispersing test.

once with suspension: t ¼ t f ¼ V GC V GM V_ Susp

ð3Þ

where VGC (m3) is the grinding chamber volume, VGM (m3) the overall solid volume of the grinding media and V_ Susp (m3 h–1) the volume ﬂow rate of the suspension. The speciﬁc energy corresponds to the ratio of the power input into the grinding chamber (power draw at stationary operation, Pstat, minus no-load power, P0) and the mass ﬂow rate of the product: Em ¼

P stat P 0 _p m

ð4Þ

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267

3. Multiple passage mode – Pendulum mode The mean residence time is nearly the grinding time multiplied with the ratio of the free grinding chamber volume and the volume of the suspension. It is also possible to calculate the mean residence time by taking the sum of the mean residence times of the single passages: t t V GC V GM ¼ V Susp

n X V GC V GM V_ Susp;i

ð5Þ

i¼1

where n is number of passages, VSusp the volume of the suspension in the vessel and V_ Susp the volume ﬂow rate of the suspension The speciﬁc energy is the sum of the speciﬁc energies of the individual passages calculated by equation (4). Alternatively the speciﬁc energy can also be determined by integration based on equation (2). 4. Circuit mode The mean residence time is approximately equal to the test time multiplied with the ratio of free grinding chamber volume and overall volume of the suspension as long as the product samples are taken from the stirred vessel. t t V GC V GM V Susp

ð6Þ

If the samples are taken directly behind the mill, the grinding time corresponds approximately to the mean residence time of the circuit deﬁned above plus the mean residence time of the last passage (calculation according to equation (3)). The speciﬁc energy can be determined using equation (2) if the sample is taken from the stirred vessel. If the sample is taken directly behind the mill, the speciﬁc energy of the last passage has to be added to the speciﬁc energy of the true circuit. If the number of circuits is high, in a ﬁrst approximation the speciﬁc energy of the last passage can be neglected. If there is high wear of grinding media (i.e. grinding ceramic materials), for the comparison of the grinding performance of different kinds of grinding media, the wear of grinding media DmGM , should be taken into account for the speciﬁc energy input and should be calculated. In this case the speciﬁc energy for a batch operation can be determined as follows: Rt ðPðtÞ P 0 Þdt EðtÞ E m;W ðtÞ ¼ ¼ 0 ð7Þ mP þ 0:5DmGM mp þ 0:5DmGM In the following the log–log scale is usually used to present the grinding results, because in this scale the lower particle size region can be better distinguished at high overall speciﬁc energy inputs. Moreover, at low speciﬁc energies differences in the product ﬁneness can be seen more clearly. Indeed it has to be considered that because of the log–log scale speciﬁc energies, which are required to obtain a ﬁner product at particle sizes below 1 mm, are easily underestimated.

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Particle size x [μm]

40 dGM = 1050 μm ρGM = 7550 kg/m3 vt = 9.6 m/s ϕGM = 0.8 cm = 0.4

10

1

Median size x50 Particle size x90

0.4 10

100 1000 Specific energy Em [kJ/kg]

10000

Fig. 11. Product ﬁneness as function of speciﬁc energy (log–log scale).

Furthermore, at large speciﬁc energies small deviations of the speciﬁc energy in the log–log scale often correspond to large differences in the absolute value of the speciﬁc energy. A typical relation between the product ﬁneness and the speciﬁc energy is depicted in Fig. 11 in log–log scale. As characteristic parameter for the product ﬁneness the median size x50 and the particle size x90 were chosen. From Fig. 11 it follows that over a wide range the relation between the product ﬁneness and the parameter speciﬁc energy can be described by one straight line, i.e. the relation can be described by a power function. The power function can be described by the following equation: X P ¼ a E bm ; X P ¼ c t d

ð8Þ

The power function according to equation (8) can be derived from the so-called ‘‘general grinding law’’, which was published by Walker et al. [17]: dE m ¼ C

dx xn

ð9Þ

Since the grinding product exists not only of one particle size x, the product ﬁneness must be described by a characteristic particle size (e.g. median size x50). Integration of equation (9) results in an equation for the speciﬁc energy input, which is required for grinding a product from particle size xF to particle size xP: ! C 1 1 Em ¼ n1 ð10Þ n 1 x n1 xF P For the case, that x F x P is valid, by rearranging equation (10) the power function described above (equation (8)) results.

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2.2. Stress models for a better description of the physical processes in stirred media mills and in order to look at a stirred media mill not as a black box, at the Institute for Particle Technology of the Technical University of Braunschweig so-called stress models were developed. Two ways exist to look on the grinding process: ﬁrst of all we can consider the performance of the mill, i.e. how the mill stresses the particles, how frequent a stress event takes place and which energy is available at these stress events. On the other side we can consider what happens with a feed particle, i.e. how the particle is stressed, how often the particle is stressed and with which intensity the particle is stressed. The two different ways to deﬁne a stress model are described in the following.

2.2.1. Mill related stress model For the characterisation of a mill characteristic numbers have to be deﬁned which are independent of the size and other properties of the product particles: the grinding behaviour of a mill is determined by the type of stress (impact or compression and shear) including number of

particles stressed at one stress event the number of stress events which are supplied by the mill per unit time, the so-

called frequency of stress events, SFM the energy which is supplied by the mill at each stress event, the so-called

stress energy, SE. The product of the frequency of stress events, SFM, and the mean grinding time, tgrind, is named the total number of stress events, SNtot (SNtot ¼ tgrind SFM). The mean grinding time, tgrind, to achieve a certain product quality and with it the total number of stress events, SNtot, are a function of the stress energy and the breakage behaviour of the product. The stress energy, SE, is deﬁned as the energy transferred to one or more product particles at one stress event. The stress energy is not constant at all stress events, so that for an exact description of the mill the frequency distribution of the stress energy must be known. The frequency distribution like shown in Fig. 12 describes which relative frequency belongs to a certain stress energy and thus, how often a certain stress energy occurs per unit time. Similar, in reality the stress frequency, SFM, can only be described by a distribution. Nevertheless, in practice it is often sufﬁcient to use characteristic parameters like average values of these distributions to describe the stress frequency and the stress energy of a certain mill at certain operating parameters.

A. Kwade and J. Schwedes

Frequency distribution sf [s-1 J-1] sf(SEj)

270

ΔSFM,j = sf(SEj) • ΔSE

0

SEj

SEmax

Stress energy SE [J]

Fig. 12. Qualitative frequency distribution of the stress energy.

2.2.2. Product related stress model By deﬁnition the stress frequency and stress energy deﬁned above are independent of whether one or more particles are really stressed during one stress event. Moreover, the size of the particles does not matter. But the result of stressing one particle depends among others on the amount of energy transferred compared to the size or mass of the particle. Therefore, in order to describe the grinding or dispersing of a product particle it has to be considered how many particles and which particles sizes are stressed at one stress event. This is considered by the so-called product related stress model: For a given feed particle the product quality and ﬁneness achieved in a grinding or dispersing process is determined by how the feed particles and the resulting fragments are stressed and thus, which

type of stress acts (e.g. impact or compression and shear) how often each feed particle and its resulting fragments are stressed and thus,

by the number of stress events of a feed particle, SNF how high the speciﬁc energy or speciﬁc force at each stress event is and thus,

by the stress intensity at each stress event, SI. In real grinding processes the feed particles and the resulting fragments are not stressed equally often with the same stress intensity, but differently often with different stress intensities. Thus, in detail number of stress events and stress intensity can only be characterised by distributions, not by single numbers. Both distributions, particularly the magnitude of SNF and SI, depend on the operating parameters. The width of the distribution of the stress number is determined

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271

Fig. 13. Speciﬁc surface and disintegration degree as function of relative stress intensity.

above all by the residence time distribution of the particles in the mill. The width of the distribution of the stress intensity depends mainly on how the stress energies differ locally and with time. The exact determination of these distributions is difﬁcult, but may be possible using numerical methods in the future. The inﬂuence of the number of stress events on the product quality or ﬁneness is obvious: with an increasing number of stress events per feed particle the product quality or ﬁneness increases. Against that the stress intensity determines how effective the speciﬁc energy transferred to the product is transposed into product quality and product ﬁneness. The principle effect of the stress intensity on the product ﬁneness when single particles are stressed follows from Fig. 13. As a measure for the product quality the increase in speciﬁc surface DSm or the disintegration degree A (for the disintegration of microorganisms) is chosen. In Fig. 13 the speciﬁc surface or disintegration degree is depicted as function of the relative stress intensity. The relative stress intensity is deﬁned as the ratio of stress intensity SI and the optimum stress intensity SIopt. The stress intensity is optimum and the energy utilisation is maximum, when the energy is just sufﬁcient to break a particle, to deagglomerate an agglomerate or to disintegrate a microorganism. As long as the stress intensity is smaller than the optimum stress intensity (SI/SIopto1) the product ﬁneness increases with the stress intensity for all three applications. But, if the stress intensity is larger (SI/ SIopt41), there are differences. In an ideal deagglomeration process the speciﬁc surface is constant because all agglomerates were already deagglomerated at optimum stress intensity. The same holds for the disintegration of microorganisms

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because all microorganisms were already disintegrated at optimum stress intensity. Against that when grinding crystalline materials the speciﬁc surface will further increase with increasing stress intensity but usually at a lower slope, because the energy utilisation is smaller than at the optimum stress intensity. The more difﬁcult the feed material is to grind, the greater the slope of the curve. Thereby, theoretically two boundary cases can be deﬁned: The upper limit can be assumed to be the case when the new surface area

increases proportional to the stress intensity. In this case the slope of the curve is one. The lower limit is equal to a deagglomeration process, in which the size of the fragments does not depend on the stress intensity as long as the stress intensity is higher than the optimum stress intensity. In this case the slope of the curve and the value of exponent is zero. Real cases of grinding crystalline materials will lie between these two boundary cases, so that the exponent of the power function will be between zero and one. Therefore, at SI4SIopt it can be stated: SI a DSm ; A / ð11Þ SI opt with a ¼ 0 for ideal deagglomeration/disintegration and 0oao1 for grinding crystalline materials. Instead of the product ﬁneness the energy utilisation can be used to show the effect of the stress intensity (Fig. 14 follows from Fig. 13). In Fig. 14 the relative energy utilisation EU/EUmax is plotted versus the relative stress intensity SI/SIopt. The energy utilisation EU is deﬁned as the ratio of the new produced speciﬁc surface DSm and the speciﬁc energy Em required to produce DSm. If a single particle is stressed once, the stress intensity corresponds to the speciﬁc energy for stressing the particle, so that the energy utilisation is equal to DSm/SI. At the optimum stress intensity SIopt the energy utilisation has its maximum value EUmax at which a certain speciﬁc surface can be produced with a minimum of speciﬁc energy. At the optimum both ratios are one. Left of the optimum the energy utilisation increases with increasing stress intensity for all three applications in a similar, but not equal mode. Right of the optimum the relative energy utilisation decreases. In case of ideal deagglomeration and disintegration the slope of the curve is –1, because the stress intensity added to the optimum stress intensity is not used at all and does not effect the product ﬁneness or the disintegration rate. Therefore, in this case the energy utilisation is inversely proportional to the speciﬁc energy and thus, to the stress intensity.

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273

Fig. 14. Relative energy utilisation as a function of the relative stress intensity.

If a single crystalline particle is stressed, right of the optimum stress intensity the relative increase in product ﬁneness or speciﬁc surface respectively is smaller than the corresponding relative increase in stress intensity, so that the energy utilisation, EU, decreases right of the optimum. The more difﬁcult the feed material is to be ground the smaller the decrease in energy utilisation. The upper limit is probable if the energy utilisation stays constant while the stress intensity increases. Thus, for SI4SIopt it can be stated: EU / EU max

SI SI opt

a1 ð12Þ

with a ¼ 0 for ideal deagglomeration/disintegration and 0oao1 for grinding crystalline materials. It has to be mentioned that the above explained trends are related only to those processes, where the product quality can be measured – directly or indirectly – by the particle size distribution, the deagglomeration degree or the disintegration degree. At those processes an identical result can be achieved by either stressing the feed material many times at low stress intensities or by stressing only a few times at high stress intensities. The only prerequisite is that the lowest used stress intensity leads to an increase in product ﬁneness. In other processes where the product quality does not or only partly depend on the particle size the just mentioned combination of stress intensity and stress number might not hold. As an example the brightness in the paint and lacquer industry is such a product quality. It might be possible that a special effect can only be obtained with high

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stress intensities. It is also not yet known if the maximum product ﬁneness depends on the stress intensity.

2.2.3. Relation between the model parameters and speci¢c energy, power input and production capacity The total energy transferred to the product particles can be determined by the summation of all stress energies of the individual stress events. If a frequency distribution of the stress energies like shown in Fig. 12 is known, the total energy transferred to the product particles can also be estimated by adding up the stress energies in an interval j multiplied with the corresponding number of stress events. The speciﬁc energy, Em,P, actually transferred to the product particles is obtained by relating the total energy to the total mass of the product. Owing to friction and other losses the speciﬁc energy consumed by the grinding device or mill, Em,M, is not equal, but proportional to the speciﬁc energy, Em,P, transferred to the product particles. If the losses are taken into account by an energy efﬁciency factor, nE , the two characteristic numbers SNtot ¼ t SFM and SE can be related to the speciﬁc energy consumed by the mill as shown in the following: tot Snj¼1 SE j DSF j SSN SN tot SE i¼1 SE i ¼ t grind ¼ ¼ E m;P ¼ nE E m;M mP;tot mP;tot mP;tot

ð13Þ

where SEi is the stress energy at stress event i, SNtot the total number of stress events to achieve a certain product quality, SEj the average stress energy of interval j, DSFj the frequency of stress events of interval j, tgrind the grinding time, Em,P the speciﬁc energy transferred to the product particles, nE the energy transfer factor of the mill and Em,M the total speciﬁc energy consumption of the mill. The speciﬁc energy transferred to the product Em,P is also named effective speciﬁc energy. This speciﬁc energy is the part of the total energy consumption of the mill, which is really used for stressing the particles. Under the assumption of a constant shape of the stress frequency distribution and the stress energy distribution based on equation (13) the product quality is already ﬁxed, if two of the three parameters total number of stress events, SNtot, mean stress energy, SE, and speciﬁc energy transferred to the product, Em,P, are set. Thereby Em,P is ﬁxed, if the energy efﬁciency factor, nE , and the speciﬁc energy consumed by the mill, Em,M, are set.

2.3. Application of the stress models on stirred media mills In a wet operated stirred media mill the product particles which are dispersed in a ﬂuid are stressed and ground by grinding media which are intensively moved by a stirrer. The power or energy consumed by the stirrer is transferred from the stirrer

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275

to the product suspension and to the grinding media. But only a part of the power or energy consumed by the stirrer is transferred to the product particles. The other part of the energy is dissipated into heat by friction processes. In order to derive characteristic parameters of the stress frequency and stress number as well as stress energy and stress intensity it must be clariﬁed in advance, where and how grinding is performed in a stirred media mill. The following three grinding mechanisms are possible in stirred media mills. The importance of the different mechanisms on the grinding effect depends on the properties of the mill: Feed particles can be stressed by grinding media (A) which are accelerated from the stirrer shaft towards the grinding chamber wall and thus take up kinetic energy (B) which are pressed against the grinding chamber wall due to the centrifugal acceleration (C) which move in a tangential direction with high velocities and collide with grinding media with lower velocities. According to Kwade [18,19] the most important grinding mechanism is (C): only in a tangential direction enough stress events take place in which enough stress energy is available to grind the product. Particularly this is valid in mills with pincounter pin and annular gap geometry. An important condition that particles are stressed between two grinding media is that the particles are captured by the grinding media and are not carried out with the displaced ﬂuid. According to the number of captured particles three cases can be distinguished: (A) Only one particle is captured, which is stressed with the entire energy or force, respectively (single particle stressing). (B) More than one particle is captured between two beads, all particles have contact to both beads during the stress event and all particles are stressed independent of each other. In this case at ﬁrst the particle is captured, which has the largest size and/or which has the smallest distance to the connection line of both bead centres. This particle is stressed with the maximum energy or force. The particles, which are captured between the two beads after the ﬁrst particle, are stressed with a considerably reduced energy or force. At the end of the stress event diverse single particle stressings with different intensities occur. (C) A particle bed is captured and stressed between two grinding media. The number of captured particles depends among others on the solids concentration of the suspension and the size of the particles. Based on the

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considerations of Kwade [18,19] and observations of the ﬂow ﬁeld between two grinding media [20] the following can be assumed and postulated regarding the number of particles captured between two grinding media: As a rule, at low and medium solid volume concentrations only one single (case

A) or several particles (case B) are captured. For case (A), the stressed volume or the stressed area equals the volume or the cross-section of the captured particle. If for the case that more than one particles are captured it is assumed that the stress intensity of the grinding media is just sufﬁcient to break the ﬁrst captured particle case (B) equals case (A). Thus, the volume being responsible for the increase in ﬁneness is only determined by the size of the ﬁrst captured particle. The number of captured particles has nearly no inﬂuence. At high solid volume concentrations, very small particles (as a rule distinctively smaller than 1 mm) and high viscosities of the suspension stressing of a layer of particles or even a bed of particles cannot be excluded (case C). In this case the active volume and thus the stressed volume of the particle bed depend besides others on the diameter of the grinding media. All in all the most important mechanism is stressing of single product particles between tangential moving grinding media. Under this condition characteristic parameters for the different stress models can be derived.

2.3.1. Estimation of number of stress events and stress frequency At low and medium solid concentrations and suspension viscosities it is most probable that at each contact of two beads and thus, at each stress event only one particle is stressed intensively. Under this condition in batch mode, the average number of stress events of each product particle, the stress number SN, is determined by the number of bead contacts, Nc, by the probability that a particle is caught and sufﬁciently stressed at a bead contact, Ps, and by the number of feed particles inside the mill, Np: SN ¼

Nc P S NP

ð14Þ

The number of bead contacts can be assumed to be proportional to the number of revolutions n, to the grinding time t and to the number of grinding media NGM in the mill: N C / n t N GM / n t

V GC jGM ð1 Þ ðp=6Þ d 3GM

ð15Þ

where VGC is the grinding chamber volume, jGM the ﬁlling ratio of the grinding media, e the porosity of the grinding media ﬁlling at rest and dGM the diameter of a grinding media.

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dGM

Xp

active volume VP, act

Fig. 15. Active volume VGM, act.

The probability that a particle is captured and sufﬁciently stressed depends besides others on the type of the grinding process. According to Stadler et al. [21] and Bunge [22] for deagglomeration of pigments and disintegration of microorganisms this probability is proportional to the surface of the grinding media, since the ﬂuid shear stresses acting between the beads are sufﬁcient for deagglomeration and disintegration. P S / d 2GM

ð16Þ

For the grinding of crystalline materials or very tough agglomerates (minerals, ceramics) the probability is proportional to the active volume between two grinding media, shown in Fig. 15. This active volume is proportional to the diameter of the grinding media. Thus, it follows: P S / d GM

ð17Þ

The number of feed particles is equal to the ratio of the overall volume of the feed particles, Vp,tot, to the average volume of the feed particles, Vp: Np /

V p;tot ð1 jGM ð1 ÞÞ cv ¼ V GC Vp Vp

ð18Þ

where cv is the solids volume concentration. Combining equations (14) to (18) the following proportionality of the stress number SN can be derived: Deagglomeration/disintegration: SN /

jGM ð1 Þ nt ð1 jGM ð1 ÞÞ cv d GM

ð19Þ

SN /

jGM ð1 Þ nt ð1 jGM ð1 ÞÞ cv d 2GM

ð20Þ

Real grinding:

Thus, the stress number SN is different for deagglomeration/disintegration and for grinding of crystalline materials. For deagglomeration and disintegration it is

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inversely proportional to the diameter of the grinding media whereas it is inversely proportional to the square of the diameter of the grinding media for grinding of crystalline materials.

2.3.2. Estimation of stress energy and stress intensity According to Bunge [22] and Reinsch et al. [23] the stress intensity in stirred media mills can be described by the quotient between the torque of the stirrer and the mass of the product in the mill. A similar deﬁnition was introduced by Winkler [24,25], who stated that the intensity during dispersing is a function of the power of the mill and thus, a function of the power density of the mill. Besides these two statements which are based on the power consumption of a mill the stress intensity and stress energy can be derived directly from the possible grinding mechanisms. Based on the considerations shown above three different stress mechanisms exist. Out of these three stress mechanisms the stressing between two grinding media which collide due to velocity differences in tangential direction was found out to be most decisive. Moreover, since the size of the product particles and with it also the mass of the product particles changes with increasing grinding time and increasing speciﬁc energy, it is not convenient to derive a characteristic expression of the stress intensity. Instead a characteristic for the stress energy is derived. The stress energy is determined by the kinetic energy of the faster one of the two colliding grinding media. If it is further assumed that only single particles are stressed intensively between two grinding media and,

therefore, the stressed particle volume does not depend on the size of the grinding media, the tangential velocity of the grinding media is proportional to the circumferential velocity of the discs the diameter of the discs is kept constant the displacement of the suspension between two approaching grinding media causes no essential decrease of the media velocities and thus, of the kinetic energy of the two grinding media the elasticity of the feed material is much smaller than that of the grinding media and, therefore, at grinding mechanism C the kinetic energy of the beads is nearly completely transferred to the feed particles and not partly consumed by the deformation of the beads,

then the stress energy is approximately proportional to the following expression, called the stress energy of the grinding media, SEGM. SE / SE GM ¼ d 3GM rGM v 2t

ð21Þ

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279

The stress energy of the grinding media is determined by the size and density of the grinding media and remains constant during a grinding process, whereas the sizes of the stressed particles and thus, the stress intensities at each stress event change with grinding time. Since the stress energy of the grinding media, SEGM, was deﬁned using the stirrer tip speed as measure for the speed of the grinding media, the stress energy SEGM is a measure for the maximum stress energy in the mill. The mean stress energy can differ because of different distributions of the stress energy although the stress energy of the grinding media, SEGM, is constant (see Section 6). At higher viscosities the displacement of the suspension between approaching grinding media cannot be neglected anymore because a greater part of the kinetic energy of the grinding media is lost due to the displacement of the suspension. In order to take this effect into account, the expression in equation (21) has to be extended by a term, which describes this displacement process. A possible way is described by Kwade and Mende [26]. If the elasticity of the feed material is about the same or higher than that of the grinding media material (e.g. grinding of ceramic materials), the modulus of elasticity of the product, YP, and the one of the bead material, YGM, have to be taken into account. The higher the modulus of elasticity of the feed material is compared to that of the bead material, the higher the deformation of the grinding media at each stress event and the less energy transferred from the grinding media to the product particle. The part of the energy which is consumed by the deformation of the grinding media can be estimated following the modelling of Becker [27]. Becker determined the energy transfer from the grinding media to the feed particles using a simple spring-mass-model without damping. Under the assumption that only linearly elastic deformations are considered, for a given kinetic energy of the bead before the collision, the deformations of a grinding bead and feed particle are calculated using the Hertz equation for the collision of two spheres. By deriving and solving the equation of motion and neglecting the Poisson’s ratios of both materials the maximum energy transferred from two grinding beads to the feed particle, EP,max, was derived. Y P 1 E P;max / d 3GM rGM v 20 1 þ ð22Þ Y GM where YP (N m–2) is the modulus of elasticity of the feed material and YGM (N m–2) the modulus of elasticity of the grinding media. If it is assumed that the velocity of the grinding media v0 is proportional to the circumferential velocity of the disc, the following proportionality of the volumerelated energy EP,max can be derived: Y P 1 E P;max / d 3GM rGM v 2t 1 þ ¼ SE P ð23Þ Y GM

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The stress energy transferred to the product particle, SEP, is a measure of the stress energy, which is transferred to the product under consideration of the deformation of the grinding media. The resulting expression corresponds to the stress energy of the grinding media, SEGM, multiplied by a factor, which describes the part of energy transferred to the stressed particle. The other part of the former kinetic energy of the grinding media is stored as elastic energy in the grinding media and cannot be used for grinding. The percentage of energy, EP,rel, which can be transferred to the feed material, is E P;rel ¼

E P;max Y GM ¼ ¼ E P;max þ E max Y P þ Y GM

1þ

YP Y GM

1 ð24Þ

This relative amount of energy EP,rel is plotted versus the ratio YGM/YP of the moduli of elasticity of both materials in Fig. 16. With increasing values of the modulus of elasticity of the grinding media and decreasing values of the modulus of elasticity of the feed material the ratio YGM/YP and, as a consequence, the relative amount of energy EP,rel being transferred to the feed material increases. Simultaneously the slope of the dependency of Fig. 16 decreases with an increasing ratio of the moduli of elasticity. Having feed materials with high moduli of elasticity like corundum or silicon carbide the ratio of the moduli of elasticity is small and the slope of the dependency is great, i.e. a relatively small change in the modulus of elasticity will produce a 1.0 YP = 410 GPa YGM = 625 GPa

EP,ref [ - ]

0.8

YP = 410 GPa YGM = 625 GPa

YP = 30 GPa YGM = 240 GPa

YP = 30 GPa YGM = 63 GPa

0.6 limestone (YP = 30 GPa) for moduli of elasticity between 63 GPa and 240 GPa

0.4

0.2

corundum (YP = 410 GPa) for moduli of elasticity between 100 GPa and 625 GPa

YP = 410 GPa YGM = 100 GPa

0.0 0

1

2

3

4 5 YGM / YP [ - ]

6

7

8

9

Fig. 16. Relative volume-related energy of the feed material EP, rel as a function of the ratio of moduli of elasticity YGM/YP.

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281

relatively great change of the relative amount of energy part which can be transferred to the feed material. An increase of the modulus of elasticity of the grinding media from 100 GPa (mixed oxide 2) to 265 GPa (Y2O3 stabilised zirconium oxide) leads to a doubling of the relative amount of energy EP,rel, when fused corundum (YGM ¼ 410 GPa) is stressed (Fig. 16). The extreme case would be if ceramic materials with high modulus of elasticity are stressed by plastic beats. In this case almost no energy will be transferred to the produced particle, i.e. Ep,relE0. If limestone as feed material (YPE30 GPa) is stressed with grinding media as just mentioned (YGM ¼ 100 or 265 GPa) the ratio of the moduli of elasticity is large and the inﬂuence of the moduli of elasticity of the grinding media on a change of the relative energy EP,rel is small (compare the two open circles in Fig. 16), i.e. a large change in the modulus of elasticity of the grinding media causes only a small change in stress intensity when stressing limestone. It has to be mentioned that the derived expressions for the stress energy consider only the inﬂuences of the parameters circumferential disc velocity as well as size and density of the grinding media. Inﬂuences coming from the geometries of the grinding chamber and the agitator as well as from the viscosity of the suspension cannot be described by the derived expressions. In this case further calculations are necessary (see Section 6 and [26]).

2.3.3. Speci¢c energy and energy transfer factor From Section 2.3 it follows that the speciﬁc energy Em,P, which is transferred to the product particles, and the speciﬁc energy Em,M, which is consumed by the stirrer, are different. If the speciﬁc energy Em is mentioned, usually the speciﬁc energy consumed by the stirrer is meant. Therefore, for simpliﬁcation the speciﬁc energy consumed by the stirrer is called speciﬁc energy and is described by the symbol Em in the following. The speciﬁc energy Em,P, which is transferred to the product particles, is much smaller than the speciﬁc energy consumed by the stirrer. This speciﬁc energy, Em,P, can be determined by multiplication of the speciﬁc energy consumed by the stirrer, Em, with the energy transfer factor, nE . The high amount of the energy, which is transferred or dissipated into heat inside the grinding chamber, is according to Kwade [20,28] mainly due to the following ﬁve phenomena (see also Fig. 17): (A) The energy consumed by the stirrer is transferred to the product suspension and to the grinding media, i.e. only a part of the energy consumed by the stirrer is transferred to the grinding media in the form of kinetic energy. The other part of the energy is given to the product suspension and transferred into heat by friction. (B) Owing to friction at the grinding chamber wall a part of the kinetic energy of the grinding media is dissipated into heat.

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(C) A part of the remaining kinetic energy has to be used to displace the suspension between the grinding media if two grinding media approach each other. (D) In the grinding chamber grinding media contacts without product particles take place. The kinetic energy transferred into heat at these grinding media contacts is not used for grinding. (E) When two beads catch and stress one or more product particles a part of the kinetic energy is used to deform the grinding media and not the product particle or particles. Based on the many different energy dissipations it can easily be deduced that only a small percentage of the power input of the stirrer is really used for the deformation and breakage of the product particles. The main percentage of the power input is transferred into heat by friction. Based on the description of the energy dissipation in a stirred media mill the energy transfer factor nE of a stirred media mill can be estimated. Using the loss factors cA to cE from the phenomena or energy dissipations (A) to (E) the part of the energy which is effective for the grinding can be determined. Thus, the energy transfer factor nE can be deﬁned as follows: nE ¼ ð1 cA Þ ð1 cB Þ ð1 cC Þ ð1 cD Þ ð1 cE Þ

(D) Grinding media contacts without stressing product particles

ð25Þ

(E) Deformation of the grinding media

(C) Displacement of the suspension during approach of two grinding media

D

Energy

(A) Energy dissipated inside the suspension

L (B) Friction at the grinding chamber wall

Fig. 17. Energy usage and energy dissipations in stirred media mills [20,28].

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283

The energy transfer factor is inﬂuenced by geometric parameters (mill geometry, type of stirrer, size) and operating parameters as well as by the product and grinding media properties. As described in Section 2.3 the product of overall number of stress events, SNtot, and the mean stress intensity, SE, corresponds to the energy which is transferred to the product particles during the process. The ratio of this energy to the stressed overall mass of the product mp corresponds to the so-called effective speciﬁc energy Em,P, which is really transferred to the product, and thus, to the product of energy transfer factor nE and the speciﬁc energy consumed by the stirrer, Em. With the derived characteristic parameter the following relation results: SN tot SN tot SE / SE GM / SN P SE GM / E m;P / nE E m mP mP

ð26Þ

Therefore, the so-called effective speciﬁc energy Em,P or the product of energy transfer factor nE and speciﬁc energy Em are constant, if the stress number SNP and the stress energy SEGM are kept constant. If SNP and SEGM are constant, according to the stress model the product quality is constant as long as the same feed material is stressed. However, in order to achieve an identical product ﬁneness it is also sufﬁcient, if beside SNP or SEGM the speciﬁc energy Em,P is constant. In case of ideal deagglomeration and cell disintegration the stress intensity and therefore the stress energy have no effect on the result as long as the stress intensity is higher than the intensity, which is necessary for breakage of the agglomerates or cell disintegration, respectively. In this case the stress number alone is sufﬁcient to describe the progress in an ideal deagglomeration or disintegration process.

3. INFLUENCE OF IMPORTANT OPERATING PARAMETERS ON THE GRINDING AND DISPERSING RESULT Based on the fundamental considerations described in Section 2, the effect of different operating parameters on the grinding and dispersing result are discussed. The stress models and particularly the stress energy and stress number are the basics for the description of the inﬂuence of the different parameters. The product quality is presented always in dependence of the stress number and the speciﬁc energy, because these parameters can describe the inﬂuence of the different parameters better than the grinding time. The grinding results shown in the following were investigated mainly at the Institute of Particle Technology at the Technical University of Braunschweig. Most of the results were obtained with a stirred media mill with disc stirrer using limestone as grinding material [2,3,18]. Although the grinding results were achieved mainly in batch grinding tests, most of the results can be transferred to a continuously operated grinding process if the effect of the residence time

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distribution (see Section 4) is considered and as long as no extreme operating conditions exist due to media packing. The results, which were achieved with other mills and other materials, will be pointed out especially.

3.1. Tip speed of the stirrer as well as size and density of the grinding media The stress intensity and stress energy in stirred media mills are determined mainly by the operating parameters; tip speed of the stirrer as well as the size, density and Young’s modulus of the grinding media as theoretically shown in Section 2. In the following regarding grinding media size always the arithmetic mean of a narrow grinding media fraction is meant. Besides the stress energy also the number of stress events depends strongly on the stirrer tip speed and the grinding media size. Therefore, these operating parameters have an essential inﬂuence on the grinding or dispersing result. As mentioned in Section 2, the inﬂuence of these parameters depends on the kind of grinding process (grinding of crystalline materials, deagglomeration and disintegration). In the following sections the different kinds of grinding processes will be distinguished.

3.1.1. Grinding of weak to medium-hard crystalline materials During stressing of a particle between two grinding media not only the particle but also the two grinding media are deformed. Because of this not all of the energy stored in the grinding media can be used for particle breakage. Part of the energy is consumed for deforming the grinding media. As shown in Section 2, this part of energy is relatively small in case of weak to medium-hard crystalline materials, and so can be neglected here. Thus, the Young’s modulus of the grinding media and of the feed material are not considered for the calculation of the stress energy. As a measure of the stress intensity the so-called stress energy SEGM can be used, if the same feed material with constant particle size distribution is ground. SI / SE GM ¼ d 3GM rGM v 2t

ð27Þ

3.1.1.1. Relation between product ﬁneness and stress number The stress number deﬁned in Section 2 is a measure for the real number of stress events in stirred media mills. The number of stress events depends besides on the grinding time on the tip speed of the stirrer as well as on the grinding media size. Since for the following considerations the solids concentration and the ﬁlling ratio of the grinding media are held constant, in the following the so-called reduced stress number SNr is used as a measure of the number of stress events.

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The reduced stress number is only a function of the grinding time, tgrind, the rotational speed of the stirrer, n, and the grinding media size, dGM. In order to get a dimensionless expression, the grinding media size is related to a characteristic particle size of the feed, x. x 2 SN / SN r ¼ n t grind ð28Þ d GM Figure 18 shows the product ﬁneness as function of the stress number for different grinding media sizes. The course of the different curves shows that the relation between product ﬁneness and stress number cannot be described by one curve in case of grinding crystalline products if the grinding media size is varied. But such a result has to be expected, because at a constant grinding time by changing the media size not only the number of stress events, but also the stress energy changes. With very small grinding media (97 and 219 mm), and thus, at very small grinding intensities nearly no progress in grinding is achieved although the stress number is very high, because the stress energy is too small to break the feed particles. Using grinding media with medium sizes (399–661 mm) and thus, medium stress intensities at the beginning, i.e. at small stress numbers, only a low progress in grinding is gained, because the stress energy is too small for a fast grinding and multiple stressing is necessary. But after the feed particles are broken the fragments are small enough to be ground further quickly at medium grinding media sizes. Using large media sizes and with it large stress

Median size x50 [μm]

100

ρGM = 2894 kg/m3 vt = 9.6 m/s ϕGM = 0.8 cm = 0.4

10

1

0.1

Grinding media diameter dGM 97 μm; 219 μm 399 μm; 515 μm 661 μm; 838 μm 1090 μm; 1500 μm 2000 μm: 4000 μm 0.1

1

10 100 Stress number SNr [-]

1000

10000

Fig. 18. Effect of grinding media size on the relation between product ﬁneness and reduced stress number SNr.

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A. Kwade and J. Schwedes 100

Particle size x [μm]

ρGM [kg/m3] dGM [μm] vt [m/s] SEGM [Nm]

10 x90

2894

2000

6.4

0.95 10-3

2894

1500

9.6

0.90 10-3

7550

1500

6.4

1.04 10-3

7550

1050

9.6

0.81 10-3

x50

ϕGM = 0.8 cm = 0.4

1

0.4 0.3

1

10 Stress number SN [-]

100

300

Fig. 19. Product ﬁneness as function of the reduced stress number at a nearly constant stress energy.

intensities, the stress energy is already at the beginning of the grinding process sufﬁcient to break the feed particles quickly. Analogous to the effect of the grinding media size, the stirrer tip speed and the grinding media density also affect the stress energy and with it the relation between product ﬁneness and stress number. Since the stress energy inﬂuences the size distribution of the fragments in case of grinding crystalline materials, only at a constant stress energy a deﬁned relationship between the product ﬁneness and the stress number can exist. For each stress energy a certain relation between the product ﬁneness and the stress number arises. Figure 19 shows the relation between the product ﬁneness (median size x50 and particle size x90) and the reduced stress number SNr at a stress energy of SEGM0.9 103 Nm. The measured characteristic particle sizes of four different grinding tests with different sizes and densities of grinding media and different tip speeds can be described by a common power function. The reduced stress number SNr, which is required to produce a median size of x50 ¼ 2 mm, is presented in Fig. 20 as function of the stress energy SEGM. The description of the measurement values by one common curve shows that at a constant ﬁlling ratio of the grinding media and a constant solids concentration for each stress energy a deﬁnite relation between the reduced stress number and the product ﬁneness exists. Thereby the required stress number increases with decreasing stress energy. At small stress energies the curve tends towards inﬁnity because nearly no grinding takes place: the stress intensities are smaller

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287

800 ρGM [kg/m3] =

700 600 500 SNr [-]

2894

7550

vt [m/s] = 6.4 vt [m/s] = 9.6 vt [m/s] = 12.8 dGM [μm] = 399 - 4000

400 300 200 100

x50 = 2 μm ϕGM = 0.8 cm = 0.4

0 0.002

0.01

0.1

1

10

20

Stress energy SEGM [10-3 Nm]

Fig. 20. Reduced stress number, which is required for a median size of 2 mm, as function of stress energy.

than the stress energy which is at least necessary to break a product particle. Against that at high stress intensities the stress number approaches a minimum value, because even at very high stress intensities each feed particle must be stressed once at least. The results shown were obtained in batch grinding experiments, in which the residence time distribution is ideally narrow. In continuous operation by a change in the parameters stirrer tip speed and grinding media size, not only the mean number of stress events, but also the residence time distribution in the mill and with it the number distribution of the stress events change. Tendency-wise the residence time distributions becomes wider with increasing stirrer tip speed and decreasing grinding media size, i.e. increasing stress number at constant grinding time (see Section 4). Thus, at continuous operation two effects (change of the mean stress number and mean stress energy as well as change of the residence time distribution) are superimposed. The residence time distribution inﬂuences above all the maximum particle size.

3.1.1.2. Relation between product ﬁneness and speciﬁc energy 3.1.1.2.1. Stirrer tip speed and grinding media density. On the basis of older investigations (among others Stehr [29] and Weit [30]) it was concluded that within the investigated parameter range the inﬂuence of stirrer tip speed and grinding media density can be described completely by the speciﬁc energy input. But a

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Median size x50 [μm]

10 - 20%

ρGM [kg/m3] = 2894 vt [m/s] = 6.4 vt [m/s] = 9.6 vt [m/s] = 12.8

7550

dGM = 1500 μm ϕGM = 0.8 cm = 0.4

1

0.4 10

100 1000 Specific energy Em [kJ/kg]

10000 20000

Fig. 21. Median size as function of the speciﬁc energy for three different stirrer tip speeds and two different grinding media densities [2,3,18].

more precise analysis of older measurement values shows that at a constant speciﬁc energy input the measured average particle sizes (mean value or median value) vary more than 25% from the ﬁtted curve. These variations are conﬁrmed by the results shown in Fig. 21, in which the median size x50 is presented as a function of the speciﬁc energy for three different stirrer tip speeds and for two media densities [2,3,18]. The size of the glass and steel grinding media is 1500 mm. The measured median sizes vary by more than 20% from the ﬁtted curve. For other bead sizes variations of more than 725% can be found. Measurement errors are one reason for these variations. Running 10 tests with identical operating parameters (dGM ¼ 1.5 mm, rGM ¼ 2894 kg m–3, vt ¼ 9.6 m s–1, cm ¼ 0.4, jGM ¼ 0.8) showed that the measuring inaccuracy amounts to less than 78%. Therefore, the different circumferential speeds of the discs and the different bead densities must be responsible for the variations exceeding 78%. Figure 21 shows that at constant speciﬁc energy inputs the product becomes coarser with increasing tip speed of the discs and increasing bead density. The reason for the inﬂuence of the stirrer tip speed and the grinding media density on the relation between the product ﬁneness and the speciﬁc energy is a change in stress energy. 3.1.1.2.2. Grinding media size. Studies published more recently (among others Joost [31], Thiel [32], Bunge [22], Mankosa et al. [33], Stadler et al. [21] and Roelofsen [34]) show that besides the speciﬁc energy input the grinding media size has a great inﬂuence on the grinding result. The speciﬁc energy consumption can be decreased considerably by accommodating the grinding media size to the grinding problem. In Fig. 22 the median size x50 is presented as a function of the

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Median size x50 [μm]

100

ρGM = 2894 kg/m3 vt = 9.6 m/s ϕGM = 0.8 cm = 0.4

10

Grinding media size dGM 219 μm 97 μm; 515 μm 399 μm; 838 μm 661 μm; 1500 μm 1090 μm; 4000 μm 2000 μm:

1

0.1 20

100

1000 Specific energy Em [kJ/kg]

6000

Fig. 22. Inﬂuence of the grinding media size on the relation between product ﬁneness and speciﬁc energy [2,3,18].

speciﬁc energy Em for different glass bead sizes [2,3,18,40]. The other operating parameters were held constant. Using glass beads with a size greater than or equal to 838 mm, smaller beads yield a ﬁner product at given speciﬁc energy inputs. For smaller glass beads (399–661 mm) the position of the curve depends on the speciﬁc energy input: for small speciﬁc energies, larger glass beads yield a ﬁner product whereas for great speciﬁc energies smaller glass beads are advantageous. For very small beads (97 and 219 mm) nearly no progression is found in the product ﬁneness. The inﬂuence of the grinding media size on the relation between product ﬁneness and speciﬁc energy is based on the inﬂuence of the grinding media size on the stress energy and the stress number: with increasing grinding media size the mass of one grinding medium and with it the stress energy increases. Simultaneously the number of grinding media and with it the stress number decreases (see Section 3). If the stress energy is large (dGM4838 mm) then a grinding time of just one minute will result in all feed particles being stressed and broken. At medium stress energy and therefore medium media sizes (399–661 mm) the stress energy is not sufﬁcient for a fast grinding and multiple stressing is necessary. Thus, it has to be ground for some time and a certain amount of energy has to be consumed before all feed particles are broken. After all feed particles are broken, the stress energy is sufﬁcient for breakage of the product particles, so that an increase in the stress number and therefore a decrease in the grinding media size results in a larger grinding progress than an increase in stress energy.

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Median size x50 [μm]

60 ρGM [kg/m3] = 2894 vt [m/s] = 6.4 vt [m/s] = 9.6 vt [m/s] = 12.8

7550

10 Em = 1000 kJ/kg ϕGM = 0.8 cm = 0.4

1 0.5 80 100

500 1000 Grinding media size dGM [μm]

5000

Fig. 23. Inﬂuence of grinding media density and stirrer tip speed on the relation between median size and grinding media size (Em ¼ 1000 kJ kg–1).

If very small grinding media are used the stress energy is so small that practically no grinding takes place. An improvement of the grinding result is only possible by increasing the stress energy. The inﬂuence of the operating parameters grinding media size, grinding media density and stirrer tip speed is shown comprehensive in Fig. 23, in which the relation between the median size obtained at a speciﬁc energy input of 1000 kJ kg–1 and the grinding media size is presented: the six curves plotted for different stirrer tip speeds and different densities of the grinding media have a characteristic shape. With increasing grinding media size the median size ﬁrst declines down to a minimum. At this minimum the corresponding grinding media size is most advantageous for a speciﬁc energy input of 1000 kJ kg–1. For grinding media sizes greater than the optimum size the median size increases with increasing grinding media size. Comparing the six different curves, it can be seen that for increasing tip speed of the stirrer and increasing grinding media density the optimum grinding media size decreases. Moreover, at grinding media sizes which are larger than the optimum grinding media size the product ﬁneness increases with decreasing stirrer tip speed and decreasing grinding media density.

3.1.1.3. Stress energy The stress energy of the grinding media, SEGM, describes the effect of the three parameters stirrer tip speed, grinding media density and grinding media size in combined form. Therefore, the speciﬁc energy and the stress energy are the two

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most important inﬂuencing variables on the grinding of crystalline materials (e.g. limestone) in stirred media mills: for each stress energy, a deﬁned relation between the product ﬁneness and the speciﬁc energy exists. As an example the relation between speciﬁc energy and median size x50 as well as characteristic particle size x90 is presented in Fig. 24 for a stress energy of approximately 0.9 103 Nm. If the speciﬁc energies required to produce a median size of x50 ¼ 2 mm are depicted as function of the stress energy, the relation shown in Fig. 25 results. At small stress intensities high speciﬁc energies are necessary to produce a median size of 2 mm, because the stress intensities are too small for an effective grinding

Fig. 24. Relation between product ﬁneness and speciﬁc energy at a stress energy of approximately 0.9 103 Nm.

Fig. 25. Speciﬁc energy required to product a median size of 2 mm as function of stress energy.

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and therefore a large number of stress events are necessary. Therefore, in this range of stress energy the speciﬁc energy required for a certain product ﬁneness decreases with increasing stress energy. If the stress energy is so high, that in average the feed particles can be broken, the speciﬁc energy reaches a minimum value. If the stress energy is increased further, the energy utilisation of each stress event decreases, so that with increasing stress energy more speciﬁc energy is required to produce a median size of 2 mm. Besides the speciﬁc energy at a constant product ﬁneness also the product ﬁneness at a constant speciﬁc energy input can be presented as a function of the stress energy. In Fig. 26 the median size and characteristic particle size x90 produced at a speciﬁc energy input of 1000 kJ kg–1 are depicted as function of the stress energy. The measurement values show that at a constant speciﬁc energy a certain product ﬁneness (median and maximum particle size) is produced with a certain stress energy SEGM. It has to be considered that the grinding results were investigated in batch grinding tests, in which the maximum particle size is not inﬂuenced by the residence time distribution. In Fig. 26 the measurement values, which form different curves in Fig. 23, can be described by one ﬁtted curve. The effect of the stress energy on the grinding result can be explained using results found for single particle stressing. Starting from very small stress intensities, at which even after a very high number of stress events practically no grinding progress is gained, with increasing stress energy the probability of fracturing, and therefore the energy utilisation (deﬁned as the produced surface area related to the introduced energy) increases up to a maximum value, so that particles can be ground in case of a sufﬁcient number of stress events. With increasing stress energy the number of stress events required to break a particle decreases if the

Fig. 26. Inﬂuence of the stress energy on the product ﬁneness at a speciﬁc energy of 1000 kJ kg–1.

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speciﬁc energy remains constant. If the stress energy is that high, that almost each stress event results in a complete particle breakage, the ﬁnest product is obtained for a given speciﬁc energy. According to Priemer [35] and Schubert [36] for single particle stressing, the maximum energy utilisation and thus, the optimum stress energy can be found at fracture probabilities of 100%. If the stress energy is increased further the stress energy becomes larger than the optimum stress energy and the energy utilisation of each stress decreases because of increasing energy losses. Since the speciﬁc energy can be described by the product of stress energy and stress number, the number of stress events decrease with increasing stress energy at a constant speciﬁc energy input. Because of the decrease in energy utilisation, with increasing stress energy increasing the number of stress events is more advantageous than increasing the stress energy. For example, the new surface achieved by stressing a particle volume only once with a certain stress energy is smaller than the new surface created by stressing this particle volume twice with half stress energy. For different speciﬁc energy inputs different relationships between the stress energy SEGM and the product ﬁneness and thus, different optimum values of the stress energy exist. In Fig. 27 the median size is shown as a function of the stress energy for six different speciﬁc energies. With increasing speciﬁc energy and, therefore, increasing product ﬁneness the optimum stress energy decreases because with decreasing particle size lower energies and lower forces of pressure are necessary to break a product particle. Therefore, the optimum and most advantageous stress energy always depends on the demanded product ﬁneness.

Fig. 27. Relation between the product ﬁneness, the stress energy and the speciﬁc energy.

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3.1.2. Grinding of crystalline materials with high modulus of elasticity If grinding materials with a high modulus of elasticity and thus, usually hard grinding materials (e.g. ceramic materials), the deformation of the grinding media during the stress event and, therefore, the modulus of elasticity cannot be neglected anymore. Additionally the grinding media wear is usually very high if such materials (e.g. white fused alumina) are ground. Thus, in a batch or charge process the concentration of the rubbed-off fragments increases with increasing grinding time, so that the average mass of the grinding media wear (average mass regarding grinding time) should be taken into account for the calculation of the mass of product particles, to which the energy input is related. The speciﬁc energy calculated with this product mass was named Em,W (W for considering the grinding media wear, see equation (7) by Becker [27]). In Fig. 28 the relation between the median size and the speciﬁc energy is shown for two different grinding media materials with different modulus of elasticity and for two different grinding media sizes (0.5 and 0.8 mm). The two grinding media materials differ not only in the modulus of elasticity, but also in further properties, particularly in the density. A change of the modulus of elasticity without changing other properties is not possible. The stirrer tip speed was identical in all grinding tests (vt ¼ 6 m s–1). At this stirrer tip speed the grinding media with sizes of 0.5 and 0.8 mm are too small to break a feed particle out of white fused alumina at a single stress event. For an effective grinding the particles have to be

Fig. 28. Inﬂuence of the modulus of elasticity of the grinding media on the relation between the product ﬁneness and the speciﬁc energy at the grinding of white fused alumina [27].

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stressed several times. Thus, the curves cannot be described by a straight line in the log–log scale at small and medium speciﬁc energies. The measurement values in Fig. 28 show that the grinding media with a Young’s modulus of 100 GPa need much more speciﬁc energy for a certain product ﬁneness at a given grinding media size than the grinding media with a modulus of elasticity of 265 GPa. For example the yttrium-stabilised zirconium oxide grinding media with a diameter of 0.5 mm produce a median size below 1 mm at a speciﬁc energy of 4000 kJ kg–1, whereas at the same speciﬁc energy the grinding media with the same size but out of mixed oxide produce only a median size of 25 mm. The reason for the different grinding results using grinding media with different moduli of elasticity is based on the fact that the energy which is transferred from the grinding media to the product particle depends on the modulus of elasticity: for the breakage of the feed particles enough energy has to be transferred from the grinding media to the feed particles. If the energy is too small (e.g. to obtain a particle breakage at most of the stress events), the particles must be stressed several times. Under the assumption of an elastic deformation in the feed particle and the grinding media the kinetic energy of the grinding media is transferred into the energies of the elastic state of stress (of the feed particle and the grinding media). Only the energy which is transferred to the feed particle can be used for the grinding of the feed particles. With increasing modulus of elasticity of the grinding media material the portion of energy which is transferred to the feed particles increases. Therefore, with increasing modulus of elasticity of the grinding media also the length of the incipient cracks increases, which are produced in the feed particles at each stress event. Thus, less stress events are necessary to break the feed particles. The number of stress events in the grinding chamber is independent on the modulus of elasticity of the grinding media. Because of their higher modulus of elasticity the grinding media out of yttrium-stabilised zirconium oxide need less stress events and thus, less speciﬁc energy for an effective grinding than the grinding media out of mixed oxide (see Fig. 28). Besides the modulus of elasticity the density of the grinding media inﬂuences the relation between speciﬁc energy and product ﬁneness. However, investigations regarding the effect of stress energy on the grinding result show that the stress energy changes by a factor of 1.6 due to the change in grinding media density and by a factor of 2 due to the change in Young’s modulus. As shown in Section 3 for the case of grinding white fused alumina (see Fig. 16), possibly only 20% of the energy made available by the grinding media is transferred to the caught particle. The remaining energy is consumed for the deformation of the grinding media. Therefore, the reduction of the stress energy due to the deformation of the grinding media has to be considered for the deﬁnition of the stress energy. As a measure for the effective stress energy, the stress energy transferred to the product particles, SEP, should be used (see

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Section 2.3.3 equation (23)):

Y P 1 SE / SE P ¼ d 3GM rGM v 2t 1 þ Y GM

ð29Þ

Since a great portion of the energy which is transferred from the stirrer to the grinding media is not consumed by stressing the product particles, but by the deformation of the grinding media during the stress events, the question arises, whether the energy consumption of the stirrer is still a good measure for the speciﬁc energy consumed for stressing the particles. If the whole energy spent for the deformation of the grinding media is transformed into heat, only the portion of the speciﬁc energy which is transferred to the feed and product particles would be of interest. Thus, in this case only this portion of the speciﬁc energy should be used for the correlation with the product ﬁneness. If against that the elastically stored energy is transformed fully into kinetic energy, the energy required for the deformation of the grinding media is not lost and can again be used at the next stress event. In this case the whole energy transferred from the stirrer to the grinding media is still a measure for the progress in grinding. According to investigations of Becker et al. [27,37] the reality lies between these two extremes, whereas according to Becker for the case of grinding white fused alumina approximately only 10% of the energy which is elastically stored in the grinding media is transferred into kinetic energy and approximately 90% of the elastically stored energy is dissipated into heat. Therefore, the following expression for the speciﬁc energy should be used: ! Y P 1 E m;W;grind ¼ j þ ð1 jÞ 1 þ ð30Þ E m;W Y GM As described above according to Becker et al. [27,37] for the grinding of white fused alumina the weighing factor j is about 0.1. If the stress energy transferred to the product particles, SEP, and the weighted speciﬁc energy Em,W,grind (equation (30)) is used for the correlation of the product ﬁneness with the speciﬁc energy and the stress number, in principal the same relations result as shown for the grinding of weak and medium-hard crystalline materials (e.g. limestone). As an example in Fig. 29 the product ﬁneness is presented as function of the stress energy SEP for a speciﬁc energy Em,W,grind of 2000 kJ kg–1. During the grinding test stirrer tip speeds of vt ¼ 6–14 m s–1, grinding media densities of rMK ¼ 2670–15,000 kg m–3 and grinding media sizes of dMK ¼ 0.35–3 mm were realised and the modulus of elasticity was varied from YGM ¼ 100 to 625 GPa.

3.1.3. Deagglomeration and cell disintegration In case of deagglomeration and cell disintegration the objective of the grinding process is to break off an agglomerate or to disintegrate a cell. If an agglomerate

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Fig. 29. Effect of the stress energy SEP on the product ﬁneness for the grinding of white fused alumina (constant speciﬁc energy Em,W,grind of 2000 kJ kg–1) [27,37].

is destroyed or a cell is disintegrated, further grinding does not cause further improvement of the product quality. While in case of cell integration at a stress event the cell wall remains either intact or is destroyed and thus, only two conditions (disintegrated or not disintegrated) are possible, in case of deagglomeration usually other effects are possible. Besides that the agglomerate remains in its original form or that the agglomerate is decomposed completely in the individual primary particles (called ideal deagglomeration), an agglomerate composed out of several primary particles can be decomposed in a few smaller agglomerates and possibly in primary particles, i.e. a partial success can be obtained. In this case more than one stress event is necessary until the agglomerate is completely decomposed into the primary particles. Moreover the primary particles can be ground further and thus, contribute to a further increase in product ﬁneness. In case of cell integration the so-called disintegration rate can be measured directly by determining the concentration of the cell content substances in the solution by measuring the concentration of proteins [22], the activity of the enzymes [22] or the oxygen consumption rate [38]. Against that the rate of deagglomeration can only determined indirectly by the particle size distribution or further product qualities (e.g. intensity of colour, gloss). As shown in Section 2 for ideal deagglomeration and cell disintegration it can be assumed that no further increase in product quality is obtained using stress intensities larger than the stress energy at which the agglomerate can be just

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decomposed or the cell can be just disintegrated. Thus, in this range of stress energy the product quality should only be a function of the stress number and not a function of the stress energy, i.e. the relation between the product quality (deagglomeration rate or disintegration rate) and the stress number should be independent on the stress energy. Since the product of stress number and stress energy is a measure for the speciﬁc energy and since each product quality belongs to a certain stress number, in this range of the stress energy the speciﬁc energy consumption required to produce a certain product quality should be proportional to the stress energy. This means for example that if the stress energy is twice as large as the optimum stress energy for the deagglomeration or disintegration process, the speciﬁc energy requirement is also twice as large as the minimum speciﬁc energy requirement. To what extent these basic considerations are valid is discussed in the following section for the disintegration of yeast cells. In case of cell disintegration the grinding task usually is to stress the cell between two grinding media by shearing. In this case the probability that an agglomerate or a cell is stressed is no longer proportional to the active volume between two grinding media, but to the surface of the grinding media. Therefore, as shown in Section 3 the following proportionality results for the stress number: SN /

nt d GM

ð31Þ

In Fig. 30 the disintegration rate is presented as a function of the stress number. The disintegration rate describes the amount of disintegrated and thus, 100

disintegration rate / %

80

60 dGM = 0.5 - 2 mm

40

CBFM = 0.1 kg/l

20

0

5

101

2

Vt

= 8 m / s-1

jGM

= 80 %

dP

= 5 mm

5 102 2 5 stress number SN ~ 1/dGM

103

2

5

Fig. 30. Relation between the disintegration rate and the stress number for high stress intensities (disintegration of yeast cells) [22].

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destroyed cells in relation to the overall amount of cells. The operating parameters (among others grinding media size and stirrer tip speed) are chosen in a way that the stress energy is high enough to destroy the cell walls in the active zones between the grinding media. Under this presumption the measured disintegration rates can be described in a ﬁrst approximation by one ﬁtted curve. If the operating parameters are chosen in such a way that the stress energy in the active zones is often not high enough for a disintegration of the cells, the measurement values lie below the ﬁtted curve shown in Fig. 30. If the disintegration rate is presented as a function of the speciﬁc energy, for each combination of operating parameters a different curve exists. This is shown in Fig. 31 for three different combinations of operating parameters. The curves are shifted to greater speciﬁc energies or smaller disintegration rates, respectively, if the stirrer tip speed and with it the stress energy is increased. If the relation between disintegration rate and speciﬁc energy is presented for different grinding media sizes, likewise different relations exist for different grinding media sizes. Thereby at a constant speciﬁc energy at ﬁrst the disintegration rate increases with increasing grinding media size and with it increasing stress energy until an optimum curve is reached. At the optimum point for a given speciﬁc energy the maximum disintegration rate is produced. If the grinding media size and with it the stress energy is increased further, the disintegration rate decreases again. This consideration is shown in Fig. 32, in which the disintegration rate is presented for a speciﬁc energy of 300 kJ kg–1 and a stirrer tip speed of 4 m s–1. Moreover, the ﬁgure shows that the solids concentration also inﬂuences the relation between the disintegration rate and the speciﬁc energy. This effect will be discussed further in Section 3.3. 100

disintegration rate / %

dGM = 2 mm 80

CBFM=0.1kg/l

60

40 Vt = 2 m/s-1 Vt = 4 m/s-1 Vt = 8 m/s-1

20

0 101

2

5

102

2

5

103

2

5

104

2

5

105

-1

specific energy / kJ/kg

Fig. 31. Inﬂuence of the stirrer tip speed on the relation between disintegration rate and speciﬁc energy (disintegration of yeast cells) [22].

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disintegration rate /%

100 CBFM = 0.1 kg/l CBFM = 0.6 kg/l Vu = 4 m/s ϕGM = 80 % E = 300 kJ-kg-1

80

60

40

20

0 0.1

0.2

0.5 1 2 grinding bead size dGM / mm

5

10

Fig. 32. Disintegration rate as function of the grinding media size at a constant speciﬁc energy input [22].

Fig. 33. Speciﬁc energy required for a disintegration rate of 60% as function of the stress energy.

If the speciﬁc energy required for a disintegration rate of 60% is depicted as a function of the stress energy of the grinding media, the relation presented in Fig. 33 is obtained. At small stress intensities a lot of speciﬁc energy is needed because only at a few stress events is the stress energy sufﬁcient for cell disintegration. At medium stress intensities the stress energy in the active zone between two grinding media is just sufﬁcient to disintegrate the cells at nearly every stress event. If the stress energy is increased further, a stress energy which is much higher than the stress energy required for destroying the cell walls

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acts on the cells, so that the speciﬁc energy consumption increases strongly with increasing stress energy. As discussed above the speciﬁc energy increases approximately proportional to the stress energy because in this range of stress energy the disintegration rate is only a function of the stress number. Therefore, the slope of the curve amounts to one in log–log scale.

3.1.4. Conclusions from the in£uence of stress number and stress energy The inﬂuence of stirrer tip speed as well as size and material of the grinding media on the result of different grinding processes can be described well by two of the three parameters; stress number, stress intensity (stress energy) and speciﬁc energy. The relationship between product quality (ﬁneness or disintegration rate), stress intensity and stress number or speciﬁc energy depends on the breakage behaviour of the material. Vice versa if the relationship among product quality, stress intensity and stress number or speciﬁc energy is known, statements regarding the breakage characteristics and grinding behaviour of the material can be found. This will be demonstrated by comparing grinding results of the following seven different materials: pigments (results published by Stadler et al. [21]), yeast cells [22], synthetically produced SiO2-aggregates (median size of feed particles about 22 mm), water basis ink (results published by Vock [39]), printing ink, limestone [18] and fused alumina (median size of feed particles about 33 mm [27]). The grinding behaviour of the seven materials can be compared by looking at the inﬂuence of the stress energy on the speciﬁc energy required for a certain product quality. Thus, in Fig. 34 the ratio of the speciﬁc energy required for a certain product quality to the minimum speciﬁc energy required for the same product quality is presented as function of the stress energy related to the optimum stress energy. The stress energy ratio is a measure of the stress intensity. In Fig. 34 only results of measurements are shown, at which the stress intensity or stress energy, respectively, is approximately equal to or higher than the optimum stress intensity. The measurement values for the different materials can be described by different approximation curves, so that the speciﬁc energy required for a certain product quality depends more or less strongly on the stress intensity. The strongest inﬂuence of the stress intensity on the speciﬁc energy exists for the ideal deagglomeration and the disintegration processes, where the measurement values can be described in a ﬁrst approximation by a straight line with a slope of nearly one. Therefore, above all for a deagglomeration and a disintegration process it is advisable that the stress intensity lies in the optimum range. For these two materials the result of a single stress event is independent of the stress intensity as long as the stress intensity is higher than the optimum stress intensity. Since at a constant product quality the stress number stays constant with increasing stress intensity, the speciﬁc energy consumption increases proportional to the stress intensity.

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Em / Em,min [-]

100 Pigments[1] x50 Yeast cells A SiO2 aggregates x50 Water basis ink [17] CI Printing ink T Limestone x50 Al2O3 x50

10

= 1 μm = 60 % = 2 μm = 140 % = 80 % = 2 μm = 2 μm

1 0.8 0.5

1

10 SE / SEopt [-]

100

1000

Fig. 34. Speciﬁc energy related to the minimum speciﬁc energy as function of the stress energy related to the optimum stress energy (constant product ﬁneness or constant disintegration rate) [40].

In case of the synthetically produced SiO2-aggregates the slope of the straight line is slightly smaller than one, so that the stress intensity has already a slight effect on the result of a stress event. Therefore, with increasing stress intensity an aggregate or agglomerate is decomposed into smaller agglomerates or primary particles. The smallest slopes can be found for grinding limestone and fused alumina. In case of these two materials at a stress event ﬁner fragments are produced with increasing stress intensity. This effect is somewhat more distinct for fused alumina than for limestone, but the slope of the approximation curve is clearly greater than 0. Therefore, the effect of the stress intensity on the product ﬁneness of one stress event is higher for fused alumina than for limestone. The relationship between the ratio of speciﬁc energy and minimum speciﬁc energy and the ratio of stress intensity and optimum stress intensity can be derived from equations shown in Section 2.2.2, especially equation (12), if the product quality and thus, the produced speciﬁc surface is set constant. At a constant speciﬁc surface the ratio EU/EUmax corresponds to Em,min/Em, so that the following relation can be given for SE4SEopt: Em SE 1a / ð32Þ SE opt E m;min with a ¼ 0 for deagglomeration/disintegration and 0oao1 for grinding crystalline materials.

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Table 1. Values of the exponent a Material

Pigments

Yeast cells

Synthetic SiO2

Water basis ink

Printing ink Limestone

Fused alumina

1a (–) a (–)

E1 0

1 0

0.77 0.23

E 0.4 0.60

0.37 0.63

0.26 0.74

0.33 0.67

From Fig. 34 the values of the exponent ‘‘a’’ can be found for the seven investigated materials. This exponent ‘‘a’’ determines the slope of the curves shown in Section 2.2.2, Fig. 13. The values are given in Table 1. It can be seen that the value of the exponent ‘‘a’’ increases with increasing grinding resistance of the material. Summing up the results in Fig. 34 and Table 1 it can be seen that depending on the feed material the speciﬁc energy required for a certain product quality depends more or less on the stress energy. For pure deagglomeration and disintegration the strongest inﬂuence of the stress intensity on the speciﬁc energy exists. In case of deagglomeration and disintegration attention should be paid to an optimum setting of the stress energy because otherwise the speciﬁc energy requirement becomes needlessly high and thus, at constant power input the production capacity needlessly low. Looking at the effect of stress number and stress energy it should be considered that in continuous operation, above all in one passage mode, this effect is superimposed by an inﬂuence of the residence time distribution. The residence time distribution and thus the number distribution of the stress events determine essentially the maximum particle size of the product. Against that the average particle size is inﬂuenced by the residence time distribution only slightly. The inﬂuence of the different operating parameters on the residence time distribution is discussed in Section 4. The basic idea that, at a sufﬁcient stress energy (probability of breakage equal to 100%) an improvement in the product quality is obtained through an increase in the number of stress events, is possibly not valid when the product quality is not determined only by the particle size distribution. Possibly in such a case by a further increase in the stress energy an increase in the product quality can be obtained, which is not possible with an increase in the stress number alone.

3.1.5. Determination of optimum operating parameters In order to achieve a maximum production capacity according to Section 2.1 the power input of the mill should be as high as possible and the speciﬁc energy requirement should be as low as possible. The power input is as high as possible if the installed motor power is fully used. Against that the speciﬁc energy is as low as possible if the stress energy is set at an optimum value. In order to achieve a

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production capacity which is as high as possible, the following procedure can be followed: 1. A grinding material has to be chosen which is most favourable with respect to wear and cost of the grinding media itself as well as to wear of the mill. ) Density rGM and Young modulus YGM of the grinding media. 2. The optimum stress energy has to be determined by a few grinding tests. ) Optimum stress energy SEGM or SEP, respectively. 3. The stirrer tip speed has to determined in a way that the power input is as high as possible for the grinding media size chosen. Therefore it must be ensured that the product temperature is not higher than the maximum product temperature and that the mechanical load of the machinery and the grinding media does not become too high. ) Stirrer tip speed vt 4. Using the optimum stress energy and the maximum stirrer tip speed determined in step 3 the optimum grinding media size can be determined: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ SE 3 d GM ¼ vt ¼ p dd n ð33Þ v 2t rGM If the grinding media size cannot be chosen as necessary according to equation (33) (e.g. because the separation device is not sufﬁcient or because grinding media of the required size are not available) the smallest grinding media size that is available should be employed. 5. In an iteration step it has to be proved, if at the grinding media size obtained in step 4 and the stirrer tip speed chosen in step 3 the power input is still as high as possible. If this is not true, steps 3–5 have to be repeated until the power input is as high as possible. 6. It has to be noted that besides a maximum power input and optimum stress energy further conditions must have also be considered. On one hand it should be veriﬁed that the operating parameters chosen are acceptable regarding wear and grinding media compression (see Section 5.2). If the product quality is determined by the maximum product particle size, the effect of the operating parameters on the residence time distribution has to be taken into account (see Section 3).

3.2. Filling ratio of grinding media Already Engels [4] and Bosse [5] stated that the ﬁlling ratio of the grinding media has a great inﬂuence on the grinding result. With increasing ﬁlling ratio of the grinding media the number of media contacts increases and the distance between the individual grinding media decreases. According to Bosse and Engels

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especially the reduction of the media distances causes an improvement of the grinding result. But above a certain ﬁlling ratio of grinding media the grinding result gets worse because of too small distances between the grinding media and with it too low freedom of motion, so that for every grinding task and every mill (type and size) an optimum ﬁlling ratio exists. For a long time the inﬂuence of the ﬁlling ratio of the grinding media on the speciﬁc energy consumption was nearly not investigated systematically. In an investigation of the cell disintegration in stirred media mills Bunge [22] found that an increase of the ﬁlling ratio in the range of jGM40.4 causes an approximately proportional increase in the disintegration rate. Kwade [18] varied the ﬁlling ratio in a wide range from 0.3 to 0.85. At higher ﬁlling ratios the motion of the grinding media is limited too much, so that ﬁlling ratios greater than 0.85 are not convenient anymore. If the product ﬁneness is presented as function of speciﬁc energy, the curves shown in Fig. 35 result. Fig. 35 shows that for each ﬁlling ratio a distinct relation between the product ﬁneness and the speciﬁc energy exists and that at a given speciﬁc energy with increasing ﬁlling ratio of the grinding media a ﬁner product and with it a smaller median size x50 is achieved. Only the curve for the ﬁlling ratio of 0.85 lies slightly higher than the one for the ﬁlling ratio of 0.8 because at a ﬁlling ratio of 0.85 the freedom of motion is already too small. Therefore, at a ﬁlling ratio of approximately 0.8 the speciﬁc energy to produce a given product ﬁneness is the lowest. 50

Median size x50 [μm]

ϕGM = 0.30 ϕGM = 0.50 ϕGM = 0.70 ϕGM = 0.80

ϕGM = 0.40 ϕGM = 0.60 ϕGM = 0.75 ϕGM = 0.85

10

dGM = 1090 μm vt = 9.6 m/s cm = 0.4

1 0.5 6

10

100

1000

6000

Specific energy Em [kJ/kg]

Fig. 35. Effect of the ﬁlling ratio of grinding media on the relation between product ﬁneness and speciﬁc energy [18].

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The inﬂuence of the ﬁlling ratio of the grinding media on the speciﬁc energy consumption can be explained as follows: the overall energy input into the grinding chamber is transferred from the stirrer on the suspension and the grinding media. The greater the amount of grinding media in the grinding chamber and thus, the greater the ﬁlling ratio, the greater the proportion of energy transferred to the grinding media and the more energy can be used for grinding. Therefore, the product out of speciﬁc energy and ﬁlling ratio of the grinding media is a measure for the effective speciﬁc energy. Thus, the ﬁlling ratio is a measure for the energy transfer factor (see Section 2.3, mechanism A). Figure 36 shows that the measured values can be described by a ﬁtted curve over a wide parameter range if the median size is presented as function of the product of speciﬁc energy Em and ﬁlling ratio jGM. This conﬁrms that the proportion of the speciﬁc energy used for grinding is proportional to the ﬁlling ratio of grinding media. From the above-mentioned considerations it follows that the speciﬁc energy is only a measure for the product of stress number and stress energy if the speciﬁc energy is multiplied by the ﬁlling ratio. The effect of a higher ﬁlling ratio cannot be described by a higher power input. But against the product of stress number and stress energy the product of ﬁlling ratio and speciﬁc energy can take into account the change of the motion of the grinding media by the power input. By the increase of the velocities of the grinding media, and with it the number and intensity of the grinding media contacts, the power input increases. The stress number and stress energy do not account for this effect, at least not completely.

Median size x50 [μm]

50 ϕGM = 0.30 ϕGM = 0.50 ϕGM = 0.70 ϕGM = 0.80

ϕGM = 0.40 ϕGM = 0.60 ϕGM = 0.75 ϕGM = 0.85

10

dGM = 1090 μm vt = 9.6 m/s cm = 0.4

1

2

10

100 ϕGM • Em [kJ/kg]

1000

8000

Fig. 36. Relation between the product ﬁneness and the product of speciﬁc energy and ﬁlling ratio of grinding media [18].

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3.3. Solids concentration and ﬂow behaviour of the suspension The solids concentration determines, among others, how many particles are located in a certain volume. If single particle stressing is assumed for wet grinding in stirred media mills, the number of stress events per particle and time decreases with increasing solids concentration and with it increasing number of feed and/or product particles. On the other hand the probability that one or more particles are captured between two grinding media increases with increasing solids concentration. Moreover, at very high solids concentrations the stressing of a particle bed cannot be excluded. In investigations performed by Kwade [18] the solids mass concentration, cm, was varied from 0.1 to 0.5 and with it the solids volume concentration, cV, from 0.04 to 0.27. No dispersing agents or other additives were used. If the median size is presented as function of the speciﬁc energy for different solids concentrations, the relations shown in Fig. 37 result. The measurement values for a solids concentration of cm ¼ 0.1 or cV ¼ 0.04, respectively, lie clearly above the other measurement values. At this low solids concentration the supply of particles in the active zones between two grinding media is not sufﬁcient. The probability that a particle is stressed intensively between two grinding media is smaller than at a higher solids concentration. Thus, the number of grinding media contacts increases at which no particles are stressed.

Median size x50 [μm]

10 dGM = 1090 µm ρGM = 2894 kg/m3 vt = 9.6 m/s ϕGM = 0.8

1 cm = 0.10; cv = 0.04 cm = 0.20; cv = 0.08 cm = 0.30; cv = 0.15 cm = 0.40; cv = 0.20 cm = 0.50; cv = 0.27 0.2 20

100

1000 Specific energy Em [kJ/kg]

10000 20000

Fig. 37. Relationship between product ﬁneness and speciﬁc energy at different solids concentrations of the product suspension [18].

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If the solids mass concentration is varied in the range from 0.1 to 0.5 and if only median sizes greater than 2 mm are considered, the result of Stehr [29] and Weit [30] that the inﬂuence of the solids concentration on the grinding result can be described in a ﬁrst approximation by the speciﬁc energy is conﬁrmed. In the range of greater speciﬁc energies and thus, smaller median sizes the curves differ. Firstly it seems that at a given speciﬁc energy a higher product ﬁneness is obtained with a greater solids concentration. But below a median size of 1 mm at high solids concentrations the median size declines only slightly, so that the curves of the high solids concentrations cut the curves of the low solids concentrations. This behaviour is due to the increasing viscosity of the product suspension: the higher the solids concentration at a given product ﬁneness, the greater the viscosity of the product suspension. If the speciﬁc energy required for a certain product ﬁneness is presented as function of the solids mass concentration, Fig. 38 results. It can be seen clearly that the minimum values of the different curves lie at different solids concentrations. While at a median size of 3 mm the minimum can be found above a solids concentration of 0.5, at a median size of 1.5 mm the minimum lies at a solid mass concentration of approximately 0.5 and at a median size of 0.6 mm at a solids mass concentration of only 0.3. According to the aspired product ﬁneness the speciﬁc energy consumption has a minimum value at different solids concentrations. The decrease of the optimum solids concentration with increasing product ﬁneness is caused by the increase in the suspension viscosity. This consideration

20000 dGM = 1090 μm

x50

Specific energy Em [kJ/kg]

10000

0.6 μm

ρGM = 2894 kg/m3

0.8 μm

vt = 9.6 m/s ϕGM = 0.8

1.0 μm

1.5 μm

1000

2.0 μm 3.0 μm 100 0.0

0.1

0.2 0.3 0.4 Solids mass concentration cm [-]

0.5

0.6

Fig. 38. Speciﬁc energy as function of the solids concentration for the production of different median sizes [18].

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25 cm = 0.10; cv = 0.04 cm = 0.20; cv = 0.08 cm = 0.30; cv = 0.15 cm = 0.40; cv = 0.20 cm = 0.50; cv = 0.27

Torque T [Nm]

20

dGM = 1090 μm

15

ρGM = 2894 kg/m3 vt = 9.6 m/s ϕGM = 0.8

10

5

0 0.4

1

10 Median size x50 [μm]

Fig. 39. Torque of the stirrer as function of the median size for different solids concentrations.

can be seen clearly from Fig. 39, in which the torque of the grinding tests discussed above is presented as a function of the median size. Tendency-wise it is valid that the torque of the mill and with it the viscosity of the suspension increase with increasing solids concentration and decreasing median size. An increase because of the decreasing median size can be found above all at solids mass concentrations of 0.4 and 0.5. If the median size at which the torque rises strongly is determined for a certain solids mass concentration, it can be shown that this median size corresponds approximately to the median size, at which the solids mass concentration under consideration is the best regarding the speciﬁc energy requirement (see Fig. 38). Therefore, the speciﬁc energy requirement becomes unfavourable when the viscosity of the suspension starts to rise strongly. Above all the reason for this behaviour is that at too high a viscosity too much energy is dissipated by the ﬂuid displacement of two approaching grinding media (loss factor see Section 2.3, mechanism B) and the stress energy and the energy transfer factor decrease. Therefore, at high suspension viscosities the stress energy depends on the solids concentration of the product suspension. Moreover, the number of media contacts and with it the stress number decreases. Thus, at high solids concentrations the use of heavier grinding media is advantageous because of the greater inertia forces. In order to be able to grind effectively at higher solids concentrations and/or at a higher product ﬁneness, it is convenient to decrease the suspension viscosity

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Median size x50 [μm]

Dispex 0.1 % Dispex 0.4 % Dispex 0.7 %

dGM = 1090 μm ϕGM = 0.8 vt = 9.6 m/s cm = 0.6

1

0.5 30

100

1000

2000

Specific energy Em [kJ/kg]

Fig. 40. Median size as function of the speciﬁc energy for different concentrations of the dispersing agent.

by using dispersing agents (additives) [23,41]. In Fig. 40 this effect is shown for the grinding of limestone at a solids concentration of 0.6. At the batch grinding tests three different concentrations of a dispersing agent were used. The measurement values show that at median size greater than 2 mm the same median size is produced independent of the concentration of the dispersing agent. At smaller median sizes at ﬁrst the measurement values for an agent concentration of 0.1%, and at even smaller median sizes the measurement values for an agent concentration of 0.4%, deviate from the straight line. The deviations from the straight line correspond to a strong increase in the torque and with it a strong increase in the suspension viscosity. Therefore, depending on the solids concentration and the demanded product ﬁneness a concentration of the dispersing agent exists at which the speciﬁc energy consumption is the lowest. Besides changing the interactions between the particles by adding dispersing agents the grinding efﬁciency is also affected by the nature of the ﬂuid phase. For example solvents can be used to improve the interactions between the particles. Frequently, particularly in the paint and ink industry, different ﬂuid components like solvents, varnishes, oil, vaselines and others are used to obtain certain characteristics of the end product. Usually the ﬂuid components have a great inﬂuence on the grinding efﬁciency: as a rule with increasing viscosity of the ﬂuid component the efﬁciency of a real grinding process decreases because a greater portion of the speciﬁc energy is consumed by the ﬂuid friction. Moreover, a larger portion of the kinetic energy is lost due to the displacement of the ﬂuid between two grinding media (see also Section 2.3).

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10

x50 [ μm ]

Water (1 mPas) Water with Alkylen oxide (100 mPas)

VGC = 5.54 l dGM = 1090 μm ϕGM = 0.8 vt = 9.6 m/s cm = 0.5

1

0.5 20

100

1000 Specific energy Em [kJ/kg]

5000

Fig. 41. Median size as function of the speciﬁc energy for two different ﬂuid phases.

This effect is shown in Fig. 41 for the grinding of limestone with two different ﬂuid phases (water and a mixture of water and Alcylenoxide). The two curves demonstrate clearly that with a viscosity of the ﬂuid phase of 100 mPa s two to three times the speciﬁc energy is required than with water. In this case an improvement should be obtained by using heavier grinding media, because they have a greater kinetic energy at the same size and velocity because of larger inertia forces, but the same resistance forces or ﬂuid forces act on the grinding media through the suspension. Besides reducing the suspension viscosity, additives are used to avoid reagglomeration of the particles after the grinding process and to produce a stable suspension, particularly at the production of submicron particles.

3.4. Construction and size of the stirred media mill In principle construction and size of a stirred media mill inﬂuence the number of stress events in a certain volume and the intensity of those stress events. Thereby not only are the average number of stress events and the average stress energy affected, but particularly the distribution of the stress number and the stress energy. The distribution of the stress number results from the transport behaviour of the mill. The narrower the residence time distribution is, the less the stress numbers for the individual particles differ. Additionally, the

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Fig. 42. Grinding chamber and stirrer geometries used for the grinding of white fused alumina [31]. (a) Disk Stirrer; (b) pin-counter–pin-stirrer; (c) annular gap geometry.

more uniform the shear gradient and thus, the power density is, the less the intensities of the individual stress events differ. Therefore, the more uniform the power density in the mill is, the narrower the distributions of the stress energy in the mill. The inﬂuence of the geometry of the grinding chamber and the stirrer on the relation between product ﬁneness and speciﬁc energy was investigated by Joost [31]. Joost carried out grinding tests with white fused alumina in a stirred media mill with disc stirrer geometry, in a mill with pin-counter pin geometry and in an annular gap mill. All three mills which are shown in Fig. 42 have similar volumes of the grinding chamber. The median sizes obtained with the three different geometries are presented in Fig. 43 as a function of the speciﬁc energy. The measurement values can be described in a ﬁrst approximation by a ﬁtted straight line. But looking in more detail it can be seen that the measurement values of the annular gap mill lie

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100

particle size x50 [ μm ]

disc geometry pin - counterpin geometry annular gap geometry

10 ϕGM = 0.7 - 0.8 vt = (6); 8 - 16 m • s-1 cm = 0.1 - 0.4 mSusp = 8.1 - 42.7 kg • h-1 dGM = 1.5 mm (steel) HGM = 750 HV 5 1 100

1000

10000

specific energy Em [ kJ • kg-1 ]

Fig. 43. Median size as function of the speciﬁc energy for three different grinding chamber geometries [31].

slightly above the measurement values obtained with the disc stirrer or the pincounter pin stirrer. Against that no systematic difference can be found between the measurement values of the disc stirrer and the pin-counter pin stirrer. Although the values for the annular gap mill lie slightly above the other values, Fig. 43 shows that the relation between the average product ﬁneness (median size) and the speciﬁc energy is only slightly affected by the grinding chamber geometry. The median sizes, which were obtained with the annular gap geometry and which at a given speciﬁc energy are slightly greater than the ones of the two other grinding geometries, are attributed to higher stress intensities and, particularly, to a lower energy transfer factor of the annular gap mill. A reason for the lower energy transfer factor is above all the friction losses at the great surface of the grinding chamber compared to the chamber volume. The lower energy transfer factor of annular gap mills can also be seen in Fig. 44 in which the median size at a speciﬁc energy weighted with the ﬁlling ratio of the grinding media of 1000 kJ kg–1 is presented as function of the stress energy of the grinding media [42]. In Fig. 44 the measurement values of an annular gap mill and a mill with disc stirrer are shown, both with the same inner diameter of the grinding chamber. The courses of the two curves show that the measurement values of the annular gap mill are displaced to greater median sizes, i.e. at the same speciﬁc energy and stress energy of the grinding media, SEGM, a higher speciﬁc energy is required to achieve a certain product ﬁneness. The reason for this is above all the lower energy transfer factor of the annular gap mill.

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Median size x50 [μm]

Annular gap mill Gap width s = 6 mm VGC = 0.97 l, ϕGM = 0.7 disc geometry VGC = 5.54 l, ϕGM = 0.8 ρGM = 2510 - 7550 kg/m3 cm = 0.4

10

vt = 5 - 15 m/s dGM = 97 - 4000 μm Em • ϕGM = 1000 kJ/kg

1 0.5 1E-4

1E-3

0.01

0.1

1

10

50

SEGM = dGM3 • ρGM • vt2 [10-3 Nm]

Fig. 44. Median size as function of the stress energy at constant weighted speciﬁc energy for annular gap mill and mill with disc stirrer [42].

3.5. Formation of nano-particles by wet grinding in stirred media mill 3.5.1. Conditions of producing nano-particles with stirred media mills Beside the production of nano-particles from liqueﬁed materials, solutions or the gas phase it is possible to produce particles below 100 nm by grinding coarser solid particles. Moreover, at the production of nano-particles by bottom-up procedures (e.g. by precipitation) often agglomerates or aggregates are formed which have to be deagglomerated or dispersed afterwards. An effective device producing nano-particles by grinding coarser particles or by dispersing agglomerates is the stirred media mill. The problem in producing nano-particles by grinding can easily be realised by considering the number of particles produced from a few coarse feed particles. For example, consider a spherical particle with a size of 1 mm. If it is assumed that by grinding spherical fragments of this particle are produced, as shown in Table 2, extremely high numbers of fragments arise depending on the size of these fragments. The extremely high number of fragments arising from one particle with a size of 1 mm shows that we have to stress an extremely high number of particles to produce particles in the nanometre size range and also that we have to deal with an extremely large solid surface area. Therefore, in order to grind and disperse

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Table 2. Number of fragments arising from a spherical particle with a size of 1 mm Size of fragments Number of fragments

10 mm 6

10 (1 million)

1 mm 9

10 (1 billion)

100 nm 12

10

(1 trillion)

10 nm 1015 (1 quadrillion)

particles down to particles sizes below a few hundred nanometres the following conditions must be fulﬁlled: a. Per unit time an extremely high number of particles must be stressed, i.e. the stress frequency must be very high. b. The intensity of the single stress events must be sufﬁcient to break the particles or destroy the agglomerates. c. The new created surface of the fragments or primary particles must be stabilised by adsorption of sufﬁcient additives so that no new agglomerates are created by van der Waals attractive forces. d. The rheology, particularly the viscosity of the suspension must be controlled in a way that the suspension can be handled and the grinding effect of the mill is preserved. A sufﬁcient stress frequency (point a) can be achieved by employing grinding media with sizes as low as possible [2,18,43]. By using small grinding media a relatively high number of grinding media and thus a high number of stress events exist in a certain volume (e.g. grinding chamber volume). The number of particles, which are stressed at each grinding media contact, can be optimised by an appropriate selection of the solids concentration. Tendency-wise the higher the number of particles stressed at one grinding media contact, the higher the solids concentration is. But if the solids concentration is too high a distinct increase of the viscosity and with that a decrease in the grinding efﬁciency can arise (see point d). The intensity of single stress events (point b) is determined above all by the kinetic energy of the grinding media, which is inﬂuenced by the density and size of the grinding media and by the stirrer tip speed. Usually in case of grinding particles in the nanometre size range the stress intensities are sufﬁcient to break the particles or to disperse the agglomerates. Exceptions to this rule include the case of grinding hard materials (e.g. ceramic materials) with a grinding media that is too soft compared to the product and thus the grinding media are more deformed than the product particles. Moreover with increasing ﬁneness the strength of the particles against breakage increases so that at an extremely high ﬁneness the strength can increase strongly and can be higher than the stress intensity supplied by the grinding media. Last but not least the suspension viscosity should

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not be too high because otherwise the kinetic energy of the grinding media becomes too small to stress the product particles sufﬁciently after displacing the suspension during the approach of the two grinding media. The stabilisation of the newly created nano-particles (point c) is important, because otherwise immediately after breakage or deagglomeration the ﬁne fragments or primary particles can reagglomerate either inside or after leaving the grinding chamber. The newly created agglomerates can be stronger than the original particles or agglomerates. The danger of reagglomeration is much lower inside the stirred media mill than outside, because inside the mill the suspension is stressed by extremely high shear forces. The stabilisation of the product suspension can be established in principle by Electrostatic stabilisation: deposition of only positive or only negative charges Steric stabilisation: adsorption of sufﬁcient polymers and tensides Electrosteric stabilisation: combination of electrostatic and steric stabilisation.

The control of the rheology and particularly the viscosity of the suspension (point d) is decisive for grinding and dispersion of particles in the nanometre size range because the high increase in the number of particles and the simultaneous increase in the attractive forces compared to the inertia forces can cause an increase in the viscosity, resulting in a change of the rheological behaviour (e.g. from a Newtonian ﬂuid to a strong intrinsically viscous ﬂuid). If the viscosity becomes too high the kinetic energy of the grinding media is no longer sufﬁcient to displace the suspension between two approaching grinding media and additionally to stress the particles with a sufﬁcient intensity. If the relative movement between the grinding media and the suspension stops, in practice no grinding or dispersion is possible.

3.5.2. Grinding of alumina down to sizes in the nanometre range In order to show the principle effect of different operating parameters on the production of nano-particles the grinding of white fused alumina with a median size of approximately 30 mm down to product sizes below 100 nm is discussed. The experiments were run at the Technical University of Braunschweig in cooperation with the Technical University of Mu¨nchen [44].

3.5.2.1. Experimental setup For the experiments a laboratory stirred media mill with a disc stirrer and a grinding chamber volume of approximately one litre was used. In order to minimize contamination by wear of the grinding chamber and stirrer to a minimum, stirrer discs out of polyurethane and a grinding chamber cylinder out of SiSiC

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were chosen. A ZrO2 sieve was employed as separation device for the grinding media. The mill was operated with a stirred vessel in a circuit. By the addition of nitric acid HNO3 or soda lye NaOH into the stirred vessel the pH value could be adjusted for electrostatic stabilisation. Owing to the wear of the grinding media the product mass increases during the grinding process. The particle size distributions of the product were measured with a ultrasonic spectrometer of the type DT 1200. This measurement device which was specially developed for particle size measurements at solids concentrations of 1–50% by volume allows an inline determination of the particle size distribution and the x-potential without time consuming preparation of diluted suspension samples and without stabilisation problems caused by the dilution.

3.5.2.2. Experimental results At ﬁrst the effect of the grinding media material and of the stabilisation on the grinding result was investigated. Figure 45 shows the results for two different grinding media materials (yttrium-stabilised zirconium oxide and aluminium oxide) at similar pH values of approximately 10 and yttrium-stabilised zirconium oxide at a smaller pH value of approximately 5. The pH value, the x-potential and the median particle size are plotted versus the speciﬁc energy. At the end of the grinding test with a pH value of 10 using grinding media out of yttrium-stabilised zirconium oxide, the pH value was reduced from 10 to 5 (see open triangular symbols). Without stabilisation and thus without the addition of nitric acid a pH value of approximately 10 is present inside the mill. Under these conditions different results can be found for the two different grinding media materials: using the aluminium oxide grinding media the product ﬁneness increases steadily up to a median particle size below 100 nm (approximately a straight line in the log–log diagram). Against that using the zirconium oxide grinding media at a median size of 350 nm no further grinding effect can be found and the median size varies in a range of 300–400 nm despite a steadily increasing energy input (see ﬁlled triangles in Fig. 45). The different behaviours are due to differences in the surface charges and due to different particle–particle interactions: using the aluminium oxide grinding media the x-potential is negative over the entire energy or ﬁneness range. This means that the interfaces of the aluminium oxide grinding media are always negatively charged and repel each other. Using zirconium oxide grinding media the x-potential drops from a small positive value to a distinct negative value. In this case the x-potential crosses the point of no charge, at which no repulsive effect exists anymore and at which the attractive forces (van der Waals forces) can act without any counter effect. From Fig. 45 it can be seen that after passing the point of no charge the product ﬁneness does not increase anymore. The

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pH - value [-]

10 9 Grinding media dGM

8

Al2O3

7 6

pH

900 μm 10

ZrO2

800 μm 10→5

ZrO2

800 μm 5

5

ζ-Potential [mV]

70; 4 60 40 Addition of HNO3

20 0 -20

Median size x50 [nm]

1300;-40 1000

100

cm = 0.2 ϕGM = 0.8 vt = 12 m/s

50 4x103

104 Specific energy Em,W [kJ/kg]

105

Fig. 45. Effect of grinding media material and stabilisation (pH value) on the grinding of fused corundum into the nanometre size range [44,45].

reason is the agglomeration of ﬁne particles due to the van der Waals attractive forces. If at a speciﬁc energy input of approximately 6 105 kJ kg–1 the zeta potential is increased to a positive value of approximately 50 mV by addition of nitric acid, a strong repulsive potential is produced, so that the agglomerates can be dispersed again. Thus, in this case a median size of less than 100 nm can be achieved with the zirconium oxide grinding media, as was found for aluminium oxide grinding media. After the increase of the zeta potential the median size drops in a very short time from a value of approximately 300 nm to value of less than 100 nm. From this it can be concluded that without stabilisation even with the zirconium

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oxide grinding media an effective grinding took place. Thus, either the further grinding of the agglomerated fragments are as effective as the further grinding of the stabilised fragments or the shear forces inside the mill are so effective that at least in the moment of a grinding media contact the fragments are not agglomerated. If, for the case of using zirconium oxide grinding media, the suspension is stabilised already from the beginning of the grinding process by adjusting the pH value to a value of 5, the median size decreases steadily down to a value of approximately 80 nm. At a pH value of 5 the zeta potential is greater than 40 from the beginning to the end of the experiment, so that relatively large repulsive forces act between the particles. These repulsive forces prevent the agglomeration of newly produced fragments. Moreover, due to the stabilisation the viscosity is kept low. Because of the lower viscosity the energy dissipation in the mill due to friction is smaller and thus the grinding is more efﬁcient regarding energy consumption, i.e. the same product ﬁneness is produced with less speciﬁc energy. In further experiments the pH value was varied systematically from 5 to 10. The results of these experiments are shown in Fig. 46. Depending on the pH value and thus on the amount of ions in the suspension different ﬁnal product ﬁnenesses could be achieved. Whereas at a pH value of 8 only a ﬁnal median size of approximately 60 nm could be achieved, at a pH value of 5 a ﬁnal median size of below 20 nm is produced. Looking at the viscosities it can be determined that at a pH value of 5 the yield stress is very small whereas at a pH value of 8 or 10 the yield stress is very high [44]. Moreover the wear of the grinding media is affected by the stabilisation and thus by the pH value. Tendency-wise the wear of the

Fig. 46. Effect of different pH values on product ﬁneness.

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dGM 1300 μm 1100 μm 800 μm 500 μm 350 μm 350 μm 200 μm

Median size x50 [nm]

500 250

100 75 50 25

GM-material: ZrO2(Y2O3) = 12 m/s vt = 0.2 cm pH

10 4x103

= 5 104

105 Specific energy Em,W [kJ/kg]

8x105

Fig. 47. Effect of grinding media size on product ﬁneness.

grinding media increases with increasing stabilisation. For example the wear of the zirconium oxide grinding media is signiﬁcantly smaller at a pH value of 10. Subsequently, also a reduction of the pH value from 10 to 5 at the end of the grinding process causes less wear compared to a pH value of 5 already at the beginning of the process. Beside the effect of the grinding media material and the pH value the effect of grinding media size on the relation between product ﬁneness and speciﬁc energy was investigated. In Fig. 47 the median size is depicted as a function of the speciﬁc energy Em,W for grinding media sizes ranging from 200 to 1300 mm. All of these experiments were run with zirconium oxide grinding media at a pH value of 5. The measurement results show that for product median sizes below 200 nm a certain product ﬁneness (e.g. 100 nm) can be produced with less speciﬁc energy if smaller grinding media are used. For example, in order to produce a median size of 25 nm (measured with the DT 1200), by using a grinding media size of 200 mm a speciﬁc energy of approximately 105 kJ kg–1 is necessary whereas by using a grinding media size of 800 mm a speciﬁc energy of 4 105 kJ kg–1 is required [46]. The reason for the better grinding effect of the smaller grinding media (especially at higher product ﬁnenesses) is that by using smaller grinding media the energy utilisation is higher at each stress event and simultaneously more stress events take place. A better energy utilisation means that at a single stress event a certain increase in product quality (e.g. production of new surface) is achieved with less speciﬁc energy. Investigations have shown that the energy utilisation is the highest and the stress intensity has an optimum value, if the stress intensity is just sufﬁcient for a complete breakage [18,43]. Therefore, it is most favourable to

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321

stress the particles as often as possible with a stress intensity which is just sufﬁcient. In case of higher product median sizes (x504200 nm) the ﬁnest product for a certain speciﬁc energy is achieved with a grinding media size of 350 mm. The lower grinding efﬁciency of the grinding media with a size of 200 mm in this size region is caused by the lower stress intensity. At the beginning of the grinding process the stress intensity of the 200 mm grinding media is obviously not sufﬁcient for an effective grinding of feed particles with a size of 30 mm. The feed particles must be stressed by the 200 mm grinding media several times before they are broken. As soon as all feed particles are broken the stress intensity is high enough for the resulting fragments and the fragments can be ground very efﬁciently with the 200 mm grinding media. The course of the curves in Fig. 47 shows that by the use of even smaller grinding media an even more efﬁcient grinding can be achieved. Indeed in a ﬁrst stage the feed particles should be ground with larger grinding media. Using grinding media below 200 mm the problem arises that a safe separation of the small grinding media from the product is very difﬁcult and in most cases not possible. Moreover with decreasing grinding media size the control of the viscosity becomes even more important, because the inﬂuence of the ﬂuid forces increases in comparison to the inertia forces. Thus, the danger of grinding media compression or blocking in front of the separation device increases.

4. TRANSPORT BEHAVIOUR AND OPERATION MODE Usually stirred media mills are operated continuously, so that the suspension ﬂows axially through the well mixed grinding chamber. Therefore, axial transport and mixing action are superimposed. Thus, the transport behaviour of the suspension lies between one of plug ﬂow and that of an ideally stirred vessel. As in a discontinuous process for the case of plug ﬂow the product particles would be stressed with the same time period and thus, on average with the same stress frequency. Because of the mixing action in the grinding chamber the product particles are stressed with different time periods and thus, some particles are stressed less frequently and others are stressed more frequently. The distribution of the relative number of stress events depends essentially on the transport behaviour and the operation mode of the mill and thus, on the residence time distribution of the particles in the mill. As a result, the particle size distribution of the product is strongly inﬂuenced by the residence time distribution. In particular, the maximum product particle size is determined by the residence time distribution: a wide residence time distribution causes a tail of coarse particles in the product. The residence time distribution and with it particularly the coarse range of the particle size distribution can be clearly improved by an appropriate

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selection of the operating parameters and/or the operation mode (one passage, multiple passage or circuit mode) [9,21].

4.1. Basic considerations The residence time distribution of the product suspension can be determined experimentally. The axial transport of the product suspension in a grinding chamber with disc stirrer was investigated [18,47,48]. The horizontally placed grinding chamber has a net volume of approximately 5.5 l and is equipped with a rotating separation gap. Under continuous operation the product suspension ﬂows through the lid of the left side into the grinding chamber and leaves the grinding chamber through the rotating separation gap. The grinding media are held back in the grinding chamber by the separation gap. Because of the axial transport of the product suspension and the mixing action of the stirrer a certain residence time distribution of the particles results. Since the product particles move like a ﬂuid phase in case that the particles are small enough (usually smaller than 10 mm [29]), the residence time distribution of the product particles corresponds to the residence time distribution of the ﬂuid phase. In case of an identical transport behaviour of the solid and the ﬂuid phase the residence time distribution of the product suspension can be measured by a pulsed injection of a salt solution at the inlet of the grinding chamber and a measurement of the conductivity at the outlet (see for example [18,22,32,48,49]). At the inlet a small amount of salt solution was injected in pulsed manner into the feed suspension. The respond of the pulse was measured continuously at the outlet pipe by a conductivity measuring instrument. By considering the residence time distribution in the inlet and outlet pipe the residence time density function E(t) can be determined directly from the measured conductivity by normalisation. Figure 48 shows a typical residence time distribution measured using this method. The residence time density function E and the corresponding cumulative function F are shown as functions of the dimensionless time Y ¼ t=t. The mean residence time t is equal to the integral mean value of the density function E and corresponds to the so-called ideal ﬁlling time tf, because no dead zones exist in the grinding chamber. The ideal ﬁlling time can be calculated as follows: tf ¼

V GC V GM V_

ð34Þ

where VGC (m3) is the volume of the grinding chamber, VGM (m3) the solid overall volume of the grinding media and V_ (m3 h–1) the volume ﬂow rate of the product suspension. The density function E(Y) can be determined directly from the measured pulse answer by normalisation, the cumulative function F(Y) is determined by integration of the density function. Already at very short residence times the density

Wet Grinding in Stirred Media Mills 1.0

1.0

0.9

0.9 0.8

Density function E(θ) Cumulative function F(θ)

0.7 0.6

0.7 0.6

dGM = 1090 μm ϕGM = 0.8 vt = 10 m/s V = 100 l/h

0.5 0.4

0.5

F (θ) [-]

0.8

E (θ) [-]

323

0.4

0.3

0.3

0.2

0.2

0.1

0.1 0.0

0.0 0.0

0.5

1.0

1.5 2.0 2.5 3.0 3.5 Dimensionless time θ = t / t [-]

4.0

4.5

5.0

Fig. 48. Typical density and cumulative functions of the residence time distribution of the product particles in the grinding chamber.

function has values greater than 0 because of the intensive mixing action of the mill. Therefore, after only a very short time the ﬁrst product particles leave the mill. The density distribution is asymmetric and has a maximum at a residence time being smaller than the integral mean value (Y ¼ 1). Moreover, the cumulative function shows that at the time Y ¼ 1ðt ¼ tÞ more than 50% by mass of the product particles have left the mill, i.e. the median value of the residence time distribution is smaller than the mean residence time and thus, than the ideal ﬁlling time. From the two distributions it follows additionally that after a dimensionless time of Y ¼ 5 approximately all product particles have left the mill. Thus, approximately ﬁve times the mean residence time is required until steady state is obtained after a disturbance or a change of the operating parameters.

4.2. Modelling the axial transport in stirred media mills Stehr [29] could describe the transport behaviour of stirred media mills by the one-dimensional dispersion model. Further investigations [18,22,32,48,49] conﬁrm the results of Stehr. Besides the one-dimensional dispersion model frequently cell models are used for modelling technical ﬂow systems. According to Heitzmann [49] and Kwade [18,47,48] cell models, particularly the cell model with backmixing, are suitable also for the description of the residence time behaviour of stirred media mills. The one-dimensional dispersion model is based on the idea that the convective axial transport being characterised by the mean axial velocity v ¼ LGC =t is

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superimposed by a mixing action which is characterised by a so-called axial dispersion coefﬁcient D: dcðt; xÞ dcðt; xÞ d2 cðt; xÞ ¼ v þD dt dx dx 2

ð35Þ

By normalisation the following differential equation can be derived for the onedimensional dispersion model: dCðy; X Þ dCðy; X Þ 1 d2 Cðy; X Þ ¼ þ dy dX Pe dX 2

ð36Þ

with C(x,t) ¼ c(x,t)/c0 being the normalised concentration, Y ¼ t=t the dimensionless time and X ¼ x/LGC the dimensionless length of the grinding chamber. The state of mixing is described in equation (36) by the dimensionless Pe-number: Pe ¼

v LGC D

ð37Þ

The Peclet-number Pe is zero in case of an ideal mixed vessel and tends to inﬁnity in case of plug ﬂow. The value of the Pe-number characterises the residence time behaviour of a ﬂow system. According to Molerus [50] for an unknown system the Pe-number or the dispersion coefﬁcient D, respectively, can be determined by the ﬁrst and second moment of the density function. Thereby it is assumed that the axial transport velocity and the dispersion coefﬁcient do not depend on position and time and that the ﬂow system is ideally closed. The measured residence time density distribution of Fig. 48 is compared in Fig. 49 amongst others with a residence time distribution, which was calculated using the one-dimensional dispersion model with the Pe-number determined as described above. The comparison of the two curves shows that the calculated residence time distribution increases a slightly later than the measured distribution and that the calculated distribution has a higher maximum value. Therefore, the dispersion model can describe the residence time behaviour of a stirred media mill with a disc stirrer only approximately. Technical ﬂow systems – mainly chemical reactors – are also frequently described by so-called cell models. The simplest form is a series of ideally mixed vessels, through which a constant ﬂow rate ﬂows (see for instance [49]). The best ﬁt between the measured and the calculated residence time distributions determines the number of cells. Having the stirred media mill with a disc stirrer the number of cells is already given by the number of discs. Thus, an adjustment of the residence time distribution by changing the numbers of cells would not be physically correct. Against that the so-called cell model with backmixing can describe the transport behaviour of vessels with multiple stirrers very well. Here the series of ideally mixed cells is not ﬂown through only in one direction, but further volume ﬂow rates

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325

1.0 Measurement Dispersion model (Pe = 2.33) Cell model (R = 2.52)

0.9 0.8

E (θ) [-]

0.7 0.6

dGM = 1090 μm ϕGM = 0.8 vt = 10 m/s V = 100 l/h

0.5 0.4 0.3 0.2 0.1 0.0 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Dimensionless time θ = t / t [-]

Fig. 49. Comparison of typical measured residence time distributions with results of calculations. R

R V

VH,1

V

C1

VH,2

V

C2 R

R VH,3

V

C3 R

R VH,4

V

C4 R

R VH,5

V

C5 R

VH,6

V

C6 R

Fig. 50. Schematic drawing of the cell model with backmixing.

ﬂow between the cells in both directions. For example, a stirred media mill with six stirrer discs can be modelled with six cells. The arrangement of the six cells is drawn schematically in Fig. 50. Through all cells which have a free volume VH,i and in which a concentration ci exists, the axial volume ﬂow rate V_ ﬂows. Ad_ (backﬂow rate) ﬂows ditionally, because of the backmixing a volume ﬂow rate R _ is assumed to be between the cells in both directions. The backﬂow rate R independent of the position in the mill. Material balances around each cell lead to differential equations. As it was shown for the one-dimensional dispersion model from equations (35) to (37) dimensionless parameters are introduced leading to a normalised differential equation for cell i: yi

dCi ¼ ð1 þ RÞCi1 ð1 þ 2RÞCi þ RCiþ1 dy

ð38Þ

where Ci ¼ ci/c0 being the normalised concentration, Y ¼ t=t the dimensionless _ V_ the dimentime, yi ¼ Vi/VGC the dimensionless volume of the cells and R ¼ R= sionless backﬂow number.

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Like the Pe-number in the one-dimensional dispersion model, the backﬂow number R characterises the mixing state of the ﬂow system and with it the residence time distribution. The residence time distribution calculated using the cell model with backmixing was ﬁtted to the measured residence time distribution by minimisation of the deviations. Thereby the optimal backﬂow rate was determined (for more details see [18]). In Fig. 49 also the residence time density function is presented, which was obtained using the cell model with backmixing. This density function corresponds very well to the measured density function. Contrary to the one-dimensional dispersion model the cell model with backmixing can describe the fast increase of the function at small residence times and the maximum value very well. The fact that the consistency between the measured and the calculated distributions is worse in the one-dimensional dispersion model is based on the assumption that the concentration changes continuously and not stepwise in ﬂow direction. Investigations with a grinding chamber manufactured out of glass allowed for the observation of the ﬂow. Changing stepwise from pure water to coloured water in a continuous process clearly indicated that the colour does not change steadily, but stepwise. Different residence time distributions result for a pin-counter pin stirrer geometry or an annular gap geometry versus a grinding chamber with disc stirrer. In case of these stirrer geometries the residence time distributions calculated with both models are very similar. Therefore, the one-dimensional dispersion model can be employed as well. To demonstrate the inﬂuence of operating parameters on the residence time distribution only one example is shown here: the inﬂuence of the stirrer tip speed and the axial transport velocity on the Pe-number [29]. In Fig. 51 the Pe-number is plotted versus the axial transport velocity for three different stirrer tip speeds. Vu = 6.4 ms-1 5

Vu = 9.6 ms-1

Peclet Number Pe / -

Vu = 12.8 ms-1 4 3 2 1 0 0.0

4.2

8.4 12.7 16.9 21.1

29.7

axial transport velocity v / cm min-1

Fig. 51. Pe-number as function of the axial transport velocity and the stirrer tip speed vu ( ¼ vt).

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An increase in the axial transport velocity and with it in the volume ﬂow rate results in all cases in a higher Pe number and with it in a narrower residence time distribution. Against that an increase in the stirrer tip speed causes a decrease in the Pe-number and with it a broader residence time distribution. In a one passage mode usually Pe-numbers of one to ﬁve are valid for grinding chambers with a disc stirrer, in extreme cases Pe-numbers of up to eight are possible. More information on the inﬂuence of the Pe-number and especially of the backﬂow number R on individual operating parameters can be found in the literature, especially in papers by Kwade [18,47,48].

4.3. Effect of the operation mode on the residence time distribution Different residence time distributions and with it different distributions of the stress number can be obtained with the same stirred media mill by different operation modes. Usually a residence time distribution which is as narrow as possible is demanded in order to achieve a narrow particle size distribution with a low content of coarse particles. An ideal narrow residence time distribution is obtained in a discontinuous process, at which the feed is ﬁlled into the grinding chamber and is stressed for a given time. Since such an operation is only possible for very small batches, and moreover causes a high expenditure of work, stirred media mills are usually operated continuously, i.e. a suspension is steadily pumped through the grinding chamber. Thereby the suspension can be pumped differently often through one or more stirred media mills. In addition, the mills can be connected differently to one or more agitated vessels. In the simplest case a batch is transported through the mill in one passage. In order to obtain a narrow residence time distribution and with it a narrow particle size distribution, often other operation modes are employed, an overview of them is given in Fig. 52. The so-called one passage mode is very simple and is used above all for the mass product production. The disadvantage of this mode of operation is that the residence time distribution is determined directly by the transport behaviour of the mill and is usually relatively broad. Thus, also a relatively wide particle size distribution is obtained. An improvement of the residence time distribution and with it a narrower particle size distribution can be achieved by a connection of several mills in series (cascade of mills). Similar to the one passage mode a multiple passage mode with a cascade of mills is most suitable for mass production because the product quality must be exactly reached behind the last mill. The use of two or more mills is meaningful if the grinding rate is relatively high, i.e. the change in particle size from the start to the end ﬁneness is relatively large. The grinding media size can be optimally adjusted to the progress in grinding, i.e. the size of the grinding media is large in the ﬁrst mill and small in the last mill.

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Fig. 52. Different operation modes with the corresponding residence time distributions (schematic presentation) [21].

Another operational mode is the pendulum mode, which is also a multiple passage mode, in which the product is pumped several times through the same grinding chamber. Opposite to a cascade of mills the suspension is transported several times through the same mill. For that two stirred vessels are required, the content of which is pumped alternating through the mill. If at pendulum operation – what is usually the case – more passages are run than with a cascade of mills, a narrower residence time distribution can be obtained. The pendulum mode is well suited for the production of small and medium sized batches, but less for the production of high mass ﬂow rates. The closed circuit mode is easier to handle than the pendulum operation, especially regarding control of the process. Another advantage of the circuit mode in comparison to the pendulum operation is that for the operation of the mill only one stirred vessel is necessary. A disadvantage is that backmixing occurs in the stirred vessel and thus, the residence time distribution is wider at an equal

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number of circuits or passages. Generally it can be stated for the pendulum and circuit operation the greater the number of circuits or passages is, the narrower are the residence time distributions and with it the particle size distributions of the product. For this reason mills are used which allow a very high volume ﬂow rate without grinding media compression or packing, respectively. Since for one passage these mills produce usually a relatively wide residence time distribution, the application of these mills is often only advantageous at a sufﬁciently high number of circuits or passages. For the production of a batch or charge different layouts and operations of the facility result from the combination of the passage and circuit mode and the use of a different number of mills. In Fig. 53 two typical examples are shown: in example (a) the facility is composed out of three stirred vessels (vessel for predispersing Solid components Fluid components Additives

M

M

M

M

(a)

Predispersing

Circuit

Let-Down

Vessel

Vessel

Vessel

Solid components Fluid components Additives M

M

M

(b)

M

M

M

Predispersing

1. Circuit

2. Circuit

Let-Down

Vessel

Vessel

Vessel

Vessel

Fig. 53. Different facility layouts. (a) Circuit mode with ﬁlling and emptying passage. (b) Circuit mode with pre-grinding and polishing.

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and eventually metering, circuit vessel and let-down vessel) and one stirred media mill. In example (b) the facility consists of three or four stirred vessels and two stirred media mills (one mill for pregrinding and one mill for polishing). Details to the layout of facilities with stirred media mills and to the advantage and disadvantage of different operation modes are discussed among others by Kwade and Schwedes [1] as well as Vock [39]. The residence time distributions of the different operation modes (one passage, multiple passage, pendulum, circuit) can be evaluated using the transport models (dispersion model, cell model with backmixing) presented above. In case of a multiple passage mode the overall residence time density function after another passage can be determined, if not an impulse, but the residence time density function at the end of the preceding passage is the starting point for the calculation of the residence time distribution of the next passage. The residence time density distribution E(t) at the end of the new passage can be determined by solving the convolution integral: Z t EðtÞ ¼ E A ðWÞ gðt WÞ dW ð39Þ 0

where E(t) being residence time density function after the new passage, EA(t) the residence time density function of the inﬂowing product pension and g(t) the transport behaviour of the mill (residence time density function). In case of the circuit mode the residence time distribution of the mill is superimposed by the residence time distribution of the stirred vessel. This can be calculated for an ideally stirred vessel as follows: 1 EðtÞ ¼ et=t t

ð40Þ

where t is the ideal ﬁlling time of the stirred vessel. Since all particles which leave the stirred vessel ﬂow back into the same vessel, and since in an ideally stirred vessel the probability that a particle leaves the vessel is independent of how long the particle has already been in the vessel, not all particles leave the vessel during one theoretical circuit. Many circuits are necessary to be sure that all particles have left the vessel at least once. The percentage of particles having left the vessel results from the cumulative function F(t) of the residence time density distribution E(t) of the ideally stirred vessel: Z t FðtÞ ¼ EðWÞdW ð41Þ 0

Solving this equation leads to the percentage of particles which have left the stirred vessel after a given number of theoretical circuits at least once (Table 3). The table clearly shows that at least six circuits should be run in order that nearly all particles are stressed in the mill at least once. In practice the rule exists that under circuit operation the number of theoretical circuits should be higher

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Table 3. Percentage y of the particles which have left the stirred vessel after the given number of theoretical circuits at least once

Number of theoretical circuits (–) 1 Percentage y (%)

2

3

4

5

6

63.21 86.46 95.02 98.16 99.32 99.96

200 180

Pe = 0.5 Pe = 1 Pe = 2 Pe = 3 Pe = 4 Pe = 6 Pe = 8 Pe = 10

resulting Peclet Number

160 140 120 100 80 60 40 20 0

0

2

4

6

8 10 12 number of passages

14

16

18

Fig. 54. Fitted Pe-number of multiple passage operation as function of the Pe-number of the mill and the number of passages [1].

than 10. Theoretically it follows from equation (41) that in every circuit process at all times particles exist which run through the mill inﬁnite times and at the same time particles exist which did not leave the vessel. Therefore it is not sufﬁcient to know the residence time density function of the stirred vessel, because at any time particles with different histories exist in the vessel and additionally, because their quantities change with time. Moreover, the residence time distribution of the particles inside the mill is of interest, rather than the residence time distribution of the entire process. From these considerations it follows that a direct calculation of the residence time distribution of the circuit operation mode is not possible. Nevertheless, estimations are possible as they are performed for instance by Kwade and Schwedes [1] as well as Vock [39]. The following two ﬁgures calculated at the Institute for Particle Technology, TU Braunschweig, show the change of the residence time distribution with increasing number of passages (Fig. 54) and number of circuits (Fig. 55). As a measure of the width of the residence time distribution the Pe-number was chosen. Parameter of the different curves is the Pe-number

332

A. Kwade and J. Schwedes 30 Pe = 0.5 Pe = 1 Pe = 2 Pe = 3 Pe = 4 Pe = 6 Pe = 8 Pe = 10

resulting Peclet Number

25

20

15

10

5

0

0

2

4

6

8 10 number of circuits

12

14

16

18

Fig. 55. Fitted Pe-number of circuit operation as function of the Pe-number of the mill and the number of circuits [1].

of one passage through the mill. The Pe-numbers were determined by comparing the residence time distribution calculated for a deﬁned number of passages or circuits with the residence time distribution which results for one passage at different Pe-numbers. From Fig. 54 it follows, for example, that the residence time distribution for 10 passages and a Pe-number of the mill of Pe ¼ 2 is similar to the residence time distribution of a Pe-number of 35. Against that the residence time distribution for 10 circuits and a Pe-number of the mill of Pe ¼ 2 is similar to the residence time distribution for a Pe-number of only 12. A comparison of the Pe-numbers at equal numbers of passages and circuits shows that at a usual Pe-number of the mill approximately three to four times more circuits than passages must be run to obtain a similar residence time distribution.

4.4. Effect of residence time distribution on the particle size distribution The residence time distribution affects directly how often each particle is stressed in the mill during the entire grinding process. As shown above due to the residence time distribution not all particles are stressed for an equal amount of time and thus, equally frequently: some particles are not stressed or stressed only once, while other particles are stressed very frequently. The narrower the residence time distribution is or the less the number of stress events per particle ﬂuctuates, the narrower the resulting particle size distribution is.

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The particle size distribution produced at a certain operation mode can be predicted if the residence time distribution of the product particles in the mill, in addition to particle size distributions obtained in batch grinding tests, are known. The operating parameters (ﬁlling ratio of grinding media, grinding media size and density, stirrer tip speed and solids concentration of the product suspension) of the batch grinding tests must correspond with the operating parameters of the continuous grinding process. According to Stehr [29] the particle size distribution Q3,cont(x) of the product from the continuous grinding process can be determined from the residence time density function E(t) of the product particles in the mill and the particle size distributions Q3,disc(x,t), which are determined in discontinuous (batch) grinding tests: Z 1 Q3;cont ðxÞt ¼ Q3;disc ðx; tÞEðtÞ dt ð42Þ 0

The calculation of the particle size distributions using the residence time distribution and the results of batch grinding tests is shown in the following for the grinding of limestone [29]. At ﬁrst batch grinding tests were carried out, the results of which are shown in Fig. 56 as the cumulative particle size distribution Q3,disc versus grinding time t. The different curves represent different particle sizes, ranging from 60 to 0.25 mm. The curves present the cumulative weight undersize, i.e. the mass percentage of the particles smaller than the size x, as function of grinding time tgrind. In a batch process all solid particles spend an equal amount of time in the mill. Therefore, with respect to the residence time distribution the batch operation contains the ideal boundary condition for continuous operation.

100

80

Q3 / %

60

40

Cv = 0.2 Vu = 9.6ms-1

20

0

0

5

10 15 20 grinding time t / min

x = 60.00 μm x = 30.00 μm x = 15.00 μm x = 8.00 μm x = 4.00 μm x = 2.00 μm x = 1.00 μm x = 0.50 μm x = 0.25 μm

25

30

Fig. 56. Result of batch grinding tests: cumulative weight Q3,disc(x,t) as function of time [29].

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A. Kwade and J. Schwedes 100 80

Q3 / %

60 batch grinding continous grinding with t = 4.5 min ± 5% Pe = 1.22 ± 30%

40 20 0 10-1

100

101

102

Xst / μm

Fig. 57. Comparison of experimental and theoretical cumulative distribution curves Q3(x) after continuous grinding.

The results of batch grinding, Q3,disc(x,t), are weighted with the experimentally or analytically determined residence time distribution E(t) of the continuous process (see equation (42)) leading to predictions of the product particle size distribution Q3,cont(t) of the continuous process. An example is presented in Fig. 57. After batch operation for 4.5 min the particle size distribution which is shown by the dashed line was obtained. With the same values for stirrer tip speed and solids concentration and with a mean residence time of 4.5 min the continuous process resulted in a distribution marked by the open symbols. This distribution is much wider and contains far more coarse particles. The mathematical prediction of the grinding results has been performed with an uncertainty of 75% for the experimental determination of the mean residence time and another uncertainty of 730% for the Pe-number. The two solid line curves result from the calculations. They can locate the experimental data well. Based on the calculations presented above the effect of the residence time distribution on the grinding results was further investigated. In Fig. 58 calculated particle size distributions are presented for a similar set of operating parameters to those in Fig. 57, but for different Pe-numbers being varied from Pe ¼ 0 to 8. For a Pe-number of inﬁnity characterising plug ﬂow or a batch operation the particle size distribution after 4.5 min of batch grinding is lined out. For this operation all particles are smaller than 6 mm. With Pe ¼ 0 the calculation results in the right-hand curve, which is most disadvantageous: the percentage of particles smaller than 50 mm is even less than 100%. The particle size distributions after continuous grinding at different Pe-numbers are located between the two limiting curves. Depending on the residence time behaviour of the mill the particle size distributions lie closer to the curve of Pe ¼ N or 0. It is remarkable that the residence time distribution has a strong effect on the size and amount of coarser

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335

100

Q3 / %

80 60

Pe = ∞ Pe = 8.00 Pe = 2.00 Pe = 0.50 Pe = 0.00

40 t = tf = 4.5 min Vu = 12.8 ms-1 Cv = 0.2

20 0 10-1

100

101

102

Xst / μm

Fig. 58. Effect of the residence time distribution (characterised by the Pe-number) on the particle size distribution (calculations).

particles, but not on the size and amount of ﬁne particles. This was conﬁrmed by experimental results and theoretical considerations [30]. According to Fig. 58 a relatively low value of the Pe-number (Pe ¼ 8) can already be looked upon as optimal. With the help of similar calculations it is possible to decide whether further improvements of the residence time behaviour of a certain stirrer geometry, of the operating parameters and of the operation mode will have a signiﬁcant effect on the particle size distribution.

5. OPERATION OF STIRRED MEDIA MILLS 5.1. Power draw The production rate is proportional to the power consumed inside the grinding chamber of the stirred media mill and inversely proportional to the speciﬁc energy which is needed to produce the required product quality. The power consumed inside the grinding chamber is the overall power draw decreased by the no-load power. The maximum production rate is gained, if: the power draw is maximal the speciﬁc energy demand for the production of the required product quality is

minimal. The optimisation of the speciﬁc energy is shown in Sections 2.2 and 2.3. In the following it will be explained how the power draw depends on the operating and geometric parameters and, thus, how the power draw can be maximised.

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5.1.1. Power-number diagram without grinding media The power draw is estimated by suitable models developed in stirring technology. The basis of this model is the transfer of power draw behaviour of geometrically similar stirring systems (in this case stirred media mills) by means of two characteristic numbers: the Power-number Ne and the Reynolds-number Re. Powerand Reynolds-number are deﬁned as follows: Power-number: P GC Ne ¼ 5 ð43Þ dd n3 rSusp Reynolds-number: Re ¼

nd 2 rSusp ZSusp

ð44Þ

where PGC (W) is the power consumed inside the grinding chamber, dd (m) the diameter of stirrer discs, n (s1) the number of revolutions, rSusp (kg m–3) the density of product slurry and ZSusp (Pa s) the dynamic viscosity of product slurry. In analogy to stirring technology it is assumed that the Power-number only depends on the Reynolds-number. Tests with a Newtonian ﬂuid but without grinding media were carried out in a stirred media mill with disc-stirrer geometry to obtain a theoretical relation between the Power-number and the Reynoldsnumber. The results are shown in Fig. 59. Three different regions can be distinguished: Reo1.2 102 1.2 102oReo3.5 104 Re43.5 104

1. A laminar region 2. A transition region 3. A turbulent region 103

d / D = 0.8 L / D = 2.5 Z=8

102

PGC

Ne =

d b5·n3·ρSusp.

1 101 2 100

10-1 100

3

101

102 Re =

103

104

105

106

d b2 ·n·ρSusp. ηSusp.

Fig. 59. Relation between Power-number and Reynolds-number without grinding media.

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In each region the results can be described by a straight line. The laminar region (1) can well be described by the theory of a cylindrical stirrer. The Powernumber Ne for this region can be calculated with equation (45). According to this the disc-stirrer geometry behaves like a cylinder with the diameter of the discs. Ne0 / k laminar Re1 Ne0 ¼ 4p3

LS 1 Re1 d d 1 ðd d =DGC Þ2

ð45Þ

where LS is the length of stirrer from ﬁrst to last disc, dd the diameter of discs, and DGC the diameter of grinding chamber In the transition region (2) the results are described by the model of rotating disc in a laminar ﬂow ﬁeld (equation (46)). The ﬂuid ﬂow ﬁeld around a disc is not affected by an adjacent disc or the grinding chamber wall. Ne0 / k transition Re0:5 Ne0 ¼ z 11:968 Re0:5

ð46Þ

where z is the number of stirrer discs. The turbulent region (3) can be described by the model of a single disc in a turbulent ﬂow ﬁeld. Based on the boundary layer theory Schlichting [51] derived equation (47) which describes this region. Ne0 / k turbulent Re0:2 Ne0 ¼ z 0:517 Re0:5

ð47Þ

5.1.2. Power-number diagram with grinding media In the previous subsection the power consumption of a stirred media mill without grinding media, i.e. only with a Newtonian ﬂuid, was discussed. In practice a stirred media mill is operated always with grinding media. In case of an operation with grinding media the question arises, whether the grinding media belong to the suspension or to the mill itself: if the grinding media belong to the suspension, for the physical characteristics of the density and the viscosity the values of the grinding media-product-suspension must be used. In this case the relationship between the Power- and Reynolds-number shown in Fig. 59 is also valid for operations with grinding media. However, in practice the viscosity of the grinding media-product-suspension cannot be measured by a viscometer. Moreover, such a grinding media-product-suspension does not have a Newtonian ﬂow behaviour. Therefore, even if the viscosity could be measured correctly, the problem arises of which characteristic shear gradient exists in the mill and at which shear gradient the viscosity should be measured.

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If the grinding media are considered part of the mill (like bafﬂes in a stirred vessel) the physical characteristics of the product-suspension can be taken for the suspension density and the suspension viscosity. But in this case the problem arises that for each ﬁlling ratio of the grinding media, for each grinding media density and for each grinding media size a different relationship between the Power- and Reynolds-number exists. Both considerations have their beneﬁts and have been used. If the grinding media are considered as part of the suspension, a predetermination of the Reynolds-number for an unknown product suspension and thus, a predetermination of the Power-number out of a Power-number diagram is not possible. The viscosity of the grinding media-product-suspension can only be determined by the following procedure: the Power-number has to be measured in a laboratory grinding test. Using the Power-number the Reynolds-number has to be determined using the Power-number diagram of the mill. Based on the Reynolds-number the viscosity can be calculated. If the viscosity of the grinding media-product-suspension is known for a certain shear gradient, at least for mills which are geometrically similar and which are operated at a similar mean shear gradient, the Reynoldsnumber can be calculated and based on this the Power-number can be determined. Particularly, this procedure can be used for scale-up (see Section 6). If the grinding media are considered as part of the mill, the relationship between the Power- and Reynolds-number must be determined for each conﬁguration of ﬁlling ratio, density and size of the grinding media. After determination of these relationships the Reynolds-number, the Power-number and thus, the power draw of the mill can be determined for different viscosities of the product-suspension and for different numbers of revolution. Figure 60 shows the Power-number as function of the Reynolds-number for a mill with disc stirrer which was operated without grinding media (continuous line) and with three different ﬁlling ratios of grinding media. For each ﬁlling ratio of the grinding media a different relationship between the Power- and the Reynolds-number exists. The greater the Power number at a constant Reynolds-number, the higher is the ﬁlling ratio of the grinding media. In detail it can be seen that the curves for the operation with grinding media can be distinguished into ﬁve regions each with a different slope: 1. Laminar region: Reo1.2 102: 2. Lower transition region: 1.2 102oReo8 103: 3. Upper transition region: 8 103oReo3.5 104: 4. Lower turbulent region: 3.5 104oReo2 105: 5. Upper turbulent region: Re42 105:

Ne0 ¼ Klaminar Re1 Ne0 ¼ Ktransition,A Re0.5 Ne0 ¼ Ktransition,B Re0.3 Ne0 ¼ Kturbulent,A Re0.2 Ne0 ¼ Kturbulent,B

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Based on the equations above, for each region of the Reynolds-number a different relationship between the power draw and the operating parameters (number of revolutions or stirrer tip speed, density of the product suspension, viscosity of the product suspension and stirrer diameter) arises: !Y n d 2d rSusp P GC / ZSusp d 5d n3 rSusp Y 3þY Y ) P GC / n3þY d 5þ2Y r1þY d 2þY r1þY d d Susp ZSusp / v t Susp ZSusp

ð48Þ

where 2

Y ¼ 1 for Reo1.2 10 ; 2 3 Y ¼ 0.5 for 1.2 10 oReo8 10 ; 3 4 Y ¼ 0.3 for 8 10 oReo3.5 10 ; 4 5 Y ¼ 0.2 for 3.5 10 oReo2 10 ; 5 Y ¼ 0 for Re42 10 . The inﬂuence of ﬁlling ratio of grinding media, jGM, and grinding media size, dGM, on power draw can be determined empirically and thus, are added to equation (47): 2þY Y P GC / jXGM v 3þY r1þY d ZGM t Susp ZSusp d d

ð49Þ

The exponent Y depends on the Reynolds-number and results from equation (48). The exponents X and Z can only be estimated from experimental results.

5.1.3. In£uence of important operating parameters The inﬂuence of important operating parameters on the power draw of stirred media mills was investigated by Weit [30]. The effect of the ﬁlling ratio of grinding media on the relation between Power- and Reynolds-number is shown in Fig. 60, in which the ﬁlling ratio was varied from 0 to 0.9. The Power-number and thus the power draw increases with increasing ﬁlling ratio of grinding media for a constant Reynolds-number. From the shape of the curve it follows for the mill with disc-stirrer geometry investigated by Weit that the exponent X of the ﬁlling ratio jGM is in the range of 2.8 for low Reynolds-numbers and 2.2 for high Reynoldsnumbers. The effect of the stirrer tip speed on the power draw can directly be derived from the relationship between the Power- and the Reynolds-number, i.e. the exponent is between 2 for laminar ﬂow and 3 for turbulent ﬂow. Against that the effect of the grinding media size on the power draw is not straight forward. According to Weit [30] and Kwade, the power draw remains either constant, decreases or increases when the media size is changed. Usually

340

A. Kwade and J. Schwedes 102

PGC

grinding media : glass dGM = 1 mm

ϕGM = 0.65 101

Ne =

d 5d ·n3·ρSusp.

ϕGM = 0.90

100 ϕGM = 0.35

10-1

ϕGM = 0.0 102

103 Re =

104

105

106

d2 ·n·ρSusp. ηSusp.

Fig. 60. Relation between Power-number and Reynolds-number of stirrer for different ﬁlling ratios of grinding media.

the power draw increases with increasing media size in the turbulent region and remains constant for a certain viscosity or turbulence range, respectively. For low Reynolds-numbers (low turbulences, high viscosity) the power draw is higher for smaller grinding media. An explanation for this behaviour can be attributed to the different mechanisms of power transfer from the stirrer to the grinding media. For example at low Reynolds-numbers the power draw is determined mainly by the effect of the media size on the viscosity of the grinding mediaproduct-suspension, which increases with smaller media sizes. Against that at high Reynolds-numbers the hits between the grinding media are more important for the power consumption of the mill. Therefore, in principle the exponent is less than zero at low Reynolds-numbers and higher than zero at high Reynoldsnumbers. The inﬂuence of the suspension density on the power draw is lower than the inﬂuence of the ﬁlling ratio and the tip speed (see [1,30]). Along the same lines of thinking as above, the value of the exponent 1+Y of the density is between 0 (low Re-numbers) and 1 (high Re-numbers). This means that the media density has only a minor effect on the power draw. The inﬂuence of viscosity (product viscosity or combined product-media-viscosity depending on how the grinding media is appointed) is also low. The calculations show an exponent from 1 (low Re-numbers) to 0 (high Re-numbers). In the turbulent regime the product viscosity does not affect the power draw.

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341

5.1.4. In£uence of mill geometry In addition to the operating parameters, the power draw is inﬂuenced by the stirrer and the grinding chamber geometry. The use of the equations above also gives information about the inﬂuence of the stirrer disc diameter dd. The resulting exponent 2+Y is 1 for the laminar regime and 2 for the turbulent regime. Another geometric parameter is the number of stirrer discs. The inﬂuence of the disc-number is shown in Fig. 61. For these experiments at the Institute for Particle Technology, Braunschweig, one grinding chamber of constant length and diameter was equipped with different numbers of discs, therefore the disc spacing, ad, also varied from 17.7 mm to 134.4 mm. Starting with one single disc, the power draw increases with increasing number of discs. However, beyond 8 discs almost no increase in power draw can be seen if the number of discs is further increased. This means that in principle an increase in power draw can be gained with an increase in number of discs, but only up to a certain limit.

5.1.5. Summary on power draw The following table gives an overview on how the different parameters inﬂuence the power draw of stirred media mills with a perforated disc stirrer. The exponents are given for the lower transition regime and the upper turbulent regime (Table 4).

Fig. 61. Inﬂuence of grinding media density on power draw. Table 4. Exponents of operating parameters regarding power consumption Inﬂuencing parameter Exponent Low number of revolutions, high viscosity (1.2 102oReo8 103) High number of revolutions, low viscosity (Re105)

jGM X 2.8

Y 0.5

2.2

0

vt 3+Y 2.5

rSusp 1+Y 0.5

ZSusp –Y 0.5

dd 2+Y 1.5

dGM Z r0

3

1

0

2

Z0

342

A. Kwade and J. Schwedes

Using the relation for the power draw and a relation for the speciﬁc energy based on equation (32) the production rate can be predicted [52].

5.2. Pressure and packing of grinding media The ﬂuid forces acting on the grinding media are directed towards the grinding chamber outlet and, consequently, cause a higher concentration of the grinding media near the outlet. If the volume near the outlet is ﬁlled entirely with a bulk of grinding media, packing of grinding media occurs. Owing to the increasing grinding media concentration near the outlet the pressure and the power draw increase. Possible consequences of grinding media packing are:

Increase of power input Increase of pressure in grinding chamber Increased grinding media wear Increased wear of mill Local overheating of product

The known relation between the grinding or dispersion result and the speciﬁc energy looses its validity when packing occurs. Grinding or dispersion results that were achieved under packing conditions are hardly reproducible, since slight changes of operating conditions have a big impact on the product quality.

5.2.1. Experimental results on media packing According to Stehr’s [53] systematic investigations with a vertical pin-counter pin stirred media mill, depending on the product an increase of power input and/or an increase in grinding chamber pressure indicate grinding media packing. At the Institute for Particle Technology of the Technical University of Braunschweig radiometric densitometry was performed for a more accurate examination of grinding media packing instead of using the indirect measurement signals. Thereby, the axial grinding media distribution in a stirred media mill with perforated disc stirrer and glass grinding chamber was measured with a radiator– detector arrangement (Fig. 62). The gamma rays emitted from the radiator are weakened by the grinding media located in the path of rays. The more grinding media are existent in the path of rays (e.g. when grinding media packing occurs), the stronger is this weakening and the weaker the detected signal. Figure 63 shows a measurement result of a stirred media mill with perforated disc stirrer and rotating gap: the increase in grinding media concentration and in local ﬁlling ratio, jGM, local, from the left to the right-hand side (ﬂow direction) is clearly recognisable. The decrease of grinding media concentration between two

Wet Grinding in Stirred Media Mills

343

radiator

detector shaft GC wall

GC wall

Fig. 62. Measurement of grinding media distribution with the aid of radiometric densitometry.

stirrer disc

measuring plane

ϕGM, local [ - ]

1.0

0.8

0.6

0.4 0.0

0.2

0.4 0.6 l / LGC [ - ]

0.8

1.0

Fig. 63. Grinding media distribution in a stirred media mill with perforated disc stirrer at existent grinding media packing [1].

discs is remarkable. The stronger the grinding media packing, the larger is this decrease. The examination of the ﬂow events in a stirred media mill gives a possible explanation for the decrease of grinding media concentration in front of a perforated disc (Fig. 64). There are radial-axial-turbulences between two discs. During continuous operation those turbulences are superimposed by the main ﬂow directed from the entrance to the outlet. This main ﬂow forces the grinding media to move to the grinding chamber outlet. In this manner grinding media packing is caused. The effect of the main ﬂow is particularly distinct at the grinding chamber wall. This is caused by the occurring centrifugal forces leading to a high grinding media density at the grinding chamber wall. Moreover, the current is less inﬂuenced by stirring elements in this area of the grinding chamber. The radial-axialturbulences cause a compensation ﬂow near the shaft. Thus, grinding media are transferred towards the grinding chamber entrance. However, the perforated

344

A. Kwade and J. Schwedes

Fig. 64. Flows and ﬂow forces in a stirred media mill.

discs act as obstacles for the moving grinding media so that the grinding media concentration is higher at the right-hand side of the disc than at the left-hand side.

5.2.2. Grinding media distribution model On the basis of investigations with a stirred media mill with a glass chamber, Thiel [32] derived a model for the grinding media distribution in the grinding chamber. This model assumes that a grinding chamber with length L can be divided in two zones (see Fig. 65) with different ﬁlling ratios of grinding media. In zone 1 the ﬁlling ratio of grinding media is lower than the average ﬁlling ratio (jGM1ojGM). The grinding media are densely packed in zone 2 with the characteristic packing ﬁlling ratio jGM2. The length of the zone with resting grinding media is characterised by the packing length lp. For a known ﬁlling ratio of the packed zone, jGM2, the ﬁlling ratio jGM1 can be calculated with the overall ﬁlling ratio jGM and the ratio of lengths according to the following equation: jGM1 ¼

L jGM l p jGM2 L lp

ð50Þ

Since the ﬁlling ratio jGM1 cannot become negative, the relative packing length lp/L is limited by the relation of the ﬁlling ratios jGM and jGM2. The relative packing length is given by: lp j GM L jGM2

ð51Þ

Thiel carried out experiments with a coal/water slurry with two different pincounter pin stirrers and could show that the relative packing length is proportional

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345

Fig. 65. Model of grinding media distribution with two zones with different ﬁlling ratios of grinding media.

to the volume ﬂow rate or ﬂow velocity, respectively, and that the viscosity is inversely proportional to the tip speed. By using an additional approach for the ﬂuid ﬂow through a bulk the inﬂuence of grinding media size could be described as well. A proportionality for the packing length (equation (51)) can be derived under certain assumptions for the ﬁlling ratio jGM2. V_ Z lp v Z / / 2 L v t d GM v t d 2GM

ð52Þ

Figure 66 shows the obtained correlation for the variation of the operating parameters mentioned in equation (52) for tests with two different pin-counter pin stirrers and a ﬁlling ratio of grinding media of jGM ¼ 80%. It has to be noted that equation (52) is valid for high viscosities. At low viscosities the quotient between ﬂow and centrifugal forces on the grinding may be the decisive factor. The model developed by Thiel [32] can be used to calculate the power input at an existent grinding media packing. According to equation (53) the Powernumber Ne1,2 which arises from the total power input is composed of the Powernumber in range (1) (length L–lp) and the Power-number in range (2) (length lp). Ne1;2 ¼

L lp lp NeðjGM1 Þ þ NeðjGM2 Þ L L

ð53Þ

346

A. Kwade and J. Schwedes 0.8 KWS 0.7

d = 0.252 m; z = 7 ϕGM = 0.8; dGM = 10mm Cm = 0.7 Vt = 5.5 - 4.0 m/s ϕGM 2 = 1.000

relative packing length Ip/L

0.6

d = 0.260 m; z = 9 ϕGM = 0.8; dGM=2mm Cm = 0.4 - 0.55 Vt = 4.0 m/s ϕGM 2 = 1.065

0.5 0.4 0.3 0.2 0.1 0 0

10

20

30 v.η vt.d2GM

40 in

50 Pas

60

70

80

m2

Fig. 66. Relative packing length lp/L in dependency of important operating parameters.

For the case of grinding media packing the total power input Ne1,2 is greater than or equal to the power input Ne (jGM) without grinding media packing. Ne1;2 NeðjGM; without grinding media packing Þ

ð54Þ

For a certain operation mode (Re ¼ const.) and a constant grinding media size the corresponding Power-numbers for the ﬁlling ratios jGM1 and jGM2 can be read off a ﬁgure similar to Fig. 67. Together with the ratio lp/L (results from Fig. 66 for the corresponding operation mode) it is possible to calculate the expected total power input. The dependencies described above show directly how and which operating parameters can be changed to react on media packing. If the product is changed very often, an automatic control to optimise the mill utilisation can be set up for each product. Thereby the power input, tip speed, outlet temperature of product and ﬂow rate are measured. Depending on these operating parameters and on the speciﬁc energy, the tip speed of the stirrer, the throughput and the ﬂow of cooling medium are controlled in a way that grinding media packing is avoided. Besides the change and control of operating parameters it might also be possible to solve a packing problem by choosing a different type of stirred media mill where media separation is improved by a centrifugal ﬁeld.

Wet Grinding in Stirred Media Mills d = 0.252 m, z =7 dGM = 10 mm Newtonian fluid

90

ρSusp.n3.d d5

Newton number Ne =

PGC

80

347

η ~ 0.7 Pas

70 Vt = 5.5 ms-1 Vt = 5.0 ms-1 Vt = 4.3 ms-1

60 50 40 30 20 10 0 0

0.2 0.4 0.6 0.8 1.0 1.2 filling ratio of grinding media ϕGM

1.4

Fig. 67. Power-number Ne depending on the ﬁlling ratio of grinding media.

5.3. Wear Wear of grinding media, of stirrer and stirrer discs, of the grinding chamber and of the separation device, which holds the grinding media back in the grinding chamber, occurs during the operation of media mills. The grinding media are by far subjected to the highest wear. Two stress mechanisms leading to wear occur in mills with loose grinding media: on one hand grinding media are stressed by contacts of two or more grinding media. On the other moving grinding media collide with the moving stirrer, the stationary grinding chamber wall or the moving or stationary media separation device. The wear process is characterised by the approach and stress of two solid bodies in presence of a liquid and solid particles (feed). The prevailing type of wear is abrasive wear.

5.3.1. Wear of mills The mill parts subjected to wear are mainly the stirrer and the discs or pins mounted on it, the grinding chamber wall as well as media separations like rotating separation gap and especially separation screens, which are placed inside the grinding chamber. The wear rate of all mill parts can be reduced by suitable

348

A. Kwade and J. Schwedes

measures. The operating parameters tip speed, grinding media size, grinding media density and ﬁlling ratio of media are supposed to be chosen moderately. Since operating parameters and feed are determined by grinding or dispersion results, the wear rate can also be reduced by using wear resistant materials. Typical materials are: different stainless steels, hard-chrome steel, different polyurethanes, ceramics as well as rubber and natural caoutchouc. In case of processing hard and abrasive materials usually polyurethane or ceramics are used to protect the stirrer and the grinding chamber walls. Because of the bad heat transfer coefﬁcient of polyurethane a typical material combination is polyurethane for the stirrer and ceramics for the grinding chamber. An operating issue might be that the mill needs to be cleaned more or less often. Prior to a cleaning process the mill needs to be rinsed. Rinsing with pure water or solvent can worsen the wear behaviour of mill parts. The velocity of grinding media in pure water or solvent is higher than in a slurry due to the reduced liquid viscosity. Moreover, the collisions are not damped. Thus, the grinding media hit the moving and stationary parts of the mill with a higher energy. Frequent rinsing due to frequent product changes can lead to a signiﬁcant increase in wear compared to normal grinding or dispersion processes. It is recommended to perform the rinsing process at reduced stirrer tip speeds.

5.3.2. Wear of grinding media Wear of grinding media is the most important wear issue. The worn grinding media particles get into the slurry. Thus, the product is contaminated since grinding media and product are usually not out of the same material. This can impair product quality considerably. Furthermore, considerable costs result due to grinding media wear. Several authors used grinding media and products of different hardnesses for grinding in ball mills [54–57]. All investigations have shown that grinding media wear decreases with increasing grinding media hardness and decreasing product hardness. Product particles weaker than the grinding media are not able to penetrate into the media surface and the wear remains low. Product particles harder than the grinding media can penetrate deeply into the media surface causing high media wear. There is a transition region between the high wear and the low wear region. This characteristic behaviour is known for different tribological systems and has been discussed by de Silva [58]. The solid concentration affects the wear [59]: wear was measured to be higher for grinding without solids than for grinding with solids if the product is weaker than the media. Thereby, grinding media wear decreases with increasing solids concentration. It becomes different if the product is harder than the grinding media: at ﬁrst grinding media wear increases with increasing solids concentration until a wear maximum was reached. Media wear decreases again for higher

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349

solids concentrations [60]. This behaviour was explained with a viscosity increase due to higher solids concentrations. The stress events are damped by the higher viscosity. There are basically two approaches to explain the inﬂuence of grinding media size on media wear: according to one approach grinding media wear is proportional to the media surface while according to the other approach grinding media wear is proportional to grinding media volume. Depending on the stress conditions one or the other dominates. Real conditions are often between both approaches [61–63]. Ceramic materials show a much more brittle behaviour than metals. The wear of ceramic materials depends more on structural properties. Small grains (primary materials particles) and low porosities are characteristic for wear resistant ceramic materials. On the other hand, rough surfaces increase wear. Furthermore, the wear of ceramic materials is inﬂuenced by the Young’s modulus and the fracture toughness (see later). High hardness is usually related with high wear resistance, but too high hardnesses and Young’s moduli lead to brittle material behaviour and hence, in some cases to reduced wear resistance [64,65]. To judge grinding media wear it is necessary to weigh the media before and after each test or production step. The mass difference is the loss of grinding media DmGM. Relating DmGM to the mass of grinding media at the beginning mGM yields a characteristic wear value enabling the comparison regarding wear of different grinding media types. w tot ¼

DmGM mGM

ð55Þ

Investigations of Joost [31] have shown that the characteristic wear value correlates with the energy input Etot into the grinding chamber. Since grinding media wear decreases with increasing grinding chamber volume he relates the total energy input Etot to the grinding chamber volume VGC leading to the volumerelated energy EV,GC. Rt PðtÞ dt E tot ¼ 0 ð56Þ E V;GC ¼ V GC V GC Table 5 investigated by Becker [27] shows a survey of frequently used grinding media in industrial processes. It also gives a valuation of wear resistance of different types of grinding media. The evaluation is done by means of mass content of media wear related to the entire mass of solids of the slurry for the grinding of Al2O3 down to a median particle size of x50 ¼ 2 mm.

5.3.2.1. Inﬂuence of operating parameters Neither Joost [31] nor Becker [27] could determine a systematic inﬂuence of the operating parameters tip speed as well as grinding media size and density on the

350

A. Kwade and J. Schwedes

Table 5. Wear resistance of different types of grinding media [27] Density (kg m–3)

Name

Material

Glass Glass

SiO2-content 72% SiO2-content 61% PbO content 24% 100 Cr 6 Al2O3-content 99.7% Al2O3-content 92% Al2O3-content 86.5% Al2O3-content. 85% ZrO2/SiO2 ZrO2/SiO2 Si3N4 SiO2/MgO TiO2 WC ZrO2, Y2O3-stabilised ZrO2, Mg-stabilised ZrO2, Ce-stabilised

Chrome-steel Alumina Alumina Alumina Alumina Mixed oxide 1 Mixed oxide 2 Silicon nitride Steatite Titanium dioxide Tungsten carbide Zirconia Zirconia Zirconia

Young’s Hardness modulus (GPa) (HV5)

2510 2894 7640 3860 3620 3300 3320 3710 3840 3250 2670 4015 15,000 6065 5745 6165

– – 760 1405 1090 900 1015 640 625 1345 505 1095 1535 1345 1040 1075

Wear resistance

72 63

– –

245 355 265 220 235 110 100 335 105 300 625 265 260

+ – – + + O O ++ O + ++ ++ + +

–

grinding media wear. In analogy to the grinding process the grinding media wear can be analysed with a machine related model. The grinding media wear depends on the frequency of wear events (number of wear events per time WNM) the amount of energy at one wear event (wear energy WE).

According to Becker [27], Stender [42] and Stender et al. [66], the energy of a grinding media at one stress event is divided in one part which is transferred to the product particle and one part which remains in the media. Both parts depend on Young’s moduli of product and media material: Y P 1 Y GM 1 E GM ¼ E GM 1 þ þ E GM 1 þ Y GM YP |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ ﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ ﬄ} transferred to product

remains in media

Y GM YP ¼ E GM þ E GM Y P þ Y GM Y P þ Y GM |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} |ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ} transferred to product

ð57Þ

remains in media

The energy remaining in the media is responsible for the media wear. Therefore, a formula for the wear energy of the grinding media WEGM can be derived which is proportional to the wear energy WE: Y GM 1 WE / WE GM ¼ d 3GM rGM v 2t 1 þ ð58Þ YP

Wet Grinding in Stirred Media Mills

351

wtot = ΔmGM / mGM [ - ]

The higher the grinding media diameter, media density and tip speed and the higher the ratio of Young’s modulus of the product material to that of the grinding media material are, the higher is the energy which remains in the grinding media at each wear event. In addition, the higher the ratio of Young’s moduli, the higher the deformation of the grinding media at each wear event. Thus, an increased wear energy and a higher deformation of the grinding media lead to increased media wear. Following these ideas the wear energy increases with increasing media size and increasing tip speed. Thus, a higher media wear should be obtained. But, as Figs. 68 and 69 show, the inﬂuences of the operating parameters on the media wear are different and not unique. Figure 68 shows the relative total media wear wtot versus the volume-related energy EV,GC for the grinding of fused corundum (Al2O3) with yttrium-stabilised zirconium oxide grinding media at different operating conditions of the stirred media mill. As expected, the media wear increases with increasing energy input. However, although different values of media size and tip speed were used leading to different wear energies, nearly no effect on the grinding media wear can be seen. Therefore, different operating conditions and different wear energies of the grinding media WEGM seem not to inﬂuence the media wear. However Fig. 69 shows different results. Different to the experiments leading to the results presented in Fig. 68 mixed oxide grinding media were used. A signiﬁcant inﬂuence of the operating conditions can be seen for these grinding media. With decreasing tip speeds and more signiﬁcantly with smaller grinding medias the relative total media wear wtot decreases considerably. From the two ﬁgures the question arises why different ceramic grinding media materials show

0.01

ϕGM = 0.8

comminuition of fused corundum: Al2O3, YP = 410 GPa grinding media : ZrO2(Y2O3), YGM = 265 GPa

cm = 0.2

dGM = 0.35 mm. vt = 6 m/s dGM = 0.35 mm. vt = 8 m/s dGM = 0.50 mm. vt = 6 m/s dGM = 0.50 mm. vt = 10 m/s dGM = 0.50 mm. vt = 14 m/s dGM = 0.80 mm. vt = 6 m/s dGM = 1.50 mm. vt = 10 m/s dGM = 1.50 mm. vt = 12 m/s dGM = 1.50 mm. vt = 14 m/s dGM = 3.0 mm. vt = 6 m/s

1E-3

1E-4 100

1000

10000

EV,GC [J/cm3]

Fig. 68. Media wear for the grinding of fused corundum with yttrium-stabilised zirconium oxide grinding media at different operating conditions.

352

A. Kwade and J. Schwedes comminuition of fused corundum: Al2O3,

0.5

ϕGM = 0.8

grinding media : mixed oxide ZrO2 / SiO2,

cm = 0.2

YGM = 100 GPa

0.1 wtot = ΔmGM / mGM [ - ]

YP = 410 GPa

0.01 dGM = 0.5 mm. vt = 6 m/s dGM = 0.5 mm. vt = 8 m/s dGM = 0.7 mm. vt = 8 m/s dGM = 0.7 mm. vt = 10 m/s dGM = 1.3 mm. vt = 6 m/s dGM = 1.3 mm. vt = 8 m/s dGM = 1.3 mm. vt = 10 m/s dGM = 1.3 mm. vt = 12 m/s dGM = 1.5 mm. vt = 12 m/s

1E-3

1E-4 50

100

1000

10000 20000

EV,GC [J/cm3]

Fig. 69. Media wear for the grinding of fused corundum with mixed oxide grinding media at different operating conditions.

different behaviour regarding the inﬂuence of the operating parameters. Since changing operating conditions change the wear energy in theory, it has to be explained why only for certain materials the grinding media size seems to have an effect. This problem is closely related to the question of how the operating conditions affect the wear energy. In addition to the above-mentioned model of the grinding and wear mechanisms the relative total media wear can also be expressed by the relative wear per wear event multiplied by the number of wear events: Relative total media wear ¼

relative media wear number of wear events ð59Þ wear event

The relative wear per wear event depends on the wear energy acting at one wear event. Low wear energy leads to a low wear per wear event, a high wear energy leads to a high wear per wear event. However, the relative total media wear is also inﬂuenced by the number of wear events. Therefore, it is possible that a low wear energy ( ¼ low wear per wear event) and a high number of wear events lead to the same relative total media wear than a high wear energy ( ¼ high wear per wear event) and a low number of wear events. Thus, the relative total media wear wtot is not a suitable measure for the identiﬁcation of the inﬂuence of the operating parameters on the media wear. According to equation (59) such a suitable measure is the relative wear per wear event which can be calculated by dividing the relative total media wear by the number of wear events. Since this number cannot be calculated reliably, the

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media wear per wear event will be calculated according to the following equation: Relative media wear Relative media wear energy ¼ wear event energy wear event

ð60Þ

The energy per wear event is the acting wear energy during one wear event (WE) which can be estimated by the wear energy of the grinding media, WEGM. The relative media wear per energy can be estimated from measurements regarding wear and energy. For this, an energy related wear rate kGM,E will be used. The calculation of kGM,E follows a procedure explained in [66]. Relative media wear / w e ¼ k GM;E WE GM wear event

ð61Þ

Equation (61) deﬁnes we as wear per wear event (being proportional to the relative wear per wear event) as the product of energy related wear rate kGM,E and wear energy of the grinding media WEGM. Figure 70 shows the relative wear per wear event we for the grinding of fused corundum with yttrium-stabilised zirconium oxide grinding media at different operating parameters (as in Fig. 68). Contrary to the relative total media wear wtot the relative wear per wear event we is inﬂuenced by the operating parameters. With increasing values of grinding media size and tip speed and therefore, increasing wear energy of the grinding media, WEGM, higher values of the relative wear per wear event we result. A similar ﬁgure results from the grinding of fused corundum with mixed oxide grinding media at different operating parameters (from Fig. 69) (Fig. 70). ϕGM = 0.8 cm = 0.2

we = kGM,E · WEGM [ - ]

1E-8

dGM [mm] 0.35 0.35 0.5 0.5 0.8 0.5 1.5 1.5 1.5 3.0

1E-9

1E-10

1E-11 50

100

1000

10000

vt [m/s] 6 8 6 10 6 14 10 12 14 6

WEGM [10-3 Nm] 0.006 0.010 0.017 0.046 0.068 0.091 1.247 1.796 2.445 3.593

30000

EV,GC [J/cm3]

Fig. 70. Relative wear per wear event for the grinding of fused corundum with yttriumstabilised zirconium oxide grinding media at different operating conditions.

A. Kwade and J. Schwedes wtot = ΔmGM / mGM [-]

354 0.1 0.01 1E-3 1E-4 1E-5 1E-6 we = kGM, E · WEGM [-]

EV, GC = 500 J/cm3

1E-7 1E-8

wtot = ΔmGM/ mGM steatit mixed oxide ZrO2 (Y2O3) we = kGM, E· WEGM steatit mixed oxide ZrO2 (Y2O3)

1E-9 1E-10 1E-11 0.01

0.1

WEGM = dGM3 · ρGM · vt2 · (1 + YGM / YP)-1

1

10 [10-3 Nm]

Fig. 71. Relative total media wear wtot and media wear per wear event we versus wear energy of the grinding media WEGM at a constant volume-related energy of EV,GC ¼ 500 J cm–3 for the grinding of fused corundum (Al2O3).

An explanation why Figs. 68 and 69 seem to show different results is given by Fig. 71. The relative total media wear wtot and the relative media wear per wear event we are plotted versus the wear energy of the grinding media WEGM for the grinding of fused corundum with steatite, mixed oxide and yttrium-stabilised zirconium oxide grinding media at a constant volume-related energy of EV,GC ¼ 500 J cm–3. For the yttrium-stabilised zirconium oxide grinding media the relative total grinding media wear wtot does not change, while the relative media wear per wear event we changes with increasing wear energy of the grinding media WEGM. If the energy input is kept constant (EV,GC ¼ const.), an increase in wear energy results in a decrease of the number of wear events. For the yttrium-stabilised zirconium oxide grinding media the increase in relative wear per wear event we and the decrease in the number of wear events are equal. Thus, a low wear energy and a high number of wear events lead to the same relative total wear than a high wear energy and a low number of wear events. An almost similar effect can be seen for the steatite grinding media. However, the values are fundamentally higher for the steatite media. For the mixed oxide grinding media the increase in wear per wear event we is higher than the decrease in number of wear events, therefore an increase in relative total wear wtot is obtained with increasing wear energy WEGM. Regarding the inﬂuence of the operating parameters on the grinding media wear the effect of the wear energy during one single wear event and the number of wear events have to be considered.

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5.3.2.2. Inﬂuence of structural constitution of ceramic media The structural constitution of ceramic media and their wear behaviour are tightly linked to each other. Ceramic materials are frequently composed of so-called primary particles or grains. Their size depends partly on the particle size of the powder used for production. Different additives are added to the powder. The forming stage is normally followed by sintering of the ceramic parts. Solid bridges are generated between the particles during the sintering process so that the mass gets a solid structure. According to zum Gahr [64] the primary particle size has a strong inﬂuence on the wear resistance of the material. He mentions wear tests, in which ceramic bodies with different primary particle sizes were cut with a diamond saw. Thereby, wear resistance increased with decreasing primary particle size. Becker [27,67] used ceramic grinding media with different primary particle sizes for grinding tests. Figure 73 shows results for the grinding of fused corundum with different ceramic grinding media, all having the same size (1.3 mm). Two groups of grinding media can be distinguished: one group is characterised by a continuous grinding progress with increasing speciﬁc energy. The grinding media made from steatite, zirconia (Y2O3-stabilised), silicon nitride and the alumina media with a Al2O3-content of 86.5% belong to this group. For the other group of grinding media the curves ﬂatten early (alumina media with an Al2O3content of 99.7% and 92%). The achievable ﬁneness is limited for these grinding media. In spite of an increasing energy input it is impossible to produce products with a mean particle size of o1 mm (Al2O3-content of 92%) and o1.3 mm (Al2O3content of 99.7%), respectively. The reason can be found in the structural constitution of the grinding media. Figure 73 shows SEM photos of the surfaces of two different types of grinding media belonging to different groups [27,67]. Both media types were used for the experiments the results of which were used in Fig. 72. The grinding media with an Al2O3-content of 99.7% are composed of relatively large primary particles (5–10 mm). The structure of the surface shows that entire primary particles are ruptured out of the media surface during the grinding process. These primary particles get into the product so that the product is permanently contaminated with wear particles of 5–10 mm size. The product particle size is limited due to the permanent supply of large particles into the product. This leads to the ﬂattening of the curves in Fig. 72. In contrast the alumina grinding media with an Al2O3-content of 86.5% belong to the group of grinding media showing a continuous progress in ﬁneness with increasing speciﬁc energy. These grinding media are composed out of smaller primary particles (up to 1.5 mm). The wear mechanism is different for this type of grinding media. The surface shows that small wear particles are ground off the primary particles. These small wear particles are a lot smaller than the primary particles and therefore, they are not able to limit the achievable particle size in the investigated particle size range.

356

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median particle size x50 [ µm]

30 grinding media: alumina (99.7 % Al2O3) alumina (92 % Al2O3) alumina (86.5 % Al2O3) steatite zirconia (Y2O3-stab.) silicon nitride

10

vt = 12 m/s ϕGM= 0.8 cm = 0.2 dGM = 1.3 mm

1

0.4 200

1000

10000 specific energy Em [kJ/kg]

50000

Fig. 72. Dependency of grinding result on grinding media material for grinding of fused corundum [27,67].

5.3.2.3. Inﬂuence of grinding media and product hardness For the group of grinding media showing a continuous increase in ﬁneness with increasing speciﬁc energy input (see Fig. 72 and Fig. 73 right) grinding media wear depends essentially on the hardness of the product and the grinding media. Figure 74 shows the dependency of the characteristic wear values on grinding media hardness for the grinding of fused corundum, silicon carbide and quartz at a constant volume-related energy of EV,GC ¼ 1000 J cm–3. This energy is chosen arbitrarily. In principal the same tendencies between grinding media wear and grinding media hardness are found for different speciﬁc energies. The grinding media wear decreases with increasing media hardness for a certain product. During the stress events product particles penetrate partly into the grinding media surface and rupture wear particles out of this surface. The ability of product particles to penetrate into the media surface decreases with increasing grinding media hardness and thus, grinding media wear is reduced as well. Different relations between grinding media wear and volume-related energy are obtained for products with different hardnesses. Different hardnesses of the products (quartz 1400 HV; fused corundum 2200 HV; silicon carbide 2650 HV) are the reason for this behaviour. Grinding media wear increases with increasing product hardness for constant grinding media hardness. The ability of product particles to penetrate into the media surface increases with increasing product hardness. Thus, higher wear is caused by harder products. For the grinding of ceramic materials Becker [27] has shown that grinding media wear decreases with

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357

Fig. 73. Surfaces of two types of grinding media belonging to different groups.

grinding media wear ΔmGM/mGM [ - ]

0.5 silicon cabide (2650 HV) fused corundum (2200 HV) quartz (1400 HV) 0.1

0.01 EV,GC = 1000 J/cm3

1E-3 400

500

600 700 800 900 1000 grinding media hardness HGM [ HV5 ]

Fig. 74. Effect of grinding media hardness on media wear for different products.

2000

358

A. Kwade and J. Schwedes

increasing ratio of grinding media hardness and product hardness HGM/HP. Hardness of product particles can normally not be changed for a certain grinding task. To reduce media wear as much as possible one is supposed to choose grinding media with a high hardness, which are composed of small primary particles. Moreover, according to Uetz [54] the ‘‘sharpness of a grain’’ is particularly important in the transition region between low and high wear. The ‘‘sharpness’’ of a grain is independent of the grain size and depends only on the shape of the grain. The sharper a grain is, the higher its ability to penetrate into the media surface and cause media wear.

5.4. Autogenous grinding Wear of grinding media and product contamination caused by media wear are a great problem, e.g. in the ceramic industry for the case that the product and grinding media are not made out of the same material. Furthermore, grinding media wear is expensive for the mass production of silicates and ores. One solution to this problem is the autogenous grinding where the grinding media are replaced by coarse particles of the product material. Kwade [18,47] investigated the autogenous grinding of limestone and fused corundum in batch tests in a stirred media mill with disc-stirrer geometry. Besides autogenous pebble grinding also fully autogenous grinding (without feed, only grinding media) has been tested. The continuous autogenous grinding of limestones has been realised, too. Tests with limestone and fused corundum show that a product smaller than 10 mm can be produced by autogenous grinding. In the following the principal course of the batch autogenous pebble grinding process is explained by discussing the changes in the particle size distribution of the hold-up. In Fig. 75 the cumulative size distribution by mass, Q3, is depicted as a function of particle size and grinding time. With the exception of the absence of ﬁne feed a fully autogenous grinding process runs comparably. Before starting the grinding process (grinding time of 0 min) the hold-up is composed of the grinding media (about 200–1600 mm) and the ﬁne feed (about 20–100 mm). The particles smaller than 20 mm are rubbed-off particles, which were ﬁlled into the grinding chamber with the grinding media. The proportion of the grinding media mass to the overall mass is 83% and the one of the feed and the rubbed-off particles is 17%. After starting the grinding process the feed particles are quickly reduced in size by the grinding media and a ﬁne product with particle sizes smaller than 10 mm results. Besides the feed particles small grinding media, which are stressed between large grinding media as well as grinding media with internal cracks, are reduced in size by complete fracture. After a grinding time of 1 min nearly all feed particles are broken so that further increase in product mass is caused by the decrease in mass of the grinding media. After a grinding time of 48 min

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359

1.0 0 min 1 min 3 min 6 min

0.9 0.8 0.7

12 min 24 min 48 min

0.6 Q3 [-]

dGM= 848 µm vt = 10 m/s ϕGM,t=0 = 0.8 cm,t=0 = 0.25

0.5 0.4 0.3 0.2 0.1 0.0 0.3

1

10 100 particle size x [µm]

1000

3000

Fig. 75. Particle size distributions of entire fractions for autogenous pebble grinding of limestone [18].

approximately 56% of the overall solids mass is product, which is ﬁner than 5 mm. Because of the increase in product mass and because the amount of water does not change, the solids concentration of the product-water-slurry rises steadily. Simultaneously, the ﬁlling ratio of grinding media drops because of the decrease in mass of the grinding media. After a grinding time of 48 min more than half of the product mass belonged originally to the grinding media. There are almost no particles in the size range between product and grinding media fraction. Therefore, this range is called the ‘‘particle size gap’’. Figure 75 shows that after one minute a particle size gap is already formed between 20 and 100 mm. This gap increases with increasing grinding time. No signiﬁcant amount of particles can be detected between 5 and 200 mm after 48 min. A sieve analysis shows that the amount of particles between 20 and 160 mm is less than 0.01% after 48 min. The particle size gap is important for the separation of product and grinding media. Since the upper boundary of the particle size gap is higher than 100 mm and the maximum product particle size is below 10 mm a product smaller than 10 mm can be obtained by separating grinding media and product at 100 mm. A separation at 10 mm is not necessary. The formation of the particle size gap during the autogenous grinding process is closely related to the change in particle size distribution of the grinding media, i.e. of the particles greater than 90 mm. Figure 76 shows the cumulative size distribution by mass of the grinding media for ﬁve different grinding times. It can be clearly seen that with increasing grinding time the size distribution moves to

360

A. Kwade and J. Schwedes 1.0 0.05 0.9 0.8

Q3 [-]

t = 0 min t = 1 min t = 3 min t = 12 min t = 48 min

0.04

0.7

0.03

0.6

0.02

0.5

0.01

0.4 0.00 80 100

0.3

200

500 dGM = 848 µm vt = 10 m/s ϕGM,t=0 = 0.8 cm,t=0 = 0.25

0.2 0.1 0.0 80

100 particle size

500 x [µm]

1000

2000

Fig. 76. Change of particle size distribution of grinding media fraction with time for autogenous grinding of limestone.

smaller particle sizes through abrasion and fracture phenomena. An exception to this trend can be found in the particle size range below 500 mm, i.e. for values of Q3 below 0.05. This area indicated by the dashed line is depicted enlarged in the upper left corner of Fig. 76. The mass of grinding media in the particle size range below approximately 300 mm decreases with increasing grinding time and becomes nearly zero for particle sizes smaller than 200 mm. Therefore, the lower part of the size distribution moves to larger particle sizes and the particle size gap becomes wider. The reason for the decrease in mass of small grinding media is that small grinding media are captured and stressed by larger grinding media. The resulting fragments are further broken quickly as was shown for the feed particles. Similar courses of the grinding process are obtained for fused corundum. Yet the feed fraction is ground signiﬁcantly slower compared to limestone as shown in Fig. 77. Furthermore, the amount of material per unit time that gets from the grinding media into the product (due to abrasion and breakage) is smaller than for limestone. After a sufﬁcient grinding time the particle size gap becomes at least as wide as for limestone. For autogenous grinding the product ﬁneness is limited due to the permanent supply of wear particles from the grinding media into the product. The maximum ﬁneness depends on the structural constitution of the material. Limestone and fused corundum are composed out of primary particles bound in a matrix. The size of these primary particles determines the maximum achievable product ﬁneness.

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Q3 [1]

1.0 0.9

0 min 1 min

0.8

3 min 6 min

0.7

12 min

0.6

24 min 48 min

vt = 10.7m/s ϕGM,t=0 = 0.8 cm,t=0 = 0.2

96 min

0.5

192 min 0.4 0.3 0.2 0.1 0.0 0.4

1

10 100 particle size x [µm]

1000 2000

Fig. 77. Particle size distribution of entire fraction for autogenous grinding of fused corundum.

The product ﬁneness is mainly determined by the speciﬁc energy and the stress energy. From grinding processes using grinding media it is known that besides the speciﬁc energy the ﬁlling ratio of grinding media affects the grinding result (see Section 3.2): at constant speciﬁc energy input the product becomes ﬁner with increasing ﬁlling ratio. This ﬁnding can be explained as follows: in stirred media mills the energy is transferred from the stirrer to the slurry and the grinding media. The larger the amount of grinding media there is and, therefore, the larger the ﬁlling ratio is, the larger the portion of the energy which is transferred to the grinding media and not to the slurry. Thus, the product of ﬁlling ratio of grinding media and speciﬁc energy, jGM. Em, is a measure for the portion of the speciﬁc energy, which can be used for grinding. Kwade [18] compared the speciﬁc energy requirement for autogenous grinding and grinding with glass media. The tests with glass media were carried out by Stehr [29]. In Fig. 78 the integral mean value of the particle size distribution, x 1.3, is depicted as function of the weighted speciﬁc energy for autogenous grinding and grinding using glass beads. The comparison shows that the mean sizes agree relatively well, although the values for autogenous grinding lie slightly higher than the ones for glass beads. Especially for fully autogenous grinding a little more speciﬁc energy is required to obtain a certain particle size. Therefore, the autogenous grinding is an economical alternative to grinding with grinding media since product contamination can be avoided and costs for grinding media are negligible.

362

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x1,3 [µm]

100 autogenous glass media dGM 848 / 907 µm 1000 µm 8 - 14 m/s 6.4 - 14.4 m/s vt ϕGM 0.7 - 0.8 0.8 cm,feed 0 - 0.54 0.23 - 0.54 cm,p 0.09 - 0.55 0.23 - 0.54 • Va 20 - 140 l/h 20 - 140 l/h

10

autogenous pebble comminution fully autogenous comminution comminution with glass media

1 0.5 10

100 ϕGM · Em,p [kJ/kg]

1000

3000

Fig. 78. Comparison of continuous autogenous grinding and continuous grinding with glass media [18,47].

6. SCALE-UP Before a new stirred media mill is bought for a production site, questions regarding the size and the type of the machine have to be answered. For small production quantities a test of the mill in the original size is possible. But often the decision for the right production mill has to be made on the basis of laboratory or small scale tests. The aim of scale-up of stirred media mills is to obtain information about production-scale mills based on one or a few tests with a lab-scale mill. Based on a desired production rate the geometry and size of the mill, mill wear and grinding media wear as well as cooling capacity have to be determined. Many of these questions are closely related to equation (62) since a stirred media mill is used most economically if the required product quality is produced with maximum _ p , at lowest possible operating costs. production rate, m _p ¼ m

mP P ¼ Em t

ð62Þ

The required power draw of the machine follows from equation (62) for a given production rate with a certain speciﬁc energy (due to the required product quality). From the value of the required power draw certain conclusions regarding the geometry and size as well as the operating parameters have to be drawn.

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To answer the above questions more or less exact methods can be used. Both types of methods will be addressed in the following sub-sections.

6.1. Practical methods – consideration of cooling area of grinding chamber 6.1.1. Stirred media mills with disc stirrer Typical grinding chamber volumes for continuously operated stirred media mills with disc or pin-counter pin stirrers are in the range from about one litre up to several thousand litres. The grinding chamber volume is proportional to the third power of the grinding chamber diameter for a constant ratio of length to diameter (L/D E constant). The cooling area increases only with the square of the diameter. 2=3

V GC / D3 ; SGC / D2 ) SGC / V GC

ð63Þ

According to Stehr [68] the product throughput and the installed motor power do not increase linearly with the mill volume, because the ratio of power to cooling area has to be constant if the product temperature needs to be constant (especially for heat sensitive products). P ¼ const: SGC

ð64Þ

Under the assumption that a constant speciﬁc energy is necessary to get the same grinding result in mills of different size, Em ¼

P ¼ const: _ m

ð65Þ

Equations (63)–(65) lead to the result that the production rate is proportional to the cooling surface of the mill and thus, to the grinding chamber volume to the power of (2/3): _ / P / SGC / V 2=3 m GC

ð66Þ

Therefore, the resulting production rate and the necessary power draw of the machine increase with the ratio of the grinding chamber volumes of production and lab-scale mill to the power of 2/3: V GC;P 2=3 _P ¼m _L m ð67Þ V GC;L V GC;P 2=3 PP ¼ PL V GC;L where P is the production scale and L is the laboratory scale.

ð68Þ

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A. Kwade and J. Schwedes

_ L or the power draw PL and the If after a laboratory test the mass ﬂow rate m speciﬁc energy Em are known for scale-up, the size of the production mill can be determined as follows: V GC;P ¼ V GC;L

3=2 3=2 _P _P m m ¼ V GC;L _L m P L =E m

ð69Þ

Judgement: Some comments have to be made on the scale-up of production rates by consideration of cooling area or a constant ratio of power to cooling: 1. The assumption that in different machine sizes the same speciﬁc energy is needed for the production of the same product quality is not correct. Owing to different media movements in the different machines and because of different energy transfer factors, the speciﬁc energy consumption changes with respect to machine size and geometry. 2. This method does not estimate the power draw of the production size mill. It estimates the necessary power draw that is needed for the required production rate. In practice the operating parameters are tuned in a way that this power draw is reached (change of ﬁlling ratio and tip speed). However, the required power draw cannot always be reached. In addition, a change of operating parameters changes the stress energy (e.g. SEGM) and therefore the needed speciﬁc energy which causes conﬂicts with the assumption of a constant speciﬁc energy (see point 1). 3. Furthermore, this scale-up procedure assumes equal cooling behaviour in laboratory and production machines. Hence, all heat ﬂows have to be proportional and all heat transition coefﬁcients have to be constant. This condition cannot be fully fulﬁlled because of different ﬂuid ﬂow behaviours and different velocities of product slurry as well as different grinding chamber wall thicknesses. However, in practice this assumption is often used with sufﬁcient success.

6.1.2. Stirred media mills with an annular gap If the consideration of the cooling is used for the scale-up of an annular gap mill with constant gap width, equations (69) and (70) show that the grinding chamber volume as well as the cooling surface is proportional to D2. i ph p ð70Þ V GC;ag ¼ D2 ðD sÞ2 L ¼ 2Ds s2 L 4 4 with s D it follows: V GC;ag

p p 2DsL DsL 4 2

ð71Þ

Wet Grinding in Stirred Media Mills

365

with D/L ¼ const. and s ¼ const. follows: V GC;ag / D2

ð72Þ

SGC;ag / D2

ð73Þ

Along the same line of thinking as for the stirred media mill with the disc stirrer, the ratio of power draw and cooling surface must be constant. P ¼ const SGC

ð74Þ

Assuming that a constant speciﬁc energy is necessary to get the same grinding result, Em ¼

P ¼ const: _ m

ð75Þ

Equations (71)–(74) lead to the result that the production rate is proportional to the grinding chamber volume and to the cooling surface of the mill chamber: _ / P / SGC / V GC m

ð76Þ

Therefore, the resulting production rate and the necessary power draw of the machine increase with the ratio of the grinding chamber volumes of production and lab-scale mill: V GC;P _ _ mP ¼ mL ð77Þ V GC;L V GC;P PP ¼ PL V GC;L

ð78Þ

_ L or the power draw PL and the If after a laboratory test the mass ﬂow rate m speciﬁc energy Em are known for scale-up, the size of the production mill can be determined as follows: _P _P m m V GC;P ¼ V GC;L ¼ V GC;L ð79Þ _L m P L =E m Judgement: 1. The assumption that in different machine sizes the same speciﬁc energy is needed for the production of the same product quality is correct for annular gap mills if the same operating parameters and a constant gap width are used. Owing to constant stress energy distributions and same energy transfer factors the speciﬁc energy need is constant. 2. This method does not estimate the power draw of the production size mill. It estimates the necessary power draw that is needed for the required production rate. However, since a certain part of the annular gap is responsible for a

366

A. Kwade and J. Schwedes

certain power draw of the machine, in theory the power draw increases proportional to the grinding chamber volume if the gap width is not changed. Therefore it is more likely that the estimated power draw can be reached without extreme changes of the operating parameters. This may not hold for special types of mill with grinding media separation devices which are responsible for a certain power draw. 3. Regarding cooling see mills with disc stirrer and pin-counter pin stirrer geometry.

6.2. Exact method based on stress model One critical point of the practical methods was the assumption of a constant speciﬁc energy need for different machine sizes. The following investigations will show why and how the speciﬁc energy consumption changes for different machines and how theses changes can be taken into account. For an exact scale-up method the description of the grinding and dispersion process with the mill related stress model is used. Therefore, the frequency of stress events and the stress energy as well as the distributions of these parameters will be addressed. In addition, the different mechanisms of energy loss will be analysed for different mill sizes.

6.2.1. Grinding behaviour of di¡erent mill sizes Systematic investigations with geometrical similar stirred media mills with perforated disc stirrers were carried out by Stender [42] and Stender et al. [69] at the Institute for Particle Technology of the Technical University of Braunschweig, Germany. The grinding chamber volumes were 0.73, 5.54 and 12.9 l. The operating parameters grinding media size, grinding media density and tip speed of stirrer discs were varied for the grinding of limestone. Figure 79 shows some of the results of these investigations. The relation between median particle size of the product and speciﬁc energy is depicted for three different grinding media sizes and grinding chamber volumes (0.73, 5.54 and 12.9 l). For small grinding media (dGM ¼ 355 mm) the best grinding result is achieved with the smallest grinding chamber for speciﬁc energies below 500 kJ kg–1. However, this behaviour changes, if 900 mm grinding media are used: a coarser product is produced in the 0.73 l grinding chamber, while the intermediate and the large grinding chamber show a similar but better result. The order of grinding chamber volumes changes again for 1360 mm grinding media. Now the ﬁnest product is produced in the largest grinding chamber. From Fig. 79 it becomes clear that different stirrer disc diameters and, with it, different grinding chamber volumes show different grinding results. Which of the grinding

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367

100 VGC= 0.73 l VGC= 5.54 l VGC= 12.9 l

10

x50 [ µm]

1 0.4 10

dGM = 355 µm

1 dGM = 900 µm

0.4 10

ϕGM= 0.8 cm= 0.4 ρGM = 2510 kg/m3 vt= 9.6 m/s

1 dGM = 1360 µm 0.4 10

100 1000 specific Energy Em [ kJ/kg ]

10000

Fig. 79. Inﬂuence of grinding chamber size on relation between product ﬁneness and speciﬁc energy.

chambers shows the best grinding behaviour depends on the operating parameters. Figure 80 gives an explanation for the different behaviour of different grinding chamber sizes. The median particle sizes for a speciﬁc energy of 1000 kJ kg–1 (see Fig. 79) are plotted versus the grinding media size. Figure 80 shows that different curves for the three different grinding chamber sizes exist. The smallest grinding chamber achieves the best grinding result with the small grinding media and the worst grinding result with the large grinding media. The reasons for these differences are different mean stress energies in grinding chambers of different sizes. Figure 70 shows that at small speciﬁc energies the stress energy of the small grinding media in the largest grinding chamber is not sufﬁcient to break the particles. In contrast it is possible to grind particles in the small grinding chamber effectively from the very beginning. Therefore, the stress energy seems to increase with decreasing grinding chamber size. This is an explanation for the different

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Fig. 80. Inﬂuence of grinding chamber size on relation between product ﬁneness and grinding media size.

grinding behaviour of different grinding media sizes in different grinding chamber sizes, too. The highest stress energy also exists in the smallest grinding chamber for the 900 mm grinding media. According to Fig. 80 the 0.73 l grinding chamber is already on the right side of the minimum (which is the optimum) for this grinding media size and thus, obtains a worse grinding result. At this grinding media size the other grinding chambers are closer to their optimum and therefore, generate similar grinding results. A similar behaviour can be recognised for the 1360 mm grinding media. The smallest grinding chamber has the highest stress energy, but is far away from its optimum and therefore, achieves the worst grinding result. In contrast the largest grinding chamber is the one that is closest to its optimum and achieves the ﬁnest product at the same speciﬁc energy. Plotting the median particle sizes of the particle size distributions versus the stress energy of the grinding media, SEGM, leads to Fig. 81. For stress energies SEGM larger than 0.1 103 Nm the ﬁnest product is obtained with the largest grinding chamber. The values for different mills are partly the same for stress energies lower than 0.1 103 Nm. Hence, the grinding chamber size is another inﬂuencing parameter, which affects the grinding or dispersion result. The reasons for different grinding behaviours of mills with different grinding chamber sizes are the change in grinding conditions and motion patterns of grinding media mainly due to different shear ﬁelds at the discs. This leads to different distributions of the number of stress events and stress energies. Since the stress energy of the grinding media, SEGM, does not take the distribution of the actual stress

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369

100

Median particle size x50

[µm]

ρGM= 2510 - 7550 kg/m3 ϕGM = 0.8 cm = 0.4 vt = 6.4 - 12.8 m/s dGM = 97 - 4000 μm Em= 1000 kJ/kg 10

1 0.5 1E-4

VGC = 0.73 l VGC = 5.54 l VGC = 12.9 l 1E-3

0.01

0.1

SEGM = dGM3 · ρGM · vt2

1

10

50

[10-3 Nm]

Fig. 81. Inﬂuence of grinding chamber size on relation between product ﬁneness and stress energy for a speciﬁc energy of 1000 kJ kg–1.

energy into account, it cannot be used to describe the grinding result for different grinding chamber volumes and/or geometries. However, these investigations show that a constant speciﬁc energy alone is not sufﬁcient to get constant grinding results, especially for scale-up. In Fig. 81 the depicted curves of the different mill volumes are shifted along the SEGM axis as well as along the median particle size axis. This means that different energy distributions as well as different grinding efﬁciencies have to be considered. In production processes the speciﬁc energy consumption for a certain product quality is usually of great interest. The inﬂuence of the grinding chamber size on the speciﬁc energy consumption is shown in Fig. 82. Here the speciﬁc energy for a median particle size of x50 ¼ 1.5 mm is plotted versus the stress energy of the grinding media, SEGM. The effect of the grinding chamber size on the speciﬁc energy consumption and grinding behaviour can be seen more clearly in Fig. 82 compared to Fig. 81. Even for optimal operating conditions in each grinding chamber, the speciﬁc energy needed for the production of a required product quality (or ﬁneness) is two times higher in the smallest grinding chamber compared to the largest grinding chamber. Again three different curves for the three different grinding chamber sizes can be depicted. The optima are shifted along the SEGM axis and along the energy axis. Owing to different stress energy distributions and energy transfer factors in a smaller grinding chamber a certain ﬁneness can only be reached with a higher speciﬁc energy.

370

A. Kwade and J. Schwedes 10000

ϕGM = 0.8 cm = 0.4

Em, x50 = 1.5μm

[kJ/kg]

ρGM = 2510 - 7550 kg/m3 vt = 6.4 - 12.8 m/s dGM = 97 - 4000 μm x50= 1.5 μm 1000

VGC = 0.73 l VGC = 5.54 l VGC = 12.9 l 100 1E-3

0.01

0.1 SEGM = dGM3 · ρGM · vt2

1

10

50

[10-3 Nm]

Fig. 82. Inﬂuence of grinding chamber size on the speciﬁc energy needed to produce a product ﬁneness of x50 ¼ 1.5 mm.

These results show that a correct scale-up of a stirred media mill with disc stirrer is only possible if, the different stress energy distributions of different machine sizes are taken into

account, the different energy transfer factors of the different machine sizes are taken into

account.

6.2.2. Calculation of stress energy distribution and mean stress energy Stender [42,69] calculated the stress energy distributions of the three different mills discussed above on the basis of the calculation of the tangential ﬂuid velocities in a stirred media mill with disc-stirrer geometry [12–14] (see Section 1.4). He assumed that the tangential velocity gradients at the discs and at the grinding chamber wall are equal in a small and a large scale mill. Moreover, he assumed equal ratios between the tangential velocities in the middle between two discs and the velocity of the disc surface for each radius. For the simpliﬁed calculations he divided the grinding chamber volume into four different volumes, in which the stress energy distribution was calculated based on a simpliﬁed ﬂuid ﬂow ﬁeld. As a result Stender achieved different stress energy distributions for the three mills of different size. Figure 83 shows the calculated stress energy distributions.

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371

SFM

ΔSE

SFM, i

ΔSFM, i

0

SEi

SEmax

Stress energy SE

Fig. 83. Stress energy distribution of different grinding chamber sizes.

1 0.7

ΣVtot, i / VGC [ - ]

0.6 0.5 0.4 VGC = 0.73 l VGC = 5.54 l VGC = 12.9l

0.3 0.2

SEVGC = 0.73 l = 0.00532 · SEGM SEVGC = 5.54 l = 0.00301 · SEGM SEVGC = 12.9 l = 0.00212 · SEGM

0.1 0.0 0

1E-5

1E-4 1E-3 ci = SEi / SEGM [ - ]

0.01

0.1

Fig. 84. Inﬂuence of the mean stress energy of different grinding chamber sizes on the product ﬁneness for a constant grinding effective speciﬁc energy Em,grind ¼ 1000 kJ kg–1.

However, often it is more practical to use certain characteristic numbers or coefﬁcients to compare different machines. For this reason a mean stress energy is deﬁned. Figure 84 shows a stress energy distribution with relevant numbers of the mill related stress model (see Section 2.2). For the different distributions a mean stress energy SE exists. The product of this mean stress energy and the

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stress frequency SFM is equal to the sum of the products of the individual stress energies SEi and the according stress frequencies DSFM,i. X SE SF M ¼ ðSE i DSF M;i Þ ð80Þ A stress energy distribution exists due to the different velocity gradients and the resulting different media velocities in the grinding chamber. Under the assumption that the stress frequency is almost the same in a certain volume, the stress frequency DSFM,i of a certain stress energy SEi can be replaced by the sum of the corresponding volumes Vtot. X SE V GC ¼ ðSE i V tot;i Þ ð81Þ with DSFM,ipVtot and SFMpVGC. The stress energy SEi of a certain volume Vi or Vtot can be expressed as function of the mean stress energy SE or of the stress energy of the grinding media SEGM which is proportional to the mean stress energy. X SE V GC ¼ SE GM ðci V tot;i Þ ð82Þ with SE i ¼ SE GM ci / SE

P SE ¼ SE GM

ðci V tot;i Þ V GC

ð83Þ

Based on equation (83) Stender calculated the mean stress energy for the different grinding chambers. The above-mentioned grinding results were explained with different stress energy distributions and different mean stress energies. Figure 84 now shows these different energy distributions and the resulting mean stress energies. A stress energy of 0.001 SEGM occurs in about 55% of the total grinding chamber volume in the smallest grinding chamber, whereas this stress energy can be found in only 40% of the total volume of the grinding chamber of the medium size mill and in only 22% of the volume of the largest mill. Therefore, substantially more stress events with higher stress energies occur in the smallest grinding chamber which leads to the highest mean stress energy of the three sizes.

6.2.3. Calculation of energy transfer factor According to the grinding results of the different grinding chamber sizes shown above, the different energy transfer factors have to be considered, too. Only part of the speciﬁc energy input into the grinding chamber is used for the grinding process, the other part is dissipated without any grinding progress. An analysis of the different energy dissipations shows that in a ﬁrst approximation only the energy transfer factor which is related to the energy dissipation at the

Wet Grinding in Stirred Media Mills

373

grinding chamber wall changes for different mill sizes. Therefore a variable nE,S is deﬁned. nE;S ¼ ð1 cB Þ

ð84Þ

Under the assumption that the factor cB is proportional to the ratio of the surface SGC and the volume VGC of the grinding chamber, nE,S can be expressed by the following expressions: cB /

SGC ; V GC

cB ¼ k

SGC V GC

nE;S ¼ ð1 cB Þ ¼

1k

SGC V GC

ð85Þ ð86Þ

Therefore the speciﬁc energy which is effective for grinding can be calculated by the energy transfer factor nE,S and the speciﬁc energy Em introduced into the grinding chamber. SGC E m;grind ¼ nE;S E m ¼ 1 k ð87Þ Em V GC The constant k describes how the energy is dissipated into heat at the surface of the grinding chamber wall. It is independent of the grinding chamber size but depends on surface conditions (e.g. type of material, surface texture) and product properties (e.g. hardness of particles, form of particles, lubrication). A rough surface texture with an abrasive material will lead to higher energy losses at the grinding chamber wall than a polished, smooth surface. Owing to the inﬂuence of mill and product properties, the value of the constant k can only be estimated from grinding or dispersion tests. The value of k is adjusted correctly if independent of the grinding chamber size the same grinding result is gained for the same grinding effective speciﬁc energy Em,grind. Table 6 shows the characteristic numbers of the three different grinding chambers investigated by Stender. With the above-mentioned grinding results the value of the constant k has been estimated to be 8.5 mm. With this value the energy transfer factor of the three machine sizes are 0.56, 0.73 and 0.83. Again, Table 6. Characteristic numbers and efﬁciency factors for the different grinding chambers

Grinding chamber volume VGC (l) Ratio of cylindrical surface and volume of SGC (mm–1) V GC the grinding chamber Energy transfer factor with k ¼ 8.5 mm nE,S (–) Necessary speciﬁc energy Em for Em (kJ kg–1) Em,grind ¼ 1000 kJ kg–1

0.73 0.052

5.54 0.031

12.9 0.021

0.56 1786

0.73 1370

0.83 1205

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A. Kwade and J. Schwedes

these energy transfer factors only take the energy ‘‘losses’’ due to the energy dissipation at the grinding chamber wall into account. If the different mean stress energies and the different energy transfer factors are taken into account, the grinding results for the different grinding chamber sizes can be plotted as shown in Figs. 85 and 86. Figure 85 shows the grinding results for a constant effective speciﬁc energy of Em,grind ¼ 1000 kJ kg–1 versus the mean stress energy. In Fig. 86 the required effective speciﬁc energy Em,grind for a product ﬁneness of x50 ¼ 1.5 mm is plotted versus the mean stress energy. Both ﬁgures show that the experimental data points can be ﬁtted by a single curve. Therefore, the mean stress energy takes the combined inﬂuence of the grinding media size, the media density and the tip speed as well as the inﬂuence of the stress energy distribution into account. In addition, the energy transfer factor takes the different energy dissipations due to different grinding chamber sizes into account. Together, the mean stress energy and the effective speciﬁc energy are the main inﬂuencing parameters for the grinding in different grinding chamber sizes of stirred media mills with a disc stirrer. Regardless of the size of the machine a deﬁned relation between the product ﬁneness and the effective speciﬁc energy exists for each mean stress energy. Figure 86 shows relations between the grinding results (product ﬁneness) and the grinding effective speciﬁc energy Em,grind for different grinding chamber sizes at almost constant mean stress energies SE ¼ 0:0018 103 Nm.

Fig. 85. Inﬂuence of the mean stress energy of different grinding chamber sizes on the needed grinding effective speciﬁc energy Em,grind for a product ﬁneness of x50 ¼ 1.5 mm.

Wet Grinding in Stirred Media Mills with κ = 8.5 mm VGC= 0.73 l: (1-κ SGC/VGC) = 0.56 ϕGM = 0.8 VGC= 5.54 l: (1-κ SGC/VGC) = 0.73 cm = 0.4 VGC= 12.9 l: (1-κ SGC/VGC) = 0.83 ρGM= 2510 - 7550 kg/m3

[kJ / kg]

10000

Em, grind, x50 = 1.5µm

375

SE = 0.00532 SEGM SE = 0.00301 SEGM SE = 0.00212 SEGM

vt = 6.4 - 12.8 m/s dGM = 97 - 4000 μm x50 = 1.5 µm

1000

VGC = 0.73 l VGC = 5.54 l VGC = 12.9 l

100 1E-5

1E-4

1E-3 SE

[10-3

0.01

0.1

Nm]

Fig. 86. Relation between ﬁneness and grinding active speciﬁc energy Em,grind for different grinding chamber sizes at similar SE.

The grinding results of the different grinding chamber sizes (median particle x50 size as well as the particle size x90) can be ﬁtted by one single line if the different efﬁciency factors are taken into account. This demonstrates that the grinding and dispersion results of one machine size can be transferred to other machines of different sizes. Therefore, the scale-up is possible with the characteristic mean stress energy and the energy transfer factor.

6.2.4. Scale-up with Newton-- Reynolds diagrams As we have seen, the speciﬁc energy required for a certain product quality is constant for annular gap mills but changes for stirred media mills with disc stirrer. However, we have investigated a method which allows for the determination of the correct speciﬁc energy requirement of the production size mill even for stirred media mills with disc stirrer. With this and equation (60) the required power draw of the machine can be estimated. However, one cannot be sure that the production size mill shows this estimated power draw. A good scale-up is only possible if the power draw of the machine can be determined. As explained in Section 5 the power draw of a stirred media mill can be described with Power diagrams. Based on this knowledge a method to calculate the power curve of a production-scale mill by means of tests on a lab-scale mill has been developed by Weit [30]. The power draw and production capacity of a production-scale mill can be predicted with this method, but in this case the

376

A. Kwade and J. Schwedes VGC 30

[l]

10

0.73 0.73 5.54 12.9

particle size x

[μm]

x90

υE, S

ρGM

dGM

vt

SEGM

SE

[-] [kg/m3] [μm] [m/s] [10-3 Nm] [10-3 Nm] 0.56 0.56 0.73 0.82

2894 1090 9.6 2510 900 12.8 2894 1090 12.8 2894 1500 9.6

0.345 0.300 0.614 0.900

0.00184 0.00159 0.00185 0.00191

x50

1 ϕGM = 0.8 cm = 0.4

0.3 10

100

1000

10000

Em, grind [kJ/kg]

Fig. 87. Ne-- Re diagrams for different mill conﬁgurations.

power curve for the same size, density and ﬁlling ratio of the grinding media must be known in lab and in production scale. In most cases this will be not the case. Therefore, Becker et al. [70] developed a method based on power curves determined without grinding media in the mill. Figure 87, in which the power curves for different mills with disc-stirrer geometry are presented, shows that different curves in the Ne–Re diagram exist for different mill sizes. According to this investigation the geometry parameters have an inﬂuence on the characteristic Power- and Reynolds-number in a way that the machines do not behave like similar systems. Therefore, mills of different size do not have similar power curves. Moreover, power curves for production size mills cannot be calculated based on a power curve of a lab-scale mill. In the following the procedure developed by Becker et al. [70] for the determination of the power draw of the production size mill is described: 1. The Ne-number NeL of the laboratory mill is calculated with the density of the product-grinding media suspension and the according power draw of the lab mill as well as the number of revolutions. 2. For this Ne-number a characteristic Re-number can be estimated from the Ne–Re diagram. This Re-number is used to calculate a viscosity of the product-grinding media suspension for the operating conditions.

Wet Grinding in Stirred Media Mills

377

3. With this viscosity the Re-number of the production size mill can be calculated (use of production machine data, e.g. bigger diameter of disc). 4. With this Re-number of the production size the Ne-number NeP can be estimated from the Ne–Re diagram. From the Ne-number the power draw is calculated through: P P ¼ NeP d 5d Z3P rSusp

ð88Þ

Using the procedure above the power draw of the production size mill can be estimated for different stirrer tip speeds. Another procedure could be to calculate ﬁrst the required power draw based on the speciﬁc energy requirement and the production rate (mass ﬂow rate) and in a second step to determine the Reynoldsnumber which is required to achieve this power draw. Based on this Reynoldsnumber the required tip speed of the production mill can be calculated using the viscosity determined from the lab-scale tests.

Nomenclature

A a ad b C cBFM cAcE ci cm CNe cv D dd DGC dGM E E(t) EA(t) Ekin Em Em, grind Em, M Em, min

desintegration degree (e.g. of microorganisms) (%) exponent (–) distance between discs (m) exponent (–) normalised concentration (–) concentration of bio mass (kg l–1) loss factors (–) normalised concentration (of cells) (–) solids concentration by mass (–) normalised power-number (–) solids concentration by volume (–) dispersion coefﬁcient (m2 s–1) diameter of stirrer discs (m) diameter of grinding chamber (m) diameter of a grinding media (m) energy during grinding time t (J) residence time density function (–) residence time density function of the inﬂowing product pension (–) kinetic energy (J) mass related speciﬁc energy (J kg–1) effective speciﬁc energy (J kg–1) speciﬁc energy consumed by the mill (J kg–1) minimum speciﬁc energy (J kg–1)

378

Em, P Em, W Em, W, grind EP, max EP, rel Etot EV, GC EU EUmax F F(t) F(Y) g(t) HGM HP kGM, E LS L lp mGM DmGM mP mp,tot _L m _P m _ Susp m n n Nc Ne NGM Np P P PGC Pstat. P0 Ps Pe Q3, cont(x) Q3, disc(x,t) r R, r

A. Kwade and J. Schwedes

speciﬁc energy transferred to the product particles (J kg–1) mass related speciﬁc energy under consideration of mass of media wear (J kg–1) effective speciﬁc energy under consideration of mass of media wear (J kg–1) maximum energy transferred to a product particle (J) relative volume-related energy (–) total energy input (J) volume-related energy transferred into grinding chamber (J m–3) energy utilisation (J m–2) maximum energy utilisation (J/m2) force (N) cumulative function of residence time distribution (–) cumulative function of residence time distribution (–) transport behaviour of the mill (density function) (–) grinding media hardness (HV) product hardness (HV) energy related wear rate of grinding media (1/J) length of stirrer from ﬁrst to last disc (m) length (m) packing length (m) mass of one grinding medium (kg) difference in mass of grinding media (kg) product mass (kg) total mass of product (kg) mass ﬂow rate of liquid (kg h–1) mass ﬂow rate of solid product (kg h–1) mass ﬂow rate of suspension (kg h–1) number of revolutions (s1) number of passages (–) number of bead contacts (–) Power number (–) number of grinding beads in the mill (–) number of feed particles inside the mill (–) Power input (power draw) (W) average power input (W) power consumed inside the grinding chamber (W) power input at stationary state (W) no-load power (W) probability that a particle is caught and sufﬁciently stressed (–) Peclet-number (–) particle size distribution of the product (continuous process) (–) particle size distributions in batch (discontinuous process) (–) radius (m) radius (m)

Wet Grinding in Stirred Media Mills

_ R R Re s SE SEGM SEopt SEP SE sf SFM DSFj SGC SGC, tot SI SIopt DSm SN SNF SNr SNtot t t tCh tf v v0 vt VGC VGC, ag VGM VGM, act VH, i Vp Vp, tot VSusp Vtot V_ V_ Susp wtot WE WEGM we WNM

379

backﬂow volume ﬂow rate (m3 s–1) backﬂow number (–) Reynolds number (–) distance (m) stress energy (J) stress energy of the grinding media (J) optimum stress energy (J) stress energy transferred to the product particle (J) mean stress energy (J) density function of stress energy – stress frequency – distribution (1/s J) frequency of stress events of the mill (1/s) frequency of stress events of interval j (1/s) surface area of grinding chamber (m2) total surface area of grinding chamber (with lids) (m2) stress intensity (J kg–1) optimum stress intensity (J kg–1) increase in speciﬁc surface by mass (m2 g–1) stress number (–) number of stress events of a feed particle (–) reduced stress number (–) total number of stress events (–) comminution time (s) mean residence time (s) production time for a batch (s) ideal ﬁlling time (s) mean axial velocity (m s–1) velocity of the grinding beads (m s–1) stirrer tip speed (m s–1) grinding chamber volume (m3) volume of annular gap grinding chamber (m3) overall solid volume of the grinding media (m3) active volume between two grinding media (m3) free volume of cells (m3) average volume of one product particle (m3) total volume of the product particles (m3) Volume of the suspension (m3) total volume (m3) volume ﬂow rate (m3 s–1) volume ﬂow rate of suspension (m3 s–1) relative total media wear (–) wear energy acting one wear event (J) wear energy of the grinding media (J) relative wear per wear event (–) number of wear events (–)

380

X X x xF xP x50 x90 y yi yGM yp Z Zd e e Z Zp ZSusp Y k k n nE nE, S rGM rP rSusp x j jGM jGM, local t

A. Kwade and J. Schwedes

dimensionless length of the grinding chamber (–) exponent (–) particle size (m) feed particle size (m) product ﬁneness for Q3 ¼ 0.9 (m) median particle size (m) particle size (m) exponent (–) dimensionless volume of the cells (–) young modulus of grinding media (Pa) young modulus of product media (Pa) exponent (–) number of discs (–) energy dissipation (J) porosity of the grinding media ﬁlling at rest (–) dynamic viscosity (Pa s–1) viscosity of product suspension (Pa s–1) viscosity of the product suspension (Pa s–1) dimensionless time (–) turbulent ﬂuctuating energy (J) dissipation factor (m) time (s) energy transfer factor (–) energy transfer factor for energy losses at the cylindrical surface (–) density of grinding media (kg m–3) density of the product particle (kg m–3) density of product suspension (kg m–3) x-potential (V) weighing factor (–) ﬁlling ratio of grinding media (–) local ﬁlling ratio of grinding media (–) ideal ﬁlling time of the stirred vessel (s)

REFERENCES [1] A. Kwade, J. Schwedes, Grinding and Dispersing with Stirred Media Mills, Short Course, Institute for Particle Technology, TU Braunschweig, 2005. [2] A. Kwade, J. Schwedes, KONA 15 (1997) 91–101. [3] A. Kwade, Powder Technol. 105 (1999) 14–20. [4] K. Engels, Farbe Lack 71 (5) (1965) 375–385 and (6) 464–472. [5] D.G. Bosse, Ofﬁcial Digest 3 (1958) 251–276. [6] H. Du¨rr, Verfahrenstechnik 12 (11) (1978), 708 and 13 (2) (1979) 64–72. [7] N. Stehr, in M.H. Pahl (Ed.), Zerkleinerungstechnik, Verlag TU¨V Rheinland, 1991 pp. 283–317. [8] B. Joost, A. Kwade, Das Keramiker-Jahrbuch (1996) 23–38. [9] J. Schwedes, in GVC-Tagung Feinmahl- and Klassiertechnik, VDI-GVC, Ko¨ln, 1993.

Wet Grinding in Stirred Media Mills [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54]

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M. Clement, J. Voigt, H. Uetz, Erzmetall 26 (1973) 583–592. A. Gangopadhyay, A.J.J. Moore, Wear 104 (1985) 49–64. M. Clement, E. Huwald, Dechema-Monographien 79 (1976) 235–252. S.R. de Silva, Proc. Symp. Attrition Wear Powder Technol., Utrecht, Netherlands, 1992, pp. 303–340. J. Voigt, M. Clement, H. Uetz, Wear 28 (1974) 149–169. I. Iwasaki, S.C. Riemer, J.N. Orlich, K.A. Natarajan, Wear 103 (1985) 253–267. L.G. Austin, R.R. Klimpel, Powder Technol. 41 (1985) 279–286. L.A. Vermeulen, Powder Technol. 46 (1986) 281–285. K. Scho¨nert, Hochschulkurs Zerkleinern, University of Karlsruhe, 1980. K.-H. Gahr, Microstructure and Wear of Materials, Tribology Series 10, Elsevier, Amsterdam, Netherlands, 1987. D. Munz, T. Fett, Mechanisches Verhalten keramischer Werkstoffe – Versagensablauf, Werkstoffauswahl, Dimensionierung, Springer, Berlin, Germany, 1989. H.H. Stender, A. Kwade, J. Schwedes, Int. J. Miner. Process. 61 (3) (2001) 189–208. M. Becker, J. Schwedes, Powder Technol. 105 (1999) 374–381. N. Stehr, in: M.H. Pahl (Hrsg.), Zerkleinerungstechnik, Fachbuchverlag Leipzig/Verlag TU¨V Rheinland, 1993. H.H. Stender, A. Kwade, J. Schwedes, Int. J. Miner. Process. 74S (2004) S103–S117. M. Becker, W. Ford, O. Gutsche, Proc. 3rd European Cong. Chem. Eng., Nu¨rnberg, 2001.

[59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70]

CHAPTER 7

Roller Milling of Wheat Grant M. Campbell Satake Centre for Grain Process Engineering,School of Chemical Engineering and Analytical Science,The University of Manchester, Manchester M60 1QD, UK Contents 1. Introduction 2. The structure of the wheat kernel 3. The international development of modern ﬂour milling 4. Roller milling of wheat kernels 4.1. Grist to the mill 5. Key issues in milling of wheat 6. Breakage of wheat kernels during First Break roller milling 7. Pearling of wheat prior to milling 8. Conclusions Acknowledgements References

383 385 387 391 394 395 396 412 415 415 416

1. INTRODUCTION The grinding of wheat into ﬂour is mankind’s oldest continuously practised industry and the parent of all modern industry; all modern particle breakage operations have wheat milling in their ancestry. In pursuing the need for efﬁcient and ubiquitous milling of wheat, millers and millwrights of old developed a practical mastery of several of the fundamental engineering disciplines: ﬂuid dynamics and aerodynamics for power generation from water wheels and windmills, mechanical engineering for the transmission of power via gearing and control mechanisms, and particle handling, breakage and separation operations. Oliver Evans, American designer of the original highly automated ﬂour mills, has been described as ‘‘the ﬁrst thoroughgoing plant engineer’’ [1], while Professor Friedrich Kick (Fig. 1), author of the ﬁrst scientiﬁc treatise on ﬂour milling in 1871 [2], also furnished comminution science with one of its most beloved laws [3–6]. Securing the grain supply motivated the construction of ancient empires and triggered the more recent development of national and international transport infrastructures and trading Corresponding author. Tel.: +44 161 306 4472; Fax: +44 161 306 4399; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12010-8

r 2007 Elsevier B.V. All rights reserved.

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Fig. 1. Professor Friedrich Kick.

systems [7–9]. Meanwhile, the anatomical difference between the structures of the wheat kernel and the rice kernel, that the former features a crease while the latter does not, required different approaches to milling that fundamentally altered the respective technological evolution of Western and Eastern civilisations. Through direct consumption and indirectly via animal feed, cereals supply more than half of our global food consumption [10]. Around 2 billion tonnes of cereals are produced annually, with wheat, maize and rice each contributing around 600 million tonnes [11,12]. Rice is mostly eaten directly by humans, while wheat and maize are also used as animal feed and increasingly as a feedstock for production of nonfood products. Wheat is the most widely grown cereal and the most extensively traded internationally [9,13,14] and has had the greatest impact on the history of the human race, both in ancient times and in recent centuries and decades [15]. The milling of cereal kernels, in particular wheat kernels, to release their multifunctional potential is an industrial activity that underpins all human society. This chapter describes how the ﬂour milling process interacts with the wheat kernel structure to separate it into its functional components. It traces brieﬂy the historical development of ﬂour milling technology leading to the introduction of roller mills a century ago, introducing the themes of wheat hardness, bread

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quality and the international grain trade that brought about modern ﬂour milling and dominate its practice still. The initial breakage of the wheat kernels is the most critical operation in ﬂour milling; the development of mathematical models to predict breakage based on the distribution of kernel characteristics in the feed is reviewed. Finally, kernel breakage is considered in the context of fractionation and bioconversion to produce non-food products as a renewable and sustainable alternative to oil.

2. THE STRUCTURE OF THE WHEAT KERNEL Understanding the ﬂour milling process begins with understanding the structure of the wheat kernel. Figure 2 illustrates the structure of the wheat kernel, which is generally around 4–10 mm in length and consists of three major parts: the germ, the endosperm and the bran [16–18]. The germ comprises around 2.5–4% of the kernel, the bran around 14–16%, and the endosperm around 81–84%. The oilrich germ is the baby plant, consisting of an embryonic axis which includes a rudimentary shoot and root and the scutellum which serves as a storage, digestive and absorbing organ. The endosperm is the storage reserve for the nutritional needs of the germinating plant until it emerges from the soil and is able to begin photosynthesising its food needs. The endosperm is therefore rich in starch and protein, for energy and construction needs, respectively. These nutritious components are also valuable for other living organisms including humans, animals, insects and microorganisms. For this reason, the germ and endosperm are

Fig. 2. Illustration of the wheat kernel, showing the germ, the endosperm, the layers making up the bran and the crease.

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covered with several protective layers known collectively as bran (a milling rather than a botanical term), which provide a physically tough barrier to intrusion. While the physiological beneﬁts of bran are now widely recognised, it is nutritionally poor in comparison with endosperm. Wholemeal ﬂour also has poor storage properties in comparison with white ﬂour and produces less appealing bread. The purpose of ﬂour milling is therefore to break open the wheat kernel and separate its structural components in such a way as to recover as much endosperm as possible, as free as possible from contamination with bran and germ, at minimum cost. Pure white ﬂour can be obtained by accepting a low yield (extraction), but as endosperm extraction rates approach the theoretical maximum of 81–84%, the ﬂour becomes increasingly contaminated with bran. The economics of ﬂour milling are therefore dictated by the balance between yield and quality, with the skill of the miller measured in terms of the amount and purity of white ﬂour that he can produce. Figure 2 also illustrates the distinctive anatomical feature of wheat kernels that makes the goal of recovering pure white ﬂour such a challenge – the crease. The crease is a deep furrow that runs the length of the kernel and extends the bran layers deep into the kernel. The rice kernel, by contrast, does not exhibit a crease, and bran and germ can be removed simply by polishing, such that rice endosperm is eaten in a pure and intact form. (Of the other major cereals, barley, oats and rye also feature a crease, while maize kernels and the spherical kernels of sorghum and millet do not have a crease.) The presence of the crease in the wheat kernel means that milling of wheat to separate bran from endosperm ultimately requires breaking open the kernel. While several of the cereals have kernels that exhibit a crease, wheat is unique in one respect. Wheat ﬂour alone has the ability to form a dough that is able to retain fermentation gases to produce a raised, highly aerated and palatable loaf of bread. This unique ability arises from the gluten proteins of wheat that form on hydration a viscoelastic, strain hardening network that is able to retain inﬂating gas bubbles [19–23]. (Rye also has gluten proteins and gives some retention of leavening gases, but this ability is vastly inferior to that of wheat and insufﬁcient to challenge wheat’s claim of uniqueness is this respect.) The desirability of raised bread (enhanced by the mysterious nature of the leavening process) has been a major socio-economic factor that has inﬂuenced trade, technology, religion and politics for centuries, and continues to do so [7,8,23–26]. Bread can therefore rightly be ascribed the status of the world’s most important food, and the reason that wheat is the world’s most important cereal. The light and palatable texture of raised bread is so appealing, and the raising process so amenable to the creation of a great and diverse range of distinctive breads, that even if wheat did not exhibit a crease, it would still be milled into ﬂour and turned into bread. A further feature to note from Fig. 2 is the aleurone layer, a single layer of cells that almost completely surrounds the starchy endosperm and the germ, separating them from the outer bran layers. Botanically, the aleurone layer is part of

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the endosperm, serving to provide starch-degrading enzymes during seed germination, but during conventional milling the aleurone layer adheres strongly to the outer bran layers and is therefore removed as a part of the bran [17,27–29]. The aleurone layer is rich in protein (although this protein, unlike gluten proteins, is non-functional with respect to breadmaking) and in vitamins and minerals. Its inclusion in ﬂour can increase milling yield and the nutritional quality of the ﬂour; millers aim as much as possible to recover aleurone material without also causing bran to enter the ﬂour. Flour quality is often indicated by measuring the ash (mineral) content of the ﬂour. Ash levels are higher in bran than in endosperm, so ash levels in ﬂour indicate the degree of undesirable bran contamination [27,29]. However, it is increasingly recognised that aleurone material is particularly high in mineral content but does not have negative consequences on ﬂour colour or functionality and indeed is nutritionally beneﬁcial [27,28]. The value of ash measurements has therefore been questioned, and the use of image analysis to quantify dark bran specks in ﬂour is being increasingly adopted [27].

3. THE INTERNATIONAL DEVELOPMENT OF MODERN FLOUR MILLING The history of ﬂour milling is a story of innovative incremental development over millennia to develop increasingly sophisticated technologies to break wheat kernels and separate the endosperm material from the bran and germ, efﬁciently and where and when needed, in order to supply the nutritional and social underpinnings of societies throughout Western civilisation. Its importance in earlier times and still now cannot be overrated. Storck and Teague [30] observe ‘‘In these two operations the story of milling y begins: the breaking up of cereal grain seeds, [and] the removal from the resulting meal of the unwanted portions y The story y is of how we men have learned to do these tasks better and better, devising improved tools and new skills as time passed; enlisting the forces of nature to help us; enlarging our mechanical arts and our mental capacities as we struggled with the twin problems of increasing the quantity and improving the quality of our product; adopting new ways of life, forming new social organizations as a result of a growing dependence on this increasing food supply. There is no other single thread of development that can be followed so continuously throughout all [Western] history, and none which bears so constant a cause-and-effect relation to every phase of our progress in civilization.’’ While separation of the distinct parts of the broken grain (originally by hand, later by sifting and air classiﬁcation) is an integral operation, the breakage aspect of the processing of wheat kernels into ﬂour dominates, such that ‘‘milling’’ refers to the entire process as well as to the speciﬁc operations of breakage. The

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Fig. 3. Wooden ﬁgure of a woman grinding grain, from Egypt, ca. 2300 BC. Figures like these were placed in tombs to ensure an eternal supply of food. Printed with permission from The Manchester Museum.

technology of breaking wheat, from prehistory to the current time, moved from pounding grains between two stones to saddlestones (illustrated in Fig. 3), slab, push and lever mills, the more sophisticated hourglass mill introduced by the Greeks and leading to the more long-lived rotary quern, eventually scaled up to the millstones of sufﬁciently recent history to be familiar to us still (and even now serving a small but growing specialist market), and mechanised over time via water, wind, steam and electrical power [24,31]. Then, in the late 1800s a revolution occurred in milling with the rapid introduction and adoption of roller milling technology. The timing and success of this transformation are attributable to an intricate combination of agricultural, social and technological circumstances occurring principally across three countries, Hungary, America and Britain. The story of this revolution bears re-telling brieﬂy here (based on Storck and Teague [24], Morgan [7] and Jones [31]), as it introduces the fundamental themes of gradual reduction, hardness, global trade and feedstock variability that underpin an appreciation of the nature and signiﬁcance of wheat breakage. Wheats are broadly divided into soft, hard and very hard, the former two arising from common wheat (Triticum aestivum) and suitable for bread and biscuit making, the latter describing durum wheat (Triticum durum) and being the wheat of choice for pasta making. Wheat kernel hardness is a genetically determined factor, but is also inﬂuenced by growing environment and agronomic practice [32]. Wheat hardness therefore exhibits a continuous spectrum rather than sharp transitions. Millstones (illustrated in Fig. 4, typically 1.5 m in diameter and operated in pairs) are adequately suited to grinding soft wheats, which tend naturally to break such that bran material remains as large fragments while endosperm

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Fig. 4. Illustration of a millstone.

shatters into small particles, facilitating separation of bran from endosperm by sifting. By contrast, the bran and endosperm layers of hard wheats tend to break together, such that separation of bran from endosperm is difﬁcult. Hard wheats tend to give ﬂour with better breadmaking properties, but in the less afﬂuent past, the absence of sufﬁcient markets for excellent bread ﬂour and the difﬁculty of milling hard wheats using millstones combined to favour the production and processing of soft wheats and the consumption of, to the modern palate, rather inferior bread. While the traditional millstones were generally operated as a single pair, over time the beneﬁts of repeated milling and sifting using a sequence of millstone pairs were recognised. This ‘‘gradual reduction’’ approach developed ﬁrst in France from the 16th century and became known as the French system or mouture eŁconomique. The concept was taken further in the 1800s by Austrian and Hungarian mills to create the mouture en in¢ni (milling to inﬁnity) system. This was an incredibly complex and labour-intense arrangement involving more than 80 intermediate product streams manually conveyed between operations by scores of men. It yielded more than 10 ﬁnal products of varying quality, of which the ﬁnest was a ﬂour whiter than had ever been previously experienced. Such a system was only conceivable in the highly stratiﬁed Hungarian society, which gave ready markets for such a reﬁned product and at the same time for the entire portfolio of lower quality ﬂours produced elsewhere in the process. It was also only applicable to the hard wheats typical of that region that could withstand the harsh treatment. In time, this superior ﬂour production developed into a lively export market that reached as far as England, raising the bar for the expectations of bread quality obtainable from the miller’s product.

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Meanwhile, in the more egalitarian and expanding American population, production efﬁciency to supply a single grade of product to a mass market, rather than differentiation of ﬂour quality to serve different social classes, was the driver for technological innovation. The American contributions to the development of ﬂour milling technology were therefore the introduction of automation within ﬂour mills, initially by Oliver Evans, along with scale-up and advances in particle separation systems based on improved sifting and air classiﬁcation. The latter advances also facilitated the processing of hard wheats and encouraged increased planting of these varieties by American farmers. America was becoming the breadbasket of Britain, with the continent’s rapid settlement driven in large part by the need back in Britain for wheat and ﬂour. Morgan [33] notes: ‘‘At the end of the eighteenth century a human need y developed on a scale that had never been experienced before in history. That need was for bread y Workers in Paris, London and Manchester in 1800 paid half their wages for bread alone. Eating bread was a badge of attainment and social status. Once achieved, it was not easily given up, as governments invariably found when they took administrative measures to lower the quality of bread to conserve wheat. It was no coincidence that the century in which bread eating ﬁnally extended downward to all elements of society in Western Europe y was also the century of revolution y An adequate wheat supply was, therefore, a prerequisite of social order and political stability.’’ By the late 1800s Britain had initiated the Industrial Revolution and was at the height of Empire. Its need for wheat to feed the growing and increasingly urbanised population was having dramatic inﬂuences globally. ‘‘Parliament, with its stroke of repeal [of the Corn Laws in 1846], had changed the world. Repeal of the protectionist system had opened England to the wheat of all the world, created incentives for the settlement of vast territories across the oceans, and established the conditions for modern international trade’’[34]. The British climate suits the production of soft wheats, while the imported hard wheats and ﬂours tended to be better suited to breadmaking. This cultivated in the public a taste for the superior breads and added to the pressure on millers to obtain foreign wheats and process them, with all their variability, into consistent quality ﬂour. As a result, as Jones [35] observes, ‘‘One essential for success in Britain was the production of high yields of good quality ﬂour from mixtures of different wheats, with the added difﬁculty that the resources varied through the cereal year. The British situation was unique in the requirement to deal with varying characteristics of the material to be processed; Hungarian and American millers had comparatively uniform raw materials.’’ The gradual reduction approach using millstones was partially successful in processing the hard wheats of Hungary and increasingly of America and of Britain’s imports, particularly when combined with air classiﬁcation to separate ﬂour stocks. However, the problems of processing the increasingly prevalent hard

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wheats prompted the successful invention in Hungary of an alternative approach to open up the wheat kernel, using roller mills. Thus, the revolution that was to replace millstones wholesale with rollers was driven by the push resulting from the availability of hard wheats and the need to process them, combined with the pull of increasing markets for the excellent breadmaking ﬂour that hard wheats, suitably processed, can yield. The new Hungarian innovation was adopted enthusiastically across Europe and particularly by British millers. This was because, in addition to allowing milling of hard wheats, the roller mills offered great advantages over millstones in terms of versatility and control. Thus, roller milling helped the miller address the particularly British problem of wheat variability and thus to deal at a practical level with Thomas Malthus’s observation a century earlier that ‘‘No two grains of wheat are exactly alike’’ [36].

4. ROLLER MILLING OF WHEAT KERNELS Manchester in the UK, Minneapolis in America and Budapest in Hungary can all lay claims of similar validity for the ﬁrst installation by about 1880 of a ‘‘modern’’ ﬂour mill, judged by three deﬁning factors [37]: the use of the gradual reduction approach, full automation, and the complete displacement of millstones by roller mills. From its invention in 1873, the roller mill triumphed rapidly over millstones, such that by the turn of the century the revolution was essentially complete. The factors of wheat hardness, international grain trade and bread quality that gave rise to its birth are still those that dominate the practice of ﬂour milling today. As noted above, rollers gave the miller vastly superior control over the process compared with the less precise and more temperamental behaviour of millstones, thereby facilitating the processing of a great variety of wheats of widely varying characteristics. The versatility of the roller mill arose in several ways: the roll gap could be adjusted, rolls could be operated at different relative speeds to give a differential cutting action, different roll surfaces and ﬂuting proﬁles could be used and successive roll pairs could be arranged in a variety of conﬁgurations. Nevertheless, the miller’s ﬁrst obligation remained: to produce ﬂour of consistently uniform quality, day after day and year after year, in the face of a constantly changing feedstock. In this respect, the very versatility of the roller mill was also its greatest challenge, as small changes in its operation could severely disrupt the intricate balance of downstream processes and hence the ﬁnal ﬂour quality. Figure 5 illustrates a typical modern ﬂour milling process ﬂowsheet employing the gradual reduction system using, in this example, 16 roll pairs with accompanying sifters. (Wheat ﬂour milling is a dry process that produces an impure starchrich product, in contrast to the wet milling process used to process maize into

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Reduction system

Wheat First break

SIZ 1

A

C

H

B

D

K

B2

E

to F Second break

3MD

SIZ 2 to C

Third break

to F

Break roll Reduction roll Fourth break

F

G Sifter Flour

Quiver sifters Bran

Fig. 5. Typical ﬂour milling ﬂowsheet with four break rolls. Reprinted with permission from [38].

cornﬂour, which is pure starch and much more expensive than ﬂour.) The process can be broadly divided into the break and reduction systems. The break system uses ﬂuted rolls of increasingly ﬁne ﬂuting to break open the wheat kernel and to scrape the endosperm material from the bran. Some ﬂour is produced at each stage, while the larger endosperm material is sent to the reduction system for further grinding and bran and germ separation. In this example, ﬂour (endosperm material smaller than about 212 mm) is produced at 21 different points in the mill. The quality of ﬂour recovered from each different stage varies [29]. The ﬂour is usually combined into a single composite product, the quality of which depends on the proportions and characteristics of the various daughter ﬂours. Figure 6 illustrates a roller mill, with each roll typically around 1 m in length and 250 mm in diameter. Break rolls are ﬂuted, as illustrated in the ﬁgure, while reduction rolls have a smooth, slightly frosted surface. Break rolls are ﬂuted with an asymmetric saw tooth proﬁle and operate with a gap between the rolls and under a differential (i.e. ratio of fast roll speed to slow roll speed) of about 2.5:1. Operating the rolls under a differential reduces the energy consumption [39]. The ﬂuting breaks open the wheat kernel such that the bran tends to stay relatively intact in large particles, while the endosperm shatters into small particles, facilitating the separation of endosperm from bran by sifting. Reduction rolls are smooth and

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Fig. 6. Illustrations of roll pairs from the ﬂour milling break system.

operate under pressure at lower differentials of about 1.25:1. This causes damage to starch granules and thereby affects the water absorption properties and baking performance of the ﬂour; the degree of starch damage in the composite ﬂour is therefore one of the factors that the miller must control. The introduction of roller mills in the late 1800s was accompanied by a frenzy of experimentation to establish the best roll designs and operational characteristics, particularly of First Break. Eventually this converged on a saw-tooth break roll proﬁle with a sharp leading edge followed by a ﬂat ‘‘land’’ and a dull following edge, which gives a cleaner cutting action than rounded proﬁles. The precise angles and depths of the sharp and dull edges and the lengths of the land portions are matters of variation and distinction between different roll manufacturers. In addition, the ﬂutes exhibit a slight twist or spiral along the length of the roll, such that the roll pair imparts a scissor-like cutting action on the wheat kernel or the later part-broken stocks. The asymmetric saw-tooth proﬁle of the break rolls and the speed differential between them allow the rolls to be operated under different dispositions: sharp-tosharp (S-S), sharp-to-dull (S-D), dull-to-sharp (D-S) and dull-to-dull (D-D). Figure 7 illustrates the ﬂuting proﬁle and the four possible dispositions. Over time ﬂutes become worn and rolls must be reground, such that frequently a roll pair will be operated D-D initially, then changed progressively as the rolls wear to D-S, S-D and ﬁnally S-S, to try to keep the breakage patterns relatively consistent over time. Returning to Fig. 5, it is immediately evident that the particle size distribution produced from First Break, the ﬁrst roller milling operation that the wheat encounters, determines the balance of stream ﬂows through the rest of the milling process. First Break is therefore a critical control point in milling, the importance of which cannot be overstated [40,41]. To a ﬁrst approximation, if the miller could maintain a constant particle size distribution from First Break in the face of a constantly changing feedstock, the rest of the mill would run more or less consistently. The detailed study of particle breakage within ﬂour milling has therefore focussed thus far almost exclusively on First Break.

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Fig. 7. The four possible break roll dispositions: sharp-to-sharp (S-S), sharp-to-dull (S-D), dull-to-sharp (D-S) and dull-to-dull (D-D).

4.1. Grist to the mill The ‘‘golden rule’’ of the ﬂour miller is to maintain consistent ﬂour quality, as consistency of raw materials is the baker’s number one priority. In pursuing this goal, the major tool employed by the miller is blending of different wheats together, known as gristing [41–44]. Pyler [45] highlights this emphasis on consistency and the contribution that gristing makes: ‘‘This is a vital operation y since correct blending of wheats constitutes the basis for the uniformity of ﬂour performance in the bakery.’’ Typically, at least half a dozen wheat varieties of widely differing origins and characteristics will be mixed together to give the desired ﬂour quality at least cost. Milling the wheats separately and blending the ﬁnal ﬂours has considerable merits, and some millers have invested in the necessary facilities and beneﬁt from fewer grist changes to enable a ‘‘just in time’’ response to customer orders (B. McGee, personal communication). However, in practice it is easier to blend wheat than ﬂour, so most millers grist prior to milling [46]. The practice of gristing exacerbates the problem of wheat variability entering First Break and of maintaining consistent performance from the mill in terms of ﬂour yield and functionality. The latter depends not only on composition, which is easily controlled through gristing, but also on the different processing histories of the daughter ﬂours from the different parts of the mill that are combined into the ﬁnal composite product. These histories depend on the grist’s initial breakage characteristics, which determine the proportions of streams ﬂowing through the

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different parts of the rest of the mill, described by Storck and Teague [47] as ‘‘that great desideratum, the balance of the mill.’’ In addition to gristing, the second major tool for altering ﬂour quality by adjusting the feedstock is conditioning (or tempering) [48–51]. Conditioning is the controlled addition of water (and/or heat) to the wheat to alter its breakage characteristics. Wheat can be stored almost indeﬁnitely provided its moisture content is maintained below about 14% (wet basis). However, when it comes to milling the wheat, it is preferable to raise its moisture content to about 16%. This has the effect of toughening the bran while making the endosperm more friable, such that the bran tends to stay more intact as larger particles, while the endosperm shatters more readily into smaller particles. Hence, conditioning facilitates the separation of bran from endosperm based on size. Conditioning also compensates for moisture loss during handling and pneumatic conveying of ﬂour stocks, thereby maintaining the moisture content of the ﬁnal ﬂour. Once again, the peculiarities of ﬂour milling in Britain have given rise to a unique mastery of conditioning practice: as Storck and Teague [52] observe, ‘‘Under stress of necessity English millers have learned y to alter [wheats’] character by removing or adding moisture or heat. The Englishman uses this conditioning constructively to improve the strength and baking quality of the ﬂour, prescribes special forms of treatment suited to individual wheats, makes mill mixes out of a half-dozen or more kinds of wheat drawn from all over the globe according to price and what they can contribute to his ﬂour, and in general is a versatile master of the possibilities of diverse wheats.’’

5. KEY ISSUES IN MILLING OF WHEAT Milling wheat into ﬂour for human consumption is a mature industry that operates on highly efﬁcient processes at low economic margins. The key issue is how to process a variable feedstock to produce a consistent quality product economically. However, in the 21st century, cereal milling will increasingly be applied to meet a new challenge, that of supplying and processing a renewable feedstock that can provide the energy and chemical needs of society [53–55]. Oil, on which the enormous technological and social strides of the 20th century were based, is a ﬁnite resource and increasingly recognised as so, as reﬂected in increasing prices and the regular featuring of oil concerns in scientiﬁc pronouncements and government initiatives. In addition, oil processing and consumption is a source of pollution and environmental hazard, including the production of greenhouse gases and their contribution to global warming. For these reasons, there is an urgent need to ﬁnd renewable alternatives to oil to supply the energy and material needs of modern society. This need is urgent for at least three reasons: (i) to address the pollution problems inherent in oil usage; (ii) to ensure ﬁnite oil

396

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supplies are conserved for as long as possible; and (iii) to ease the transition from an oil-proﬂigate to an oil-depleted global village. And in terms of production levels, chemical density, infrastructure and ease of processing, cereals present themselves as the most promising candidate to form the basis of a sustainable chemical industry. In the UK, of the 22 million tonnes of cereals produced annually, wheat contributes 14–15 million tonnes and is the UK’s largest crop [56]. A sustainable UK chemical industry based on benign processing of renewable feedstocks is likely to be dominated, at least in the medium term, by wheat. At present, the economics of cereal processing for non-food uses are generally unfavourable compared with oil. With increasing oil prices, in time the economics will become favourable, but as noted above, there is an urgent need to develop technologies and approaches to accelerate the economic feasibility of cerealbased products. To meet this need, the concept of the cereal bioreﬁnery, analogous to oil reﬁneries in co-producing a range of added value products from cereals, is the way forward. In order to make cereal bioreﬁneries economic, the key issue is to maximise the value that can be extracted from every last component of the cereal grain. Again, analogous with oil reﬁning, the necessary approach is one of fractionation followed by extraction and/or conversion into added value products. In both conventional ﬂour milling for human food and novel wheat (or other cereal) processing for non-food products, the key issue is fractionation of the grain into its compositionally distinct structural components prior to further processing of each component. Thus, breakage of the wheat kernel is critical, particularly the initial breakage which determines the subsequent processing.

6. BREAKAGE OF WHEAT KERNELS DURING FIRST BREAK ROLLER MILLING Hardness is, as implied in the foregoing discussion, a major aspect of wheat quality and, not unexpectedly, a major factor affecting breakage [32]. Pomeranz and Williams [57], in a comprehensive survey of wheat hardness research up until 1990, note ‘‘Kernel texture is the most important single characteristic that affects the functionality of a common wheat y a parameter of great signiﬁcance in both the wheat and ﬂour industry and in domestic and world trade [that] affects every aspect of wheat functionality except gluten strength and its associated factors.’’ Numerous hardness tests have been developed over the last century, broadly divided into the power or time required to grind a sample to a given particle size, or the particle size resulting from grinding a sample under standardised conditions. These are empirical tests that give relative rather than absolute indications of wheat kernel hardness. Pomeranz and Williams note that over 100 different

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397

methods for measuring wheat hardness have been documented [57]. Near-infrared reﬂectance (NIR) spectroscopy of wheat samples milled under standardised conditions is the most common method currently used for measuring hardness. The Stenvert test, measuring the time to produce a given volume of ﬂour through a 2 mm screen mesh using a standardised hammer milling procedure, has also been widely used [57–59]. Hardness tests are mostly applied to bulk samples of wheat, reporting an average hardness value for the sample. In recent years, however, the trend has moved to the measurement of the hardness of each individual kernel in a sample and the reporting of the entire distribution of hardness values. The most well developed commercial example of this wheat testing philosophy is the Perten Single Kernel Characterisation System (SKCS), developed by the USDA Research Centre and commercialised by Perten Instruments, Sweden. The SKCS crushes usually 300 individual kernels within 5 min and reports the distributions of their hardness values, along with the kernel weight, diameter and moisture content distributions [32,60–66]. The challenge remains to relate these distributions of kernel characteristics to milling performance and ﬂour quality. Breakage of wheat kernels during First Break roller milling depends on the characteristics of the wheat (including the distributions of kernel size, hardness and moisture content) and on the design and operation of the roller mill (including roll speeds and differential, roll disposition, ﬂuting proﬁle, number of ﬂutes, ﬂuting spiral, roll gap, the degree of roll wear and the feed rate). The effects of these factors are manifest in the particle size distribution exiting First Break, the compositional distribution of those particles (as large particles tend to be richer in bran, while small particles are pure endosperm), the power required to mill the wheat and the rate of roll wear. Figure 8 illustrates these interrelationships. Work in our laboratories has introduced the breakage equation for First Break roller milling of wheat, based on the paradigm of ﬂour milling as ‘‘the evolution of the particle size distribution.’’ Previous work on wheat breakage has tended not to focus so strongly or quantitatively on the particle size distribution, preferring to report break releases (the amount of ﬂour produced by individual break rolls) or, if straying into attempts to apply fracture mechanics to wheat breakage, stresses on the wheat kernel or its components under slow uniaxial compression or tension [67–73]. The breakage equation approach offers a practical basis for relating the distributions of single kernel information measured by the SKCS to breakage during First Break and hence for relating wheat variability to initial breakage patterns and their consequences for mill balance and ﬂour yield and quality. The breakage equation for roller milling of wheat kernels in terms of the size of the input and output particles is given in its cumulative form by Z

D¼1

P 2 ðxÞ ¼

Bðx; DÞr1 ðDÞ dD D¼0

ð1Þ

398

G.M. Campbell

Roller mill design • •

•

roll diameter fluting • no. flutes • flute profile • spiral pressure/rigidity

Roller mill operation • speed • differential • gap • disposition • sharp to sharp • dull to dull • sharp to dull • dull to sharp • roll wear

• • • •

Feed characteristics particle size distribution particle shape distribution particle density distribution particle hardness distribution

Process performance • power • roll wear

• •

Output characteristics particle size distribution distribution of other quality factors e.g. composition, starch damage

Fig. 8. Feed, design and operational factors affecting particle breakage during First Break roller milling of wheat. Reprinted with permission from [74].

where r1(D) is the particle size distribution of the feed entering the roller mill, P2(x) the cumulative particle size distribution of the output, and B(x,D) the cumulative breakage function describing the proportion of material smaller than size x in the output, originating from an input particle initially of size D. Campbell et al. [41] explain that, in contrast with similar breakage equations for once-through milling operations such as those reviewed by Austin [75], the lower limit of the integration in equation (1) is D ¼ 0 rather than D ¼ x. The reason is that D and x are measuring different things and are not directly comparable. D is kernel thickness as measured by the SKCS (or by image analysis or using slotted sieves), while x is the smallest square aperture through which a particle will pass, as measured by sieve analysis. It is therefore meaningless in this context to write D ¼ x. Previous workers studying breakage have performed the integration from D ¼ x on the basis that they were measuring the size distributions of their input and output material in the same way, e.g. using the same sieve analysis procedure. They therefore argued that the breakage process implied that output particles of size x could only have arisen from inlet particles originally larger than x, a reasonable argument that simpliﬁes the mathematics. However, this approach renders the breakage equation less general and less ﬂexible, by excluding the possibility that inlet and outlet particles might be measured differently, and that the particular dimension chosen to characterise an outlet particle size might be larger in magnitude than the dimension used to characterise the size of the

Roller Milling of Wheat

399

original inlet particle. In the case of wheat, the roller milling process tends to open up the initially compact wheat kernel to create large bran ﬂakes along with ﬁner endosperm material. Thus, it is quite possible for a kernel, initially of 2 mm diameter, to yield a bran ﬂake measuring 3 mm across when measured by sieve analysis. Hence, the appropriate range for the integration in equation (1) is from zero to inﬁnity. Campbell and Webb [76] also discuss the issues of selection functions and normalisation that feature strongly in other breakage equations but that are, in our view, inappropriate or unnecessary for a breakage equation describing First Break roller milling of wheat kernels. First, all wheat kernels are broken during First Break, so there is no concept of selection. The notion of a selection function has arisen in previous work from the observation that following breakage, some outlet material is still of the same size range as the inlet material, implying that a portion has not been selected for breakage. However, in wheat breakage, even if some of the outlet material is of the same ‘‘size’’ as the original inlet material, the sizes are, as noted above, not actually comparable physical entities. Second, much of the literature on breakage functions assumes that normalised functions apply, in which the daughter distribution scales directly with inlet particle size, implying that larger inlet particles break to give larger outlet particles. In roller milling, the ﬁnite size of the gap between the rolls requires more breakage for a larger kernel to pass through than for a smaller one, such that the output particles resulting from breakage of a large wheat kernel are smaller than those resulting from breakage of a small kernel. Thus, the common concepts of normalisation and selection that are applied to other breakage processes are inappropriate for describing roller milling of wheat. Some may argue that the relationship between inlet and outlet particle size distributions for roller milling of wheat could indeed be modelled using repeated selection within the breakage zone of the roll pair, with normalised breakage functions describing each breakage event. However, in our view this introduces unveriﬁable assumptions about the details of the wheat kernel breakage process (such as the number of separate breakage events occurring during a single passage through the roll pair, and the appropriate normalised function to apply on each occasion). Campbell et al. [74] derived the form of the breakage function that describes the particle size distribution resulting from breakage of an individual wheat kernel as a function of kernel size and roll gap, from which the breakage of a mixture containing a distribution of kernel sizes could be predicted. This work showed that wheat kernel breakage is determined by the ratio of roll gap to kernel thickness (the third longest dimension, which equates to the diameter reported by the Perten SKCS), called the milling ratio, G/D. This work also led to the important conclusion that underpins the breakage equation, that wheat kernels break independently during roller milling, such that a small, hard kernel

400

G.M. Campbell

will break as a small, hard kernel even if it is surrounded by large, soft kernels, and vice versa. Bunn et al. [77] demonstrated that this form of the breakage function was adequate for a wide range of wheat varieties. Fang and Campbell [78] investigated the effect of the four roll dispositions on the breakage function, and Fang and Campbell [79] added a term to account for kernel moisture content as well as size, thereby accounting for two of the four SKCS parameters. Campbell et al. [41] completed the work by extending the breakage function to include SKCS hardness and investigated the possibility of adding the fourth SKCS parameter, kernel mass, in order to allow for the effects of kernel shape on breakage. Based on Fang and Campbell [78] and Campbell et al. [41], an empirically determined cumulative breakage function cubic in x and quadratic in G/D is appropriate to describe breakage under S-S, S-D, D-S and D-D roll dispositions: Bðx; DÞ ¼ a0 þ b0 x þ c0 x 2 þ d 0 x 3

G þ ða1 þ b1 x þ c1 x þ d 1 x Þ D 2 G þ ða2 þ b2 x þ c2 x 2 þ d 2 x 3 Þ D 2

3

ð2Þ

leading to 3 G Z D¼1 6 a0 þ b0 x þ c0 x þ d 0 x þ ða1 þ b1 x þ c1 x þ d 1 x Þ D 7 7 6 P 2 ðxÞ ¼ 7r1 ðDÞ dD 6 2 5 D¼0 4 G 2 3 þða2 þ b2 x þ c2 x þ d 2 x Þ D ! ! 1 1 1 1 2 2 þ b0 þ b1 G x þ a2 G þ b2 G ¼ a0 þ a1 G D D D2 D2 ! 1 1 2 þ c2 G þ c0 þ c1 G x2 D D2 ! 1 1 þ d 2 G2 2 þ d0 þ d1G ð3Þ x3 D D 2

2

3

2

3

where

1 Dn

Z

D¼1

¼ D¼0

N X 1 1 pi n n r1 ðDÞdD D D i i¼1

ð4Þ

Roller Milling of Wheat

401

N is the number of discrete size fractions into which kernels are separated, and pi the proportion of kernels in size fraction i. Thus, knowing ð1=DÞ and ð1=D2 Þ and the 12 coefﬁcients of equation (2) for the particular wheat sample and roll disposition, the particle size distribution of the output from First Break can be predicted for any roll gap. Figure 9 illustrates the effects of roll disposition and roll gap on wheat breakage, for a typical UK hard wheat, Hereward, and a typical UK soft wheat, Consort. Under S-S milling, the particle size distribution is relatively even over the size range 200–2000 mm, with larger roll gaps tilting the balance towards larger particles. Moving through S-D and D-S through to D-D milling, the particle size distribution becomes progressively more ‘‘U-shaped’’, indicating that D-D milling produces greater proportions of both large and small particles, with fewer in the mid-size range. As the large particles are bran-rich and the small particles starchrich, this distribution will be easier to separate by size into compositionally distinct fractions for further processing; D-D milling is therefore preferred by millers. As noted above, as rolls wear down, millers may change from D-D to D-S, S-D and ﬁnally S-S milling to counteract the effect of roll wear. Fang and Campbell [80,81] analysed the fracture mechanics of wheat breakage during roller milling to explain the effects of roll disposition on the observed breakage patterns. Figure 10 presents the same data in its cumulative form, to facilitate comparison with predictions based on the breakage equation (the coefﬁcients being derived from independent data). The predictions are in excellent agreement with the experimental data and would give a good ﬁrst approximation for the breakage behaviour of these wheats. Fang and Campbell [79] added a correction function to account for the effect of moisture content, to give

Z

mmax

Z

PðxÞ ¼

1

Bðx; D; mÞ r1 ðDÞ r1 ðmÞ dD dm 0

ð5Þ

0

where m is the moisture content of an individual kernel, and r1(m) the probability density function describing the distribution of individual kernel moisture contents in the sample. Equation (5) assumes that the size distribution and moisture distribution in a sample are independent, i.e. that large kernels have the same moisture distribution as small kernels (or, equivalently, that wetter kernels have the same size distribution as drier kernels). Thus, if the extended cumulative breakage function B(x,D,m) is known, then the outlet particle size distribution could be predicted for any size distribution and moisture distribution of a given wheat sample. They then simpliﬁed the cumulative breakage function by separating it into two components, the cumulative breakage function at a nominal moisture content m0 (say 16%) and a correction term to account for the variation

402

Hereward 0.3 mm

0.12

0.5 mm

0.5 mm

0.10

0.7 mm

ρ (x)

0.06

0.08

0.6 mm

0.7 mm

0.06

0.7 mm

0.08 0.06

0.06

0.04

0.04

0.04

0.02

0.02

0.02

0.02

0.00

0

600

1200

1800

0.00

0

2400

Particle size x (μm)

600

1200

1800

2400

0.5 mm

0.10

ρ (x)

ρ (x)

0.6 mm

0.7 mm

0.08

0.04

0.04

0.04

0.04

0.02

0.02

0.02

0.02

Particle size x (μm)

2400

0.00

0

600

1200

1800

Particle size x (μm)

2400

0.00

0

600

1200

1800

Particle size x (μm)

2400

0

600

1200

1800

2400

Particle size x (μm)

Fig. 9. The particle size distribution resulting from First Break roller milling under four roll dispositions of Hereward (a hard wheat) and Consort (a soft wheat). Reprinted with permission from [81].

G.M. Campbell

0.00

1800

0.7 mm

0.08

0.06

1200

0.5 mm

0.10

0.6 mm

0.06

600

0.3 mm 0.4 mm

0.06

0

2400

Roll gap 0.12

0.06

0.00

1800

Dull to Dull

0.4 mm

0.7 mm

1200

Particle size x (μm)

0.3 mm

0.6 mm 0.08

600

0.14

0.12

0.5 mm

0.10

0.6 mm 0.7 mm

0

Dull to Sharp

0.4 mm

0.4 mm

0.08

2400

Roll gap

0.3 mm

0.12

0.5 mm

1800

0.14

Roll gap

Roll gap 0.3 mm

1200

Particle size x (μm)

0.14

0.10

600

Consort

Sharp to Dull

Sharp to Sharp 0.12

0.00 0

Particle size x (μm)

0.14

0.7 mm

0.08

0.04

0.00

0.5 mm

0.10

0.6 mm

ρ (x)

ρ (x)

0.08

0.4 mm

0.5 mm

0.10

0.6 mm

0.6 mm

0.3 mm

0.12

0.4 mm

ρ (x)

0.10

0.3 mm

0.12

0.4 mm

0.4 mm

Roll gap

Roll gap

Roll gap

0.3 mm

0.12

Dull to Dull 0.14

0.14

0.14

Roll gap

ρ (x)

0.14

ρ (x)

Dull to Sharp

Sharp to Dull

Sharp to Sharp

Sharp to Dull

100

R2=0.9964

90

0.3 mm 90

Dull to Sharp 90

60 50 0.3 mm

40

0.4 mm 30

80

0.5 mm

80 0.5 mm

0.6 mm

60

R2=0.9905

90

0.4 mm

% smaller than x

70

% smaller than x

% smaller than x

80

70

R2=0.9969

0.3 mm

0.4 mm 80

Dull to Dull 100

100

R2=0.9962

0.7 mm

50 40 30

70

% smaller than x

Sharp to Sharp

100

0.6 mm

60

0.7 mm

50 40 30

70 60 50 0.3 mm 40 0.4 mm 30

0.5 mm

0.5 mm 20

20

20

20

10

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10

0

0

0

0.6 mm

0.6 mm 10

0.7 mm

0 0

500

1000

1500

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2000

500

Particle size x (μm)

1000

1500

2000

0

500

Particle size x (μm)

1000

1500

Roller Milling of Wheat

Hereward

2000

0.7 mm 0

Particle size x (μm)

500

1000

1500

2000

Particle size x (μm)

Consort R2=0.9971

90

Sharp to Dull

100

Dull to Sharp

90 80

70

70

80

0.4 mm 30

80 0.5 mm

% smaller than x

0.3 mm 40

% smaller than x

% smaller than x

50

R2=0.9964

90

0.4 mm

0.6 mm

60

R =0.9942

90

0.5 mm

60

2

0.3 mm

0.4 mm

80

Dull to Dull 100

100

R2=0.9963

0.3 mm

0.7 mm

50 40 30

70

% smaller than x

Sharp to Sharp

100

0.6 mm 60 0.7 mm 50 40 30

70 60 50 0.3 mm 40 0.4 mm 30

0.5 mm

0.5 mm 20 0.6 mm 10

0.7 mm

0 0

500

1000

1500

Particle size x (μm)

2000

20

20

20

10

10

10

0

0 0

500

1000

1500

Particle size x (μm)

2000

0.6 mm 0.7 mm

0 0

500

1000

1500

Particle size x (μm)

2000

0

500

1000

1500

2000

Particle size x (μm)

403

Fig. 10. Comparison between experimentally determined cumulative particle size distributions and predictions based on the breakage equation, for First Break roller milling under four roll dispositions of Hereward and Consort. Reprinted with permission from [78].

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G.M. Campbell

in breakage that occurs at moisture contents other than 16%: Bðx; D; mÞ Bðx; DÞm0 þ Kðx; m m0 Þ

ð6Þ

leading to Z

mmax

Z

PðxÞm ¼ 0

0

¼ PðxÞm0 þ

1

Z

Bðx; DÞm0 þ Kðx; m m0 Þ r1 ðDÞr2 ðmÞdD dm

mmax

Kðx; m m0 Þr2 ðmÞdm

ð7Þ

0

They found that an equation cubic in x and quadratic in (m16%) was appropriate to describe the effect of a departure from 16% moisture on breakage of the wheat. This implied that the effect of adding moisture to wheat was to turn an initially inverted ‘‘U-shaped’’ distribution at low moisture contents into a progressively more U-shaped distribution at higher moisture contents. In other words, at low moisture, kernels broke to give a large proportion of particles of much the same mid-range size, with little size distinction, while increasing moisture gave larger proportions of both large and small particles, with fewer in the mid-size range. Figure 11 illustrates this effect of increasing moisture content. As noted above, the main purpose of conditioning (adding moisture to the wheat) is to toughen the bran particles and soften the endosperm, such that the former stay more intact and the latter shatter more readily. The increasingly U-shaped distribution on adding moisture conﬁrms this effect, as large particles are predominantly composed of bran while small particles are essentially endosperm material. The breakage equation with the moisture correction factor allows this effect to be described quantitatively. The above breakage equations need to be determined for each wheat sample individually, preventing their practical application for predicting the milling behaviour of a grist of several wheat varieties. As noted above, blending wheats of widely varying origins and properties is the major tool employed by the miller to produce consistent ﬂour quality. The ability to predict and control the breakage of today’s particular grist during First Break would help the miller to control the performance of the whole mill. When different wheats are blended and milled, it is the differences in their kernel hardnesses that have the major effect on breakage. Campbell et al. [41] therefore extended the above work to include, in addition to kernel size and moisture content, the effects of wheat kernel hardness on breakage. They also investigated the effect of the fourth SKCS parameter, mass, which when combined with kernel size can give an indication of kernel shape. This work thereby demonstrated the potential to construct a ‘‘universal’’ breakage

Roller Milling of Wheat

405

Hereward G/D=0.128

0.14

G/D=0.224

0.14

9.5%

9.5%

10.6%

0.12

10.6%

0.12

12.6%

12.6%

14.6%

0.10

14.6%

0.10

16.2%

ρ(x, D)

ρ(x, D)

16.2% 18.0%

0.08

20.1% 0.06

20.1% 0.06

0.04

0.04

0.02

0.02

0.00

18.0%

0.08

0.00

0

500

1000

1500

2000

2500

0

500

1000

1500

2000

2500

S ize x (μm)

S ize x (μm)

Consort G/D=0.139

0.14

G/D=0.267

0.14

9.7%

9.7%

10.6%

0.12

10.6%

0.12

12.3%

12.3%

14.4%

0.10

14.4%

0.10

16.6%

18.1%

0.08

ρ(x, D)

ρ(x, D)

16.6% 20.0% 0.06

18.1%

0.08

20.0% 0.06

0.04

0.04

0.02

0.02

0.00

0.00 0

500

1000

1500

2000

2500

S ize x (μm)

0

500

1000

1500

2000

2500

S ize x (μm)

Fig. 11. Effect of moisture content on the particle size distribution of stocks exiting First Break, following S-S milling of Hereward and Consort at different milling ratios.

equation that allows prediction of the breakage of an unknown mixture of wheat kernels directly from SKCS data. The effect of kernel hardness on wheat breakage could be adequately described by linear functions, leading to an extended breakage equation and function as follows: Z

H¼1

Z

D¼1

P 2 ðxÞ ¼

Bðx; D; HÞ r1 ðDÞ r1 ðHÞ dD dH H¼0

D¼0

ð8Þ

406

G.M. Campbell

Bðx; D; HÞ ¼ ða01 þ b01 x þ c01 x 2 þ d 01 x 3 Þ þ ða02 þ b02 x þ c02 x 2 þ d 02 x 3 ÞH G þ ða11 þ b11 x þ c11 x 2 þ d 11 x 3 Þ D G þ ða12 þ b12 x þ c12 x 2 þ d 12 x 3 Þ H D 2 G 2 3 þ ða21 þ b21 x þ c21 x þ d 21 x Þ D 2 G þ ða22 þ b22 x þ c22 x 2 þ d 22 x 3 Þ H D

ð9Þ

where B(x,D,H) is the extended breakage function describing the proportion of material smaller than size x produced by breakage of an inlet particle originally of size D and hardness H. Equation (8) assumes that there is no interaction between kernel size and hardness with respect to their effects on breakage; the good agreement between predictions and independent experimental data conﬁrmed the adequacy of this assumption. The coefﬁcients of equation (9) were determined by milling 19 wheat varieties of widely varying hardness at six different roll gaps under both S-S and D-D roll dispositions. For a given roll disposition, knowing the 24 coefﬁcients, the particle size distribution in the range 200–2000 mm resulting from breakage of an unknown mixture of wheat kernels varying in size and hardness, at any roll gap in the range 0.3–0.8 mm, could be predicted directly from the distribution of SKCS characteristics. Figure 12 illustrates the good agreement between the cumulative particle size distribution 100

80

80

70

70

60 50 40 30

0.3 mm

20

500

1000

1500

Particle size x (μm)

50 40 30

0.3 mm 0.5 mm

10

0.7 mm 0

60

20

0.5 mm

10 0

Consort/Spark, Dull-to-Dull

90

% smaller than x

% smaller than x

100

Consort/Spark, Sharp-to-Sharp

90

2000

0

0.7 mm 0

500

1000

1500

2000

Particle size x (μm)

Fig. 12. Comparison of predicted (lines) and experimental (symbols) cumulative particle size distributions for a 50:50 mixture of Consort (a soft wheat) and Spark (a hard wheat) milled at different roll gaps under S-S (left) and D-D (right) roll dispositions. Adapted from [41].

Roller Milling of Wheat

407 16

24

14 20

12

Frequency

Frequency

16

12

10 8 6

8

4 4

2 0

0 1

1.5

2

2.5

3

3.5

Diameter (mm)

4

4.5

5

0

10 20 30 40 50 60 70 80 90 100 110

Hardness

Fig. 13. SKCS hardness distribution for a 50:50 mixture of Consort and Spark. Adapted from [41].

predicted from equations (8) and (9) and that resulting from independent milling of a 50:50 mixture of a hard and a soft wheat variety. Figure 13 shows the SKCS diameter distribution and the bimodal hardness distribution used to predict the breakage of the mixture. Figures 14 and 15 illustrate the effect of wheat hardness on breakage (with all wheat samples conditioned to 16% moisture prior to milling). Campbell et al. [41] present this data and discuss its implications more fully, and also consider the effects of kernel shape on breakage. Brieﬂy, Fig. 14 shows the percentage smaller than x vs. average kernel hardness for each aperture size used in the sieve analysis and for selected roll gaps. The ﬁtted extended breakage functions are also shown as solid lines, illustrating the linear effect of kernel hardness on breakage. Under both S-S and D-D milling there is a divergent pattern, such that the percentage smaller than 2000 mm increases with hardness (implying the percentage of large particles decreases), while the percentage smaller than 212 mm decreases with increasing hardness. This demonstrates that soft wheats tend to break to produce relatively larger proportions of both larger and smaller particles, with fewer in the mid-size range, compared with hard wheats which produce fewer particles at the extremes and more in the middle. The divergent patterns are more pronounced under D-D than S-S, indicating that D-D milling is more sensitive to wheat hardness than S-S. Figure 15 illustrates the effect of wheat hardness with the particle size distributions resulting from breakage of Claire, a soft wheat, and Mercia, a hard wheat, at different roll gaps and under the two dispositions. Under S-S, Claire gives a relatively straight-line distribution across the range, while Mercia gives a pronounced peak in the distribution, with increasing roll gap increasing the

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Roller Milling of Wheat

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proportion of larger particles and moving the peak to the right. Under D-D the peak has disappeared for Mercia, while Claire has moved to a pronounced U shape with large proportions of both large and small particles and few in the midsize range. This indicates that soft wheats tend to shatter easily into numerous small endosperm particles, while leaving the bran material relatively intact as large particles. Hard wheats, by contrast, transmit the stresses throughout the kernel, such that the endosperm resists shattering into numerous small particles and breaks together with the bran [57]. D-D milling gives more of a crushing action, which encourages shattering of the brittle endosperm but leaves the bran layers relatively intact, while the shearing action of S-S milling cuts through both

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the bran and the endosperm material, slicing the kernel into smaller particles but not shattering it to the same extent as D-D [78,80,81]. Thus, wheat hardness and roll disposition have similar effects, such that a soft wheat such as Claire under D-D gives a pronounced U shape, while at the other extreme, a hard wheat under S-S gives quite a peak or an inverted U. As noted above, adding moisture also tends to favour U-shaped distributions by toughening the bran while making the endosperm more friable. In summary, soft wheats high moisture contents D-D milling

tend to give good fractionation of bran and endosperm into large and small particles, respectively, while hard wheats S-S milling low moisture contents

tend to give broken particles of more uniform size and composition. (However, despite these differences in the initial breakage patterns that would appear to favour soft wheats for clean bran separation, the bran from hard wheats is in fact easier to ‘‘clean up’’ subsequently [29,57]. Also, ﬂour stocks from hard wheats ﬂow and sift more easily [82], so that hard wheats are generally easier to process into ﬂour than soft wheats.) This work also demonstrates that SKCS hardness is meaningful in relation to breakage of wheat during roller milling. This is a surprising ﬁnding, as the breakage mechanism in the SKCS, involving a single rotor with a relatively ﬁne sawtooth proﬁle crushing kernels against a smooth stationary crescent with a large gap between [61], is very different from the breakage action occurring during First Break roller milling. This surprising but convenient ﬁnding means that distributions of SKCS data could be used directly to predict breakage during First Break roller milling, either off-line or as part of an automatic control system. This would aid millers in delivering consistent quality ﬂour to bakers in the face of a constantly varying feedstock. The breakage equation describing roller milling of wheat is a simple example of a population balance model. Future work will formulate the population balance explicitly in order to allow extension of the breakage equation to predict particle composition as well as size following First Break milling and to include the energy of breakage. The quantitative nature of the breakage equation approach could also be applied to subsequent break operations and to reduction milling. This would form a basis for developing complete ﬂour milling simulations and for

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clarifying more directly the mechanistic relationships between wheat kernel characteristics and the various facets of milling performance and the baking functionality of ﬂour. Other workers have also studied particle breakage during roller milling of wheat kernels and of ﬂour stocks, giving results that are generally consistent with those presented above and giving additional insights into effects on particle composition as well as size following breakage. Cleve and Will [83] investigated effects of roll gap and corrugation (ﬂute proﬁle) on breakage, concluding that the corrugation proﬁle of First Break was more important than of subsequent breaks. Hsieh et al. [40] studied the effects of conditioning moisture, feed rate, roll gap, differential and roll speed on First Break milling of a Canadian wheat sample under a D-D disposition. First Break release (the amount of ﬁne material produced) increased with moisture content and roll differential and decreased as roll gap was increased, while feed rate and roll speed had little effect. They also investigated the effects on the composition of the size fractions exiting First Break. This demonstrated that adding moisture not only increased the break release but also improved the purity of the released ﬂour, by maintaining the bran more intact as large particles and avoiding production of bran powder. Increasing roll differential similarly increased break release but at the cost of greater bran contamination in the break ﬂour due to the increased scraping of bran particles at higher differential. Scanlon and Dexter [84] present one of the few studies on reduction roller milling of ﬂour stocks. They examined the effects of roll speed, differential and feed rate on size reduction, energy requirements, starch damage and the degree of bran contamination in the ﬁnal ﬂour as indicated by ash levels and colour. Increasing roll velocity or differential, or decreasing feed rate, gave greater breakage and greater energy consumption. Increasing roll differential also gave greater starch damage due to the greater shear exerted on the starch granules within endosperm particles. Pujol et al. [85] described a micromill designed to measure accurately the mechanical energy consumption during milling of small quantities of wheat. Speciﬁc milling energy under the conditions of their study ranged from 13.2 kJ/kg for a soft wheat to 19.6 kJ/kg for a hard wheat, and correlated well with NIR hardness. This work underlines the importance of including the energy consumption in models of wheat breakage during roller milling and relating this to the particle size distribution produced. Yuan et al. [86] presented a unit operations-based analysis of the break subsystems in a pilot-scale ﬂour mill. They highlighted the need for models of particle breakage throughout the milling process and the beneﬁts of incorporating such models into computer simulations to aid mill control and optimisation as well as the training of millers. Al-Mogahwi and Baker [87] investigated breakage in both break and reduction roll systems in a commercial mill and suggested some alternative approaches for

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characterising the particle size distribution relationships of ﬂour stocks from these operations. Their approaches might allow simpler forms of the breakage function to be developed, as well as facilitating extension of the breakage equation approach developed here for First Break to the rest of the milling process. Greffeuille et al. [29] presented a study that focussed on the fate of the aleurone layer during milling of hard and soft wheats, using biochemical markers to identify the aleurone content in ﬂours from different stages of the mill. This study gave additional insights into how kernel hardness affects initial breakage patterns and hence the fractionation of the kernel into compositionally distinct components at different stages of the milling process. Fistes and Tanovic [88] similarly focussed on the composition of ﬂour particles following breakage. They demonstrated that the breakage matrix approach (the discretised form of the breakage equation) can be extended to include prediction of the compositional distribution of ﬂour stocks as well as their size distribution.

7. PEARLING OF WHEAT PRIOR TO MILLING Just over 100 years since the revolution that saw the rapid and comprehensive replacement of millstones with roller mills, another revolution has been proceeding quietly in the milling industry, particularly in the UK, over the last decade or so. That revolution is the introduction and rapid adoption of pearling technology within ﬂour milling, arguably the most signiﬁcant advance within wheat milling for several decades [89,90]. Pearling wheat prior to milling gives superior breadmaking performance in terms of organoleptic quality (larger loaf volume, more uniform crumb texture, extended shelf-life), nutrition (inclusion of the aleurone layer in ﬂour), safety (removal of undesirable surface-borne contaminants) and consistency, particularly in evening out year-to-year variations and allowing greater use of UK-grown wheat [89,91,92]. All the major UK millers have invested in the new technology, such that it has very quickly come to account for the majority of UK premium ﬂour production and has dramatically changed the entire wheat-to-bread industry. As highlighted earlier, the anatomical difference between wheat and rice, that the former has a crease and the latter does not, has required different approaches to milling these two cereals to separate bran from endosperm. Rice can be simply pearled or polished to remove the bran, but pearling is unable to remove crease bran from wheat, which can ultimately only be removed by a more demanding process involving breaking the wheat kernel open. However (following earlier work by Tkac [93]), the Satake Corporation of Japan, the rice milling engineers who in 1991 moved into ﬂour milling through the purchase of the UK-based Simon Robinson Group, applied their rice pearling technology to

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wheat to develop and commercialise the PeriTec system [14,94–96]. This system was initially intended to simplify the milling process and increase mill capacity, but the bakers quickly noticed the superior bread resulting from the ﬂour supplied by those millers who had invested in the new technology. The demands from the baking industry for this superior ﬂour has caused the rapid uptake in the technology across the UK and increasingly elsewhere and the development of similar systems by the other major milling engineers. The reasons for the superior performance of ﬂour from pearled wheat have not yet been elucidated, but this revolution in the milling industry has consequences for wheat breeding programmes and agriculture (to develop and grow varieties that perform well in pearling systems), for bakery ingredient functionality (ingredients may respond or perform differently in pearled ﬂour doughs compared with conventional ﬂours) and for non-food uses. The change also demands a revisiting of ﬂour milling and breadmaking processes to understand and exploit fully the beneﬁts of the new process. This is a challenging task, as the distance between the applied change (pearling of wheat at the beginning of milling) and the observed effect (better bread) is separated by two highly complex and interacting processes. Understanding the consequences of pearling wheat kernels starts with the effects of pearling on First Break roller milling. The different breakage patterns produced at this point determine the ﬂows through the rest of the ﬂour milling process and therefore the proportions, compositions and characteristics of the numerous streams making up the ﬁnal composite ﬂour. As yet unreported work from our laboratories has therefore investigated the effect of pearling on breakage of wheat kernels and has developed adjuncts to the breakage equation for roller milling of wheat to account for the effects of pearling. Figure 16 illustrates 0.012

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the effect of pearling a hard wheat to different extents on the subsequent particle size distribution following milling at a roll gap of 0.3 mm under both S-S and D-D dispositions. Clearly, the pearling reduces the proportion of particles at the larger end of the size range (as pearling removes bran material, and it is bran than forms the larger particles following breakage), boosts the proportion in the 500–1000 mm range and has little effect on the proportion of small particles. A cumulative pearling function, c(x,G,z) of the form cðx; G; zÞ ¼ Az3 þ K 1 Az2 þ K 2 Az

ð10Þ

where z is the degree of pearling and A ¼ ðe4 G þ f 4 Þx 4 þ ðe3 G þ f 3 Þx 3 þ ðe2 G þ f 2 Þx 2 þ ðe1 G þ f 1 Þx

ð11Þ

was developed for both a hard and a soft wheat under both S-S and D-D milling. This allows the effect of pearling to different degrees on subsequent milling to be predicted, relative to breakage without pearling as predicted by the equations developed above. This also allows the effects of pearling on the subsequent evolution of ﬂour quality through the mill to begin to be understood. As with the successful introduction of roller mills a century before, the principal driver for the current rapid uptake of pearling technology in ﬂour milling is the superior bread that results. Once again this will have implications right back through the supply chain as breeders and farmers develop and grow new varieties speciﬁcally suited to the new process. Already it is giving a new impetus to understand the interactions within and between the ﬂour milling and breadmaking processes that give rise to bread quality, in order to maximise the beneﬁts of the new technology. Pearling is also being successfully applied to durum wheat, where it gives superior pasta [14,90], as well as to other cereals including barley and oats [55,97]. The successful application of pearling also has implications for non-food uses of cereals as a renewable and sustainable alternative to oil. As with oil, economic processing of cereals for non-food products involves precise fractionation followed by conversion. In contrast with oil, cereal fractionation is undertaken on an initially solid raw material. Particle breakage is therefore a key issue, for which pearling offers an additional means of generating precise and selective separations of the kernel components. Bioconversion and extraction are then key technologies for maximising the value that can be derived from the fractions. A cereal bioreﬁnery, analogous to an oil reﬁnery, will fractionate, extract and convert cereal components to produce a varying portfolio of added value products, in response to the variable feed characteristics and the changing market opportunities. Using the empirical expertise and mathematical models developed for ﬂour milling, the cereal fractionation technologies of pearling and roller milling are poised to play a key role in bringing about sustainable chemical and energy industries in the coming century.

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8. CONCLUSIONS Breakage of wheat kernels during roller milling has been modelled in terms of the distribution of kernel characteristics (hardness, size, moisture) and roller mill operation (roll gap, disposition). A correction function to account for the effects of pearling kernels prior to breakage has been demonstrated. The breakage equation for roller milling of wheat allows the effects of a constantly changing feedstock to be predicted and controlled. It also illustrates how multiple feed particle characteristics can be included in breakage equations. International interactions and innovations at the end of the 19th century triggered a revolution that swept away crude millstones for the more precise and versatile roller mills. With the introduction of pearling technology, the ancient distinctions between wheat and rice milling have come together to create a new revolution for the 21st century. Once again this milling revolution is being driven by improved bread quality, but this time with opportune prospects for contributing to the urgent need for sustainable chemical and energy industries in an oil-depleted world.

ACKNOWLEDGEMENTS The author is grateful to the Satake Corporation of Japan, the EPSRC and the EU for support of the studies reviewed in this chapter and to Philip Bunn, Chaoying Fang, Ida Muhamad, Lili Yang, RuoHang Wang and Prasan Choomjaihan for previous and ongoing research in this area. Special thanks go to Fernan MateosSalvador for preparing several of the ﬁgures and, along with Bryan McGee, for helpful comments on the manuscript. Dr Christina Riggs, Curator of the Egyptology Section of the Manchester Museum, is gratefully acknowledged for providing Fig. 3, along with Eva Reineke of the Deutsches Museum Bibliothek for searching out information regarding Professor Friedrich Kick.

Nomenclature

an–dn ann–dnn A B(x,D)

B(x,D,H)

coefﬁcients in the breakage function coefﬁcients in the extended breakage function coefﬁcient in the cumulative pearling function cumulative breakage function describing the proportion of material smaller than size x in the output, originating from an input particle initially of size D extended cumulative breakage function describing the proportion of material smaller than size x in the output, originating from an input particle initially of size D and hardness H

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c(x,G,z) D D-D D-S e1–e4, f1–f4 G H K(x,mm0) K1, K2 m mmax n N pi P2(x) S-D S-S x z

G.M. Campbell

extended cumulative breakage function describing the proportion of material smaller than size x in the output, originating from an input particle initially of size D and moisture content m pearling function, describing the change in breakage patterns of pearled wheat kernels relative to unpearled kernels size of input particle, SKCS diameter (mm) dull-to-dull roll disposition dull-to-sharp roll disposition coefﬁcients in the cumulative pearling function roll gap (mm) SKCS kernel hardness correction term to account for the variation in breakage that occurs at moisture contents other than a nominal moisture content m0 coefﬁcients in the cumulative pearling function moisture content (%) maximum moisture content in the feed kernels (%) an integer taking values of 0, 1 or 2 number of discrete size fractions into which kernels are separated for the purpose of calculating average values proportion of kernels in size fraction i cumulative particle size distribution of the output sharp-to-dull roll disposition sharp-to-sharp roll disposition size of output particle (mm) percentage of kernel material removed by pearling (%)

Greek symbols r1(D) r1(H) r1(m) r2(x)

particle size distribution of the feed (mm1) hardness distribution of kernels in the feed moisture distribution of kernels in the feed particle size distribution of the output (mm1)

REFERENCES [1] J. Storck, W.D. Teague, Flour for Man’s Bread, University of Minnesota Press, Minneapolis, USA, 1952, p. 163. [2] F. Kick, Flour Manufacture: A Treatise on Milling Science and Practice, 2nd edition, 1888 translation by HHP Powles of 1878 German edition and 1883 supplement, Crosby, Lockwood and Son, London, UK, 1878. [3] F. Kick, Recent Progress in Flour Manufacture: Supplement to the Treatise on Milling, 2nd edition, 1888 translation by HHP Powles, Crosby, Lockwood and Son, London, UK, 1883, p. 14. [4] F. Kick, Dinglers Polytech J. 247 (1883) 1–5. [5] F. Kick, Dinglers Polytech J. 250 (1883) 141–145.

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[6] J.F. Richardson, J.H. Harker, J.R. Backhurst, Coulson and Richardson’s Chemical Engineering, vol. 2, Particle Technology and Separation Processes, 5th Edition, Butterworth-Heinemann, Oxford, UK, 2002, p. 100. [7] D. Morgan, Merchants of Grain, Weidenfeld and Nicolson, London, UK, 1979. [8] R. Tannahill, Food in History, Penguin Group, London, UK, 1988, pp. 43, 45, 72–73. [9] M. Atkin, The International Grain Trade, 2nd Edition, Woodhead Publishing Ltd., Cambridge, UK, 1995. [10] L.T. Evans, Crop Evolution, Adaptation and Yield, Cambridge University Press, UK, 1993. [11] P.C. Morris, J.H. Bryce (Eds.), Cereal Biotechnology, Woodhead Publishing Ltd., Cambridge, UK, 2000. [12] D.A.V. Dendy, B.E. Brockway, Introduction to Cereals, in: D.A.V. Dendy, B.J. Dobraszczyk (Eds.), Cereals and Cereal Products, Chemistry and Technology, Aspen Publishers Inc., Maryland, USA, 2001, pp. 1–22. [13] N.L. Kent, A.D. Evers, Kent’s Technology of Cereals, 4th Edition, Elsevier Science Ltd., Oxford, UK, 1994. [14] B.C. McGee, Assoc. of Operative Millers Bulletin, March 1995, pp. 6521–6528. [15] H. McGee, On Food and Cooking: The Science and Lore of the Kitchen, Harper Collins Publishers, London, UK, 1984, p. 234. [16] ibid., pp. 230–232. [17] A.D. Evers, S. Millar, J. Cereal Sci. 36 (2002) 261–284. [18] E.S. Posner, A.H. Hibbs, Wheat Flour Milling, 2nd Edition, American Association of Cereal Chemists Inc., Minnesota, USA, 2005, pp. 3–6. [19] H. McGee, On Food and Cooking: The Science and Lore of the Kitchen, Harper Collins Publishers, London, UK, 1984, pp. 234, 291–297. [20] Z. Gan, P.R. Ellis, J.D. Schoﬁeld, J.Cereal Sci. 21 (1995) 215–230. [21] C.E. Stauffer, Principles of dough formation, in: S.P. Cauvain, L.S. Young (Eds.), Technology of Breadmaking, Blackie Academic and Professional, London, UK, 1998, pp. 262–295. [22] S.P. Cauvain, in: S.P. Cauvain, (Ed.), Breadmaking: Improving Quality, Woodhead Publishing Ltd., Cambridge, UK, 2003, pp. 8–28. [23] G.M. Campbell, Bread Aeration, in: S. Cauvain, (Ed.), Breadmaking: Improving Quality, Woodhead Publishing Ltd., Cambridge, UK, 2003, pp. 352–374. [24] J. Storck, W.D. Teague, Flour for Man’s Bread, University of Minnesota Press, Minneapolis, USA, 1952. [25] H.E. Jacob, Six Thousand Years of Bread: Its Holy and Unholy History, Lyons Press, New York, USA, 1944, reprinted in 1997. [26] H. McGee, On Food and Cooking: The Science and Lore of the Kitchen, Harper Collins Publishers, London, UK, 1984. [27] A.D. Evers, Grain and Feed Milling Technology, Nov–Dec 2004, pp. 6–9. [28] R.C. Buri, W. von Reding, M.M. Gavin, Lebensmittel-Technologie 37 (2004) 1–2. [29] V. Greffeuille, J. Abecassis, C. Bar l’Helgouac’h, V. Lullien-Pellerin, Cereal Chem. 82 (2005) 138–143. [30] J. Storck, W.D. Teague, Flour for Man’s Bread, University of Minnesota Press, Minneapolis, USA, 1952, p. 5. [31] G. Jones, The Millers – A Story of Technological Endeavour and Industrial Success, 1870–2001, Carnegie Publishing Ltd., Lancaster, UK, 2001. [32] K.M. Turnbull, S. Rahman, J. Cereal Sci. 36 (2002) 327–337. [33] D. Morgan, Merchants of Grain, Weidenfeld and Nicolson, London, UK, 1979, pp. 24, 27–28. [34] ibid., p. 30. [35] G. Jones, The Millers – A Story of Technological Endeavour and Industrial Success, 1870–2001, Carnegie Publishing Ltd., Lancaster, UK, 2001, p. 23.

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[36] T.R. Malthus, An essay on the principle of population, Penguin, London, 1798, 1970. [37] J. Storck, W.D. Teague, Flour for Man’s Bread, University of Minnesota Press, Minneapolis, USA, 1952, pp. 241–242. [38] G.M. Campbell, C. Fang, P.J. Bunn, A.A. Gibson, F. Thompson, A. Haigh, Wheat Flour milling: A case study in processing of particulate foods, in: W. Hoyle, (Ed.), Powders and Solids – Developments in Handling and Processing Technologies, Royal Society of Chemistry, Cambridge, UK, 2001, pp. 95–111. [39] E. Haque, Cereal Foods World 36 (1991) 368–375. [40] F.H. Hsieh, D.G. Martin, H.C. Black, K.H. Tipples, Cereal Chem. 57 (1980) 217–223. [41] G.M. Campbell, C.-Y. Fang, I.I. Muhamad, On predicting roller milling performance VI. Effect of kernal hardness and shape on the particle size distribution from First Break milling of wheat, Trans. IChemE, Part C, Food Bioprod. Proc. 85 (2007) 7–23. [42] J.F. Lockwood, Flour Milling, The Northern Publishing Co., London, UK, 1945, p. 110. [43] J.H. Scott, Flour Milling Processes, Chapman and Hall Ltd., London, UK, 1951, pp. 21–25. [44] G. Jones, The Millers – A Story of Technological Endeavour and Industrial Success, 1870–2001, Carnegie Publishing Ltd., Lancaster, UK, 2001, pp. 58–59. [45] E.J. Pyler, Baking Science and Technology vol. I, Siebel Publishing Company, Chicago, USA, 1973, p. 299. [46] P. Catterall, Flour milling, in: S.P. Cauvain, L.S. Young (Eds.), Technology of Breadmaking, Blackie Academic and Professional, London, UK, 1998, pp. 296–329. [47] J. Storck, W.D. Teague, Flour for Man’s Bread, University of Minnesota Press, Minneapolis, USA, 1952, p. 258. [48] ibid., pp. 52, 232, 236. [49] J.F. Lockwood, Flour Milling, The Northern Publishing Co., London, UK, 1945, pp. 189–224. [50] J.H. Scott, Flour Milling Processes, Chapman & Hall Ltd., London, UK, 1951, pp. 152–181. [51] E.S. Posner, A.H. Hibbs, Wheat Flour Milling, 2nd Edition, American Association of Cereal Chemists Inc., Minnesota, USA, 2005, pp. 163–177. [52] J. Storck, W.D. Teague, Flour for Man’s Bread, University of Minnesota Press, Minneapolis, USA, 1952, p. 266. [53] G.M. Campbell, C. Webb, S.L. McKee (Eds.), Cereals: Novel Uses and Processes, Plenum Press, New York, USA, 1997. [54] B.K. Kamm, M. Kamm, P.R. Gruber, S. Kromus, Bioreﬁnery systems–an overview, in: B. Kamm, P.R. Gruber, M. Kamm (Eds.), Bioreﬁneries – Industrial Products and Processes, Vol. 1, WILEY-VCH Verlag GmbH and Co., Weinheim, Germany, 2006, pp. 3–40. [55] A.A. Koutinas, R. Wang, G.M. Campbell, C. Webb, A whole crop bioreﬁnery system: A closed system for the manufacture of non-food products from cereals, in: B. Kamm, P.R. Gruber, M. Kamm (Eds.), Bioreﬁneries – Industrial Products and Processes, Vol. 1, WILEY-VCH Verlag GmbH and Co., Weinheim, Germany, 2006, pp. 165–191. [56] Anonymous, http://statistics.defra.gov.uk/esg/statnot/cpssur.pdf, 2006, accessed on 21st March 2006. [57] Y. Pomeranz, P.C. Williams, Wheat hardness: Its genetic, structural and biochemical background, measurement and signiﬁcance, in: Y. Pomeranz, (Ed.), Advances in Cereal Science and Technology, Vol. 10, American Association of Cereal Chemists, St. Paul, Minnesota, USA, 1990, pp. 471–544. [58] N.L. Stenvert, Flour Anim. Feed Mill. 156 (1974) 24–25, p. 27 [59] S.C.W. Hook, FMBRA Bull. 1982/1, 1982, pp. 12–23. [60] C.R. Martin, R. Rousser, D.L. Brabec, Trans. Am. Soc. Agric. Eng. 36 (1993) 1399–1404. [61] C.R. Martin, J.L. Steele, Trans. Am. Soc. Agric. Eng. 39 (1996) 2223–2227. [62] C.S. Gaines, P.F. Finney, L.M. Fleege, L.C. Andrews, Cereal Chem. 73 (1996) 278–283.

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[63] B.G. Osborne, Z. Kotwal, A.B. Blakeney, L. O’Brien, S. Shah, T. Fearn, Cereal Chem. 74 (1997) 467–470. [64] J.B. Ohm, O.K. Chung, C.W. Deyoe, Cereal Chem. 75 (1998) 156–161. [65] M.J. Sissons, B.G. Osborne, R.A. Hare, S.A. Sissons, R. Jackson, Cereal Chem. 77 (2000) 4–10. [66] B.G. Osborne, R.S. Anderssen, Cereal Chem. 80 (2003) 613–622. [67] P.C. Arnold, A.W. Roberts, J. Agric. Eng. Res. 11 (1966) 38–43. [68] G.M. Glenn, F.L. Younce, M.J. Pitts, J. Cereal Sci. 13 (1991) 179–194. [69] G.M. Glenn, R.K. Johnston, J. Cereal Sci. 15 (1992) 223–236. [70] Y. Haddad, J.C. Bent, J. Abecassis, Cereal Chem. 75 (1998) 673–676. [71] F. Mabille, J. Gril, J. Abecassis, Cereal Chem. 78 (2001) 231–235. [72] S. Peyron, M. Charand, X. Rouau, J. Abecassis, J. Cereal Sci. 36 (2002) 377–386. [73] B.J. Dobraszczyk, M.B. Whitworth, J.F.V. Vincent, A.A. Khan, J. Cereal Sci. 35 (2002) 245–263. [74] G.M. Campbell, P.J. Bunn, C. Webb, S.C.W. Hook, Powder Technol. 115 (2001) 243–255. [75] L.G. Austin, Powder Technol. 5 (1972) 1–17. [76] G.M. Campbell, C. Webb, Powder Technol. 115 (2001) 234–242. [77] P.J. Bunn, G.M. Campbell, C. Fang, S.C.W. Hook, Proc. 6th World Chemical Engineering Congress, University of Melbourne, Melbourne, Australia, 2001. [78] C. Fang, G.M. Campbell, J. Cereal Sci. 37 (2003) 21–29. [79] C. Fang, G.M. Campbell, J. Cereal Sci. 37 (2003) 31–41. [80] C. Fang, G.M. Campbell, Cereal Chem. 79 (2002) 511–517. [81] C. Fang, G.M. Campbell, Cereal Chem. 79 (2002) 518–522. [82] D.V. Neel, R.C. Hoseney, Cereal Chem. 61 (1984) 262–266. [83] H. Cleve, F. Will, Cereal Sci. Today 11 (1966) 128–132. [84] M.G. Scanlon, J.E. Dexter, Cereal Chem. 63 (1986) 431–435. [85] R. Pujol, C. Le´tang, I. Lempereur, M. Chaurand, F. Mabille, J. Abecassis, Cereal Chem. 77 (2000) 421–427. [86] J. Yuan, R.A. Flores, D. Eustace, G.A. Milliken, Trans. IChemE, Part C, Food Bioprod. Proc. 81 (2003) 170–179. [87] H.W.H. Al-Mogahwi, C.G.J. Baker, Trans. IChemE, Part C, Food Bioprod. Proc. 83 (2005) 25–35. [88] A. Fistes, G. Tanovic, J. Food Eng. 75 (2006) 527–534. [89] J. Bradshaw, Grain and Feed Milling Technology, July–Aug 2004, pp. 10–13. [90] J. Bradshaw, Grain and Feed Milling Technology, July–Aug 2005, pp. 14–17. [91] E. Buckley, World Grain, September 2003, pp. 52–57. [92] W. Eugster, World Grain, April 2004, pp. 75–77. [93] J.J. Tkac, US Patent 5,082,680, 1992. [94] S. Satake, T. Ishii, Y Tokui, US Patent 5,390,589, 1995. [95] J.E. Dexter, P.J. Wood, Trends Food Sci. Technol. 7 (1996) 35–41. [96] D.E. Forder, Flour milling process for the 21st Century, in: G.M. Campbell, C. Webb, S.L. McKee (Eds.), Cereals: Novel Uses and Processes, Plenum Press, New York, USA, 1997, pp. 257–264. [97] D.A. Gray, R.H. Auerbach, S. Hill, R. Wang, G.M. Campbell, C. Webb., J.B. South, J. Cereal Sci. 32 (2000) 89–98.

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CHAPTER 8

Air Jet Milling Alain Chamayou and John A. Dodds Centre RAPSODEE, Ecole des Mines d’Albi-Carmaux, 81013 Albi, France Contents 1. Introduction 2. The different types of air jet mill 2.1. Fluid impact mills 2.2. Opposed jet mills 2.3. Spiral jet mills, pancake mills 2.4. Oval chamber jet mills 2.5. Fluidized bed opposed jet mills 3. Modelling 3.1. Parametric modelling 3.2. Population balances 4. Examples of application 4.1. Mineral industries 4.1.1. Talc grinding 4.1.2. Changes in particle morphology 4.1.3. Pharmaceutical industry 4.1.4. Co-grinding and formulation 4.2. Toner production References

421 422 422 425 425 426 427 427 429 430 431 431 431 431 431 433 433 435

1. INTRODUCTION Air jet milling, or more correctly ﬂuid energy milling, uses high velocity jets of gas to impart energy to particles for size reduction. Albus [1] gives a brief history of air jet mills. The ﬁrst versions, developed in the late 19th century, used a jet of compressed air to project solids against a target. Such devices have the severe disadvantage of target wear and subsequent contamination of the products but have lead to the development of other versions with opposed jets, tangential intersecting jets and more recently opposed jets in a ﬂuidized bed. The carrier ﬂuid is usually compressed air, but nitrogen is often used in the pharmaceutical industry for inerting. In the mineral industry the ﬂuid energy can be brought by steam. Corresponding author. Tel.: (33) 05 63 49 31 22; Fax: (33) 05 63 49 30 25; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12011-X

r 2007 Elsevier B.V. All rights reserved.

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Air jet mills have mainly been developed for producing ﬁne particles of below 50 mm and have the following common features: There are no moving parts in the grinding chamber and energy for size reduction is brought by the carrier gas. The main grinding action is by particles hitting other particles making for little or no product contamination. Two types of nozzles are used (Albus [1]). The most common is the converging nozzle, which comes to an abrupt stop leading to sonic velocities and an exit pressure about 50% of the inlet pressure. The Laval type convergent–divergent nozzle, where gas expands in the divergent section increasing velocity to supersonic has been examined by Voropayev et al. [2]. The energy in the jet is controlled by the speed of sound, which is inversely proportional to the square root of the molecular weight of the gas used. Thus, replacing nitrogen (Mw 14) with helium (Mw 2) gives an energy gain of 3. The fragmentation mechanism leads to an intrinsically wide product particle size distribution but being air swept makes it simple to have integral product classiﬁcation and re-cycle of coarse particles, turning this to an advantage. The main features of air jet mills include:

Produce very ﬁne particles, typically less than 40 mm Good control of particle size and distribution by integral classiﬁcation Applicable to a wide range of material hardness Autogeneous action giving low contamination Mechanically simple with few, if any, moving parts Relatively costly in energy.

Other features are that the adiabatic pressure release at the nozzles and high ratio of transport gas to solids loading makes for good cooling capacity allowing for processing of heat sensitive materials. There is also the possibility of simultaneous drying or re-hydration, of operating in an inert atmosphere or on the contrary providing for gas–solids chemical reactions. Finally, most models are available in many sizes up from lab-size versions (as low as to 1 gram/h) up to industrial versions (up to 6 tonne/h) making for easy testing and scale-up. However, a disadvantage is that air jet mills require ancillary air compressors which can assure high ﬂow rates at pressures up to about 10 bars.

2. THE DIFFERENT TYPES OF AIR JET MILL 2.1. Fluid impact mills In a ﬂuid impact mill particles are projected against a ﬁxed target by a jet of gas. It was historically the ﬁrst type of jet mill introduced in 1882 by Goessling [3]. In this

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Fig. 1. Fluid impact mills [1].

Fig. 2. Micronjet mill (Hosokawa).

version, the lid of the mill served as the impact anvil and separation and recirculation was done by a sieve at the bottom (Fig. 1). This type of mill is now not in general industrial use as it has only a low throughput and there is excessive wear of the target. A more modern variant is the Coldstream impact process introduced in 1962, which produced grinding by impact on a labyrinth of replaceable balls. Another is the Alpine Micron Jet in which input material is accelerated and impacted on a target ring slowly rotating round the vertical axis of an integral classiﬁer (Fig. 2). The air pressure used and the design of the jets leads to velocities up to the speed of sound. The integral classiﬁer recycles the coarse

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Fig. 3. Multi-impact of PMMA spheres: (a) mean particle size number of impacts [6]; (b) aspect of the particles [6].

particles in the grinding zones thus limiting over-grinding by removing the ﬁnes. The grinding action is both by impact and by attrition. Impact on a target is also used in many different forms as a laboratory test method for impact grinding [4–6]. The projection at and subsequent deﬂection of the jet of gas and particles at the target means that there is selective breakage; only particles greater than a certain size having sufﬁcient inertia to leave the streamlines hit the target [7]. Particles smaller than this size follow the jet stream round the target. A recent example is that of Mebtoul [5] and Lecoq [6] shown in Fig. 3. Particles are accelerated in a convergent–divergent nozzle and for impact at a target which can be angled. The test method gives information on the changes in particle size distribution as a function of impact velocity, impact angle, air pressure type of nozzle, etc. The particles are only subject to one single impact but can be collected and recycled for investigating the effect of multiple impacts. An interesting result is shown in Fig. 3 where it was found that with ductile

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polymethylmethacrylate (PMMA) spheres up to 10 successive impacts at 283 m/s were required to fragilize the particles before fragmentation.

2.2. Opposed jet mills Projecting one jet of particles against another is one way to overcome the problem of wear on the impact anvil. This method was introduced by Willoughby in 1917 [8] and is the basis of the Majac mill. The feed product is swept round the grinding chamber and through the two phase (air+particles) nozzles to impact each other until the particles are ﬁne enough to leave through the centrifugal classiﬁer located in the grinding chamber (Fig. 4). The test rig of Mebtoul [5] and Lecoq. [6] mentioned above was also modiﬁed to study impacts of particles in two opposed jets. One signiﬁcant result was that the collision probability of particles in low concentration jets was very low indicating the importance of high concentration in inter-jet impacts.

2.3. Spiral jet mills, pancake mills Tangential jet mills were introduced in 1934 by the Micronizer Company. Two names are used for the same mill: either ‘‘spiral jet’’ mills or ‘‘pancake’’ mills. In this type of mill, feed particles are injected by a venturi into a ﬂat disk-shaped chamber fed with high-speed peripheral inclined ﬂuid jets. This produces a high-speed ﬂat ﬂuid vortex in rotation in the chamber where comminution occurs by particleparticle impact at what is called the grinding circle. This has been visualized by Rumpf and co-workers [9] using triboluminescence. The rotating vortex in the grinding chamber means that there is simultaneous centrifugal separation and

Fig. 4. Opposed jet mills: (a) the Majac mill; (b) the Trost mill.

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Fig. 5. Pancake or spiral jet Mill.

acts as an integral classiﬁer. Thus, ﬁne product particles leave by the central outlet and coarse particles are pushed outwards by centrifugal force and remain in rotation in the mill chamber to be subjected to further impacts. These mills are now in extensive use in all sectors of industry due to their high capacity for ﬁne grinding of substances with a Moh’s hardness of less than 3.5. They are especially favoured in the pharmaceutical for their simplicity and ease of cleaning. Several companies propose such mills with different variants such as, the shape of the grinding chamber (ﬂat cylinder Hosokawa, octagonal, Jet Pharma, Elliptical FPS), the number of nozzles and whether their angle can be adjusted or not and separate outlets for air and ground particles or just one single outlet with air–particle separation. Some equipment manufacturers make small single use spiral jet mills for small quantities of high value active ingredients. Recent studies reported by Nackach et al. [10] have shown that for organic crystals, the product particle size obtained correlates with the speciﬁc energy proportional to the grinding air pressure and the solids ﬂow rate [11] (Fig. 5).

2.4. Oval chamber jet mills Torus chamber mills have many similarities with tangential jets mills. The main differences are that the grinding chamber is oval or ‘‘bean’’ shaped rather than circular and is ﬁxed vertically rather than horizontally. Also, the ﬂuid jets are fewer and mainly located at the base of the mill near the product entry (Fig. 6). This type of mill was ﬁrst introduced in 1941 by the Jet-O-Mizer Company. One derivative was a double side-by-side version which added opposed jet impact to the ﬂuid jet grinding action. As with spiral jet mills the product particle size distribution is a function of the speciﬁc energy. In general, these mills have higher capacities than spiral jet mills and grinding rates of up to 6 tonne/h are reported.

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Fig. 6. Jet-O-Mizer Mill.

2.5. Fluidized bed opposed jet mills In ﬂuidized bed opposed jet mills, air jets are used to give high-energy impacts between particles which are in suspension in a ﬂuidized bed. Thus, these mills have high particle concentrations in the grinding chamber but the nozzles only carry air and are less subject to wear than the two phase opposed jets in the mill described in Section 2.2. The feed particles are introduced in the mill by a screw feeder and the outlet of the mill is attached at the top by means of an integral centrifugal classiﬁer. The speed of rotation of the classiﬁer deﬁnes the upper size of the particles which can leave the grinding chamber. Particles greater than this size are returned to recirculate in the grinding zone. Vogel [12] has given a description of the development of such mills by the Alpine company. Recent studies are by Berthiaux et al. [13] and Godet et al. [14] (Fig. 7). These mills are in use in many industries and are available as standard in sizes from a chamber diameter of 10 cm, nominal capacity 10 kg/h, up to 1.25 m diameter, nominal capacity 2 t/h. The operating capacity depends strongly on the type and particle size of the material being ground as shown in Table 1 giving typical performance of Alpine ﬂuidized bed opposed jet mills [15].

3. MODELLING There are two main theoretical approaches to describe the operation of air jet mills. The ﬁrst is by correlations based on parametric studies of operating variables and

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Fig. 7. Fluidized bed opposed jet mill (Alpine).

Table 1. Operating information on Alpine ﬂuidized bed opposed jet air mills [15]

Process requirement Extreme purity

Material

Product ﬁneness

Speciﬁc air consumption (Nm3/kg)

Electronics

99%o20 mm 97%o20 mm 99.9% o10 mm 99.5%o5 mm 99.9%o20 mm Allo10 mm 95%o75 mm 99.9%o10 mm 99%o20 mm 99%o50 mm 99.99%o 63 mm 99.7%o10 mm 99.9%o 20 mm 99%o125 mm 99.7%o50 mm

2.6 1.7 10.6 34 3 21.3 7.3 14.2 7.8 11.6 16.4 3 10.7 5.6 24.4

Alumina Abrasive powder

Clogging powder

Corundum Quartz Spinel Silicon carbide

API Pigment Herbicide Heat sensitive Toner powder Artiﬁcial carbon Polypropylene wax SelectiveSiderite–bauxite Sideriteo1% separative Foundry sand Glow loss 0.4% grinding (binder-quartz)

1.2 1

Operating pressure (bar) 3 4 6 6 10 8 3 6 6 6 6 6 6 6 10 1.4 1.2

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speciﬁc grinding energy, the second by population balances. A useful performance parameter is the grinding ratio deﬁned as the ratio of feed speciﬁc surface to the product speciﬁc surface.

3.1. Parametric modelling Considering spiral jet mills, Midoux et al. [16] reports that the main factors governing operation for a given material are (1) the design of the mill: chamber diameter, number and angle of the nozzles; and (2) the operating variables: solids ﬂow rate, grinding pressure and injector pressure. The diameter of the grinding chamber conditions the capacity of the mill. Taking the volumetric ﬂow rate as proportional to the square of the chamber diameter and the feed rate proportional to the volumetric ﬂow rate to the power 1.470.1 leads to the following relation between the grinding capacity and the diameter of the chamber: Qsolid / Dm

ð1Þ

Here m is a factor which depends on the material being ground and may be taken to be m ¼ 2.870.2. The number and angle of the nozzles is an important design feature. Experiments in which the number of nozzles was varied whilst keeping constant the total cross section of nozzles (to maintain the same air ﬂow rate) showed that the greater number of nozzles gave the best results. This could be attributed to the greater regularity of the grinding circle which is also improved by the corresponding lessening of the nozzles diameter so that each nozzle less perturbs the ﬂow in the chamber. The penetration angle of the jet also affects the grinding circle and it is found that the optimum is between 521 and 601. The injector pressure is ﬁxed higher than the nozzle pressure to avoid back ﬂow but only slightly higher so as to disturb the grinding circle as little as possible. A value of grinding pressure+0.5 bar is often recommended. A speciﬁc energy consumption in air jet milling can be deﬁned as the kinetic energy of the gas ﬂow divided by the solids ﬂow rate. This allows expressing operating conditions independently of different mill types and working conditions. It is found that the speciﬁc surface of the ground product is a power function of the speciﬁc surface energy. Ek ¼

1 Mgv 2 2

E sp ¼

Ek P / Qsolid Qsolid

S0p / E Xsp

ð2Þ

Experimental results indicate that in given conditions with a given type of mill there exists an optimum feed rate above which there is a coarser product. This may also be expressed in terms of a critical value of speciﬁc energy required to obtain narrower product size distribution. For example, Schurr et al. [17] found

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Fig. 8. Speciﬁc energy as a function of product particle size for three types of jet mill: (a) spiral jet mill; (b) oval chamber mill; (c) ﬂuidized bed opposed jet mill (Alpine).

3600 kJ/kg for sand and Midoux et al. found 400–800 J/kg for crystalline pharmaceutical products. Comparisons have been made of the speciﬁc energy for the three main types of jet mill. These are shown in Fig. 8 where it can be seen that speciﬁc energy rises sharply as the product particle size is less than 10 mm and that the spiral jet mill requires more energy than the ﬂuidized bed opposed jet mill with the Majac mill lying between these. This is obviously not the only criterion for selection and the nature of the powder being ground can be important. For example, ﬂuidized bed opposed jet mils and pancake mills do not tolerate well highly cohesive powders which can form deposits in the grinding chamber.

3.2. Population balances The population balance method has been used to model a ﬂuidized bed opposed jet mill by Berthiaux et al. [18]. They divided the grinding chamber into three zones: (1) a perfectly mixed zone where grinding took place, (2) a plug ﬂow transport zone and (3) a classiﬁer zone. The grinding kinetics in zone 1 were established by tests using the mill in batch mode with the classiﬁer at maximum speed. The particle size distributions in the chamber at different grinding times were analyzed using Kapur’s approximate solution of the batch grinding equation

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thus allowing determination of the selection matrix S. An approximation was proposed for the b matrix allowing it to be calculated from the S matrix. Combining this with the overall three-zone ﬂow model made it possible to predict the product size distribution in steady state regime (Fig. 8). This approach has been taken further by de Vegt et al. [19] who introduced a correlation which linked the S function to the mechanical properties of the material to be ground and the speciﬁc energy of the mill.

4. EXAMPLES OF APPLICATION 4.1. Mineral industries 4.1.1. Talc grinding Fluidized bed opposed jet mills are used for the industrial production of talcum powder, a lamellar structure magnesium phyllosilicate. The ﬁrst stage of grinding of mineral from the quarry is done by rotating pendular mills to about 50 mm which feed ﬂuidized bed opposed jet mills to give a ﬁnal product characterized by particle size and whiteness. Information on industrial production conditions is not available but tests in a 100AFG Alpine ﬂuidized bed opposed jet mill have been reported by Godet [20] and Godet et al. [21]. The main conclusions from this work is that there is a shallow optimum in the particle size feed rate curves indicating that running the mills at the ﬂooding limit may not be optimal. The other is that the product quality from the mill in terms of mean particle size and tightness of the distribution depend mainly on the characteristics of the integral classiﬁer. Typical results are shown in Fig. 9 and 10.

4.1.2. Changes in particle morphology It is well known that the type of grinding can have a strong effect on the physico–chemical characteristics of the ground product. Palaniandy et al. [22] report results of ﬁne grinding of silica for use as ﬁller in paints. Silica has several advantages over other ﬁller minerals for paints such as calcium carbonate, talc etc. but it can be abrasive and deteriorate paint mixing equipment. The aim was to produce ﬁne silica particles with a high sphericity and smooth surface so as to reduce abrasivity. Tests in a ﬂuidized bed opposed jet mill showed that the circularity of the ground particles depended on the classiﬁer speed and that a speed of 10 000 rpm gave particles with narrow size distribution and high circularity.

4.1.3. Pharmaceutical industry The pharmaceutical industry has speciﬁc requirements for milling processes. Nakach et al. [10] evaluated six different types of mill including three types of air

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Fig. 9. Product talc size distribution as a function of classiﬁer speed (Malvern).

Fig. 10. Product median particle size as a function of: (a) hold up in mill; (b) feed rate [21].

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jet mill for pharmaceutical grinding using vitamin C crystals. These were: Pancake mill, Oval Chamber mill, Fluidized bed opposed jet mill. They found that the product quality in terms of speciﬁc surface determined by a permeability method depended mainly on the speciﬁc energy in the pancake mill and the oval chamber mill and on the characteristics of the integral classiﬁer in the ﬂuidized bed opposed jet mill. At equal speciﬁc energy the pancake mill gave a ﬁner mean particle size with a tighter distribution and was found to be more energy efﬁcient than the oval chamber mill. The ﬂuidized bed opposed jet mill gave good product quality but lower production rate but was deemed difﬁcult to clean due to mechanical complexity and required more maintenance than the simple pancake and oval chamber mills.

4.1.4. Co-grinding and formulation Drug administration by dry powder inhalation has many advantages for the patient but requires careful control of particle size to have drug substance reach the inner lung for exchange with the blood stream. This requires conciliating large particle sizes for reproducible inhaler dosing and ﬁne particle sizes of 2–7 mm for the drug. In practice, the drug substance is carried from the administration device on larger particles o100 mm and detaches before capture in the upper larynx mucous layer. Formulation for such applications requires free ﬂowing powders with controlled particle size and good mixing for ﬁxing the drug substance on the carrier particles. Giry et al. [23] have shown that ﬁxing fusafungine (API) on a lactose carrier is conveniently done by a two-step process. The ﬁrst step is by co-grinding lactose with the API in a spiral jet mill to a particle size of 2–7 mm. The second step is blending this lactose-API mixture on coarse lactose carrier particles of lactose. This method gives more reproducible results than simple grinding of the API and mixing with lactose carrier. The spiral jet mill has therefore the role of adjusting the particle size to be in the alveolar fraction and providing a ﬁne powder compatible with lactose for easy, stable mixing. Co-micronization of drug substances with tensioactives can improve the rate of dissolution and enhance dispersion. Boullay [24] and Godet et al. [21] have also investigated the use of additives in the grinding air in ﬂuidized bed opposed jet mils for talc grinding. They found that in certain circumstances, the chamber ﬂooding limit was reduced and the production rate could be almost doubled.

4.2. Toner production The market for toners for photocopiers and printers both black and white and coloured is in constant increase. These products are high-value powders with very tight speciﬁcations for physical characteristics, in particular particle size and distribution. After formulation and mixing, the material is extruded in the form of

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A. Chamayou and J.A. Dodds

Fig. 11. Toner production plant [25].

granules of about 2 mm particle size, which is then micronized in jet mills either target type or ﬂuidized bed opposed jet mills to obtain a powder of about 6–10 mm. The process usually involves a second stage of classiﬁcation followed by blending and then ﬁnal separation by sieving. The primary user quality is the mean particle size and the sharpness of the distribution together with absence of metal contamination. A recent tendency is to use in-line particle size analysis for product quality control and process optimization. Typical plant capacities are of the order of 10 tonnes/day. Figure 11 shows an example of a toner production line involving extrusion, micronization/classiﬁcation in a ﬂuidized bed opposed jet mill followed by second stage of classiﬁcation re-cycle, storage and blending, then ﬁnal separation by ultrasonically assisted sieving. In this plant the extrudates are fed directly to the ﬂuid mill. In other cases, a pre-grinding step may be used before micronization. Also this plant has in-line particle size analysis at two points in the circuit: at the ﬂuid mill outlet and the outlet of the ﬁnal classiﬁer. This allows close control of product quality and also optimum start-up especially useful for efﬁcient management of changes from one product formulation to another.

Nomenclature

D Ek Esp Mg

diameter of mill chamber (m) gas kinetic energy (J) speciﬁc energy consumption (J kg1 s) mass solid (kg)

Air Jet Milling

P Qsolid S0p V X, m

435

gas pressure (Pa) solids ﬂow rate (kg s1) speciﬁc surface of product (m1) velocity (m s1) constants (–)

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25]

F.E. Albus, Chem. Eng. Prog. 60 (1964) 102–106. S. Voropayev, D. Eskin, Miner. Eng. 15 (2002) 447–449. J.W. Gosset, Chem. Process. July (1966) 29–64. J.A. Dodds, O. Lecoq, A. Chamayou, Conveying and handling of particulate solids, Plenary Conf., 4th Int. Conf., Budapest, Hungary, May 2003. M. Mebtoul, Doctorat thesis, UTC Compiegne, 1994. O. Lecoq, Doctorat thesis, UTC Compiegne, 1997. J.A. Laitone, Wear 56 (1979) 239–246. A. Willoughby, N. Andrews, American Patent, 1917. H. Kurten, H. Rumpf, Chemie-Ing. Techn. 38 (1966) 331–342. M. Nakach, J.R. Authelin, A. Chamayou, J.A. Dodds, Int. J. Min. Proc. 74 (2004) 173–181. H.J.C. Gommeren, D.A. Heitzmann, J.A.C. Moolenaar, B. Scarlett, Powder Technol. 180 (2000) 147–154. A. Vogel, Powder Handl. Process. 3 (1991) 129–132. H. Berthiaux, J.A. Dodds, Powder Technol. 106 (1999) 78–98. L. Godet-Morand, A. Chamayou, J.A. Dodds, Powder Technol. 128 (2002) 306–313. P. Garnier, personal communication, Alpine, SA. N. Midoux, P. Hosek, L. Pailleres, J.R. Authelin, Powder Technol. 104 (1999) 113–120. G.A. Schurr, Q.Q. Zhao, 8th European Symp. on Comminution, May 1994. H. Berthiaux, J.A. Dodds, Chem. Eng. Sci. 51 (1996) 4509–4516. O. de Vegt, H. Vromans, F. Faasen, K. van der Voort Maarschalk, Part. Part.Systems Charact. 22 (2005) 133–140; 261–267. L. Godet-Morand, Doctorat thesis, INP, Toulouse, 2001. L. Godet-Morand, A. Chamayou, J.A. Dodds, Powder Technol. 128 (2002) 306–313. S. Palaniandy, K.A.M. Azizli, E.X. Hong, S. Shashim, H. Hussin, Azojomo Online J. Mater. (2006) 1–9, http://azom.comm K. Giry, J.M. Pean, L. Giraud, S. Marsas, H. Rolland, P. Wuthrich, Int. J. Pharm. 321 (2006) 162–166. G. Boullay, Personal communication, Lab-Services, SA. M. Mebtoul, Conf. Infovrac, 2004.

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CHAPTER 9

Breakage and Morphological Parameters Determined by Laboratory Tests Meftuni Yekeler Cumhuriyet University, Department of Mining Engineering, Sivas 58140,Turkey Contents 1. Introduction 1.1. Formulation of the problem for grinding circuits 2. Operation of ball mills 2.1. Breakage parameters 2.1.1. Speciﬁc rate of breakage 2.1.2. Cumulative progeny fragment distribution 2.1.3. Size–mass rate balance modelling 2.2. Slowing down phenomena in ball milling 2.3. Grinding aids 2.4. Morphological parameters 2.5. Breakage parameters of minerals 2.5.1. Quartz 2.5.2. Calcite 2.5.3. Barite 2.5.4. Zeolite 2.5.5. Coals (lignite and anthracite) 2.5.6. Clinker 2.5.7. Chromite 2.5.8. Ceramic raw materials 2.6. Simulation of ball milling products using the breakage parameters 3. Morphological parameters of minerals ground by different mills 3.1. Materials and methods employed 3.1.1. Quartz 3.1.2. Calcite and barite 3.1.3. Talc 4. Summary 5. Concluding remarks References

438 438 439 439 439 439 440 441 442 442 444 444 447 452 454 455 455 456 457 459 463 463 470 472 479 482 483 485

Corresponding author. Tel.: +346 2191010/1579; Fax: +346 2191173; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12012-1

r 2007 Elsevier B.V. All rights reserved.

438

M. Yekeler

1. INTRODUCTION Size reduction of solids and minerals by crushers and grinding mills is an important industrial operation involving many aspects of mineral, metallurgical, power and chemical industries. It is already known that machines for breakage of large lumps are called crushers and those for smaller sizes are called mills. Size reduction by crushers does not create problems due to having high energy consumption and capital cost per ton per hour; however, ﬁne grinding by mills consumes a lot of energy and causes high abrasive wear. Therefore, many scientiﬁc and technical problems are related to ﬁne grinding operations and its associated problems [1,2]. As mineral particles are reduced to ﬁner product sizes, their surfaces become more important. Surface characteristics and properties affect any of the ﬁne particle processing operations.

1.1. Formulation of the problem for grinding circuits The aim in mill circuit design is to select a mill that will produce a desired tonnage per hour of required product from a speciﬁed feed; therefore, the capital costs need to be minimized which are related to correct mill conditions including rotational speed, ball load and sizes [2]. In general, the mill employed for grinding circuits should be operated efﬁciently in terms of high mill capacity and low energy consumption, subject to lifter wear, maintenance costs, product contamination that will cause problems in the beneﬁciation stage. Another problem is the oversize product from the mill, which needs to be recycled by devising several stages of grinding with a classiﬁer that splits the product into coarser and ﬁner sizes. The coarse particles are recycled back to the mill feed to be reground as seen in Fig. 1. In order to design the efﬁcient mill circuits, the following factors need to be considered: mill type and size, mill power, efﬁcient grinding conditions, recycle and classiﬁcation efﬁciency, mill circuit behaviour under different conditions and economic constraints [2]. Fine Product Feed

Mill product Mill Classifier

Recycle

Fig. 1. Mill circuit design with recycling of coarser particles. Redrawn from [2].

Breakage and Morphological Parameters

439

2. OPERATION OF BALL MILLS Coarse feed particles enter one end of the mill, pass down the mill receiving breakage actions because of the heavy balls, and exit as an end product with a ﬁner size distribution. Here, energy input is converted to mechanical breakage action to form the broken ﬁner size particles. The key point in designing the mill circuit is to size a mill to produce a desired tonnage per hour of a required product from a speciﬁed feed [2].

2.1. Breakage parameters 2.1.1. Speci¢c rate of breakage The breakage of a given size fraction of material usually obeys a ﬁrst-order breakage pattern [3]. Rate of breakage of size i ¼ Si w i W

ð1Þ

where wi is the weight fraction of material of size i, W the total material charge in the mill and Si the speciﬁc rate of breakage of size i. Now, equation (1) becomes

dw 1 ðtÞW ¼ S1 w 1 ðtÞW dt

ð2Þ

integrating equation (2) log w 1 ðtÞ log w 1 ð0Þ ¼ S1 t=2:3

ð3Þ

where t is the grinding time. Figure 2 shows the typical ﬁrst-order plots for breakage of one-size fraction feeds for relatively large top sizes [4].

2.1.2. Cumulative progeny fragment distribution The particles of a given size produce a set of primary daughter fragments which are mixed into the bulk of the powder and then fractured again. For the general size interval i, a size–mass balance equation becomes [5] i1 X dw i ðtÞ ¼ Si w i ðtÞ þ bi; j Sj w j ðtÞ; n4i4j41 dt j¼1

ð4Þ

i41

where bi,j is the fraction of material broken from larger sizes j which reports to smaller size i, and n is the sink size. The accumulation of wi(t) values gives the fraction of charge below size j P j ðtÞ ¼ n

j X n

w i ðtÞ

440

M. Yekeler

Weight Fraction Remaining in Top Size, [w1(t) / w1(0)]

1.00

−d+e −c+d

0.10

−b+c

−a+b

0.01

0

4

2

6

8

10

Grinding Time, minutes

Fig. 2. First-order plots for breakage of various one-size feeds (which a4b4c4d). Redrawn from [4].

The cumulative primary daughter fragment distribution Bi,j is also deﬁned as Bi; j ¼

i X

bk; j

n

and ﬁts the empirical equation below x i1 b x i1 g þ 1 Fj noioj Bi; j ¼ F xj xj

ð5Þ

where parameters F, g and b deﬁne the distribution and also Bi,j can be estimated from Bj;i ¼

log½ð1 P i ð0ÞÞ=ð1 P i ðtÞÞ log½ð1 P 2 ð0ÞÞ=ð1 P 2 ðtÞÞ

ð6Þ

where Pi(t) is the fraction less than upper size of interval i at time t. Figure 3 shows how to obtain the Bi,j parameters of any material ground [6].

2.1.3. Size-- mass rate balance modelling The concept of ‘‘size–mass balance’’ used in the formulation of grinding equations is simply a rate–mass balance on each particle size interval; it is considered as a population balance but it is usually mass that is experimentally measured

Breakage and Morphological Parameters

441

1.00

Cumulative Breakage Function (Bi,j)

φ

0.10

γ β

0.01 0.01

0.10 Relative Size (xi/xj)

1.00

Fig. 3. The cumulative primary daughter fragment distribution of any material ground. Redrawn from [6].

rather than numbers of particles, because it is more convenient to work in terms of mass. Also, ﬁrst-order breakage leads to a simpler solution with physical reality for many cases, as ﬁnal balance equation given in equation (4), which is the fundamental size–mass rate balance for fully mixed batch grinding and this set of n differential equations explain the grinding process: this equation means that ‘‘the rate of production of size i material equals the sum of the rate of appearance from breakage of all larger sizes minus the rate of its disappearance by breakage’’ [2].

2.2. Slowing down phenomena in ball milling The computed results do not agree with the experimental results at ﬁner degrees of grinding, indicating a slowing down of breakage rate [5]. One possible reason for the slowing down phenomena is that air might be trapped between particles in a ﬁne charge and the well-known slow movement of air through beds of ﬁne particles might change the mechanics of the breakage action by blowing away particles or by absorbing impact like a hydraulic shock absorber [7]. Grinding time required to reach a given size distribution is longer due to the slowing down effect in the mill. The other possibility is that a bed of cohesive ﬁne particles develops almost liquid-like properties, so that particles ﬂow away from the ball–ball collision region and insufﬁcient stress is transmitted to individual particles for fracture to occur.

442

M. Yekeler

The instantaneous value of Si at time t is reported by Si ðtÞ ¼ k Si ð0Þ

ð7Þ

where k is also a reduction factor (0rkr1) which is a function of the ﬁneness of grinding. K ¼ dy=dt

ð8Þ

knowing the variation of y with t enables k to be determined by graphical differentiation.

2.3. Grinding aids Laboratory and industrial grinding tests show that using chemicals during grinding process can signiﬁcantly improve the efﬁciency of powder production. The term ‘‘grinding aid’’ or ‘‘grinding additive’’ is used for a substance which is mixed into the mill and causes an increase in the rate of size reduction [2]. A mechanism which is often explained by the Rehbinder effect [8] which is the adsorption of additive on the surface of a solid lowers the cohesive force that bonds the molecules of the solid together. This allows more efﬁcient breakage between media and particles by reducing the slowing-down effect. These grinding aids used are ethylene glycol, propylene glycol, triethanolamine, oleic acid, and aminoacetates. The most important one for tumbling mills is water. The efﬁciency ranges from 1.2 to 2 times greater than that of grinding. In the case of wet grinding, chemical aids can be effective if (1) the mill is operated at high slurry density, (2) solids have sufﬁcient adsorption sites to adsorb enough chemicals, (3) the chemicals act consistently to lower viscosity over the considered pH range, and (4) the chemicals do not adversely affect downstream processing operations. The chemicals that meet these conditions are low molecular weight, water-soluble anionic polymers [2].

2.4. Morphological parameters The particle shape distribution of feed, product and gangue products of mineral processing unit operations need to be characterized in order to predict how shape affects the mineral recoveries for designing the equipment and circuits. Therefore, shape and surface properties of particles play an important role in many aspects of mineral processing and wettability based operations, mainly ﬂotation and ﬂocculation. The shape and morphological characterization of particles are based only on the analysis of the particles or their projections. Assuming that the projection of the particles has an ellipse-like shape [9], the major axes length (L) and width (W) of

Breakage and Morphological Parameters

443

P L A w

Fig. 4. Measurement of length (L), width (W), perimeter (P) and area (A) of a particle projected [10].

each particle are measured. The mean values of ﬁve lines drawn on the particle projection for each axis were taken as the real length (L) and width (W) values of that particle in mm considering the scale of the image. Thus, the area (A) and perimeter (P) of the particle projection can be calculated on the basis of the measured length and width as given in equations (9), (10) and in Fig. 4 [10]. Q LW A¼ ð9Þ 4 Q pﬃﬃﬃﬃﬃﬃﬃﬃ 3 ðL þ W Þ LW ð10Þ P¼ 2 2 From these basic measurements and calculations, four shape factors given in equations (11)–(14) were used for the shape characterization of the particles ground by any of the mills: elongation, ﬂatness, roundness and relative width [9,11–14]. Elongation ðEÞ ¼

Flatness ðFÞ ¼

4

Roundness ðRÞ ¼

L W

P2 Q 4

ð11Þ

ð12Þ

A

Q P

A

2

Relative width ðRW Þ ¼

W L

ð13Þ

ð14Þ

From equations (12) and (13), it can be seen that ﬂatness is actually the inverse of roundness. Roundness has a maximum of 1.0 for a circle. On the other hand, ﬂatness has a minimum of 1.0 for a circle. The values of relative width and roundness increase with decreasing elongation. That is, the higher the values of the parameter, the rounder the shape of the particle projection. However, the value of ﬂatness increases with increasing elongation, i.e., the higher the value of the function, the more elongated the particle. Surface roughness occurs due to ﬂuctuations around a smooth and sharp interface, but probably represents a lower free energy state [15]. Roughness is

444

M. Yekeler

also the most important property of solid surfaces [16] and is of great interest to researchers in materials science and applications [17]. The physical structure of a solid surface, i.e., its roughness and topography will affect the reactivity of the solid. Interfacial reactions are increased by the larger speciﬁc surface areas associated with the rough surfaces, while the presence of surface features will generally tend to increase the density of high energy sites relative to a ﬂat surface of the same substance. The atomic structure of the layers have a major effect on the surface properties that are variable depending on the type of the mill employed, even for the samples that have identical composition and conditions [18]. Surface roughness values of particles are determined by two different techniques. In the ﬁrst, a three-dimensional approach is used by a portable stylus type roughness-measuring instrument called Surtronic 3+ HB-103 and expressed by Ra. The second method is based on the gas adsorption technique and expressed by the calculated roughness (RBET) values.

2.5. Breakage parameters of minerals 2.5.1. Quartz Dry grinding of quartz mineral was carried out in the laboratory ball mill, as described in Table 1, at a low ball load of 20% of the mill volume ﬁlled with the ball bed and a low powder load corresponding to a formal interstitial ﬁlling of the void spaces of the ball bed of U ¼ 0.5 [19]. These conditions were chosen because it is known [2] that both dry and wet grinding give normal ﬁrst-order grinding kinetics under these loading conditions. Figure 5 shows the initial grinding results plotted in ﬁrst-order form, which was given in equation (3). As shown in Fig. 6, the values of Si can be ﬁtted to the expression (15), Si ¼ aT ðx i =x 0 Þ

1 1 þ ðx=mÞL

ð15Þ

with aT ¼ 0.6 min1, a ¼ 0.80, m ¼ 1.9 mm, l ¼ 3.7 and x0 ¼ 1 mm. The cumulative primary breakage distribution function determined using the BII procedure given by equation (6) is shown in Fig. 7. Since experience has shown that the best grinding conditions in ball mills are produced by a slurry density which makes grinding more efﬁcient than dry grinding, wet grinding of the same quartz mineral was also studied [20]. In addition, using a dispersing agent improves grinding even further in terms of producing a ﬁner size distribution. Fig. 8 shows the ﬁrst-order disappearance plot for the wet grinding of 600+425 mm quartz using a dispersing agent. The breakage parameters (when S value is ﬁtted in equation 15) were obtained to be aT ¼ 0.94 min1 and a ¼ 0.80.

Breakage and Morphological Parameters

445

Table 1. Ball mill characteristics and test conditions used for quartz grinding [19]

Mill

Inner diameter (mm) Length (mm) Volume (cm3) Operational speed (rpm) % critical speed Net mill power (watts) Number Cross-section Radius (mm) Material Diameter (mm) Number Speciﬁc gravity Average ball weight (g) Fractional mill ﬁlling (J) Quartz Speciﬁc gravity Powder weight (g)

Lifters

Media (balls)

Mineral

a

ﬃﬃﬃﬃﬃﬃﬃﬃ rpm Calculated from N c ¼ p42:3 Dd

b

balls=ball density 1:0 Calculated from J B ¼ mass ofmill volume 0:6

194 175 5170 75a 76 10 6 Semi-circular 10 Alloy steel 25 74 7.8 66.2 0.02b 2.65 320

Weight Fraction Remaining in Top Size, [w1(t) /w1(0)]

1.00

Feed mesh size 0.10

40x50

30x40

20x30

0.01

0

5 Grinding Time, minutes

10

Fig. 5. First-order plots of quartz for dry grinding in a laboratory ball mill whose conditions are given in Table 1 [19].

446

M. Yekeler

Specific Rate of Breakage (Si), min-1

1.0 aT = 0.60 min-1

Experiment Si = 0.60 xi0.80 0.1 0.1

1 1+(xi/1.9)3.7

1.0 Sieve Size (xi), mm

Fig. 6. Variation of ﬁrst-order speciﬁc rates of breakage with particle size (see Fig. 5) [19].

Cumulative Breakage Function (Bi,j)

1.00 φ = 0.68

0.10

γ = 1.24 β = 5.35

0.01 0.02

0.10

1.00

Relative Size (xi/xj)

Fig. 7. Primary breakage distribution function of 600+425 mm quartz using sodium metaphosphate as a dispersing agent [20].

The primary breakage distribution function determined using the BII calculation procedure is shown in Fig. 9. These parameters ﬁt the equation (5) with F ¼ 0.74, g ¼ 1.20 and b ¼ 6.46. These values were not signiﬁcantly different from those obtained for dry grinding of the same quartz.

Breakage and Morphological Parameters

447

Weight % remaining in top size

100

Dry Si = 0.40 min-1 Slope = Si/2.3

10

Wet Si = 0.60 min-1 1

0

1

2

3 4 5 6 Grinding time, minutes

7

8

9

Fig. 8. First-order plot for wet grinding of 600+425 mm quartz using sodium metaphosphate as a dispersing agent [20]. 1.00

Cumulative Breakage Parameters

φ

β 0.10

γ

Dry Wet γ 1.24 1.20 β 5.35 6.46 φ 0.68 0.74 0.01 10

Dry grinding Wet grinding 100 Sieve Size, µm

1000

Fig. 9. Primary breakage distribution function of the 600+425 mm quartz feed [20].

2.5.2. Calcite Dry grinding of calcite mineral used in all experimental testing was 850+600, 600+425, 425+300, 300+212 mm single-sized feed fractions obtained from

448

M. Yekeler 100

Weight %Remaining in Top Size, W1(t)

-850+600µm, S=1.25 min-1 -600+425µm, S=1.00 min-1 -425+300µm, S=0.80 min-1 -300+212µm, S=0.64 min-1

10

1 0

1

2 3 4 Grinding Time,minutes

5

6

Fig. 10. First-order plots of breakage of 850+600, 600+425, 600+425, 425+300 and 300+212 mm feed of calcite ground with 12.8 mm ball diameter [21].

Turkey. The experiments were performed in a similar laboratory size ball mill whose characteristics and test conditions were outlined in Table 1 with the exception of mill diameter, D ¼ 209 mm and ball diameters of d ¼ 46, 26 and 12.8 mm used separately in the tests [21]. Figure 10 shows the ﬁrst-order plots of calcite mineral for varying feed size fractions with a ball diameter of 12.8 mm. The results of the other tests for the ball diameters of 26 and 46 mm on these feed size fractions are also similar [21]. Figure 11 is a combination of different ball sizes studied against the feed size fractions and shows the variation of the speciﬁc rates of breakage (Si) with particle size [21]. The effect of ball diameter on the aT values is also given in Fig. 12, which means as the ball sizes increase, the aT values (or Si values) decrease. As a result, initial grinding results obey the ﬁrst-order breakage form as given in equation (3). The primary breakage distribution function of calcite is shown in Fig. 13 for 425+300 mm feed fraction. The Bi,j values of the other feed size fractions were also very close to these values. The parameters obtained were F ¼ 0.65, g ¼ 0.92 and b ¼ 4.25 for dry grinding of calcite ground in the laboratory size ball mill used. Calcite mineral was also ground in a laboratory-scale ceramic ball mill to obtain the breakage parameters [22]. The grinding tests were performed dry, wet and wet with a grinding aid (sodium oleate). The sample mineral used in all experimental testing were sieved to 1180+850, 850+600, 600+425, 425+300 and

Breakage and Morphological Parameters

449

10.0

Specific Rate of Breakage (Si), min-1

d=12.8 mm; aT=1.5 min-1; α=0.72 d=26 mm; aT=1.00 min-1; α=0.73 d=46 mm; aT=0.7 min-1; α=0.74

1.0

0.1 100

1000 Size (Xi), min

Fig. 11. Variation of the Si values of calcite with particle size for different ball diameters [21].

aT , min-1

10.0

log a

T

1.0

= -0.5

9 log

d+1

.93

0.1 10

100 Ball diameter (d), mm

Fig. 12. Variation of aT value with ball diameter [21].

300+212 mm single-sized fractions. The experimental ceramic mill characteristics and test conditions are outlined in Table 2. Figure 14 shows the initial grinding results for the calcite mineral, the surface of which had been treated with sodium oleate, plotted in the ﬁrst-order form for

450

M. Yekeler 1.0

Primary Breakage Distribution (Bij)

φ = 0.65

γ = 0.92

0.1 β = 4.25

0.0 10

100 Size, µm

1000

Fig. 13. Primary breakage distribution function of calcite for 425+300 mm feed size fraction ground in the mill [21].

Table 2. Ceramic ball mill characteristics and test conditions [22]

Ceramic mill

Media (balls)

Material

Water Grinding aid

Inner diameter (D) (mm) Length (mm) Volume (cm3) Critical speed (Nc) (rpm) Operational speed (rpm) Material Diameter (d) (mm) Number Speciﬁc gravity Average ball weight (g) Fractional ball ﬁlling (JB) Calcite Speciﬁc gravity Powder weight (g) Pulp density (%) (by volume) Distilled water Sodium oleate, sodium dodecyl sulphate (SDDS)

128 212 2500 132 92 Ceramic 25.3 37 3.75 30.4 0.2 2.69 193 40

Breakage and Morphological Parameters

451

Weight % Remaining in Top Size

100.0

10.0 slope = -S i /2.3

1.0 Dry grinding (Si= 0.26 min-1) Wet grinding (Si= 0.32 min-1) Wet grinding with sodium oleate (3.5x10-4 M) -1 (Si= 0.33 min )

0.1 0

2

4

6 8 10 12 14 Grinding Time, minutes

16

18

Fig. 14. First-order plots for dry, wet and wet with sodium oleate aided grinding of 600+425 mm calcite in a ceramic mill [22]. Table 3. The breakage values of calcite mineral ground in a ceramic mill [22]

Grinding type

Feed size (mm)

Si

g

F

b

Dry

600+425 425+300 300+212 600+425 425+300 300+212 600+425 425+300 300+212 600+425 425+300 300+212

0.26 0.20 0.17 0.32 0.25 0.22 0.35 0.28 0.23 0.33 0.27 0.23

1.14 1.01 1.15 0.92 0.90 0.99 0.95 0.87 0.97 0.99 0.86 0.98

0.53 0.56 0.57 0.53 0.60 0.60 0.53 0.56 0.57 0.53 0.56 0.53

5.84 4.38 5.09 5.02 4.26 6.24 5.63 4.33 6.06 5.97 4.30 4.61

Wet

Wet with SDDS aided

Wet with Sodium oleate

600+425 mm feed ground in the ceramic mill. The results for the other feed sizes ground dry, wet and wet with sodium dodecyl sulphate (SDS) addition under the same experimental conditions are also given in Table 3. The values of Si for all feed sizes with sodium oleate treated calcite are given in Fig. 15, including the largest sizes (1180+850 and 850+600 mm) that show

M. Yekeler Specific Rate of Breakage (Si), min-1

452 1.0

α

Dry 0.60

Wet 0.54

Sodium oleate 0.50

α Dry grinding Wet grinding

0.1 100

Wet grinding with sodium oleate (3.5x10-4 M)

1000

2000

Sieve Size (xi), µm

Fig. 15. Variation of ﬁrst-order speciﬁc rate of breakage (Si) values of different feed sizes of 1180+850, 850+600, 600+425, 425+300, 300+212 mm fractions (plotted at upper size interval) of calcite [22].

abnormal breakage behaviour for the sizes coarser than 650 mm when ground in this mill with a ball diameter of 25.3 mm. The SDS-treated calcite also gave similar results and abnormal behaviour in terms of Si values plotted against size. The primary breakage distribution function determined using the BII calculation method is shown in Fig. 16 for 600+425, 425+300 and 300+212 mm feeds that gave normal breakage behaviour. The obtained Bi,j parameters (g, F and b) are also given in Table 3.

2.5.3. Barite Figure 17 shows the grinding results of barite mineral (from Turkey) plotted in ﬁrst-order form for the feed size fractions of 850+600, 600+425 and 425+300 mm ground with a ball diameter of 26 mm (same test conditions as in calcite mineral) [21]. Figure 18 is a variation of Fig. 17 to show the speciﬁc rates of breakage (Si) with particle feed size for the ball diameter of 26 mm. As the feed size increases, the Si values increase as well. The initial grinding results obey the ﬁrst-order breakage form. When the values of Si are ﬁtted to equation (15), aT value is obtained to be 1.1 min1 by inserting a ¼ 0.60, xi ¼ 850 mm and x0 ¼ 1000 mm. The primary breakage distribution function of barite determined were F ¼ 0.69, g ¼ 0.85 and b ¼ 3.73 for the 425+300 mm feed size fraction [21]. Table 4 outlines the breakage parameters of barite mineral that were ground in ceramic mill whose characteristics and test conditions were already given in

Breakage and Morphological Parameters

453

Cumulative Breakage Parameters (Bi,j)

1.00

Dry γ 1.10 φ 0.55 β 5.10

0.10

Wet 0.94 0.58 5.17

Sodium oleate 0.94 0.54 4.96

Dry grinding Wet grinding Wet grinding with sodium oleate (3.5x10-4 M)

0.01 0.01

0.10 Relative Size (x i /x j)

1.00

Fig. 16. Primary breakage function of calcite for dry, wet and sodium oleate aided grinding [22].

Weight % Remaining in Top Size, W1(t)

100 -850+600µm, S=0.99 min-1 -600+425µm, S=0.78 min-1 -425+300µm, S=0.66 min-1

10

1

0

1

2

3 4 5 6 Grinding Time, minutes

7

8

Fig. 17. First-order plots of breakage of 850+600, 600+425 and 425+300 mm feeds of barite ground dry [21].

454

M. Yekeler

Specific Rate of Breakage (Si), min-1

10.00

aT =1.1 min-1

1.00

α = 0.60

α

0.10 100

1000 Particle size, µm

Fig. 18. Variation of the Si values of barite with particle size [21].

Table 4. The breakage parameters of barite mineral obtained from ceramic mill tests [23]

Grinding type

Feed size (mm)

Si (min1)

g

Dry grinding

600+425 425+300 300+212 600+425 425+300 300+212 600+425 425+300 300+212

0.332 0.275 0.215 0.356 0.316 0.261 0.367 0.328 0.279

0.76 0.82 0.98 0.63 0.66 0.83 0.64 0.67 0.78

Wet grinding

Wet grinding with SDDS (1 103 M)

Average g

a

0.85

0.62

0.71

0.45

0.70

0.40

Table 2 [23]. The grinding of barite mineral, which were 600+425, 425+300 and 300+212 mm fractions, was also performed as dry, wet and wet with chemical aid SDS. The primary breakage distribution function for dry, wet and wet grinding with SDS is also given in Fig. 19 for the ceramic-milled barite mineral [23].

2.5.4. Zeolite The zeolite sample from Turkey was also tested in our experimental study [24]. The feed size fractions were 850+600, 600+425 and 425+300 mm for all tests. The grinding experiments were performed in a steel laboratory ball mill of 200 mm internal diameter in conditions very similar to those given in Table 1, at a

Breakage and Morphological Parameters

455

CumulativeBreakage Parameters (Bi,j)

1.00 φ

γ φ β

Dry 0.85 0.56 4.90

Wet SDDS 0.70 0.71 0.52 0.50 5.04 5.18

γ β

0.10

Dry grinding Wet grinding

0.01 0.01

-3 Wet grinding with SDDS (1x10 M)

0.10 Relative Size (xi/xj)

1.00

Fig. 19. Primary breakage distribution function for ceramic-milled barite mineral [23].

low ball load of 20% of the mill volume and a low powder load corresponding to a formal interstitial ﬁlling of the void spaces of the ball bed of U ¼ 0.5. Figure 20 shows the ﬁrst-order plots for dry grinding of various feeds of zeolite. Figure 21 is given to show the primary breakage distribution function for 600+425 mm zeolite.

2.5.5. Coals (lignite and anthracite) Figure 22 shows the ﬁrst-order disappearance plot for dry and wet grinding of 600+425 mm lignite using the same laboratory size ceramic ball mill as used before (in Table 2 [23] and thereafter). The grinding results obey the ﬁrst-order breakage behaviour [25]. Table 5 outlines the overall breakage parameters for various feed sizes of lignite fractions in different mill environments and also includes breakage parameters of anthracite under the same experimental conditions [25,26].

2.5.6. Clinker Figure 23 shows the ﬁrst-order breakage plots of 1000+600, 420+250 and 106+75 mm feed size fractions of cement clinker ground in the laboratory size of steel ball mill used extensively in our studies [27]. Table 6 outlines the Si and Bi,j values of cement clinker studied.

456

M. Yekeler

Weight % Remaining in Top Size

100.0

slope = -Si /2.3

10.0

-1

S -850+600 µm = 0.85 min

S -600+425 µm = 0.76 min-1 S -425+300 µm = 0.65 min-1

1.0 0.0

1.0

2.0 3.0 4.0 Grinding Time, minutes

5.0

Fig. 20. First-order plots for dry grinding of zeolite for various feed size fractions [24]. 1.00

Cumulative Breakage Function (Bi,j)

φ = 0.61

β = 4.25 γ = 0.84

0.10

0.01 10

100 Sieve Size, µm

1000

Fig. 21. Primary breakage distribution function for 600+425 mm zeolite feed [24].

2.5.7. Chromite The run of mine chromite mineral from Turkey was the feed fraction of 425+250 mm for grinding tests. The laboratory steel ball mill whose

Breakage and Morphological Parameters

457

Specific rate of breakage, S i (min-1)

1.00

0.10

α x Si = αT ( x i )α 0

Dry grinding (α= 0.43) Wet grinding (α= 0.35)

0.01 100

2

3

4

5

6

7 8 9

1000

Particle size (µm)

Fig. 22. First-order plots for batch grinding of 600+425 mm lignite ground as dry and wet grinding [25]. Table 5. The breakage parameters of lignite and anthracite obtained from the grinding tests [25,26]

Dry Coal Lignite

Feed size (mm) Si (min1) g

600+425 425+300 300+212 Anthracite 425+300

0.123 0.102 0.091 0.294

Wet F

b

Si (min1) g

0.131 1.58 0.58 7.11 0.113 0.103 1.63 0.63 4.7 0.315

F

b

1.45 0.63 6.45 1.30 0.59 5.6

characteristics were described before was used to determine the Si and Bi,j values [28]. Dry and wet grinding were carried out in the ball mill. The Si value of the 425+300 mm feed fraction for dry grinding is 0.28 min1, and it was 0.56 min1 for wet grinding, which indicates that wet grinding is two times faster at breaking the largest particles compared to dry grinding. The primary breakage distribution function determined using the BII calculation procedure were F ¼ 0.75, g ¼ 1.14 and b ¼ 7.9.

2.5.8. Ceramic raw materials Ceramic raw materials (minerals) used in tests which were quartz, kaolin from Turkey and potassium–feldspar from Egypt were the feed fractions of 3,350 +2,360, 2,000+1,400, 0,850+0,425, 0,500+0,355 and 0,300+0,212 mm

458

M. Yekeler

Fig. 23. First-order plots of various feed sizes of cement clinker [27].

Table 6. Si and Bi,j values of cement clinker [27]

Size (mm)

Si (min1)

g

F

b

1000+600 420+250 106+75

0.62 0.37 0.08

1.00 1.26 1.71

0.69 0.30 0.42

4.22 14.46 11.97

ground in a laboratory Bond mill. Test conditions and Bond mill characteristics are given in Table 7 [29]. Figures 24, 25 and 26 show the ﬁrst-order plots for dry grinding of quartz, kaolin and potassium–feldspar, respectively. The initial grinding results obey the ﬁrst-order breakage kinetics. The overall breakage parameters including Bi,j values determined by BII calculation method are given in Table 8. Since these minerals are the major raw materials for ceramic industries, the binary mixtures (1:1) of quartz, kaolin and potassium–feldspar were also ground in the same Bond mill as used for individual grinding of the ceramic raw materials [30].

Breakage and Morphological Parameters

459

Table 7. Bond mill characteristics and test conditions for grinding of ceramic raw materials [29]

Mill

Diameter (D) (cm) Length (L) (cm) Volume (V) (cm3) Speed (rpm) Critical speed (Nc) Diameter (d) (mm) Number Total mass (g) Speciﬁc gravity (g/cm) Fractional ball ﬁlling (J) Fractional powder ﬁlling (f) Powder-ball loading ratio (U) Quartz powder weight (g) Kaolin powder weight (g) Feldspar powder weight (g)

Media charge

Weight remaining in top size (w1(t)), %

Material charge

30.5 30.5 22272 70 86.55 30.1,31.75,25.4,19.05,12.7 285 22648 7.79 0.22 0.08 1.0 3110 3029 3052

100.00

10.00

1.00 -3.350+2.360 mm -2.000+1.400 mm

0.10

-0.850+0.600 mm -0.500+0.355 mm -0.300+0.212 mm

0.01 0

2

4 6 Grinding time (min)

8

10

Fig. 24. First-order plots for dry grinding of different feed sizes of quartz [29].

Figures 27, 28 and 29 show the ﬁrst-order plots for these binary mixtures and Table 9 outlines the characteristic breakage parameters of the mixture minerals in 1:1 ratios.

2.6. Simulation of ball milling products using the breakage parameters There are programs for the computer design of mineral processing circuits, and these programs contain computer simulation models for ball mill design. These

460

M. Yekeler

Weight remaining in top size (w1(t)), %

100.00

10.00 -3.350+2.360 mm -2.000+1.400 mm -0.850+0.600 mm -0.500+0.355 mm -0.300+0.212 mm

1.00 0

2

4 6 Grinding time (min)

8

10

Weight remaining in top size (w1(t)), %

Fig. 25. First-order plots for dry grinding of different feed sizes of kaolin [29].

100.00

10.00

-3.350+2.360 mm

1.00

-2.000+1.400 mm -0.850+0.600 mm -0.500+0.355 mm -0.300+0.212 mm

0.10 0

2

4 6 Grinding time (min)

8

10

Fig. 26. First-order plots for dry grinding of different feed sizes of potassium–feldspar [29].

Table 8. The overall breakage parameters for the ceramic raw minerals studied [29]

Minerals

Si (min1)

aT

a

m

l

F

g

b

Quartz Kaolin K–feldspar

0.52 0.34 0.52

0.69 0.52 0.75

1.46 1.07 1.34

1.91 1.37 1.55

3.74 2.13 2.93

0.60 0.51 0.49

1.54 0.93 1.18

7.37 4.08 5.36

Breakage and Morphological Parameters

461

Mass Fraction of Feed Remaining

100.0

10.0

-3.350+2.360 mm

1.0

-2.000+1.400 mm -0.850+0.600 mm -0.500+0.355 mm -0.300+0.212 mm

0.1 0

2

4

6

8

10

Grinding Time, min.

Fig. 27. First-order plots for dry grinding of different feed sizes of quartz–kaolin binary mixtures [30].

Mass Fraction of Feed Remaining

100.0

10.0

-3.350+2.360 mm

1.0

-2.000+1.400 mm -0.850+0.600 mm -0.500+0.355 mm -0.300+0.212 mm

0.1 0

2

4

6

8

10

Grinding Time, min.

Fig. 28. First-order plots for dry grinding of different feed sizes of quartz–potassium–feldspar binary mixtures [30].

models need the input of characteristic breakage parameters for the mineral of interest and these parameters are often determined in a small-size laboratory ball mill [21]. The simulator used for product size distributions of the minerals studied was the PSUSIM program that consists of a ball mill model (with a number of options for the residence time distribution) plus linking algebra to enable two mills to be connected in any desired circuit [31]. The program assumes that the rates of

462

M. Yekeler

Mass Fraction of Feed Remaining

100.0

10.0

-3.350+2.360 mm

1.0

-2.000+1.400 mm -0.850+0.600 mm -0.500+0.355 mm -0.300+0.212 mm

0.1 0

2

4

6

8

10

Grinding Time, min.

Fig. 29. First-order plots for dry grinding of different feed sizes of kaolin–potassium–feldspar binary mixtures [30]. Table 9. Characteristic breakage parameters for the binary mixtures of minerals obtained from the laboratory test [30]

Mineral mixtures

aT

a

m

l

F

g

b

Quartz–kaolin Quartz–K–feldspar Kaolin–K–feldspar

0.64 0.67 0.54

1.45 1.32 1.13

1.44 1.71 1.49

2.88 2.90 2.39

0.50 0.55 0.49

1.00 1.40 0.94

4.70 7.71 4.64

breakage are ﬁrst-order. PSUSIM is written in IBM Basic, and is constructed in modular form, to allow easy modiﬁcation of mill models, classiﬁer models, etc. The mill models used are not perfect descriptions of every ball mill, but they are sufﬁciently accurate for design and to demonstrate operating trends, and when used in conjunction with plant data or extensive laboratory data on a given material they will give good simulation of full-scale circuit behaviour. Grinding results for ﬁrst-order breakage times (depending on mineral type) were simulated using the characteristic parameters of Si and Bi,j (i.e., a, m, l, b, g, F) in the PSUSIM simulator [31]. B values were dimensionally normalized (i.e., F constant, irrespective of breakage size), breakage was ﬁrst-order and values of S and B for small sizes could be obtained from equations (5) and (15). The simulations gave good agreement with the experimental data for the ﬁrst-order grinding region. However, the predicted size distributions after the ﬁrst-order breakage times were ﬁner than those observed experimentally. This was treated using the false time concept [2] by making the simulator produce a match to a speciﬁed point on the product size distribution and designing the grinding time necessary to achieve this match as the false time y, where yrt.

Breakage and Morphological Parameters

463

Cumulative Weight % Finer Than Size

100.00

10.00

1.00 Real time (t), minute False time (θ), minute t=1, θ=1 t=2, θ=2 t=4, θ=4 t=8, θ=7 t=16, θ=12 t=32, θ=23 t=64, θ=35 t=128, θ=50 t=256, θ=82 t=512, θ=116 Simulation

0.10

0.01

1

10 100 Sieve Size, µm

1000

Fig. 30. Simulated and experimental product size distributions of wet ground quartz [20].

Experimental and simulated product size distributions were given in Figs. 30–38 for quartz, calcite, barite, zeolite, lignite, clinker, chromite, kaolin and potassium– feldspar, respectively.

3. MORPHOLOGICAL PARAMETERS OF MINERALS GROUND BY DIFFERENT MILLS Since shape is also a factor in the behaviour of powders, the shape difference between the products of different mills has a particular signiﬁcance, as in any other industry where particles are involved. For example, gravity concentration is affected by shape properties of liberated particles.

3.1. Materials and methods employed Quartz, calcite, barite and talc samples from Turkey were used in our experimental working. The analyses of these samples showed that they are pure enough for our

464

M. Yekeler

Cumulative Weight % Finer Than Size

100

29.8

128

23.6

64

16.9

32

11.1

16

7.6

8

4

4

2

2

10 experimental simulated 1

1

false time real time θ t 1 10

100 Sieve Size, µm

1000

Fig. 31. Simulated and experimental product size distributions of dry ground calcite [21].

Cumulative Weight % Finer Than Size

100

27.9 18.9 12.7

128 64 32

8.2

16

5.8

8

3.2

4

2

2

1

1

10 false time real time θ

1 10

experimental

t

simulated

100

1000

Sieve Size, µm

Fig. 32. Simulated and experimental product size distributions of wet ground barite [21].

Breakage and Morphological Parameters

Cumulative Weight % Finer Than Size

100.0

465

8 16.0 12 64 13.4 32 10.6 16 8.8

8 5.6 4

10.0

3.3 False time, minutes

2

1

1/2.5

Grinding time, minutes

Sieving Simulation

1.0 10

100 Sieve Size, µm

1000

Fig. 33. Simulated and experimental product size distributions of dry ground zeolite [24].

studies. The materials were prepared by crushing to 850+600 mm sieve fractions for ball milling, 4.75+3.35 mm fractions for rod milling, 10+1 mm fractions for autogenous milling to obtain the appropriate amount of ﬁne mill products (250+45 mm fraction) for determining the morphological parameters. Grinding tests by autogenous, ball and rod mills are outlined in Table 10 [32]. The characterization of shape properties was carried out utilizing the twodimensional (2D) measurement technique using a Jeol JSM-6400 scanning electron microscope (SEM) [32]. The size of the particles for SEM measurement was 250+45 mm fraction. The representative sample taken from each mill product was mounted in epoxy resin and coated by gold to provide the conductivity. Since the samples used in SEM were not subjected to any polishing process, the risk of changing the particle shape was discarded. Four pictures from different locals of the sample of each mill product were taken by making proper magniﬁcation. Each micrograph was transformed into digital images by a scanner. From the micrographs, the axes of particles were measured by using a computer program, Corel Draw 10. After each image was imported, the magniﬁcation quantity was taken as 400%. Particles with no overlapping and no border out of the picture frame were chosen for the axes measurements. The shape and morphological characterization of particles is based only on the analysis of the silhouettes of the particles or projection. Assuming that the projection of the particle has an ellipse-like shape [9] as shown in Fig. 39 [32], the major axes length (L) and width (W) of each particle

466

M. Yekeler

Cumulative weight finer than size (%)

100.0 64

10.0

37

32

22

16

13

8 4

6

1.0

False time (minutes)

2 1 Grinding time (minutes)

Sieving Simulation 0.1 1

10 100 Sieving size (µm)

1000

(a) 100.0 Cumulative weight finer than size (%)

64 39

32

22

16

10.0 13 False time (minutes)

8 4 2 1

1.0

Grinding time (minutes)

Sieving Simulation 0.1 1

10 100 Sieving size (µm)

1000

(b)

Fig. 34. Simulated and experimental product size distributions of (a) dry and (b) wet ground lignite [25].

were measured in mm and used as input data for MS Excel program. The mean values of the ﬁve lines drawn on the particles on the particle projection for each axis were taken as the real length (L) and width (W) values of that particle in mm considering the scale of the image. The same procedure was followed for about 100 particles for each mill product. Thus, the area (A) and perimeter (P) of the

Breakage and Morphological Parameters 100.0

467

40

30

44.6 31.9

Cumulative Weight % Finer Than Size

18.2

7 3

6.2

10.0

20

1 Grinding time, minutes

2.6

1 False time, minutes

1.0

Sieving Simulation 0.1 10

100

1000

Sieve Size, µm

(a) 100.0

40 30 20

47.0 35.7

Cumulative Weight % Finer Than Size

24.2 7

10.0

10.4 3 5.3

1

3.0 False time, minutes

1.0

Grinding time, minutes

Sieving Simulation 0.1 10

100

1000

Sieve Size, µm

(b) 100

40 32.4

Cumulative Weight % Finer Than Size

23.5

30 20

16.5

7 7.8

10

3 5.0 3.5 False time, minutes

1 Grinding time, minutes

Sieving Simulation 1 10

(c)

100

1000

Sieve Size, µm

Fig. 35. Simulated and experimental product size distributions of (a) 1000+600, (b) 420+250, and (c) 106+75 mm feed fractions of dry ground clinker [27].

M. Yekeler

Cumulative Weight %Finer than Size

468

36 min.

27

in.

18 m

14 9 min. 8

6 min.

k. 6 3 da

3 1 min.

1

Experimental Simulation

t=real time θ = False time

10

100

1000

Sieve size, µm

(a) 100 Cumulative Weight % Finer than Size

36 min. 31

18 min.

15

9 min.

8

10

6 3

6 min. 3 min.

1 min.

1

1

10

t=real time

Experimental

θ=False Time

simulation

100

1000

Sieve Size,µm

(b)

Fig. 36. Simulated and experimental product size distributions of (a) dry and (b) wet ground of chromite [28].

particle projection can be calculated on the basis of the measured length and width as given in equations (9) and (10) [10,32]. From these basic measurements and calculations, four shape factors (equations (11)–(14)) were used for the shape and morphological characterization of the particles ground by different mills. In order to determine the surface roughness of particles in pelletized form, a portable stylus type roughness-measuring instrument called Surtronic 3+ HB-103, which has a microprocessor, was used as given in Fig. 40 [33]. It measures the

Breakage and Morphological Parameters

469

Fig. 37. Simulated and experimental product size distributions of dry ground kaolin [29].

Fig. 38. Simulated and experimental product size distributions of dry ground potassium–felsdpar [29].

average roughness (Ra) values directly by traversing across the surface of the pellets formed. The surface roughness of measurement is based on the mechanical sensing of surface topology in combination with electronic ampliﬁcations of the signal obtained. Such measurements represent a highly accurate method (reproducibility within 10 1A of vertical resolution) [34]. The main advantage of the method of the surface roughness is found in the direct mechanical nature of the technique. For each mill product of the minerals that are in pelletized form, three

470

M. Yekeler

Table 10. Grinding test conditions of quartz mineral for the mills employed [32] Mill type

Parameters

Items

Values

Ball mill

Mill

Inner diameter (mm) Length (mm) Critical speed (rpm) Operational speed Diameter (mm)

200 184 102 76 30, 26

Average ball weights (g) Speciﬁc gravity (g/cm3) Fractional ball ﬁlling Total mass (kg) Speciﬁc gravity (g/cm3) Total powder weight (g) Fractional powder ﬁlling Powder-ball loading ratio Feed size (mm) Optimum time

118.18, 68.19 7.90 0.2 5.475 2.65 367.3 0.04 0.5 800+600 16

Inner diameter (mm) Length (mm) Critical speed (rpm) Operational speed Diameter (mm) Average rod weights (g) Speciﬁc gravity (g/cm3) Total mass (kg) Total powder weight (g) Feed size (mm) Optimum time

200 280 102 51 29, 24, 19 1521, 1041, 669 7.90 22600 745.5 4750+3350 16

Inner diameter (mm) Length (mm) Critical speed (rpm) Critical speed (rpm) Operational speed Diameter (mm) Total mass (kg) Total powder weight (g) Feed size (mm) Optimum time (min)

420 225 71.5 71.5 28 80+50 3000 2000 10000+1000 256

Speed Media (stainless steel)

Mineral (quartz)

Grinding time Rod mill

Mill Speed Media (steel rods) Mineral (quartz) Grinding time

Autogenous mill

Mill Speed Media (lump ore)

Grinding time

measurements were made and the roughness values (Ra) were determined (in mm) by taking the average of these values.

3.1.1. Quartz Figure 41 shows the product size distributions of quartz mineral ground dry in ball, rod and autogenous mills [32]. In order to carry the shape and morphological measurements, appropriate amounts and proper size of samples from the grinding tests were saved, i.e., the grinding time to produce 100% passing at 250 mm was enough to save the samples for further tests. These grinding times were 16 min for ball and rod mills, and 256 min for an autogenous mill as given in Fig. 41.

Breakage and Morphological Parameters

471

Fig. 39. Measurement of axes of particles on a SEM micrograph [32].

Considering the proper size and amount of feed material used in these tests (SEM and Surtronic 3+), very ﬁne material (45 mm ) was removed by screening from the ground material ﬁner than 250 mm. Table 11 shows the determined shape properties of the particles ground in different environments by means of SEM techniques. After measuring (at least 100) particles from each mill product, the area and perimeter values were calculated using equations (9) and (10) followed by the shape properties such as elongation, ﬂatness, roundness and relative width using equations (11)–(14), respectively. Figure 42 illustrates the comparison of the results of the shape properties determined by SEM techniques. As shown in Fig. 42, elongation and ﬂatness values were higher in rod-milled product and lower in the ball-milled product. Although roundness and relative width values were higher in the ball-milled product, they were lower in the rod-milled product of quartz. In other words, ball mill produces particles having higher roundness, while rod mill products have more elongated shape as shown in Fig. 43(a)–(b). These statements were also supported by the representative image taken from the SEM micrograph as shown in Fig. 43(c)–(d). This could be explained by the fact that autogenous, rod and ball milling are different according to the mechanism of breakage employed during grinding. Table 12 outlines the results of the measured surface roughness (Ra) values by the Surtronic 3+ instrument. The surface roughness values of the pelleted samples from grindings were in the range 4.24–4.49 mm. While the higher values of Ra were observed in the product of ball mills, lower values were seen in the

472

M. Yekeler

Fig. 40. Surtronic 3+ instrument used in the direct roughness measurements [32]. (a) Surtronic 3+ instrument; (b) The stylus probe (which traverses across the surface) of the Surtronic 3+ instrument.

rod-milled product. Figure. 44 illustrates that the rod mill produces particles having smoother surfaces than the other mills. This was also attributed to the different breakage mechanism that occurs in different mills [32].

3.1.2. Calcite and barite To characterize the morphology of the mineral surface area, the surface roughness factor is useful and it is deﬁned as the ratio of the real surface area of a particle of a certain diameter to the surface area of a sphere of the same diameter [35–37]. The surface area of the particles measured by BET and other gas adsorption techniques also quantitatively characterizes the surface

Breakage and Morphological Parameters

473

Cumulative weight % finer than size

100

10

Grinding time, minutes

1 2 4 8 16 32

1 10

(a)

100

1000

10000

Particle size, (μm)

Cumulative weight % finer than size

100

10

Grinding time, minutes

1 2 4 8 16 32

1 10

(b)

100

1000

10000

Particle size, (μm)

Cumulative weight % finer than size

100.0

10.0

1.0 Grinding time, minutes

0.1

0.0 10

(c)

1 2 4 8 16 32 64 128 256

100

1000

10000

Particle size, (μm)

Fig. 41. Particle size distribution of quartz mineral ground in different mills: (a) ball-milled, (b) rod-milled, and (c) autogenous-milled products [32].

474

M. Yekeler

Table 11. Shape characteristics of quartz particles ground in different mills calculated from SEM measurements based on 100 particles counted [32]

Mill product L (mm) W (mm) A (mm2) Ball 118.53 73.34 Rod 106.43 65.13 Autogenous 89.87 55.42

P (mm)

1.094

1.635

1.093

1.630

1.092

Flatness

Elongation

R

RW

Flatness

1.640

1.625 1.620 1.615 1.610

1.091 1.090 1.089 1.088

1.605

1.087 Ball mill

(a)

Rod mill

Autogenous mill

Ball mill

(b)

Grinding type

Rod mill

Autogenous mill

Grinding type

Roundness

Relative width

0.920

0.620

0.919

0.618 Relative width

Roundness

F

6823.997 305.473 1.616 1.089 0.919 0.619 5441.452 273.310 1.634 1.093 0.915 0.612 3909.777 231.358 1.622 1.090 0.917 0.617

Elongation

0.918 0.917 0.916 0.915

0.616 0.614 0.612 0.610

0.914

0.608

0.913 Ball mill

(c)

E

Rod mill Grinding type

Autogenous mill

Ball mill

(d)

Rod mill

Autogenous mill

Grinding type

Fig. 42. Comparison of the shape properties calculated by SEM measurement of quartz ground by different mills [32]:(a) elongation; (b) ﬂatness; (c) roundness; (d) relative width.

roughness of particles. The calculated roughness (RBET) values are obtained by [38]: RBET ¼ ABET d

D 6

ð16Þ

where, ABET is the BET surface area measured, d the density of a solid and D the average particle diameter tested in the equipment. From an experimental point of view, the BET equation is easy to apply, and the surface areas obtained are reasonably consistent. The determined shape properties of the calcite particles ground in different environments by means of SEM techniques are given in Table 13 and the same properties for barite particles are also given in Table 14 [39]. After measuring (at least 100) particles from each mill product, the area and perimeter values

Breakage and Morphological Parameters

475

(a) product of ball milling

(b) product of rod milling

(c) representative image taken from (d) representative image taken from the SEM micrographs of ball mill. the SEM micrographs of rod mill.

Fig. 43. Examples of SEM micrographs representing ball and rod mill product of quartz ground in different mills ( 100) [32]: (a) product of ball milling; (b) product of rod milling; (c) representative image taken from the SEM micrographs of ball mill; (d) representative image taken from the SEM micrographs of rod mill.

476

M. Yekeler

Table 12. The surface roughness values (Ra) of quartz mineral in ball, rod and autogenous mills [32]

Mill product

Surface roughnessa (Ra) (mm)

Ball mill Rod mill Autogenous mill

4.49 4.24 4.30

a

Treated chemically by 5 104 M sodium dodecyl sulphate.

Fig. 44. Comparison of the surface roughness (Ra) values determined by Surtronic 3+ instrument for different mills [32].

Table 13. Shape characteristics of calcite particles ground in different mills calculated from SEM measurements based on over 150 particles counted [39]

Mill product L (mm) W (mm) A (mm2) Ball 78.81 Rod 79.15 Autogenous 86.15

54.98 56.04 53.59

P (mm)

E

F

R

RW

3401.231 211.727 1.434 1.049 0.953 0.698 3481.919 213.810 1.412 1.045 0.957 0.708 3624.171 222.411 1.608 1.087 0.920 0.622

Table 14. Shape characteristics of barite particles ground in different mills calculated from SEM measurements based on over 150 particles counted [39]

Mill product L (mm) W (mm) A (mm2) Ball 77.99 Rod 72.50 Autogenous 71.22

54.73 51.30 51.04

P (mm)

E

F

R

RW

3350.913 209.990 1.425 1.048 0.954 0.702 2916.611 195.802 1.413 1.045 0.956 0.708 2853.529 193.265 1.395 1.042 0.960 0.717

Breakage and Morphological Parameters

477

were calculated using equations (9) and (10) followed by the shape properties such as elongation, ﬂatness, roundness and relative width using equations (11)–(14), respectively. As shown in Table 13, the calcite particles ground by the autogenous mill had the highest ﬂatness and elongation ratio with the lowest relative width and roundness. However, the highest roundness and relative width along with the lowest elongation ratio and ﬂatness belonged to rod milling. Table 14 indicates that ball milling produced the highest elongation ratio and ﬂatness with lowest roundness and relative width. The lowest elongation ratio and ﬂatness, the highest roundness and relative width were obtained by autogenous milling. The overall SEM results for both minerals show that the shape properties are dependent on the milling type. The surface roughness of particles (calcite and barite) was calculated from the BET measurements using equation (16). The size of the particles for the BET measurements was 250+45 mm. The D values in equation (16) were taken from the average of the 45–250 mm fraction, which is 148 mm [40]. The product size distributions of calcite and barite minerals are shown in Fig. 45. The surface areas of particles measured by the BET technique for these minerals ground in different mills are also summarized in Table 15. Table 15 shows that the lowest RBET value was obtained by autogenous milling of calcite, whereas the highest value was found in the rod-milled product. Conversely, the autogenous mill product had the highest roughness factor for the barite mineral, while the lowest value was obtained by ball mill grinding. These ﬁndings were attributed to the type of mill used for comminution of such minerals. While abrasion is the main mechanism of comminution in autogenous mills, impact predominates in ball and rod mills [41]. It should be noted that by increasing surface roughness, an increase in the surface area occurs [42]. It is evident from Table 15 that the surface roughness values are highest for calcite and the lowest for barite mineral. This is due to the brittle breakage characteristics of the mineral being ground [38]. Every material performs differently in the milling process and exhibits a different grinding response. Therefore, grinding of the particles can also be approached from the standpoint of the fracture mechanics [43]. The results of surface roughness (Ra) values measured by the Surtronic 3+ stylus instrument are given in Table 16 [44]. The surface roughness values of the pellets of the ground minerals were in the range 2.28–3.54 mm. The lowest roughness values were observed for barite compared to calcite. The roughest product for calcite is the rod-milled product, while the smoothest one belongs to the autogenously-milled product. For barite, ball milling results in the lowest roughness value, but autogenous milling gives the highest. Figure. 46 shows the SEM micrographs of calcite mineral to illustrate the surface roughness from different mills.

478

M. Yekeler 100 Cumulative weight % finer than size

Cumulative weight % finer than size

100

10 Grinding time, minutes 1 2 4 8 16 32

1

10

100 1000 Particle size, (µm) (a)

10 Grinding time, minutes 1 2 4 8 16 32

1 10

10000

Cumulative weight % finer than size

Cumulative weight % finer than size

10 Grinding time, minutes 1 2 4 8 16 32

1 10

100 1000 Particle size, (µm) (c)

10 Grinding time, minutes 1 2 4 8 16 32

1 10

10000

100.00

100 1000 Particle size, (µm) (d)

10000

100.00

10.00

1.00

Grinding time, minutes 1 2 4 8 16 32

10

100 1000 Particle size, (µm) (e)

10000

Cumulative weight % finer than size

Cumulative weight % finer than size

10000

100

100

0.10

100 1000 Particle size, (µm) (b)

10.00

1.00

0.10

Grinding time, minutes 1 2 4 8 16 32

10

100 1000 Particle size,(µm) (f)

10000

Fig. 45. Product size distributions of calcite and barite minerals in different mills: (a) ballmilled calcite, (b) ball-milled barite, (c) rod-milled calcite, (d) rod-milled barite [40], (e) autogenous milled calcite and (f) autogenous milled barite.

Breakage and Morphological Parameters

479

Table 15. The calculated surface roughness factors based on the BET surface area measurements of calcite and barite minerals studied [40]

Grinding type

Mineral

Measured BET surface area (ABET) (cm2/g)

Ball

Calcite Barite Calcite Barite Calcite Barite

2700 1100 3000 1300 2400 1400

Rod Autogenous

Calculated surface roughness factor (RBET) 17.98 12.21 19.98 14.43 15.98 15.54

Table 16. The surface roughness values (Ra) of calcite and barite minerals ground in ball, rod and autogenous mills [44]

Grinding type

Calcite

Barite

Ball mill Rod mill Autogenous mill

3.24 3.54 2.90

2.28 2.68 2.77

3.1.3. Talc Table 17 shows the values determined by the SEM technique for the shape properties of talc mineral ground in different mills [45]. After measuring for about 139 particles from each mill product, the area and perimeter values were calculated using equations (9) and (10) followed by the shape properties such as elongation, ﬂatness, roundness and relative width using equations (11)–(14). As shown in Table 17, the rod-milled product has the highest elongation and ﬂatness values, whereas the ball-milled product has the lowest value. Although ball milling produces the highest roundness and relative width values, rod milling produces the lowest values. In other words, ball mill products have more roundness while rod mill products have a more elongated shape as shown in Fig. 47 (a) and (b). The SEM micrographs support this theory in Fig. 47 (c) and (d). This could be explained by the fact that autogenous, rod and ball milling have different breakage mechanisms in action during grinding. Table 18 outlines the results of the measured surface roughness (Ra) values by the Surtronic 3+ instrument. The surface roughness values of the pelleted samples from grindings were in the range 0.55–0.68 mm [45]. While the highest value of Ra was observed in the ball-milled product, the lowest value was seen in the rodmilled product of talc mineral; this was due to the different breakage mechanisms of the mills employed.

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Fig. 46. Examples of the scanning electron microscope (SEM) micrographs representing the surface roughness of the calcite, particles ground by different mills ( 500): (a) rodmilled product, (b) autogeneous-milled product [33].

Breakage and Morphological Parameters

481

Table 17. Shape characteristics of talc particles ground in different mills calculated from SEM measurements based on 100 particles counted [45] Mill Product

L (mm)

W (mm)

A (mm2)

P (mm)

E

F

R

RW

Ball Rod Autogenous

103.62 95.07 98.97

67.54 59.01 62.55

5493.818 4403.913 4859.600

271.740 245.265 256.852

1.534 1.611 1.582

1.070 1.088 1.081

0.934 0.920 0.925

0.652 0.621 0.632

Fig. 47. Examples of SEM micrographs representing ball and rod mill product of talc ground by 16 and 32 min, respectively ( 100): (a) product of ball milling, (b) product of rod milling, (c) representative image taken from the SEM micrographs of ball mill, (d) representative image taken from the SEM micrographs of rod mill [45].

Table 18. The surface roughness (Ra) values of talc mineral produced by different mills employed [45] Product of mill

Surface roughness (Ra) (mm)

Ball mill Rod mill Autogenous mill

0.68 0.55 0.66

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4. SUMMARY When dry ball milling quartz to ultra-ﬁne sizes, it is necessary to prevent the charge of ﬁne powder from adhering the mill walls and forming a dry cake by taking necessary action. The shape of particles after longer grinding times becomes more rounded. The change in breakage character appears to be a slower chipping or abrasion mechanism that gives more ﬁne products with rounder larger sizes. The Si values of quartz for wet grinding in the normal grinding region are higher than the dry values by a factor of 1.5. The Bi,j values for wet grinding are essentially the same as those of dry grinding. The Si values of calcite and barite increase, as feed size fractions increase. The ball size has an inverse effect on the Si value, which means that the smaller the ball size, the higher the Si value. The Si values for barite are higher than those for calcite. More ﬁnes are produced in barite grinding. Zeolite is broken faster than quartz and slower than calcite and barite in terms of Si values. Also, zeolite produces ﬁner material than quartz and calcite. The wet Si values of lignite are higher than the dry Si values of lignite when ground in the same conditions. However, the dry Bi,j values are different from the wet values. The Si values of anthracite are higher than those of barite, calcite, lignite and quartz. Clearly, for wet grinding Si values are always greater than the dry values. The wet Bi,j values for anthracite are similar to the dry values. Faster breakage occurs in the order of quartz, K–feldspar and kaolin for ceramic raw materials. The Bi,j values were different for each material. When compared to binary mixtures of ceramic raw materials, faster breakage is in the order of quartz–kaolin, quartz–K–feldspar and kaolin–K–feldspar. The Bi,j values are also different from each other. The Si value of cement clinker is found to be very close to quartz mineral. The Si value of wet ground chromite is higher than that of dry ground chromite by a factor of 2. The slowing down effect is seen for all minerals after a certain time of grinding in the mill. This effect is treated by the false time concept in the simulation approach of product size distributions. The simulations of product size distributions for all materials are in good agreement with the experimental size distributions. When particle shapes of quartz and talc are measured by SEM techniques, ball milling produced particles that have higher roundness and relative width, while the rod mill produced more elongated and ﬂat particles. When the surface roughness of pelleted samples of quartz and talc minerals are measured by Surtronic 3+ instrument, ball mill produced the roughest surfaces, as the rod mill produced the smoothest surfaces. When particle shapes of quartz are measured by SEM techniques, ball mill produces higher roundness and relative width, while rod milling produces higher elongation and ﬂatness. Measuring the surface roughness of calcite and barite particles by BET surface area measurement, the highest roughness is obtained for calcite by rod milling,

Breakage and Morphological Parameters

483

while it is obtained for barite by autogenous milling. The lowest roundness and relative width with highest ﬂatness and elongation are obtained by rod milling of talc. The surface roughness of pelleted samples was measured by the Surtronic 3+ instrument to characterize the mineral powders. Obtaining the two different roughness results from the measurements is due to the techniques employed for the measurement of surface roughness that have different approaches; the surface roughness measured by the Surtronic 3+ instrument is a three-dimensional measurement method, while the BET technique produces a calculated surface roughness based on the total surface area measurement by the gas adsorption technique.

5. CONCLUDING REMARKS A grinding circuit should produce a desired quantity of product with quality speciﬁcations for speciﬁed feed material that is ground. The process engineer who designs the grinding circuits faces many choices such as mill type, size and power, selectivity (classiﬁcation), the amount of coarse product to recycle, mode of operation, circuit conﬁguration, the selection of efﬁcient and stable operating conditions within reasonable cost limitations. Particle breakage depends on a range of factors: particle size, mill diameter and size of grinding media. Breakage of one particle produces a complete range of ﬁner breakage product sizes, known as a primary breakage distribution function. By knowing how fast each size is broken and how those broken sizes are distributed down to ﬁner sizes gives rise to the concept of size–mass balance approach to characterize the breakage behaviour. This method combines the concepts of speciﬁc rates of breakage, grinding time, classiﬁcation and the relationship between size–mass balance and mill conditions and sizes. The grinding circuits are simulated, compared and optimized from a process engineering viewpoint. It is possible to obtain S and B parameters with laboratory tests to characterize the mineral for scale-up grindings using the size–mass balance approach. Slowing down phenomena that occurs mostly in longer grinding times in the mills could be predicted by simulation studies. The morphological parameters (elongation, ﬂatness, roundness, relative width and surface roughness) could be obtained easily for any mineral to be able to characterize the powders produced in different mills. The particle morphology plays a very important role in many aspects of powder technology and enables us to predict how the minerals may behave when they are ground and to determine how those minerals may respond to processing. The breakage parameters of quartz, calcite, barite, zeolite, lignite, anthracite, ceramic raw minerals, cement clinker and chromite were presented in this work. Also, product size distributions were simulated with good agreement using the

484

M. Yekeler

parameters obtained from our laboratory studies. Morphological parameters of quartz, calcite, barite and talc minerals were given as various examples of the materials to show the characterization of mineral powders. As a result, there is a correlation between small-scale grinding results and large-scale results. Therefore, accurate prediction of plant-scale results from laboratory size milling requires detailed analysis using the size–mass balance approach. Obtaining the breakage and morphological parameters by laboratory studies helps us to better understand the breakage behaviour and predict the process outputs in desired unit operations to solve complex problems.

Nomenclature

aT a g f b Bi,j D d I JB U Nc Si T W wi(t) xi x0 Pi(0) Pi(t) P2(0) P2(t) y k Ra L V L

speciﬁc rate of breakage at x0 ¼ 1000 mm, particle size (min1) characteristic constant characteristic constant characteristic constant characteristic constant cumulative primary breakage function of size j; fraction broken to less than size xi in one breakage mill diameter (m) ball diameter (m)pﬃﬃﬃ integer denoting 2 size interval fraction of mill volume ﬁlled with ball bed fraction of void spaces in ball bed ﬁlled with powder critical speed of the mill (rpm) speciﬁc rate of breakage of material of size i (min1) time of grinding (min) total powder mass in the mill fraction of mill charge in size interval i size of particles (mm) standard size (1 mm) cumulative weight fraction of time 0 for size interval i cumulative weight fraction of time t for interval i cumulative weight fraction of time 0 for the second interval cumulative weight fraction of time t for the second interval false time (min) slowing down factor: ratio of speciﬁc rate of breakage at time t to normal speciﬁc rate of breakage at time zero surface roughness (mm) length of mill (mm) volume of mill (cm3) length of axes of particle projection (mm)

Breakage and Morphological Parameters

W A P E F R RW

485

width of axes of particle projection (mm) area of the particle projection (mm2) perimeter of the particle projection (mm) elongation ﬂatness roundness relative width

REFERENCES [1] M.E. Fayed, L. Otten (Eds.), Handbook of Powder Science and Technology, Van Nostrand Reinhold Co., New York, 1984, pp. 562–606. [2] L.G. Austin, R.R. Klimpel, P.T. Luckie (Eds.), Process Engineering of Size Reduction: Ball Milling, Society of Mining Engineers, New York, 1984. [3] L.G. Austin, Powder Technol. 5 (1972) 1. [4] C. Tangsathitkulchai, L.G. Austin, Powder Technol. 42 (1985) 287. [5] L.G. Austin, P.S. Bagga, Powder Technol. 28 (1981) 83. [6] L.G. Austin, Min. Eng., June (1984) 628. [7] S.G. Malgham (Ed.), Ultraﬁne Grinding and Separation of Industrial Minerals, Society of Mining Engineers, New York, 1983, pp. 9–19. [8] P. Rehbinder, N. Kalinkovskay, J. Technol. Phys. 2 (1932) 726. [9] E. Forsberg, H. Zhai, Scand. J. Metall. 1/14 (1985) 25. [10] W.H. Beyer (Ed.), Handbook of Mathematical Sciences, CRS Press, Florida, 1978, pp. 7–30. [11] H. Heywood (Ed.), The Scope of Particle Size Analysis and Standardization, Institution of Chemical Engineers, London, 1947, p. 25. [12] H.H. Hausner, Pulvermetallurgie 14/2 (1996) 75. [13] T.H. Hagerman, K. Black, M. Lillieskold, Swedish Conc. Build. Res., D26 (1980) 63–72. [14] J. Serra (Ed.), Image Analysis and Mathematical Morphology, Academic Press, New York, 1982. [15] I. Szleifer, A.B. Shaul, W.M. Gelbert, J. Chem. Phys. 85/9 (1986) 5345. [16] J.J. Bikermann (Ed.), Physical Surfaces, Solid Surfaces, Academic Press, New York, 1970. [17] K. Salama (Ed.) Proceedings of the 7th International Conference on Fracture, Pergamon Press, Oxford, 1989, p. 3391. [18] P. Somasundaran (Ed.), Fine Particle Processing, Society of Mining Engineers, New York, 1980, pp. 492–524. [19] L.G. Austin, M. Yekeler, T.F. Dumm, R. Hogg, Part. Part. Syst. Charact. 7 (1990) 242. [20] M. Yekeler, A. Ozkan, L.G. Austin, Powder Technol. 114 (2001) 224. [21] E. Teke, M. Yekeler, U. Ulusoy, M. Canbazoglu, Int. J. Miner. Process. 67 (2002) 29. [22] M. Yekeler, A. Ozkan, Part. Part. Syst. Charact. 19 (2002) 1. [23] M. Yekeler, A. Ozkan, Powder Technol. 134 (2003) 108. [24] A. Ozkan, M. Yekeler, Part. Part. Syst. Charact. 20 (2003) 276. [25] M. Yekeler, A. Ozkan, Indian J. Eng. Mater. Sci. 9 (2002) 383. [26] A. Ozkan, M. Yekeler, S. Aydogan, Indian J. Eng. Mater. Sci. 10 (2003) 269. [27] H. Ko¨se, V. Aslan, M. Tarıverdi (Eds.), 3.Endu¨striyel Hammaddeler Sempozyumu, TMMOB Maden Mu¨h. Odası, Ankara, Turkey, 1999, pp. 201–207. [28] P. Massacci (Ed.), Proc. XXI Int. Miner. Process. Congress, Rome, Italy, 2000, pp. C4 16–21. [29] H. Ipek, Y. Ucbas, M. Yekeler, C. Hosten, Trans. Inst. Min. Metall. C 114 (2005) C213.

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[30] H. Ipek, Y. Ucbas, M. Yekeler, C. Hosten, Ceram. Int. 31 (2005) 1065. [31] L.G. Austin, K. Yildirim, P.T. Luckie, H.C. Cho, Two Stage Ball Mill Circuit Simulator: PSUSIM, Pennsylvania State University, PA, USA, 1989. [32] U. Ulusoy, M. Yekeler, C. Hic- yilmaz, Miner. Eng. 16 (2003) 951. [33] U. Ulusoy, M. Yekeler, Int. J. Miner. Process. 74 (2004) 61. [34] R.T. Habson, Surtronic 3+ Operating Instructions, RTH-HB-103, 1992. [35] J.J. Bikerman (Ed.), Surface Chemistry, Academic Press, New York, 1958, pp. 181–183. [36] M.J. Jaycock, G.D. Parﬁtt (Eds.), Chemistry of Interfaces, Wiley, New York, 1981, pp. 156–161. [37] C. Anbeek, Geochim. Cosmochim. Acta 56 (1992) 1461. [38] C. Hicyilmaz, U. Ulusoy, S. Bilgen, M. Yekeler, Int. J. Miner. Process. 75 (2005) 229. [39] U. Ulusoy, C. Hicyilmaz, M. Yekeler, Chem. Eng. Process. 43 (2004) 1047. [40] M. Yekeler, U. Ulusoy, Trans. Inst. Min. Metall. C 113 (2004) C145. [41] M. Digre, Autogenous grinding in relation to abrasion conditions and minerological factors, Seminar, Paper A-1, Trondheim, Norway, May 1979. [42] P. Molitor, V. Barron, Int. J. Adhes. Adhes. 21 (2001) 129. [43] L. Vogel, W. Peukert, Powder Technol. 129 (2003) 101. [44] U. Ulusoy, M. Yekeler, Chem. Eng. Process. 44 (2005) 557. [45] M. Yekeler, U. Ulusoy, C. Hicyilmaz, Powder Technol. 140 (2004) 68.

CHAPTER 10

Selection of Fine Grinding Mills Toyokazu Yokoyama and Yoshiyuki Inoue 1-9, Shodai-Tajika, Hirakata 5731132, Japan Contents 1. Introduction 2. Classiﬁcation of ﬁne grinding mills 3. Features and selections of ﬁne grinding mills 3.1. Impact mills 3.2. Ball media mills 3.3. Air jet mills 3.4. Roller mills 3.5. Other mill types 4. Selection after the particle size of feed and product 5. Selection after the feed properties 5.1. Hard materials 5.2. Heat-sensitive materials 5.3. Flammable and explosive materials 5.4. Fibrous materials 6. Fine grinding operation and mill selection 6.1. Wet and dry milling 6.2. Batch and continuous operation 6.3. Open- and closed-circuit grinding system 7. Applications of ﬁne grinding mills to particle modiﬁcation 8. Conclusions References

487 488 489 489 491 494 494 496 497 498 499 499 500 500 500 500 501 501 506 507 508

1. INTRODUCTION For the purpose of size reduction of solid particles, a number of different types of grinding mills are used in various engineering and industrial ﬁelds. The variety of the grinding machines is attributable to the diversity in the requirements for the grinding and the properties of the materials to be ground. As for the material properties, the strength, toughness, etc. are widely different with each feed material and the environmental conditions of the grinding are also to be considered to select the grinding mill. Corresponding author. Tel.: +81 72-855-2307; Fax: +81 72-855-2561; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12013-3

r 2007 Published by Elsevier B.V.

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Nowadays, the requested product particle size is becoming ﬁner and ﬁner because of the advantages of ﬁne particles with their large speciﬁc surface area and high activity of the particle surface and so forth. The subject of selection of grinding machines here is limited to the ﬁne grinding mills, which in principle produce ﬁne and ultra-ﬁne particles.

2. CLASSIFICATION OF FINE GRINDING MILLS The ﬁne grinding mills are classiﬁed often into ﬁve major groups (i) impact mills, (ii) ball media mills, (iii) air jet mills, (iv) roller mills, and (v) shearing attrition mills from the viewpoints of grinding machines. Table 1 shows typical types of ﬁne grinding mills in each group. The major grinding mechanisms are expressed in terms of impaction, shearing, compression and attrition, which are different combinations of the mechanical Table 1. Classiﬁcation of ﬁne grinding mills

Group

Type/Model

Impact mill

High-speed rotation disc type Hammer type Axial ﬂow type Annular type

Roller mill

Roller tumbling type Roll type

Ball media mill

Vessel drive

Agitator drive

Tumbling type Vibration type Planetary type Centrifugal ﬂuidized-bed type Tower type Agitation vessel type Tubular type Annular type

Air jet mill

Target collision type Fluidized-bed type Attrition type

Other type mills (shearing attrition mill, etc.)

Mortar and pestle Stone mill Powder-bed attrition-type mill Wet high-speed shearing mill

Selection of Fine Grinding Mills

489

forces having different strength, direction and speed. The impaction is caused by predominantly the normal force at high speed to pulverize the feed materials. The shearing is exerted by the tangential force to cut the materials and the compression is performed principally by the normal force between two plates or rolls at rather lower speed to crush lumps or particles. Attrition is carried out by the shearing force under compression to grind them. In most mills, these grinding mechanisms usually take place simultaneously.

3. FEATURES AND SELECTIONS OF FINE GRINDING MILLS 3.1. Impact mills The impact mill is one of the most popular types of mills for the ﬁne and intermediate grinding. They have different mechanical structures, which give different grinding performance, and are classiﬁed into some typical types such as highspeed rotating disk, hammer, axial ﬂow and annular. The rotating disk type has grinding pins or blades on the disk, which rotates at high speed up to nearly 150 m/s. The hammer mill has hammers, which are ﬁxed on the rotor or set to swing freely and usually have the screen or slot to separate the ground products for discharge. The axial ﬂow type has rather longer residence time for the grinding and the annual type has the annual grinding zone with narrow space between the beater and the casing in the mill. The hammer mills are the most popular ones and often used for general grinding purpose but the high-speed rotating disk-type mills generating greater impaction are used for the ﬁner grinding in many applications. Some of them have air separation mechanism integrated in the mill to produce ﬁne product less than some decades of mm. Figures 1, 2 show examples of a hammer mill with screens and a high-speed rotating disk-type mill called ACM Pulverizer with an integrated air classiﬁer. The grinding tips are ﬁxed at the periphery of the rotating disk. The coarse particles rejected by the integrated air separator are recycled to the grinding zone in the mill. Figure 3 shows the schematic diagram of one of the axial ﬂow-type impact mills called a Super Micron Mill E. This mill has a larger processing volume and operated at a middle-range speed but is capable of the ﬁner grinding by the attrition mechanism and separation nozzle to discharge foreign materials, which often cause the reduction in the grinding capacity and the product quality. The annular-type impact mills have some unique performance characteristics due to the concentrated mechanical energy generated in the narrow grinding zone between the blade tips and the casing. Because of the constant tip speed of the rotating blades in the annular part, high capacity and narrower particle size distribution could be expected compared with a simple impact mill. Figure 4

490

T. Yokoyama and Y. Inoue Lining Plate

Hammer

Hopper

Screw Screen

Fig. 1. Hammer-type impact mill.

Guide ring

Separating blade

Liner Grinding disc Feed screw

Fig. 2. High-speed rotating disk-type impact mill (ACM Pulverizer).

shows an example of the relationship between the grinding capacity and the product ﬁneness of different kinds of mills [1]. The annular-type impact mill succeeded in the grinding of calcium carbonate down to several microns, which was not possible by the conventional high-speed rotating impact mill. In addition, the capacity was considerably higher than the jet mill, though the jet mill was capable of ﬁner grinding then the annular impact mill. However, when such ﬁneness in the range of several mm is not required, high-speed rotating disk-type is more suitable for the general use because of wide applicability and easy control.

Selection of Fine Grinding Mills

491

Feed Air inlet Grinding blade

Fine product Nozzle (for separating foreign material) Nozzle product

Fig. 3. A new axial ﬂow-type impact mill with attrition mechanism and nozzle for coarse discharge.

Material : Calcium carbonate (x50 = 200 μm)

Capacity per power [kg/kWh]

10

Annular type impact mill

5

High speed impact mill Jet mill 0

0

5

10

15

Product particle size [μm]

Fig. 4. Comparison of grinding capacity of high-speed rotating disk-type and annular-type impact with a jet mill.

3.2. Ball media mills The ball media mills using balls or beads as grinding media have been used for a long time [2]. They are classiﬁed into two groups as shown in Table 2. The ﬁrst includes mills in which the balls are driven by the movement of the mill casing,

492

Table 2. Classiﬁcation of ball media mills

Driving method

Machine name

Vessel drive

Tumbling ball mill (rotating vessel)

Mechanism

Maximum acceleration atum ¼ 1 g

ωR

avibo30 g

Vibrating ball mill 2rv ωv=2π f

Planetary ball mill

aplao150 g

ωRT d

ωRV

Agitator drive

Agitating ball mill

rs

ωs=2πΝ

aagto Hundreds of g

T. Yokoyama and Y. Inoue

G/2

Selection of Fine Grinding Mills

493

P9

1:Feed bin 2:Screw feeder 3:Weighing bin

1

8

13 2

12

4:Screw feeder (High accuracy) 14 5:Ball hopper 6:Ball feeder

10

11

7:Backet elevetor 8:Agitation ball mill (ATR, HOSOKAWA/ALPINE)

7

9:Pump

5 6

3 4

10:Discharging screw 11:Sieving machine 12:Classifier (ATP, HOSOKAWA/ALPINE) 13:Product collector 14:Blower

Fig. 5. An example of dry grinding system of ball agitation mill.

and the second contains those having the ball agitation mechanism inside the mill. In the former, there are tumbling, vibration, planetary ball mills and the like. The mills in the latter group are further classiﬁed after the shape of the mill casing into tower type, agitation vessel type, tubular type and annular type. The ball media mills can be used for dry and wet grinding and in batch and continuous operation. Figure 5 shows an example of agitation ball mill in a closed circuit together with an air separator operated in dry state [3]. The grinding rate of the ball media mills depends upon the intensity and frequency of the collisions of the ball media against the balls, the casing or the agitator. The intensity is controlled by the mechanical conditions of the mill as well as the properties of balls. The balls in the tumbling mill fall under gravitational acceleration g. They are agitated by the vibration acceleration of the vibration mill avib and by the rotation acceleration of the planetary mill apla calculated by the rotation and revolution speeds. avib is a few tens of times, apla up to 150 times and the rotation acceleration of the agitating-type mills aagt could be several hundred times as large as g. The grinding rate tends to increase with the grinding intensity. Therefore the vibration, planetary or agitation mills are applied to accelerate the grinding but the tumbling mills are still widely used because of their simple structure and the mild grinding condition. In use of the ball media mills, the properties of ball media are very important. They need to be made of strong and tough material enough to grind the materials. However, it has been found that the smaller the beads that are used, the ﬁner the product obtained because of the larger number of contact points for better dispersion to avoid the reaggregation by excessive force. However, to produce the collision force necessary for the breakage of the feed particles, sufﬁcient acceleration intensity is required. In this way, recent grinding to nanosized particles

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has been realized by the agitation-type ball mills with small beads having diameters as small as 20–30 mm [4].

3.3. Air jet mills The jet mills grind the feed materials using the ﬂuid energy in a different manner from the other mechanical grinding mills. As far as the dry grinding is concerned, the jet mill is often used for the ﬁnest grinding owing to the large impact force by the jet and the dispersing effect to avoid reaggregation. However the energy consumption, which is greatly affected by the feed ﬁneness, is much greater than for the mechanical grinding mill and therefore is usually applied at the ﬁnal stage of the ﬁne grinding system after the intermediate or ﬁne grinding with other mechanical mills. The typical types of the jet mill are target collision type, ﬂuidized-bed type and attrition type. In the target collision type, the feed particles are accelerated by the air jet stream in the nozzle and broken by the collision against the hard target material. In the ﬂuidized-bed type as shown in Fig. 6, the feed is injected into the ﬂuidized material and the grinding takes place by mainly the collision between the particles. The attrition type has a variation in the structure. Some of them are panshaped with the tangential jet injection from the sidewall and an exit in the center. Others have tubular ring structure, where the particles are mainly ground by the attrition against the particles and the tube wall. Each type of jet mill has its own advantages and disadvantages. The target type shows generally high power in particle crushing, but has a problem of wear of the target, especially when the feed contains the components with high hardness. Therefore, the ﬂuidized-bed type is widely used for the usual ﬁne grinding. The pan-shaped jet mill is often applied for the ﬁne grinding of pharmaceutical materials, since the cleaning is easy because of its simple structure.

3.4. Roller mills Roller mills are classiﬁed into the mills using roller tumbling on the table or in the vessel (called hereafter ‘‘roller-type mill’’) and those where the feed is ground between the cylindrical rolls (called hereafter ‘‘roll-type mill’’). The roll-type mills used to be applied for coarse or intermediate grinding to treat the large lumps of brittle feed materials as well as grains. However, it was found that this type of mill could be used for ﬁne efﬁcient grinding in combination with a ball mill or other ﬁne grinding mill as shown in Fig. 7 [5]. The concept is to give high-pressure on the feed material to generate micro cracks inside it and then disintegrate it into ﬁne particles. With this type mill, the mechanical energy is given directly and

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Fig. 6. Fluidized-bed type jet with an integrated air separator.

effectively to the materials resulting in efﬁcient grinding, though the performance depends upon the material properties. Figure 8 shows an example of comparison of grinding performance between the high-pressure roll mill system and the tumbling ball mill system [6]. Especially in the coarser particle size range, the former indicates higher performance than the latter. The discrepancy of the required speciﬁc energy around 40 mm in the ﬁgure is due to the use of different separator for recycling. The roller-type mill has been used for a long time to grind grains, gunpowders and so on. The shape of the roller plays a big role with this roller-type mill. Since attrition effects caused by the difference in the tangential speed at the different radial points of the roller is expected, it is advantageous for the ﬁne grinding of larger throughput to use larger rollers.

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T. Yokoyama and Y. Inoue Raw material

Recycle of coarse particles

Compressive milling of feed particles of powder bed (Roll mill) Disintegration of agglomerated particles (e.g. Ball mill)

Classification (eg. Sieving, Air separator) Product

Fig. 7. The grinding system of high-pressure roll mill combined with machines for disintegration and separation of coarse fraction.

Specific grinding energy [kWh/t]

200

100

50

30 20

High-pressure roll mill system Ball mill system

10 5

10

20

30

40 50

70

100

Product particle size x97 [μm]

Fig. 8. Comparison of grinding performance between a high-pressure roll mill system and a ball mill system.

3.5. Other mill types There are some other type mills available for ﬁne grinding such as the powderbed attrition mill, stamp mill, stone mill and so forth. In the Angmill, as an example

Selection of Fine Grinding Mills

Press head

Rotating direction

497

Powder bed

Rotating vessel

Fig. 9. Principle of powder-bed attrition-type mill (Angmill).

of the ﬁrst, the feed material is ﬁxed on the inner sidewall of the rotating chamber by the centrifugal force and receives strong shearing under compression between the chamber wall and the stationary press head with a different radius of curvature from that of the chamber (Fig. 9). In this way, continuous dry grinding of mineral materials down to the submicron range has been performed with this machine [7]. In this case, the ﬁne particles are generated generally not by the volumetric crushing but by the attrition of the surface of the larger feed particles and they should be taken out as the ﬁnal ﬁne product. The stamp mill gives the impact force to the powder bed of feed material by the falling pestle. It is usually used for the intermediate grinding of various kinds of materials like metallic powders or corns. The stone mill is one of the oldest mills but is capable of ﬁne grinding by the shearing under compression between the rotating and the stationary parts.

4. SELECTION AFTER THE PARTICLE SIZE OF FEED AND PRODUCT The range of particle size of raw materials to be ground is so wide ranging from tens of cm down to a few microns. However, the requirement for the product particle size is also different depending upon the application and purpose of the grinding. It is usually difﬁcult to produce ﬁne powders from large bulky material using a single machine in one step but some different types of mills are to be used step-by-step to optimize the whole grinding process. As for the selection of ﬁne grinding mills, each mill has its own advantageous particle size range for the individual feed materials. Impact milling has generally its own grinding limit size of around 10 mm with typical minerals at even higher tip speed of 100 m/s [8]. Compression milling with roll mills is usually effective in the range of several tens of mm but high-compression milling is capable of efﬁcient ﬁner grinding combined with an impact mill or ball mill. When the ﬁnest grinding is required by dry grinding, the jet milling or the attrition milling with a roller mill, a ball media mill or a powder-bed attrition mill would be useful. Figure 10 shows a

T. Yokoyama and Y. Inoue Specific grinding capacity [kg/kWh]

498 60 40

A1

Material: Talc

20 10 6 4 2

A2 C

B

1 0.6 0.4 0.2

C’ 0.1 0.4 0.6 0.8 1

2

3 4

6 8 10

20

30

Product particle size [μm] (measured by sedimentation method) A1: Axial flow impact mill 1 (Super Micron Mill) A2: Axial flow impact mill 2 (Fine Micron Mill) B: Air jet mill (Micron Jet) C: Powder bed attrition mill (Angmill) C’: C + Classifier

Fig. 10. Comparison of grinding performance between impact mills, air jet mill and powderbed attrition mill.

comparison of grinding performance between the powder-bed attrition-type mill and an air jet mill as well as impact mills equipped with attrition mechanism. The former is capable of submicron range grinding, though the capacity is reduced considerably in such a ﬁne range. In practice, the jet mills, the roller mills or the ball media mills are often used for the grinding to get the ﬁneness of around a micron or so. When ﬁner grinding down to submicron range is required, wet milling is applied if a proper liquid media is available. Nowadays, the ultra-ﬁne grinding to obtain the ﬁnal product having an average particle size of less than 100 nm has been realized using wet agitation ball mills [9]. However sufﬁcient attention should be paid considering the material properties, when dry product is desired, because drying generally causes the agglomeration of ﬁne particles. The ultra-ﬁne grinding is accomplished by the wet grinding using an agitation ball mill with ﬁne beads and well-dispersing condition obtained by the control of pH or zeta potential of the feed slurry [10].

5. SELECTION AFTER THE FEED PROPERTIES The feed materials to be ground are so diverse from the inorganic materials such as minerals and ceramics to the organic materials like resin, food and pharmaceuticals, and the metallic materials and so on. Their properties are also very

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different. Recently, there has been an increase in the desire to grind more composite materials, which makes it even more complicated to select the most suitable grinding machine and the operating conditions than before. There are many properties of the feed materials that affect the grinding performance directly and indirectly such as strength, toughness, hardness, cohesiveness, ﬂowability, wetability and so on. Although various relationships [11] between the grindability and material property have been proposed, it is not yet realistic to predict the grinding performance only from the data of the material properties for each mill in most cases. In practice, the performance of the actual grinding plant is estimated from the data with a lab-size machine of the same type and the scale-up is to be made to meet the requirement. Then it is important to select the proper type of grinding machine in consideration of the material properties and the requirements at the beginning. In general, for the grinding of hard and brittle inorganic materials like minerals, mills that are based on the main grinding mechanisms of compression, impaction and attrition are used. However, the size reduction of elastic resin and the ﬁbrous materials is conducted usually by mills with a shearing mechanism. Furthermore, ductile metallic powders are difﬁcult materials for ﬁne grinding and are usually treated with ball media mills under controlled atmosphere for the size reduction and/or making amorphous structure. The basic principle of mill selection according to the feed properties is introduced in the following sections.

5.1. Hard materials When the hardness of the feed material is high, hard material is used for the mill body construction as well as the grinding media, or the impact speed must be reduced to minimize the wear of the mill. The mill parts contacting powder material can be coated with hard metal or ceramics. The anti-abrasive zirconia balls are often used to minimize the ball wear. From the aspect of grinding mechanism, a roll mill can be more suitable than an impact mill because of the reduced impact speed. As for the air jet mills, the ﬂuidized-bed type is more suitable than the target impact type or attrition type because more impaction and attrition between the particles rather than the collision against the mill body or the target can be expected.

5.2. Heat-sensitive materials The grinding of heat sensitive materials needs special care since grinding can cause the generation of considerable amount of heat. To minimize the grinding temperature, there are three ways, namely (i) to reduce the heat generation, (ii) to remove the generated heat promptly, or (iii) to cool actively using cooling media. From the viewpoint of grinding mechanism, shearing or impact milling generates

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less heat than attrition grinding. When impact milling is applied, enough air volume is required to remove the heat from the system. The air jet mill is most suitable for processing heat-sensitive materials because of the Joule–Thompson effect as well as the air ﬂow itself. The active cooling is made by cooling media in cryogenic grinding using liquid nitrogen with a boiling point of 1961C. The cryogenic grinding mill needs to be equipped with the shaft bearing and sealing to work at the low temperature and good insulation to minimize energy losses. Table 3 shows some examples of grinding performance on different kinds of materials using a cryogenic grinding mill [12].

5.3. Flammable and explosive materials Flammable or explosive material should be ground in a wet state or in an inert gas. When dry grinding is necessary for the practical production line, nitrogen is usually used as an inert gas and recycled to reduce the operational costs. The jet mill can be also applied using nitrogen to minimize the explosion risks. In special cases, some metal powders are ground under a vacuum to avoid oxidization.

5.4. Fibrous materials The ﬁne grinding of organic ﬁbrous materials like wooden tips, pulps or dried ﬁsh is often difﬁcult. They are ground with a cutter mill or high-speed rotating disk mill based on the cutting, shearing and impact grinding mechanism for intermediate grinding and with the annular type or attrition-type impact mill or powder-bed attrition mill based on mainly the attrition mechanism for the ﬁner grinding. A new axial ﬂow impact mill with attrition mechanism shown in Fig. 3 is used also for the ﬁne grinding of materials containing organic ﬁbrous substances like crude vegetable drug and fresh soybeans with husk down to 10–30 mm [13]. The ﬁne grinding of these ﬁbrous materials is realized by the attrition mechanism between the rotor blades and casing in addition to the impaction. Besides, the nozzle assembly to reject the foreign material or the hard substances for grinding is quite effective for the ﬁne grinding of these ﬁbrous materials.

6. FINE GRINDING OPERATION AND MILL SELECTION 6.1. Wet and dry milling The grinding operation is conducted in either the dry or wet state. It is generally admitted that the wet grinding is capable of ﬁner grinding at higher grinding rate than the dry grinding [14] because of the reduction of particle strength [15] as

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501

well as the dispersing effect of the liquid media to avoid the particle reaggregation [16]. However, the wet grinding needs careful selection of the liquid media and more complicated system than the dry grinding and besides additional drying process after the grinding, when dry powder is required as the ﬁnal product. The wet grinding is usually performed with the ball media mill and partially with the roller mills spraying the liquid on the materials. The air jet mills and the highspeed rotating type mills are used for solely the dry grinding.

6.2. Batch and continuous operation The grinding is carried out in batch, semi-batch or continuous operation. For mass production, the continuous grinding operation is desirable. However, when the required capacity is small or when a long residence time or special atmosphere such as a vacuum is required, the batch system is applied because of the simplicity of the system. When both the mass production and batch-wise treatment are needed, the semi-batch system, where more than two lines of batchgrinding system are connected in parallel, is installed and operated by changing the line after a certain running time. Most grinding mechanisms can be applied for both batch and continuous operations in principle but every mill is designed and operated preferably in either way. For the continuous grinding systems, the separation mechanism is also important to obtain the product with required particle size.

6.3. Open- and closed-circuit grinding system The grinding mills are also used in either open circuit without returning a coarse part of the ground material to the feed or the closed-circuit grinding system with recycling. In the latter case, the separation process plays an important role in the improvement of the grinding performance. Various combinations of the grinding mills and separators are used in the industrial ﬁelds [16]. When the grinding is difﬁcult for the required particle size, the recycling ratio tends to increase and therefore the lager size of separator compared with the grinding mill itself is needed. It is often troublesome to ﬁnd the grinding capacity of a closed system under some operational conditions, since there are many factors related with each other and affecting the grinding performance of the whole system. It is easier to grasp the grinding characteristics of the mill in the open grinding circuit. Then an example of the method to predict the grinding capacity of a closed circuit system from the results of the open circuit is demonstrated in the following [17] (Fig. 11).

502

Table 3. Example of cryogenic grinding for different kinds of materials

Temperature Capacity (1C) (kg/kWh)

Feed materials Resin

Beans

100 100 120 120 100 100 50 100 40 100 100 100 160 100 140 140

0.7 2.0 0.1 1.1 0.1 0.3 18.8 18.2 10.7 8.7 22.4 16.1 8.7 13.4 12.7 33.5

Product particle size Product status

12.6 5.0 87.2 9.2 120.0 30.5 0.4 1.9 2.2 2.8 3.5 1.9 3.4 2.2 2.5 1.5

100 mm495.3% 100 mm483% 150 mm499.5% 150 mm474% 150 mm494% 150 mm472% 150 mm445% 47 mm485% 47 mm491% 47 mm497% 14 mm 15 mm 44 mm499% 44 mm499% 500 mm 44 mm

Powder (RT) Powder (RT) Powder (RT) Powder (RT) Powder (RT) Powder (RT) Powder (RT) Powder (RT) Powder (RT) Powder (RT) Paste (RT) Powder (RT) Paste (RT) Paste (RT) Paste (RT) Paste (RT)

T. Yokoyama and Y. Inoue

Corns

Nylon 12 Nylon 12 Polyethylene Polyethylene Polypropylene Polypropylene Polyvinylchloride(PVC) Husked rice Polished rice Polished sticky rice Soybeans (raw) Coffee beans (roasted) Almond Cacao beans Peanuts Sesame

LN2 consumption (kg-LN2/kg-feed)

Fish/sea weed

Meat

Crude drug Vegetables

Ground tea Natural leaf tea Mandarin orange Kelp Shrimp Clam Bonito Chicken Beef Pork Puerariae radix Rehmanniae radix Dried mushroom Spinach Cabbage Onion Sweet corn

100 100 100 100 100 120 100 100 100 100 80 100 100 50 80 100 100

1.6 2.4 15.0 5.9 14.2 10.3 16.1 23.1 15.8 12.7 8.6 13.4 4.7 14.3 10.7 10.1 16.1

11.0 7.5 2.2 3.7 1.8 2.7 1.6 1.7 1.9 2.8 2.7 2.2 2.3 2.0 2.8 2.5 2.3

35 mm498% 35 mm498% 74 mm 74 mm497% NRT NRT NRT NRT NRT NRT 27 mm490% 150 mm499% 74 mm 74 mm 74 mm NRT NRT

Powder (RT) Powder (RT) Powder (RT) Powder (RT) Paste (RT) Paste (Frozen) Paste (Frozen) Paste (RT) Paste (RT) Paste (RT) Powder (RT) Powder (RT) Powder (RT) Powder (RT) Powder (RT) Paste (RT) Paste (Frozen)

Selection of Fine Grinding Mills

Leaves

Note: NRT, No roughness with tongue; RT, Room temperature.

503

504

T. Yokoyama and Y. Inoue To be processed in the same open grinding system repeatedly. R

W

γ

1- γ

W

G

G

CLASSIFIER

CLASSIFIER

1- γ F 1-βn

F MILL M

βn

MILL D

L

M

(a) Closed grinding system

(b) Open grinding system

Fig. 11. Flow diagrams of closed and open grinding systems with a mill and a separator.

It is assumed that all the feed particles are coarser than the required product particle size xp (mm) and the separation is ideal. The fraction of particles coarser than xp in the ground product after one pass through the mill is indicated by b0. It is presumed that the fraction of the coarser particles than xp in the ground product is bn with the throughput Ln, when the coarse material from the separator is ground n times in the open-circuit grinding system and that these factors are independent of the recycling ratio. Furthermore, a fraction 1g of the coarse particles from the separator is rejected from the closed-circuit grinding system. Under these conditions, the throughput of the mill of the closed grinding system D is given in Pn k Q k k¼0 g Lk i¼0 bi1 D ¼ lim P ð1Þ Q n k k b n!1 k¼0 g i¼0 i1 and the capacity of the closed grinding system M becomes !1 n k X Y kþ1 M ¼ D 1 þ lim g bi n!1

k¼0

ð2Þ

i¼0

The recycling of the coarse R is R ¼DM

ð3Þ

These values could be obtained from the simple open circuit grinding tests and calculation. Table 4 shows how to calculate M and D of the closed grinding system from the data of the open grinding tests using the same mill and separator with the assumption of constant L and b after repetition of grinding tests with the open circuit system sometimes. Figure 12 shows some examples of the change of capacity of open grinding system L and unground fraction of the coarse material b for the case of grinding calcium carbonate using an axial ﬂow impact mill combined with an air separator

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Table 4. Estimated throughput of mill and capacity of closed grinding system from the data of pen grinding system

Number of regrinding 1 2

Mill throughput (D) n o gb0 L1 L0 þ 1gb

D¼

1gb1 1gb1 þgb0

D¼

ðL0 þgb0 L1 Þð1gb2 Þþg2 b0 b1 L2 ð1þgb0 Þð1gb2 Þþg2 b0 b1

1

2

3

3

3 ÞðL0 þgb0 L1 þg b0 b1 L2 Þþg b0 b1 b2 L3 D ¼ ð1gb ð1gb Þð1þgb þg2 b b Þþg3 b b b 3

Number of regrinding 1

M¼

2

M¼

3

M¼

0

0 1

0 1 2

Product capacity (M) 1gb1 1gb1 þgb0 D 1gb1 D 1gb2 þgb0 g2 b0 b1 þg2 b0 b1 1gb3 D 1gb3 þgb0 ð1gb3 þgb1 g2 b1 b3 þg2 b1 b2 Þ

60 50

CASE B

40 30 20

CASE A

10 0

0

1

2

3

Number of regrinding, i [-]

Unground fraction of coarse, βi [-]

Grinding capacity of coarse fraction L [kg/h]

70

1.0 CASE A 0.5 CASE B

0

0

1 2 3 Number of regrinding, i [-]

Fig. 12. The change of capacity of open grinding systems L unground fraction of the coarse b. (Case A) Axial ﬂow-type impact combined with an air separator (cut point: 11 mm). (Case B) Hammer-type impact mill combined with a screening machine (cit point: 150 mm).

(case A) and the high-speed hammer mill combined with a screening machine (case B). In these cases, the tendency of change of these values is completely different depending on the grinding performance of the mills as well as the required ﬁneness. In this way, fairly good agreement between the estimated and the actual capacity of the closed grinding system has been obtained by this simple estimation method.

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7. APPLICATIONS OF FINE GRINDING MILLS TO PARTICLE MODIFICATION Most ﬁne grinding mills have other functions for particle processing than size reduction under carefully chosen operating conditions. Some of them have been redesigned to be applied for particle shape control and particle surface modiﬁcation by making composite particles by mechanical force. For example the Angmill introduced in the previous section can be used for particle composing as a MechanoFusion system [18]. Figure 13 shows an example of composite particles made by this system. Usually, the core particles in the range from 1 to 700 mm are coated by the ﬁner nanosized particles by the mechanical bonding. The treated composite particles have various unique characteristics and are used to create new materials and to improve the product properties.

Fig. 13. An example of composite particles made by the dry particle bonding system.

Specific surface area (m2/g)

14

High-speed shearing mixer (Cyclomix®)

12 10 8

Particle composing machine (Mechanofusion®)

6 4

Particle composing machine (NOBILTA®)

2 0

0

2

4 6 8 Specific input energy (J/kg)

10

12

Fig. 14. The change of speciﬁc surface area of the mixture of core particles (silica, x50 ¼ 28 mm) and ﬁner shell particles (TiO2, x50 ¼ 15 nm) with the energy input per unit mass of the mixture.

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This mechanical particle bonding is advantageous in the respect that no liquid binder is required and the combination as well as applicability of the materials is very wide. Furthermore, the system is much simpler compared with the wet system. The selection of the machine and the operational condition should be made appropriately considering the material properties and the required function of the composite particles. There are several methods to evaluate the degree of particle composing from electro microscopic picture, particle size change, spectrometer, screen analysis and speciﬁc surface area. For the purpose of quantitative evaluation of the degree of particle composing, the speciﬁc surface area is practically used [19], since it tends to decrease as the particle bonding proceeds and the ﬁne particles are densiﬁed on the surface of core particles. Figure 14 shows the change of speciﬁc surface area of the mixture of core particles (silica, x50 ¼ 28 mm) and ﬁner shell particles (TiO2, x50 ¼ 15 nm) with the energy input per unit mass of the mixture. It is seen that each machine has different performance in terms of the degree of particle composing and the processing speed. In this way, the choice of the correct machine and operating conditions should be made considering the particle properties and applications.

8. CONCLUSIONS There are a number of grinding mills having different structure, size and performance. Each machine and method has its own features. It is important to understand the characteristics of the machine and the system including the related devices such as particle separation equipment and to make the best use of them. For the selection of the grinding machine, the properties of the feed materials as well as the requirements for the product and capacity should be taken into consideration from the technical and economical viewpoints in practical use.

Symbols and deﬁnitions

aagt apla atum avib D F F g G L

acceleration of agitation ball mill (m s2) acceleration of planetary ball mill (m s2) acceleration of tumbling ball mill (m s2) acceleration of vibrating ball mill (m s2) vessel diameter of planetary mill (m) frequency of vibrating mill (Hz) ﬁnes ﬂow rate (kg h1) acceleration of gravity (m s2) diameter of revolution of planetary mill (m) throughput (kg h1)

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M N R rv T W x50 x97 xp b

capacity of the closed grinding system (kg h1) number of regrinding (–) recycling ﬂow rate (kg h1) revolution radius of vibrating mill (m) throughput of the mill in the closed grinding system (kg h1) rejecting ﬂow rate (kg h1) median particle size (mm) particle size at 97% cumulative undersize (mm) required product particle size (mm) fraction of the coaser particles than xp in the ground product (–) subscript 0 data for raw material without regrinding recycling ratio (–) angular velocity of rotating motion (rad s1) angular velocity of vibrating motion (rad s1) angular velocity of rotation of planetary mill (rad s1) angular velocity of revolution of planetary mill (rad s1)

g oR oV oRT oRV

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19]

M. Inoki, Powder Sci. Eng. 28 (7) (1996) 67; (in Japanese). G. Jimbo, Kagaku Koujou 26 (10) (1982) 23; (in Japanese). J. Stein, Verfahrenstechnik 28 (11) (1994) 28; (in German). M. Inkyo, T. Tahara, J. Soc. Powder Technol. Japan 41 (2004) 578–585; (in Japanese). K. Schoenert, F. Fluegel, European Symposium Particle Technology, Amsterdam, 1980, Preprints A, p. 82. K. Toyotate, Preprint of the 25th Symposium on Powder Technology, 11, Osaka, 1991, (in Japanese) T. Yokoyama, K. Urayama, T. Yokoyama, KONA 1 (1983) 53. T. Yokoyama, Chem. Eng. Japan 50 (7) (1986) 467; (in Japanese). K. Kugimiya, The Micromeritics (Funsai) 36 (1992) 177; (in Japanese). S. Mendle, F. Stenger, W. Peukert, J. Schwedes, Proc. of 10th European Symposium on Comminution, Heidelberg, Germany, 2002. E.L. Piret, Chem. Eng. Prog. 49 (1953) 56. N.P. Chopey, Chem. Eng. 80 (9) (1973) 54. Y. Inoue, Powder Sci. Eng. 38 (2) (2006) 72–76; (in Japanese). P. Somasundaran, I.J. Lin, Ind. Eng. Chem., Process Des. Dev. 11 (1972) 321. O. Imanaka, S. Fujino, K. Shinohara, Bull. Japan Soc. Prec. Eng. 2 (1966) 22. H. Schubert, Aufbereitungs-Technik 3 (1988) 115; (in German). T. Yokoyama, T. Yamaguchi, Proc. Inter. Sympo. Powder Technol. 81 (1982) 420. T. Yokoyama, K. Urayama, M. Naito, M. Kato, T. Yokoyama, KONA 5 (1987) 59. M. Naito, M. Yoshikawa, T. Tanaka, A. Kondo, J. Soc. Powder Technol., Japan 29 (1992) 343; (in Japanese).

CHAPTER 11

Fine Grinding of Materials in Dry Systems and Mechanochemistry Qiwu Zhang, Junya Kano and Fumio Saito Institute of Multidisciplinary Research for Advanced Materials,Tohoku University, Katahira 2-1-1, Aobaku, Sendai 980-8577, Japan Contents 1. Introduction 2. Fine grinding mills 3. Rate process of grinding phenomena 4. Simulation of media motion during milling 5. Mechanochemistry and nano-particles formation by dry grinding 5.1. Mechanochemical phenomena 5.2. Phase change 5.3. Solid-state reactions 5.4. Material processing 5.5. Formation of nano-particles References

509 510 511 512 516 516 517 518 523 526 527

1. INTRODUCTION Milling is known as one of the important unit operations and has been widely used in many material processing operations such as mining, food and medicine, chemicals and building materials. New operations have replaced the milling operation in some areas; nevertheless, milling is still widely used and has also found new applications. Recent milling technology covers shape and size distribution control besides size reduction. Of course, the milling efﬁciency has been improved, and this leads to the prevention of wear, resulting in a change in the design of milling devices. Furthermore, it is noted that simulation of media (balls and beads) motion as well as mechanochemistry are particularly impressive. Regarding the ﬁne size reduction, it is noticeable that the surrounding technology has been very advantageous, especially with regard to ﬁne beads having a diameter smaller than 0.05 mm in Japan. In fact, recent reports have shown high possibility in preparation of ﬁne particles being less than 10 nm diameter by Corresponding author. Tel.: +81 22 217 5200; Fax: +81 22 217 5596; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12014-5

r 2007 Elsevier B.V. All rights reserved.

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the use of a bead-mill. This technology is able also to disperse agglomerates which are synthesized by CVD processes to produce mono-dispersed nanosize particles, leading to the creation of new engineering applications for nanotechnology. The ﬁeld of mechanochemistry has attracted more than before and has had a great deal of work done on this topic in recent years. The science of mechanochemistry is related to the use of solid state reactions to produce useful compounds, and this is related to separation and recovery of useful chemical species and components from minerals and waste materials and to dry coating of other components or metals on the surface of small particles or plate under ambient conditions. Understanding the media motion during milling by tumbling, planetary ball mills and the like is a very important issue in controlling milling performance, including mechanochemical efﬁciency. The motion can be seen visually from outside of a mill device through a transparent mill wall, but the view is limited to near the wall. Considering the fact that the main events take place inside the milling device, it is necessary to understand all the events of media motion during milling and the mechanochemical phenomena. The Discrete Element Method (DEM) was invented in 1979 [1], and it has been growing along with the advancement in computer performance. This enables us to simulate particulate systems such as particle motion in mills, mixing devices and ﬂuidized beds as well as the discharge behaviour of powder in a bin. It has been proven to be useful to understand various phenomena occurring in such devices and mechanochemical reaction induced by this operation. This chapter focuses on recent development of ﬁne grinding in relation to media motion simulated by the use of DEM and mechanochemistry.

2. FINE GRINDING MILLS Milling can be operated under dry and wet conditions, and due to ﬁnal size of product, it can be called as coarse, middle and ﬁne millings. Several types of mills have been invented and applied to ﬁne to coarse milling operations. The mill types that may be used for ﬁne grinding are summarized as follows. 1. Roller mills: 2. Impact mills: 3. Ball mills: 4. Agitation mills: 5. Jet mills: 6. Shear-type mills: 7. Colloid mills:

ring roller mill, centrifugal roller mill, ball bearing mill. hammer mill, cage mill, pin mill, turbo mill, screen mill, disintergrator mill, ﬁne micron mill, super micron mill. pot mill, tube mill, vibration ball mills, planetary ball mills. tower mill, attritor, DYNO mill, sand grinder, basket mill. jet mill with various types. raymond mill. mortar.

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3. RATE PROCESS OF GRINDING PHENOMENA Milling operations can be depicted by a rate process, and the rate is deﬁned by counter balance of decreasing rate of initial mass and increasing rate of ground mass coming into certain particle size section per unit time. Thus, the milling rate can be deﬁned as this change within a speciﬁc period of time and it can be formulated by the following expression. Change in mass of particles in range (x-x+dx) in interval (t-t+dt), represented as @2 Dðx; tÞ=@x@t, results from the difference between the mass transferred from the size range (x-x+dx) to smaller range and the mass into this range transferred from the larger range the initial size. Z x max @2 Dðx; tÞ @Dðx; tÞ @Dðg; t Þ @Bðg; x Þ ¼ Sðx; t Þ þ Sðg; t Þ dg ð1Þ @D@t @D @r @x x S ( ¼ @P=@t) denotes a selection function, whereas P is a breakage probability. B(g, x) a breakage function, indicating the mass ratio of particles smaller than x to the milled sample from particles with size g. Therefore, (qB/qx) is also called a distribution function, and this is the basic expression for grinding rate. There are other kinetics on grinding, particularly, aiming at covering the whole characteristics of ground products, such as increasing rate in speciﬁc surface area and decreasing rate in average particle size. In these cases, selection function and breakage function are the basic parameters in milling phenomena. Both functions vary with the types of mill and operational conditions. In the ﬁeld of ﬁne grinding, both functions become small and milling rate becomes slow correspondingly. Figure 1 shows schematic change in residue (R) and fraction (f ) percentage of the ground sample with particle size. R can be represented by the following formula at an arbitrary time t, as shown in Fig. 2: R ¼ expðK P tÞ ð2Þ

Fraction percentage, f

Residue percentage, R

100

0

Particle diameter, D

Fig. 1. Size distribution curves of the ground product.

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Residue percentage, log R

512

Grinding time,t

Fig. 2. Residue percentage logR as a function of grinding time t.

Fig. 3. Simulation model of interactive forces between two balls: (a) normal force, (b) tangential force.

4. SIMULATION OF MEDIA MOTION DURING MILLING The DEM has been used for simulating three-dimensional motion of media (balls or beads) in a media mill [2–4], in which some amount of powder is present [5–6]. The model in the DEM is shown as interactive forces between two balls (i-th and j-th) colliding each other; therefore, it involves a slider, a spring and a dashpot, as shown in Fig. 3. The interactive forces in the normal and tangential directions acting on the contact surface between two balls are given by the following

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equations. The interactive force in the normal direction, fn, the interactive force in the tangential direction, fs, where u is the relative displacement of the gravitational centre between i-th and j-th ball. j is the relative angular displacement and is taken as a positive value in a counter clockwise direction. Z the dashpot coefﬁcient, K the spring coefﬁcient, t the time and d the diameter of the ball. The subscripts n and s denote the normal and tangential directions, respectively. f n ¼ Zn

dun þ K n un dt

d d d us þ f þ K s us þ f f s ¼ Zs dt 2 2

ð3Þ

ð4Þ

The following condition is considered in the calculation of tangential direction [6], in which the friction coefﬁcient of the balls plays a signiﬁcant role in their movement, where, m is the friction coefﬁcient and SIGN the signum. dun d d d f s ¼ m Zn us þ f þ K s us þ f þ K n un SIGN Zs dt 2 2 dt d d d du n at Zs us þ f þ K s us þ f 4m Zn þ K n un ð5Þ dt 2 2 dt The motion of the ball is considered to be divided into a translational motion of the gravitational centre of the ball and its rotational motion around the gravitational centre. The translational motion is conducted on the basis of Newton’s law of motion, which is generally applied to the rigid balls. The translational motion of i-th ball is given by the following equations of motion, where m is the mass of i-th ball, g the gravitational acceleration. xi, yi and zi are x-, y- and z-coordinates of the gravitational centre of i-th ball in the Cartesian coordinate system. Fx, Fy and Fz are the resultant interactive forces, fn and fs, between i-th ball and j-th balls in x-,y- and z-directions. m

m

m

d2 x i

¼ Fx

ð6Þ

¼ Fy

ð7Þ

¼ F z mg

ð8Þ

dt 2 d2 y i dt 2

d2 zi dt 2

The rotational motion of i-th ball is given by the following equations, where I is the moment of inertia of i-th ball, xi, ci and zi denote angular velocity of i-th ball around the x-, y- and z-axis. Mx, My and Mz are the resultant moments around x-,

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y- and z-axis. I

dxi ¼ Mx dt

ð9Þ

I

dci ¼ My dt

ð10Þ

I

dzi ¼ Mz dt

ð11Þ

These are obtained from the interactive forces in the tangential direction, fs, acting on the surface between i-th ball and j-balls. Material properties and physical constants of the milling in this simulation are tabulated in Table 1. As veriﬁcation of this simulation method, we have chosen a tumbling ball mill as a model mill for grinding sample powders with balls and for the visualization of their movement. The mill consists of a cylindrical tube made of stainless steel and is sandwiched by two lids made of transparent acrylic acid resin. The cylindrical tube was 0.15 m in inner diameter and 0.153 m in length. Steel balls were made of diameter 0.025 m. Forty eight balls were charged into the mill, which corresponds to about 50% in ball ﬁlling. Rotational speed of the mill was adjusted in the range 1.1–2.2 s1. The critical speed of the mill is calculated to be 2.0 s1. Four kinds of powder samples were used as model samples in the experiment: They are kaolinite (ASP-200, Tsuchiya Kaoline Industry, Co. Ltd., Japan), aluminium hydroxide (Wako Pure Chemical Industries, Ltd., Japan), glass beads (Toshiba Barotini, Co. Ltd., Japan) and silica. 50 g of the sample was charged into the mill pot. Movement of the balls in the vicinity of the transparent mill lid was observed and recorded by a video camera. Figure 4 shows balls trajectories calculated by the simulation method under different friction coefﬁcients for the simulation of balls motion in a mill. As can be seen from the ﬁgure, the movement of balls is inﬂuenced by the friction coefﬁcient, and the rising ratio increases with an increase in this value. This result is found to correspond to the experimental one. The Table 1. Material properties and physical constants

Density of ball Normal stiffness Tangential stiffness Normal damping coefﬁcient Tangential damping coefﬁcient Diameter of ball Number of balls Time step

r Kn Ks Zn Zs d N Dt

7.91 103 1.32 1012 5.02 1011 2.43 107 1.50 107 3.0 102 48 3.0 106

(kg/m3) (N/m) (N/m) (Ns/m) (Ns/m) (m) (–) (s)

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Fig. 4. Balls trajectories in the mill at different values in friction coefﬁcient of balls. 1.8x10-3

1.5x10-3

Mill diameter Ball-filling ratio

KP [s-1]

1.2x10-3

dB [mm] 4.8 6.4 7.9 10.2 12.7 15.8 19.1 25.4 31.7

9.0x10-4

6.0x10-4

3.0x10-4

0.0 0

10

20

30 40 EW [J/(s.kg)]

50

60

70

Fig. 5. Relation between impact energy of the balls Ew and grinding rate constant Kp.

frictional coefﬁcient used in the simulation is dependent on the kind of material to be ground and could be correlated with the angle of repose of the sample [5]. Figure 5 shows the relation between the grinding rate constant, KP and the impact energy, Ew [7]. The impact energy can be calculated by using the DEM

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simulation, as follows, where, m is the mass of a ball, W is the sample mass loaded into the mill pot, and vj is the relative velocity at the collision: Ew ¼

n X 1 mv 2j 2W j¼1

ð12Þ

As can be seen in Fig. 5, the impact energy is well correlated with the grinding rate constant, irrespective of the ball diameter, ball ﬁlling ratio and pot diameter. This result indicates that the grinding rate constant could be predicted from the relation. Thus, we have proposed a method for simulating approximately the motion of balls in a tumbling ball mill in the presence of a small amount of sample powder, and as has been shown, it is based on the DEM with a suitable friction coefﬁcient of balls colliding in the mill. The three-dimensional movement of balls in the mill with the sample powder can be simulated by choosing the suitable coefﬁcient of friction of balls. The suitable coefﬁcient of friction is a key to express the motion of balls in the mill with and without sample powder and is correlated with the angle of repose of the sample powder. The impact energy calculated from the DEM simulation also would be a key factor in predicting the grinding rate constant.

5. MECHANOCHEMISTRY AND NANO-PARTICLES FORMATION BY DRY GRINDING 5.1. Mechanochemical phenomena Milling induces mechanochemical effects such as phase transformation and solid state reaction. Some amount of powder is trapped between the two when two balls collide inside a mill pot, and particles deform plastically, and are repeatedly ﬂattened, cold-welded, fractured and rewelded. The force of the impact acts on the powder particles, leading to crystallographic bonds broken and new surface being produced. The new surfaces created enable the particles to weld together easily and this leads to an increase in the rate of dissolution of solid material. With continued mechanical deformation, fragments generated by this mechanism may continue to reduce in size, and with the increase of surface energy of the particle, other profound changes affecting the surface as well as the chemical, physicochemical and structural properties may also take place. This is manifested by the presence of a variety of crystal defects such as increased number of grain boundaries, dislocations, vacancies and interstitial atoms, stacking faults and deformed and ruptured chemical bonds. The presence of this defective structure enhances the diffusivity of solute elements. Additionally, the slight rise in temperature during milling further aids the diffusion behaviour. Consequently, grinding a mixture of two or more solid substances results in micro-homogenization of

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starting components, and sometimes, it induces formation synthesis of new ﬁne ceramic powders. A typical summarization of mechanochemical phenomena is given below: 1. Formation of dislocations and point defects in the crystalline structure, 2. Mechanical activation of solid materials, 3. Polymorphic transformation, amorphization, crystallization.

5.2. Phase change Polymorphic transformations are the crystallographic structural changes from one to other phases, and this transformation does not involve an alteration of chemical composition. With an increase in milling operation, the polymorphic transformation can take place with mechanical activation. The transformation is accompanied with change in crystallite size and lattice distortion, and the crystallite size decreases and lattice distortion increases with an increase in grinding time, releasing the stored strain energy. At the same time, the lattice distortion decreases while the crystallite size is kept constant. In general, the transformation shifts from unstable to stable forms. Many reports have been published on polymorphic transformations and alterations in physical properties of the bulk phase: calcite (CaCO3) to aragonite, anatase (TiO2) to rutile, massicot (PbO) to litharge. Lin [8] has pointed out that this kind of polymorphic transformation depends largely on the conﬁguration of the pressure–temperature phase diagrams. Table 2 shows the available results on these polymorphic transformations. Mechanical activation of materials ground can be understood from various aspects. A typical example is the leaching behaviour of a speciﬁc element from Table 2. Polymorphic phase transformations of materials by the grinding

Co TiO2 ZrO2 Dy2O3 Al2O3 Fe2O3 MgCl2 GeO2 PbO CaCO3

Starting phase

Final phase

Fcc Anatase Baddaleyite c-type g g a Quartz Massicot Calcite

hcp Rutile Cubic b-type a a d Rutile Litharge Aragonite

518

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the milled sample, and this leaching enables us to effectively extract the chemical species due to the mechanical activation. This is accelerated by intense milling causing densities in dislocations, vacancies and interstitial atoms, deformed and ruptured chemical bonds. In fact, milled phosphate rock results in a much increased solubility in a 2% solution of citric acid and the mechanochemically activated sample can be used as phosphate fertilizer without any other treatments [9]. There have existed other successful examples of practical application of milling operation in metallurgical process [10–11]. Amorphization is a kind of mechanical activation induced by a milling operation, and it may be attributed to the increase in the stored energy through the grain boundaries and disordering to a high level over the amorphous state. Broadening of the X-ray diffraction (XRD) patterns is generally seen, due to drastic crystal size reduction or amorphization or both. As an example of particle size reduction, ﬁne grinding of indium oxide in the presence of alumina powder as a grinding aid enables us to extract effectively indium from some industrial wastes in its leaching operation [12]. It has been found that sub-nanometric ﬁne particles of indium oxide are produced by the grinding for 15 min; thus, milling of the indium oxide results in easy dissolution of indium in the acid leaching stage. When clay and hydrated silicate minerals are subjected to dry milling, their crystal forms are changed into dehydrated state and amorphization [13–14]. According to the XRD analysis, the OH base in the clay sample was released and caused the change in bonding states of Mg octahedral local structure in talc and serpentine, and of Al octahedral in kaolinite and pyrophyllite. These changes lead ﬁnally to the destruction of the whole crystalline structure. This implies that Mg can be dissolved into dilute acid solution when the ground talc is subjected to acid leaching, due to the formation of an amorphous phase. When the ground talc mineral is subjected to heating, re-crystallization takes place at low temperature to form enstatite. Thus, this phenomenon can be applied to improvement of ﬁne ceramic processing [15–17].

5.3. Solid-state reactions Extensive work on mechanochemical reaction has recently been seen [18–20], and it is difﬁcult to complete solid state reactions mechanically using the milling operation. It depends on the crystalline structures of both initial and ﬁnal products as well as the thermodynamic feasibility with negative change in Gibbs free energy change. Table 3 shows the solid state reactions between solid alkalis such as NaOH, KOH, LiOH and sulphate of alkaline earth elements such as calcium, strontium, barium. This reﬂects clearly a reaction induced mechanochemically depending on the negative change in Gibbs free energy [21]. The observed reactions take place

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Table 3. Gibbs free energy changes of the reactions

Reaction formula

DG298 [kJ/mol]

Results

CaSO42H2O+2LiOH ¼ Ca(OH)2+Li2SO4+2H2O SrSO4+2LiOH ¼ Sr(OH)2+Li2SO4 BaSO4+2LiOH ¼ Ba(OH)2+Li2SO4

19.25 21.99 58.95

Yes No Yes

CaSO42H2O+2NaOH ¼ Ca(OH)2+Na2SO4+2H2O SrSO4+2NaOH ¼ Sr(OH)2+Na2SO4 BaSO4+2NaOH ¼ Ba(OH)2+Na2SO4

85.59 44.70 7.75

Yes Yes Yes

138.76 96.30 59.74

Yes Yes Yes

CaSO42H2O+2KOH ¼ Ca(OH)2+K2SO4+2H2O SrSO4+2KOH ¼ Sr(OH)2+K2SO4 BaSO4+2KOH ¼ Ba(OH)2+K2SO4

at negative free energy change, while no reaction occurs when the free energy change is positive. Sulphates, tungstate, carbonate and phosphate of alkaline earth elements are very important mineral resources so that in general, it is important to extract useful components from them. The discovered reactions described above open a new route for processing Sr and Ba-sulphates and extracting W (tungsten) from Ca-tungstate [22–23]. Most cases of reaction can be achieved at negative free energy change, thus, negative free energy change is a necessary condition but not a sufﬁcient one for stimulating a mechanochemical reaction. Perovskite-type complex oxides are good examples to be used to interpret the dependence on the crystalline structures of starting materials in the solid reactivity of trivalent metal oxides, M2O3 with La2O3 to form LaMO3. It is well known that Al2O3 exists as various types of structures depending on the preparing conditions. Alpha-alumina of corundum structure does not react with La2O3 by milling, while gamma-alumina of cubic structure reacts with La2O3 to form LaAlO3 during 2 h milling with a planetary ball mill [24]. Similar results have been conﬁrmed with other trivalent M2O3 by categorizing them into two groups; one is corundum structure and the other non-corundum. It is interesting to note that oxides with non-corundum structure generally exhibit phase transformation into corundum structure when they are subjected to milling without any additives, and oxides with corundum structure remain unchanged during milling [25]. Of course, the phenomena may not be simple, and when the milling is applied to a complex system, A2O3 reacts with B2O5 to form ABO4 [26]: For example, CaO reacts with ABO4 to form Ca2ABO4, a double perovskite-type oxide, where the crystalline structures of starting materials are versatile and the phase transformation between these samples are also versatile. Our preliminary experimental results have clearly shown that the reactivity between these oxides is also closely related

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520 Table 4. Conﬁrmed MC reactions with ﬂuorides

KF+MF2 ¼ KMF3 [M ¼ Zn, Ni, Co, Fe, Mn, Ca, Mg] AF+YF3 ¼ AYF4 A ¼ K, Na, NH4 NaF+MF3 ¼ NaMF4 [M ¼ Pr, Nd, Gd, Ho, Er, Y] 3AF+GaF3 ¼ A3MF6 [A ¼ NH4, K, Na, Li]

to their crystalline structures. Ion radii and coordination numbers are particularly key factors in understanding the reaction mechanism. It is indicative when oxides tend to react with other compositions when the radius of metal ion in the oxide is large and its coordination number becomes high. This understanding may give an indication for the synthesis of new compounds and detailed work. This ﬁnding is based on the experimental results with oxides, and similar results have been obtained with other substances such as ﬂuorides, chlorides and even hydrides. Table 4 is shown for other examples on the mechanochemical syntheses of complex ﬂuorides such as ABF3, ABF4 and A3BF6 [27–28]. Recently, formation of complex hydrides ABH3 with perovskite structure has been reported for possible application to hydrogen storage [29]. It is noted that there are other routes through mechanochemical reactions between oxides and ﬂuorides to form oxyﬂuorides [30]. Mechanochemical reactions occur in systems of halogen-containing organic polymers and inorganic oxides with metal halides or oxyhalides. For example, La2O3 reacts with polytetraﬂuoroethylene (PTFE) or other ﬂuoro-polymers to form LaOF of nanoscale particles [31–32]. Oxides of alkaline earth elements can also react with these polymers with ﬂuorides, deﬂuorinating the polymers. A new process based on the mechanochemical reactions has been developed to decompose halogen-containing polymers which have a potential to generate many toxic substances when it is burned without careful operation. The process has been found to exhibit high feasibility for applications in waste treatment and environmental protection. The reaction mechanism seems quite complicated and varies with the changes in compositions of both polymers and added inorganic oxides. The decomposition behaviours of polymers with different compositions by co-grinding with CaO are examined as an effort to understand the mechanism of the decomposition reaction. As to the compositions of polymers, whether or not hydrogen and benzene rings exist is used for comparison. PTFE with a linear structure and

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hexabromobenzene (HBB) with a benzene ring are typical examples without hydrogen in the compositions. The decompositions induced by co-grinding with CaO are quite similar, with Ca-ﬂuoride and bromide as well as Ca-carbonate, carbon as the main reaction products, irrespective of the difference in the structures [33–34]. Figure 6 shows the XRD patterns of the PTFE and SrO mixture ground for different periods of time. Figure 7 shows the TG-DTA curves of the HBB and CaO mixture ground for different times. The detailed interpretations for the experimental results can be referred to from the provided references. The results shown in both ﬁgures clearly indicate the progresses of the decomposition reaction induced mechanochemically with an increase in grinding time. Polyvinyl chloride (PVC) with a linear structure and trichlorobenzene (TCB) with benzene ring are used as model samples with hydrogen in the composition. A difference in decomposition behaviours has been observed: When PVC is milled with CaO or Ca-hydroxide, decomposition of PVC takes place through dehydro-chlorination, and this is induced by cutting both hydrogen and chloride off from the chain structure. On the contrary, when TCB is subjected to milling with CaO, effective decomposition is observed, but the mechanism is different from the case of PVC and CaO, and it is induced through dechlorination, rather than dehydrochlorination [35]. The more impressive difference is that no observable decomposition is obtained with Ca-hydroxide. With the dehydrochlorination mechanism, both oxide and hydroxide exhibit high ability to react with polymers based on their neutralizing ability, while dechlorination of TCB proceeds in a different way. It is found that charge separation, radical formation and charge

Sr F2 P TFE

30 min 40 min

60 min

240 min 10

20

30 40 2θ [degree, C u K α]

50

60

Fig. 6. XRD patterns of the PTFE–SrO mixture ground for different periods of time.

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0

Weight change [%]

-20

-40

-60

-80

HBB

3h

1h

4h

2h

6h

-100 200

400 600 Temperature [°C]

800

1000

Fig. 7. TG curves of the HBB–CaO mixture ground for different periods of time.

transfer from inorganic to organic are necessary for the disconnection of C–Cl bonding, and these occur with the CaO sample but not with Ca-hydroxide. Similar phenomena happen with other oxides when hydroxides of magnesium, aluminium and lanthanum are used [36]. Similar results have been obtained from the mechanochemical treatment of monochlorobiphenyl (BP-Cl), a model sample of very toxic polychlorinated biphenyl (PCB) [37]. Figure 8 shows the ESR spectra of the 6h-ground mixtures of BP-Cl+CaO, BP-Cl+CaO+SiO2 and BP-Cl+Ca(OH)2+SiO2. There are several peaks observed when CaO is ground itself, and unpaired electrons are produced in the ground sample. When BP-Cl is ground with CaO, quite different patterns with high intensity are seen in the ESR spectrum. The addition of SiO2 does not change the pattern of the ESR spectrum except for the intensity of peaks. The sharp peak is assigned to a trapped electron (e) in an oxygen vacancy on the CaO powder surface and the broad signal is attributed to aromatic hydrocarbon radicals [38]. On the other hand, when BP-Cl is ground with Ca(OH)2, there is no ESR signal observed. The effective dechlorination is achieved with CaO where ESR signal is detected. On the other hand, no effective dechlorination is achieved with Ca(OH)2 where none of the ESR signal is observed. It can be deduced that the dechlorinating reaction is closely related to radical formation. This information is particularly useful for applications involving waste materials containing halogens and, in particular, PCB and dioxins. Many other reactions are induced mechanically, but it is still very difﬁcult to interpret all the results by a common concept. It is hoped that the understanding of the mechanism of disordering and dissociation by milling operations may help

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CaO + BP-Cl

CaO + SiO2 + BP-Cl

Ca(OH)2+ SiO2 + BP-Cl

3450

3500 Magnetic field

3550

Fig. 8. ESR spectra of the three kinds of mixture ground for 6-h.

to realize the nature of mechanochemical reactions and to design new reactions for further applications.

5.4. Material processing It is important to choose carefully starting materials with crystalline structures when we apply mechanochemical reactions to material synthesis and processing. We can expect synthesis of many functional materials including oxides, halides, sulphides, hydrides and oxyhalides by milling operations, and the effect makes the sample change into a kind of ideal precursor for material processing. Milling a mixture promotes the mixing state and becomes a well mixed precursor, so that it allows a formation of target sample by heating at lower temperature than that for the unground mixture. Thus, materials with ﬁne grains (on the nano-scale) can be prepared by low-temperature heating. Other advantages of low-temperature heating achieved by milling are the prevention of appearances of intermediate phases even with complicated compositions and obvious volatilization loss where volatile compositions are necessarily used for materials processing or the depression of grain growth for the cases where small grains are required. A good example is the preparation of single-phase perovskite 0.9Pb(Mg1/3Nb2/3)O3-0.1PbTiO3 from a stoichiometric mixture of starting materials to a precursor [39]. A stoichiometric mixture of PbO, TiO2, Mg(OH)2, and Nb2O5 was milled for 60 min and heated at temperatures as low as 8501C for 4 h

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to obtain a single phase without evident evaporation of PbO and formation of intermediate phase of pyrochlore. In contrast, poor densiﬁcation and coexistence of the pyrochlore phase were observed in the samples from the unground mixture. Another attraction from the mechnaochemical treatment is to dope oxides with non-metallic elements such as nitrogen, sulphur and ﬂuorine [40–41]. Normally, oxide is more stable than the corresponding nitride, sulphide and ﬂuoride, and it is difﬁcult to dope these elements. High-temperature heating leads to decomposition and low-temperature heating gives insufﬁcient doping into the oxide when milling is not applied. On the contrary, milling oxide with sulphur enables us to dope into the structure. In fact, a sulphur-doped TiO2 sample in rutile structure has been synthesized by milling the mixture of TiO2 and sulphur. The bonding force between Ti–S is enhanced by heating the milled sample at 4001C in an inert gas ﬂow, and the product has shown high photocatalytic reactivity under visiblelight irradiation [42]. Figure 9 shows the relationship between the wavelength of the light source and the remaining percentage of NO gas for the S-doped TiO2 samples (b) and (c), and the sample (a) as a reference. It is clearly observed that the reference sample (a) exhibits low photocatalytic activity by irradiation of light with wavelength over 510 nm. On the contrary, about 20% NO has been removed for sample (b), while nearly 40% NO is decomposed for sample (c). This means that sample (c) has a high photo-reactivity under visible-light irradiation. These results indicate that mechanochemical method allows an easy preparation of rutile phase TiO2 doped with sulphur, with high photocatalytic activity under visible-light illumination. Similarly, the N-doped TiO2 samples can be synthesized > 510 nm

100

> 400 nm

> 290 nm

a

Residual NO concentration [%]

b 80 c 60

40

20

0

0

10

20

30

Time [min]

Fig. 9. Decomposition behaviours of NO gas over the S-doped TiO2: (a) 120 min ground TiO2; (b) co-grinding with sulphur for 20 min; (c) co-grinding with sulphur for 120 min.

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from a mixture of TiO2 and nitrogen-containing compounds by the same doping method [43]. Of course, this mechanochemical approach can be applied to the non-metal element doping to other oxides or even complex oxides. Amorphization induced by milling has provided wide applications in materials processing: Preparation of solid state electrolytes is particularly worth discussing. Investigations done by the Tatsumisago group have shown that a high lithium ion conducting glass can be prepared in the Li2S-based sulphide and oxysulphide systems with a wide range of compositions by a mechanochemical procedure [44–45]. For example, amorphous materials in the systems Li2S–SiS2 and Li2S–SiS2–Li4SiO4 were synthesized by a mechanical milling (MM) technique, from crystalline starting materials at room temperature. These amorphous materials prepared by the MM technique exhibited high lithium ion conductivities in the order of 10–4 S/cm at room temperature. Other systems such as 80SnS 20P2S5 [mol%], 80Li2S (20 – x)P2S5 xP2O5 [mol%], 75Li2S xP2S3 (25 – x)P2S5 [mol%] and Li4.4GexSi1–x alloys have been reported to exhibit similar behaviours as solid state electrolytes with high lithium ion conductivity. Besides the solid state electrolyte preparation, there have been reports on the application of the mechanochemical method to the preparation of negative electrode materials for rechargeable lithium cells. For example, oxide glassy powders in the systems SnO–B2O3–P2O5, SnO–B2O3, and SnO–P2O5 prepared by MM technique exhibit high speciﬁc capacity for lithium secondary batteries [46]. Mechanical activation has been applied to materials for hydrogen storage, a highly attractive topic in the ﬁeld of renewable energy. Mechanical alloying such as formation of Mg2Ni has been fully studied and alloying needs a special section. A cautious approach must be given to some compounds. For example, it has been reported that heating the milled mixture of LiNH2 and LiH emits hydrogen gas together with Li2NH formation, while NH3 gas is emitted for the mixture without milling treatment. Mechanical activation changes the decomposition mechanism and makes the LiNH2 and LiH mixture a potential source for hydrogen storage [47]. Similar phenomena have been observed with other hydride systems. The effect of mechanochemical processing on the morphology and performance of a Ni–YSZ cermet anode is another contribution to energy-related materials. A Ni–YSZ cermet has been usually used as an anode material for SOFC. In this case, the electrochemical activity of the cermet anode strongly depends on a three-phase boundary composed of Ni grains, YSZ grains and pores. It has been reported [48] that the mechanochemical processed powder achieved better homogeneity of NiO and YSZ particles, where submicron NiO particles were covered with ﬁner YSZ particles. A Ni–YSZ cermet anode fabricated from the NiO–YSZ composite particles showed the porous structure in which Ni and YSZ grains of less than several hundred nanometres as well as micron-size pores were uniformly dispersed. The cermet anode achieved high electrical

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performance at low temperature operation (o800 1C) and lower values of the electrochemical polarization. It has been demonstrated that the activation of chemical reactions by milling can lead to many interesting applications in material processing, from ﬁne ceramic particles of advanced materials with novel microstructures and enhanced mechanical properties to energy-related materials. More and more interesting reports will come out and much contribution to material processing can be expected.

5.5. Formation of nano-particles There are two routes for producing nanoparticles using grinding operation: (a) grinding a single phase powder and controlling the balance point between fracturing and cold welding, so that particles larger than 100 nm will not be excessively cold-welded; and (b) producing nanoparticles using mechanochemical reactions. The former includes bead milling and other elaborate efforts. It has been reported that a number of transition metals and ceramics, such as Fe, Cu, Co, Ni, Al2O3, ZrO2, Fe2O3, Gd2O3, CeO2, Ce2S3, and ZnS have been produced by the latter method involving the mechanical activation of solid-state displacement reactions [49–51]. Grinding of precursor powders leads to the formation of a nanoscale composite structure of the starting materials that react during grinding or subsequent heat treatment to form a mixture of separated nanocrystals of the desired phase within a soluble salt. For example, ultraﬁne ZnO powder was synthesized by the grinding and subsequent heat treatment of a ZnCl2 and Na2CO3 mixture [52]. The displacement reaction, ZnCl2+Na2CO3-ZnO+ 2NaCl+CO2, was induced in a steady-state manner during milling, forming ZnO nanoparticles within a NaCl surrounding substance. Removal of the NaCl by-product with a simple washing with water resulted in separated ZnO particles of about 5 nm in size. A similar method has been applied to the synthesis of complex oxides such as LaCoO3 [53].

Nomenclature

d Fx Fy Fz fn fs g

diameter of a ball (m) resultant interactive force in x-direction (N) resultant interactive force in y-direction (N) resultant interactive force in z-direction (N) interactive force in the normal direction (N) interactive force in the tangential direction (N) gravitational acceleration (m/s2)

Fine Grinding of Materials in Dry Systems and Mechanochemistry

I Kn Ks Mx My Mz m R T t un us W xi yi zi r zi Zn Zn m xi f ci

527

moment of inertia (kg m2) normal stiffness (N/m3/2) tangential stiffness (N/m) resultant moment around x-axis (N m) resultant moment around y-axis (N m) resultant moment around z-axis (N m) mass of a ball (kg) rising ratio of a ball (–) milling time (min) time, simulation time (s) relative displacement of the gravitational centre in the normal direction (m) relative displacement of the gravitational centre in the tangential direction (m) sample load (kg) x-coordinates of the gravitational centre of i-th ball (m) y-coordinates of the gravitational centre of i-th ball (m) z-coordinates of the gravitational centre of i-th ball (m) density of ball (kg/m3) angular velocity of i-th ball around z-axis (rad) normal damping coefﬁcient (N s/m) tangential damping coefﬁcient (N s/m) coefﬁcient of friction (–) angular velocity of i-th ball around x-axis (rad) relative angular displacement (rad) angular velocity of i-th ball around y-axis (rad)

REFERENCES [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]

P.A. Cundall, Geotechnique 29 (1979) 47–65. B.K. Mishra, R.K. Rajamani, KONA 8 (1990) 92–98. H. Ryu, H. Hashimoto, F. Saito, R. Watanabe, MMIJ, Japan 108 (1992) 549–555. T. Yokoyama, K. Tamura, G. Jimbo, Kagaku-Kogaku Ronbunshu 17 (1991) 1026–1033. J. Kano, N. Chujo, F. Saito, Adv. Powder Technol. 8 (1997) 39–51. J. Kano, F. Saito, Powder Technol. 98 (1998) 166–170. J. Kano, H. Mio, F. Saito, M. Miyazaki, Miner. Eng. 14 (2001) 1213–1223. I.J. Lin, S. Navid, Mater. Sci. Eng. 39 (1979) 193–209. H.H. Lim, R.J. Gilkes, P.G. McCormick, Nutr.Cycl. Agroecosys. 67 (2003) 177–186. Q. Zhang, F. Saito, Hydrometallurgy 45 (1997) 323–331. J. Ficeriova, P. Balaz, E. Boldizarova, Int. J. Miner. Proces. 76 (2005) 260–265. Y. Murakami, D. Shindo, Q. Zhang, F. Saito, Mater. Sci. Eng. A 332 (2002) 64–69. J.M. Filio, K. Sugiyama, F. Saito, Y. Waseda, Powder Technol. 78 (1994) 121–127. I. Sondi, M. Stubicar, V. Pravdic, Colloid. Surf. A 127 (1997) 141–149. A. Ahniyaz, T. Fujiwara, S.W. Song, M. Yoshimura, Solid State Ionics 151 (2002) 419–423. O.K. Tan, W. Cao, Y. Hu, W. Zhu, Solid State Ionics 172 (2004) 309–316. Y. Kameshima, M. Irie, A. Yasumori, Solid State Ionics 172 (2004) 185–190.

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[18] E.M. Gutman (Ed.), Mechanochemistry of Solid Surfaces, World Scientiﬁc, Singapore River Edge, NJ, 1994. [19] V.V. Boldyrev (Ed.), Reactivity of Solids: Past, Present, and Future, Cambridge Blackwell Science, 1996. [20] P. Balaz, J. Mater. Sci. 39 (2004) 5097–5102. [21] Q. Zhang, F. Saito, Adv. Powder Technol. 8 (1997) 129–136. [22] Q. Zhang, F. Saito, J. Chem. Eng. Japan 30 (1997) 724–727. [23] Q. Zhang, F. Saito, Chem. Eng. J. 66 (1997) 79–82. [24] Q. Zhang, F. Saito, J. Am. Ceram. Soc. 83 (2) (2000) 439–441. [25] Q. Zhang, J. Lu, J. Wang, F. Saito, J. Mater. Sci. 39 (2004) 5527–5530. [26] T. Tojo, Q. Zhang, F. Saito, J. Solid State Chem. 179 (2006) 433–437. [27] J. Lee, Q. Zhang, F. Saito, Chem. Lett. (2001) 700–701. [28] J. Lu, Q. Zhang, F. Saito, Chem. Lett. (2002) 1176–1177. [29] K. Ikeda, Y. Kogure, Y. Nakamori, Scripta Mater. 53 (2005) 319–322. [30] J. Lee, Q. Zhang, F. Saito, J. Am. Ceram. Soc. 84 (2001) 863–865. [31] J. Lee, Q. Zhang, F. Saito, J. Alloys Compd. 348 (2003) 214–219. [32] J. Lee, Q. Zhang, F. Saito, Indus. Eng. Chem. Res. 40 (2001) 4785–4788. [33] Q. Zhang, H. Matsumoto, F. Saito, Chem. Lett. (2001) 148–149. [34] Q. Zhang, H. Matsumoto, F. Saito, M. Baron, Chemosphere 48 (2002) 787–793. [35] Y. Tanaka, Q. Zhang, F. Saito, J. Phys. Chem. B 107 (2003) 11091–11097. [36] Y. Tanaka, Q. Zhang, F. Saito, T. Ikoma, S. Tero-Kubota, Chemosphere 60 (2005) 939–943. [37] Q. Zhang, F. Saito, T. Ikoma, S. Tero-Kubota, K. Hatakeda, Environ. Sci. Technol. 35 (2001) 4933–4935. [38] T. Ikoma, Q. Zhang, F. Saito, K. Akiyama, S. Tero-Kubota, T. Kato, Bull. Chem. Soc. Japan 74 (2001) 2303–2309. [39] J.G. Baek, T. Isobe, M. Senna, J. Am. Ceram. Soc. 80 (1997) 973–981. [40] J. Wang, S. Yin, Q. Zhang, F. Saito, T. Sato, J. Mater. Chem. 13 (2003) 2348–2352. [41] J. Wang, S. Yin, M. Komatsu, Q. Zhang, F. Saito, T. Sato, Appl. Catal. B: Environ. 52 (2004) 11–21. [42] Q. Zhang, J. Wang, S. Yin, T. Sato, F. Saito, J. Am. Ceram. Soc. 87 (2004) 1161–1163. [43] S. Yin, Q. Zhang, F. Saito, T. Sato, Chem. Lett. 32 (4) (2003) 358–359. [44] M. Tatsumisago, Solid State Ionics 175 (2004) 13–18. [45] M. Tatsumisago, Electrochemistry 69 (2001) 793–797. [46] A. Hayashi, M. Nakai, M. Tatsumisago, J. Electrochem. Soc. 150 (2003) A582–A587. [47] P. Chen, Z.T. Xiong, J.Z. Luo, Nature 420 (6913) (2002) 302–304. [48] T. Fukui, K. Murata, S. Ohara, H. Abe, M. Naito, K. Nogi, J. Power Sources 125 (2004) 17–21. [49] P.G. McCormick, T. Tsuzuki, Mater. Sci. Forum 386-3 (2002) 377–386. [50] P.G. McCormick, T. Tsuzuki, J.S. Robinson, Adv., Mater. 13 (2001) 1008–1010. [51] T. Tsuzuki, P.G. McCormick, Nanostruct. Mater. 12 (1999) 75–78. [52] T. Tsuzuki, P.G. McCormick, Scripta Mater. 44 (2001) 1731–1734. [53] T. Ito, Q. Zhang, F. Saito, Powder Technol. 143–144 (2004) 170–173.

CHAPTER 12

Comminution Energy and Evaluation in Fine Grinding Yoshiteru Kanda and Naoya Kotake Yamagata University, 4-3-16 Jonan,Yonezawa,Yamagata 992-8510, Japan Contents 1. Introduction 2. Laws of comminution energy 2.1. Laws of comminution energy 2.1.1. Rittinger’s law 2.1.2. Kick’s law 2.1.3. Bond’s law 2.1.4. Holmes’s law 3. Crushing of single particles 3.1. Fracture of spheres 3.2. Variation of strength with particle size 3.3. Variation of fracture energy with particle size 4. Crushing resistance and grindability 4.1. Hardgrove grindability index 4.2. Bond’s Work index 4.3. Grindability in ﬁne grinding 5. Ball mill grinding 5.1. Variation of optimum grinding condition with rotational mill speed 5.2. Rate constant of feed size reduction 5.3. Expression of ﬁne grindability References

529 530 530 530 530 531 531 532 532 533 533 535 535 535 536 539 540 542 546 550

1. INTRODUCTION Comminution is the oldest mechanical unit operation for size reduction of solid materials and an important operation in the ﬁeld of mineral processing, the ceramic industry, the electronics industry and so on. The purposes for comminution are to liberate minerals for concentration processes, to reduce the size, to increase the surface area, and to free the useful materials from their matrices. There are also more recent technologies, resulting in the need to modify the Corresponding author. Tel.:+81 238 26 3163; Fax: +81 238 26 3414; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12015-7

r 2007 Elsevier B.V. All rights reserved.

530

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surface of solids, prepare composite materials, and to recycle the useful components of industrial wastes. Comminution has a long history, but it is still difﬁcult to control the mean particle size and its distribution. Hence, fundamental analysis and optimum operation have been investigated. A demand for ﬁne or ultra-ﬁne particles is increasing in many kinds of industries. The energy efﬁciency of comminution is very low and the energy required for comminution increases with a decrease in feed or produced particle size. Research and development to ﬁnd energy saving and the energy required in comminution processes have been performed.

2. LAWS OF COMMINUTION ENERGY In design, operation and control of comminution processes, it is necessary to correctly evaluate the comminution energy of solid materials. In general, the comminution energy (i.e., the size reduction energy) is expressed as a function of the particle size of feed and product [1].

2.1. Laws of comminution energy 2.1.1. Rittinger’s law Rittinger assumes that the energy consumed is proportional to the produced fresh surface. The speciﬁc surface area is inversely proportional to the particle size, then the speciﬁc comminution energy E/M is given by E ¼ CR ðSp Sf Þ M

ð1Þ

where Sp and Sf are the speciﬁc surface areas of product and feed, respectively and CR is a constant which depends on the characteristics of the materials.

2.1.2. Kick’s law Kick assumes that the energy required for comminution is related only to the ratio between the size of the feed particle and the product particle: E xf ¼ CK ln ð2Þ M xp where xp and xf are the particle sizes of product and feed, respectively and CK is constant. Equation (2) can be derived by assuming that the strength is independent of the particle size, the energy for size reduction is proportional to the volume

Comminution Energy and Evaluation in Fine Grinding

531

of particles, and the ratio of size reduction is constant at each stage of size reduction.

2.1.3. Bond’s law Bond [2] suggests that any comminution process can be considered to be an intermediate stage in the breakdown of a particle of inﬁnite size to an inﬁnite number of particles of zero size. Bond’s theory states that the total work useful in breakage is inversely proportional to the square root of the size of the product particles: 10 10 W ¼ W i pﬃﬃﬃﬃ pﬃﬃﬃﬃ ð3Þ P F where W (kWh t1) is the work input and F and P are the particle size in microns at which 80% of the corresponding feed and product passes through the sieve. Wi (kWh t1) is generally called Bond’s Work index. The Work index is an important factor in designing comminution processes and has been widely used.

2.1.4. Holmes’s law Holmes [3] proposes a modiﬁcation to Bond’s law, substituting an exponent r, in place of 0.5 in equation (3) as follows: 10 10 W ¼ Wi ð4Þ Pr Fr Values of r, which Holmes determined for materials, are tabulated in Table 1 [4]. Table 1. Values of r which Holmes determined

Material

Holmes exponent, r

Amygdaloid Malartic Springs Sandstone Morenci East Malartic Chino Nevarda consolidated Real Del Monte La Luz Kelowna exploratory Utah Copper

0.25 0.40 0.53 0.66 0.73 0.42 0.65 0.57 0.34 0.39 0.50

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3. CRUSHING OF SINGLE PARTICLES In principle, the mechanism of size reduction of solids is based on the fracture of a single particle and its accumulation during comminution operations.

3.1. Fracture of spheres In a system composed of an elastic sphere gripped by a pair of rigid parallel platens, the load–deformation curve can be predicted by the theories of Hertz as summarized by Timoshenko and Goodier [5]. The elastic strain energy, E (J), input to a sphere up to the instant of fracture is given by the integral of the load acting through the deformation: 2=3 1 v2 E ¼ 0:832 x 1=3 P 5=3 Y

ð5Þ

where Y (Pa) is Young’s modulus, v (–), Poisson’s ratio, x (m) the diameter of the sphere (particle size), and Pn (N) is the fracture load. In this system, the stress analysis concerning the compression of point loading on spherical specimens was conducted by Hiramatsu et al. [6]. The tensile strength S (Pa) of the specimen is given by S¼

2:8P px 2

ð6Þ

Substituting equation (6) into equation (5), the speciﬁc fracture energy E/M (J kg1) is given by 2=3 E 1 v2 ¼ 0:897r1 p2=3 S5=3 M Y

ð7Þ

where r (kg m3) is the density of sphere. The relationship between the speciﬁc fracture energy and the strength for borosilicate glass and feldspar are shown in Fig. 1 [7]. However, when two spherical particles, 1 and 2, collide with each other, the maximum stress Smax, generated inside the particles is expressed by a function of particle size, x, relative velocity, u(m s1) and mechanical properties [8]: Smax ¼ 0:628

m1 m2 m1 þ m2

1=5

u2=5

2 2 þ x1 x2

3=5

1 v 21 1 v 22 þ Y1 Y2

4=5

where m1 (kg) and m2 (kg) are the mass of particles 1 and 2, respectively.

ð8Þ

Comminution Energy and Evaluation in Fine Grinding

103

E/M[J/kg]

5

533

Borosilicate glass X=0.5cm 1.0 1.5 2.0 2.5 3.0

102 5

Feldspar X=1.0cm 1.5 2.0 2.5

10 5

1 106

5

107

5

108

5

S[Pa]

Fig. 1. Relationship between strength, S and speciﬁc fracture energy, E/M (x is diameter of sphere).

3.2. Variation of strength with particle size Strength is a structure-sensitive property and changes with specimen volume. From a statistical consideration of the distribution of the presence of minute ﬂaws [9], Weibull [10], and Epstein [11] showed that the mean strength of the specimen, S, is proportional to the (1/m) power of the specimen volume, V (m3): 1=m

S ¼ ðS0 V 0 ÞV 1=m

ð9Þ

where S0 (Pa) is the strength of unit volume V0 (m3) and m is Weibull’s coefﬁcient of uniformity. Experimental data lines determined by the least-squares method for quartz are shown in Fig. 2 [12].

3.3. Variation of fracture energy with particle size From equations (7) and (9), the relationship between speciﬁc fracture energy, E/M or fracture energy of a single particle, E, and particle size, x is obtained as follows: 2=3 E 1 v2 1=m 5=3 5=m ¼ 0:897ð6Þ5=3m r1 pð2m5Þ=3m S0 V 0 x M Y

ð10Þ

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109

5

10-2

X [cm] 10-1

5

5

1

5

S [Pa]

m=2.57

10

Quartz

108

m=21.3

m=6.49

107

106 -8 10 10-7 10-6 10-5 10-4 10-3 10-2 10-1 V [ cm3 ]

1

m=4.56

10

102

103

Fig. 2. Variation of strength, S with volume of specimen, V.

10-1

10-2

Feldspar

10-3

10-4

103

E [J]

E/M [J/kg]

104

Marble 102

10-5

10

10-6 Gypsum

1 5 3 10-2

10-1 X [ cm]

10-7 1

10

Fig. 3. Relationship between particle size, x and speciﬁc fracture energy, E/M, or fracture energy, E.

E ¼ 0:15ð6Þ

5=3m ð5m5Þ=3m

p

1 v2 Y

2=3

1=m

ðS0 V 0 Þ5=3 x ð3m5Þ=m

ð11Þ

The calculated result for feldspar, marble, and gypsum are shown in Fig. 3 [12]. It is important to note that the speciﬁc fracture energy increases rapidly for smaller particle size less than approximately 100–500 mm: namely the requirement of large amounts of energy in ﬁne or ultra-ﬁne grinding can be presumed. The strength and the speciﬁc fracture energy increase also with an increase in loading rate [13].

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4. CRUSHING RESISTANCE AND GRINDABILITY Importance of crushing resistance or grindability of solid materials and energy efﬁciency have been recognized as key parameters in optimising comminution processes in a variety of industries. The grindability is obtained from a strictly deﬁned experiment; two typical methods include:

4.1. Hardgrove grindability index The machine to measure the grindability consists of a top-rotating ring with eight balls of 1 inch. diameter. A load of 6470.5 lb (pounds) is applied on the top rotating ring. Fifty grams of material sieved between 1.19 and 0.59 mm is ground for the period of 60 revolutions. The Hardgrove Grindability Index (HGI) is deﬁned as HGI ¼ 13 þ 6:93w

ð12Þ

where w (g) is the mass of ground product ﬁner than 75 mm.

4.2. Bond’s Work index Bond’s Work index Wi deﬁned in equation (3) [14] is given by Wi ¼

0:82 P 0:23 1 Gbp

1:1 44:5 pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ 10= P 0 10= F

ð13Þ

where P1 is the sieve opening in microns for test grindability, Gbp (g rev1) is the ball mill grindability, P0 is the product size in mm (80% of product ﬁner than size P1 passes) and F is the feed size in mm (80% of feed passes). A standard ball mill is 12 in. (305 mm) in internal diameter and 12 in. in internal length charged with 285 balls as tabulated in Table 2. Table 2. Composition of steel balls for measurement by Bond’s Work index

Diameter (mm)

Number of balls

36.5 30.2 25.4 19.1 15.9

43 67 10 71 94 285 (Total)

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The lowest limit of the total mass of balls is 19.5 kg. The amount of feed material is 700 cm3 bulk volume, composed of particles ﬁner than 3,360 mm. The mill is rotated for a number of times so as to yield a circulating load of 250% at 70 rev min1, where the circulating load is deﬁned as the component ratio of the oversize to the undersize. The process is continued until the net mass of undersize produced per revolution becomes constant Gbp in equation (13). Table 3 shows Work indexes measured by wet process [15]. In ﬁne grinding, when P in equation (3) is smaller than 70 mm, the Work index, Wi, is multiplied by a factor f to account for the increased work input. The factor f is found from the following empirical equation [16]: f ¼

P þ 10:3 ðP 70 mmÞ 1:145P

ð14Þ

Bond [16] proposed a relationship between Work index, Wi, and Hardgrove grindability index (HGI): Wi ¼

435 ðHGIÞ0:91

ð15Þ

4.3. Grindability in ﬁne grinding When the particle size of products is submicron or micronized, it will be difﬁcult to estimate the comminution energy by equations (3), (13) and (14). Bond [2] had proposed equation (16) for measurement of Wi before equation (13) P 1 0:5 0:82 W i ¼ 1:1 16 Gbp ð16Þ 100 Equation (16) is simpler than (13). There was not a great difference [17] between Wi, calculated by equation (13) and Wi, by equation (16). Figure 4 shows the relationship between mass fraction, Qxc (–), ﬁner than a particle size, xc (mm), and grinding time, t (min), in ball mill grinding of silica glass [18]. In early stage of grinding, a zero-order increasing rate is applicable as shown in following equations: Qxc ¼ k xc t

ð17Þ

W xc ¼ Qxc W s ¼ k xc W s t

ð18Þ

where Wxc is the mass of product ﬁner than a size xc and Ws, the mass of feed. From equations (16), (17) and (18), the following equations can be obtained: 0:82 0:82 W i / P 0:5 / x 0:5 1 Gbp c ðk xc W s Þ

ð19Þ

Comminution Energy and Evaluation in Fine Grinding

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Table 3. Average work indexes

Average Material

Number tested

Speciﬁc gravity

Work index

All materials tested Andesite Barite Basalt Bauxite Cement clinker Cement raw material Coke Copper ore Diorite Dolomite Emery Feldspar Ferro-chrome Ferro-manganese Ferro-silicon Flint Fluorspar Gabbro Glass Gneiss Gold ore Granite Graphite Gravel Gypsum rock Iron Ore Hematite Hematite- specular Oolitic Magnetite Taconite Lead ore Lead-zinc ore Limestone Manganese ore

1211 6 7 3 4 14 19 7 204 4 5 4 8 9 5 13 5 5 4 4 3 197 36 6 15 4

2.84 4.50 2.91 2.20 3.15 2.67 1.31 3.02 2.82 2.74 3.48 2.59 6.66 6.32 4.41 2.65 3.01 2.83 2.58 2.71 2.81 2.66 1.75 2.66 2.69

14.42 18.25 4.73 17.10 8.78 13.56 10.51 15.18 12.73 20.90 11.27 56.70 10.80 7.64 8.30 10.01 26.16 8.91 18.45 12.31 20.13 14.93 15.05 43.56 16.06 6.73

56 3 6 58 55 8 12 72 12

3.55 3.28 3.52 3.88 3.54 3.45 3.54 2.65 3.53

12.93 13.84 11.33 9.97 14.60 11.73 10.57 12.54 12.20

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Table 3. Continued Average Material Magnesite Molybdenum ore Nickel ore Oilshale Phosphate rock Potash ore Pyrite ore Pyrrhotite ore Quartzite Quartz Rutile ore Shalc Silica sand Silicon carbide Slag Slate Sodium silicate Spodumene ore Syentite Tin ore Titanium ore Trap rock Zinc ore

Number tested 9 6 8 9 17 8 6 3 8 13 4 9 5 3 12 2 3 3 3 8 14 17 12

Speciﬁc gravity

Work index

3.06 2.70 3.28 1.84 2.74 2.40 4.06 4.04 2.68 2.65 2.80 2.63 2.67 2.75 2.83 2.57 2.10 2.79 2.73 3.95 4.01 2.87 3.64

11.13 12.80 13.65 15.84 9.92 8.05 8.93 9.57 9.58 13.57 12.68 15.87 14.10 25.87 9.39 14.30 13.50 10.37 13.13 10.90 12.33 19.32 11.56

0:82 W i;c ¼ x 0:5 c ðk xc W s Þ

ð20Þ

Wi,c is proportional to Wi which was proposed by Bond. Wi,c could be estimated by the examination of the zero-order increasing rate constant of the mass fraction less than a sieving size using an arbitrary ball mill. Figure 5 shows the relationship between sieving size, xc and Wi,c for silica glass [18]. It was presumed that the Work index could be approximately constant to a sieving size of 20 mm and increased in the range of a size less than 20 mm. It was also found that large amounts of energy are necessary to produce ﬁne or ultra-ﬁne particles. Similar results have been reported for other solids in ball, vibration, and planetary mill grinding, or wet grinding by ball mills [19–22].

Comminution Energy and Evaluation in Fine Grinding

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Fig. 4. Variation of mass fraction ﬁner than size xc, Qxc with grinding time, t.

Fig. 5. Variation of corresponding Work index, Wi,c with sieving size, xc.

5. BALL MILL GRINDING Comminution processes generally consist of several stages in series. Various types of crushing and grinding equipments have been used industrially as a mechanical way of producing particulate solids. The working phenomena in these equipments are complex and different principles are adopted in the loading, such as compression, shear, cutting, impact, and friction; in the mechanism of force transmission or the mode of motion of grinding media, such as rotation, reciprocation, vibration, agitation, rolling, and acceleration due to ﬂuids; and in the operational method, such as dry, wet or grinding aid system, batch or continuous operation and so on. However, in practice, it is most common to classify comminution processes into four stages by the particle size produced. Although the sizes are not clearly deﬁned, they are called primary, intermediate, ﬁne, and ultraﬁne according to the size of the ground product.

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On the basis of the above classiﬁcation, ﬁne grinding equipments produce particles ﬁner than about 10 mm. There are many kinds of equipments in this category. They are roughly classiﬁed into three types: ball-medium type, medium agitating type, and ﬂuid-energy type. Ultra-ﬁne grinding equipments produce particles ﬁner than 1 mm. The ball mills are widely using in ﬁne and ultra-ﬁne grinding equipment. In a ball mill, the grinding energy is transferred to materials through media such as balls, rods, and pebbles by moving the mill body. A tumbling mill or a ball mill is most widely used in both wet and dry systems, in batch and continuous operations, and on small and large scales. The optimum rotational speed is usually set at 65–80% of critical speed, Nc (rpm) when the balls are attached to the wall due to centrifugation: 42:3 N c ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Dm d b

ð21Þ

where Dm and db are the mill diameter and the ball diameter in metres, respectively. It is desirable to reduce the ball size corresponding to the smaller size of feed materials. The grindabilities of solids in ball milling are mentioned below brieﬂy.

5.1. Variation of optimum grinding condition with rotational mill speed Various laboratory studies, pilot plant works, and full scale plant works show that the mill rotational speed, which is one of the operating variables, can affect grinding efﬁciency and ﬁneness of ground products. The optimum grinding conditions in a ball mill are also affected by mill rotational speed. Grinding conditions investigated in ball mills include the feed size, the feed mass of the material, and the ball mass. These conditions are evaluated by the following equations (22) and (23), and the optimum condition is determined by maximum values of K1 and k0: dR ¼ K 1 R dt

ð22Þ

dW x ¼ k0 dt

ð23Þ

where R (–) is the mass fraction of feed size particles, Wx (g) the mass ﬁner than a size x, (x5xf), K1 (min1) the decreasing rate constant (selection function), k0 (g min1) the increasing rate constant. The material used in the experiment is quartz. The ball mill is made of alumina with an inside diameter of 144 mm and an inner volume of 2,100 cm3. The grinding ball is also alumina and of diameter 20 mm. The rotational speed of the mill is

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Fig. 6. Relationship between grinding rate constant, K1 and feed size, xf at 84 rpm.

Fig. 7. Variation of grinding rate constant, k0 with ball mass, Wb at 84 rpm.

varied between 74 and 108 rpm, which corresponds to 63–90% of the critical speed calculated by equation (21). Figure 6 shows the relationship between K1 and feed size, xf when varying feed mass, Ws and ball mass, Wb at 70% of the critical mill speed. The rate constant K1 increases with increasing feed size up to a certain size and then decreases with increasing feed size. There is an optimum feed size, feed mass, and ball mass at which K1 takes a maximum value. Figures 7 and 8 show the variation of k0 with ball mass and feed mass, respectively. Optimum conditions of ball mass and feed mass can be also found from these ﬁgures. The optimum conditions are tabulated in Table 4 when varying the mill rotational speed (the ratio of critical mill speed). From this table, the optimum feed

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Fig. 8. Variation of grinding rate constant, k0 with feed mass, Ws at 84 rpm. Table 4. Variation of optimum grinding condition with rotational mill speed

Mill rotational speed, N (rpm) Ratio of critical speed, fc (–) Feed size, xf (mm) Feed mass, Ws (g) Ball mass, Wb (g) Ball-ﬁlling volume fraction, J (–) Particle-ﬁlling volume fraction, fc (–)

75 0.63 2.01.7 280 2,800 0.62 0.10

84 0.70 2.01.7 260 2,600 0.57 0.095

95 0.79 2.01.7 230 2,300 0.51 0.084

108 0.90 2.01.7 200 2,000 0.44 0.073

J ¼ [(ball mass)/{(ball density) (10.4)}]/(mill volume). fc ¼ {(feed mass)/(bulk density)}/(mill volume). fc ¼ N/Nc.

size is constant irrespective of the mill speed, and the optimum feed mass and ball mass increase with decreasing the mill speed.

5.2. Rate constant of feed size reduction The grinding rate constant, K1 (selection function) in equation (22) has been investigated by many researchers using a wide variety of grinding mills under different conditions [23–27], and this item is of great interest when considering the grinding efﬁciency, the design of the circuit of the grinding, and classiﬁcation processes [11]. Figures 9(a)–(c) show the relationship between the mass fraction of feed size particles and grinding time in ball mill grinding (alumina mill: diameter 144 mm, volume 2,100 cm3, mill speed 108 rpm, ball mass 2,000 g, feed mass 200 g). These ﬁgures indicate that the breakage of a feed size of material follows the ﬁrst order law (equation (22)) irrespective of feed sizes, ball diameters, and materials.

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Fig. 9. First-order plots: (a) quartz, db ¼ 20 mm; (b) quartz, db ¼ 20 mm; (c) talc, db ¼ 20 mm.

Figure 10 shows the relationship between the grinding rate constant K1 and the feed size xf for quartz, with varying ball diameter. The tendency in the variation of the rate constant with feed size is independent of the ball diameter and there is an optimum feed size xm at which K1 takes a maximum value, Km. Figure 11 shows the results obtained when K1 and xf are normalized by Km and xm. The dimensionless rate constant, K1/Km and the dimensionless size, xf/xm lies fairly well along a convex curve, irrespective of the ball diameter. This relation is expressed by using equation (24) that is revised Snow’s equation [28]. K1 ¼ Km

xf xm

a

xf xm exp c xm

ð24Þ

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Fig. 10. Variation of grinding rate constant, K1 with feed size, xf (Quartz).

Fig. 11. Variation of dimensionless grinding rate constant, K1/ Km with dimensionless feed size, xf/xm (Quartz).

Figure 12 shows the relationship between the dimensionless rate constant and the dimensionless size for ﬁve kinds of solid materials. Equation (24) can follow the experimental results well with by choosing parameters, a ¼ 1.23 and c ¼ 1.08. Figures 13 and 14 show the relationship between the optimum feed size, xm and the ball diameter, db, and the relationship between the maximum value of K1, Km and the ball diameter, db, respectively for solid materials. Equations (25) and

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Fig. 12. Variation of dimensionless grinding rate constant, K1/ Km with dimensionless feed size, xf/xm for various solid materials.

Fig. 13. Relationship between optimum feed size, xm and ball diameter, db.

(26) are obtained from these ﬁgures: x m ¼ Ad Bb K m ¼ A0 d Bb

ð25Þ 0

ð26Þ

where A and B, A0 and B0 are constants for each material, respectively. Substituting equations (25) and (26) into equation (24), the grinding rate constant for the each material can be expressed by the ball diameter and the

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Fig. 14. Relationship between maximum rate constant, Km and ball diameter, db.

Table 5. Values of constants in equation (27)

Material

C1 (–)

C2 (–)

m (–)

n (–)

Silica glass Quartz Limestone Gypsum Talc

0.36 0.28 0.22 0.70 2.1

8.3 4.9 6.0 6.4 6.4

0.35 0.47 0.80 0.32 0.30

0.84 0.60 0.51 0.60 0.48

feed size: xf a K 1 ¼ C1 d m x exp C 2 n ða ¼ 1:23Þ b f db

ð27Þ

where C1, C2, m and n are constants, respectively. The values of the above constants for materials are summarized in Table 5.

5.3. Expression of ﬁne grindability The demand for ﬁne or ultra-ﬁne particles is increasing in many industries. This means that it is very important to evaluate and express the ﬁneness of the ground product and the progress of ﬁne or ultra-ﬁne grinding. There are usually two methods (or ways) to evaluate and express the ﬁne grindability of solids. The ﬁrst method consists of increasing the rate of ﬁne particles. The other consists of increasing the rate of speciﬁc surface area of ground product. Fine grinding tests were performed on silica glass particles using an alumina ball mill. Effects of the feed size and the feed mass of silica glass, and the ball

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diameter on the ﬁne grindability were investigated when the ball mass and the mill rotational speed were constant. Grinding conditions are shown in Table 6. Figure 15 shows the relationship between the speciﬁc surface area by BET method and the under size fraction at a size x by photo extinction method. From this ﬁgure, equation (28) is obtained: Sw ¼ 4:65 105 x 1:1 QaðxÞ aðxÞ ¼ 1:1x 0:12 ð28Þ x where Sw (m2 kg1) is the speciﬁc surface area and Qx (–) is the mass fraction ﬁner than a size, x (mm). Figure 16 shows the comparison of experimental results and calculated results of Sw obtained by using equation (28) for all grinding conditions. From this result, the experimental values agree approximately with the calculated results, and the validity of equation (28) is conﬁrmed in evaluating the grindability for each grinding condition. Table 6. Grinding conditions

Ball diameter, db (mm) Feed size, xf (mm) (Maximum size) Feed mass, Ws (g) Ball mass, Wb (g) Mill speed, N (rpm) Mill diameter, Dm (mm) Mill volume, Vm (cm3)

10 1.0

15 1.7

20 2.0

25 2.8

100,200 300,400

200,300 400 2,000 108 144 2,100

100,200 300,400

200

Fig. 15. Relationship between speciﬁc surface area, Sw and mass fraction ﬁner than size x, Qx (Silica glass).

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Fig. 16. Comparison of observation value and those calculated by equation (28).

Nomenclature

A A0 B B0 c C1 C2 CK CR Dm db E E/M F f fc Gbp J K1 k0 Km

constant in equation (25) (–) constant in equation (26) (–) constant in equation (25) (–) constant in equation (26) (–) constant in equation (24) (–) constant in equation (27) (–) constant in equation (27) (–) constant in equation (2) (J kg1) constant in equation (1) (J m2) mill diameter (m) ball diameter (m) comminution (fracture) energy (J) speciﬁc fracture energy (J kg1) feed size (80% of feed passes) (mm) adjustment factor (–) particle-ﬁlling volume fraction ( ¼ {(feed mass)/(bulk density)}/ (mill volume)) (–) ball mill grindability (g rev1) ball-ﬁlling volume fraction ( ¼ [(ball mass)/{(ball density) (10.4)}]/(mill volume)) (–) decreasing rate constant (selection function) (min1) increasing rate constant (g min1) maximum value of rate constant K1 (min1)

Comminution Energy and Evaluation in Fine Grinding

kxc M m m m1 m2 N n Nc P P P0 P1 Qx Qxc R r S S0 Sf Sp Sw Smax t V V0 Vm u W w Wb Wi Wi,c Ws Wx Wxc x x1 x2 xc xf xm xp Y

549

increasing rate constant (g min1) mass of particles (kg) Weibull’s coefﬁcient (–) constant in equation (27) (–) mass of particle 1 (kg) mass of particle 2 (kg) mill rotational speed (rpm) constant in equation (27) (–) critical value of mill rotational speed (rpm) fracture load in equations (5), (6) (N) product size (80% of product passes) in equation (3) (mm) product size (80% of product ﬁner than a size P1 passes) (mm) grindability test sieve opening (mm) mass fraction ﬁner than a size, x in equation (28) (–) mass fraction ﬁner than a size, xc in equation (17) (–) mass fraction of feed size particles (–) exponent in equation (4) (–) strength of the specimen (Pa) strength of unit volume V0 (Pa) speciﬁc surface areas of feed (m2 kg1) speciﬁc surface areas of product (m2 kg1) speciﬁc surface area in equation (28) (m2 kg1) the maximum stress in equation (8) (Pa) grinding time (min) specimen volume (m3) unit volume (m3) mill volume (cm3) relative velocity (m s1) work input (kWh t1) mass of ground product ﬁner than 75 mm (g) ball mass (g) Bond’s Work index (kWh t1) corresponding Work index (mm0.5 (g min1)0.82) feed mass (g) mass of ﬁner than a size x (g) mass of product ﬁner than a size xc (g) diameter of the sphere (particle size) (m), (mm) diameter of sphere particle 1 (m) diameter of sphere particle 2 (m) sieving size for evaluating grindability (mm) particle size of feed (m), (mm) optimum feed size xm at which K1 takes a maximum value, Km (mm) particle size of product (m) Young’s modulus (Pa)

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Young’s modulus of particle 1 (Pa) Young’s modulus of particle 2 (Pa)

Greek letters a fc v v1 v2 r

constant in equation (24) (–) ratio of critical speed, ( ¼ N/Nc) (–) Poisson’s ratio (–) Poisson’s ratio of particle 1 (–) Poisson’s ratio of particle 2 (–) density of sphere particle (kg m3)

REFERENCES [1] W.H. Walker, W.K. Lewis, W.H. McAdams, E.R. Gilliland, Principles of Chemical Engineering, McGraw-Hill, New York, 1937, p. 254. [2] F.C. Bond, Trans. AIME 193 (1952) 484. [3] J.A. Holmes, Trans. Inst. Chem. Eng. 35 (1957) 125. [4] G.C. Lowrison, Crushing and Grinding, Butterworth, London, 1974, p. 54. [5] S. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1951, p. 372. [6] Y. Hiramatsu, T. Oka, H. Kiyama, J. Mining Inst. Japan 81 (1965) 1024. [7] Y. Kanda, S. Sano, S. Yashima, Powder Technol. 48 (1986) 263. [8] H. Rumpt, Chem. Ing. Technol. 31 (1959) 323. [9] A.A. Grifﬁth, Proc. 1st Int. Congr. Appl. Mech, 1924, pp. 55–63. [10] W. Weibull, Ing. Vetenshaps Akad. Handle. 151 (1939) 1–45. [11] B. Epstein, J. Appl. Phys. 19 (1948) 140. [12] S. Yashima, Y. Kanda, S. Sano, Powder Technol. 51 (1987) 277. [13] S. Yashima, F. Saito, H. Hashimoto, J. Chem. Eng. Japan 20 (1987) 257. [14] F.C. Bond, Trans. AIME 217 (1960) 139. [15] F.C. Bond, Br. Chem. Eng. 6 (1961) 543. [16] F.C. Bond, Br. Chem. Eng. 6 (1961) 378. [17] T. Ishihara, J. Miner. Metal. Inst. Japan 80 (1964) 924. [18] N. Kotake, H. Shoji, M. Hasegawa, Y. Kanda, J. Soc. Powder Technol. Japan 31 (1994) 626. [19] N. Kotake, T. Yamada, M. Hareyama, Y. Kanda, J. MMIJ 114 (1998) 29. [20] N. Kotake, N. Shimoi, Y. Kanda, J. Soc. Powder Technol. Japan 35 (1998) 792. [21] N. Kotake, T. Yamada, F. Kawasaki, T. Kanda, Y. Kanda, J. Soc. Powder Technol. Japan 37 (2000) 505. [22] N. Kotake, Y. Kanda, J. MMIJ 116 (2000) 901. [23] D.F. Kelsall, K.J. Reid, C.J. Restarick, Powder Technol. 1 (1968) 291. [24] L.G. Austin, K. Shoji, P.T. Lukie, Powder Technol. 14 (1976) 71. [25] Y. Kanda, H. Gunji, H. Takeuchi, K. Sasaki, J. Soc. Mater. Sci. Japan 27 (1978) 663. [26] Q.Q. Zhao, G. Jimbo, J. Soc. Powder Technol. Japan 25 (1988) 603. [27] S. Nomura, K. Hosoda, T. Tanaka, Powder Technol. 68 (1991) 1. [28] R.H. Snow, Proc. 1st Int. Cof. Particle Technol. IITRI, Chicago, 1973, p. 28.

CHAPTER 13

Enabling Nanomilling through Control of Particulate Interfaces Marc Sommer and Wolfgang Peukert Institute of ParticleTechnology, Friedrich-Alexander-Universitaºt Erlangen-Nuºrnberg, CauerstraX e 4, D-91058 Erlangen,Germany Contents 1. Introduction 2. Particle interactions in suspensions 2.1. Van der Waals interactions 2.2. Electrostatic interactions 2.2.1. The origin of surface charges in aqueous media 2.2.2. The origin of surface charges in organic liquids 2.2.3. The electrical double layer 2.2.4. Electrostatic double layer interaction 2.3. Born interactions 2.4. Solvation, structural and hydration interactions 3. Stabilization of particles against aggregation 3.1. Introduction 3.2. Electrostatic stabilization 3.2.1. The DLVO-theory 3.3. Steric stabilization 3.4. Electro-steric stabilization 3.5. Summary of stabilization methods 3.6. Coagulation in stirred media mills 4. Inﬂuence of particle interactions on suspension rheology in stirred media mills 4.1. Suspension rheology 4.2. Rheology of electrostatically stabilized suspensions 4.3. Rheology of sterically stabilized suspensions 5. Experiments in nanomilling 6. Mechano-chemical effects during nanomilling 7. Summary References

552 552 552 554 555 556 557 560 560 561 561 561 562 562 564 572 572 574 577 578 581 583 586 593 598 601

Corresponding author. Tel.: +49-9131/85-29401; Fax: +49-9131/85-29402; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12016-9

r 2007 Published by Elsevier B.V.

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1. INTRODUCTION Nanoparticles are increasingly used in many areas of the chemical and pharmaceutical industry as well as in the ceramic and microelectronic industry. Applications for sub-micron particles are, for example, pigments, nanocomposites, drug delivery and ceramic materials. Besides the direct synthesis of these materials by chemical methods, wet grinding in stirred media mills is a suitable method for the production of sub-micron particles. The manufacturing of ﬁne particles in stirred media mills is inﬂuenced, besides machine parameters (e.g. design, function and size of the mill) and process parameters (e.g. rotational speed, ﬁlling degree, etc.) by interparticle interactions. These interactions inﬂuence the stability of the milling suspension against coagulation as well as the rheology of the suspension. Particles in the order of 1 mm and below feature a high mobility due to Brownian diffusion. This leads to a high collision frequency between the particles. If non-stabilized particles collide, agglomerates strong enough to withstand the grinding process may be formed. This effect has limited the milling process in the past. A grinding limit was postulated for particle sizes of around 0.5 mm. By producing particles smaller than a median particle size of 1 mm, a steady state between breakage and agglomeration exists in the milling process. This equilibrium is controlled by interparticle interactions as well as the milling conditions. The more the particles decrease in size the more the interparticle forces between the particles become dominant. Attractive forces lead to agglomerates when the particles collide, thus acting against the comminution process. To prevent this agglomeration process, the particles in the mill must be stabilized by increasing the repulsive forces in the suspension. The stabilization will move the steady state to smaller particle sizes. To study the grinding limits of particle sizes below 1 mm, a detailed understanding of the agglomeration process and its mechanism is needed. In the present contribution, the properties of particle surfaces in the liquid phase are discussed. Based on this, possibilities to inﬂuence the interparticle interactions and with it the stabilization of the particles are described. Closely connected to the particle–particle interactions is the ﬂowability of the suspension, which inﬂuences the energy consumption of the mill as well as the stressing mechanism of the particles between the milling beads and the amount of attrition. Finally, experimental results of nanomilling in stirred media mills are presented and mechanochemical effects during nanomilling are discussed.

2. PARTICLE INTERACTIONS IN SUSPENSIONS 2.1. Van der Waals interactions Stirred media mills can be used for the comminution of particles down to the nanometre size range [1–8]. The behaviour of particles below 10 mm is increasingly

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inﬂuenced by surface effects. The forces between particles due to van der Waals interactions can be several orders of magnitude larger than the particle weight. The van der Waals interaction affects the physical behaviour of the suspension increasingly with decreasing particle size and so inﬂuences the performance of the mill. Agglomeration of particles can occur in the mill due to the van der Waals attraction, which can inhibit the progress of the milling process. At the same time the ﬂowability of the suspension and with it the energy consumption are inﬂuenced. On the microscopic level the stress of particles between impacting milling beads is affected, too. The origin of the mostly attractive van der Waals forces is the interaction between induced dipoles, which arise due to orientational effects between molecules with permanent dipoles or induction effects in neutral particles. Dispersion forces are quantum mechanical in origin and depend on the ﬂuctuation of electron clouds. This results in an instantaneous dipole, which generates an electric ﬁeld that polarizes any other nearby atom, inducing a dipole moment in it. The resulting interaction between the two dipoles gives rise to an instantaneous force between the two atoms. In the classic calculation method according to Hamaker [9], the intermolecular forces are accumulated pairwise while retardation effects are neglected. For two spheres with radius a1 and a2 at a surface to surface distance h, the interaction FvdW becomes A 2a1 a2 2a1 a2 FvdW ¼ 2 þ 2 6 h þ 2hða1 þ a2 Þ h þ 2hða1 þ a2 Þ !# h2 þ 2hða1 þ a2 Þ þ ln 2 ð1Þ h þ 2hða1 þ a2 þ 4a1 a2 Þ In this equation, A is the material speciﬁc Hamaker constant, which is very difﬁcult to quantify precisely. Hamaker constants can be measured, e.g., from adsorption of small molecules interacting only through van der Waals forces [10,11] or by means of Lifshitz theory as a function of frequency-dependent dielectric constants. The Hamaker constants between different materials may be approximated from known Hamaker constants of the individual materials using combining relations. A frequently used combining law is given by Israelachvilli [12] pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ A123 ð A11 A33 Þ ð A22 A33 Þ ð2Þ where A123 is the non-retarded Hamaker constant for media 1 and 2 interacting across medium 3. Hamaker constants can also be approximated using the Liftshitz theory. Here, the problem of additivity is completely avoided since the atomic structure is ignored and the forces between large bodies are treated as continuous media. The Hamaker constant can be calculated using dielectric constants and refractive indices. A calculation requires that the dielectric and optical properties of the

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Table 1. Hamaker constants for some typical materials in aqueous phase [16]

Material

Hamaker constant A/1020 J

Metals Graphite Oxides

15–30 14 1–5 4.2–5.3 [12]; 3.11 [17] 2.75a [18] 4.52 [19] 1.7 [19]; 1.6a [18] 0.85 [19] 13 [12] 0.3–1.4 0.33 1.30

Al2O3 Fe2O3 SiO2 (crystalline) SiO2 (fused) ZrO2 Polymers PTFE PVC a

Calculated from full spectral data.

materials are known at all wavelengths. Since these full spectra data are difﬁcult to obtain for most materials, a number of approximation equations were developed [13–16]. Recent Lifshitz calculations for a number of solids (titania, silica, graphite) interacting across water did show that the maximal error induced by equation (2) is less than 25% [17]. In this case, the full spectra given in literature were used. The following approximation can be used, if the absorption frequencies of all three media are assumed to be the same [12] 3 1 3 2 3 A kT 4 1 þ 3 2 þ 3 ðn21 n23 Þðn22 n23 Þ 3hv e h i þ pﬃﬃﬃ 8 2 ðn2 þ n2 Þð1=2Þ ðn2 þ n2 Þð1=2Þ ðn2 þ n2 Þð1=2Þ þ ðn2 þ n2 Þð1=2Þ 1

3

2

3

1

3

2

ð3Þ

3

Table 1 gives Hamaker constants for some typical materials interacting in aqueous phase.

2.2. Electrostatic interactions Interfaces in solvents carry often electrostatic charges. The origin of these charges is discussed in the following sections. There are excellent monographs discussing the subject in much greater detail than we can do it in this contribution [19,20,24]. However, most of the work is devoted to an aqueous medium; solvents with low dielectric constant (sayo10) are very sparse. The next section summarizes only the most important topics.

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2.2.1. The origin of surface charges in aqueous media For the formation of charges in colloidal systems, the following mechanisms are known [16,21]: The dissociation of surface groups in liquids with a high dissociation constant

explains for many cases the origin of surface charges. Surfaces that carry groups, such as sulphates, carboxyl, hydroxyl and oxides will acquire charges if the free energy of the system favours their dissociation. The formation of counter ions in the liquid maintains the electrical neutrality. Solids that have some solubility in their surrounding medium can acquire charge by dissolution of either a cation or anion from the particle surface. This mechanism acts mainly for inorganic solvents in aqueous media. In case of crystals (e.g. AgJ) the charge arises by dissolution or adsorption of the crystal building ions. In dependency of the composition of the solvent Ag+ or J-ions dissolve from the surface. This leads to a negative (in case of Ag+-dissolution) or a positive (in case of J-dissolution) surface charge. These ions are referred to as potential-determining ions. Special cases are mineral oxides, such as Al2O3, TiO2 or SiO2. The surface charges of these materials are produced by adsorption or desorption of protons (H+) in the surface area. Hence, the surface charge changes with the pH value or the concentration of potential determining ions of the dispersion, as demonstrated in Fig. 1. In summary: In many systems, the adsorption of potential determining ions from solution

leads to surface charge. The net surface charge is determined by the surface equilibrium constants

involved in complex systems including several equilibria. Adsorption of polymers with polar end groups leads to surface charges.

OH2+

OH

O-

Ag+ Ag+

+ σ0

-

J-

J-

+ σ0

point of zero charge

pH

Fig. 1. Inﬂuence of potential determining ions.

-

point of zero charge

pAg

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The point where the surface charge is zero is deﬁned as the point of zero charge (pzc). The pzc is material dependent and can be determined experimentally by charge titration. It is important to distinguish the isoelectric point (IEP), where the measured z-potential is zero, from the pzc.

2.2.2. The origin of surface charges in organic liquids Although the origin of the charge is fairly well known for aqueous dispersions, the mechanisms operating in non-aqueous liquids are not as clearly understood or agreed on. Because of the low dielectric constants of most organic liquids, dissociation of surface groups and ionization of the electrolytes are limited. Often ionic strengths are much lower than 106 M. Charges on the particle surface in organic media can arise by the following mechanisms: Verwey [22] proposed, in 1941, for the surface charge of inorganic oxides

suspended in organic liquids an acid–base mechanism involving protons or hydroxyl ions as potential determining ions. Acid–base interactions involving proton transfer are responsible for the creation of surface charge. Donor–acceptor interactions with electron transfer between surface and solvent can lead to surface charges. The sign of the surface charge depends on the direction of the electron transfer [23]. In non-aqueous solvents, the adsorption of impurities (even in traces [16]) can lead to charging of the particles. Traces of water can change the surface charge, because water can act directly as a base or can transfer a proton to the solvent. In this case OH-groups of the surface can be bonded. In case of the acid–base and the donor–acceptor mechanism, the surface charge correlates with the donor properties of the solvent, i.e. the Gutmann donicity series [25]. The donor number DN is a qualitative measure of Lewis basicity and is deﬁned as the negative enthalpy value for the reaction of the organic solvent with the standard Lewis acid SbCl5 in 1,2-dichlorethane. The higher the donor number the easier is the electron transfer between the particle surface and the solvent and the higher is the surface charge. The sign of the surface charge depends on the direction of the electron transfer. Likewise the acceptor number AN was introduced by Gutmann as a scale for the acceptor strength of organic solvents [26]. Extensive tabulation of the donor and acceptor numbers of most organic solvents is given in the literature [27,28]. Labib showed [21], that the z-potential for a-Al2O3 and TiO2 particles suspended in organic liquids are positive for low donor numbers (DNo10 kcal/mol) and acceptor numbers higher than 20. In this case, the acceptor strength of the liquids outweighs their donicity and they behave as acceptors (acids). In solvents with donor numbers around 10 kcal/mol or higher the potential became negative up to 40 mV. With increasing donor

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557

Table 2. Dissociation constants of different solvents

KW ¼ [H+][OH] ¼ 1014(mol2/l2) K ¼ [H+][CH3O] ¼ 1017(mol2/l2) 15.9 K ¼ [H+][CH (mol2/l2) 3 CH2O ] ¼ 10 17.6 K ¼ [H+][CH (mol2/l2) 3 (CH2)3 O ] ¼ 10

H2O CH3OH CH3 Butanol

numbers the negative charge was declining. In contrast to a-Al2O3 and TiO2 no pzc could be found for SiO2 particles. The charge for SiO2 particles was in all solvents negative. According to [21] the mechanism of charging in protonic organic liquids that have a ﬁnite dissociation constant should be similar to that of water. The mechanism is based on the fact that the particle (P) will act for the solvent (S) as a proton donator or acceptor. þ PHþ 2 þ S Ð PH þ SH Ð P þ SH2

ð4Þ

Protonic liquids are liquids that can react by loss or gain of a proton. Examples for dissociation constants are summarized in Table 2. In aprotic liquids, whose molecules cannot accept or donate protons or electrons, i.e. non-polar organic solvents, saturated hydrocarbons or crude oil fractions, charging of particles can only occur by adsorption [16,21]. In those media only little screening of charges occurs and the Debye–Hu¨ckel screening parameter k is relatively small.

2.2.3. The electrical double layer Since the system as a whole is electrically neutral, the surface charges of the particles are electrically equalized by charges in the dispersion medium of the opposite sign. These charges are carried by counter ions, which move freely under the inﬂuence of electrical and thermal forces in the so-called diffuse layer. The distribution of the counter ions in dependency of the distance from the particle can be described with the Gouy–Chapman-model with the following simplifying assumptions. Ions in solution are point charges. The electrolyte is an ideal solution with uniform dielectric properties. Surface charges and potentials are evenly spread over the particle surface.

Starting from the Poisson equation, which describes the charge density at a location x in an electrical ﬁeld with the electrostatic potential j and the dielectric constant e and a Boltzmann distribution for the ions in the potential ﬁeld, the one-dimensional Poisson–Boltzmann distribution can be formulated for a

558

M. Sommer and W. Peukert

symmetrical electrolyte d2 j 2F z c0 zFj ¼ sinh dx 2
ð5Þ

where z is the valence of the ions, F the Faraday constant and c0 the concentration of ions in the bulk phase. Linearization of the sinh function for small potentials (jr25 mV) leads to the Debye–Hu¨ckel approximation d2 j ¼ k2 j dx 2

ð6Þ

where k is the so-called Debye–Hu¨ckel-parameter k2 ¼

2F 2 2F 2 ðz2 c0 Þ ¼ I r 0 < T r 0 < T

ð7Þ

with I being the ionic strength. The Debye–Hu¨ckel parameter k has the dimension of a reciprocal length. k1 is a measure of the extension of the electric double layer. Solving equation (6) leads to j ¼ j0 expðk xÞ

ð8Þ

for a ﬂat surface where j0 denotes the electrostatic potential at the surface (x ¼ 0). Hence the potential decays exponentially with the distance from the surface. Stern (1924) and Graham (1947) improved the model from Gouy–Chapman by assuming a ﬁnite expansion of the hydrated ions. Accordingly, the concentration of ions at the particle surface is limited and cannot exceed a saturation value. Part of the counter ions are adsorbed on the particles surface in a layer of thickness js. The ions in the so-called Stern layer are not necessarily connected to the particle surface; especially hydrated ions will be separated from the surface by water molecules. The electrostatic potential in the Stern layer decays in the simplest case linearly from the surface potential j0 to a potential js the Stern-potential. Outside the Stern layer, the counter ions are building a diffuse double layer according to the Gouy–Chapman model. If more counter ions are adsorbed as necessary to neutralize the surface charge, a change of the sign of the surface charge will occur. This is especially common for polyvalent ions, which are not bound equivalent but equimolar [16]. Figure 2 shows the assembly of the electrical double layer according to the Stern–Graham model including charge enhancement due to speciﬁc adsorption of ions and the resulting potential curve. Furthermore, Fig. 2 shows the inﬂuence of electrolyte concentration on the potential development. With increasing ion concentration the screening of the surface charge is enhanced and the decay of the electrostatic potential is steeper. The valence of ions in solution has a similar effect. Polyvalent ions are able to screen charges of the surface more effectively. This leads to a compression of the double layer. Experimental examinations of charged particle surfaces exploit mostly titrations or electrokinetic phenomena. A charged particle will move with a certain velocity in

Enabling Nanomilling through Control of Particulate Interfaces Stern layer

diffuse layer

Gouylayer

shear layer

inner Helmholtzlayer

δS

559

1/κ

potential ϕ

counter ion ϕ0

specific adsorbed ion

ϕS ζ

solvent molecule c counter

distance from particle surface h

Fig. 2. Assembly of the electrical double layer according to the Stern–Graham model including charge enhancement due to speciﬁc absorption of ions [1].

an electric ﬁeld. This phenomenon is called electrophoresis. The particle’s mobility is related to the dielectric constant and viscosity of the suspending liquid and to the electrical potential at the boundary between the moving particle and the liquid. This boundary is called the shear plane and is usually deﬁned as the point where the Stern layer and the diffuse layer meet. The Stern layer is considered to be rigidly attached to the colloid, while the diffuse layer is not. As a result, the electrical potential at this junction is related to the mobility of the particle and is called the z-potential. Experimentally the z-potential can be determined with the methods of electrophoresis, streaming potential, electroosmosis or electroacoustics [24]. A new promising technique to measure surface electrostatic potentials and surface charge densities directly is the method of second harmonic spectroscopy (SHG) [29,104,105]. This method has some advantages over traditional techniques, since no assumptions have to be made to determine the surface charge. To obtain the surface potential using traditional techniques from the z-potential assumptions about the distance between the shear layer and the interface have to be made, since it is not precisely known where the shear surface is located.

560

M. Sommer and W. Peukert

Table 3. Approximate forms of the electrostatic energy

Constraint Constant potential

Constant charge

Superposition

Electrostatic energy 2 kT j ez2 Fel ¼ 2pr 0 a s lnð1 þ ekh Þ ze kT

Fel ¼ 2pr 0 Fel ¼ 32pr 0

kT ze

2 kT aq2 lnð1 þ ekh Þ ze

2

a tanh2

ð9Þ

ð10Þ

j ez s expðk hÞ ð11Þ 4kT

2.2.4. Electrostatic double layer interaction If two charged particles in an electrolyte solution are approaching each other so closely that their double layers overlap, an electrostatic repulsive force is generated. This force is due to the repulsion of identically charged surfaces and due to an osmotic pressure, which is based on dilution effects of the salt concentration in the gap between the two approaching spheres. The repulsive force can be obtained from the numerical solution of the Poisson–Boltzmann equation. However, a variety of approximations exist. The Derjaguin approximation [30], for example, is applicable for small separations compared to the radius of the spheres. It is assumed that each sphere is divided into inﬁnitesimal elements, which are considered to be parallel planar elements at the same separation. The total interaction is the sum over the interactions of the inﬁnitesimally small elements. An analytical solution of the differential equation requires further assumptions, which are mostly the two borderline cases: constant surface potential or constant surface charge. Both conditions are barely applicable to real systems. The error for constant potential boundary remains small for small separations and is larger for longer distances. Conversely, the constant charge boundary condition is invalid at separations smaller than the Debye thickness. A further method to approximate the repulsive energy is the linear superposition of single sphere potentials. This method is a good compromise between the two borderline cases. For a detailed discussion of the technique, see [31]. Table 3 lists approximate forms of the electrostatic energy for two spheres with radius a, a surface-to-surface distance h and a symmetric electrolyte.

2.3. Born interactions If two particles approach each other to very small distances the electron clouds of the atoms overlap and a resulting strong repulsive force – the Born repulsion

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561

arises. This repulsive force is characterized by having short-range efﬁcacy and increases very sharply if two molecules come closely together. Strictly, the distance dependency of the Born interaction should be described by quantum mechanical theory. Instead, simplifying approximations are established. The three most common of such potentials are the hard sphere potential, the inverse power-law potential (exponent-12) and the exponential potential [12].

2.4. Solvation, structural and hydration interactions At small separations between two particle surfaces (closer than a few nanometres), the continuum theories for attractive van der Waals and repulsive double layer interactions often fail to describe the interactions. This is either because the continuum assumption of one or both types of interactions is not valid or because additional non-DLVO forces come into play [32]. These forces can be attractive, repulsive or oscillatory and they can be much stronger than the DLVO forces at small separations [12]. These repulsive forces appear due to a structural effect of the suspending ﬂuid near particle surfaces, because at small separations the solvent molecules have to be squeezed out of the gap between the two approaching surfaces leading to an oscillating force with exponential increasing amplitude. These oscillatory forces are dependent on the type of solvent molecules. While linear chain molecules (e.g. n-decane) show signiﬁcantly high oscillations, molecules with an irregular structure, which cannot be ordered into regular layers, do not. An ordered layer around the particle can also be disrupted by surface roughness, even if the roughness is in the order of only a few Angstroms. In aqueous suspensions an additional repulsive interaction was observed for the material SiO2 [33]. Owing to hydrogen bonding between silanol-groups and surrounding water molecules a structured layer of water molecules on the silica surface is formed, leading to the additional hydration force. The density of silanolgroups increases as the pH-value decreases. This is leading to a surprisingly large hydration force at low pH values and to very stable suspensions near the IEP [34].

3. STABILIZATION OF PARTICLES AGAINST AGGREGATION 3.1. Introduction Under stabilization different topics can be understood. A distinction can be drawn between stabilization against chemical reactions (e.g. oxidation), stability against phase changes,

562

M. Sommer and W. Peukert

++ + + + + +++

++ + + + + + + +

a) electrostatic stabilization

b) electrosteric stabilization

c) steric stabilization due to adsorption of polymers

Fig. 3. Possibilities for stabilization.

stability against sedimentation, stability against coagulation, and stability against ripening of primary particles into a hard agglomerate.

In the following only the stabilization against coagulation is considered. The relative motion of particles (especially of nanoparticles) as well as attractive interactions can lead to coagulation. Thus agglomerates are built, which can affect the handling and the properties of the ﬁnal product considerably. Typical undesirable phenomena include

sedimentation, building of solid bridges (Ostwald ripening), difﬁculties by redispersing of particles, and defect formation during manufacturing of compact moulds, e.g. ceramics.

Thus ﬁnely dispersed systems, especially with particle size below 1 mm, have to be stabilized often against coagulation. Applications where deﬁned agglomerate structures are desired are not discussed here. Fundamentally, ﬁne particles can be stabilized electrostatically due to unipolar charging of particles, by the adsorption of molecules (sterically) or by a combination of both (electrosterically) (Fig. 3).

3.2. Electrostatic stabilization 3.2.1. The DLVO-theory The superposition of the van der Waals attraction and electrostatic repulsion can be traced back to the independent research of Derjaguin and Landau [35] as well as Verwey and Overbeek [36] and is referred to as DLVO theory. Mostly the repulsive Born interaction is added so that the net interaction energy consists of Ftot ¼ FvdW þ Fel þ FB

ð12Þ

Enabling Nanomilling through Control of Particulate Interfaces

+

563

Born repulsion ΦB

total interaction energy Φtot

Φtot,max

electrostatic repulsion Φel

distance h Φtot,min2

Φtot

+

-

Φtot,min1

van der Waals attraction ΦvdW

celectrolyte -

ϕ0

Fig. 4. Typical energy vs. distance proﬁles of DLVO theory.

A typical plot of the total interaction energy is shown in Fig. 4. Negative values denote an attraction and positive values a repulsion of two approaching particles. At very small separations the Born repulsion is dominating. The primary minimum Ftot,min1 is dominated by the van der Waals interaction and Born repulsion. Particles in the deep primary minimum are thermodynamically stable and are often bound irreversibly since in addition to the van der Waals forces chemical bonds can be formed, e.g. by hydrolysis of hydroxyl-groups on the surface of oxides. The energy maximum Ftot,min depends on the superposition of van der Waals attraction and electrostatic repulsion. This maximum represents an energy barrier for dispersed particles, which has to be overcome to agglomerate approaching particles. Stability criteria are usually based on either the potential barrier or the maximum force (equation (13)). The height of the potential barrier indicates how stable the system is against Brownian agglomeration. To avoid agglomeration the energy barrier should be at least 15 kT [16]. Next to the height of the energy barrier, the slope of the total interaction energy from the energy maximum is important. The steeper the slope the higher the repulsive force, F F¼

dF dh

ð13Þ

564

M. Sommer and W. Peukert

In sheared systems this force F must be larger than the hydrodynamic force to ensure stability. At larger distances from the surface a secondary minimum might be reached. This secondary minimum suggests loose reversible aggregation. To disperse particles agglomerated in the secondary minimum, a low energy input is sufﬁcient. Furthermore, the inset of Fig. 4 shows the inﬂuence of surface potential and electrolyte concentration on the total interaction energy. With decreasing surface potential as well as increasing electrolyte concentration, the inﬂuence of the repulsive electrostatic interaction and with it the height of the energy barrier is reduced. Increasing electrolyte concentration causes a compression of the electric double layer and thus a reduction of the range of the electrostatic repulsion. Therefore, a high surface potential and low ion concentrations are preferable to effectively stabilize the suspension electrostatically against agglomeration.

3.3. Steric stabilization For many applications electrostatic stabilization is insufﬁcient. For example, particles cannot be dispersed in the presence of high salt concentrations in polar or non-polar solvents. Also dispersions with high percentage of solid phase and a low dielectric constant cannot be dispersed efﬁciently with electrostatic stabilization. In these cases, one has to consider different stabilization mechanisms, e.g. the steric or electrosteric stabilization mechanism. The steric stabilization mechanism is based mainly on the adsorption or chemisorption of polymer chains on the surface of particles [37]. The polymers act as spacers between the particles. Coagulation is prevented by a reduction of the strongly distance dependent van der Waals force. The concentration of the polymer must be high enough to reach a sufﬁcient coverage of the particle surface by adsorbed polymer chains. If the polymer concentration is low, e.g. if less than half of the particle surface is covered with polymer, bridging ﬂocculation may occur [16]. In bridging ﬂocculation, one polymer chain adsorbs on the surface of two particles at the same time. The inﬂuence of the polymer on the stability depends strongly on the interactions between polymer and solvent. The quality of a solvent can be expressed in reference to the Theta point. This point is deﬁned as the temperature (Theta-temperature) or solvent composition (Theta-solvent) where interactions between solvent molecules and segments of the polymer chain are energetically equal to the interactions between polymer segments [38]. The Flory–Huggins parameter w records the interactions of the macromolecules with the solvent. w is equal to 0.5 at the Theta point. In a good solvent where polymer solvent contacts are favoured over polymer contacts among each other, then w is less than 0.5 holds. In a poor solvent w is greater than 0.5. The thickness of the adsorbed polymer layer is inﬂuenced by the quality of the solvent. The better the solvent the longer the polymer chains stretch into solution and the stronger the steric repulsion force. The thickness of the adsorbed layer

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565

increases with increasing polymer concentration mainly by building longer brushes. If two particles coated with a protective shielding of polymer are approaching each other, one can distinguish three distance areas [39]. At large distances between the particles the polymer layers do not overlap. In this case no repulsive forces are acting. At closer distances, the polymer chains start to overlap, a densiﬁcation of the polymer chains is detected and solvent molecules are depleted out of the overlap area into the solvent. In a good solvent the entanglement of polymer chains is thermodynamically disadvantageous, because the entanglement means a separation of polymer segments and solvent molecules. A good solvent prevents the entanglement and increases the stability of the dispersion; a poor solvent increases the entanglement and decreases the stability of the dispersion. The chemical potential of the solvent molecules is in the overlap area higher than in the bulk solvent. Owing to this potential difference an osmotic pressure arises, which is in fact the reason for the repulsion. If particles approach each other even closer the polymer layer is compressed. In this case one has to consider, in addition to osmotic forces, forces due to the compression of the polymer layers. The steric interactions are a function of the segment density distributions, which substantially depend on the conformation of the macromolecules on the particle surface. This is determined by the chemical constitution of the polymer, the chemical nature and the geometric shape of the interface, the composition of the solvent, the conformation of the polymer on the surface (loops, tails or trains) and the density of the macromolecules on the surface [16]. The number of loops is higher in case of adsorption from a poor solvent than in the case of adsorption from a good solvent. However, the ratio of brushes is widely independent of the quality of the solvent. This is why the quadratically averaged thickness and the hydrodynamic thickness of the adsorbed layer are independent from the solvent quality, because the tails reach longer into the solution than the loops. With increasing polymer concentration, the number of tails increases and the number of trains decreases while the number of loops is constant [16]. At volume fractions cv4102, the ratio of tails increases conspicuously. The tails consume almost two-thirds of a polymer chain at the transition to the melt (cv-1) [16]. While the Flory–Huggins parameter w describes the interactions between the polymer segments and the solvent, the adsorption energy parameter ws accounts for the difference between the free transfer energy of a segment from the solvent to the surface and a solvent molecule from the pure solvent to the surface. The segment is preferably adsorbed onto the particle surface if ws40.5. Otherwise, a solvent molecule is preferably adsorbed on the surface. The difference between the adsorption of small molecules and the adsorption of polymers is that ws has to exceed a critical value before it comes to adsorption. The reason is that the loss of conformation entropy by the adsorption of a ﬂexible macromolecule has to be compensated by the gain of adsorption energy.

566

M. Sommer and W. Peukert

Evans and Napper [41] calculated the total interaction potential Ftot ¼ Fsteric þ FvdW with the following equation: Ftot ¼

ð2pÞ5=2 2 Aeff a hr i n2p ða2 1Þ a S k T 27 12hr 2 i1=2 d0

ð14Þ

Here, the root mean square end-to-end distance of the polymer chains hr 2 i1=2 and the expansion factor a refer to the structure of the polymer in solution. np is the number of polymer chains sticking to the surface per unit surface. The exact value of np must be detected from adsorption measurements. According to Evans and Napper this can be estimated with np hr 2 i a is the particle radius, Aeff the effective Hamaker constant and d0 ¼ h=hr 2 i3=2 the normalized distance between two particles. S is a geometric constant, which depends on the polymer conformation and on d0. Values of S are tabulated from Evans and Napper [41]. The osmotic term Fosm steric dominates for distances doho2d. Vincent used for this term simpliﬁed equations [42]. The osmotic term is dependent on the segment density distribution in the polymer cover. For different segment density distributions the following equations apply in the area doho2d and h5a. Constant segment density: Fosm steric

4pakT a 2 h 2 f2 ð0:5 wÞ d ¼ n1 2

ð15Þ

Linear decreasing segment density Fosm steric ¼

16pakT a 2 h 4 f ð0:5 wÞ d 2 2 3n1 d2

ð16Þ

Fosm steric ¼

32pakT a 2 h 6 f ð0:5 wÞ d 2 2 5n1 d4

ð17Þ

Homopolymer

With tails Fosm steric

" # 32pakT a 2 h 4 h 5 h 6 2 f2 ð0:5 wÞ 10d d ¼ 12d d þ d 2 2 2 5n1 d4 ð18Þ

a the median volume fraction of the here n1 is the molar volume of the solvent, f 2 segments on the adsorbed layer, d the thickness of the polymer layer, h the distance between the particle surfaces and w the Flory–Huggins parameter. For distances Hod, the polymer chains are not only overlapping they are being compressed. This leads to a high repulsive force, which can be calculated

Enabling Nanomilling through Control of Particulate Interfaces

567

δ
Φ/kT

20

0

-20 van der Waals attraction ΦA -40 0

5

10 H /nm

15

20

Fig. 5. Interaction potential for steric stabilized a-Al2O3.

according to [19] Fosm comp

¼ np kT

" # p2 ns I 2s 3h2

þ ln

3h2 8p I 2s ns

!6 ð19Þ

An estimation of the total interaction energy of steric stabilized a-alumina in water is shown in Figs. 5 and 6. In this approximation, the energies are calculated with equations (15) and (19) with the assumption that water is a good solvent. This means that the segment density in the polymer layer is constant, that the polymer chains are fully stretched and that the polymer density at the particle surface can be calculated with np ¼ l2 s [19]. As Fig. 6 shows, a molecular weight of Mn ¼ 5000 g/mol is not sufﬁcient to effectively stabilize the suspension. In this case, a weak coagulation is expected even though a ‘‘better than Theta solvent’’ is assumed in the calculation. At molecular weights Mn ¼ 10,000 and 15,000 g/mol the repulsive force is mainly inﬂuenced by the compression of the polymer chains, since the force of the osmotic term is relatively low. The molecular weight of the polymer chains must be high enough to effectively stabilize the particles. The estimation shows that a polymer weight of 20,000 g/mol is necessary to stabilize a dispersion if an ideal solvent is assumed. For this case a layer thickness of the polymer of 15 nm is estimated. This thick adsorbed layer on the particles surface may lead on the one hand to a strong quenching of van der Waals forces. On the other thick polymer layers may adsorb the mechanical

568

M. Sommer and W. Peukert Mn= 5 000 10 000 15 000 20 000

φ /kT

40

Mn

20

0

-20 0

5

10 H /nm

15

20

Fig. 6. Interaction potential for steric stabilized a-Al2O3 with the chain length as parameter.

energy needed for grinding. In summary, more research is needed to fully understand these complex phenomena. In the following the steric stabilization of Al2O3 is exempliﬁed. In this example the polymer DAPRAL is used as dispersing agent. This polymer can be used in wet milling with organic solvents of different polarities and water. The adsorption of the polymer on oxide surfaces in organic media depends according to Somasundaran [77] mainly on the interactions between hydroxyl groups of the oxide surface and the polar regions of the surface-active macromolecules. The hydrophilic side chains of the polymer are responsible for the adsorption of DAPRAL on the Al2O3 surface and the steric stabilization mechanism is archived by the polymer chains reaching in the solution. The structure of the polymer DAPRAL is shown in Fig. 7. Figure 8 shows adsorption isotherms for DAPRAL on a-alumina dispersed in different solvents. For the measurements alumina (AKP-30) with a median particle size of 350 nm was used. Alumina particles of 6.25 wt% were added to the respective solvents, which contained different polymer concentrations, ranging from 0.6 to 2.5 mg/ml. The suspensions were allowed to stand over night under gentle stirring to reach the adsorption equilibria. To separate particles from free and not adsorbed polymer chains, the suspensions were centrifuged at 3500 rpm for 30 min in a Heraus Labofuge 400 and washed with pure solvent for ﬁve times. The centrifuged particles were dried and the adsorbed polymer amount was thermogravimetrically determined with a heating rate of 10 K/min under nitrogen atmosphere. Based on the speciﬁc surface area of the alumina particles (10.1 m2/g),

Enabling Nanomilling through Control of Particulate Interfaces

569

CH3 CH2

COOH CH C

9

CH

CH2

CH2

O

CH

O

O CH2

H 7-8

n

Fig. 7. Structure of the polymer DAPRAL.

2.5

2-butanol

adsorption amount / mg / m2

saturation

2.0

complete coverage ethanol

1.5

toluene water

1.0 α-Al2O3 in 2-butanol α-Al2O3 in ethanol

0.5

α-Al2O3 in toluene

T=25°C

α-Al2O3 in H 2 O Al(OH) 3 in H 2O

0.0 0.00

0.01

0.02 0.03 CDAPRAL /Calumina / -

0.04

0.05

Fig. 8. Adsorption isotherm of DAPRAL on alumina particles in different solvents.

the amount of adsorbed polymer per unit area can be calculated. In Fig. 8, the adsorbed amount of polymer is plotted against the dimensionless concentration cDAPRAL /calumina. At a dimensionless concentration cDAPRAL /calumina between 0.02 and 0.025, a plateau region is reached independent of the kind of the solvent. That means from this ratio onwards a complete coverage of the particle surface with polymer chains is attained. However, the amount of adsorbed polymer depends on the quality of the solvent. The lowest amount of adsorbed polymer of 1.2 mg/m2 was measured for alumina particles in water and the highest amount was

570

M. Sommer and W. Peukert 700 650

diameter x50,3 / nm

600 550 500 450

primary particle size

400 350 300 0.00

in water in ethanol

0.01

0.02 CDAPRAL /Calumina / -

0.03

0.04

Fig. 9. Dispersing experiments for a-Al2O3 stabilized with DAPRAL in water and ethanol.

detected in 2-butanol. The adsorbed amount in toluene (1.5 mg/m2) and ethanol (1.7 mg/m2) lies between these values. A hydroxide layer around the a-alumina particles has no inﬂuence on the amount of adsorbed polymers in aqueous phase. Figure 9 shows dispersing experiments for a-Al2O3 in water and ethanol. In this diagram the median particle size x50,3 is plotted against the dimensionless concentration cDAPRAL =cAl2 O3 . The samples were shaken for 24 h at room temperature to reach the adsorption equilibrium. Subsequently, the samples were stressed with ultra sound for 3 min to destroy weak agglomerate structures. Afterwards the particle size measurements were performed with dynamic light scattering. As it can be seen in Fig. 9, the agglomerates can be dispersed to their primary particle size if the concentration of DAPRAL is in the plateau region of the adsorption isotherm. In this region the surface is covered completely with polymer. The dispersion starts to ﬂocculate at a certain polymer concentration due to depletion effects if more polymers are in solution than can be adsorbed to the particle surfaces. However, a dispersion destabilized by depletion ﬂocculation can be stabilized again at higher polymer concentrations. However, the stabilization of dispersions at higher polymer concentrations plays a minor role in comparison to the steric stabilization, where polymers are adsorbed to the surface. The kinetics of the stabilization depend on the material transport (convection and diffusion) from the free solvent to the particle surface as well as the structure formation kinetics at the surface. The material transport to the surface can be correlated with the diffusion coefﬁcient and the Reynolds number in a ﬂowing system. The diffusion coefﬁcient depends on the molecular weight. The following

Enabling Nanomilling through Control of Particulate Interfaces

571

equation for spherical polymer particles with the hydrodynamic radius rH applies according to Stokes Einstein D¼

kT 6p Z r H

ð20Þ

Herein, Z denotes the dynamic viscosity, T the temperature and k the Boltzmann constant. Typical diffusion coefﬁcients in a diluted solvent are in the order of 1010 –1011 m2/s. With the viscosity law of Einstein (see equation (34)) the hydrodynamic radius rH can be estimated for the borderline case of inﬁnite dilution (c-0) jZj ¼ limc!0

Zrel 1 2:5 p N A 8 3 ¼ rH c 6M

ð21Þ

Little is known on the adsorption kinetics of polymers. Investigations of the adsorption of polyethylene oxide (PEO) in water on SiO2 surfaces showed that in this case the adsorption kinetics is, up to 75% surface coverage, independent of the allocation kinetics [40]. At higher coverages an inhibition of the adsorption kinetics in dependence of the molecular weight was detected. As far as this result can be generalized, the stabilization kinetics in a stirred media mill will depend only on the material transport assuming that a coverage of 75% is sufﬁcient. A selection of polymers is carried out mostly empirically. Amphipathic block and graft copolymers are optimal synthetic particle stabilizers. Usually the anchor part of the polymer is insoluble in the solvent while the tails, which reach in the suspension, are soluble. In this case the anchor adsorbs on the particle surface and the tails reach in solution and cause steric stabilization. The anchor of the surfaceactive substrate is bound to the particle surface either by physical adsorption, by van-der-Waals forces, dipole interactions, hydrogen bridges or by chemisorption. The chemisorption produces relatively strong covalent bonds. A reasonable preselection of polymers can be made due to the following guidelines based on the speciﬁc interactions of the particular groups with the surface and the solvent [43]. For cationic and anionic surfaces, respectively, a polymer with an oppositely

charged ionic group is necessary. For polar and non-polar surfaces, a polymer with corresponding polar or non-polar

groups has to be chosen. The same applies for lipophilic or lipophobic surfaces. The following hints are especially for hydrophobic chains. The longer the hydrophobic chain length of the polymer, the worse the solubility

in water and the better in organic solvents. Entanglements in the hydrophobic chain enhance the solubility of the polymer

in water as well as in organic solvents. Aromatic rings within the hydrophobic chain enhance the adsorption behaviour

of the polymer on polar surfaces.

572

M. Sommer and W. Peukert CH3 CH2

CH

a)

C

C

O

C

O- M+

CH

b)

O P

O

O- M+

n

polyacrylic acid

CH2

O

O- M+

n

n

polyphosphoric acid

polymethacrylic acid

CH2 + X-

NH3

n

polyvinylamine O c)

R

CH

C

OH

OHH+

O R

NH3+ long-chain amino acid

CH

C

O

NH3+

-

OHH+

O R

CH

C

O-

NH2

Fig. 10. Chemical structure of some polyelectrolytes.

3.4. Electro-steric stabilization Polyelectrolytes carry ionic groups along the polymer chains. The charge of these groups depends on its nature and on the pH value. In principle polyelectrolytes can be classiﬁed according to the kind of the charge in cationic, anionic or zwitterionic polyelectrolytes. Some important polyelectrolytes are shown in Fig. 10 [44]. The effectiveness of the polyelectrolytes depends strongly on the dissociation degree of the polyelectrolyte. This is inﬂuenced by the pH value and can be determined, for example, by potentiometric titration. Neutral particles can be stabilized by polyelectrolytes electrostatically. The non-polar ends of the polymers adsorb favourably on hydrophobic particles in polar surroundings, while the polar ends reach into the polar solvent. This leads to the development of a double layer. This means the principle of the DLVO theory can be applied. Consequently, the ionic strength should not be too high to avoid the compression of the double layer. The stabilization is strengthened by the polymer layer due to the steric effect. If polyelectrolytes are used at high ionic strengths, they only act as steric stabilizers.

3.5. Summary of stabilization methods A comparison of the steric and electrostatic mechanisms is shown in Table 4. Both mechanisms feature advantages as well as disadvantages. Therefore, a selection of the stabilization mechanism must be adjusted carefully to the

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573

Table 4. Comparison between electrostatic and steric stabilization

Property

Electrostatic

Steric

Kinetic

Fast, diffusion of ions

Contamination

Ions, pH value

Measuring technique

Online detection (electroacoustic, pH, conductivity) Low ionic strengths and polar solvents High particle concentration can lead to gel formation

Slow, diffusion of macromolecules Due to polymers and their fragments Difﬁcult to measure

Application range

Handling properties

Yield stress can be controlled

Post-processing

Irreversible solid bridges can be formed during drying

Works in aqueous and in non-aqueous dispersions Effective at low and high solid concentration Viscosity is considerably lower at high solid concentrations in comparison to electrostatically stabilized suspensions Good redispersing behaviour, because solid bridges are prevented due to polymer coverage

particular process and the product properties. The advantages of steric stabilization are [16]: The stabilization can be carried out as well in aqueous as in non-aqueous

dispersions. Steric stabilization is possible in non-polar solvents and in aqueous solutions

with high salt concentrations. The steric stabilization is effective at low and high solid concentrations. While electrostatically stabilized dispersions at high solid concentrations often

builds gels, the viscosity for steric stabilized dispersions is considerably lower. Despite these advantages also signiﬁcant disadvantages exist for the application in stirred media milling. High molecular polymers can be degraded due to the high-energy input during the grinding process. Moreover, on the particle surface

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adsorbed polymers can be desorbed by the high shear forces and by resiliencies in the mill. Furthermore, the stabilizing polymers can contaminate the product. This may be prohibitive, for example, for pharmaceutical applications. However, the largest disadvantage lies in the slower and mostly unknown adsorption kinetics for high molecular weight molecules.

3.6. Coagulation in stirred media mills Besides the breakage kinetics, further two kinetic processes are important in nanomilling: the agglomeration kinetics and the stabilization kinetics. It has to be assured that the stabilization process of newly built particles is fast enough to certainly stabilize the suspension. Therefore, it should always be checked if the fast electrostatic stabilization is feasible. Three fundamental issues govern the type of clusters formed during aggregation: the mode of motion that brings particles into contact, the probability of the particles sticking upon contact and the mobility of the subsequent aggregate. The aggregation event can be modelled by a second-order kinetic process of the particle concentration N. The kinetic constant b0 is dependent on the transport mechanism. b0 is a measure of the probability of a collision of two particles due to the acting transport mechanism within a time unit in the absence of interparticle forces. As transport mechanisms, Brownian motion for particles o1 mm (perikinetic agglomeration) and laminar or turbulent shear (orthokinetic agglomeration) are considered. In the literature a number of equations exist to describe the aggregation kernel. In Table 5, the most common collision kernels are presented. All described equations disregard interparticular and hydrodynamic interactions. Furthermore, in equation (25) it is assumed that the particles are small compared to the Kolmogorov-vortex size, which is admittedly the case for stirred media mills. Stenger [1] estimated the inﬂuence of the different agglomeration mechanisms in a stirred media mill. According to his calculations, the turbulent agglomeration kernel dominates for particles larger than approx 80 nm, while smaller particles are mainly inﬂuenced by Brownian agglomeration. The collision probability and adhesion of particles are described in the agglomeration literature mostly by a sticking probability. The particle sticking probability leads to two universal limiting aggregate regimes: diffusion and reaction limited aggregation. In diffusion limited colloidal aggregation (DLCA), the absence of a strong repulsive barrier leads to a high sticking probability and the particles stick on contact [47], whereas a strong repulsive barrier leads to the reaction limited regime (RLCA) [48]. Several light scattering studies [47–54] of the agglomeration kinetics have shown that the average agglomerate size grow in a characteristic way for each regime. In the DLCA regime the agglomerate size growth follows

Transport mechanism Brownian diffusion [45]

Sedimentation [45]

Laminar shear [45]

Turbulent shear [46]

Collision kernel 2kT 1 1 ðx 1 þ x 2 Þ þ b0 ¼ 3Z x1 x2 2 2 2 x1 x2 b0 ¼ pg þ 2 2

b0 ¼

r 1 F rp

! x 1 2 x 2 2 2rp 2 2 9Z

3 x 1 x 2 3 g_ þ 4 2 2

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8p E rF x 1 x 2 3 b0 ¼ þ 15Z 2 2

ð22Þ

ð23Þ

ð24Þ

ð25Þ

Enabling Nanomilling through Control of Particulate Interfaces

Table 5. Collision kernels for different transport mechanisms

575

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M. Sommer and W. Peukert

power law behaviour, while in the RLCA process the agglomerate size growth exponentially. Whether coagulation is slow or rapid and whether it occurs by a perikinetic (diffusion) or an orthokinetic (shear-inﬂuenced aggregation) mechanism [55,56], the structure of the resulting aggregates will be different [19]. Fast coagulating suspensions will form loose and open aggregate structures. Slowly coagulating systems, however, form dense very compact aggregates. The degree of openness of a colloidal aggregate is best represented by its mass fractal dimension dF with 1rdFr3 (where dF ¼ 3 is the limit of a compact spherical aggregate). The fractal dimension dF reﬂects the internal structure of the ﬂocs and depends on the mode of aggregation. For rapid irreversible Brownian ﬂocculation without subsequent rearrangement, a value of 2.5 is found for dF in computer simulations [19], if the ﬂocs grow by adding one particle at a time. For diffusionlimited aggregation, in which cluster–cluster interactions are dominating, the fractal dimension is around 1.7–1.8. The fractal dimension for reaction-limited aggregation will be between 2.0 and 2.2 [19]. If electrostatic and hydrodynamic interactions are accounted for, not every theoretical collision leads to sticking of particles. The collision frequency b is reduced by a collision efﬁciency a or a stability factor W ¼ 1/a. The stability factor is deﬁned as the ratio of the number of collisions between hard spheres and the particles with interparticle interactions that result in coagulation [57,58]. Z 1 b Ftot;max dr W ¼ 0 ¼ 2a ð26Þ exp 2 r GðrÞ b kT 2a where r is the particle centre to centre distance and G(r) a correction accounting for the effect of hydrodynamic interaction according to Honig [59]. In the strict sense, computation of W requires the numerical integration of the overall interaction potential. However, a value of the stability factor W can be estimated according to equation (27) as long as Ftot,maxckT [60].

exp Ftot;max =kT ð27Þ W ¼ 2a k For large energy barriers, Ftot,max/kTZ3 the stability ratio W can also be approximated according to Prieve and Ruckenstein [61] using an empirical relation, which relates the stability ratio and the energy barrier Ftot;max W W 1 þ 0:25 exp 1 ð28Þ kT where WN is the rapid coagulation value of the stability ratio, which is in most cases slightly larger than 1 [50]. In practice, however, the estimate of the stability ratio W requires a good characterization of the particle surface since the total interaction energy is very sensitive to surface charge, hydration, hydrophobicity, chain adsorption, etc. Small uncertainties in these surface properties can lead to large errors in the

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577

computed value of W [19,50]. The stability ratio W can be measured experimentally by determining the rate of doublet formation [50,62,63]. Several techniques can detect doublet formation for limited ranges of particle sizes. The coagulation rate for perikinetic agglomeration, b, can be directly estimated, for example, from dynamic light scattering data for the very initial state of the aggregation using a method from Lattuada et al. [49]. The aggregation rate constant b0 for rapid coagulation can be calculated for initially monodisperse suspensions using the Brownian kernel. b0 ¼

8kT 3Z

ð29Þ

Whether the aggregation process is dominated by a perikinetic or orthokinetic mechanism is dependent on the Peclet number Pe. Pe ¼

3pZa3 kT

ð30Þ

For Pe51 the aggregation process is controlled by Brownian motion while for Pec1 the process is dominated by convection [19,64]. Agglomeration in a stirred media mill is rather complex and will depend on the spatial turbulence and thus shear stress distribution in the mill. The overall agglomeration process in a 1 l Drais mill was estimated to be in transition regime between perikinetic and orthokinetic agglomeration considering particles with a median particle size of 100 nm and stirrer velocities between 6 and 10 m/s [1].

4. INFLUENCE OF PARTICLE INTERACTIONS ON SUSPENSION RHEOLOGY IN STIRRED MEDIA MILLS Particle–particle interactions play a dominant role in the nano size range strongly inﬂuencing the suspension rheology, which in turn is often the limiting factor of the production process due to various rheological, non-Newtonian phenomena including shear thinning, shear thickening, yielding and thixotropy. Thus, the control of the ﬂow behaviour of these suspensions is critically important to the unit operation itself and the following process steps. Thus, it is necessary to control the rheological properties of the milling suspension throughout the whole milling process especially at high solid loading. This can only be achieved by understanding the different inﬂuencing parameters on the rheology of concentrated suspensions and further by inﬂuencing them in a way that the suspension displays a well-deﬁned and desired ﬂow-behaviour. In the following sections, a short review of different rheological models describing suspension rheology under various ﬂow and suspension conditions is presented.

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M. Sommer and W. Peukert

4.1. Suspension rheology The suspension rheology is inﬂuenced next to the solid loading by the particle size distribution and particle shape, which determines the interparticle interactions as well as the hydrodynamics of the system. To determine the rheological behaviour of suspensions, the shear stress function tð_gÞ or alternatively the viscosity function Zð_gÞ has to be measured in stationary shear experiments. Both the functions are connected by the shear stress law. tð_gÞ ¼ Zð_gÞ_g

ð31Þ

where g_ is the shear rate, which is applied by the rheometer and Zð_gÞ a characteristic material function which describes the ﬂow behaviour of the ﬂuid for given ﬂuid mechanical conditions. In Fig. 11 some typical ﬂow characteristics are shown. Dilute suspensions with particles interacting through Brownian and hydrodynamic forces alone (Brownian hard spheres), show Newtonian behaviour and can be described by the well-known equation of Einstein [65]. Concentrated suspensions show in general non-Newtonian ﬂow behaviour. One typical ﬂow behaviour of concentrated suspensions is shear thickening, which is deﬁned as a strong increase of viscosity with increasing shear rate, whereas a decrease in viscosity with higher shear rates is called shear thinning. During shear thinning structures formed by the solvents and particles are destroyed or degraded with increasing shear rates. This leads to lower viscosities. Both rheological behaviours can be observed in suspensions in dependency of the particle concentration and the particle size. Shear thickening mostly occurs at high solid loadings and high shear rates. Hereby, an abrupt rise of the viscosity and respectively the shear stress in order of several tens of magnitude occurs at a certain shear rate [66]. This is caused by the transition of the suspension from an ordered to a disordered state [67,68] or through a reversible hydrodynamic cluster formation [69,70]. In general shear thickening is undesired, because the extremely high stress may lead to a failure of the apparatus.

Fig. 11. Common ﬂow behaviour: (a) Newtonian, (b) Bingham, (c) shear thinning and (d) shear thickening.

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579

The ﬂow curves of suspensions can be described by an empirical constitutive equation according to Ostwald de Waele [20]. tð_gÞ ¼ Z0 g_ n

ð32Þ

Herein, Z0 is the viscosity for g_ ! 0 and n the so-called ﬂow index. Both parameters have to be determined experimentally. For shear thinning the values of n are in the range of 0–1 whereas shear thickening can be described with n41. A suspension is called a Bingham ﬂuid, if a certain shear stress has to be applied before the suspension starts to ﬂow with a Newtonian behaviour. This behaviour can be described, according to Herschel and Bulkley, cited in [20] tð_gÞ ¼ t0 þ K g_ n

ð33Þ

where n ¼ 1 describes a Bingham ﬂuid. Suspensions with high particle concentrations tending to agglomeration are best described by this approach [71,72]. Furthermore, there are many other empirical correlations in the literature, which specify the rheological behaviour of suspensions. Detailed information is given, for example, in the work of Macosco [73]. The ﬂow behaviour of suspensions can qualitatively be classiﬁed into two regimes: at low shear rates particle–particle interactions dominate (if they are larger than the hydrodynamic interactions). In this case the suspension exhibits a yield stress (Bingham-ﬂuid). When attractive forces are present, a three-dimensional particulate network structure exists whose strength is determined by the sum of the overall interactions. The yield stress increases with increasing particle concentration (see Fig. 16) and decreasing particle size. The maximal yield stress can be found at the IEP (z ¼ 0) since, here, the repulsive forces vanish and only attractive forces occur. Often the rheology of concentrated suspensions is described by a relative viscosity Zr. This relates the viscosity of the suspension Z to the viscosity of the matrix ﬂuid Z0. Some examples for empirical equations for this are listed in Table 6. At higher shear rates hydrodynamic forces dominate over the interparticle interactions. In this case the ﬂow behaviour of the suspension can be described with the stress equivalent inner shear rate model according to GleiXle and Baloch [74]. The ratio of the forces between the particles and the hydrodynamic forces determine the border of the two regimes. The stress equivalent inner shear rate model is valid in the hydrodynamic determined shear rate range or for hard spheres, respectively. In this model, it is assumed that particles dispersed in the matrix ﬂuid reduce the space of the matrix ﬂuid in the shear gap (see Fig. 12). This leads to an increase of the inner shear rate of the ﬂuid between the particles. By introducing a shift factor B a master curve in the hydrodynamic regime can be build.

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M. Sommer and W. Peukert

Table 6. Empirical equations

Author

Relative viscosity (Zr) Zr ¼ 1 þ 2:5 cv ;

Einstein [65] Batchelor [76]

cv 5vol%

ð34Þ

Zr ¼ 1 þ 2:5 cv þ 6:2 c2v

ð35Þ

1 Zr ¼ p½Z 1 cpv

ð36Þ

Krieger and Dougherty [75]

p ¼ maximal packing density [Z] ¼ intrinsic 1 viscosity ¼ limcv !0 ðZr 1Þ cv

Fig. 12. Model of the concept of a stress equivalent inner shear rate model.

For a pure liquid (cv ¼ 0) one can write g_ 0 ¼

v ; t0 ¼ ZM g_ 0 s

ð37Þ

where ZM is the viscosity of the matrix ﬂuid. The particles are assumed to be concentrated in z rigid plates of thickness h. This reduces the gap for the pure liquid by zh. Accordingly, the shear rate enhances g_ c ¼

v szh

ð38Þ

The shift factor can then be calculated by B¼

g_ susp ðtÞ 1 1 ¼ ¼ zh 1 cv g_ 0 ðtÞ 1 s

ð39Þ

This shift factor relates the shear rates of the suspension g_ susp ðtÞ to the shear rate of the matrix ﬂuid g_ m ðtÞ at an equal shear stress and is only dependent on the solid concentration. The advantage of this model is that the ﬂow behaviour of the suspension can be described by a single master curve and the shift factor B.

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4.2. Rheology of electrostatically stabilized suspensions Electrostatically stabilized suspensions can exhibit a high degree of order, because a regular arrangement of particles is energetically favourable due to the mutual repulsion of particles. Energy must be applied to destroy this ordered structure leading to a yield stress. A further reason for the yield stress can be a colloidal network structure due to agglomeration, which has to be destroyed by shear. The necessary energy consumption is reﬂected in the yield stress. This is demonstrated in Fig. 13, which shows the inﬂuence of the pH value on a milled alumina suspension. Owing to the adjustment of different pH values, the particle interactions were modiﬁed. As discussed earlier the z-potential changes with the pH value (see Fig. 14). Higher absolute values of the z-potential are leading to a stronger

shear stress τ / Pa

10 pH = 5.5 pH = 6.4 pH = 6.87 pH = 7.32 pH = 8.75 pH = 9.47 pH = 9.96 pH = 11.0

1

0.1

0.01 0.1

1 10 100 shear rate γ /1/s

1000

Fig. 13. Inﬂuence of the pH value on a milled alumina suspension (x50,3 ¼ 218 nm, vt ¼ 10 m/s, dMK ¼ 0.7–0.9 mm, cm ¼ 0.2) [3]. 60 40

1.5

20 1.0

yield stress ζ-potential

0

0.5

ζ-potential / mV

yield stress τ0 / Pa

2.0

-20

0.0

-40 5

6

7

8 9 pH- value / -

10

11

12

Fig. 14. Correlation of the yield stress with the pH value and the z-potential for the milled suspension of Fig. 13 [3].

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M. Sommer and W. Peukert

repulsion, which reduces the yield stress. The highest yield stress can be found at the IEP (z ¼ 0) since there only attractive forces are acting. The yield stress can be correlated with the square of the z-potential if particle size and volume concentration of the solids are constant [3]. The sign of the potential has no inﬂuence since all particles are equally charged. As can be seen, the rheological behaviour of the suspension can be controlled by changing the interparticle interactions by several orders of magnitude. This is a good example for the control of macroscopic product properties by regulation of the microscopic particle interface. Figure 15 shows typical ﬂow curves for an alumina suspension in dependence of the milling progress. The initially Newtonian ﬂuid develops a distinct yield stress in the low shear rate regime with increasing milling time and respectively decreasing particle size. Here the interparticle interactions are determining the system. With decreasing particle size the interactions between the particles are increased due to smaller particle–particle distances leading to a higher yield stress. At higher shear rates the hydrodynamic interactions dominate. In this regime the ﬂow rates can be shifted to a single master curve. The shift factor of suspensions, which do not show a yield stress, is only dependent on the volume fraction of the particles (30 and 60 min). With increasing speciﬁc energy input the shift factor increases. The reason for the increasing shift factor is that the particles are not totally dispersed even at high shear rates. Hence, the volume fraction of the particles cv increases to an effective volume fraction cv,eff due to immobilized solvent in the cavities of the agglomerates. The agglomerate size increases with increasing bond strength of the agglomerates, which increases with decreasing particle size.

30 min (x50,3 = 1,965 µm) 60 min (x50,3 = 0,602 µm) 120 min (x50,3 = 0,411 µm) 240 min (x50,3 = 0,307µm)

10

6 5

shift factor B /-

shear stress / Pa

100

1

0.1 480 min (x50,3 = 0,207 µm)

0.01 0.1

1406 min (x50,3 = 0,161 µm) 2875 min (x50,3 = 0,133 µm)

water

1

10

100

1000

inner shear rate B γ / 1/ s

10000

4 3 2 1 0 10

100

1000

4000

Em /kJ/kg

Fig. 15. Rheology of an alumina suspension in dependence of the milling time and the particle size, respectively [3].

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583

4.3. Rheology of sterically stabilized suspensions In this section the case is discussed where polymer chains are adsorbed on single particles. The phenomenon where a single polymer chain is adsorbed onto different particles, which is desirable for ﬂocculation, is not covered here. If the polymer concentration is sufﬁciently low and the adsorption equilibrium is preferable, then the polymer chains are adsorbed on the particle surface. An example for this case is shown in Figs. 16 and 17. Here the rheological behaviour of sterically stabilized a-Al2O3 suspended in water and ethanol, respectively, is

10

T=25°C c /c = 0.03 x = 150 nm

9

1 increasing wt%

0.1 c 0 0.1 0.2 0.3 0.4 0.5

0.01 water

0.001 1

10

γ B=

8 shift factor B /-

shear stress / Pa

10

τ /Pa 0 0.005 0.04 0.3 0.7 1

100

η /mPa s 1.0 1.14 1.47 3.19 6.66 9.0

n 1 0.995 0.957 0850 0.785 0.745

1000

(τ)

γ 0(τ)

7 6 5 4 3

B=

2

1 1-c v

1

0 10000 0.00

0.06

inner shear rate B γ / s-1

0.12

0.18

0.24

cv /-

Fig. 16. Flow curves of sterically stabilized a-Al2O3 in water in dependency of the weight fraction. 5 T=25°C c /c = 0.03 x , = 150 nm

B= increasing wt%

0.1

0.01 ethanol

0.001

γ

4

1

1

cm /0 0.2 0.3 0.4 0.5

τ0 η0 /Pa /mPa s 0 1.19 0.015 1.3 0.04 1.4 0.07 1.5 0.4 6.6

10 100 1000 inner shear rate B γ / s-1

n 1 1.02 1.05 1.07 0.93

shift factor B /-

shear stress τ / Pa

10

(τ)

γ 0(τ)

3

2

1 B=

1 1-c v

0

10000 0.00 0.04 0.08 0.12 0.16 cv /-

Fig. 17. Flow curves of sterically stabilized a-Al2O3 in ethanol in dependency of the weight fraction.

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M. Sommer and W. Peukert

shown in dependency of the mass concentration of alumina. The experiments are carried out on a Physica rheometer at 251C using a double-gap geometry. To effectively stabilize the particles sterically, the polymer concentration was selected from the plateau region of the adsorption isotherm (see Fig. 8). Thus, the DAPRAL to alumina concentration was chosen to be cDAPRAL =cAl2 O3 ¼ 0:03. The parameters used to model the data according to the model of Herschel and Bulkley are shown in the legend of the plot. All suspensions show non-linear ﬂow curves and exhibit a yield stress. With increasing wt% of alumina particles the yield stress increases. The ﬂow curves of Al2O3 particles in ethanol are similar to the ﬂow curves in water. However, the zero shear viscosity and the yield stress are conspicuously smaller than for Al2O3 particles in water, especially for high mass concentrations. The ﬂow curves are shifted to a master curve at high shear rates according to the concept of shear equivalent inner shear rate from GleiXle and Baloch, where the system is determined by hydrodynamic interactions and not by interparticle interactions. The shift factor is calculated by equation (39) with the measured shear stress at a shear rate of 1000 s1. The so determined shift factors are much larger than the theoretical shift factors determined by the volume fraction. The reason for this could be adsorbed polymer chains on the particles, which increase the volume fraction of the particles cv to an effective volume fraction cv,eff. In this case the polymer cover around the particles is added to the particle volume. The layer thickness of adsorbed polymer can be estimated with the dependency of the shift factor from the particle volume concentration. B1 B rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3 6 c v;eff V tot ¼ pN

cv;eff ¼

x pþ2d

d¼

x pþ2d x p 2

ð40Þ

ð41Þ ð42Þ

According to this a layer thickness of about 50–55 nm is estimated. A different method to estimate the layer thickness rheologically is shown in Figs. 18 and 19. Here, the elastic and viscous moduli obtained from oscillatory measurements are plotted over the weight fraction of alumina particles stabilized with polymer. The experiments were performed at a constant frequency of 1 Hz and a strain of 0.01 at 251C. The crossover point at which the elastic modulus G0 is equal to the viscous modulus G00 can be taken as an indication of the volume fraction at which the adsorbed layers just overlapped [78,79], because this point characterizes the transition between liquid- and solid-like behaviour. The crossover point occurs at a weight fraction of 0.38 in water and 0.35 in ethanol. An approximation of the median distance between the particles at this volume

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585

102 101

moduli G', G'' /Pa

100

water T=25°C /c = 0.03 c x1,2 = 150nm ω = 1 Hz

10-1

overlapping of polymer chains

10-2 10-3 10-4 10-5

elastic modulus G' viscous modulus G''

10-6 10-7 0.0

0.1

0.2 0.3 weight fraction of alumina /-

0.4

0.5

Fig. 18. Plot of moduli as a function of weight fraction at a constant frequency of 1 Hz for steric stabilized alumina in water. 1e+1 ethanol T = 25°C c /c = 0.03 x = 150nm

moduli G',G'' /Pa

1e+0

ω = 1 Hz

1e-1

1e-2 overlapping of polymer chains

1e-3

1e-4

elastic modulus G' viscous modulus G''

1e-5 0.0

0.1

0.2 0.3 weight fraction of alumina /-

0.4

0.5

Fig. 19. Plot of moduli as a function of weight fraction at a constant frequency of 1 Hz for steric stabilized alumina in ethanol.

fraction provides a chain length of 42 nm in water and 53 nm in ethanol. By this approximation it is assumed that the particles are ordered in a cubic densest packing of spheres. The estimated chain length corresponds to the used DAPRAL with the molecular weight of 40,000—50,000 g/mol.

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Hence, the milling experiments should be carried out below a weight fraction of 0.35, because an overlap of the polymer chains would restrict the free motion of not yet adsorbed polymers in the suspension. The position of the crossing-point in dependence of the weight-fraction of particles may also serve as a ﬁrst estimate of the maximal achievable mass concentration and determines, thus, the economy of the process.

5. EXPERIMENTS IN NANOMILLING By producing particles smaller than a median particle size of 1 mm a steady state between breakage, deagglomeration and agglomeration exists in the milling process. This equilibrium is controlled by interparticle interactions as well as the milling conditions. As particles decrease in size the interparticle forces between the particles become dominant. Attractive forces lead to agglomerates when the particles collide, thus acting against the comminution process. To prevent this agglomeration process, the particles in the mill must be stabilized by increasing the repulsive forces in the suspension. This will move the steady state to smaller particle sizes. To produce particle sizes below 1 mm in stirred media mills, a detailed understanding of the agglomeration process and its mechanism is needed. In the following example, we demonstrate the importance of controlling agglomeration. We consider in this study primary SiO2 nanoparticles of around 30 nm diameter and observe the agglomeration and dispersing process under perikinetic, orthokinetic and under milling conditions. Figure 20 shows the evolution of the hydrodynamic diameter over time for a 5 wt% Ludox (SiO2) suspension with 0.6 M KNO3 under perikinetic and under milling conditions. To slow down the agglomeration kinetics the temperature was set to 121C. The curve with the ﬁlled circles shows the perikinetic agglomeration of the Ludox suspension. Under milling conditions, the agglomeration of the particles is accelerated until the steady state between agglomeration and deagglomeration is reached. This steady state is strongly dependent on the energy input. As can be seen in Fig. 20, the steady-state particle size decreases with increasing stirrer tip speed. Similar results were found from other authors as well [55,56,80–85]. However, these authors studied particles in the size range of several microns. None of them used a stirred media mill, instead they worked in Couette ﬂow systems. The data in Fig. 20 show that these ﬁndings can be transferred to stirred media milling and that agglomeration plays a signiﬁcant role by producing particles in the nano size range. The agglomeration kinetics increase for an increasing stirrer tip speed. However, the steady-state size between agglomeration and dispersing is smaller for higher tip speeds.

Enabling Nanomilling through Control of Particulate Interfaces

587

1000 hydrodynamic diameter xDLS /nm

5wt% Ludox; pH8.84; T=12°C; 0.6M KNO3

vt

100 perikinetic agglomeration milling experiments:

vt

4 m/s 8 m/s

30 0

50

100

150 time /min

200

250

300

Fig. 20. Inﬂuence of the milling time on the agglomeration under milling conditions in a 1 l Drais mill. 1000 hydrodynamic diameter xDLS /nm

5wt% Ludox; pH8.84; T=12°C; tip speed 4m/s

100 0.2M KNO3

c

0.3M KNO3 0.6M KNO3 0.9M KNO3

30 0

50

100 150 time /min

200

250

Fig. 21. Inﬂuence of different salt concentrations on the agglomeration under milling conditions.

Figure 21 shows the evolution of the hydrodynamic diameter over time for a 5 wt% Ludox suspension with the salt concentration as parameter. The experiments were carried out at a constant tip speed of 4 m/s. Under milling conditions, the agglomeration of the particles is accelerated until the steady state between

588

M. Sommer and W. Peukert time /h (SiO2 Mikrosil LS500) 0

5

10

15

20

1100 hydrodynamic diameter xDLS /nm

pH8.84; T=12°C

1000

Ludox TM50 25 4 m/s 0.6M KNO ; 5wt% 8 m/s SiO2 Mikrosil LS500 8 m/s, 0 M KNO ; 20wt% 8 m/s, 0.6 M KNO ; 20wt% 8 m/s, 0.6 M KNO ; 5wt%

100

30 0

50

100 150 200 time /min (Ludox)

250

300

Fig. 22. Evolution of the hydrodynamic diameter over time for milling experiments with Ludox and SiO2.

agglomeration and deagglomeration is reached. This steady state is dependent on the salt concentration. With increasing salt concentration the median particle size is larger and the agglomeration kinetics are faster. Figure 22 shows that the steady state between breakage and agglomeration can also be reached by real breakage of solid particles. In this diagram the hydrodynamic diameter is plotted over the milling time. The lower abscissa is for the agglomeration and dispersing data of Ludox particles. Because the time scale of real breakage of SiO2 is different from the agglomeration and dispersing data (ﬁlled symbols) a second abscissa for the grinding experiments with SiO2 Mikrosil LS500 (open symbols) is introduced. The data with the ﬁlled symbols are for milling experiments with 5 wt% Ludox in 0.6 M KNO3. For this system the agglomeration dominates the process and the steady state is reached starting from small particles. To show that the steady state can also be reached from larger particle sizes, SiO2 Mikrosil (quartz, Westdeutsche Quarzwerke) with a median particle size of 6 mm was used as feed material. The experiments were performed at pH 8.84, there the maximal z-potential of 35 mV was measured. The data with the halfﬁlled circles were measured for a 20 wt% SiO2 suspension at a tip speed of 8 m/s. In this experiment no salt was added to show were the minimal median particle size x50,3 under stable conditions lays for these milling parameters. A median particle size of 100 nm was reached. In a further experiment 0.6 M KNO3 was added to the feed suspension at the starting time of the experiment. The addition of salt increases the agglomeration rate and the steady state moves in favour of agglomeration. The median particle

Enabling Nanomilling through Control of Particulate Interfaces

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pH = 8 (ζ = 29 mV) pH = 10 (ζ = -27mV)

0.2

particle size x50,3 /μm

pH = 5 (ζ = 47mV) pH = 4 (ζ = 68mV)

ZrO2 /94.5% cm = 0.2

0.1

dGM = 0.8mm ϕGM = 0.85 vt =10m/s 105 specific energy Em / kJ/ kg

Fig. 23. Inﬂuence of different pH values on the grinding result of an alumina suspension.

size of the steady state for a 20 wt% SiO2 suspension is 500 nm. To compare these experiments with the agglomeration and dispersing experiments for Ludox, an experiment under 5 wt% SiO2 was performed. The steady state at these conditions is with 390 nm lower than for the experiments with 20 wt%. However, the steady state for the milling experiments with Ludox lies below this result at the same conditions. The reason for the difference may be due to the different manufacturing techniques of the solids and associating to this, different impurities, which inﬂuence the stability of the particles. A further example for the inﬂuence of different suspension stability conditions on the milling result is shown in Fig. 23 for an alumina suspension. The suspension stability and agglomeration behaviour of alumina is strongly inﬂuenced by the pH value according to the DLVO theory [6]. With decreasing pH values, the absolute value of the z-potential and with it the stability is increasing. During a grinding experiment a certain pH value was kept constant by titrating 10 M HNO3 to the suspension. As grinding media yttrium stabilized ZrO2 grinding media with a diameter of dGM ¼ 0.8 mm at a ﬁlling ratio of fGM ¼ 0.85 was used. The solid concentration was 0.2 and the tip speed of the discs was set to 10 m/s. Figure 23 shows the median particle size x50,3 measured by dynamic light scattering plotted over the speciﬁc energy input. The particle size decreases with increasing speciﬁc energy. The ﬁnal particle size depends on the pH value and suspension stability. The higher the z-potential the higher is the stability and agglomeration. By inﬂuencing the z-potential due to adjustment of the pH value the steady state between breakage, deagglomeration and agglomeration can be shifted to the breakage side. The same ﬁneness is achieved no matter at what stage of the

590

M. Sommer and W. Peukert 1 vt = 10 m/s, dGM = 0.4-0.8 mm,

particle size x50,3 / μm

ZrO2, cm = 0.1- 0.2, DT 1200

0.1

TiO2 SiO2 SnO2 Al2O3 0.01 103

104 105 specific energy Em / kJ/ kg

106

Fig. 24. Nanomilling results for different oxides.

process the pH value is adjusted to the desired suspension stability. This is demonstrated with an experiment that starts at pH 10. During the experiment at a speciﬁc energy input of 170,000 kJ/kg, the pH value was changed from pH 10–pH 5. The result is a shift in particle size to the particle size achieved with an experiment where the pH value was set to pH 5 from the very beginning. This means that there are small particles at a low z-potential present. However, these particles are agglomerated. The agglomerates are so strong that they cannot be dispersed with ultrasound at pH 3 for external particle size analysis [4]. This means the suspension stability does not inhibit the production of primary particles, but inﬂuences the ﬁnal dispersed state of the particles. In Fig. 24, nanomilling results for different oxides are compared. The milling conditions were for all experiments the same. In the ﬁgure particle sizes, measured with an ultrasonic spectrometer, are plotted with respect to of the speciﬁc energy. The ﬁnest particle sizes in the range of 10–20 nm were achieved for SnO2 and Al2O3 particles. As discussed in the next chapter, milling of alumina particles in water is strongly inﬂuenced by mechano-chemical changes. During milling alumina hydroxide is build, which dissolves and inﬂuences in that way the grinding behaviour in the sub-micron size range. This fact might explain the possibility to ‘‘grind’’ alumina particles down to median particles sizes below 10 nm. However, by milling SnO2 no inﬂuence of mechano-chemistry was detected at all. Crystallite sizes obtained von Rietveld analysis of the XRD spectrum (line broadening) was almost similar to BET based diameter, i.e. almost single crystal were obtained for SnO2. The reasons when a material is inﬂuenced by mechano-chemistry are not yet understood. The milling conditions were carefully optimized for alumina and tin

Enabling Nanomilling through Control of Particulate Interfaces

591

1000

x1,2 /nm

Al2O3 Ethanol cm = 0.2 /c c

= 0.03

100

UPA UPA (experiment repeated) BET

10 102

103

104 Em /kJ/ kg

105

106

Fig. 25. Milling results for steric stabilized Al2O3 particles in ethanol.

oxide, but not yet for titania and silica, so that even ﬁner sizes may be possible for the latter materials. Figure 25 shows milling results of sterically stabilized alumina particles in ethanol. The Sauter mean diameter x1,2 is plotted in this diagram over the measured speciﬁc energy input. The experiments were carried out on a Netzsch mill (Labstar) at a rotary speed of 2000 min1 (vt ¼ 8.5 m/s). Yttrium stabilized ZrO2 milling beads (dGM ¼ 0.5–0.6 mm), at a ﬁlling ratio of 0.8, were used and the solid mass concentration of the alumina particles was set to cm ¼ 0.2. The starting particle size was 800 nm measured by dynamic light scattering. As can be seen in Fig. 25 a ﬁnal particle size of x1,2 ¼ 60 nm was reached after 24 h milling time. The particle size measured by BET was x1,2 ¼ 40 nm for the ﬁnal product. The BET surface area was determined by a standard 4-point analysis (p/p0 ¼ 0.1, 0.2, 0.3 and 0.4). In contrast to milling experiments in water [2], no inﬂuence of mechano-chemical changes could be detected by milling of alumina particles in ethanol. Because no hydroxide layer develops on the surface of the particles no inﬂuence of the outgasing temperature was detected. Furthermore, no micro pores could be measured using the V–t method according to de Boer [89]. In contrast to this Stenger [2] showed that micropores are formed in water due to the phase transformation of the surface layer from bayerite to g-Al2O3 leading to an increase of the speciﬁc surface area. Figure 26 shows the corresponding rheological ﬂow curves in dependency of the milling time. The graph shows that in the ﬁrst 13 h the rheological behaviour of the milling suspension is Newtonian and the ﬂow behaviour is determined by the

592

M. Sommer and W. Peukert Al2O3 T = 25°C 10 cm= 0.2 /c c

= 0.03

1 increasing milling time

0.1 0.01 0.001 100

ethanol

101

ethanol 170 min (x1,2=124 nm) 5 h (x1,2=120 nm) 9 h (x1,2= 94 nm) 13 h (x1,2= 98 nm) 17 h (x1,2= 90 nm) 20 h (x1,2= 70 nm) 22 h (x1,2=60 nm) 24 h (x1,2=60 nm)

102 103 inner shear rate B γ / s-1

104

17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0

shift factor B /-

shear stress / Pa

100

105

0

40000 80000 120000 Em / kJ/ kg

Fig. 26. Flow curves of steric stabilized a-Al2O3 in ethanol in dependency of the milling time (for the calculation of the shift factor, see equation (39)).

hydrodynamic interactions. The interactions between the particles can almost be neglected. Stenger [1] showed that the shift factor for Newtonian suspensions without a yield stress is only dependent on the volume fraction and not on the median particle size. After 9 h the particle size decreases below 100 nm and a yield stress develops. The development of the yield stress can be traced back to the decrease of particle size and the associated increasing interparticle interactions. With increasing speciﬁc energy input the shift factor increases. The reason for the increasing shift factor is that the particles are not totally dispersed even at high shear rates. The volume fraction of the particles cv increases to an effective volume fraction cv,eff due to immobilized solvent in the cavities of the agglomerates. The agglomerate size increases with increasing bond strength of the agglomerates, which increases with decreasing particle size. In addition, the speciﬁc surface area of the particles increases with increasing speciﬁc energy input. Hence, more polymers can be boned on the particle surface, which increases the effective volume fraction and leads to a higher shift factor. As described in this chapter, the grinding behaviour of particles in the nano size range and the minimum achievable particle size is strongly inﬂuenced by the suspension stability and thus the agglomeration behaviour of the suspension. Therefore, an appropriate modelling of the process must include a superposition of the two opposing processes in the mill i.e. breakage and agglomeration which can be done by means of population balance models. The use of population balance equations to model grinding processes was ﬁrst proposed by Epstein [90] and is now widely accepted. Many publications describe the modelling of grinding processes of different dry mills [91–93], wet mills [94,95] and agglomeration processes [96–98] by population balances. Modelling must now include the inﬂuence of colloidal surface forces and hydrodynamic forces on particle

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593

aggregation and breakup. The superposition of the population balance models for agglomeration and grinding with the appropriate kernels leads to a system of partial differential equations, which can be solved in various ways numerically. For solving the complex system of partial differential equations there are generally ﬁve methods known from literature that are described by Ramkrishna [99]. Besides the method of successive approximations and the method of Laplace transformation that can both only be applied to problems with certain kernels (e.g. constant or sum kernel), the Monte Carlo simulation method is restricted to systems with only small numbers of particles due to the computational effort. The most successful approach so far is done by discretization of the particle space domain that divides the continuous size range into discrete size classes. Hounslow et al. [100] and Litster et al. [101] developed a discretization procedure for the aggregation problem that was extended to the breakage problem by Vanni (2000). Kumar and Ramkrishna [102] developed a new method of discretization (pivot technique) that involves a selective reﬁnement of a relatively coarse grid while the number of sections is kept to a minimum that was successfully applied to agglomeration/breakage problems. Numerical problems with the conservation of mass and/or the correct total number of particles are overcome by correction. Recently, it was shown that sub-micron grinding in stirred media mills could be well modelled by a superposition of grinding and agglomeration [1,103]. Numerically derived results (PARSIVAL, (CiT GmbH, Rastede, Germany)) from a population-balance model that accounts for agglomeration and breakage are in reasonable agreement with experimental observations. This opens the possibility to determine optimal stability conditions necessary to archive a desired particle size. A parameter study shows that the superposition of grinding and agglomeration leads to a steady-state PSD that is mainly inﬂuenced by the width of the breakage function produced in a breakage event and the stability factor. The initial width of the PSD has no inﬂuence on the ﬁnal width of the PSD.

6. MECHANO-CHEMICAL EFFECTS DURING NANOMILLING It is well known that the mechanical activation of solids by applied stresses, e.g. in mills, can induce several solid-state reactions. Various experimental investigations did study the inﬂuence of different milling devices on texture, structure, phase transformation and thermal behaviour of various types of alumina and their hydroxides [86,87]. However, the reported mechano-chemical phenomena are very complex in nature and are often inconsistent with each other due to the different intensities and types of mechanical stresses that are induced by different milling devices. The focus of the following section is on mechano-chemical changes of alumina during nanomilling and the inﬂuence of different suspension properties on it.

594

M. Sommer and W. Peukert A

pH = 8, vt = 10m/s

diffraction intensity / cps

A

A

A

A

A

G B B

B

G 0 min 60 min 120 min 240 min 480 min

20

30 40 diffraction angle 2Θ / °

50

60

Fig. 27. XRD patterns for different milling times (A ¼ a-Al2O3, B ¼ Al(OH)3-bayerite, G ¼ Al(OH)3-gibbsite) [4].

Figure 27 shows XRD-patterns for a-Al2O3 suspensions in dependency of the milling time. The experiments were conducted at pH 8 and a stirrer tip speed of 10 m/s. The measurements of the XRD patterns show a decrease in the diffraction intensity and the characteristic peaks for a-Al2O3 with increasing milling time. This can be attributed to the decreasing crystal size of alumina. Furthermore, characteristic peaks that could be identiﬁed as those of pure Al(OH)3 (Bayerite) appeared after a milling time of 120 min, which indicates the existence of a second phase. With increasing milling time the diffraction peaks of this phase increases. This shows the increasing amount of bayerite produced during the milling process. Additionally after 240 min a second hydroxide phase could be detected, which could be attributed to Gibbsite (Figs. 28 and 29). Corresponding TGA experiments were performed under nitrogen atmosphere at a heating rate of 10 K/min. The results show a loss of mass of up to 20 wt% at temperatures between 200 and 3001C. The volatile component could be detected as water in a mass spectrometer (Blazer QME 125). This dehydration process is expected to be due to the transformation of alumina hydroxide to alumina. Measurements in a DSC cell (TA Instruments) display an endothermic peak at a temperature of around 2301C that was measured for pure bayerite as well. The temperature, where the heat ﬂow peaks appear, corresponds to the conversion temperature from Al(OH)3 to Al2O3. The areas under the curves correlate with the conversion enthalpies known in literature. The amount of the generated hydroxide can be calculated by comparing the conversion enthalpies of the milled sample to the enthalpy of pure bayerite. The pure bayerite as a reference sample was precipitated from a 10 wt% suspension of Al(NO3)3 and a 4 M NH4OH solution,

Enabling Nanomilling through Control of Particulate Interfaces

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mass loss Δm/m / -

0,0

-0,1 60 min 120 min -0,2

240 min 480 min

-0,3

100

bayerite

200

300

400 500 600 temperature / °C

700

800

900

Fig. 28. TG data for various milling times [2].

0.5

heat transfer / W/g

0.0

-0.5

60 min 240 min 120 min

-1.0

480 min -1.5

pH = 8, vt = 10m/s

-2.0

-2.5 100

bayerite

150

200 250 temperature / °C

300

350

Fig. 29. DSC curves for various milling times [4].

aged in mother liquid for 24 h, washed, ﬁltered and dried at 451C under vacuum. These results clearly show the change in structure and phase formation due to the applied stresses in a wet grinding process. Figure 30 shows the amount of hydroxide formation during the grinding process. The three methods agree very well. The values determined from XRD patterns by means of Rietveld analysis are

596

M. Sommer and W. Peukert specific energy Em / kJ/kg 104

103

105

amount of Al(OH)3 /-

0.16

DSC

0.12

TG XRD

0.08

0.04

0.00 10

100 milling time / min

1000

Fig. 30. Amount of hydroxide produced during milling.

amount of Al(OH)x3-x /Altot /%

100

80

Al3+ Al(OH)2+

60

Al(OH)3,aq Al(OH)4-

40

Al(OH)2+ 20

0

Al(OH)3,s

3

4

5

6

7 8 pH-Wert /-

9

10

11

12

Fig. 31. Calculated aluminium phase equilibria of bayerite in dependence on solution pH.

somewhat lower than values determined from the other methods, because the XRD method only accounts for the crystal and not the amorphous phase. The hydroxide phase, which is developed during milling, dissolves below pH 5 and above pH 9. In the intermediate pH range a hydroxide layer is formed around the particles. This is supported by calculations of the thermodynamical equilibria for different hydroxyl complexes in dependence of the pH value in Fig. 31. The data

Enabling Nanomilling through Control of Particulate Interfaces

597

steric stabilization in ethanol and toluene

0.0

heat flow / W/g

-0.5

-1.0

-1.5

-2.0

sample

ΔHtrans /kJ/kg

steric in ethanol and toluene

0

0%

steric in water

136.7

14.8%

electrostatic in water

132.4

14.3 %

923.6

100 %

bayerite

-2.5 100

150

electrostatic stabilization in water

%changed to Al(OH)3

200 250 temperature / °C

steric stabilization in water

bayerite 300

350

Fig. 32. DSC-measurements for milled alumina in different solvents.

are calculated with the software program MEDUSA [88] based on tabulated equilibrium constants. The observations and calculations show that the bayerite phase dissolves at pH values lower than 5, which in turn can inﬂuence the grinding behaviour in the sub-micron size range. Owing to suspension stability and thus suspension viscosity reasons, it is best to set the pH value of the alumina suspension to pH 5 or lower. The fact that median particle sizes below 10 nm can be reached might be explained by this dissolution process. The grinding mechanism is at such pH values determined by a combination of grinding, agglomeration, deagglomeration, mechanical activation, hydroxide formation and dissolution. To examine if the solvent and the stabilization mechanism has an inﬂuence on the amount of aluminium hydroxide produced during milling, milling experiments for a-alumina in water, toluene and ethanol were performed under otherwise similar milling conditions. Calorimetric measurements of the product samples were accomplished in a DSC cell. In Fig. 32, the measured heat ﬂow is plotted over the temperature. The results of the DSC analysis show endothermic heat ﬂow peaks for the samples milled in water. The temperature at which the heat ﬂow peaks appear corresponds to the conversion temperature from Al(OH)3 to Al2O3. The areas under the curves correlate with the conversion enthalpies. The amount of the generated hydroxide can be calculated by comparing the conversion enthalpies of the milled sample to the enthalpy of pure Bayerite. For the electrostatically as well as for the sterically milled sample in water, 14% of alumina hydroxide was formed. In contrast to this, no mechano-chemical changes could be determined for the samples milled in ethanol and toluene.

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M. Sommer and W. Peukert

The particle sizes obtained by BET analysis for Al2O3 in organic solvents are around 40 nm. Thus, nanoparticles can be obtained in organic solvents as well.

7. SUMMARY The postulated grinding limit at around half a micron that has existed for decades does not exist. Particle sizes down to 10 nm can be obtained. Nanomilling in aqueous and non-aqueous media becomes possible when the particulate interfaces are carefully controlled to prevent agglomeration of the broken fragments. Particle stability ensures also low suspension rheology. Any inﬂuence of the type of mill is not pronounced, if it exists at all. Important features are small milling beads, a high energy density in the mill, i.e. stirrer speeds in the range of 6–12 m/s and inhibition of agglomeration. Stabilization must be faster than the breakage kinetics. The latter can be estimated from population balance models. Ionic stabilization is preferable wherever it works because this mechanism is rather simple and can be detected on-line during milling by means of electroacoustic spectroscopy. Even though stabilization can be achieved by the steric mechanism more research is necessary to understand details of the stabilization process. Nanomilling is a neat example which shows that real breakthroughs become possible even in very old and traditional technologies by fundamental studies of the underlying mechanisms. Nanomilling became, thus, an option for those who look for simple and versatile processes to nanoparticles.

Nomenclature

A a A123 B c c0 cm cv d D dF dGM DN e

Hamaker constant (J) radius of primary particles (m) non-retarded Hamaker constant for media 1 and 2 interacting across media 3 (J) shift factor (–) concentration (M) concentration of ions in the bulk phase (M) mass concentration (–) volume concentration (–) pressure wave decay parameter for acoustophoresis (m) particle diffusivity (m2/s) fractal dimension (–) diameter of the grinding media (m) donor number (kcal/mol) elementary charge (e ¼ 1.6602 10–19)C (C)

Enabling Nanomilling through Control of Particulate Interfaces

Em E F g G00 G0 h h I K k KW ls Md Md,0 mMG Mn n n N nr ns p P Pe q r rH /r2S3/2 S < SE Sm SMP SN St T vt W WN x x x1,2 x50,3 z

599

mass speciﬁc energy (J/kg) mean mass speciﬁc power input (W/kg) Faraday constant (F ¼ 9.6485 107 C/kmol) (C/mol) acceleration of gravity (m2/s) viscous modulus (Pa) elastic modulus (Pa) thickness (m) sphere surface to surface separation (m) ionic strength (mol/l) constant (Pa s1/n) Boltzmann constant (k ¼ 138,066 1023 J/K) (J/K) dissociation constant (mol2/l2) segment length (m) torque (N m) idle torque (N m) milling charge mass (g) molecular weight (g/mol) exponent (–) refractive index (–) particle concentration (l1) rotary speed (s1) number of segments in a polymer chain (–) pressure (N/m2) power input (W) Peclet number (–) charge (C) particle centre to centre distance (m) hydrodynamic radius (m) quadratic averaged end to end distance of the polymer chains (m) geometric constant (–) gas constant 8.3144 103 (J/(kmol K)) stress energy (Nm) speciﬁc BET surface area (m2/g) BET surface area of micro pores (m2/g) stress number (–) total speciﬁc BET surface area (m2/g) temperature (K) tip speed (m/s) stability ratio (–) stability ratio for rapid coagulation (–) distance from particle surface (m) particle diameter (m) Sauter mean diameter [m] mean value of the Q3 distribution (m) electrolyte valence (–)

600

a ac b b0 w ws d d0 ds DmGM e e0 er j0 jGM js a f 2 F FB Fel Fsteric Fosm steric Ftot Ftot,max Ftot,min1 Ftot,min2 Fvdw g_ Z Z0 Zm Zr k l m n n1 ne np rF rp tð_gÞ t0 o

M. Sommer and W. Peukert

expansion factor (–) collision efﬁciency (–) agglomeration kernel with interparticle interactions (s1) agglomeration kernel without interparticle interactions (s1) Flory–Huggins parameter (–) adsorption energy parameter (–) thickness of the polymer layer (m) normalized distance between two particles (m) Stern layer thickness (m) grinding media wear (g) absorption frequency (s1) permittivity of the vacuum (8.854 1012/AsV/m) (AsV/m) dielectric constant (–) electrostatic surface potential (V) ﬁlling degree of grinding media (–) Stern potential (V) median volume fraction of the segments on the adsorbed layer (–) interaction energy (J) Born interaction energy (J) electrostatic repulsion energy (J) steric interaction energy (J) osmotic steric interaction energy (J) total interaction energy (J) maximum of the total interaction energy (J) primary minimum of the total interaction energy (J) secondary minimum of the total interaction energy (J) van der Waals energy (J) shear rate (1/s) dynamic viscosity (kg/(ms)) dynamic viscosity g-0 (kg/(ms)) matrix viscosity (kg/(ms)) relative viscosity (kg/(ms)) inverse Debye screening length (m1) wavelength (m) electrophoretic mobility (m2/s/V) kinematic viscosity (m2/s) molar volume of the solvent (–) mean electronic absorption frequency in the UV (s1) polymer density at the particle surface (m2) density of the ﬂuid (kg/m3) particle density (kg/m3) shear stress (Pa) yield stress (Pa) frequency (s1)

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REFERENCES [1] F. Stenger, Grenzﬂa¨cheneffekte bei der Nanozerkleinerung in Ru¨hrwerkskugelmu¨hlen, PhD thesis, TU Munich, Shaker Verlag, Aachen, ISBN 3-8322-3910-3, 2005. [2] F. Stenger, M. Go¨tzinger, P. Jakob, W. Peukert, Part. Part. Systems Charact. 21 (2004) 31–38. [3] F. Stenger, W. Peukert, Chem. Eng. Tech. 26 (2) (2003) 177–183. [4] F. Stenger, S. Mende, J. Schwedes, W. Peukert, Powder Technol. 156 (2–3) (2005) 103–110. [5] F. Stenger, S. Mende, J. Schwedes, W. Peukert, Chem. Eng. Sci. 60 (16) (2005) 4557–4565. [6] S. Mende, F. Stenger, W. Peukert, J. Schwedes, Chem. Ing. Tech. 74 (2002) 994–1000. [7] S. Mende, F. Stenger, W. Peukert, J. Schwedes, Powder Technol. 132 (1) (2003) 64–73. [8] S. Mende, Mechanische Erzeugung von Nanopartikeln in Ru¨hrwerkskugelmu¨hlen, Dissertation, Cuvillier Verlag Go¨ttingen, Braunschweig, 2004. [9] H.C. Hamaker, Physica 4 (1937) 1058–1072. [10] W. Peukert, C. Mehler, M. Go¨tzinger, Part. Part. Systems Charact. 18 (5–6) (2002) 229–234. [11] W. Peukert, C. Mehler, M. Go¨tzinger, Appl. Surf. Sci. 196 (1–4) (2002) 30–40. [12] J. Israelachvilli, Intermolecular and Surface Forces, Academic Press, London, 1991. [13] D.B. Hough, L.R. White, Adv. Colloid Interface Sci. 14 (1) (1980) 3–41. [14] D.C. Prieve, W.B. Russel, J. Colloid Interface Sci. 125 (1) (1988) 1–13. [15] D. Tabor, R.H.S. Winterton, Nature 219 (5159) (1969) 1120–1121. [16] G. Lagaly, O. Schulz, R. Zimehl, Dispersionen und Emulsionen, Steinkopff, Darmstadt, 1997. [17] M. Go¨tzinger, Zur Charakterisierung von Wechselwirkungen partikularer Feststoffoberﬂachen, PhD thesis, TU Munich, Shaker Verlag, ISBN 3-8322-4242-2, 2004. [18] H.D. Ackler, R.H. French, Y.M. Chiang, J. Colloid Interface Sci. 179 (2) (1996) 460–469. [19] A.S. Russell, D.A. Saville, W.R. Schowalter, Colloidal Dispersions, Cambridge University Press, Cambridge, 1989. [20] R.J. Hunter, Foundations of Colloid Science, Oxford University Press, New York, 1986. [21] M.E. Labib, Colloids Surf. 29 (1988) 293–304. [22] E.J.W. Verwey, Recl. Trav. Chim. Pays-Bas 60 (1941) 625. [23] M.E. Labib, D.R.M. Williams, Colloid Polym. Sci. 264 (1986) 533–541. [24] J. Lyklema, Fundamentals of Interface and Colloid Science, Academic Press, London, 1993. [25] V. Gutmann, The Donor–Acceptor-Approach to Molecular Interactions, Plenum, New York, 1978. [26] V. Gutmann, Electrochim. Acta 21 (1976) 661. [27] R. Schmid, J. Solution Chem. 12 (1983) 135. [28] J.E. Huheey, E.A. Keiter, R.L. Keiter, Anorganische Chemie, Walter de Gruyter, New York, 1995. [29] E.C.Y. Yan, Y. Liu, K.B. Eisenthal, J. Phys. Chem. B 102 (33) (1998) 6331–6336. [30] B. Derjaguin, Kolloid-Zeitschrift 69 (1934) 155–164. [31] G.M. Bell, S. Levine, L.N. McCartney, J. Colloid Interface Sci. 43 (1970) 335–359. [32] B.W. Ninham, Adv. Colloid Interface Sci. 83 (1–3) (1999) 1–17. [33] L. Gu¨nther, W. Peukert, Colloids Surf. A-Physicochem. Eng. Asp. 225 (1–3) (2003) 49–61.

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CHAPTER 14

Analysis of Milling and the Role of Feed Properties Mojtaba Ghadiri, Chih Chi Kwan and Yulong Ding Institute of Particle Science and Engineering, University of Leeds, Leeds LS2 9JT, UK Contents 1. Introduction 2. Energy laws revisited 3. Evaluation of milling rate 4. Population balance models 5. Role of feed properties and deﬁnition of dimensionless groups describing the breakage propensity 5.1. Young’s modulus (E) 5.1.1. Flexure testing 5.1.2. Compression testing 5.1.3. Indentation testing 5.2. Hardness (H) 5.2.1. Indentation testing 5.3. Yield stress (sy) 5.3.1. Compaction studies 5.4. Critical stress intensity factor (Kc) 5.4.1. Single-edge notched beam (SENB) 5.4.2. Double-torsion testing 5.4.3. Radial edge cracked tablets 5.4.4. Indentation fracture test 5.5. Discussion 5.6. Derived parameters 6. Analysis of milling rate based on the input power and material properties 7. Conclusions References

605 606 608 610 613 614 614 615 615 616 616 616 617 618 618 619 620 621 622 623 624 629 632

1. INTRODUCTION Particle size reduction by mechanical means, such as milling and grinding, is a very important industrial operation in powder processing. It is achieved by a combination of body fracturing and surface damage under the action of applied Corresponding author. Tel: +44 113 343 2406; Fax: +44 113 343 2405; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12017-0

r 2007 Elsevier B.V. All rights reserved.

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stresses. In any size reduction process, the feed material generally comprises of a range of particle sizes. The crushing strength of the material usually varies with particle size with the larger particles tending to be weaker as compared to their smaller counterparts. The composition, structure and mechanical properties may vary from particle to particle even for the same particle size of a material. Besides, in a size reduction process various stressing conditions exist, depending on the machine design, such as compression, impact, bending, shearing, torsion and so on. It is however difﬁcult to measure such stresses directly [1]. Consequently, in order to obtain an understanding of the milling processes, the interactions of the material properties (material function) and the mill hydrodynamics (mill function) should be analysed [2].

2. ENERGY LAWS REVISITED The objective of milling is to reduce particle size with minimum energy consumption. Early attempts to derive models to predict energy consumption for size reduction are due to Rittinger [3], Kick [4] and Bond [5]. These models are empirical ﬁts to experimental results and express the speciﬁc energy, Em, required to reduce particle size as a function of particle size, X, and can be presented in a general form as given by dX dE m ¼ C n1 ð1Þ X where C and n1 are constants. Rittinger [3] has related the energy required for size reduction directly to the new surface area produced. Surface area in turn is proportional to the square of diameter, i.e. n1 ¼ 2, therefore the energy required is inversely proportional to the particle diameter as follows [6]: 1 1 1 1 Em ¼ C K Rf c ð2Þ Xp Xf Xp Xf where Xp and Xf are particle sizes after and before milling, respectively, KR the Rittinger’s constant and fc the milling strength of the material. Rittinger’s model is generally more applicable to ﬁne milling where the increase in surface area per unit mass is large [6,7]. However, milling is a very ineffective process. The energy applied is mostly dissipated in forms other than size reduction. Hence, in practice, the actual energy requirement is considerably higher than that required for the sole creation of new surface area. Kick [4] suggests that the speciﬁc energy required for size reduction is directly proportional to the particle size reduction ratio as expressed by Xf Xf E m ¼ C log ¼ K K f c log ð3Þ Xp Xp where KK is the Kick’s constant. For this case: n1 ¼ 1.0.

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Kick’s model implies that for a given material the speciﬁc energy is independent of particle size, for example, the energy required to reduce the particle size from 500 to 100 mm is the same as that of 50–10 mm. It therefore oversimpliﬁes the effect of particle size. It is well-known that for brittle material failure as the particle size is reduced, the number of ﬂaws present in a particle decreases, hence the energy required to initiate fracture is higher. Bond [5] has proposed a model that is intermediate between Rittinger’s and Kick’s models with n1 ¼ 1.5 and the speciﬁc energy required for size reduction is written as sﬃﬃﬃﬃﬃﬃ ! 1 1 1 1 E m ¼ 2C pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ ¼ 2C 1 pﬃﬃﬃ ð4Þ Xp q Xf Xp where q ¼ Xf/Xp. Letting C ¼ 5Ei results in sﬃﬃﬃﬃﬃﬃﬃﬃ 100 1 1 pﬃﬃﬃ Em ¼ Ei Xp q

ð5Þ

where Ei is referred to as the Bond work index, which represents the energy required to reduce unit mass of material from an inﬁnite particle size to 80% of the sample passing 100 mm [6]. Extensive investigations on the Bond work index have made this model useful for preliminary mill sizing. The approach of analysing the milling process by energy laws (equations (2), (3) and (5)) has been reported in a number of publications [8–13]. More recently, Chen et al. [14] analysed the milling behaviour of a-lactose monohydrate (aLM) in an oscillatory single-ball mill, as shown in Fig. 1, and examined the applicability of the three energy laws to describing the milling behaviour of aLM. Effects of milling frequency and powder loading on the milling behaviour of aLM were examined. Chen et al. [14] report that the milling results for low powder loadings at high milling frequencies are best described by using Rittinger’s [3] model. This corresponds to impact motion of the milling ball on the particles, where fragmentation is the dominant size reduction mechanism. However, for high powder loadings at low milling frequencies, Kick’s [4] model is found to ﬁt the milling data better than the other two models [14]. Here the milling ball has a rolling motion in

50 mm (11 ml)

12 mm

Jar movement

Fig. 1. The oscillatory single-ball mill used by Chen et al. [14].

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the mill, where surface attrition of the powder is the prevailing size reduction mechanism.

3. EVALUATION OF MILLING RATE Bowdish [15] has made an analogy between the particle breakage rate in a mill and chemical reaction kinetics. He carried out size reduction experiments with narrow feed size ranges of rock in a mono-sized ball mill and assessed the validity of zero-, ﬁrst- and second-order rate equations for describing the milling rate of materials. By analogy with the theory of chemical reaction kinetics, the rate equation can be written as

dRi ¼ k x Rai 1 AbB1 dT R

ð6Þ

where Ri is the concentration of the feed material, TR the number of revolutions of the mill, AB the surface area of the ball charge and kx, a1 and b1 are constants. The ﬁrst-order rate equation can be written as the following, whereby the effect of area of the ball charge is indirectly taken into consideration in the rate constant, k1.

dRi ¼ k 1 Ri dT R

ð7Þ

By conducting batch milling experiments using different sizes of mono-sized balls on various narrow size ranges of rock, Bowdish [15] observed that plots of the percentage of feed material remaining, Ri, with respect to the revolutions of the mill, TR, ﬁt linearly on the semi-logarithmic scale under the experimental conditions investigated, indicating that the ﬁrst-order rate equation is applicable. The milling rate constant, k1, that can be established from the plots, represents the effectiveness of the combination of mill, balls and rock charge in breaking the feed materials [15]. Deviations from the ﬁrst-order rate equation are observed when very large or very small balls are used, and at the very beginning or after prolonged milling. It has been widely reported that prolonged milling in a batch mill will not necessarily improve the size reduction rate [16,17]. Quite often, prolonged milling can have a reverse effect on size reduction. Larsson and Kristensen [18] reported an increase in the average particle diameter of Ibuprofen after prolonged milling in a Micros Ring Mill. Similarly, Kano et al. [19], observed a decrease in the speciﬁc surface area of talc powder after prolonged milling in all the milling equipment types tested, namely planetary, vibrating and tumbling mills. The particle surface properties tend to predominate as the particle surface to mass ratio increases. Hence as milling progresses, agglomeration due to Van der Waals and other short-range forces begins to take effect. This could explain the deviation from the ﬁrst-order rate equations as stated by Bowdish [15]. Despite these deviations, most of the experimental results ﬁt the semi-logarithmic plots with a long straight

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portion, indicating that the ﬁrst-order equation is generally valid to describe the size reduction of particles in ball mills [15]. Recently, Kanda and co-workers studied the milling rate of silica glass, quartz, limestone and gypsum with different feed sizes and various milling ball diameters [20,21]. By milling narrow size ranges of different feed materials, milling rates were found to be related to the mass fraction of component i (Ri) by the following equation:

dRi ¼ k i Ri dt

ð8Þ

where ki is the rate constant. This equation is similar to that stated by Bowdish [15], except Bowdish [15] uses the number of mill revolutions, TR, to quantify milling duration. Kotake et al. [20,21] have concentrated their effort in decoupling the milling rate constant from other parameters such as feed size. From all the materials tested, it is found that ki can be expressed by the feed size (Xf) and the ball diameter (dB) in the following form. ! Xf C4 C5 k i ¼ C2 d B X f exp C3 C ð9Þ dB6 where C2, C3, C4, C5 and C6 are constants determined experimentally. Saito and co-workers extended this approach further and correlated the milling rate constant, determined experimentally, with the milling energy, the latter as established by computer simulation of the ball movement in media mills [19,22–27]. They ﬁrst ﬁtted the size reduction rate of various inorganic materials milled under different conditions in various types of media mills, such as planetary, tumbling and vibrating ball mills, using a ﬁrst-order rate equation Dl Dt Dl ¼ 1 ð10Þ exp K p t þ D0 D0 D0 where Dt is the median diameter of the milled sample at time t, Dl the median diameter at the milling limit of the sample where no further size reduction is possible even with further milling, D0 the median diameter of the original material, Kp the milling rate constant of the material and t is the milling time. Kp was then related to the milling energy as described below. Saito and co-workers have simulated three-dimensional ball motion in various types of mills by using discrete element method (DEM) following the work of Cundall and Strack [28]. They use a simple contact model based on a frictional slider and the Voigt model that consists of a spring and a dashpot [22]. The actual ball motion in the mill is obviously dependent on the type of feed material selected. Kano et al. [22] account for the effect of various feed materials on the ball motions indirectly by selecting an appropriate coefﬁcient of friction for the colliding balls. The coefﬁcient of friction is in turn inferred from the angle of repose of the feed material.

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Once the ball motions are simulated, the speciﬁc impact power of the balls, Pne, can be calculated from the sample mass (W), mass of the ball (m) and the relative velocity of ball to ball and/or ball to wall (vj) at every time step as the following: P ne ¼

n X 1 mv 2j 2W j¼1

ð11Þ

where n is the number of collisions between balls and balls to walls within 1 s. The milling rate constants, Kp, as determined experimentally for various inorganic materials have been correlated with their respective simulated speciﬁc impact energy, Ene [26]. Jimbo and co-workers have also attempted to relate the energy in a vibration ball mill to experimental milling results [29]. They conducted experiments on 350–590 mm glass beads with various fractional ball ﬁlling ranging from approximately 0.57–0.97. The milling rate of 350–590 mm glass beads with various ball ﬁllings was found to follow the ﬁrst-order rate equation, with the milling rate constants increasing with increasing fractional ball ﬁlling to 0.9. Yokoyama et al. [29] also developed a model for two-dimensional ball movements based on DEM. The simulated ball motions were veriﬁed experimentally by video recording of the milling process under the same operating conditions. In a study by Yashima and Saito [30], the single-particle compressive strength of 350 mm glass beads is reported to be 60 N. Yokoyama et al. [29] utilised this result in their simulation by discarding any ball collisions that result in a force smaller than 60 N. They then calculated an e¡ective breaking collision frequency and found that the simulation results relate well to the experimental milling rate.

4. POPULATION BALANCE MODELS Broadbent and Callcott [31] applied population balance modelling to milling processes and characterised the related selection and breakage functions. The selection function describes the breakage probability of a particle in a mill, and is a function of mill dynamic. The breakage function expresses the size distribution of broken products and reﬂects the material behaviour in response to mechanical stresses. Both the selection and the breakage functions are determined experimentally and then incorporated in the population balance model to account for the mass fraction of each product size range [32]. Krogh [33] subjected individual particles to impact by a falling object. From this testing method, three basic milling characteristics were quantiﬁed, i.e. crushing probability function, energy function and breakage function. Particles of the same material with the same size do not necessarily have the same strength, hence the crushing probability function as stated in Krogh [33] is a measure of the strength distribution of particles for a given size.

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The energy required to break a particle of a given size is calculated from the height and the mass of the falling object. The energy function (Ef) and the crushing probability function (Fc) are combined to provide the following equation: F c 0:5 c E f ¼ k 2 d exp ð12Þ b1 where d is the particle size and b1, k2 , c are characteristic constants of a material determined experimentally. If Ef is proportional to the surface area of the particle (pd2), c will be equal to 2. If Ef is proportional to the volume of the particle (pd3), c will take the value of 3. From the experimental results reported by Krogh [33], it is found that c ranges from 2 to 3 for all materials tested. The breakage function is used to describe the size distribution of the broken material when Ef corresponds to a crushing probability of 0.5 [33]. During the process of milling, excessive impact energy may result in extreme particle size reduction. However, if the impact energy is too low, the stress generated in a particle may not be sufﬁcient to cause particle fragmentation. Somewhere between these two extreme cases lies an optimum level of impact energy [34]. By using a testing device similar to that of Krogh [33], Pauw and Mare´ [34] have conducted impact testing on four size ranges of ore samples at different energy levels. Each size range of particles is impacted with a known load until all the particles are broken to smaller than their lower feed size. For instance, ore particles with feed size range of 9.5–12.7 mm are reduced in size until all the material can pass through the 9.5 mm screen. The number of impacts (N) required to achieve this is recorded. The total energy (ET) for each test is simply taken as the product of number of impacts (N) and energy per impact (Ej), with the latter determined from the mass (Ml) and the initial height (hl) of the dropweight. The speciﬁc breakage energy (Et) is deﬁned as the total energy divided by the total mass of sample tested (MT) as follows: Et ¼

E T NE j NM l ghl ¼ ¼ MT MT MT

ð13Þ

where g is the gravitational acceleration. By repeating the experiments with the same feed size range of materials, but at different impact energy levels, the correlation between the speciﬁc breakage energy (Et) with respect to the input energy per impact (Ej) is established. By plotting Et versus Ej for each feed size range of materials, a minimum Ej required for breakage can be clearly identiﬁed, as shown in Fig. 2, hence demonstrating the existence of an optimum impact energy for particle breakage [34]. It is also found that the energy requirement for breakage is highly dependent on the particle size. Kapur et al. [35] present a model which takes into consideration the variation of particle resistance to breakage (selection function) in relation to impact energy. By utilising the single-particle impact results for ore as given by Pauw and Mare´

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Fig. 2. Energy requirements for the breakage of ore feed particles between 9.5 and 12.7 mm [34].

[34], the constants in the model of Kapur et al. [35] can be quantiﬁed. Owing to variations in particle strength, some particles may not be broken during the ﬁrst impact event. They may, however, become progressively weakened and eventually fragment with repeated impacts. Kapur et al. [35] take account of the effects of repeated impact energy on the strength of particles and their model provides a useful tool to study the breakage function and the total energy required to break all particles in an ensemble. Pauw [36] considers two types of energy in a milling system, i.e. applied energy and required energy. The applied energy is the energy that is provided by the milling machinery for size reduction. The required energy, however, is what is needed to overcome the feed strength to cause fragmentation. To achieve size reduction, the applied energy should obviously match the required energy. There are many variables that can inﬂuence the applied energy and the required energy. By taking media milling as an example, the mass, composition of construction material, size and trajectory of the milling media can affect the applied energy for size reduction. However, the required energy by the feed material to undergo size reduction is governed by the following variables [2,36]: (a) mechanical properties, number, shape, size distribution of the particles; (b) interaction of particles with water in wet milling, and with air in dry milling; and (c) mechanisms of breakage. The analysis provided by Krogh [33], Pauw and Mare´ [34] and Kapur et al. [35] have concentrated on the required energy for size reduction. The simulation models developed by Saito and co-workers [19,22–27] and Jimbo and co-workers [29] have focussed on the ball movement in media mills, and hence

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on the applied energy. Apart from the work by Yokoyama et al. [29] on relating the energy applied in a vibration ball mill to the crushing strength of glass beads, there seems to be a little investigation on linking the energy applied in a mill to match the energy required for size reduction.

5. ROLE OF FEED PROPERTIES AND DEFINITION OF DIMENSIONLESS GROUPS DESCRIBING THE BREAKAGE PROPENSITY The feed material properties that inﬂuence the breakage of solid particles depend on the failure mode, i.e. brittle, semi-brittle and ductile. For the brittle failure mode, the size and number of pre-existing ﬂaws affect the strength and particle breakage occurs by crack propagation from the ﬂaws, to which the linear elastic fracture mechanics applies. It is difﬁcult to quantify the size, number and position of the ﬂaws, and hence the analysis approach for brittle failure is largely empirical. For the semi-brittle failure mode, the prevailing stresses reach the plastic yield, and hence plastic deformation precedes the crack propagation. In fact the extent of crack propagation depends on the plastic zone size. For this mode of failure, indentation fracture mechanics has been used successfully to describe the particle breakage. The material mechanical properties that govern the particle breakage in this failure mode are the elastic modulus, represented by Young’s modulus (E) in the case of linear elastic deformation, hardness (H), representing the yield stress (sy) and fracture toughness, also known as the critical stress intensity factor (Kc). These properties represent the resistance of the material to elastic deformation, plastic deformation and crack propagation, accordingly. Methods for characterisation of these properties have been developed (see e.g. Ghadiri [37]). For ductile failure, size reduction can be achieved by cutting, tearing and necking, as the yield stress of the material is too low to allow sufﬁcient elastic strain energy to be built up in the body of the particle to sustain crack propagation. The properties that are inﬂuential here are the yield stress of the particle, the interfacial friction and shear strength on the cutting tool and possible work-hardening features. For this failure mode, the slip-line ﬁeld theory of plastic deformation [38] can be used to describe material removal. The approach is by numerical analysis using ﬁnite element analysis. Most particulate solid materials fail in the semi-brittle failure mode by virtue of their size and shape, because the contacts are non-conforming, and the contact size is small. Therefore, the contact stresses during mechanical loading quickly reach the plastic yield causing plastic deformation followed by crack propagation. Here, the analysis of breakage is based on the principles of indentation mechanics for which the characterisation of Young’s modulus, hardness and toughness is essential.

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5.1. Young’s modulus (E) Young’s modulus (E), is a measure of the material stiffness and is deﬁned by the ratio of the stress over the accompanying strain within the elastic region of the stress–strain curve. Young’s modulus directly relates to the inter-atomic binding energy for the case of inorganic materials or inter-molecular binding energy for organic solids [39]. The stronger the bond of a material, the higher the resistance to bond stretching and hence a higher Young’s modulus of the material. Young’s modulus of a material can be obtained by several methods. The ﬂexure testing, compression testing and indentation technique are some commonly used methods.

5.1.1. Flexure testing During ﬂexure testing, a rectangular beam is subjected to a transverse load and its central deﬂection due to bending is measured. Two common conﬁgurations of this test exist with variations in the applied load and support to the beam and these are known as four- and three-point bending, as shown in Fig. 3. Young’s modulus of the beam can then be determined from the applied force, central deﬂection, geometry of the beam and the force loading conﬁguration as shown in equations (14) and (15) for the four-point [40] and three-point [41] beam bending, respectively. ! F 6a l 2 al a2 E¼ þ þ ð14Þ x h3 b 8 2 3 E¼

Fl 3

ð15Þ

4xh3 b

where F is the applied load, x the deﬂection of the midpoint of the beam, h the beam thickness, b the beam width, l and a are the distance between loading points. Beams used in the ﬂexure testing are usually prepared under different compression loads in order to achieve a range of beam porosities. Generally, Young’s F/2

(i) a

F/2 l

F

(ii) a

h

h b

b l

F/2

F/2

F/2

Fig. 3. Geometry for (i) four-point and (ii) three-point beam bending.

F/2

Analysis of Milling and the Role of Feed Properties

615

modulus decreases with increasing specimen porosity. Spriggs [42] and Spinner et al. [43] have derived exponential and second-order polynomial empirical curveﬁtting correlations, as shown in equations (16) and 17), respectively, to describe the relationship between Young’s modulus and the beam porosity. The elastic modulus of the particles is inferred from extrapolations to zero porosity. Bassam et al. [44] have shown that the loading rate and beam height generally have no signiﬁcant effect on Young’s modulus of various pharmaceutical excipients, for example, different grades of microcrystalline cellulose (MCC) and lactose powders. However, the orientation of the beam was reported to have a small effect on Young’s modulus [44]. E P ¼ E 0 eb2 P 1

ð16Þ

E P ¼ E 0 ð1 f 1 P 1 þ f 2 P 21 Þ

ð17Þ

where EP is Young’s modulus of beam compacted at porosity P1, E0 the Young’s modulus at zero porosity, b2, f1 and f2 are constants.

5.1.2. Compression testing Samples in single crystal form or compacted specimen have previously been subjected to compression testing. For a single crystal, Young’s modulus can be determined from a single-compression measurement, provided the geometry and loading are well deﬁned. As for compacted specimens, a series of tests on specimens with different porosities are required. The compressive Young’s modulus is determined from the ratio of stress to strain during compression testing [45], as shown in equation (18). Young’s modulus of a material is then extrapolated from the results gathered from the compacted specimens by using the equation proposed by Spriggs [42] or Spinner et al. [43]. EP ¼

L ðX 1 C1 Þ A

ð18Þ

where L is the specimen length, X1 the slope of the stress–strain curve, C1 the machine constant determined by loading the compression instrument without a specimen and A the specimen cross-sectional area.

5.1.3. Indentation testing Both single crystals and compacted specimens have been tested by the indentation technique to determine Young’s modulus. Duncan-Hewitt and Weatherly [46,47] have determined Young’s modulus of 1–4 mm sucrose and various pharmaceutical crystals by using a Vickers indenter. More recently, Bentham et al. [48] have reported measurements of Young’s modulus of Paracetamol and aLM

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crystals with sizes as small as 300 mm. Young’s modulus of aLM measured by indentation technique was found to be six times larger than that obtained by fourpoint beam ﬂexure testing [48]. Bentham et al. [48] have attributed the discrepancies to the correlations used for ﬂexure testing which do not consider the mechanisms like bond interactions and size reduction due to fragmentation during compaction of aLM.

5.2. Hardness (H) Hardness represents the resistance of a material to plastic deformation. The earliest attempt to quantify hardness is probably due to Mohs [49], who derived a hardness scale according to the scratch resistance of a material. However, the scientiﬁc basis for characterisation of hardness has been developed by Tabor [50], based on the indentation technique. For ﬁne particulate solids, the standard indentation method based on Vickers indenter is not possible due to size constraint and the most appropriate technique is the nanoindentation [37]. However, there are also a number of indirect methods based on bulk testing and these are brieﬂy described in the following.

5.2.1. Indentation testing In an indentation test, the hardness of a material is determined from the applied load, the indentation projected area and the shape of the indenter. Commonly used indenters include Brinell (10 mm sphere of steel or tungsten carbide), Vickers (square-based diamond pyramid), Knoop (also a diamond pyramid), Rockwell (diamond cone) and Berkovitch (three-sided diamond pyramid) [37]. Ridgway et al. [51,52] are among the earliest researchers who compared the hardness of crystals and compacted specimen for particulate solids. They reported that the Vickers hardness values of aspirin and sodium chloride show the occurrence of work hardening in compacted specimens, as tablets had greater hardness values than their parent crystalline substances [51]. Bentham et al. [48] have however measured the hardness of Paracetamol and aLM using a Berkovitch indenter. The hardness value of aLM by Bentham et al. [48] compares well with that obtained from the Vickers indentation technique extrapolated to zero porosity from compacted specimen using the equation proposed by Spriggs [42].

5.3. Yield stress (ry) The yield stress of a material can be inferred from its hardness provided the constraint factor, f, as given by equation (19), is known. The constraint factor is

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strongly inﬂuenced by anisotropy, the rate of work-hardening, indenter shape and interfacial friction with the indenter [37]. Generally, f for glasses and polymers would be smaller than three, for a wide range of metals would be approximately three, and for ionic single crystals would be usually higher than three [37]. sy ¼

H f

ð19Þ

5.3.1. Compaction studies The yield stress of particulate solids has been studied extensively by using the Heckel analysis of the compaction process [53]. Heckel [53] proposed that for materials undergoing plastic deformation, the relative density of a material (rr) could be related to pressure (P) during powder compaction as follows: 1 ln ð20Þ ¼ K H P þ AH 1 rr where KH and AH are constants. At both low and high compaction pressures, considerable deviations do occur due to particle rearrangement and work hardening, respectively. Generally, over the middle pressure range, a linear relation exists between ln[1/(1 rr)] and P, as illustrated in Fig. 4. Hersey and Rees [54] suggested that the reciprocal of KH could be considered as numerically equal to the mean yield stress of the powder. Roberts and Rowe [55] have used this method extensively to characterise the yield stress of pharmaceutical powders and have stated that the yield stress determined from the reciprocal of KH is comparable to that deduced from indentation hardness provided a very low compression speed is used with lubricated punches and dies on materials that do not undergo fracture during testing [39]. However, Hassanpour and Ghadiri [56] have recently evaluated the Heckel [53] analysis using DEM and showed that there is a critical ratio of Young’s modulus to the yield 8 Particle rearrangement

7

1

ln

1−r

6 5

Plastic deformation

4 3

σy =

2

1 KH

Strain hardening

1 0 0

50

100 150 200 Compaction Pressure (MPa)

Fig. 4. Schematic diagram of the Heckel plot.

250

300

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stress of individual particles (E/sy), above which the Heckel analysis does reﬂect the effect of the yield stress, but below which the reciprocal of KH is in fact a measure of Young’s modulus and not the yield stress as proposed by Hersey and Rees [54]. Thus, Heckel [53] analysis should be used with caution [56].

5.4. Critical stress intensity factor (Kc) The material resistance to crack growth is represented by Kc. However, the measurement of Kc for ﬁne powders is very difﬁcult. Recently, Bentham et al. [48] used the nanoindentation technique to quantify Kc, but this can only be done if cracks can be generated in ﬁne particles. For materials which deform extensively before crack propagation, the indentation depth can easily exceed the maximum allowable displacement by the mechanical testing machine, making it difﬁcult to generate cracks by a nanoindenter. Kwan [57] and Kwan et al. [58–61] have proposed an alternative method where the group H/K2c is evaluated by singleparticle impact testing using the model of Ghadiri and Zhang [62,63], and provided H can be quantiﬁed separately by, for example, nanoindentation, then the value of Kc can be determined. Kwan [57] and Kwan et al. [58–61] have characterised a number of pharmaceutical ingredients in this way, to which the milling rate of powders has been correlated, as will be discussed later. In literature, the Kc values of pharmaceutical materials are reported to have been determined mainly by conducting tests on compacted specimens with induced notches and/or cracks. The single-edge notched beam (SENB), double torsion and radial edge cracked tablets testing methods all involve measuring Kc values over a range of beam porosities. The Kc value for particles is then deduced by extrapolating to zero porosity from the correlation of Kc with respect to beam porosity by using either the exponential or polynomial empirical curve ﬁtting equation derived by Spriggs [42] or Spinner et al. [43], respectively. However, this approach can be misleading as the crack propagation may well occur along the interfaces of the compacted particles, rather than through their body, as the fracture surface energy is usually lower for the former case as compared to the latter. Nevertheless, the state of the art for all the techniques is summarised below.

5.4.1. Single-edge notched beam (SENB) In this test, a pre-notched rectangular beam is subjected to a transverse load and the load at fracture is measured [39]. Kc can then be derived from the applied force, geometry of the beam, and the geometry of the notch. Similar to that of the ﬂexure testing as described previously, the loading can either be three- or fourpoint and an example of a four-point loading setting on a SENB is illustrated in Fig. 5. Mashadi and Newton [64,65] were among the ﬁrst to measure Kc of

Analysis of Milling and the Role of Feed Properties F/2

619

F/2 l2

h c

b

l2 F/2

F/2

Fig. 5. Geometry for four-point single-edge notched beam.

pharmaceutical materials using this method, which was later adopted by York et al. [66] and Roberts et al. [67] for determining the Kc of various materials. York et al. [66] have reported that straight-through notches are more desirable than other types of notches as straight-through notches seem to have less inﬂuence over the measured values of Kc. Furthermore, York et al. [66] observed that for MCC, the loading rate has a negligible effect on the value of Kc, as a 100-fold increase in the loading rate only results in approximately 10% rise in the Kc value. In contrast, the beam porosity was found to have a profound effect on the measured Kc value, as a decrease in porosity leads to an increase in Kc, an indication of higher resistance to crack propagation. Major disadvantage of the SENB testing method lies in the difﬁculties in inserting sharp cracks into specimens. More importantly, the geometry and dimension of notches are reported to inﬂuence the measured values of Kc [67]. York et al. [66] have commented that Kc is not a unique material constant but is dependent on the specimen geometry and loading rate. However, the measurement of Kc using compacted specimen is affected by the structure, history of compaction (residual stresses) and the crack path (i.e. interfacial or internal) and should be treated with caution.

5.4.2. Double-torsion testing This test is carried out on a rectangular specimen with a narrow groove extending to its full length as shown in Fig. 6. Kc is deduced from the applied force, Poisson’s ratio of the material and the geometry of the beam. Only MCC and sorbitol have been tested using the double-torsion method by Mashadi and Newton [68]. Kc values measured in this way are higher than those reported by using the SENB method [39]. This may be attributed to the addition of shearing mode of fracture [68]. Shortcomings of this testing method include the

M. Ghadiri et al.

620 F

h

hn

Wn

F/2

F/2

Fig. 6. Geometry for the double-torsion method.

(i)

(ii) F

d

F

c

F F

Fig. 7. Geometries for (i) edge opening and (ii) diametral compression method for measuring critical stress intensity factors for radial edge cracked tablets.

requirement of large amount of material for beam preparation, and a very large compaction pressure required to obtain a specimen with low porosity for testing.

5.4.3. Radial edge cracked tablets In this test, the compacted specimen takes the form of a ﬂat-faced tablet rather than the rectangular beam. Roberts and Rowe [69] have investigated two different methods with a variation in the direction of applied load, namely edge opening method and diametral compression method, as illustrated in Fig. 7. In addition to these two methods, Kendall and Gregory [70] have experimented with pin-loaded method, which is similar to the edge opening method, except that a pin is inserted to either side of the crack. After extensive testing by Kendall and Gregory [70] on polymethylmethacrylate (PMMA) samples (ICI Perspex), and Roberts and Rowe [69] on MCC, these two groups of researchers both concluded that edge opening is the preferred method

Analysis of Milling and the Role of Feed Properties

621

over diametral compression method. The edge opening method gives the most stable crack propagation and furthermore the effect of crack length is minimal [39].

5.4.4. Indentation fracture test The critical stress intensity factor can be obtained from indentation cracking based on the principles of indentation fracture mechanics. The length of the cracks in this system is determined by the tensile stress ﬁeld around the indentation zone and it scales with the size of the plastic zone. Hence the crack length is a function of the hardness and the critical stress intensity factor, with the latter relevant to the radial cracks which develop as a result of indentation. Hence Kc measured here would differ from those measured by other methods described above, but it is in a way more relevant to fracture of particles by virtue of their shape and size. As particles are stressed during milling, the plastic deformation acts as an indenter, generating radial and median vent cracks. This occurs even in spherical particles, but is obviously more prevalent in particles with asperities and sharp corners and edges. Duncan-Hewitt and Weatherly [46] have determined the Kc value of laboratory grown large sucrose crystals (1–4 mm diameter) by using a Vickers indenter. Bentham et al. [48] have evaluated the Kc values of laboratory grown Paracetamol and aLM with average crystal size of 600 mm by using a Berkovitch indenter. Details of the measurement technique has been described by Ghadiri [37]. A shortcoming of this testing method is that the surfaces of the crystals for testing need to be smooth and clean in order for surface traces of the cracks to be measured accurately (Fig. 8).

Fig. 8. SEM micrograph of a nanoindentation on the (100) face of a lab-produced a-lactose monohydrate (applied force: 100 mN; indenter: Berkovitch pyramid) [48].

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5.5. Discussion There are often discrepancies in the literature over the measured values of mechanical properties of material. These are presumably due to artefacts arising from sample preparation, measurement method and inadequate assumptions. For the case of the sample preparation method, the main difference in the tested samples is its form, i.e. a compacted specimen or a single crystal. The main disadvantage of conducting tests on compacted specimens is the difﬁculty in distinguishing the intrinsic particulate properties from the assembly properties. For instance, the correlations of Spriggs [42] and Spinner et al. [43] used for determining Young’s modulus of individual particles by extrapolating data obtained from compacted specimen with a range of porosities, have originally been developed for sintered compacts. These correlations do not account for effects like bonding mechanism and size reduction due to fragmentation, which can take place during the preparation of a compacted specimen from powder. Some materials may work-harden during compaction and consequently the extrapolation from the compacted specimen will unavoidably give rise to unrealistic results. Besides, the preparation of a compacted beam will require a large amount of material and furthermore several beams at various porosities will be required for testing. Comparing to compacted specimens, the preparation procedure for single-crystal samples prior to testing is relatively simple. Only a small amount of material is required to generate information on the mechanical properties of a material. During indentation testing of single crystals, the hardness, Young’s modulus and Kc, for instances where cracks are formed, can be determined concurrently. Bentham et al. [48] have determined these three mechanical properties simultaneously for 500–710 mm aLM and Paracetamol particles. The effect of particle size on the mechanical properties of these two materials has also been reported in Bentham et al. [48]. However, there are also a number shortcomings here. It is difﬁcult to determine the mechanical properties for rough and damaged surfaces for which the data spread may be too wide. Moreover, cracks may not develop within the range of displacements available. For structured solids like crystals, the deformation ﬁeld could be inhomogeneous and anisotropic with prevailing slip planes and cleavage planes, and careful attention needs to be paid to crystal habit and structure [46]. Indentation fracture mechanics is the most pertinent technique, especially for the case of milling, as particulate solids are usually in point contact with each other. Hence, the local stresses can easily exceed the plastic yield stress and result in plastic ﬂow which initiates fracture. Besides, considering the uncertainty surrounding the particulate mechanical properties extrapolated from compacted specimen, indentation testing on single crystals is a way forward to elucidate the role of mechanical properties on the milling behaviour. However, as mentioned above, in cases where indentation does not generate cracks or where the material response is strain rate sensitive, then the approach developed by Kwan et al. [59]

Analysis of Milling and the Role of Feed Properties

623

based on impact testing can be used to quantify the group H/K2c , as will be discussed in greater detail below.

5.6. Derived parameters Lawn and Marshall [71] have suggested that the resistance of a material to plastic deformation (H) and to crack propagation (Kc) can be used to deﬁne an index representing the brittleness of a material as given in equation (21). For a brittle material failure, the hardness will be large and toughness will be low, thus the brittleness index will be very large and vice versa for a semi-brittle material. Brittleness Index ¼

H Kc

ð21Þ

Roberts et al. [67] have commented that although the drugs Ibuprofen and Paracetamol have very similar Kc values, Ibuprofen possesses a much lower H value, hence a smaller brittleness index, as compared to Paracetamol. This indicates that although both materials are weak, Ibuprofen is more ductile than Paracetamol. These suggestions are in agreement with the compaction behaviour of these materials [67]. Roberts and Rowe [55] have listed the ratios of E/sy and H/sy on the expected material behaviour in the form given in Table 1. These parameters provide a simple method in ranking different materials according to their tendency to fail by plastic deformation and/or fracture. However, the reaction of a material to applied load is not only dependent on the mechanical properties of a material but also on the mode of loading. Ghadiri and Zhang [63] proposed a mechanistic model, based on lateral crack propagation, to express the extent of breakage (R*) by impact for semi-brittle material failure due to chipping as shown by equation (22). Their proposed equation takes into consideration the physical and mechanical properties and is found to describe well the extent of impact breakage of a variety of pharmaceutical powders [48,58] and polymers [72]. R ¼ a

rdHv 2s K 2c

ð22Þ

Table 1. Ratios of material properties [55]

E/sy

H/sy

Descriptions

10–25

1.5–2.0

25–30 25–30 4150

2.0–2.2 42.2 43.0

Very elastic materials able to accommodate plastic strains, typical of many polymers Brittle materials typical of glasses A tendency towards a reduction in brittle behaviour Rigid, plastic materials, typical of metals

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M. Ghadiri et al.

where a is a proportionality constant, r the particle density, d the particle diameter and vs the particle impact velocity. The above group may be viewed as a dimensionless group, which describes the breakage propensity of particulate solids. Its application to the analysis of milling is addressed in the following section.

6. ANALYSIS OF MILLING RATE BASED ON THE INPUT POWER AND MATERIAL PROPERTIES There are very few publications on quantifying the milling behaviour of materials according to their mechanical properties. Zhao et al. [73] studied the milling behaviour of more than 20 different types of materials in a planetary ball mill. All the samples were milled under the same operating conditions and product size distributions were analysed by Rosin–Rammler distribution as shown in Fig. 9. They broadly characterised the milling behaviour of materials into ﬁve different types according to their structure and the descriptions of these categories are provided in Table 2. They also suggested that materials with a higher hardness value and a higher melting point resulted in ﬁner milling limit. However, no speciﬁc comparison between hardness value and milling limit of any materials was provided in the study of Zhao et al. [73]. For brittle material failure, an empirical approach based on the Weibull analysis [74] is widely used to describe the probability of fracture, S, when a brittle material specimen is subjected to stress, s, as otherwise it is difﬁcult to characterise the size, number and position of ﬂaws to carry out a more deterministic analysis. Weibull’s equation is given by m1 s S ¼ 1 exp z ð23Þ ss

Fig. 9. Classiﬁcation of material types according to Zhao et al. [73].

Analysis of Milling and the Role of Feed Properties

625

Table 2. Description of the ﬁve types of materials in Zhao et al. [73]

Type

Descriptions

Examples

A

Crystal materials with weakly bonded crystal structure Naturally formed minerals with many ﬂaws and coarse cracks Man-made hard ceramics with controlled ﬂaws and ﬁner crack distributions Materials with dominating cleavage planes Plastic and polymeric materials

Sugar, common salt

B C D E

Quartzite, silica sand, limestone Alumina, titanium carbide, silicon carbide Graphite –

where z is the number density of ﬂaws in the specimen and ss the characteristic strength. Using this concept, Vogel and Peukert [2,75,76] developed a model for quantifying the impact breakage behaviour of various materials. They made use of the dimensional analysis of Rumpf [77] and the fracture mechanics model of Weichert [78] to relate the stresses generated on impact to the incident kinetic energy and deduced the following equation: S ¼ 1 expff mat X f NðW m;kin W m;min Þg

ð24Þ

where fmat is a material parameter, which in comparison with Weibull’s model, represents the ﬂaw density. Xf is the initial feed size, N the impact number, Wm,kin the mass-speciﬁc kinetic impact energy and Wm,min is the mass-speciﬁc threshold energy for particle breakage. Here, again Wm,min represents a characteristic impact strength when compared to ss in Weibull’s model. The breakage behaviour of various materials including ﬁve polymer samples, limestone and glass spheres, has been studied by Vogel and Peukert [2]. Particles sieved to a narrow size range were subjected to impacts in a single-particle impact device developed by Scho¨nert and Marktsheffel [79], as shown in Fig. 10. It is observed that the extent of breakage of particles subjected up to three impacts can be ﬁtted into a single plot of S against N(Wm,kin – Wm,min). This observation also applies to fatigue testing of two different PMMA samples. The extent of breakage of all these materials can be uniﬁed with a plot of S versus fmatXfN(Wm,kin – Wm,min), which Vogel and Peukert [2] called the mastercurve as shown in Fig. 11. In Vogel and Peukert’s subsequent publications [75,76], they have also conducted milling experiments on four polymers using a Fritsch laboratory scale impact mill (Pulverisette 14). In order to overcome the uncertainty of the stressing number and intensity on the particles in the Fritsch mill, Vogel and Peukert

626

M. Ghadiri et al.

Fig. 10. Single-particle impact device [79].

Fig. 11. Mastercurve for the breakage probability of various materials [2].

[75,76] matched the breakage probability, S, of a well characterised PMMA sample in the single-particle impact device shown in Fig. 10 with that obtained from the Fritsch mill. They found that a ﬁtting factor of 1.4 provided a good uniﬁcation between the single-particle impact results with the results obtained from the Fritsch mill for the four polymers tested, generating a trend similar to that shown in Fig. 11.

Analysis of Milling and the Role of Feed Properties

627

As the model proposed by Vogel and Peukert [2,75,76] is based on Weibull’s model, it is in principle relevant to the breakage of brittle materials. However, it appears that the model also ﬁts well particle breakage due to the semi-brittle failure mode, although here the crack propagation is determined by plastic deformation, rather than pre-existing ﬂaws. It is interesting to note the similarities between the model of Vogel and Peukert [2,75,76] and the model of Ghadiri and Zhang [63]. The latter is given by equation (22) for quantifying the impact breakage of semi-brittle materials, and is based on the indentation fracture mechanics for crack propagation, from which the breakage is quantiﬁed in terms of H/K2c . Nevertheless, the extent of breakage in both the cases is proportional to feed size and impact kinetic energy. However, in the model of Vogel and Peukert [2,75,76], the mechanical and physical properties that are relevant to particle breakage are implicit in the best ﬁt parameter (fmat). The model of Ghadiri and Zhang [63] has been used for analysis of particle size reduction in the jetting region of ﬂuidised beds [48,80,81]. The use of this approach allows the development of methods for relating the milling behaviour to physical properties of the material and the mill hydrodynamics, as shown in equation (25). For example Bentham et al. [48] reported that by coupling the single-particle impact breakage data with the jet hydrodynamics model, the breakage propensity of particles of aLM in a ﬂuidised mill as a function of jet velocity can be predicted successfully. Bentham et al. [81] also tried to extend this approach to predict the milling behaviour of Paracetamol in a ﬂuidised mill. The predicted breakage propensity of Paracetamol as a function of jet velocity failed to match the milling experimental results. This was attributed to the uncertainty of the effects of particle shape and surface morphology on the jet angle and the lateral gas ﬂow rate in the jetting region of the ﬂuidised mill, which were considered to be crucial parameters in the hydrodynamic model. Ra / R W s

ð25Þ

where Ra is the size reduction rate in the ﬂuidised bed jetting region, R* the extent of impact breakage determined by single-particle impact testing and is related to the mechanical and physical properties of the test material according to equation (22) and Ws the solid entrainment rate as predicted by the jet hydrodynamics model proposed by Donsı` et al. [82]. Kwan [57] and Kwan et al. [58–61] have studied the milling behaviour of various pharmaceutical powders, including aLM, MCC and starch. The milling behaviour of these samples in a single-ball mill, as shown in Fig. 1, was quantiﬁed using the concept of milling rate constant (Kp), which corresponds to the change in sample size with respect to the milling duration. Subsequently, to model the milling behaviour of these powders in the single-ball mill, Kwan [57] and Kwan et al. [58–61] quantiﬁed the mechanical properties of the materials tested by conducting

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single-particle impact testing. In parallel, they analysed the ball mill hydrodynamics using DEM. The milling rate constant was then related to the mill hydrodynamics and single-particle mechanical properties. Single-particle impact testing was conducted in the rig developed by Yu¨regir et al. [83], as shown in Fig. 12, whereby individual particles were accelerated to the desired velocity and eventually impacted on a ﬁxed target. After each impact experiment, the sample was sieved and the extent of breakage was determined by gravimetric analysis. The model of Ghadiri and Zhang [63] was used to express the impact breakage results of aLM, MCC and starch particles in terms of the mechanical and physical properties of the test materials according to equation (22) [57,59]. The DEM simulations were performed using the mechanical and physical properties of samples tested following the approach of Saito and co-workers [19,22–27] to investigate the milling ball and powder motion in the single-ball mill and to quantify the impact milling power (Pn) [57,60]. The modelling results agree well with the experimental observations using a transparent milling jar and highspeed video recording of the milling process [57,60]. The milling rate constant of the three materials tested at different milling frequency in the single-ball mill were uniﬁed by coupling the group, which represents the material mechanical properties, i.e. H/K2c with the mill power, Pn, as shown in Fig. 13 [57,61]. Figure 13 suggests that the size reduction behaviour of a material could be established by conducting a few single-particle impact experiments (to obtain H and Kc) and by simulating the operation of a milling system (to ﬁnd Pn).

Manual feeding

Glass tube

Photodiodes

Target Collection chamber

PI Filter Vacuum line

Fig. 12. Single-particle impact test rig [83].

Milling Rate Constant, K p (s-1)

Analysis of Milling and the Role of Feed Properties

629

0.06 MCC-18 Hz 0.05

MCC-25 Hz

0.04

αLM-25 Hz

y= 0.1137x R2 = 0.9833

Starch-25 Hz 0.03 0.02 0.01 0 0

0.1

0.2 PnH /

0.3 K c2

(m2

0.4

0.5

s-1)

Fig. 13. The functional relationship between the milling rate constant and the coupled group incorporating the input power Pn and material properties H and Kc for aLM, MCC and starch at 18 and 25 Hz of milling frequencies.

7. CONCLUSIONS Little attempt has been made so far to correlate the milling behaviour of particles with their mechanical properties. The main reasons are lack of hydrodynamic models of mills and breakage functions that are based on the mechanical and physical properties of particles. There is also lack of information on inﬂuencing factors like particle size, strain rate, moisture content and temperature, all known to play a role in the mechanical properties of materials and hence their breakage. Furthermore, structural complexity (as prevailing e.g. in agglomerates), anisotropy and the existence of defects within a material can inﬂuence the breakage behaviour. The application of indentation technique to the analysis of mechanical properties of particles circumvents the ambiguity of extrapolating results from beam testing or compacted specimens to single-particle mechanical properties. However, current indentation techniques are generally limited to low strain rate conditions. In contrast, many industrial systems, such as milling, mixing and pneumatic conveying, operate well beyond the quasi-static strain rate regime. Thus, the mechanical properties determined from indentation may not be applicable to the actual process if the material is sensitive to the strain rate. There is also the limitation of requiring smooth surfaces for conducting a good indentation. Owing to these shortcomings, recent developments are based on inferring the mechanical properties from the impact breakage propensities of various materials, as adapted by Peukert and co-workers [2,75,76] as well as Ghadiri and coworkers [48,58,62,63]. A huge variety of milling equipment exists with each type having different efﬁciency, loading mode and operational functions. Hence, most of the empirical

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equations developed for describing the process of size reduction in speciﬁc milling equipment tend to have a limited use and are not easily transferable to other mills and with different feed materials. A better understanding of the mill hydrodynamics and associated energy/power input to a mill can provide a more fundamental and generic base for predicting the size reduction process. In recent years, signiﬁcant progress has been made on the computational capability to determine ball motion in various media mills, thus, allowing the energy provided by a mill to be predicted. Coupling the material properties determined by singleparticle impact testing with the mill hydrodynamics provides an approach that is analogous to reaction engineering, where the kinetics of a reaction is coupled with ﬂuid ﬂow dynamics to predict the conversion rate.

Nomenclature

a a1 A AB AH B b1 b2 C C1 C2 C3 C4 C5 C6 d dB D0 Dl Dt E E0 Ef Ei Ej Em EP Et

distance between loading points (m) constant in equation (6) (–) cross-sectional area of the specimen (m2) area of ball charge (m2) constant in equation (20) (–) beam width (m) constant in equation (6) (–) constant in equation (16) (–) coefﬁcient in equation (1) (variable) machine constant (m N1) constant in equation (9) (–) constant in equation (9) (–) constant in equation (9) (–) constant in equation (9) (–) constant in equation (9) (–) particle diameter (m) ball diameter (m) mass median diameter of feed (m) mass median diameter at the milling limit of the sample (m) mass median diameter of milled sample at time t (m) Young’s modulus (Pa) Young’s modulus at zero porosity (Pa) energy function (J) Bond work index (J kg1) energy per impact (J) milling energy per unit mass (J kg1) Young’s modulus of beam compacted (Pa) speciﬁc breakage energy (J kg1)

Analysis of Milling and the Role of Feed Properties

ET f1 f2 fc fmat F Fc g h hl H k1 k2 ki kx Kc KH KK Kp KR l L m m1 Ml MT n n1 N P P1 Pn Pne R* Ra Ri S t TR vj vs W Wm,kin Wm,min Ws

631

total energy (J) constant in equation (17) (–) constant in equation (17) (–) milling strength of material (N m2) material parameter (kg J1 m1) force (N) crushing probability function (–) gravitational acceleration (m s2) beam thickness (m) initial height of drop weight (m) hardness (Pa) rate constant in equation (7) (–) characteristic constant in equation (12) (–) rate constant in equation (8) (s1) constant in equation (6) (variable) critical stress intensity factor (Pa m1/2) constant in equation (20) (Pa1) Kick’s constant (m3 kg1) milling rate constant in equation (10) (s1) Rittinger’s constant (m4 kg1) distance between loading points (m) specimen length (m) mass (kg) Weibull exponent (–) mass of drop weight (kg) total mass of sample tested (kg) number of collisions between balls and ball to wall within 1 s (–) constant in equation (1) (–) number of impacts (–) pressure (Pa) porosity (–) mill power (J s1) milling power per unit mass (J kg1 s1) extent of breakage (–) size reduction rate (g h1) mass fraction of feed size (–) breakage probability (–) time (s) number of revolutions of the mill (–) relative velocity (m s1) particle impact velocity (m s1) sample mass (kg) mass-speciﬁc kinetic impact energy (J kg1) mass-speciﬁc threshold energy for particle breakage (J kg1) solid entrainment rate (kg m2 s1)

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X Xf X1 Xp z a b1 f r rr s ss sy x c

particle size (m) characteristic linear dimension of feed (m) slope (m N1) characteristic linear dimension of product (m) number density of ﬂaws (–) proportionality constant in equation (22) (–) characteristic constant in equation (12) (–) constraint factor (–) particle density (kg m3) relative density (–) stress (Pa) characteristic strength (–) yield stress (Pa) deﬂection of the midpoint of beam (m) characteristic constant in equation (12) (–)

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[60] C.C. Kwan, H. Mio, Y.Q. Chen, Y.L. Ding, F. Saito, D.G. Papadopoulos, A.C. Bentham, M. Ghadiri, Chem. Eng. Sci. 60 (5) (2005) 1441–1448. [61] C.C. Kwan, S. Gittins, W. Yang, Y.Q. Chen, Y.L. Ding, M. Ghadiri, Analysis of the milling behaviour of organic powders, in: R. Garcia-Rojo, H.J. Herrmann, S. McNamara (Eds.), Powders and Grains 2005, A.A. Balkema Publishers, Stuttgart, 2005, pp. 1431–1435. [62] Z. Zhang, M. Ghadiri, Chem. Eng. Sci. 57 (2002) 3671–3686. [63] M. Ghadiri, Z. Zhang, Chem. Eng. Sci. 57 (2002) 3659–3669. [64] A.B. Mashadi, J.M. Newton, J. Pharm. Pharmacol. 39 (Suppl.) (1987) 67P. [65] A.B. Mashadi, J.M. Newton, J. Pharm. Pharmacol. 39 (1987) 961–965. [66] P. York, F. Bassam, R.J. Roberts, R.C. Rowe, Int. J. Pharm. 66 (1990) 143–148. [67] R.J. Roberts, R.C. Rowe, P. York, Int. J. Pharm. 91 (1993) 173–182. [68] A.B. Mashadi, J.M. Newton, J. Pharm. Pharmac. 40 (1988) 597–600. [69] R.J. Roberts, R.C. Rowe, Int. J. Pharm. 52 (1989) 213–219. [70] K. Kendall, R.D. Gregory, J. Mater. Sci. 22 (1987) 4514–4517. [71] B.R. Lawn, D.B. Marshall, J. Am. Ceram. Soc. 62 (7–8) (1979) 347–350. [72] D.G. Papadopoulos, M. Ghadiri, Adv. Powder Technol. 7 (3) (1996) 183–197. [73] Q. Q. Zhao, G. Jimbo, Y. Kaneko, Fifth World Congress of Chemical Engineering San Diego, USA, 1996, pp. 187–191. [74] W. Weibull, J. Appl. Mech. 18 (1951) 293. [75] L. Vogel, W. Peukert, Int. J. Miner. Process. 74S (2004) S329–S338. [76] L. Vogel, W. Peukert, Chem. Eng. Sci. 60 (2005) 5164–5176. [77] H. Rumpf, Powder Technol. 7 (1973) 145–159. [78] R. Weichert, Zement-Kalk-Gips 45 (1) (1992) 1–8. [79] K. Scho¨nert, M. Marktsheffel, Sixth European Symposium Comminution, Nuremberg, Germany, 1986, pp. 29–45. [80] M. Ghadiri, R. Boereﬁjn, Kona 14 (1996) 5–15. [81] A.C. Bentham, C.C. Kwan, R. Boereﬁjn, M. Ghadiri, Powder Technol. 141 (3) (2004) 233–238. [82] G. Donsı` , L. Massimillar, L. Colantuoni, in: J.R. Grace, J.M. Matsen (Eds.), Fluidisation, Plenum, New York, 1980, pp. 297–304. [83] K.R. Yu¨regir, M. Ghadiri, R. Clift, Powder Technol. 49 (1986) 53–57.

Part III: Modelling

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CHAPTER 15

Monte Carlo Method for the Analysis of Particle Breakage Barada Kanta Mishra Institute of Minerals and MaterialsTechnology, Bhubaneswar, Indiay Contents 1. Introduction 2. Particle breakage and grinding 2.1. Random nature of breakage 2.2. Batch grinding 2.3. Conventional solution method 2.4. Stochastic solution method 2.4.1. Typical example of application 3. Monte Carlo method 3.1. Distribution functions 3.2. Random number generation 3.3. Sampling 3.3.1. The most useful sampling technique 4. Monte Carlo approach to batch grinding 4.1. Computational procedure 5. Numerical results and discussion 6. Summary References

637 639 639 640 642 643 644 645 647 648 648 649 650 651 653 658 659

1. INTRODUCTION The breakage of particulate materials is achieved by the application of compressive stress. All materials resist breakage and energy must be extended for the breakage process. The quantitative aspect of energy size relationship is of interest in ore dressing operations. Therefore, there has been a lot of research interest in studying the fundamental aspects of brittle particulate fracture and the associated energy. The precise mechanisms of fracture that occur when a brittle particle is subject to impact are unknown. Application of fracture mechanics met with limited success as the key parameter – fracture toughness is rather difﬁcult to interpret when brittle materials such as rocks and ores are considered. Corresponding author. Tel.: 91-512-2597263; Fax: 91-512-2590007; E-mail: [email protected] y

On deputation from Indian Institute of Technology Kanpur

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12018-2

r 2007 Elsevier B.V. All rights reserved.

638

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Although some progress has been made to study crack propagation in brittle materials, no quantitative relationships have yet been made regarding the progeny size distribution, the speciﬁc energy input, the strain rate, and the fracture toughness of the material. Single-particle breakage research has led to a better understanding of the fundamental aspects of brittle fracture. A variety of drop weight apparatus has been developed to probe the micro-scale events leading to fracture [1–3]. The ultra fast load cell (UFLC) developed at the University of Utah permits the accurate measurement of the energy absorbed during the fracture of a single particle or particle bed. It has been possible to measure the strain and the corresponding force experienced by the particle. These types of tests provide information on two important aspects of the fracture process. First, the tests allow determination of the impact energy when the particle fractures during the impact process, and second, the associated size distribution of the progenies after fracture. For identical particles the impact energy needed for fracture and the size distribution of the progenies varies. Armed with this information, it is possible to ﬁnd a relationship between the probability of fracture and the impact energy. The behaviour of particulate-bed fracture is also probabilistic in nature; however, it may appear to be deterministic since the experiments are typically done with a large population of particles. The probabilistic/stochastic nature is clearly seen when a small population of particles is tested. Observation of large population yields the mean value of the quantity being sought, but the probabilistic nature is hidden. For example, the breakage rate of 0.1/sec assigned to a 1 mm particulate population subjected to a breakage environment means that one out of every 10 particles will break during a one second period. However, a single test, on a small particle population may not show the desired breakage rate but as the number of particles and the number of tests increase the fraction of particles that break in one second would be very close to 0.1. Thus the mean quantity is brought to focus. This suggests that if the particulate process can be modelled for small samples using stochastic theory then, in principle, one would expect to be able to predict the behaviour of a large particulate system by focusing on the response of mean quantities. Furthermore, this type of approach provides clues to conducting statistical or Monte Carlo simulations as discussed later. The information gathered through single-particle as well as multi-particle breakage provides a quantitative description of the size reduction phenomena which is used in the modelling of comminution processes. In comminution devices, breakage involves reducing the particle size from feed to product. The rate at which particles break depends on a number of factors such as particle size, mill environment, etc. It should be recognized that breakage of one speciﬁed particle of

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639

a given size produces a complete range of progeny particles. Knowing how fast each size breaks and in what size the progenies appear gives rise to the concept of population balance which can be used to mathematically model the breakage process.

2. PARTICLE BREAKAGE AND GRINDING 2.1. Random nature of breakage The nature of particle breakage in a grinding environment is quite complex but the analysis of broken fragments can be conveniently used for modelling the grinding process. It is for this reason that a vast amount of research work has been carried out on single-particle fracture [3–6]. The results have clearly established the probabilistic nature of particle fracture and the distribution of the particle fracture strengths. The fracture strength of a particle is considered to be one of the key parameters in relation to its resistance to breakage. Using a twin pendulum impact device Narayanan [1], Narayanan and Whiten [7], and Lira and Kavetsky [8] estimated the ore-speciﬁc and energy-dependent breakage distribution function. They used these data for scale-up and modelling of industrial ball mills. Pauw and Mare [9] and Pauw [10,11] proposed optimum grinding paths or impact energies for the most efﬁcient utilization of the energy invested based on the extensive data on drop weight breakage of single particles. King and Bourgeois [6] studied single particle as well as particle bed breakage behaviour using the UFLC. The results of their experiments paved way for the development of microscopic population balance model of grinding. Single-particle fracture tests allow one to determine at what level of impact energy a particle fractures. It also allows one to determine the size distribution of the progenies. Many factors relating to the constitution of the particle at microscopic level determine whether the particle will fracture during an impact event. Nonetheless, the single particle tests provide information to establish the relationship between probability of fracture and the impact energy. Figure 1 shows a typical fracture behaviour of many individual particles of same type over a narrow size range [12], which indicates that while all particles will break after an impact at about 550 J/kg, a smaller proportion will break after impacts at lower speciﬁc impact energies. For example, 50% of the particles will break after an impact of about 120 J/kg. This distribution represents the probability of breakage for particles. This aspect of the breakage behaviour of particle is exploited in the stochastic modelling of breakage process such as grinding.

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Cum. amount broken (%)

100 90 80 70 60 50 40 30 20 10 0 10

100 Specific impact energy (J/kg)

1000

Fig. 1. Fracture probability and fracture energy distribution of particles.

2.2. Batch grinding There are two distinct ways to look at the grinding process: deterministic and probabilistic. Nonetheless, all studies of the grinding process either as deterministic or as stochastic must end up drawing similar conclusions. In the deterministic approach, the population of particles is quite large in the entire size range of interest and then the statistical averages of key parameters governing the process are considered to arrive at a deterministic model. Typically, the entire size range is divided into a number of size classes for computational simplicity. For a given size class, a couple of parameters are assumed to represent the resistance of particles to fracture and the speciﬁc milling conditions. These are basically the average characteristics of the group of particles. Once the average parameters are identiﬁed, then physically meaningful descriptive equations could be developed for simulation purposes. The model equation, however, lumps together the entire spectrum of breakage events which prevail in the system under a given set of operating conditions. In the stochastic approach the grinding of material is treated as a random process. When several single particles of a given material and size are subjected to impact under identical conditions, some break and some do not, reﬂecting the inherent randomness of the fracture process. If the data is made available over a statistically sufﬁcient number of tests, then it is easy to estimate the probability of breakage of the particle by simply considering the ratio of the number of particles that break to the total number of particles tested. This way, it is possible to calculate the distribution of the number of particles over a size range by determining the probability of breakage of a particle. Finally, by assuming that the grinding rate is proportional to particle size, a complete stochastic description of the grinding process can be obtained.

Monte Carlo Method for the Analysis of Particle Breakage

641

Traditionally, the grinding process is described by introducing two statistical functions: Breakage function and the Selection function. The selection function (S), gives the rates of breakage of particles, and the breakage function (B), describes the instantaneous size distributions of fragments produced when the particles are broken. In the population balance framework, the average behaviour of the particles during grinding is considered. The rate of breakage S and the breakage distribution function B are used to describe the average behaviour of many particles. The size continuous mass balance equation governing particle grinding leading to breakage is an integro-differential equation of the following form: Z 1 @Fðx; tÞ ¼ Sðx 0 ÞBðx; x 0 Þf ðx 0 ; tÞ dx 0 ð1Þ @t d where F(x, t) is the cumulative size distribution function which is the fraction of particles in the population by mass whose size is less than x at time t, B(x,x0 ) is the breakage distribution function which is the cumulative amount of particles by mass less than size x broken out of the parent size x0 , S(x) is the speciﬁc rate of breakage of particle of size x, and f(x,t) and b(x,x0 ) are the corresponding density functions. The governing equation has been developed on the basis of the assumption that the breakage is of ﬁrst order and the selection function does not depend on time. There is no straightforward analytical solution of the population balance equation. However, assuming certain functional forms for the parameters involved one can arrive at the solution [13]. For the special case, where it is assumed that Sðx 0 Þbðx; x 0 Þ ¼ ak 0 x a1

ð2Þ

the analytical solution of equation (1) is Fðx; tÞ ¼ 1 ½1 Fðx; 0Þ exp½k 0 x a t

ð3Þ

A general close-form analytical solution of the integro-differential equation of grinding depends on the model hypothesis and in most cases it cannot be solved. Therefore, there arises a need for discretization in order to make the model equation less intractable. In the discretized form, it is customary to present the breakage distribution function (Bi,j) as the cumulative fraction by weight of the material broken out of size j reporting to sizes below the size fraction i. Then bi,j is the fractional weight of the material of size j reporting to size fraction i on breakage. Using bij and Si which is the breakage rate function the size discretized grinding equation is obtained as i1 X dmi ¼ Si mi þ bij Sj mj dt j¼1

ð4Þ

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where mi is the fraction of total particle population in size class i. Solutions to this discretized form of the equation that involves a ﬁnite set of ordinary differential equations are easy to generate provided that models for the selection and the breakage functions are known [14,15]. Various techniques that are successfully used to solve the population balance equation are well documented in the literature [16].

2.3. Conventional solution method The size discretized grinding equation may be put in matrix form as follows: d½mðtÞ ¼ ½I BSmðtÞ dt

ð5Þ

where S is a diagonal matrix containing the values of breakage rate for each size class i, B a lower triangular matrix containing the values of bij, mðtÞ a column matrix of the values of mass fraction in each size class, and I the unit matrix. Assuming that the breakage and selection matrix are invariant with time, equation (5) can be solved by standard matrix manipulation. mðtÞ ¼ expð½I BSt Þmð0Þ

ð6Þ

Further simpliﬁcation is possible by assuming that no two selection function values are equal. Thus, mðtÞ ¼ TJðtÞT 1 mð0Þ

ð7Þ

where T ij

P bik Sk T kj Si Si ¼0 ¼1 ¼

i4j ioj i¼j

ð8Þ

and J ij ðtÞ ¼

expðSi tÞ i ¼ j 0

iaj

ð9Þ

The selection and breakage function matrices can be determined from laboratory batch grinding tests in which only the size distribution data on the mill feed and product are required. In fact, there are several methods of estimating these parameters but the popular methods use optimization procedures to generate a ‘‘best’’ ﬁt of the grinding function with respect to the batch test results at regularly spaced grinding intervals [17,18].

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2.4. Stochastic solution method According to many experts – Austin [19], Williams [20], and Berthiaux [21], the grinding process can be modelled as a discrete time Markovian process. In such a model, the evolution of the process is only considered at regular time intervals. A process is said to be Markovian when the evolution of the system at a given time is dependent only on the preceding time, but not on its previous history. In other words, if at any time t the system status is known, no further information about the system status could be obtained at t+Dt by knowing what was the system status at any time earlier than t. It is in this sense that the process follows a Markov chain without memory. Stochastic models that include Markov chains have been applied to numerous real-world systems under uncertainties [21–25]. For such systems, the method requires that the dynamics of the process be represented as a Markov chain given by a matrix of transition probabilities. The approach to the grinding problem represented as a Markov chain is based on the technique devised by Berthiaux [21] for batch grinding. It is conceptually simple and easy to implement. The ﬁrst question that is addressed relates to the calculation of the probability distribution associated with the transition in the state of a particle as a consequence of breakage. The evolution of size distribution during grinding changes through a ﬁnite number of states over time and behaves randomly according to certain probabilities. If the probability that the particulate system will be in a particular state during a given time period depends only on its state during the previous time period, then the grinding process can be treated as a Markov chain. Such a process can be modelled using the transition matrices. A transition matrix is an n n square matrix T, where n is the number of states of the process, whose elements pij are non-negative (representing probabilities). The elements of a transition matrix must lie between 0 and 1. In addition, since each transition must take the chain somewhere, the row sums must be equal to 1. The elements in the transition matrix include transition probabilities in terms of observable attributes (e.g. breakage and selection parameters as in the case of grinding). The Markov chain model is then used in a sequential simulation to generate ‘‘realizations’’ of particle size distributions. For the simulation of the grinding process it is assumed that the transition matrix is constant and does not depend on time or the number of passages. Under this assumption, the system can be represented as a Markov chain. It is a Markov chain if, whenever it is in state i, the probability of being in state j later is pij pij ¼ PfX nþ1 ¼ jjX n ¼ i; X n1 ¼ i n1 ; :::; X 0 ¼ i 0 g ¼ PfX nþ1 ¼ jjX n ¼ ig

ð10Þ

where X represents a stochastic process and Xn is the state of the process at the instant n. When the Markov chain takes a ﬁnite number of values r, the probability

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transitions can be put together in matrix form, as 3 2 pð1; 1Þ pð1; 2Þ pð1; rÞ 6 .. 7 6 pð2; 1Þ pð2; 2Þ . 7 7 6 6 . .. 7 .. .. 7 6 . 4 . . 5 . . pðr; 1Þ

ð11Þ

pðr; rÞ

where pij 0;

and

r P

pij ¼ 1;

i ¼ 0; 1; . . . ; r

ð12Þ

j¼0

Once the transition probabilities are known then the evolution in the state of the system is easy to determine. Let ln be a distribution vector representing the probabilities of each state of the system at time step n. After one step the chain is lnþ1 ¼ ln T

ð13Þ

Then, by iterating this, after m steps the chain becomes lnþm ¼ ln T m

ð14Þ

In the context of grinding, the initial particle size distribution is represented by l0 and as a consequence of grinding it changes. The change in the grinding state is estimated by deriving a probability transition matrix corresponding to the grinding system if a small time interval y is chosen. Berthiaux [21] showed that the transition matrix can be built by using the breakage and selection function as follows. 3 2 1 yS1 yb21 S1 ybn1 S1 6 0 1 yS2 yb32 S2 ybn2 S2 7 7 6 7 6 .. .. 7 6 ð15Þ T ¼6 . 0 1 yS3 . 7 7 6 7 6 .. .. .. 4 . 0 . ybnn1 Sn1 5 . 0 1 ySn Berthiaux [21] also demonstrated that the grinding process in a stirred bead mill can be represented by a Markov chain and in the following the same approach is adopted to illustrate the idea in the case of a ball mill.

2.4.1. Typical example of application To predict the size distribution in a ball mill one needs to obtain the breakage and selection functions. There are several proven techniques available in the literature. For example, the breakage and selection parameters can be obtained from short time batch grinding data on monosize feed material [15]. Here a software package known as ESTIMILLr [26] based on the method proposed by Grandy

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645

Table 1. Short time interval grinding data

Time (min)

Size (mm) 0.25

0.5

1.0

1.5

2.0

3.0

4.0

1.0000 0.9394 0.7595 0.6122 0.4657 0.3742 0.2903 0.2240 0.1829 0.1418 0.1107 0.0881

1.0000 0.9802 0.9096 0.7945 0.6271 0.5119 0.3919 0.3123 0.2479 0.1972 0.1498 0.1171

Cumulative fraction passing (–) 1680.0 1200.0 850.0 600.0 420.0 300.0 210.0 150.0 105.0 75.0 53.0 38.0

1.0000 0.3293 0.1259 0.0769 0.0548 0.0423 0.0320 0.0247 0.0189 0.0141 0.0105 0.0085

1.0000 0.4649 0.2313 0.1504 0.1018 0.0799 0.0596 0.0473 0.0373 0.0293 0.0226 0.0179

1.0000 0.6606 0.3980 0.2708 0.1865 0.1465 0.1095 0.0859 0.0667 0.0531 0.0394 0.0299

1.0000 0.7706 0.5048 0.3624 0.2577 0.2036 0.1552 0.1205 0.0902 0.0686 0.0508 0.0382

1.0000 0.8770 0.6645 0.4960 0.3576 0.2836 0.2136 0.1683 0.1316 0.1045 0.0817 0.0614

et al. [27] is used. A set of grinding data carefully prepared by Hosten and Avsar [28] is considered. These data are presented in Table 1. Using ESTIMILLr the breakage and selection functions are determined as follows: xðiÞ SðiÞ ¼ 1:0746 exp 0:9468 ln xð1Þ 0:7418 0:8870 xðiÞ xðiÞ þ 0:5798 ð16Þ Bði; jÞ ¼ 0:4202 xðj þ 1Þ xðj þ 1Þ The discrete values of the breakage and selection functions were used in equation (15) to determine the transition matrix. For simulation purposes the size distribution at t ¼ 0.25 min was treated as the initial state of the system. Figure 2 shows a comparison between the experimental and simulated data. The agreement between the data shows that the grinding in a ball mill can be simulated by treating it as a Markov process. It can be modelled by the probability transitions of a Markov chain that take into account the breakage and selection parameters and a characteristic grinding time that corresponds to the time interval between two breakage events.

3. MONTE CARLO METHOD Another approach to solving the stochastic model for any population of particles undergoing breakage is the Monte Carlo simulation method. The Monte Carlo

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Cum. Mass fraction finer (-)

1

0.8

0.6

Simulation t = 1 min t = 2 min t = 4 min Feed

0.4

0.2 0 10

100

1000

10000

Size (micron)

Fig. 2. Comparison between experimental and simulation results.

method is a numerical simulation technique for solving problems by means of random sampling. This technique is generally applied to analyze physical systems where direct experimentation is impossible or mathematical problems that cannot be solved by direct means. In the context of population balance, it appears that the Monte Carlo method can be used on physical considerations alone to solve the governing equation as long as the physical processes are amenable to quantiﬁcation. Mathematical description of particulate systems involves integro-differential equations resulting from population balance whose solution is complex and timeconsuming. Spielman and Levenspiel [29] proposed a Monte Carlo technique to simulate population balance models. They simulated the effect of drop mixing on the conversion of chemical reactions in a liquid–liquid dispersion. Shah et al. [30] used the Monte Carlo method to solve problems involving particulate systems using an ‘‘event-driven’’ approach. They used the concept of quiescent interval distribution proposed by Kendall [31]. Later, Rod and Misek [32] proposed a ‘‘timedriven’’ approach to simulate the dispersion and formation of drops in agitated liquid–liquid systems. Rajamani et al. [33] provided a connection between these two approaches by considering the dispersion of bubbles in liquid–liquid systems. The two approaches differ in the manner in which the random waiting times are decided. In the time-driven approach the time step involved in the simulation is pre-speciﬁed. The time step in the event-driven approach is decided by considering a random waiting time, which is generated by a known distribution function. A detailed mathematical treatment of quiescent interval distributions for monoparticle, bi-particle, and multi-particle events is described by Shah et al. [30]. Gooch and Hounslow [34] apply a Monte Carlo technique similar to Shah et al. to model breakage and agglomeration during crystallization. Bandyopadhyaya et al. [35] used the stochastic approach to model precipitation in micellar systems.

Monte Carlo Method for the Analysis of Particle Breakage

647

Particle breakage during grinding in ball mill can also be made amenable to Monte Carlo analysis. In a ball mill particles are constantly broken within a size class and reappear in another size class. The population balance models can describe this change of particle size with time during milling. However, except for some special cases, analytical solution of the governing population balance equation is not feasible. Therefore, numerical techniques are used to arrive at a solution. Here the time-driven Monte Carlo approach is used to simulate the particle breakage process in a ball mill. The algorithmic detail of the numerical method and its computer implementation is discussed. Computational results of the simulations are compared with the analytical solution for a very special case of the grinding process where the analytical solutions of the population balance model are available. A brief review of the essential components of the Monte Carlo method is needed before applying the technique to analyze any particulate system undergoing breakage. These are (i) probability distribution functions, (ii) random number generator (RNG), and (iii) sampling. Other issues relating to error estimation, ease of computation, nature of randomness, etc., may assume overriding importance depending on the nature of the problem.

3.1. Distribution functions The concept of a random variable and the associated probability density function is the key to statistical simulation. A random variable takes real values and it is deﬁned as a real number xi that is assigned to a random event Ei. Random variables are useful because they allow the quantiﬁcation of random processes and facilitate numerical manipulations. In the Monte Carlo simulations one of the requirements is that the physical system be described by probability density functions (pdfs). By describing the process as a pdf, one can sample an ‘‘outcome’’ from the pdf, thus simulating the actual physical process. This is quite convenient as the breakage process can be described by distribution functions. For example, the distribution function with respect to size deﬁnes quantitatively how the fragment size is distributed among particles in the population. This is the particle size distribution function which is very commonly used to describe particle population. For modelling purposes it is convenient to deal with the density function which is derived by differentiating the distribution function. Thus if F(x) is the distribution function deﬁned as mass fraction of the particle population that consists of particles of size less than or equal to x, then the distribution density function f(x) is deﬁned by f ðxÞ ¼

dFðxÞ dx

ð17Þ

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In this case unlike probability density function, f(x) dx is regarded as the mass fraction of the particle population that consists of all the particles in the size range (x, x+dx). However in analogy to the probability density function the following holds: Z 1 f ðxÞ dx ¼ 1 ð18Þ 0

The above description of the distribution function clearly suggests that the cumulative form of the distribution function is always uniformly distributed on [0,1], independent of the pdf. Any value for the cumulative distribution function is equally likely on the interval [0,1]. As will be seen later, this result has important ramiﬁcations for sampling from an arbitrary pdf.

3.2. Random number generation The ability to generate random variables (u[0,1]) is of fundamental importance since they are the building blocks for generating other random variables. A RNG is actually a program that produces, once its initial state is chosen, a deterministic and periodic sequence of numbers. The sequence of numbers is random if there is no correlation between different numbers. There are several algorithms available for sequential RNG [36–37]. A computer generates only pseudo-random numbers. The term ‘‘pseudo-random’’ is used because an algorithm is used to generate the sequence of numbers and in that sense the sequence can be predicted. The standard RNG on calculators and computers generates uniform distribution between 0 and 1 (uniform meaning that each number has an equal chance of being generated). Spreadsheets and most computer languages also have pseudo-RNGs. Most RNGs in use today are linear congruential generators. They produce a sequence of integers according to Zi ¼ ðaZ i1 þ cÞ mod m. Here m, a, c, and Z0 are non-negative integers. Also m is referred as modulus and Z0 is the seed. If the same seed is fed to the algorithm, it will generate the same sequence of numbers. In order to obtain uniform random numbers, u[0,1] is set to Zi /m. One can stretch the range of random numbers from [0,1] to [a,b] by performing the following operation: uab ¼ (ba) u+a.

3.3. Sampling The Monte Carlo method is about describing a physical or mathematical system by random sampling from the pdf’s and by performing the necessary computations needed to describe the evolution of the system. In this case, one is often interested in generating numbers with a non-uniform distribution. A RNG with uniform distribution can be used to generate numbers with a non-uniform

Monte Carlo Method for the Analysis of Particle Breakage

649

1 (a) u F(x)

0 -1

0

x=F (u)

x

1 (b) Cxh(x)

f(x)

0

Fig. 3. (a) Inverse transformation method using continuous function, (b) acceptance– rejection method.

distribution. The standard approach is to deﬁne the cumulative distribution function. Let the desired probability density function be f(x) and its cumulative distribution function be F(x). If a is chosen with probability density f(a), then the integrated probability F(a), is itself a random variable which will occur with uniform probability density on [0,1]. If x can take any value, then a unique x can be found from the pdf for a given u by setting u ¼ F(x). Thus if the inverse exists then x ¼ F1(u). This method is known as the inverse transformation method and is shown in Fig. 3a.

3.3.1. The most useful sampling technique Very commonly an analytic form for F(x) is unknown or too complex to work with, so that obtaining an inverse is impractical. An alternative to using the cumulative distribution function is to use the acceptance–rejection method which is computationally much more efﬁcient. In this method for any given value of x the probability density function f(x) can be computed and further that enough is known about f(x) that it can be enclosed inside a shape which is C times an easily generated distribution h(x) as illustrated in Fig. 3b. Frequently h(x) is uniform or is a normalized sum of uniform distributions. It should be recognized that both f(x) and h(x) must be normalized to unit area and therefore the proportionality

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constant C41. To start with, a test value for x is generated according to h(x) and then f(x) and the height of the envelope C h(x) are calculated. Then a uniform random number u is used to test if uC h(x)rf(x). If it is, then x is accepted, if not it is rejected and a new trial is made. Basically by considering x and uC h(x) as the x–y coordinate point in a two-dimensional plot it can be shown that all such points will populate the entire area Ch(x). It is only the ones that fall under f(x) are accepted. The efﬁciency is the ratio of areas which is equal to 1/C and therefore C must be kept as close as possible to 1.0. A speciﬁc application of this method is explained later.

4. MONTE CARLO APPROACH TO BATCH GRINDING The goal of the Monte Carlo method in the grinding context is to simulate the evolution of the size spectra by random sampling from known functions that describe the grinding process. These are selection and breakage functions. The ideal method for selecting a particle for breakage and subsequently determining the daughter size is sketched in Fig. 4. A uniform random number u[0,1)] is generated, and the abscissa corresponding to the distribution function value of u is read which gives the size or size class of the particle to be selected. This approach turns out to be more time consuming, as always the cumulative distribution function needs to be updated after every calculation cycle. An alternative to using the cumulative distribution function is to use the acceptance–rejection method which is computationally much more efﬁcient. In the acceptance–rejection method, given a density function f(x) (breakage or selection function) the limiting values of the argument and value of the maximum

….

Uniform random number

F(x)

0.0 1

Selected Particle 2

3

4

………….. N

Particle Identifier

Fig. 4. Method of drawing random sample from the cumulative distribution function.

Monte Carlo Method for the Analysis of Particle Breakage

651

of the function fM are determined. Here, size x is considered as the argument, which is bounded between 0 and xmax. A pair of random numbers is generated and the following inequality is examined u2

f ðx max u1 Þ fM

ð19Þ

If the above is fulﬁlled, then the corresponding size (xmax u1) is accepted since it is going to follow the density function f(x). In the opposite case, the trial is rejected and the procedure is repeated.

4.1. Computational procedure In order to apply the Monte Carlo technique to simulate the grinding process certain assumptions are made. First, monosize particles are used as the starting material. When these particles break, progeny particles are generated that appear in lower size classes. It is assumed that at any given time a single particle can only break. Furthermore, after any breakage event, the entire mass of the parent particle is shifted from its size class to a size class below, which is again randomly chosen. A schematic of this breakage process is shown in Fig. 5. In addition to above, functional forms for Si and bij are assumed that are treated as probability density functions. A time driven Monte Carlo technique is applied to compute the size distribution of the particles during grinding. The algorithmic details of this technique are described in the following steps.

Fig. 5. Schematic of the breakage process for a single particle.

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B.K. Mishra

Start

Initially, all the breakage frequencies are summed over all the particles to obtain Sb. The critical time step for the simulation, Dt, is calculated as Dt ¼ 1=rSb where r is an arbitrary constant termed the tuning factor. The physical signiﬁcance of the above equation is that larger the particle size, smaller is the value of Dt. In other words, larger particles break at a faster rate.

Event Selection

In the time driven Monte Carlo approach, at any time it is essential to ascertain whether a breakage event occurred. This is done by generating a random number, u, such that if u Sb Dt

: breakage will occur calculation proceeds with increment of time

else

: simply increment the simulation time

Identi¢cation of Particle for Breakage

A particle is chosen for breakage by using another random number and the selection of this particle is biased towards the larger particle size. This particle is selected based on the following condition: if Sj =Smax else

: accept the particle for breakage : reject the particle for breakage

where Sj is the breakage rate for a particle in the jth size class and Smax breakage rate of the largest particle. As before, the inequality stated above physically means that the probability of breakage of larger particle is more than that of the smaller particle. Identi¢cation of Daughter Particle

Once the particle is selected for breakage, the size of the daughter particle is determined by using a pair of random numbers and the breakage density function in the following way: Let; x i ¼ u x j If u bi =bmax else

: accept the size class corresponding to x i as daughter size class : choose another size class

Monte Carlo Method for the Analysis of Particle Breakage

653

Update Data

The total number of particles in each size class and the overall breakage frequency Sb is updated Sb ¼

N X

Si

i¼1

and the calculation cycle is repeated from the event selection stage (step 2). In short, once the breakage event is chosen, the program searches for the parent and the daughter size classes. Once these classes are identiﬁed, then the mass of the particle broken from the parent size class is simply added to the daughter particle size class. Accordingly, total mass of both the size classes is updated. At the end of a calculation cycle, the maximum size of the particles present in the system and the amount of material in each size class are also updated. Another cycle of breakage events is initiated and this process is repeated until the total time exceeds the desired time of simulation. Breakage of very small particles is ignored. These particles are thought of as those that would remain in the smallest size class. A ﬂow diagram of the calculation cycle is given in Fig. 6.

5. NUMERICAL RESULTS AND DISCUSSION The simulation program is written in a modular fashion in structured C language. It uses the intrinsic pseudo RNG – ran3 – available in the numerical recipe [37]. The two important components of the program that deal with the selection and the breakage of particles merit a detailed discussion. The algorithms of these components are shown in the ﬂow diagram within the dotted rectangular boxes. The top rectangular box shows the selection algorithm. It deals with the rate at which particles are broken and it factors as S(x) in the grinding equation. The variation in the breakage rate with size x can be represented algebraically as SðxÞ ¼ kðx=x max Þc

ð20Þ

where k and c are constants. This algebraic equation can be used to draw samples to arrive at the overall selection behaviour. The kinetics of the grinding process is such that the breakage of monosize particles is similar to one that is obeyed by the molecules participating in a ﬁrst order chemical reaction. Thus, the ﬁrst order hypothesis as applied to the top size material in batch grinding reduces the grinding equation to dm1 ðtÞ ¼ S1 m1 ðtÞ dt

ð21Þ

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Start

Input initial parameters Tmax, bij, Si, Nclass, Nparticle Calculate Sb & Δt t=t+Δt

Is u1<Δt ×Sb? yes t=t+Δt j=u2×Nclass

Is u3< Sj / Smax

xi=u4 × xj

Is u5
Mi + m Mj - m Update sum of breakage frequencies Sb

Is t
Calculate the cumulative size distribution by mass

Fig. 6. Flow chart of the calculation cycle.

Monte Carlo Method for the Analysis of Particle Breakage

655

whose solution is

m1 ðtÞ S1 t log ¼ mð0Þ 2:3

ð22Þ

where m1(t) is the mass of the total material in the top size interval at time t, m(0) the initial mass, and S1 the breakage rate constant. It is always interesting to ﬁnd out whether the grinding follows the ﬁrst-order breakage hypothesis. The selection algorithm is tested to check the ﬁrst-order breakage hypothesis. It is argued that if there are N monosize particles, all having the same rate constant or selection constant S then an event is selected in the following way: PðN ! N 1Þ ¼ SNðtÞDt PðnullÞ ¼ 1 SNðtÞDt

ð23Þ

It essentially states that the selection of a particle for breakage from a monosize particle system is proportional to the total number of particles present in the system. In a real situation, particles from different size classes are to be selected for breakage. It can be easily done once the probability density function for the selection process is known. There are several different types of algebraic equations established for the process that relates the breakage rate to size. One of the algebraic equations can be used to draw samples to accumulate the overall selection behaviour in a multi-size particle environment. It should be recognized that during the simulation the selection function in the cumulative form has to be updated each time the particle population is changed. This increases the number of calculations involved in selecting a particle for breakage. Alternatively, the acceptance–rejection method, as outlined earlier, is the most appropriate algorithm for drawing random sample that results in reduced computation time. It is clear from the above that as time progresses N(t) decreases and the selection becomes slower. The above algorithm has been tested using 100 particles. The results of simulation are shown in Fig. 7 for two breakage rate constants: 1.0/min and 0.5/min that shows an excellent correlation between the analytical and simulated results. The next most important component of the main program is the breakage module. This module is tested using 100 monosize particles as before. It is assumed that the single particle breakage follows the relationship which is termed as the cumulative breakage distribution function. Bðx; x 0 Þ ¼ Qðx=x 0 Þa1 þ ð1 QÞðx=x 0 Þa2

ð24Þ

where a1, a2, and Q are constants whose values depend on the physical properties of the ore. The corresponding density function is bðx; x 0 Þ ¼ Q

a1 x a1 1 a2 x a2 1 þ ð1 QÞ a1 x 0 a2 x0

ð25Þ

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Fraction remaining [-]

1

0.1

0.01 Analytical Monte Carlo 0.001

0

1

2 3 4 Time [minute]

5

6

Fig. 7. A comparison between Monte Carlo and analytical results that shows ﬁrst-order breakage kinetics.

The density function is used in the Monte Carlo simulation to draw statistical samples for the evolution of the size spectra. In this test, a sample of 100 particles is allowed to break one at a time. Each breakage event results in a daughter fragment whose size is decided by the breakage density function according to the algorithm shown in the lower rectangular dotted box of Fig. 6. This algorithm incorporates in itself the acceptance–rejection method. The results of the simulation are presented in Fig. 8. Again, excellent agreement is obtained between the simulated and analytical result. Finally, the entire grinding process is simulated incorporating the selection and breakage parameters. These two parameters can be represented through their corresponding density functions. Now, it becomes a matter of sequentially drawing off samples using these density functions to describe the evolution of the overall system. At ﬁrst, the grinding process is simulated considering the functional forms for the breakage and selection parameters for which an analytical solution of the grinding equation is available. The solution to the grinding equation under these conditions is given in equation (3). This allows a comparison between the analytical results and those of the Monte Carlo simulation. It is possible to carry out the Monte Carlo simulation either by counting the change in the number of particles in each size class or by mass. The latter approach is considered due to computational simplicity. Moreover, even with 100 starting particles the sheer number of particles towards the end of the simulation, particularly in the lower size classes, become overwhelmingly large and unmanageable. The grinding process is simulated using 100 monosize particles of 2000 mm nominal diameter. Other parameters such as k0 and a of equation (3), are 1.5 107 and 2.0 respectively. Using these values, the process was

Monte Carlo Method for the Analysis of Particle Breakage

657

Fraction undersize [-]

1

0.1

Analytical Monte Carlo 0.01 10

100

1000

10000

Size [micron]

Fig. 8. Breakage characteristics of single particles – comparison between Monte Carlo and analytical results.

Mass fraction passing [-]

1

0.1

0.01

10 min 5 min

0.001 Monte Carlo

1 min

Analytical 0.0001 10

100

1000

10000

Size [micron]

Fig. 9. A comparison between Monte Carlo and analytical solution.

simulated for 10 min. Figure 9 shows a comparison of results at 1, 5, and 10 min. It is observed that the analytical results agree very well with the Monte Carlo simulation results. The Monte Carlo simulation method can be easily extended to incorporate the complete functional forms for both selection and breakage function.

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B.K. Mishra

6. SUMMARY Theoretical study of the grinding process as a deterministic or stochastic system is possible. These two main realms of modelling are brieﬂy discussed with particular emphasis on stochastic simulations using Markov chains. An alternate method such as the Monte Carlo method is also discussed in detail. This method is used to solve the batch grinding equation. The time-continuous and sizecontinuous batch grinding equation has not been solved incorporating the most appropriate functional forms for the breakage and selection function. Although the model equations can be solved by complicated numerical techniques, the utility of such a model is diminished when employed for simulation purposes. The Monte Carlo method poses no such problem in solving such equations. The results of the Monte Carlo simulation agree quite well with known solutions of the governing batch grinding equation where simpliﬁed forms of the breakage and selection functions are assumed. Even with the true functional forms of breakage and selection function (e.g. equation (16)), the Monte Carlo technique is powerful enough to simulate the grinding process without any computational difﬁculty. Lastly, it is worth mentioning that the scope of this technique as a simulation tool to study real systems is largely limited by the availability of computing power. To attain reasonable computational speed for bigger computational problems, parallelization and vectorization of the code is needed.

Nomenclature

U(0,1) pij xi; xj f(x), F(x) bij Bij, B(x, x0 )

Si Smax Sb S B h (x)

any random number uniformly distributed between 0 and 1 the probability of the system in state j from the state i daughter and parent size respectively distribution density function by mass/number and cumulative distribution function with respect to size breakage distribution density function – fraction by weight of the material of size class j that appear in size class i upon breakage cumulative breakage distribution function – cumulative fraction by weight of the material broken out of parent size j or x0 that are of size below i, or x speciﬁc rate of breakage of particle of size i breakage rate of the largest particle summation of breakage frequencies over all the particles diagonal matrix containing the values of breakage rate for each size class i a lower triangular matrix containing the fractional values of breakage density a known distribution function that can be easily generated

Monte Carlo Method for the Analysis of Particle Breakage

fM Dt (sec) Tmax mðtÞ mð0Þ m1(t) m(0) Mi+m Mjm T I N k, k0, c j l0 ln

659

maximum value of the function f(x) simulation time interval maximum time for simulation a column matrix of the values of mass fraction in each size class at time t a column matrix of the values of mass fraction in each size class at time t ¼ 0 total mass of particles in the top size interval at time t total mass of the particle at t ¼ 0 total mass of ith daughter size class after breakage total mass of jth parent size class after breakage transition matrix unit matrix no. of states of the process; total number of particles constants small time interval between two consecutive states initial particle size distribution particle size distribution at nth time step

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S.S. Narayanan, Int. J. Miner. Process. 20 (1987) 211. A. Hoﬂer, J.A. Herbst, 7th European Symp. on Comminution, Ljubljana, 1990, p. 1. A. Datta, R.K. Rajamani, Int. J. Miner. Process. 64 (4) (2002) 181. D. Behrens, Chemie-lng.-Techn. 37 (1965) 473. S.R. Krogh, Powder Technol. 27 (1980) 171. R.P. King, F. Bourgeois, XVIII Int. Miner. Process. Congress, Sydney, 1993, p. 81. S.S. Narayanan, W.J. Whiten, Trans. Inst. Min. Metall. Sect. C 97 (1988) C115. B.B. Lira, A. Kavetsky, Miner. Eng. 3 (1990) 149. O.G. Pauw, M.S. Mare, Powder Technol. 54 (1988) 3. O.G. Pauw, Powder Technol. 55 (1988a) 247. O.G. Pauw, Powder Technol. 56 (1988b) 251–257. R.P. King, Modeling and Simulation of Mineral Processing Systems, ButteworthHeinemann, Oxford, 2001. R.P. King, J. South, Afr. Inst. Min. Met 127 (1972) 132–136. K.J. Reid, Chem. Eng. Sci. 20 (1965) 953. L.G. Austin, R.R. Klimpel, P.T. Luckie, Process Engineering of Size Reduction: Ball Milling, SME-AIME, New York, 1984. D. Ramakrishna, Rev. Chem. Eng. 3 (1985) 49. J.A. Herbst, G.A. Grandy, T.S. Mika, D.W. Fuerstenau, 3rd European Symp. on Comminution, Cannes, Dechema Monographien, 1972, p. 69. H. Berthiaux, J. Dodds, Powder Technol. 94 (1997) 173. L.G. Austin, Powder Techol. 5 (1971) 1. M.M.R. Williams, Phys. A. Math. Gen. 28 (1995) 1219. H. Berthiaux, Chem. Eng. Sci. 55 (2000) 4117. C.L. Chiang, In: E. Robert (Ed.), An Introduction to Stochastic Processes and Their Applications, Krieger Publishing, New York, 1980. R. Nassar, J. Schmidt, A. Luebbert, Chem. Eng. Sci. 47 (1992) 3657. H.M. Taylor, S. Karlin, An Introduction to Stochastic Modeling, Academic Press, Boston, 1998.

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[25] L.T. Fan, R. Nassar, S.H. Hwang, S.T. Chou, AIChE J. 31 (11) (1985) 1781. [26] J.A. Herbst, R.K. Rajamani, D.J. Kinneberg, ESTIMILL – A program for grinding simulation and parameter estimation with linear models, Comminution Center, University of Utah, Salt Lake City, UT, USA, 1988. [27] G.A. Grandy, G.D. Gumtz, J.A. Herbst, T.S. Mika, D.W. Fuerstenau, in: A. Weiss, (Ed.), A Decade of Digital Computing in Mineral Industries, AIME, New York, 1969, p. 765. [28] C. Hosten, C. Avsar, Scand. J. Met. 33 (2004) 286. [29] L. Spielman, O. Levenspiel, Chem. Eng. Sci. 20 (1965) 247. [30] B.H. Shah, D. Ramakrishna, J.D. Borwanker, AIChE J. 23 (1977) 897. [31] D.G. Kendall, J. R. Stat. Soc. Ser. B 11 (1949) 230. [32] V. Rod, T. Misek, Trans. Inst. Chem. Eng. 60 (1982) 48. [33] K. Rajamani, W.T. Pate, D.J. Kinneberg, I & EC Fundam. 25 (1986) 746. [34] J.R. Gooch, M.J. Hounslow, AIChE J. 42 (7) (1996) 1864. [35] R. Bandyopadhyaya, R. Kumar, K.S. Gandhi, D. Ramkrishna, Langmuir 13 (1997) 3610. [36] D. Knuth, The Art of Computer Programming, Volume 2, Seminumerical Algorithms, Addison-Wesley, 1981. [37] W.H. Press, S.A. Teukolsky, W.T. Vetterling, B.P. Flannery, Numerical Recipes – The Art of Scientiﬁc Computing, Cambridge University Press, Oxford, 1992.

CHAPTER 16

Numerical Investigation of Particle Breakage as Applied to Mechanical Crushing Chunan Tanga, and Hongyuan Liub a

Center for Rock Instability and Seismicity Research, Dalian University of Technology, Dalian and 116023, People’s Republic of China b School of Civil Engineering,The University of Technology, Sydney and NSW 2006, Australia Contents 1. Particle breakage modelling 1.1. Background and literature review 1.2. Methodology 2. Realistic failure process analysis (RFPA) code and veriﬁcation 2.1. Brief introduction of RFPA2D 2.2. Validation of RFPA2D by simulating Brazilian test 3. Particle breakage under static loading 3.1. Breakage of single particle under diametral loading without conﬁnement 3.2. Breakage of a single particle under diametral loading with conﬁnement 3.3. Discussions 3.3.1. Conﬁnement effect and energy release 3.3.2. Failure modes 3.3.3. Cracks and crack branching 3.3.4. Effect of particle shape 4. Particle breakage under dynamic loading 4.1. Brief description of RFPA2D-dynamics code and numerical models 4.2. Inﬂuence of heterogeneity on stress wave propagation 4.3. Inﬂuence of pressure stress wave amplitude on fracture process and failure pattern 5. Single-particle breakage under various loading conditions 5.1. Point-to-point loading 5.2. Plane-to-plane loading 5.3. Point-to-plane loading 5.4. Multi-point loading 5.5. Discussions 5.5.1. Failure modes and mechanisms under various loading conditions 5.5.2. Effect of particle shape, size and loading conditions 6. Inter-particle breakage 6.1. Fragmentation process of a rock particle assembly in a container

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Corresponding author. Tel.: +86-411-8740-3588; Fax: +86-411-8740-3588; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12019-4

r 2007 Elsevier B.V. All rights reserved.

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6.1.1. Numerical model 6.1.2. Stress ﬁeld 6.1.3. Inter-particle breakage process 6.1.4. Resultant force and displacement relationship 6.1.5. Energy transformation 6.1.6. Size distribution 6.2. Discussions 6.2.1. Inﬂuence of the particle shape on the inter-particle breakage 6.2.2. Two kinds of fracture pattern in the inter-particle breakage process 7. Summary Acknowledgements References

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1. PARTICLE BREAKAGE MODELLING 1.1. Background and literature review Mechanical crushing is widely used in both the aggregate-producing industry and the mineral industry [1]. On one hand, during the last few years the quality requirements for roads and railroads have become more accentuated due to the desire to minimize maintenance costs. This has, in turn, led to increasing requirements in the preparation of ballast materials and pavement aggregates. As the supply of natural gravel in many countries is decreasing (due to a lack of availability or legal restrictions), the use of crushed rock material is becoming more and more important. On the other hand, in the mineral industry crushing is the ﬁrst mechanical stage in the process of comminution. The purpose of crushing is to reduce the particle size of rock materials or to liberate valuable minerals from ores. However, although widely used, mechanical crushing is a complex process that has not been understood very well in spite of the research effort that has been devoted to this topic for more than a century [2]. Traditionally, the crushing process has been understandably studied from an energy efﬁciency point of view [3–5] because of the large energy consumption. It is reported that about 3.3% of the world’s electrical energy was consumed by crushing and grinding in 1976 and that the energy cost will continue to rise in the coming years [3]. Recently, some researchers [1] focused attention on the comminution process from the size reduction point of view. However, in order to ensure that the comminution technology will develop in such a way that will lead to improved energy efﬁciency and size reduction, it is clear that a more detailed and fundamental understanding of the basic particle fracture process is required. This awareness has led to a resurgence of interest in fracture physics from a mechanics point of view as it can be applied to the particle breakage mechanisms. At the most fundamental level, all industrial mechanical crushing reduces to the breakage of individual particles that occurs through contact with other particles or with the grinding media, or with the solid walls of mill [6]. Thus, researchers have

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identiﬁed that particle breakage is the elementary process of all industrial crushing. Understanding this elementary fracture process is indispensable to the development of improved energy utilization and size-reduction techniques. It is obvious that without breakage there would be no size reduction at all, and, on the other hand, if we could control breakage there would be possibilities of improving the overall performance of the total reduction process. Particle breakage is quite a complex process and the results are affected by both the loading conditions and the rock properties. On the basis of previous research [7], two breakage modes have been identiﬁed in mechanical crushing: single-particle and inter-particle breakage. Singleparticle breakage occurs when the distance between the chamber walls is equal to or smaller than the particle size, which is relatively simple. Inter-particle breakage occurs when a particle has contact points shared with other surrounding particles, and this is believed to be an important breakage mode in mechanical crushing. Traditionally, the understanding of single-particle breakage is based on the knowledge obtained in indirect tensile strength tests of particles [8]. Moreover, the rock particles are often obtained from bedrock. Thus, mechanical tests on the parent rock can also give an indication of the likely breakage properties of the particles. Such mechanical tests of rock are usually determined according to the International Society of Rock Mechanics (ISRM) suggested methods, such as Brazilian tensile test [9], uniaxial compressive test [10] and triaxial compressive test [11]. Briggs and Bearman’s [12] and Liu et al.’s [13] studies have provided some bases for those kinds of mechanical tests of single particle, where the two loading geometries of a single particle: quasi-uniaxial compression and quasitriaxial compression are identiﬁed in the particle breakage process in mechanical crushing. In addition, in real comminution industry, the particle is heterogeneous and has irregular shape and size. It is also under different loading conditions. Actually, the breakage behaviour of a single particle depends generally on its composition, structure, size, shape and the loading conditions. Therefore, a better understanding of the breakage behaviour calls for the understanding of particles with heterogeneous material properties, irregular shape and size, taking the different loading conditions into consideration. Based on Wang et al.’s [14] and Tang et al.’s [15] work, in mechanical crushing, a single particle is generally subjected to one of the four instant loading conditions irrespective of whether it is loaded directly by the rollers or by other mineral particles: (1) point-to-point loading, (2) plane-to-plane loading, (3) point-to-plane loading and (4) multi-point loading. Under point-to-point loading conditions, the particle is loaded between two points. In fact, the failure mode for particles loaded between two points is rather similar to that induced in the conventional Brazilian test in rock mechanics. Under plane-to-plane loading conditions, the particle is loaded between two approximately parallel planes. The plane-to-plane loading is similar to the

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conventional uniaxial compression test in rock mechanics. Under point-to-plane loading conditions, the particle is loaded in a mixture of point-to-point and plane-toplane loading. The point-to-plane loading condition is rather similar to the indentation test conducted by Lindqvist [16]. Multi-point loading conditions (especially the three-point loading condition) are similar to the conventional three-point bending test used to obtain the Mode I fracture toughness in fracture mechanics. However, the breakage behaviour of a single particle cannot adequately represent the effects of stressing a large number of particles, which induces much more complicated loading conditions for particle surfaces. It is obvious that the inter-particle breakage process is very complex. To facilitate an in-depth study of this process, different kinds of particle bed models have been developed. For example, an ideal particle bed model is characterized by Scho¨nert [17] as follows: (1) it possesses a homogeneous structure (statistical homogeneity); (2) homogeneous compaction is possible; (3) the volume or mass of the stressed particles is known; and (4) the wall effect is negligible in respect to the overall sizereduction effect. Previously, the ideal particle bed model was widely used for fundamental research on inter-particle breakage and to deliver basic information for comminution [17,18]. In those studies, comminution within the particle bed is characterized by the breakage probability and the breakage function. The breakage probability is deﬁned as the mass fraction of the progeny smaller than the lower bound of the initial size fraction, and the breakage function describes the particle size distribution of the broken progeny. In the past 20 years, theoretical studies, experimental investigation and numerical modelling have been conducted to understand the single-particle breakage process and the inter-particle breakage process in mechanical crushing identiﬁed above. Theoretically, the understanding is based on the knowledge obtained in indirect tensile strength tests of particles [8] and the physical point of departure is the Grifﬁth theory of brittle fracture. In experimental studies [1,2,12,14,19–21], different test apparatus were applied in investigating the breakage mechanism and characterizing the comminution potential of rock. Recently, with the rapid development of computing power, interactive computer graphics and topological data structure, the use of computer simulations seems to be the appropriate tool to obtain some clariﬁcations of the single and interparticle breakage process. It is desirable that, from a mechanics point of view, numerical models developed for understanding the particle breakage mechanisms should take into account the growth and interaction of microcracks, which culminate in the formation of progeny particles under typical loading conditions. This will require the consideration of the material properties, particle shape and particle size. With such complex requirements, for many years it has been thought that a general numerical approach from a mechanics point of view for calculating the stressing intensity, breakage probability and breakage function seems to be almost impossible, or at least very difﬁcult [22]. The investigation of

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such behaviour requires good modelling of the breakage conditions of a single particle subjected to multiple loads from neighbouring particles [23]. However, the calculation of the stress distribution inside the particle with multiple contact forces and the prediction of its fracture condition remain a difﬁcult task. So far a limited amount of investigation has been carried out in the ﬁeld of numerical analysis on these problems from the mechanics point of view, such as the studies of Cundall and Strack [24], Ghaboussi and Barbosa [25], Bardet and Proubet [26], Rothenburg and Bathurst [27], Iwashita and Oda [28], Tsoungui et al. [23]. Of those numerical simulations, it seems that the discrete element method (DEM) is particularly well suited to the study of the detailed microstructures of particle breakage because it is capable of describing the motion of individual particles as well as the forces acting on them. However, it must be pointed out that the results of these simulations based on DEM depend on the speciﬁc particle interaction models, which can be difﬁcult to verify experimentally, particularly because of the inability to look at how particles behave inside a container. Moreover, most of the numerical models developed to investigate the particle problems inside particle packing from a mechanics point of view mentioned above are based on the granular material model. In these numerical simulations, the particles are assumed to be completely rigid, and the overall deformation is caused only by relative displacements at the contact points [29]. However, it is not adequate to use the rigid granular material model to study the particle breakage problem in crushing, in which the process of particle breakage is involved. Recently, a granular material model based on the molecular dynamics method with elastic interactions between grains has been implemented by Tsoungui et al. [23] into a two-dimensional computer simulation code to study the crushing mechanisms of grains inside a granular material under diametric compression. Compared with the previous granular material models, their model has made a big step forward, since the model has deﬁned well the breakage conditions of a single grain subjected to multiple loads from neighbouring grains and the grains are represented by elastic discs. However, in their model, when a particle fulﬁls the fracture criterion, it is artiﬁcially replaced with a set of 12 smaller discs of four different sizes, and the particle breakage is not based on mechanical principles. More recently, Kou et al. [30] used the rock failure process analysis (RFPA) model [31] to investigate the inter-particle breakage process of a particle assembly in a container. However, since the post-failure process is not related to conﬁning conditions, they have difﬁculty in modelling the conﬁnement from the neighbouring particles and the chamber walls after some particles fail. Liu et al.’s [13] work extended Kou et al.’s [30] research and mainly concentrates on the inter-particle breakage process under conﬁned compression in mechanical crushing. From the numerical studies reviewed above, it seems that the realistic failure process analysis (RFPA) code [31–34] developed on the basis of the ﬁnite element method is a promising method to investigate the particle breakage process

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in mechanical crushing. Therefore, this chapter will use the RFPA code as the main modelling methodology to investigate the particle breakage process in mechanical crushing. In the following sub-chapters, the RFPA is ﬁrstly brieﬂy outlined. Then the breakage process of single particles with different heterogeneity, particle shape and size and under various loading conditions is investigated. After that, the inter-particle process in a conﬁned particle bed is modelled. Finally, various topics relating to the particle breakage in mechanical crushing are discussed. For example, the inﬂuence of particle textural properties on the particle breakage is investigated using RFPA-DIP (digital image processing). The particle breakage behaviour under impact loading is commended using RFPA dynamics. The three-dimensional particle breakage is conducted using RFPA3D. The work presented herein is a continued development of Tang et al.’s [15,35], Kou et al.’s [30] and Liu et al.’s [13] research, which forms part of an on-going investigation into the fundamental aspects of rock breakage, in order to improve the design of rock fragmentation equipment from the mechanics point of view. Of special interests are how the particle breakage can be modelled using RFPA, how the particle breakage behaviour depends on heterogeneous material property, irregular shape and size and the various loading conditions, and how the inter-particle breakage develops in the particle bed.

1.2. Methodology The objective of this chapter is twofold. One aim is to apply a numerical model based on mechanics that can be used for analysis and prediction of the breakage behaviour of particles. The other aim is to get a better understanding of the interparticle breakage process of relevance in a cone crusher. Of special interest is the understanding of how breakage behaviour depends on the loading conditions. The realistic failure process analysis code (RFPA2D) (RFPA User’s Guide [36]), developed in Mechsoft, P.R. China, is used to fulﬁl such an approach. Since we can handle the irregular shaped particles in RFPA2D, the current model enables us to investigate the breakage mechanisms of particles in crushing in a way that cannot be achieved in continuum and discrete element approaches. One of the advantages of RFPA2D is that it can take heterogeneity into account in the model. Smear method with small elements (SMSE) has proved to be suitable for simulating fracturing processes in rocks, since this heterogeneous material, in spite of their relatively small grain size, develop long fracture process zones due to the bridging and interlocking of the heterogeneous local materials in the wake of the fracture. These process zones constitute the main energy dissipation mechanism for this kind of material [37–39]. RFPA2D with SMSE applied to heterogeneous rock has successfully explained the dependency of some of their properties and loading methods on the heterogeneity, shape and size of the

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samples under static loading conditions [32,33]. Furthermore, SMSE is also feasible for handling the dynamic fracture appearing in rocks under dynamic loading. Another feature of RFPA2D is the explicit treatment of fracture and fragmentation. It tracks individual fractures as they nucleate, propagate, branch and possibly link up to form fragments. It is reliant upon the mesh to provide a rich enough set of possible fracture paths since the model allows fracture to occur within the elements only. However, almost no mesh dependency is expected as long as the element size is adequately small to resolve the fracture process zone of the rock. The simulations in this chapter give good prediction of dependence of fracture initiation on the stress waveforms and come out with fracture patterns very similar to the actual ones observed in experiments.

2. REALISTIC FAILURE PROCESS ANALYSIS (RFPA) CODE AND VERIFICATION 2.1. Brief introduction of RFPA2D In this chapter, we omit details of how RFPA2D works. Instead, the reader can refer to typical RFPA literature [31–33] and RFPA User’s Guide [36] to be available in the RFPA software package, also available at www.mechsoft.cn. Here we give brief descriptions of RFPA in order that those readers who are not familiar with RFPA2D can get a general idea of what RFPA2D is like. SMSE is used for simulating fracture processes in rocks. According to this SMSE concept, the stress in a ﬁnite element is limited by its strength. After the strength limit is reached, the stress in the element decreases. Initially, the stress was assumed to drop suddenly to zero, but it was soon realized that better and more realistic results can be obtained if the stress is reduced gradually [40], or is reduced to certain residual strength, and an elastic damage theory can be used to describe this stress decrease [31], i.e. the strain softening is due solely to the degradation of the elastic module and no other inelastic behaviour takes place. However, after this concept had been implemented in ﬁnite element codes and widely applied, it was discovered that there exists a convergence problem and that the calculation results may depend on the mesh size [40]. Much work has been done to avoid this mesh size effect. For example, by specifying the energy dissipated by fracturing per unit length of the fracture or the fracture band, the overall energy dissipation is forced to be independent of the element size [40]. From this idea it transpires that, in a ﬁnite element formulation with a free element size, the constitutive law must be adjusted according to the element size, so whatever the element size is, the calculations would yield macroscopically consistent results. That is to say, the element size can be arbitrary since it can be replaced by any other one without a noticeable effect. Soon, however, it was

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realized that this approach needs complicated mathematical solutions, and it is difﬁcult to consider the heterogeneity in the model. To overcome this difﬁculty, by keeping the continuum mechanics formulation, which seems more convenient for structural analysis, Tang [31] developed a rock failure process analysis code based on the smear method under conditions that elements with a ﬁxed size must be relatively small compared to the model scale – SMSE. The effort was devoted to develop a numerical model that can give a consistent description of failure process in brittle materials without further particular hypotheses regarding the size effect consideration. We found that the size effect problem can be prevented if a proper element size is used in the model. This element size is deﬁned as the characteristic size which represents a reference size and is treated as a material property. That is to say, the ﬁnal results will converge if the element size is close or equal to this characteristic size. In fact, it is true that the width of the fracture tip in real material is never zero but has a ﬁnite size. This was also noted by Bazant and Lanas [40] in their fracture modelling studies, in which localization of softening into arbitrary small regions must be prevented in order to make strain softening an acceptable constitutive relation. Their modelling was achieved by applying some mathematical concept, called the localization limiter, such as the fracture band model. In RFPA2D, we rely on FEM to perform the stress analysis of the model. The model is discretized into a large number of small elements to take into account the local variations of the material heterogeneity. During simulation, the model is loaded in a displacement control mode, i.e. a quasi-static fashion. At each loading increment, the stress and strain in the elements are calculated, then the stress ﬁeld is examined and those elements which are strained beyond the pre-deﬁned strength threshold level are broken irreversibly. To break an element means to reduce the element stiffness and its strength. It is important to mention that, in RFPA2D, the broken elements may restore their stiffness when they are highly compressed. This treatment of broken elements is different from the lattice model in which to break a lattice spring means to remove it from the lattice by changing its force-deformation constant to zero [41]. Then the model with new parameters for some of its elements moves to a new equilibrium. The next load increment is added only when there are no more elements strained beyond the strength threshold level at an equilibrium strain ﬁeld. A Mohr–Coulomb criterion with a tensile cut-off [42] is used so that the elements may fail either in shear or in tension. The discontinuity feature of the initiated fracture is automatically induced by using element with very small stiffness when the tensile strain of the failed elements reaching certain values. Stress distribution including failure induced stress redistribution is one of the most important concerns in RFPA2D. Any investigation into the behaviour of particle breakage in practical problems requires knowledge of stress and strain distributions in the particle. Although a variety of successful testing apparatuses

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have been devised during the last few decades to study the stress–strain behaviour of geomaterials under a general stress state including principal stress rotation, little advance has been achieved in the technology of visualizing the stress distribution within a micro-structure developed during non-recoverable damage of the media. Conventional techniques using load cells cannot be applied for solving these problems since a large number of sensors needed to obtain a clear picture of the stress distribution will signiﬁcantly distort the behaviour of the materials. For this investigation, the use of computer simulation seems to be the appropriate tool to obtain some clariﬁcations. The heterogeneity in geomaterials is another important concern in RFPA2D. Many papers dealing with modelling aspects of geomaterial as ideal CHILE1 media have been published over the past several decades. In some research ﬁelds, the ideal CHILE media was successfully used to predict the general characteristics of geomaterials. It should be noted, however, that natural geomaterials such as rocks are usually a strong DIANE2 media. Any conclusions, if they are obtained from the study of an ideal CHILE media, must be carefully checked whenever they are generalized to geomaterials. Such an ideal CHILE media sometimes behaves in a totally different manner from the real DIANE one. We use a Weibull distribution law (other laws are also available in RFPA code) with a homogeneous index m to statistically describe the mechanical properties of the model elements. The higher the index value of m, the more homogeneous the material.

2.2. Validation of RFPA2D by simulating Brazilian test Since tensile failure is considered to be one of the main characteristics of particle breakage in crushing [43], in order to validate RFPA2D code to show if it is an appropriate tool to be used to investigate the behaviour of particle breakage, we ﬁrst carry out a numerical simulation on a Brazilian test which is a commonly used test for evaluating tensile strength of rock, or rather a measure of the tensile stress at failure. As mentioned above, in RFPA2D models, the heterogeneity in rocks can be taken into account. To show the difference between the models with and without considering the material heterogeneity, we ﬁrst study the inﬂuence of heterogeneity on the stress ﬁelds by numerically simulating two model discs with different homogeneous indices, as shown in Fig. 1. The diameter of each disc is 1 CHILE – Continuous, Homogeneous, Isotropic and Linearly Elastic, as deﬁned by J.A. Hudson and J.P. Harrison in Engineering Rock Mechanics: An introduction to the principles, Elsevier, Oxford, 1997. 2 DIANE – Discontinuous, Inhomogeneous, Anisotropic and Not-Elastic, as deﬁned by J.A. Hudson and J.P. Harrison in Engineering Rock Mechanics: An introduction to the principles, Elsevier, Oxford, 1997.

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180 mm. Each sample (including the disc and the loading platens) is discretized into 36000 FE elements. The elements are deﬁned mainly by three independent parameters: the Young’s modulus, the Possion’s ratio and the strength. The average Young’s modulus of the elements is 60 MPa, the average Possion’s ratio is 0.25 and the average compression strength is 200 MPa with the ratio of compression strength to tensile being 10. The mechanical parameters are the same for all the elements in the homogeneous disc, whereas in the heterogeneous disc, as mentioned above, randomly distributed mechanical parameters statistically following Weibull’s law [31] are assigned to the elements. A brittle-elastic constitutive law with residual strength, 10% of the element strength, is used for local elements. The model properties are listed in Table 1. It is important to mention that, although the average compression strength of the elements of the material in the heterogeneous disc is 200 MPa, the global compression strength as simulated with RFPA2D is only 54 MPa, which is only about 25% of compression strength of the material in homogeneous disc.

Fig. 1. The numerical models with 32,000 elements in each disc (generated from RFPA2D).

Table 1. Material properties for the models

Setting

Sample

Platen

Young’s modulus Poison’s ratio Compression strength Tension strength Homogeneity index

60,000 MPa 0.25 200 MPa 20 MPa 3

210,000 MPa 0.20

200

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For a Brazilian test to be valid, failure of the disc should take place with the development of a vertical crack, from the centre of the specimen, which proceeds upward and downward along the load axis. In practice, however, the conditions are somewhat different [44]. Experimental and theoretical studies show that failure may take place not only with the development of tensile cracks at the centre, but also with the formation of small cones at the contact surfaces. When it is a line contact, high shear stresses are developed which cause local crushing. Addinall and Hackett [45] found experimentally that the origin of failure in a given disc, under given loading conditions is a function of the contact area. Mellor and Hawkes [46] proved quite conclusively that, in a properly conducted test, cracking starts at the centre of the specimen. Most researchers agree that this condition can be achieved by applying a distributed load over a short strip of the circumference at each end of the diameter. Therefore, in our numerical approach, we apply a compressive load through the two contact areas of width 2a between loading platens and the disc. Since the stress distribution within a Brazilian test sample can be illustrated by a close form solution to a disc compressively loaded along its diameter [47], it is useful to make a comparison between the numerical and analytical results of the stress calculation (only for a homogeneous disc). Figure 2 shows the numerically and analytically calculated horizontal stress sy and vertical stress sr on the loaded diameter. We can then observe that the numerical results have a very good agreement with the analytical results. Figure 3 shows the RFPA2D code generated fringe patterns in both homogeneous and heterogeneous discs shown in Fig. 1. These two ﬁgures clearly show that heterogeneity of rock has strong inﬂuence on the stress ﬁelds inside the discs. Since there is no experimental method available at the moment to obtain 1

Normalized Radius

0.8 0.6

Analytical

0.4

Numerical

0.2 0 -5

-0.2 0

5

10

15

-0.4

20

σr

-0.6

σθ

-0.8 -1 Normalized Stress

Fig. 2. Normalized stress distributions along the loading axis inside elastic discs under diametric compression calculated analytically from Hondros’s expression and numerically by RFPA2D code.

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Fig. 3. The fringe contours in discs under diametrical loading (obtained from RFPA2D modelling).

1 0.8

Normalized Radius

0.6

-5

0.4

Homogeneous

0.2

Heterogeneous

0 0 -0.2

5

10

15

20

-0.4 σr

-0.6 -0.8 σθ

-1 Normalized Stress

Fig. 4. Normalized stress distributions along the loading axis inside homogeneous and heterogeneous discs under diametric compression (obtained from RFPA2D modelling).

the fringe patterns in heterogeneous materials, the numerical results are of great signiﬁcance in improving our understanding of the stress ﬁelds in a heterogeneous material. For a quantitative evaluation of the heterogeneity effects, we have shown in Fig. 4 the normalized stress distributions, sy and sr, along the loading axis inside the heterogeneous and homogeneous discs, respectively. Again the inﬂuence of

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material heterogeneity on the stress distribution in the discs is illustrated very clearly. We can then observe that the stresses sy and sr are in compression near the contact areas, which implies that no cracks can easily emanate from the contact area. Further from contact areas, Fig. 4 shows us that only the stresses sy perpendicular to the compression axis become negative, which is the reason why crack initiates along the diameter line parallel to the loading direction. Figures 5 and 6 visually show the splitting process of this quasi-static simulation of the heterogeneous disc (Fig. 1b) with the principal stress ﬁelds displayed, respectively. Referring to the original model shown in Figs. 1b, 5 and 6 show how the heterogeneity state in material properties affects the local stress or strain variation. It is seen that the crack starts at some points around the loading diameter depending on the local variation of the disc heterogeneity as can be seen in Step 61 in Figs. 5 and 6. It is interesting to ﬁnd that during the crack propagation, another isolated crack initiates in location along the loading diameter close to the bottom half of the disc, immediately after Step 61. These two cracks soon coalesce and propagate radially outward giving rise to a diametral fracture plane with many small braches, following tortuous paths which depend on the distribution of the heterogeneity state in the material (Figs. 5 and 6). It can be seen from Fig. 6 (also Fig. 4) that the failure in or close to the loaded diametral plane is the result of the tensile stress normal to this plane. We observed in Step 65–Step 150 in Figs. 5 and 6 that several fractures branching from the diametral plane appear to be wedges near the contacts, and some ﬁnes are generated from the zones near the load points. It is found that these subsidiary effects start by shear fracture in the region of high compressive stresses near the contacts. At the end of the simulation, the major fracture splits the disc into two pieces. Yao and Kim [41] have done a simulation of tensile failure in a disc sample using a random lattice network model. With a distributed load on the end particles at the top and bottom of the disc, their simulation indicates that cracks initiate from the loading sites and propagate forward to the opposite side of the disc. This result is contrary to those experimental results (such as given by Mellor and Hawkes [46]) that failure originates at the centre of the disc if a distributed load is applied in the contact area between disc and loading platen. We ﬁnd that the difference between the numerical simulation by Yao and Kim [41] and the general experiments is a result of the difference in the contact conditions between the simulation and the experiments. In the experiments, since the load is applied by a loading platen to the circumference of the disc as a pressure distributed over small strip, biaxial compression exists near the contacts. This decreases the likelihood of failure by shear fracture at the contacts but has virtually no effect on the stresses in the body of the specimen [46]. In the numerical simulation by Yao and Kim [41], however, only the vertical distributed load is applied to the disc and

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Fig. 5. Failure mode and associated maximum principal stress ﬁeld in the disc under diametral loading (obtained from RFPA2D modelling).

no boundary tangential stress (shear stress) is adopted in the model, which may be one of the reasons why the failure initiated at the loading points. According to Hondros [47], the tensile strength of the disc can be approximately calculated from the following equation: st

Pm 1481 ¼ 5:24 ðMPaÞ ¼ ð3:1416 90 1Þ ðpRtÞ

ð1Þ

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Fig. 6. Numerically simulated failure mode and associated minimum principal stress ﬁeld in the disc under diametral loading (obtained from RFPA2D modelling).

where Pm ¼ 1481 N is the numerically obtained maximum load acting on the disc, R ¼ 90 mm is the radius of the disc and the t ¼ 1 mm the thickness of the disc. In conclusion, the numerical simulation of Brazilian test shows that RFPA2D is an appropriate tool to study particle breakage, not only the stress ﬁeld but also the failure process.

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3. PARTICLE BREAKAGE UNDER STATIC LOADING 3.1. Breakage of single particle under diametral loading without conﬁnement An important step towards understanding particle crushing is to ﬁrst understand the breakage of a single particle subjected to simple loading conditions. The numerical simulation of the Brazilian test shown above is a good start. However, in real crushing industry, the particles are of irregular shapes as well as irregular sizes. They are also under different loading conditions. It has been shown that the breakage behaviour of a single particle depends generally on its size, shape and the loading conditions [23]. A better understanding of the breakage behaviour and the strength characteristics of such particles calls, therefore, for simulating particle with irregular shape, size and under different loading conditions. A 2D irregular shaped particle model is established with the RFPA2D tool, as shown in Fig. 7. As in the numerical Brazilian test, the model (including the particle and the loading platens) is discretized into 36,000 elements with randomly distributed mechanical properties following Weibull’s law. All other mechanical parameters are the same as in the Brazilian test. To show the complex stress ﬁeld in an irregular shaped particle, the RFPA2D code generated fringe patterns in a homogeneous particle with the same irregular

Fig. 7. Two-dimensional irregular shaped particle model and the loading conditions (without conﬁnement) established with RFPA2D.

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Fig. 8. The fringe contours in particle under diametral loading without conﬁnement (obtained from RFPA2D modelling).

shape is given in Fig. 8. Since the loading condition for this irregular shaped particle, termed point-to-plane loading, is different from the Brazilian test, termed point-to-point loading, the fringe patterns shown in Fig. 8 are very different from the ones shown in Fig. 3. Figure 9 shows the numerically calculated results of normalized stress distributions, sx and sy, along the loading axis inside the heterogeneous particle. Again the inﬂuence of material heterogeneity on the stress distribution in the particle is illustrated very clearly in this ﬁgure. The ﬁrst impression of these numerical results is that the primary reason for the simulated failure mode of the irregular shaped particle is the contact condition between the loading platens and the particle. Owing to the irregular shape of the particle, the contact area between the bottom platen and the particle is larger than the upper one. As a result, the distributed stresses in the vicinity of the upper contact point are higher than that in the bottom area, as shown in Fig. 9. This will undoubtedly result in different fracture behaviour compared to the Brazilian disc. Figure 10 shows plots of load and failure released energy versus the respective applied displacements. Figures 11 and 12 show a sequence of deformed conﬁgurations corresponding to the steps shown in Fig. 10. Initially, the particle is elastic and exhibits a stiff and stable response where the particle deforms essentially non-uniformly due to the different contact conditions between platens and the irregular shaped particle. It appears that due to this asymmetrical stress

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-5

0.4

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0.2

Heterogeneous

0 0 -0.2

5

10

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20

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Fig. 9. Numerically calculated results of normalized stress distributions, sx and sy, along the loading axis inside the irregular shaped particle under diametral loading without conﬁnement (obtained from RFPA2D modelling). 1400

0.0015

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Fig. 10. Load–displacement and energy–displacement curves of particle under diametral loading without conﬁnement (obtained from RFPA2D modelling).

ﬁeld, the crack does not originate from the centre but from the location between the upper loading point and the centre the particle, as seen in Figs. 11 and 12. It is seen that the elastic regime is terminated by the onset of instability involving a tensile-type mode of particle splitting in which the particle shatters into several pieces (Step 51a–d in Figs. 11 and 12). It is seen from Fig. 10 that the fracture takes place with considerable violence in a brittle manner (big load drop and energy release in Step 51). Continued loading after the maximum load is reached results in the formation of fascinating failure patterns seen in Step 70 to Step 148 in Figs. 11 and 12. Some inﬂuences from contact effects which are unavoidable in such a test are

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Fig. 11. Failure mode and associated maximum principal stress ﬁeld in the particle under diametral loading without conﬁnement (obtained from RFPA2D modelling).

simulated. We ﬁnd that the zones of localized deformation with very high maximum principal stresses seen clearly in Fig. 10 are at least partly due to the constraint effects from the loading platens, which was suspected in a real test. Despite the presence of some zones of localized deformation and failure which were initiated at the contacts between platens and particle, the overall trend seems to be more towards a splitting pattern which can be considered as

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Fig. 12. Failure mode and associated minimum principal stress ﬁeld in the particle under diametral loading without conﬁnement (obtained from RFPA2D modelling).

tensile-type failure. Owing to the irregular shape, the stress concentration in the bottom contact ﬁrst occurs in the left side, which results in the splitting failure ﬁrst occurring between the upper loading point and the bottom left corner of the particle (Step 51d in Figs. 11 and 12). Then the stress concentration location moves to the bottom right corner of the particle (Step 70), which results in another splitting fracture in the right side as shown in Step 111–148.

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3.2. Breakage of a single particle under diametral loading with conﬁnement A natural extension of the above work regarding the single-particle breakage under axial load without conﬁnement is the investigation of the particle breakage under biaxial compression. To illustrate the richness and complexity of the mechanisms of biaxial crushing, in this section we present the results from simulation involving the same particle discussed above. The numerical model and the boundary conditions are shown in Fig. 13. The numerical particle sample is compressed by relative vertical motion between two rigid platens as in the simulation described above but is conﬁned in the horizontal direction by ﬁxed points that are friction free in vertical direction. This modelling represents particle crushing in the vertical direction while the particle is restrained from expansion in the horizontal direction. The Poisson’s ratio of the particle is 0.25 and, thus, a compressive stress develops in the horizontal direction under this constraint. Figure 14 shows the RFPA2D code generated fringe patterns in a homogeneous particle with the same irregular shape and loading condition as shown in Fig. 13. It strongly differs from the one under diametral loading without conﬁnement shown in Fig. 8. Figure 15 shows the numerically calculated results of the normalized stress distributions, sx and sy, along the loading axis inside the heterogeneous particle under diametral loading with conﬁnement. Comparing the previous stress

Fig. 13. Two-dimensional irregular shaped particle model and the loading conditions (with conﬁnement) established with RFPA2D.

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Fig. 14. The fringe contours in particle under diametral loading with conﬁnement (obtained from RFPA2D modelling).

1 0.8 Normalized Distance

0.6 0.4 0.2 0 -5-0.2 0

5

10

15

20

25

30

35

40

-0.4 Horizontal Stress

-0.6

Vertical Stress

-0.8 -1 Normalized Stress

Fig. 15. Numerically calculated results of normalized stress distributions, sx and sy, along the loading axis inside the irregular shaped particle under diametral loading with conﬁnement (obtained from RFPA2D modelling).

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distribution shown in Fig. 9 for the particle under diametral loading without conﬁnement, we observe similar effects: all the stresses sx and sy being in compression near the contact areas. However, one can note that the relative value of stresses sx comparing to sy is smaller than that of diametral loading without the conﬁnement condition (as shown in Fig. 9). Conﬁned crushing response in terms of load–displacement and energy release curves for the particle under diametral loading with conﬁnement is shown in Fig. 16. The response in the horizontal direction is also included in this ﬁgure. Initially, the material is elastic and exhibits a stiff and stable response. After the maximum load is reached, a load drop with a high energy release is observed. The curve of the plateau load shows a strong saw-like undulation that indicates a complex stress response induced by the micro-fractures. Much more energy is released in the later crushing process. Figures 17 and 18 show a sequence of deformed conﬁgurations corresponding to the steps shown in Fig. 16 as indicated in Section 4, the instability, once resulting in the splitting of the particle loaded under uniaxial, involves a tensile-type failure. However, the particle loaded biaxially now involves more crushing failure. Subsequently, the fractures are mostly initiated by progressive destabilization in the vicinity of the upper loading point and then in a very wide shear band between the upper loading point and the bottom left loading point, which is affected by the presence of the conﬁnement and as a result, the failure mechanisms of events are more or less the shear mode. Besides, there are three larger fragments formed at a later stage of crushing which result from the side cracks between the neighbouring loading points, as seen in Step 300 to Step 438 of Figs. 17 and 18.

0.02

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Energy Release (J)

Vertical Load-Vertical Displacement

1000 0.005 500 0

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Fig. 16. Load–displacement and energy–displacement curves of particle under diametral loading with conﬁnement (obtained from RFPA2D modelling).

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Fig. 17. Failure mode and associated maximum principal stress ﬁeld in the particle under diametral loading with conﬁnement (obtained from RFPA2D modelling).

3.3. Discussions Careful study of the breakage behaviour arising from diametral compression of particles has shown a wide variety of forms of failure, depending largely on the loading conditions with respect to conﬁnement. However, a number of distinct categories of characteristics of particle breakage have been identiﬁed, and are described in this discussion section.

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Fig. 18. Failure mode and associated minimum principal stress ﬁeld in the particle under diametral loading with conﬁnement (obtained from RFPA2D modelling).

3.3.1. Con¢nement e¡ect and energy release Because of the difference between the loading conditions, the compressive response under diametral loading with conﬁnement considerably differs from the one without conﬁnement. As shown in Fig. 16, the lateral constraint increases the initial stiffness of the particle a little (5%) compared to the particle without

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conﬁnement as shown in Fig. 10. This, of course, is expected because the side constraint stiffens the particle. As also expected, numerical simulations on particle breakage indicate that there is a brittle-ductile transition between the two loading conditions since the load–displacement curve of the particle with conﬁnement is more ductile than the one without conﬁnement (as shown in Figs. 10 and 16). The maximum load needed to break the particle that failed in a more ductile mode is 14% higher than that required for the unconﬁned particle that fails by more brittle manner with a bigger load drop. It is important to note that, due to the conﬁnement, the plateau load level is about 60% higher than that of the corresponding unconstrained compression test, and the energy release is even hundreds of times higher (as shown in Figs. 10 and 16). In addition, Figs. 17 and 18 indicate that with conﬁnement, highly stressed zones evolve at the centre and near the contact areas, leading to extremely high volume of crushed ﬁne particles that consumes most of the failure induced energy release. If we deﬁne a conﬁnement index I ¼ Px/Py, where Py and Px are the vertical load and the conﬁnement induced horizontal load, respectively, we ﬁnd that conﬁnement index I varies with the loading steps or the particle breakage process. Figure 20a shows this variation with respect to the loading steps. The variation of the index includes three stages: (1) before the maximum vertical load is reached, the index approximately keeps constant; (2) the index has a sudden increase as soon as the maximum load is reached, and keeps the increase tendency until the collapse of the particle breakage by forming three big chips due to the side crack development; and (3) the index decreases with a high rate indicating the ﬁnal breakage of the particle. It is seen clearly from Fig. 20a that the energy release increases drastically during the third stage. An important step forward of this investigation is that in our model the material heterogeneity inside the particle is taken into account. The randomly distributed mechanical parameters imply that the particle is statistically isotropic. Despite this, the crushing response is sensitive to local variations of the mechanical properties. The spreading of fractures is affected by the presence of the heterogeneity inside the particles and as a result, the failure mechanisms of events show a more or less statistical fashion. The load plateau that is traced during the spreading of the collapse is also relative to heterogeneity characteristics of the particle. As we know from experiments of rock loaded in compression, fracture did not just occur at the maximum load but also occurred before the load reached its maximum. Evidence of this is provided by the acoustic emission (AE) recorded at low loading stress levels [49]. Both Figs. 10 and 16 indicate that continuous fracture events start fairly early before the maximum load is reached. The occurrence of fracture events implies that they are the direct result of the heterogeneity of the particle.

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3.3.2. Failure modes When a particle is broken by compression or crushing the products fall into two distinct size ranges. Coarse particles are products resulting from the induced tensile failure and ﬁnes from compressive or shear failure near the points of loading. Whiten [48] proposed that two breakage functions could be used describing two different modes of breakage. The ﬁrst mode of breakage was identiﬁed as ‘‘catastrophic’’ breakage of particle that produces a small number of relatively large particles. The second mode was the production of ﬁne material (‘‘ﬁnes’’) at the points of contact between particles and the crusher liners and also between neighbouring loading points. These conclusions are consistent with our numerical results. It is found that the single-particle breakage may be categorized into two limiting cases [50]. In the ﬁrst case, purely elastic deformation occurs. Under diametral loading without conﬁnement, curved fracture trajectories are observed to form between contact zones (as shown in Fig. 19a and b). These fractures approximately follow those principal stress trajectories that are perpendicular to the maximum tensile stresses, as expected from the fracture mechanics theories. The fractures separate the disc or particle into two main parts with some ﬁner fragments formed along the fracture zones as shown in Fig. 19a and b. In the other limiting case, inelastic deformation, or in fact micro-fracture process, is the primary mode, in this case, cone-shaped regions at the points of loading deform with very high compressive stress. With increasing external load, localized shear mode failures occur in this cone-shaped zone, forming the so-called crushed zone (as shown in Fig. 19c). Although all the samples show a pronounced increase in intensity of shear stresses immediately adjacent to the regions of the point contact areas, the diametral loading without conﬁnement produces more coarse particles (as shown in Fig. 19a and b), whereas the diametral loading with conﬁnement generates

Fig. 19. Failure modes of the three numerical samples (obtained from RFPA2D modelling).

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0.5 Energy for Confined Particle Energy for Unconfined Particle Energy Release (J)

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Fig. 20a. Accumulated energy release and the conﬁnement index (obtained from RFPA2D modelling).

many more ﬁnes, particularly in the vicinity of contact points (as shown in Fig. 19c). It is also observed from Figs. 17 and 18 that even though under conﬁnement loading conditions, tensile fractures may also occur. The three big chips formed between neighbouring loading points by side cracks are the examples of tensile fracture. Figure 16 and Fig. 20a indicate that the formation of these chips results in big load drop, big conﬁnement index drop and big increase in the energy released. As mentioned above, the energy analysis reveals that much more energy is consumed during crushing stage for the particle with conﬁnements (as shown in Fig. 16). It can be predicted that the higher the conﬁnement stress, the higher is the failure load and so is the remaining energy imparted to the particle. The higher the remaining energy, the higher the number of generated ﬁnes.

3.3.3. Cracks and crack branching As observed by Colback [51], our simulations reveal three crack types: the primary crack, the secondary and tertiary cracks. It is found from Figs. 5 or 6 and Figs. 11 or 12 that, for samples under diametral loading without conﬁnement, the primary crack starts at some points around the loading line depending on the contact conditions (point-to-point as in the Brazilian disc or point-to-plane as in the irregular particle). Owing to the sample heterogeneity, more isolated cracks may initiate near the same line. The cracks propagate radially outward giving rise to a main diametral fracture plane with some accompanied branches.

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The secondary and tertiary cracks, starting near the load points, are associated with the formation of wedges. Although crack branching has been observed experimentally, there has been little qualitative, let alone quantitative description of this process in single-particle breakage [50]. The extent to which this occurs and the conditions in which it occurs is of importance in understanding the formation of progeny fragments. A number of theories have been proposed to predict the onset of crack branching and crack branching angle, however, there are still considerable uncertainties surrounding those postulations. Our simulations on the disc or particles all demonstrate the crack branching phenomena as shown in Fig. 19a–c. It is seen that most of these are dependent on the heterogeneity of the materials and some are relative to the non-uniform stress distribution due to the contact conditions. One of the examples relative to this stress dependent is the crack branching occurred in particle under diametral loading without conﬁnement as shown in Figs. 11 and 12. In this case, the main fracture branched into two from the upper loading point to the two ends of the bottom contact.

3.3.4. E¡ect of particle shape Hess and Schonert [50] noted that particle breakage strength depends on the particle shape. They suggested that the more spherical the particles become, the higher the breakage strength that may be expected. Although our modelling is only two-dimensional, the result obtained seems to support the conclusion obtained by Hess and Schonert. Comparison of the load calculations between the Brazilian disc and the irregular particle, which have the same distance between the loading platens, indicates that the maximum load, 1481 N, of the disc is 21% higher than that, 1176 N, of the irregular particle. Presumably a disc particle has a more regular stress distribution than the irregular one. The areas with this irregular stress distribution in irregular particle may be expected to deform more extensively and subject more crack initiation compared to those in a disc particle. This extent to which this is true has yet to be examined closely in the future investigations.

4. PARTICLE BREAKAGE UNDER DYNAMIC LOADING 4.1. Brief description of RFPA2D-dynamics code and numerical models The dynamic mechanical response of rock materials is important to grant the required level of particle breakage. This is why some of the scientiﬁc research in

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this ﬁeld has focused on the determination of their dynamic properties and the development of accurate constitutive models and failure criteria. One of the important features recognized for rocks or other brittle materials is their rate-dependence, i.e. their properties (ultimate strength, Young’s modulus, fracture energy) are highly dependent on the loading rate. The general trend for rate effects is an increase in dynamic strength as the loading rate increases [52]. Much effort has been made to improve the knowledge of the constitutive relationship for a wide range of strain rates by developing a more realistic material law [53–61]. Although such work has received considerable attention in the past few decades [53–61], it still remains poorly understood because of the complexity associated with the dynamic response of rocks that are fully heterogeneous. Lots of facts demonstrate that this heterogeneity feature plays a signiﬁcant role in fracture patterns of the rocks under dynamic loadings. Experimental investigations under dynamic loading provide good opportunities for examining the ultimate failure patterns of rock samples. However, it is found that sometimes analyzing the time history of failure is more important than simply examining the ﬁnal outcome. While most of the post-experimental observations can reveal fracture patterns and their relative proportions, they do not indicate the sequential order of the events or the conditions for fracture initiation, propagation and coalescence. Additionally, the post-experimental observations under dynamic loading conditions do not provide sufﬁcient information about microfracture nor evolution of the stress ﬁeld. The focus of this contribution is to numerically investigate the failure mechanisms and the fracture patterns of rocks using Brazilian samples under different stress wave amplitudes. The study is conducted by using an RFPA-dynamics code (RFPA User’s Guide [36]). This includes how the pressure wave in heterogeneous rocks affects the dynamic fracture propagation and patterns. Different pressure stress waves in terms of peak values are employed to consider the waveform variation of the applied dynamic loading. As in the statistical modelling, in RFPA2D-dynamics, rock is also numerically divided into small elements with ﬁxed size. The heterogeneity of the rock is taken into account by assigning different properties to the individual elements according to statistical distribution function. Again, random numbers satisfying Weibull’s distribution were generated to give the spatial distribution of the microscopic strengths and Young’s modulus. A homogeneity index m is introduced to represent the heterogeneity of the rock [31]. The physical phenomena occurring on the element scale can be described at the level of the constitutive relation. We also assume that the constitutive relation is elastic-brittle with residual strength and failure is obtained for elements in which the stress exceeds a certain threshold. A Mohr–Coulomb criterion with a tensile cut-off [42] is used so that the elements may fail either in shear or in tension. The discontinuity feature of the

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initiated fracture is automatically induced by using elements with very small stiffness when the tensile strain of the failed elements reaches certain values. A standard dynamic ﬁnite element algorithm was used to implement the aforementioned rock failure process analysis model. To evaluate the inﬂuence of heterogeneity on stress wave propagation, three samples with different homogeneity indices, m ¼ 2, 5 and 10, subjected to a pressure wave input were used for the simulations. The model with sample and bars is divided into 400 15 elements with 5 mm as the length scale of the element, as shown in Fig. 20b. The rock sample was positioned between two transmitter bars. The parameters and calculation conditions are listed in Table 2. To investigate the inﬂuence of the stress waveform in terms of peak value on fracture process and failure pattern of rocks, 2D Brazilian disc samples are used for the simulations, as shown in Fig. 21. The radius of the disc is 80 mm. A section of the ﬁnite element layout for the disc is also illustrated in Fig. 21. The model is

A

B

Fig. 20b. Numerical model of the sample and transmitter bars (400 15 elements with 5 mm length scale for the element).

Table 2. Material properties for the models

Setting

Sample

Bar

Young’s modulus Poison’s ratio Homogeneity index Density

60,000 MPa 0.25 2,510,100 2.5 10–6 kg m–3

210,000 MPa 0.20 200 7.8 10–6 kg m–3

Fig. 21. The numerical Brazilian disc sample and the loading conditions (the sample with 160 160 elements).

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divided into 25,600 square elements. The parameters and calculation conditions are listed in Table 3. The modelling system consists of an incident boundary and a transmitter platen to transfer the incident compressive pulse that propagates towards the sample. The pulse is partially reﬂected in the border of the transmitter platen and partially transmitted through the sample.

4.2. Inﬂuence of heterogeneity on stress wave propagation Rock is a heterogeneous material, and the heterogeneity plays a signiﬁcant role in the fracture process and the failure pattern. To demonstrate this inﬂuence, the stress wave in samples that did not fracture was simulated using the model shown in Fig. 20b. Figure 22 shows the stress wave propagation along the bars Table 3. Material properties for the models

Setting

Sample

Bar

Young’s modulus Poison’s ratio Compression strength Tension strength Homogeneity index Density

37,500 MPa 0.25 205 MPa 18 MPa 3 2.5 10–6 kg m–3

210,000 MPa 0.20

200 7.8 10–6 kg m–3

Fig. 22. Numerically obtained stress wave propagation along the bars and the sample with m ¼ 2 (obtained from RFPA2D-dynamic modelling).

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693 incident 100 10 5 2

stress (MPa)

30 20 10 0 -1 0

0

100

200

300

400

time (us)

Fig. 23. Numerically obtained stress wave–time curves in the A and B points in the two bars (obtained from RFPA2D-dynamic modelling).

and the sample. The samples with m ¼ 2, 5, 10 and 100, which corresponds to relatively heterogeneous, medium homogeneous and relatively homogeneous rocks, respectively, are used for the simulations. The numerically obtained stress waves at the medium point in the two bars are shown in Fig. 23. These compressive stress waves caused by the incident pressure reached point A in the ﬁrst bar at 5 ms and reached point B in the second bar at 200 ms. Figure 23 shows that, after passing through the heterogeneous samples, the three stress waveforms differ largely. This implies that rock heterogeneity has a big inﬂuence on dynamic stress wave propagation. After the stress wave travels through the sample, the peak value becomes lower for heterogeneous rock than for homogeneous rock. On the other hand, the pulse length becomes longer for heterogeneous rock than for homogeneous rock. This inﬂuence will surely cause the failure pattern to be different when a sample failure modelling is conducted.

4.3. Inﬂuence of pressure stress wave amplitude on fracture process and failure pattern The Brazilian samples shown in Fig. 21 are used to investigate the inﬂuence of stress wave amplitude on fracture process and failure pattern. Figure 24 shows the applied pressure waveforms with three peak values of stress, with the rise time t0 up to the peak stress being constant. Figure 25 shows the stress history obtained in the transmitter platen during one of the simulations of a sample. The dashed line averages the oscillations in the plateau of the incident pulse. Selected results of the numerically obtained fracture processes and failure patterns are presented in Fig. 26. The main features of these results are discussed next. It should be carefully noted that in these plots, in order to aid visualization, displacements have been magniﬁed by a factor of 5. Also shown in the ﬁgures are the level contours of stress magnitude, deﬁned as the relative

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Load (N)

Fig. 24. Applied pressure waveforms with three peak values of stress.

1800 1600 1400 1200 1000 800 600 400 200 0

0

200

400 Time (us)

600

800

Fig. 25. Stress history obtained in the transmitter platen during the simulation of sample with inputted peak stress 150 MPa (obtained from RFPA2D-dynamic modelling).

value of maximum shear stress. A fully fractured surface is shaded in black, whereas the zones that are intact or failed but not fully fractured remain in the colour of level contours of stress magnitude. The comparison between the three cases shown in Fig. 26 displays graphically the good prediction of the dependency of dynamic fracture patterns on the transmitted stress wave amplitude. Snapshots of the ﬁrst line in Fig. 26 show the failure pattern and shear stress evolution for the case of stress waveform with lower amplitude (peak value is 75 MPa). The fractures start nucleating and propagating at about 120 ms when the peak value of the compressive pulse reﬂected from the bottom and reaches the centre of the sample. Distinct features that the formation of double major fractures and branching are observed. Owing to the heterogeneity, the fractures well developed in the bottom area in the right side when the sample transmits the

Numerical Investigation of Particle Breakage Fig. 26. Fracture sequence and failure patterns for the three cases with different stress wave amplitude, with peak stress being 75, 100 and 150 MPa, respectively (obtained from RFPA2D-dynamic modelling). 695

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maximum load. The development of these fractures generates relief waves which temporarily halt the failure process. The stress waves subsequently travel from the left side towards the centre of the disc, inducing a further fracture growth as well as some microfracturing in the centre area part, which ﬁnally forms a main fracture. This through fracture is clearly seen in the snapshots in the ﬁrst line in Fig. 26. In contrast to the ﬁrst case shown in the ﬁrst line of Fig. 26, the third case shown in Fig. 26 reveals that, for the case of stress waveform with higher amplitude (peak value is 150 MPa), the fractures start nucleating and propagating at about 20 ms, which is much earlier than that in the ﬁrst case, shown in Fig. 26. The fractures around the loading areas nucleate due to the incident wave, not the reﬂected wave as in the ﬁrst case. In the two sides of the bottom support areas, however, the compressive stress waves are reﬂected as tension waves when they reach the free surface, and are then superimposed upon the tail of the compressive waves. The superimposed stress waves developed an increasing amount of tension. When the tension stresses are high enough, they induce opening fractures parallel to the surfaces. The process is also accompanied by the formation of new wedge-shaped inclined fracture zone due to the intensity of the pressure stress waves near the loading area. The sequence of fracture patterns in terms of location closely followed what we expected based on our analysis. It showed that the fracture processes and the failure patterns are markedly affected by the stress wave amplitudes that are applied to the samples. The nucleation of fractures and failure patterns when p0 ¼ 75 and 150 MPa shown in ﬁrst line and third line in Fig. 26 differed signiﬁcantly. For lower p0, the fractures start from the vicinity of the bottom compressive zone approximately after t ¼ 120 ms. Whereas for higher p0, the fractures occur almost immediately after the stress wave front enters into the sample, which is 100 ms earlier than that for lower p0. It is seen from these snapshots that the failure patterns are enriched by the random distribution of the elemental stiffness and strength. The snapshots of the third line in Fig. 26 further reveal the development of profuse fracturing at the loading area which even leads to some fragmentation. Secondary fractures, parallel to the main diametral fracture, also appear as the load decreases, leading to the typical columnar failure of Brazilian tests [39]. For smaller stress wave amplitude, the model predicts the formation of the principal fracture that nucleates in the centre of the sample and grows towards the bearing areas, as well as some secondary fracturing parallel to the main fracture and near the loading areas. For higher load stress wave, the simulation reports the formation of radial fractures starting in the circular border and growing to the centre. The ability of RFPA-dynamics to account for such complex fracture patterns with relative ease is a remarkable feature of SMSE.

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5. SINGLE-PARTICLE BREAKAGE UNDER VARIOUS LOADING CONDITIONS As introduced above, in the comminution process, the particle is subjected to one of the four instant loading conditions shown in Fig. 27, irrespective of whether it is loaded directly by the roller or by other mineral particles. The ﬁrst case is termed point-to-point loading. That is, the particle is loaded at two points (Fig. 27a). The second case is termed plane-to-plane loading. In this case, the piece is loaded between two approximately parallel planes (Fig. 27b). The third case, pointto-plane loading is a mixture of the ﬁrst and second cases (Fig. 27c). The fourth loading condition is the multi-point loading, which involves at least three or more loading points. Here three loading points are considered (Fig. 27d). In this section, the particle breakage process under the above-mentioned four loading conditions

Fig. 27. Numerical particles with irregular shapes under various loading conditions (RFPA models).

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Fig. 28. Quasi-photoelastic stress fringe patterns (obtained from RFPA2D modelling).

will be numerically conducted by the RFPA2D code and the simulated results will be visually shown. The ﬁrst requirement of any soundly based simulation of particle breakage is a detailed knowledge of the stress ﬁeld within the loaded system. Our foremost aim here is to investigate the distribution of the elastic component primarily responsible for the operation of the breakage process. To show the complex stress ﬁelds in irregularly shaped particles under various loading conditions, the stress fringe patterns generated by the RFPA2D code in the homogeneous particles with the same irregular shapes and sizes are given in Fig. 28. The simulated stress fringe pattern is deﬁned as the quasi-photoelastic stress fringe attern by Liu et al. [62]. As shown in Fig. 28a, under the point-to-point loading conditions, the large stresses were concentrated around the contact points, which is expected to induce compressive failure resulting in local crushing at these contact points. The stress distribution along the line connecting the two contact points is quite uniform, where tensile fracture is expected to be induced to cause splitting fracture between the contact points. The Brazilian tensile strength test can also be explained as a particle with regular shape under point-to-point loading. The stress distribution or fringe pattern in a Brazilian disc also shows a uniform distribution

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along the loaded diameter according to the analytical results given by Hondros [47] and the numerical results by Tang et al. [35]. Because of the irregular shape and size used in this study, the uniform stress ﬁeld is disturbed slightly compared with that induced in the Brazilian disc [35]. Under the plane-to-plane loading conditions (Fig. 28b), the stress ﬁeld is relatively complicated. There are stress concentrations around the irregular corners. It is difﬁcult to say how the particle will fracture just according to the elastic stress distribution. Under the point-toplane loading conditions (Fig. 28c), due to the irregular shape of the particle, the contact area between the bottom plate and the particle is larger than that between the upper platen and the particle. As a result, the distributed stresses in the vicinity of the upper contact point are higher than those in the bottom area. Large stress concentration around the upper loading point is expected to induce local crushing failure. With the distance increasing from the loading point, the stress decreased rapidly and the stress ﬁeld is similar to the wave propagation caused in water by throwing a stone. This will undoubtedly result in a different fracture behaviour but it is difﬁcult to say how the particle will fail in the region near the bottom loading platens. Under the multi-point loading conditions, similar to previous cases, the stress concentrated around the loading points, which is expected to induce local crushing. In the interior of the particle, the stress ﬁelds emerging from the loading points interacted with each other to induce a combined stress ﬁeld. It is also difﬁcult to predict how the fracture will develop there. Since the elastic stress distribution is not enough to predict the particle fracture pattern even in the simplest homogeneous material, it is necessary to perform breakage process analysis. In the following the fracture process induced in the particle with heterogeneous material property, irregular shape and size, and under different loading conditions will be simulated and visually shown.

5.1. Point-to-point loading Since the particle breakage under the point-to-point loading condition has been investigated in detail in Section 3, here just a brief description is given in order to make comparison with the particle breakage processes induced under the other three types of loading conditions. Figure 29 depicts the force–displacement curve and associated AE counts in loading the irregular particle under point-to-point loading conditions, which indicates typical brittle failure characteristics. Figure 30 visually shows the particle breakage process in terms of the evolution of the major principal stress and the spatial evolution of AE. The letters in Fig. 30 correspond to those in Fig. 29, which indicates the different loading level. The major principal stress evolutions in Fig. 30a show how the heterogeneity state of the material properties affects the local stress or strain variation. In Fig. 30a, the brighter the element is, the higher the

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major principal stress. At the initial loading stage, tensile failure randomly occurs along the line connecting the two loading points because of the heterogeneity. It is seen that at the peak load (point A in Fig. 29), the macroscopic crack starts at some point near the central line of the particle instead of the contact point, depending on the local variation of the particle heterogeneity, as can be seen in Fig. 30a. Then, the formed crack unstably propagates along the loading direction giving rise to a diametrical fracture plane with lots of small branches, following tortuous paths which depend on the distribution of the heterogeneity state in the particle, as shown in Fig. 30a in B-1, B-2, B-3 and B-4. We use the terms ‘‘unstable crack propagation’’ because the particle was split into two halves by crack propagation induced at the same loading level (point B in Fig. 29). In Fig. 30b, each circle represents one AE event, the diameter of the circle represents the relative magnitude of the AE released energy and the colour represents the failure type. The red colour represents the tensile failure induced at the current load level, the blue colour represents the shear failure at the current load level, and the black colour represents the failure including tensile and shear failure accumulated at the previous load levels. From Fig. 30b it can be seen that tensile splitting along the loading direction, marked by the red colour, is the dominant mode of failure. In the particle near the loading points, there are a few shear fractures marked by the blue colour, which causes some ﬁnes generating from the zones near the loading points.

5.2. Plane-to-plane loading Under plane-to-plane loading, the particle is loaded between two approximately parallel planes at the ends of the particle sample. Figure 31 depicts the

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Fig. 30. Fracture process under point-to-point loading conditions (obtained from RFPA2D modelling).

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force–displacement curve in loading particle under plane-to-plane loading conditions, which indicates a more ductile behaviour compared with the curve shown in Fig. 29 obtained under point-to-point loading conditions. The corresponding fracture process in terms of the major principal stress distribution and the associated AE spatial evolution is visually shown in Fig. 32. As previously mentioned, the letters in Fig. 32 indicate the different load levels as shown in Fig. 30. It is found that the fractures did not only occur at the maximum load but also occurred before the load reached its maximum. Evidence of this is provided by the AE recorded at low stress levels, as shown later in Fig. 33. In the following let us discuss the fracture process in detail. The force–displacement response has the general features common to many particle materials, i.e. the linear elastic deformation stage, the non-linear deformation stage and the post-failure stage. Initially, the force–displacement response is relatively stiff and nearly linear. The particle deforms elastically. Then the curve begins to soften. At the peak load (point A as shown in Fig. 31) macroscopic cracks are initiated at the geometry irregular locations as shown in Fig. 32(a)A, because of the local stress concentration after the random initiation of local failures. As the loading displacement increases, the macroscopic cracks propagate downward, shown in Fig. 32(a)B, and correspondingly a big load decrease (point B in Fig. 31) is induced. Further loading displacement (point C in Fig. 31) induced the continuous crack propagation and surface splitting, shown in Fig. 32(a)C. After the surface splitting of the irregular parts of the particle, the plane-to-plane loading is similar to the conventional uniaxial compression strength (UCS) test in rock mechanics. Therefore, with further loading displacement, the particle demonstrates the behaviour similar to that observed in the conventional UCS test. Further increases in the loading displacement (point D in Fig. 31) results in the formation of shear fractures, which tend to emanate from

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Fig. 32. Fracture process under plane-to-plane loading conditions (obtained from RFPA2D modelling).

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Fig. 33. AE counts and accumulated AE counts in loading particle under point-to-plane loading conditions (obtained from RFPA2D modelling).

the core of the particle and propagate outward till the left-bottom end and the right-upper end of the particle at an angle inclined to the loading direction, shown in Fig. 32(a)D. At the ﬁnal stage (point E in Fig. 31) the internally inclined fracture forms through-going faulting and dominates the failure process of the particle, shown in Fig. 32(a)E. Figure 32(b) also shows visually the spatial distribution of the AE events and the failure mechanisms. The symbols in the ﬁgures follow the aforementioned regulations. It can be seen that at the ﬁrst loading stage, the AE events randomly distributed in the particle except a few concentrations in the geometrically irregular parts, shown in Fig. 32(b)A. Then during the surface splitting process, the AE events concentrated on the lateral surface of the particle to cause tensile crack propagation, which is marked by the red colour (light grey in a black-white picture) in Fig. 32(b)B and C. At the same time, the shear fractures are initiated in the interior of the particle, which are marked by the blue colour as shown in Fig. 32(b)C. After that, more and more AE events emitted by shear failure distributed along the line connecting the left-bottom end and the right-upper end of the particle, shown in Fig. 32(b)D. There are also some AE events caused by tensile failure in the other parts of the particle. Finally the shear AE events form a throughgoing fracture inducing the completely failure of the particle, shown in Fig. 32(b)E. We have also been interested in the frequency of the AE events and the associated energy in loading the particle. The numerically obtained AE event rate and the energy release are shown in Figs. 33 and 34. The event rate shows the following expected features: (1) during the initial deformation or linear elastic phase (as shown in Fig. 31), little elastic energy was released, though some fracture events occurred; and (2) an increasing rate of fracture events accompanied the inelastic phase and the load plateau. This agrees with the understanding

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Fig. 34. Elastic energy release (ENR) and accumulated ENR in loading particle under point-to-plane loading conditions (obtained from RFPA2D modelling).

that the AE events are generated by the micro-fractures that result in the particle non-linear deformation behaviour [32]. It is important to note that, as shown in Figs. 33 and 34, although there are a lot of AE events occurring before the peak load (point A in Fig. 31), little elastic energy is dissipated during this stage. A comparison between Figs. 31, 33 and 34 shows a good relationship between the load curve, AE event rate and energy release. Note that most of the large load drop (or stress drop) on the load curve shown in Fig. 31 corresponds to a high event rate (Fig. 33) and large event energy release (Fig. 34), e.g. point B, C and E in Fig. 31. It is also found that the fracture-released energy seems to have a closer relationship with the load drop.

5.3. Point-to-plane loading The point-to-plane loading is a mixture of the point-to-point loading and the planeto-plane loading. Therefore, it is expected that the fracture characteristics are also a mixture of characteristics observed in the previous two loading conditions. Figure 35 shows the plots of the load and associated AE events versus the loading displacement in loading particle under point-to-plane loading conditions. Figure 36 shows the sequence of a fracture process in terms of the major principal stress distribution (Fig. 36a) and the AE event spatial evolution (Fig. 36b) corresponding to the load levels labelled by alphabetical letters, as shown in Fig. 35. Similar to the previous cases, initially, the particle is elastic and exhibits a stiff and stable response although the particle deforms non-uniformly due to the different contact conditions between the platens and the irregularly shaped particle. It is found that the zones of localized deformation with very high maximum principal stress, seen clearly in Fig. 36(a)A, are located at the upper contact points. After the peak load, due to the asymmetrical ﬁeld, the major tensile crack does not originate from the centre as

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under point-to-point loading conditions, but instead from the location near the upper loading point, as seen in Fig. 36(a)B. With increasing loading displacement (Fig. 35C), the formed major tensile crack propagates out of the localized deformation zone and the crack system shows an actinomorphic distribution with a long crack extending out of the actinomorphic distribution zone, as shown in Fig. 36(a)C. Further loading (Fig. 35D) resulted another unstable propagation of the fascinating tensile crack, shown in Fig. 36(a)D. Owing to the irregular shape and heterogeneity, with continuous loading displacement (Fig. 35E), the splitting crack propagation deviated from the central line of the particle until reaching the bottom loading platen, as shown in Fig. 36(a)E. From the simulated fracture pattern, it can be seen that some inﬂuences of contact effects, which are unavoidable in such a test, are simulated. Despite the presence of some zones of localized deformation and failure, which were initiated at the contact between the platens and the particle, the overall trend seems to be more towards a splitting pattern, which can be considered a tensile-type failure. Figure 36(b) shows the spatial distribution and mechanisms of the AE events in loading the particle under point-to-plane conditions. It can be seen that the failure events around the upper loading points are marked by the blue colour, which indicates that the failure events are induced by the compressive stress and that shear fracture is the mechanism. The failure events propagating downward till the bottom loading platen are caused by tensile stress, which is marked by the red colour.

5.4. Multi-point loading The particle breakage under multi-point loading conditions depends on the number of contact points. For a particle under three-point loading conditions, the

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Fig. 36. Fracture process under point-to-plane loading conditions (obtained from RFPA2D modelling).

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Fig. 37. Force–displacement curve and AE counts in loading particle under multi-point loading conditions (obtained from RFPA2D modelling).

resultant force–displacement curve and associated AE counts are shown in Fig. 37. Correspondingly, the fracture process in terms of the distribution of the major principal stress and the spatial evolution of the AE event are visually shown in Fig. 38. It can be seen that, as the load is applied on the specimen till the non-linear deformation stage (Fig. 37A), high compressive stresses intensify near the supporting loading points and tensile stresses intensify near the central parts between the two bottom supporting points to cause random tensile fracture initiated there, as shown in Fig. 38(a)A. It has been shown that very few AE events are recorded before the load reaches its peak load. At the peak load (Fig. 37B), the randomly distributed tensile fractures coalesce to form a macroscopic tensile crack at the central part of the particle between the two bottom supporting points, as shown in Fig. 38(a)B. Then as the loading displacement increases (Fig. 37C), the crack unstably propagates, following a tortuous path until reaching the upper supporting loading point, as shown in Fig. 38(a)C-1, C-2 and C-3. Because of the irregular shape of the particle, the crack propagation deviated from the central line to the right. Figure 38(b) records the spatial locations of the AE events corresponding to the loading levels marked by the letters in Fig. 11, where the temporal distributions of the failure event counts are plotted along with the loads in the form of a histogram. The symbols in the ﬁgure follow the aforementioned regulations. It can be seen that the AE events causing the local crushing near the supporting loading points are compressive failure, which is marked by the blue colour (black in a black-white picture) in Fig. 38(b). The major crack initiating from the lower central parts of the particle and propagating till the upper supporting loading points along the loading direction in tortuous path is caused by tensile failure events, which is marked by the red colour in Fig. 38(b). The characteristics of AE events imply that not many

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Fig. 38. Fracture process under multi-point (three-point) loading conditions (obtained from RFPA2D modelling).

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microcracks are created within the particle before the initiation of the main tensile fracture. In other words, the specimen under a bending load is not damaged to any large extent with the exception of the creation of the main fracture.

5.5. Discussions From the simulated results presented above, it is clear that the particle breakage behaviour depends largely on the heterogeneous material properties, irregular shape and size, and the different loading conditions. According to the simulated results, a number of distinct categories of characteristic particle breakage will be discussed in this section.

5.5.1. Failure modes and mechanisms under various loading conditions The ﬁnal failure modes of the particles under various loading conditions are summarized in Fig. 13a. Compared with the experiential results (Fig. 39b) presented by Wang et al. [13], the simulated failure modes are reasonable. Under point-to-point loading conditions (Fig. 39A), the particle is loaded between two points. A main tensile crack is created through the piece from one loading point to the other since the tensile strength of rock-like material is much lower than its compressive strength. The particle is broken into two or more smaller progenies. Minor cracks cluster around the loading points. In fact, this kind of failure mode is rather similar to that induced in the conventional Brazilian test in rock mechanics. Laboratory Brazilian experiments [29] have shown that a rock disc under a diametrical compression will split into two or several fragments with fractures along the loaded diameter. Therefore, the simulated failure modes are reasonable. Under plane-to-plane loading conditions (Fig. 39B), the particle is loaded between two approximately parallel planes. At ﬁrst, a number of tensile cracks are initiated at the irregular corners. Then the tensile cracks propagate to form surface splitting and the irregular parts of the particle are spalled. This observation from numerical tests is directly in line with the photographs of real rock tests illustrated in Hudson et al. [63]. Finally, in the central portion of the particle, shear cracks propagate to form through-going localized faulting and the remaining particle is fragmented into two parts. According to the experiential result predicted by Wang et al.’s [14], as shown in Fig. 39(b)B, just tensile splitting fractures are induced in the particle under plane-to-plane loading conditions and there is no shear fracture. However, in a usual UCS test, shear failure is often observed. In fact, after the surface chipping, as shown in Fig. 32(a)C, the plane-to-plane loading is similar to the conventional uniaxial compression test in rock mechanics. As we know in the uniaxial compression test, in the ﬁnal stage the through-going shear fracture band is always formed and the specimens are completely divided

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Fig. 39. The comparison of failure modes between numerical simulated results and experiential results.

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into two parts by the major shear band. Therefore, the results obtained in these numerical simulations are reasonable. Under point-to-plane loading conditions (Fig. 39C), the particle is loaded in a mixture of point-to-point and plane-to-plane loading. The fracture pattern shows an actinomorphic distribution with numerous minor cracks around the vicinity of the upper loading points and a major splitting crack extending till the bottom loading platen, as can be seen from Fig. 39(a)C. In fact, the point-to-plane loading condition is rather similar to the indentation test conducted by Lindqvist [16]. There is a local crushing zone around the upper loading points. A few long cracks propagated out of the local crushing zone to form Hertzian crack and median cracks. Under multi-point loading conditions (Fig. 39D), especially under three-point loading conditions in this simulation used, the major tensile fracture starts at the middle of the lower surface between the two bottom loading points and propagates upwards. The three-point loading condition is similar to the conventional three-point bending test used to obtain the Mode I fracture toughness in fracture mechanics. For a three-point bending specimen, the stress is tensile in the lower part of the cross-section, but compressive in the upper part according to the beam theory. The tensile fracture on the loading plane is usually the only important visible fracture for this type of loading. However, under multi-point loading conditions, depending on the locations of the loading points, cracks also develop around the contact points and tend to propagate along the direction connecting contact points. On the basis of the above discussions, it can be seen that although the particles are mainly loaded by compression, a major catastrophic tensile splitting crack is also induced under various loading conditions. Moreover, around the loading points there are localized crushing caused by the high compressive stress. Therefore, in particle breakage processes, the catastrophic tensile splitting and progressive compressive crushing are the main mechanisms. As a result of these mechanisms, the particle is broken into two kinds of progenies with two distinct size ranges: coarse particles and ﬁne particles. The coarse particles are products resulting from the induced catastrophic tensile splitting failure and the ﬁne particles are products resulting from progressive compressive or shear failure near the points of loading. Whiten [48] proposed that two breakage functions could be used to describe progenies caused by two different breakage mechanisms. Our numerical results are consistent with Whiten’s conclusions [48]: the ﬁrst mode of breakage was identiﬁed as catastrophic breakage of particles, which produces a small number of relatively large particles. The second mode was the production of ﬁne material at the points of contact between the particles and the crusher lines and also between neighbouring loading points.

5.5.2. E¡ect of particle shape, size and loading conditions On the basis of the simulated results, it is observed that the catastrophic tensile splitting cracks are always induced under the various loading conditions. The

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major tensile cracks are formed between the loading contact zones and propagate in curved fracture trajectories but approximately follow the direction of the major principal stresses that are perpendicular to the maximum tensile stresses. Finally, the fractures separate the particle into two main parts with some ﬁner fragments formed along the fracture zones. Therefore, the observed fracture phenomena in loading particles under the various loading conditions are particularly related to the conventional Brazilian test. The stress distribution within the particle can be illustrated by a close form solution to a Brazilian disc compressively loaded along its diameter, as shown in Fig. 40a. The axial stress sr and the lateral stress sy along the loading diameter can be explicitly expressed as [47]: 2 P ½1 r2 sin 2a 1 1 þ r sr ¼ þ tan tan a pRta 1 2r2 cos 2a þ r4 1 r2

sy ¼

2 P ½1 r2 sin 2a 1 1 þ r tan tan a pRta 1 2r2 cos 2a þ r4 1 r2

ð2Þ

ð3Þ

where r ¼ r/R, t is the thickness of the disc, R is the radius of the disc, P is the load and a is the angle at which P is applied. Figure 14b shows the analytically calculated normalized vertical stress sr and normalized horizontal stress sy along on the loaded diameter according to above two equations. It can be seen from Fig. 14b that the vertical stress is always compressive and that although the horizontal stress in the vicinity of the loading points is compressive, it becomes tensile and distributes fairly uniformly over the central part of the loaded diameter,

Fig. 40. Diametrical compression of a Brazilian disc.

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which is the tensile strength of the disc and can be calculated as Hondros [47] st ¼

Pm pRt

ð4Þ

where Pm is the maximum load in diametrically loading the disc. Moreover, according to Cauwellaert and Eckmann [64], the elastic modulus of the disc can be calculated according to the following equation: 2P 4 4 1 sin a E¼ ð1 vÞ ln 1 þ tan þ ð5Þ pDwt sin a 2 sin2 a where E is the elastic modulus of the disc, Dw is the diametrical compression displacement and v is the Poisson’s ratio. On the basis of the above equations, data obtained by the RFPA2D code were used to calculate the following fundamental characteristics such as particle strength and particle stiffness, associated with the deformation and the fracture of particles under various loading conditions. The calculated results are summarized in Table 1. It must be pointed out that in our numerical simulation, the particle has an irregular shape and the loading condition is also different from that in diametrical compression of the Brazilian disc. However, on the basis of the work conducted by Hu et al. [8], the equations can be expanded to be somewhat valid for an irregular particle and to obtain some interesting results. The particle tensile strength is approximately calculated using Eq. (5) on the basis of the simulated peak load, as shown in Figs. 29A, 31A, 35A and 37B. According to the input parameters (compressive strength s0 ¼ 200 MPa and elastic modulus E0 ¼ 60 GPa) used to construct the numerical models for the particles, the tensile strength should be st ¼ 20 MPa if the particle has a regular circle shape of 54 mm and is loaded under diametrical compression and the elastic modulus is E0 ¼ 60 GPa if the particle has a regular cylindrical shape with a height–width ratio of 2.5 and is loaded under uniaxial compression. An index of the tensile strength Ks is deﬁned as a ratio between the particle tensile strength and the tensile strength from the Brazilian test, whose values are shown in Table 4. This index of strength reﬂects the inﬂuences of the particle irregular shape, size and the loading condition, which have been omitted in studies up to now except that Hu et al. [8] used this index to investigate stress distribution in Brazilian disc under the three loading conditions of two opposite concentrated forces, two opposite uniform pressures and two opposite Hertz’s pressures. It can be seen that the particle under point-to-point loading is most related to the Brazilian disc under diametrical compression, and then it is in the order of multi-point loading, point-to-plane loading and plane-to-plane loading. The particle stiffness (Table 4) is calculated using Eq. (6) according to the simulated vertical compression displacement corresponding to the peak load. Similarly, an index of stiff KE can also be deﬁned as the ratio

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Table 4. Parameters to measure fracture characteristics

Loading conditions Peak load (N) Particle tensile strength (MPa) Index of strength, Ks Particle stiffness (MPa) Index of stiffness, KE Worka (J) Elastic energy releasea (J) Energy utilization ratio (%)

Point-topoint

Plane-toplane

5488.0 23.3

25097.0 106.5

1.165 63.0 1.050 0.934 0.127 13.6

5.325 62.9 1.048 7.524 0.482 6.4

Point-toplane 4533.4 37.0 1.850 44.7 0.745 1.847 0.155 8.4

Multipoint 8323.5 35.3 1.765 67.3 1.122 1.887 0.211 11.2

a

Only the work and the elastic energy release before the onset of the shearing and slipping region are calculated.

between the particle stiffness and the stiffness obtained from uniaxial compression, which reﬂects the inﬂuences of the particle irregular shape and the loading condition on the particle stiffness. It seems that in those simulations, the inﬂuence on the particle stiffness is not as big as the inﬂuence on particle strength. Moreover, the comparison between force–displacement curves obtained under various loading conditions shows a brittle to ductile transition in the order of point-to-point loading, multi-point loading, point-to-plane loading and plane-to-plane loading. The energy consumption is also one of the important considerations in the particle breakage process during comminution. Early investigations on the comminution process have been concerned with the relationship between the applied energy and the energy consumed for particle size reduction, e.g. Rittinger’s surface theory, Kick’s volume theory and Bond’s third theory [65]. In this study, by integration of the resultant force P over the displacement of the loading platen (point) u, it is possible to obtain the applied work, which corresponds to the area below the force-deformation curve calculated as Z

umax

W ¼

P du

ð6Þ

0

where u is the particle deformation and umax is the deformation at fracture. As an example, the elastic energy release (ENR) for the plane-to-plane loading condition is recorded as shown in Fig. 34. The ENR corresponds to the work dissipated in breaking the particles. Combining the applied work and the accumulated ENR, the energy utilization ratio can be calculated. In loading the particle with irregular shape and size under various loading conditions, the applied work, elastic energy and energy utilization ratio are summarized in Table 4. It can be seen that the

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point-to-point loading is the most efﬁcient and then in sequence of efﬁciency it is multi-point loading, point-to-plane loading and plane-to-plane loading.

6. INTER-PARTICLE BREAKAGE In the production of ballast materials and pavement aggregates, mechanical crushing is a method widely used to liberate valuable minerals from ores or to reduce the particle size of rock materials. However, it is energetically very expensive, with a cost close to 2% of the world energy production [23]. Facing this cost, understanding the crushing mechanisms inside rock particles under compression may provide an effective key to better fragmentation efﬁciency. On the basis of previous research [7,30,35], two breakage modes have been identiﬁed in mechanical crushing: single-particle and inter-particle breakage. Singleparticle breakage occurs when the distance between the chamber walls is equal to or smaller than the particle size, which is relatively simple. Inter-particle breakage occurs when a particle has contact points shared with other surrounding particles, and this is believed to be an important breakage mode in mechanical crushing. It is obvious that the inter-particle breakage process is very complex. To facilitate an in-depth study of this process, theoretical models have been developed. In the present section, the inter-particle breakage process under conﬁned conditions in mechanical crushing is numerically investigated and is discussed in terms of the two loading geometries: quasi-uniaxial compression and quasitriaxial compression. The work presented herein forms part of an on-going investigation into the fundamental aspects of rock breakage, in order to improve the design of rock fragmentation equipment from a mechanics point of view.

6.1. Fragmentation process of a rock particle assembly in a container The calculation of the stress distribution inside particles with multiple contact forces and the prediction of the inter-particle fracture process in a particle bed have been the biggest challenges in the mechanical crushing industry. So far a limited amount of research has been carried out in the ﬁeld of numerical analysis of these problems from the mechanics point of view. The fragmentation process of particles subjected to multiple loads from neighbouring particles or machine walls in a rock particle assembly will be investigated in this section.

6.1.1. Numerical model Fandrich et al. [18] developed the experimental set-up for particle bed breakage shown in Fig. 41a to imitate the operating principle of mechanical crushing

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Fig. 41. The principle of mechanical crushing: (a) a schematic diagram of the experimental set-up for particle bed breakage [18]; (b) a crushing chamber [7].

equipment. The pot contains the sample (1), a cylinder (2), a piston (3) and a base (4), with a plate (5) on top to locate the three LVDTs (linear variable displacement transducers) (6) that are ﬁxed to the cylinder. The entire pot is placed in a loading frame. The load and LVDT signals are logged by data acquisition software running on a PC (7). Evertsson and Bearman [7] described the sample (1) in more detail to simulate the conditions to which a volume of material is subjected in a real crushing chamber, as shown in Fig. 41b. Correspondingly, a similar numerical model, shown in Fig. 42, is constructed to investigate the interparticle breakage process in a particle assembly. In practice, the shape of the working surface may be a plane, cylindrical or spherical. Any combinations of these shapes are possible for investigating a particular conﬁguration in a mill. Fundamental studies on the breakage behaviour should preferably use two parallel plane surfaces [17]. Therefore, a parallel plane surface is used in Fig. 42. The numerical test corresponds to the compressing part of the machine cycle of the crusher, when the liners move towards each other. The material is then locked between the chamber walls and can only deform elastically or break into smaller particles. For the rock materials of interest, brittle fracture is the only fracture mechanism of importance. Since the maximum radial velocity of the mantle relative to the concave in normal operating conditions is below 0.5 m s–1, it is assumed that the breakage is independent of the strain rate at this level [30]. In the present work we simply treat the breakage process as quasi-static. The numerical model (Fig. 42) consists of a crushing chamber and 27 randomly placed rock particles with radii following a Weibull distribution within the chamber, where the individual particles are subjected to an arbitrary set of contact forces. The model consists of a steel container measuring 180 mm in width and height. The thickness of the container walls is 5 mm. A steel platen measuring 170 mm in

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3

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Fig. 42. Numerical simulation model for a crushing chamber containing 27 rock particles.

width is used as a cover on the top for transferring a compressive load down to the rock particles in the vertical direction. The chamber contains 27 particles, which are numbered from 1 to 27 for convenience in the following discussion. The particle bed is loaded under form conditions with the assumption of plane strain. In the form-conditioned case, the size reduction and applied force are a function of the displacement. In the simulation, the axial load is increased by moving the upper loading platen downwards step by step in a displacement control fashion. In the model, the walls that have the same modulus as the loading platen impose a horizontal constraint against the particles inside. This provides the necessary conﬁned condition for inter-particle breakage. Similar numerical models have also been used by other researchers [23,30]. Compared with Tsoungui et al.’s model [23], the numerical model in the present paper regards the rock as breakable particles and can deal with irregularly shaped particles, even though circular particles are used here for comparison with Tsoungui et al.’s experimental results [23]. The inﬂuence of an irregular shape on the particle breakage process will be discussed in Section 5.1. Compared with Kou et al.’s model [30], the residual strength of the element after failure relating to the conﬁnement and the ability of the contact point to resist compressive stress but not tensile stress are the main features of the present numerical model. The RFPA2D code randomly generates different circular particles by adjusting the overlapping elements between neighbouring particles to satisfy the deﬁned percentage (approximately 85%). The different particles consist of different amounts of elements. The elements are deﬁned according to the heterogeneous

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material model [62] with the homogeneous index m ¼ 2 and the following elemental seed parameters: the elastic modulus E0 ¼ 60 GPa, the compressive strength s0 ¼ 200 MPa, etc. The steel container and the steel platen are simulated as homogeneous materials whose elastic modulus and strength are ﬁve times higher than those of rock in order to prevent them from the permanent deformation.

6.1.2. Stress ¢eld In a conﬁned particle bed, no particles can escape stressing by moving sideways. The general loading process is that contact forces act on a particle, deform it, and may cause inelastic deformation and breakage. A contact force in general is directed obliquely and generates always a pressure and a shear. A contact can arise between two neighbouring particles or between a hard surface and a particle. Both contact situations cause different effects on the deformation and the stress distribution in the contact volume and thus on the breakage. Therefore, knowledge of the deformation and the stress distribution in the interior of the particle skeleton is helpful in understanding the breakage behaviour. Since stress or strain cannot be measured systematically in situ in the interior of a particle assembly in the container, physical model tests, such as photoelastic tests, are often the only way to investigate a stress ﬁeld under idealized conditions. However, in applying the optical method, some demands have to be met concerning the test technique. One of them is that a transparent and optically sensitive material such as glass or epoxy resin has to be used [66]. Therefore, although optical stress measurements in photoelastic materials open new perspectives in research on stress ﬁelds, this method makes it impossible to investigate stresses in materials, such as rocks, which are by no means transparent. To overcome such a difﬁculty, the RFPA2D code is used to obtain the full ﬁeld stress information for the particles. In order to obtain clear pictures that resemble the photoelastic test, a numerical model which is the same as that illustrated in Fig. 42 is constructed, but the material in this model is considered to be homogeneous. Numerically generated fringe patterns in each particle and in the wall of the container are shown in Fig. 43a. This ﬁgure indicates that the overall load produces contact forces between the particles. These contact forces create stress distribution in the particles. The stress distribution in the model can be visualized, and the interaction between the particles, as well as between the particles and the container walls, can be examined in more detail. Figure 43b and c shows the numerically obtained major and minor principal stress ﬁelds. The larger the stress, the brighter the element becomes. Figure 43b and c indicates how the load transfers from one particle to another through contacts. The compressive stresses in the particles are attenuated from the top to the bottom, which can be seen from Fig. 43b. This is because part of the load that acts on the top cover is borne by the container wall.

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Fig. 43. Elastic stress distribution in a crushing chamber before breakage (obtained from RFPA2D modelling).

6.1.3. Inter-particle breakage process Figure 44 shows the inter-particle progressive breakage process under compression in conﬁned conditions. At the beginning of the simulation, one can observe that in the polydisperse particle bed, the grain fragmentations are located ﬁrst on the small grains, such as the grains numbered 1, 4, 6, 7, 12, 13, 16, 18, 19, 20, 21, 23, 24 and 25 (refer to Fig. 42 for the particle numbers) in Fig. 44A, B, C and D. In those grains, the splitting macroscopic cracks are initiated and propagate along the lines between the two highest stressed contact points. The reasons for this fragmentation are purely geometric. As a matter of fact, with respect to the rest of the packing, the small grains have few contact points with the neighbouring grains or the walls of the crushing chamber, i.e. they are grains under an almost quasiuniaxial compression. In Section 3, we have shown that under uniaxial compression the fragmentation processes develop very quickly and the particle collapses over a very small strain range. In this case, axial splitting between the loading points is the prominent characteristic. For the large grains, the fragmentation is more difﬁcult, because their large number of surrounding contacts create a dominant hydrostatic effect around the grains, i.e. quasi-triaxial compression, for example the grains numbered 2, 5, 8, 10, 11 and 15 in Fig. 44A, B, C and D. As the loading displacement increases, with more and more small-sized grains fragmenting, the large grains also undergo failure. Although the splitting cracks are still initiated and propagate along the lines between the two most highly stressed contact points (e.g. the grains numbered 8 and 15 in Fig. 8E, F, G, H and I), because the previous failures release the conﬁnement, grain crushing has also become an important failure mechanism (e.g. the grains numbered 2, 5, 10 and 11 in Fig. 8G, H and I). A large number of Hertzian cracks are initiated from the highly stressed contact points to form chips. The development of the major principal and minor principal stress ﬁelds that initiate material failure are visually shown in Figs. 45 and 46, corresponding to

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Fig. 44. Simulated progressive fragmentation process for rock particles inside a crushing chamber (obtained from RFPA2D modelling).

loading displacements of 0.015, 0.08, 0.145, 0.16, 0.27, 0.36, 0.405, 0.47, 0.48 and 0.53 mm, respectively. Again, it is seen that the RFPA2D code yields valuable information about stresses and deformations in the deforming particle assembly. The stresses are made visible by these ﬁgures, and the effect of variations in the failure process on the stress redistribution is clearly visualized. This is of great signiﬁcance in understanding failure mechanisms of heterogeneous materials, since, currently, the numerical method is the only method of obtaining detailed ﬁeld information about the stresses in systems with a complex stress distribution in heterogeneous materials. As shown in Figs. 45 and 46, the contact forces are

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Fig. 45. The major principal stress distribution of rock particles inside a crushing chamber during the inter-particle breakage process (obtained from RFPA2D modelling).

displayed through the contact points with a high stress ﬁeld. The externally applied pressure is carried by the particles, but it is not uniformly distributed on all the particles. At the beginning of the simulation, one can observe that the particle breakage starts ﬁrst with the small-sized particles because of the small number of contact points with the neighbouring particles or the walls of the crushing chamber. After the small-sized particles have fractured, one begins to observe that the large particles begin to break in the middle of the assembly. This result agrees well with experimental tests [23], as shown in Fig. 47: the grain fragmentation begins ﬁrst in the small-sized grains, particularly in the grains with uniaxial

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Fig. 46. The minor principal stress distribution of rock particles inside a crushing chamber during the inter-particle breakage process (obtained from RFPA2D modelling).

Fig. 47. Experimental results for inter-particle breakage inside a particle assembly [23].

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geometric conﬁgurations, and the large grains are always difﬁcult to break because of their hydrostatic environment.

6.1.4. Resultant force and displacement relationship Figure 48 records the total crushing force (F) and the displacement (S) curve in the simulation. The sequence of failure in Figs. 44–46 corresponds to the equilibrium states identiﬁed on the response curve by the letters A, B, etc. The response has the general features common to many brittle materials. Initially, it is relatively stiff and nearly linear (AB in Fig. 48). Each particle deforms elastically. At a load of approximately 1364 N (point B in Fig. 12), the response begins to soften, mainly due to the breakage of particles 1, 6, 7, 12, 13, 16 and 24 (Figs. 44–46B), and eventually a limit load develops at 1858 N (point C in Fig. 48). The fracture is localized in particles 1, 4, 6, 7, 12, 13, 16, 18, 19, 20, 21, 23, 24 and 25 (Figs. 44–46C and D) beyond the limit load (point D in Fig. 48). These particles collapse with a contact location dependence, a tensile-type mode under quasi-uniaxial compression, while other particles next to them remain more or less elastically deformed because of the conﬁnement from neighbouring particles, covers or container walls. As the collapse of these particles progresses, the failures spread to the neighbouring particles. Eventually the whole assembly collapses. The spreading of the collapse from particle to particle continues, creating an undulating load plateau, as shown in Fig. 48 after point C. One can observe the fragmentation regime, characterized by the irregular saw-toothed curve (point D, E, F, G and H) of the load, as a function of the loading step. At a loading displacement of 0.36 mm, corresponding to point F in Fig. 48, almost all the particles have collapsed (Figs. 44–46F), except a few bigger ones in the middle of the Force in X direction

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particle bed. After this point the load starts to increase again (point G in Fig. 48), which implies a material-hardening characteristic. The propagation of the collapse involves the shear-type mode as well as the tensile-type mode in the large grains (Figs. 44–46G, H and I), which causes the initial instability. Furthermore, each load undulation corresponds to the collapse of one or more particles. Therefore, the particle assembly exhibits a great many load hills and valleys across the plateau. In this case, the maximum load (point G in Fig. 48) extends much higher than the assembly strength at point C in Fig. 48. The average load of the plateau, which will be called the propagation load, is about 2000 N. Simply stated, the inter-particle breakage appears to consist of two phases. Phase I, consisting of stiff particles, behaves as an elastic structure that dissipates little energy and imposes local deformation on phase-II particles. Phase II acts like a brittle-and-plastic material that dissipates energy at the contacts and serves as a restraint on phase I. As the applied displacement increases, certain changes take place in the material structure in that some phase-I particles become phase-II particles; i.e. more and more particles progressively fail, and the particle assembly, therefore, becomes less able to carry the load. The load is transferred from new chains that develop from other particles nearby. An anisotropic fabric is established corresponding to the direction of the applied loading. Owing to the direction changing, a new fabric develops. Moreover, the side-walls of the crushing chamber are kept at the same places in the vertical direction. There is no movement in the horizontal direction. The resultant force displacement response on the side-walls shows the same features as that in the loading plates (the lower curve in Fig. 48).

6.1.5. Energy transformation Energy is one of the important considerations in the breakage process during comminution. The energy is supplied to the particle bed by moving the loading platen. The force ﬂows from the loading platen through the bed towards the bottom platen and the conﬁnement wall. By integration of the crushing force (F) over the displacement of the loading platen (S) in Fig. 48, the energy consumption (W) can be obtained Z Smax W ¼ F dS ¼ 1:003 J ð7Þ 0

where S is the displacement of the loading platen, F is the applied force and Smax is the maximum displacement of the loading platen when loading the particle bed. Figures 49 and 50 show the numerically obtained fracture event rate and the ENR inside the particle during the inter-particle breakage process. The failure event rate shows the following expected features: (1) during the initial deformation or linear elastic phase (line AB in Fig. 48), little elastic energy (Fig. 50) was

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Fig. 49. Relationship between the failure event rate and the accumulated failure event rate during the inter-particle breakage process (obtained from RFPA2D modelling).

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Fig. 50. Relationship between the elastic energy release (ENR) and the accumulated ENR during the inter-particle breakage process (obtained from RFPA2D modelling).

released, although some fracture events (Fig. 49) occurred; (2) an increasing rate of fracture events (Fig. 49) accompanied the inelastic phase and the load plateau (point C in Fig. 48). This agrees with the understanding that the fracture events are generated by micro-fractures that result in non-linear deformation behaviour [22,23]. It is important to note that, although nearly 60% (59.2% in fact) of the fracture events (Fig. 49) occur before point F in Fig. 48, less than 35% (34.3% in fact) of the elastic energy is dissipated during this stage, as shown in Fig. 50. A comparison among Figs. 48, 49 and 50 shows a good relationship between the load curve, failure event rate and energy release. Note that most of the large load

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drop on the load curves shown in Fig. 48 corresponds to a high failure event rate (Fig. 49) and a big energy release (Fig. 50). The applied work corresponds to the energy consumed in breakage of the particles and energy losses. According to the accumulated energy release shown in Fig. 50 and the applied work shown in Fig. 48, about 18% of the applied work is consumed in breaking particles. Compared with the single-particle breakage conducted by Tang et al. [35], inter-particle breakage has a lower energy utilization ratio because of local crushing at contact points. The applied energy is mainly consumed by (a) acoustic emission, (b) the formation of new surfaces, and (c) local crushing at contact points, etc.

6.1.6. Size distribution The size distribution is another important consideration in the inter-particle breakage process during comminution. A quantitative description of the size reduction would be helpful for a better understanding of the comminution in mechanical crushing and especially for modelling the crushing performance. We have studied the distribution of fragment size corresponding to the inter-particle breakage process, as shown in Fig. 51, in which the letters correspond to those in Fig. 44. An image analysis program, Particle2D developed by Wang et al.,3 has been used to measure the fragment size. At present, the way in which the degree of reduction is deﬁned has not been standardized. Here the diameter of the equivalent circle area (DECA) of a particle has been used to measure the size of the particle. In Fig. 51, the abscissa is the DECAs of the particles (mm) and the y-coordinate is the cumulated weight distribution (%). As can be seen from the ﬁgure, before crushing (Step A), less than 8% of the DECAs of the particles are smaller than 12 mm. With the loading displacement increasing, the size reduction effect increases rapidly before Step F (the onset of the material-hardening regime) in Fig. 51: from less than 8% of the DECAs of the particles being smaller than 12 mm to more than 70% of the DECAs of the particles being smaller than 12 mm, as shown in Fig. 51 (Step B, C, D and E). After Step F, local crushing at contact points becomes the important failure mechanism and the size reduction effect increases slowly, as shown in Fig. 51 (Step F, G, H and I). In practice, in order to reduce the ﬁnes or to control the microcracks within the reduced particles, a careful design of the normal stroke is important. With the assumed mechanical properties, and the size and shape of the assembled rock particles, as well as the height of the container, the present simulation indicates that a normal stroke between 0.3 and 0.4 mm (Fig. 48F) may be a good choice to avoid over-breakage (Step F in Fig. 51). After crushing (Step I), over 90% of the DECAs of the particles are below 12 mm. Besides, the size distribution also depends on 3

The program can be downloaded from the Internet at: http://www.imenco.se/particle.htm

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Fig. 51. The fragment size distributions corresponding to the slides in Fig. 44 (obtained from RFPA2D modelling).

parameters such as the bed height, the stroke and the initial size of the crushed particle.

6.2. Discussions In mechanical crushing, the comminution effect is achieved by two surfaces gripping the material between them and forcing the material with a compressive force imposed by the surfaces. Understanding the breakage mechanism inside the material is rather important for crusher design. However, so far studies have concentrated on the behaviour of cohesionless particles with particular emphasis on the macroscopic response. Few studies have been conducted to investigate the behaviour of a particle assembly at the levels of inter-particle breakage. Recently, Tsoungui et al. [23] developed a granular material model based on a molecular dynamics method to simulate the quasi-static evolution of a packing during compression and grinding. The particles are modelled as elastic discs. When a particle fulﬁls the fracture criterion, it is replaced with a set of 12 small discs of four different sizes, which are ﬁtted into its original volume and are then treated as new independent particles. In Section 4, we have presented the results of numerical simulation of inter-particle breakage in a 2D particle assembly resembling real crushing. Compared with the work of Tsoungui et al. [23], our investigation has taken a step forward in that the particle breakage in our model is completely based on mechanical principles. When the particle breaks depends on the strength criterion, how it is broken depends on the stress distribution and redistribution inside the particle, and where it is broken depends on the

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heterogeneous distribution in the particle. It is seen from Figs. 44–46 that the shapes of the small particles originating from the larger one by breaking are by no means regular. Moreover, the raw materials in the crushing chamber are different types of rock materials that are crystalline, heterogeneous and often irregularly shaped. The granular material model has difﬁculties in simulating the breakage process of real heterogeneous particles with various irregular shapes, which may cause a completely different particle bed arrangement and conﬁnement, and then inﬂuence the inter-particle breakage process. In the following, the inﬂuence of the particle shape and the conﬁnement on the inter-particle breakage process will be discussed.

6.2.1. In£uence of the particle shape on the inter-particle breakage In order to investigate the inﬂuence of the particle shape on the inter-particle breakage process, a numerical model similar to that illustrated in Fig. 42 was constructed. The particle bed consists of irregularly shaped particles, which have the same conﬁguration as those used by Kou et al. [30]. However, in Kou et al.’s [30] model, since the post-failure process is not related to conﬁning conditions, it shows difﬁculty in modelling the conﬁnement from the neighbouring particles and the chamber walls after some particles fail. If the particles in Fig. 42 are polydispersed, we can regard the particles in Fig. 52 as monodispersed, which means that the irregularly shaped particles have an approximately equal area. The irregular shape determines the number of contacts among neighbouring particles and the walls of the crushing chamber. Figure 52 shows the elastic stress distribution of irregularly shaped particles in a steel container. Similar to Fig. 43a, Fig. 52a indicates that the overall load produces contact forces between the particles. These contact forces create stress distribution in the particles. Figure 52b and c show the numerically obtained major

Fig. 52. Elastic stress distributions of irregular particles in a crushing chamber before breakage (obtained from RFPA2D modelling).

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and minor principal stress ﬁelds, respectively. The load is transferred from one particle to another through the contacts. In contrast to the stress ﬁelds induced in the regular circular disc, there are also stress concentrations around the geometric heterogeneities because of the irregular shapes. Since the stress chains in Fig. 52b are aligned in the direction of the largest forces, the bright stripes coincide with the major principal stress direction. It is seen that in the current particle bed arrangement, large forces are carried by chains of particles that are more or less aligned in the direction of the major compression. These chains become shorter and more kinked as an increasing load is applied. Figure 53 visually displays the progressive inter-particle breakage process of irregularly shaped particles. In contrast to the disc particle breakage process, where the fragmentation starts in the small-sized particles, one can observe that at the beginning of the simulation, the grain fragmentation starts ﬁrst with a few of the grains contacting the two rigid walls, as shown in the particles numbered 8 and 4 (please refer to Fig. 52 for the particle numbers) in Fig. 53B. The reason for this fragmentation near the walls is also purely geometric. As a matter of fact, with respect to the rest of the packing, the grains in contact with the walls present geometric conﬁgurations close to those of grains submitted to two opposed forces, i.e. quasi-uniaxial compression, which favours their fragmentation much more. Previously, we have shown in the polydisperse circular particle bed that the fragmentation starts with the small particles. In fact, irrespective of whether it is a small-sized circular particle or an irregular particle contacting the rigid walls, a particle is selected to break ﬁrst if two criteria have been fulﬁlled: (1) the particle is located in such a way that quasi-uniaxial compression can be achieved, i.e. in such a way that splitting failure easily occurs; and (2) the stress level of the particle has reached a critical value. After the ﬁrst few grains have been fractured, one begins to observe that the grain fragmentations are mainly located on the grains under an almost quasi-uniaxial compression, as shown in the particles numbered 4, 6, 8, 9, 12 and 13 in Fig. 53C, D and E. At the same time, because of the failure releasing the conﬁnement, the particles loaded at ﬁrst under quasitriaxial compression begin to fail also, as shown in the particles numbered 1, 5 and 10 in Fig. 53C, D and E. After that, local grain crushing at the contact points becomes an important breakage mechanism, as shown in the particles numbered 1, 4, 5, 7, 8, 9, 12 and 13 in Fig. 53F, G, H and I, although a splitting failure of the quasi-uniaxial compression type (particle 14 in Fig. 53) still occurs. Compared with the inter-particle breakage process simulated by Kou et al. [30], at the ﬁrst stage, the fragmentation develops in the same way. However, after some particles fail, it seems that in the current simulation the particles are more difﬁcult to break than those in Kou et al.’s [30] simulation. It is reasonable since the particle post-failure process in Kou et al.’s [30] model is not related to conﬁning conditions and the conﬁnement from neighbouring particles and the chamber wall after some particles fail is difﬁcult to simulate there. It is because the particle

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Fig. 53. Simulated progressive fragmentation process for irregularly shaped rock particles inside a crushing (obtained from RFPA2D modelling).

post-failure process in the model is related to the conﬁnement that the particles show the trend to be more difﬁcult to fragment in the current simulation. It was noted that the particle breakage strength depends on the particle shape. Our results suggest that the more spherical the particle becomes, the higher is the breakage strength that may be expected (particle 7, 10, 11, 14 and 15 in Fig. 53). Presumably, a disc particle has a more regular stress distribution than an irregular particle. In fact, in order to achieve the same propagation load as is shown in Fig. 48, a higher percentage (approximately 90%) of irregularly shaped

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particles (Fig. 53) is ﬁlled into the crushing chamber compared with the percentage (approximately 85%) of circular particles (Fig. 44).

6.2.2. Two kinds of fracture pattern in the inter-particle breakage process With a view to achieving a better understanding of comminution, the fracture patterns in the inter-particle breakage process are discussed herein in terms of conﬁnement from the neighbouring particle, the loading plate or the side-wall. As shown by previous simulations, no particle can escape the stress caused by moving the loading plate in a conﬁned particle bed. The general course of events is that at ﬁrst the overall load produces contact forces between the particles, and then contact forces act on a particle, deform it, and cause inelastic deformation and breakage. A contact force in general is directed obliquely and generates always a pressure and a shear. A contact can arise between two neighbouring particles or between a hard surface and a particle. Both contact situations cause different conﬁnements on the deformation and the stress distribution in the contact volume and thus on the breakage. In fact, the fracture patterns of the particles in a conﬁned particle bed are rather similar to those of a rock specimen with different conﬁnement. A careful comparison between the breakage processes of the particles and the rock specimen in uniaxial and triaxial compression can verify that statement. It is clear that the particles numbered 13, 19, 20 and 21 (please refer to Fig. 42 for the particle numbers) in Fig. 42D and numbered 4, 8, 9 and 12 (please refer to Fig. 52 for the particle numbers) in Fig. 53D were loaded similarly to a rock specimen loaded in uniaxial compression. For example, particle number 8 in Fig. 53 was loaded by the particle above; number 5 and the particle below; number 12. The conﬁnement is provided via the container wall from the left and particle number 9 from the right. However, in this case, conﬁnement from the lateral sides did not have any important inﬂuence on the particle breakage process; i.e. the particle is loaded in quasi-uniaxial compression. One can note from Fig. 53D that tensile cracks initiated from somewhere along the line connecting the two vertical contact points and split the particle into two halves, which is the typical failure pattern in uniaxial compression. This kind of loading mode also explains why the fragmentation starts from the small particle in the polydispersed disc particle bed (Fig. 44), while in the monodispersed irregular particle bed (Fig. 53), the fragmentation starts from the particles near the loading plate or the container wall. As a matter of fact, in the polydispersed disc particle bed, with respect to the rest of the packing, the small grains have fewer contact points with the neighbouring grains or the walls of the crushing chamber; i.e. they are grains under a quasi-uniaxial compression. Therefore, the fragmentation processes in the small grains develop very quickly and the particles collapse over a very small loading displacement. In the monodispersed irregular particle bed, the grains in contact with the walls present

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generally geometric conﬁgurations close to those of grains submitted to two opposed forces, i.e. quasi-uniaxial compression, which favours their fragmentation much more. The particles numbered 2, 5, 8, 10, 11 and 15 in Fig. 44 and numbered 1, 2, 7, 10, 11 and 15 in Fig. 53 can be regarded as being loaded in triaxial compression. They do not fail ﬁrst because of conﬁnement provided by the neighbouring particle or a hard surface. For example, particle number 2 in Fig. 44G was loaded via the relative movement of the upper loading cover and the particle below; number 8. The conﬁnement is provided via the neighbouring particles numbered 1, 3 and 9. In this case, conﬁnement from the lateral sides does have an important inﬂuence on the particle breakage process. In fact, the surrounding contacts create a dominant hydrostatic effect around the grain. Therefore, it does not fail ﬁrst. As the loading displacement increases, fragments are torn off from the main loading points and a large piece is preserved. In this case, grain crushing around the contact points becomes the important fragmentation mechanism. With the loading displacement increasing, chips, ﬁnes and crushed zones are formed, particularly in the vicinity of contact locations. Therefore, according to the above simulations of the inter-particle breakage process, two loading geometries and correspondingly two kinds of failure patterns can be recognized: quasi-uniaxial compression and quasi-triaxial compression. In the quasi-uniaxial compression case a particle is loaded between diametrically opposed surfaces and the resulting stress ﬁeld consists of a zone between the loading points, which is in indirect tension. The resulting fragmentation is usually two pieces of rock approximately half of the original size of the particle and a collection of ﬁne rock particles generated in the compression zone. For high reduction ratios, it has been seen that thin particles with one dimension equal to the original particle size often generate from the zone between the loading points [12]. Thus, in the rock fragmentation industry, it is suggested to arrange the particle bed so that the quasi-uniaxial compression is easily achieved to facilitate its fragmentation. In the case of quasi-triaxial compression, the stress ﬁeld is different. The loading is not diametrically opposed, so that a simple major ﬁeld cannot develop. Depending on the location of the loading points, the stress ﬁeld at the points is more direct tension. A large piece is preserved and in addition there are other fragments that have been torn off at the loading points. The effect of tearing off protruding sections is the physical reason why quasitriaxial compression improves the shape without destroying the overall size of the rock [12]. Thus, in the rock aggregate industry, it is suggested to arrange the particle bed so that the quasi-triaxial compression is easily achieved to obtain the even particle products. Moreover, in this case grain crushing around the loading points becomes an important fragmentation mechanism. Our numerically simulated results are consistent with those observed by Briggs and Evertsson [12].

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7. SUMMARY (1) Breakage is the way a particle is broken into smaller fragments. The numerical modelling of such a process should be closely physically related. The numerical approaches presented in this paper have shown that the realistic failure process analysis code (RFPA2D) provides a powerful tool in quantitatively simulating the micro- as well as macro-behaviour of particle breakage observed in laboratory tests. From the single-particle breakage studies, we demonstrate that the behaviour of particle breakage is strongly dependent on the loading conditions, and we have thus obtained the following conclusions: (a) The mechanical behaviour of the particle has shown a brittle-ductile transition depending on loading conditions with respect to conﬁnement; (b) The lateral constraint increases the initial stiffness and the maximum breakage strength of the particle; (c) Three crack types are observed: the primary crack, the secondary and tertiary cracks. The tortuous crack propagating paths and the crack branching behaviour are found to be dependent on the heterogeneity of the material and the stress distributions; (d) The diametral loading without conﬁnement produces more coarse particles due to splitting parallel to the loading diameter, whereas the diametral loading with conﬁnement generates many more ﬁnes due to shear failure, particularly in the vicinity of contact points in which a crushed zone forms; (e) Most of the energy release is spend in the crushed zones during the crushing process; (f) The regular shaped particle has larger breakage strength compared with the irregular one. (2) The particle breakage processes under various loading conditions are also numerically investigated with RFPA2D. Of special interest is the understanding of how the breakage behaviour depends on the heterogeneous particle material properties, irregular particle shape and size, and the various loading conditions. The simulated results are consistent with the general description and experimental results in the literature on particle breakage. On the basis of the simulated results, it is demonstrated that the behaviour of particle breakage is strongly dependent on heterogeneous particle material properties, the irregular particle shape and size, and the various loading conditions. Depending on the loading conditions, the mechanical behaviour of the particle has shown a brittle-ductile transition, the ﬁnal fracture modes change from the dominantly catastrophic tensile splitting to the co-exist of catastrophic tensile splitting and progressive compressive crushing, and correspondingly the size of the progenies changes from coarse to ﬁne in a sequence of pointto-point loading, multi-point loading, point-to-plane loading and plane-to-plane loading.

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(3) The numerical simulations have shown that the use of RFPA2D allows us to obtain some clariﬁcations on the inter-particle breakage when under compression. From this numerical investigation, we have thus obtained the following conclusions: (a) A simple description and qualitative model of the inter-particle breakage process in a particle assembly can be summarized as follows. At ﬁrst the overall load produces contact forces between the particles, those contact forces act on a particle, deform it, and cause inelastic deformation and breakage. In more detail, ﬁrstly the particles deform elastically, which is the elastic deformation regime. Large forces are carried by chains of particles that are more or less aligned in the direction of the major compression, depending on the particle bed arrangement. These chains become shorter and more kinked with an increasing load. As the loading displacement increases, the elastic structure gradually fails and the fragmentation regime begins. The fracture is at the beginning initiated in particles located in such a way that quasi-uniaxial compression is easily achieved, i.e. the connection line of the two highest contact points runs parallel to the major principal stress. With a further increase of the loading displacement, the particles, being loaded at ﬁrst in quasi-triaxial compression because of the conﬁnement from the neighbouring particles, loading plate or container wall, fail progressively, and therefore the particle assembly becomes less able to carry the load. The crushing propagates in a particle-by-particle fashion, while the average load remains relatively constant. Finally, as the loading displacement increases, local grain crushing at the contact points becomes an important mechanism and the densiﬁed assembly recovers a signiﬁcant stiffness because of a re-compaction behaviour, which is the assembly hardening regime. (b) A combination of the numerically obtained force–loading displacement curve, the fragment size distribution and the progressive breakage process makes it possible to estimate the necessary stroke and input energy needed to effectively breaking particles to a desired extent. With the assumed mechanical properties, and the size and shape of the assembled rock particles, as well as the height of the container, the present simulation indicates that a normal stroke between 0.3 and 0.4 mm may be a good choice. (c) The particle shape has an important inﬂuence on the particle bed arrangement and then inﬂuences the contact conditions between particles. A comparison between the breakage processes of particles with different shapes shows that in a particle bed consisting of circular particles, the fragmentation starts ﬁrst with small-sized particles, but in a particle bed consisting of irregular particles, the fragmentation starts ﬁrst with a few of the grains contacting the two rigid walls. However, in both cases, the

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fragmentation starts from the particles which are loaded in a quasiuniaxial compression condition. Besides, in current particle bed arrangements, it seems that quasi-uniaxial compression is more easily achieved in an irregular particle bed than in a circular particle bed. (d) Two kinds of loading geometry and correspondingly two kinds of fracture patterns are recognized: the quasi-uniaxial compressive fracture pattern and the quasi-triaxial compressive fracture pattern. In the quasi-uniaxial compressive fracture pattern, a particle is mainly loaded between diametrically opposed points and the resulting fragmentation is usually two pieces of rock with the main fracture connecting these two opposed points and are produced by axial splitting, which mainly occurs in the ﬁrst stage of the inter-particle breakage process. In the quasi-triaxial compressive fracture pattern, the local crushing at the contact points becomes the important failure mechanism. Depending on the location of the loading points, the stress ﬁeld at the points is more direct tension and small fragments are torn off at the loading points with a large piece preserved. Thus, in the rock fragmentation industry, it is suggested to arrange the particle bed so that the quasi-uniaxial compression is easily achieved to facilitate fragmentation. In the rock aggregate industry, it is suggested to arrange the particle bed so that the quasi-triaxial compression is easily achieved to obtain more uniform particle products. (4) We used the dynamic version of RFPA2D with the advantage of direct simulation of fracture and fragmentation, to investigate the processes of tensile failure and compressive crushing in rocks subjected to dynamics loading. This numerical approach accounts explicitly for the development of macroscopic fractures, and uses a mixed mode of failure strength criterion to control the failure and fragmentation processes. Numerical simulations of three samples with different heterogeneity demonstrate that the heterogeneity of rock has a big inﬂuence on stress wave propagation. The effective dynamic behaviour of samples under three different loading conditions is predicted as an outcome of the calculations. In particular, the simulations capture the reasonable stress wave amplitude sensitivity of the dynamic fracture patterns of the rock samples, i.e. a transition from a small number of primary fractures in the load plane to profuse secondary fractures near the load and support areas as well as along planes parallel to the load plane. At lower peak load of the stress wave, only a few larger micro-fractures form along the diametrical line under the applied load, which eventually causes the sample to split. At higher peak load, large numbers of micro-fractures are generated, which leads to higher fracture energy dissipation compared to those obtained at a lower peak load of the stress wave. For lower load stress waves, the fractures did not start to develop until the stress wave reached the bottom support areas. For higher load stress waves, however, fractures form as soon as the pressure stress

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wave travels into the sample. In the latter case, the simulation also captures the inclined fractures leading to the formation of the chips due to the reﬂected tension waves from the two bottom sides of the disc. It is concluded that although the rise time is the same for all three cases, the fracture processes and failure patterns are strongly dependent on the stress wave amplitude. The outcomes obtained from the simulations, which are very rich in information concerning fracture initiation and kinetics as well as the stress ﬁeld observations, make this method an ideal candidate for the analysis of material failure under a fully dynamic framework. The simulations not only allow identiﬁcation of model parameters but also explain the different failure mechanisms of rocks as a function of loading waveforms. It is seen that the model is suitable for simulating fracture processes and the failure patterns in rock materials under dynamic loading conditions. ACKNOWLEDGEMENTS The work presented here is also partially from results obtained during my visit to Prof. P.A. Lindqvist and Prof. S.Q. Kou as a collaboration. The contribution of my students, W.C. Zhu, T.H. Yang, W.H. Li and P. Lin to this research is also appreciated.

Nomenclature

st sr sy P Pm R t a E s0 E0 W

tensile strength (MPa) axial stress (MPa) lateral stress (MPa) load for Brazilian test (N) maximum load for Brazilian test (N) radius of Brazilian disc (mm) thickness of Brazilian disc (mm) angle at which load P is applied (1) elastic modulus (MPa) average strength for elements (MPa) average elastic modulus (MPa) work done by the applied load (J)

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CHAPTER 17

The Cohesion of Fractal Agglomerates: An Elementary Numerical Model Emile Pefferkorn Institut Charles Sadron (CNRS), 6 Rue Boussingault, 67083 Strasbourg,Cedex, France Contents 1. Introduction 2. Universality of the fractal structures 2.1. Characteristics and synthesis of fractal aggregates 2.1.1. Diffusion-limited cluster–cluster aggregation 2.1.2. Reaction-limited cluster–cluster aggregation 2.2. Fractal aggregates in industrial processes 2.2.1. Carbon black 2.2.2. Aerosils 2.2.3. Ceramics 2.3. Fractal aggregates in food processing 2.4. Fractal aggregates in agriculture and environment 3. The cohesion concept 3.1. Aggregate cohesion and additives 3.2. Aggregate assemblies: scaling approach 3.3. Cohesion against break-up rate 4. Background and literature review 4.1. From experimental observation towards modelling 4.2. Determination of the rate of aggregate breakage 4.3. Recent models of fragmentation 4.3.1. The model of Horwatt et al. 4.3.2. Fragmentation controlled by the number v of intra-agglomerate connections 5. The cohesion of fractal agglomerates: an elementary model 5.1. Aggregate formation 5.2. Agglomerate formation 5.2.1. Number of conﬁgurations 5.2.2. Connection frequency 5.2.3. Agglomerate cohesion 5.3. Fragmentation of DLA and RLA agglomerated systems confronted to the cohesion model

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Corresponding author. Tel.: +33 3 88 414026; Fax: +33 3 88 414099; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12020-0

r 2007 Elsevier B.V. All rights reserved.

742 5.3.1. Fragmentation of aggregates formed under conditions of diffusion-limited aggregation 5.3.2. Fragmentation of aggregates formed under conditions of reaction-limited aggregation 6. Forward look 6.1. Aggregate cohesion and aggregate growth 6.2. Aggregate growth in the presence of connection constraints 7. Conclusion Acknowledgements References

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1. INTRODUCTION Powder agglomeration induced by physical forces or mediated by additives, and agglomerate fragmentation intervene in a huge number of industrial, pharmaceutical, environmental and agricultural processes [1–9]. In a great number of situations, powder presents fractal structure or surface so that prediction of the total interaction energy between solids of irregular shapes is impossible. In the classical theory of colloid aggregation, colliding colloids are considered to oppose ﬂat or spherical surface areas or at least surface areas of simple geometry [10–13]. For such systems, attractive van der Waals forces and repulsive (electrostatic, hydration y) forces may be calculated according to well-known theories. The present study aims to present an elementary model able to quantitatively determine the internal cohesion of a three-dimensional agglomerate constituted of a couple of clusters on the basis of purely geometrical considerations. Threedimensional clusters generated on cubic lattices may represent very ﬁne grains when the grains develop very high surface roughness or powder of larger size when the fabrication method consists of the random agglomeration of smaller particles. Several techniques were found to lead to fractal aggregates. The model of cluster formation in the laser vaporization source starting from the atomic vapour may be applied to carbon clusters [14]. Heat released in cluster fusion allows small- and medium-size clusters to attain the lowest energy conﬁgurations in the diffusion- and reaction-limited aggregation (RLA) regimes [15]. For vapourphase silica aggregates of different sizes, electrical light scattering measurements indicate the fractal dimension to be close to 2.5 [16]. For industrial Aerosils OX-50 (Degussa), combined electric light scattering and sedimentation techniques show the fractal dimension to be close to 1.76 [17]. Carbonaceous soot, which is a common by-product of fossil fuel combustion, is composed of ramiﬁed aggregates of roughly spherical primary particles of radii 10–30 nm. Each aggregate combines from 1 to 100 particles and has a fractal dimension close to 1.8 in good agreement with diffusion-limited cluster aggregation [18]. For aggregates of Cab-O-Sil (Cabot Co) silica particles produced by ﬂame hydrolysis

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of SiCl4, the fractal dimension was determined to be close to 2.5 [19]. In the formation of fumed silica, primary spheres cool very rapidly and form strong agglomerates as they collide with each other in the cooler part of the ﬂame. These primary aggregates then agglomerate to form loose assemblages that can be dispersed by relatively mild shear forces [20]. Fractal character concerns organic materials too. Polymer grains like agglomerated PVC ﬁne particles and rubber crumbs used in the recycling of rubber manifest the fractal structure [21]. The industry of ceramics may be concerned by the present problem of aggregate assemblage when the primary particles are polymorphic and/or the powder contains fractal aggregates [22]. The manufacture of solid compounds by powder sintering includes several distinct processes. In one case, the powder composition and the thermal treatment are ﬁxed in order that a sufﬁcient quantity of liquid is formed to ﬁll all empty spaces between the particles of the powder. This sintering by viscous ﬂow is a form of consolidation for materials like clays. The second process is that of sintering in liquid phase. It rests on the use of compositions and processing temperature since liquid is produced but in insufﬁcient quantity to ﬁll all space between the grains. It is thus necessary that the grains change form during the process so that porosity is reduced to the point where the quantity of the liquid phase becomes sufﬁcient to ﬁll the residual volume. A more complex method is the sintering in a solid state without liquid phase. The ﬁrst problem relates to the reliability of the ceramic components and in particular their mechanical behaviour. The mechanical failure occurs in general in ceramics according to a fragile mode, with sudden and total failure of a component by fracture. This fracture has its origin in a crack or a defect of the microstructure. The fracture comes not from an average defect of the structure but of the most severe defect, identical to the weakest link in a chain. It is therefore very important to avoid any accidents during processing that could result in structural defects and thus in the long term, to induce the rupture of the material. Flocculation clustering was shown to induce weakness of ceramics [23]. For example, dried TiO2 powder of less than 2 mm in diameter was die-pressed at high pressure to produce a compacted pellet with a packing fraction of 0.55. On redispersing this pellet in water, the powder was seen to contain a peak of stable aggregates close to 60 mm in size and the existence of such defects was conﬁrmed by ﬂaw-size measurements on sintered ceramic specimens. Several models were elaborated to interpret agglomerate cohesion phenomena on the basis of fragmentation (bulk fracture) or attrition (surface erosion) processes. The cohesion of agglomerated solids depends both on the packing of constituent particles within the agglomerate and the strength of the inter-particle interaction [24]. Previous overly simpliﬁed models treat the structure of the agglomerates as uniformly porous or uniformly assembled spheres [25,26]. These

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models were improved taking into account that a variety of fragments can be produced from the rupture of a parent cluster. Fragments will form according to the spatial distribution of internal packing heterogeneities corresponding to both weak and strong portions of the agglomerates. This observation led Horwatt et al. [24] to investigate the effect of the fractal nature of agglomerates on their cohesion. The authors ﬁnally concluded that the different fracture models could be applied taking into account not only the fractal dimension but also the void/pore size frequency. Agglomerate attrition phenomena were also investigated with the aid of numerical simulation by Bortzmeyer [27]. Simulation of the impact fracture and fragmentation of agglomerates were carried out by Thornton et al. [28]. The bond strength of the primary particles and the velocity impacts constituted the main variables of this investigation. Investigations relative to the attrition or fragmentation of powder and grains are developed taking into account a great number of models [29]. In the present model, the grains were considered to be fractal agglomerates only differing by the number of internal breakable links. The internal cohesion of the fractal agglomerate resulting from the encounter of two solid aggregates was investigated taking into account (i) the number of its breakable inter-particle links, (ii) the aggregate and the agglomerate masses, and (iii) the fractal dimension of the aggregates. The model rests on the following assumptions. The two pre-existing aggregates are non deformable and unbreakable and only the newly formed interparticle links resulting from the sticking of the two aggregates can be disrupted. Aggregate sticking generates a certain number of inter-particle contact areas, which intuitively are functions of the aggregate geometry and/or masses. The internal cohesion of given agglomerate is measured by the number of generated contact areas. For agglomerates of a given mass, the average cohesion is derived from the average number of the newly generated contact areas and the agglomerate break-up depends on the number of bonds broken or to be broken [30]. These points have been developed in the original references [31,32]. The richness of the present model rests on the universality of fractal structures of grains and powders.

2. UNIVERSALITY OF THE FRACTAL STRUCTURES Mandelbrot introduced and developed the fundamentals of fractals in his book ‘‘Fractal Geometry of Nature’’ [33]. The fractal concept has become an important tool for understanding irregular complex systems in various scientiﬁc disciplines. An important phenomenon in a wide variety of physical, chemical, medical, biomedical, environmental and industrial processes is the formation of large

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clusters by the union of separate small elements. Gelation and polymerization processes in polymer science, ﬂocculation and coagulation processes in aerosol and colloidal physics, percolation and nucleation in phase transition and critical phenomena, agglutination and antigen–antibody aggregation in immunology, crystallization and dendritic growth processes in material science are all examples [34]. The concept of fractals was later applied to fragmentation phenomena and fracture mechanisms [35–38]. In the present context, the cohesion of solid agglomerates is considered as far as the process leading to powder and grains combining random collisions between smaller systems and sticking. These colliding systems grow up to the moment where aggregation and break-up cancel each other.

2.1. Characteristics and synthesis of fractal aggregates From a qualitative point of view, a fractal is a rough object whose bumps show up at any length scale. Observed with the naked eye, a magnifying glass or using a microscope, it never presents smooth contours [39]. Aggregates or agglomerates in which we are more particularly interested have connected structures, generally built with particles of identical or similar geometry. Since the structures of a wide variety of colloidal aggregates were described in terms of the concepts of fractal geometry, much of our understanding of the structure and properties of fractal aggregates has come from computer simulations. Flocs or aggregates frequently have a complex random structure with a low average density. Fractal geometry does not provide us with a complete description of the aggregate structure but provides a base for developing a better understanding of many of the most important properties [40]. The fractal dimension Df gives a quantitative description of the spatial mass repartition within the object. The total mass contained in a sphere of radius r, centred at a point of the object, varies as r Df . The density of the matter within this sphere then varies as r ðDf dÞ , where d is the dimension of the space in which the fractal object is imbedded. Alternatively, for aggregates composed of identical particles, the density of matter contained in a sphere of radius r centred on a particle of the object is proportional to the probability to ﬁnd a particle at a distance r. This is the correlation function C(r) between particle positions. These deﬁnitions characterize self-similar fractals. In the on-lattice diffusion-limited aggregation (DLA) model of Witten and Sander [41] particles are added to a growing cluster via random walk trajectories, one at a time. Figure 1 shows two different issues of the random walk trajectory of particles in the presence of an already formed aggregate. The particles are considered to come from the inﬁnity but practically the walk starts at a given position on a circle enclosing the cluster. One trajectory brings the random walker

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E. Pefferkorn

Fig. 1. Representation of the random walks of particles and aggregates on the square lattice. Particle (a) and aggregate (c) ﬁnally stick to the large aggregate at the grayish positions while (b) ﬁnally remains as a single particle at the end of the walk.

(a) at a free position adjacent to the cluster, and then the cluster grows. The other trajectory moves the random walker (b) far from the central cluster: the walk is stopped at a given distance and a new walker starts its walk. The procedures are repeated many times until a large cluster has grown. From similar off-lattice tri-dimensional simulations, a fractal dimension of 2.52 is derived [40]. Many aspects of this model are of considerable interest including the effects of lattice anisotropy and ﬁnite particle concentrations. However, this concept does not lead to the formation of less dense structures resembling those formed in most two- or three-dimensional aggregation processes. The following models reveal a better analogy between experimental and numerical aggregates.

2.1.1. Di¡usion-limited cluster--cluster aggregation On-lattice DLA models were developed simultaneously but independently by Meakin [42] and by Kolb et al. [43]. In these models, a small fraction of the sites on a lattice were selected at random and ﬁlled to represent the aggregating particles. Particles connected via nearest-neighbour occupancy are irreversibly combined to form clusters. As the simulation proceeds the single particle (a) or (b) or the cluster (c) is selected at random and randomly moved by one lattice unit in one of the possible directions on the lattice network (Fig. 1). After a cluster has been moved, its perimeter is examined to see if any other clusters have been contacted. If two particles, one cluster and a particle or two particles did contact, they are joined permanently and their component particles continue to move in concert. In the simplest version of the cluster–cluster aggregation model, clusters are selected with equal probabilities corresponding to a mass independent cluster diffusion coefﬁcient. In general, the cluster diffusion coefﬁcient of a cluster

Cohesion of Fractal Agglomerates

747

containing i particles is not a constant and depends on their mass as given by Di ¼ D1 i g

ð1Þ

Both off- and on-lattice models have been investigated. In general, on-lattice models allow larger structures to be realized. Off-lattice models are more realistic and may give better information concerning short-range structures [40].

2.1.2. Reaction-limited cluster--cluster aggregation In reaction-limited cluster–cluster aggregation, many encounters between pairs of clusters are required to occur before aggregation is realized. In the simplest models, once bonding occurs the clusters are rigid and irreversibly bound. Computer models usually combine the DLA model but imposed some probability conditions before encounters between clusters succeeded in permanent cluster sticking. The collision efﬁciency Pij between clusters of masses i and j towards deﬁnite assemblage is mass-dependent according to P ij ¼ P 11 ðijÞs

ð2Þ

and assemblage required Pij to be greater than a threshold value. Aggregate structure can generally be characterized by light scattering or image analysis [44–47] and expressed in terms of the fractal dimension, which provides an insight into the arrangement of particles within the aggregate as well as the degree of compactness of aggregates. It is an important measure since increasing compactness often means greater inter-particle bonding that implies that the aggregate is stronger. The two cluster–cluster aggregation models lead to aggregate conformation differing by their fractal dimension Df. In the diffusionlimited model, fractal dimensions of 1.44 and 1.78 were determined for planar and cubic systems whereas in the reaction-limited models, fractal dimensions of 1.55 and 2.04 were determined for the two systems, respectively [39]. A lower Df indicates that the aggregate is open and more tenuous in structure. Accordingly, the density of the fractal aggregate is not linear with respect to size. As the aggregate size increases, the aggregate density is in fact decreasing. As a consequence, a large aggregate will have a lower density compared to a smaller one with the same fractal dimension. This aspect leads us to ﬁrst consider the aggregate strength. Aggregate strength is not an independent property; both the cluster structure and density are interrelated with aggregate strength. For example, aggregates with compact structure tend to be stronger and more resistant to break up, as there are usually more inter-particle bonds within the aggregates compared to aggregates of lower consolidation. However, in some cases, strong bonds between the individual particles will enable the aggregates to grow large with an open structure. Let us now imagine the cohesion of the agglomerate resulting from the sticking of two fractal unbreakable aggregates when the

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cohesion of the agglomerate only results from the variable number of links, which may be realized from the assemblage of two small aggregates, two large aggregates or aggregates of very uneven masses. Since this aspect will be presented and discussed in this chapter, we are lead to describe the fractal nature of products that are present in industry, food manufacturing, agriculture and environment.

2.2. Fractal aggregates in industrial processes 2.2.1. Carbon black Industrial carbon black is used in a wide variety of applications, including printing inks, toners, coatings, plastics, paper and building products. Purposes for using industrial carbon black include pigmentary, electrical, UV absorption and rheological properties. The selection of a speciﬁc carbon black application depends on the end-product requirements, as well as on processing conditions. Carbon black is the predominant reinforcing ﬁller used in rubber compounds. Through the addition of carbon black to the rubber compounds, durability and strength are signiﬁcantly improved. Improvement in rubber properties is a function of the physical and chemical characteristics of carbon black. Carbon black’s most important fundamental characteristics are aggregate size and shape, particle size and porosity, all of these characteristics being common to fractals [48–53]. Carbon black does not exist as primary particles. During manufacturing, the primary particles fuse to form aggregates and agglomerates. The shape and degree of aggregate branching is referred to as structure. Increasing structure typically increases modulus, hardness, electrical conductivity and compound viscosity and improves dispersion of carbon black. The particle size of typical primary carbon blacks ranges from 8 nm for furnace blacks to 300 nm for thermal blacks. Increasing primary particle porosity ultimately leads to decreasing aggregate density. Higher porosity allows compounders to increase carbon black loading, while maintaining compound speciﬁc gravity. Finally, the physical properties of carbon black rubber compounds depend strongly on the degree to which the carbon black is dispersed [54–57].

2.2.2. Aerosils Like carbon blacks, Aerosils products may be used as reinforcing ﬁller in elastomers where they considerably improve their mechanical properties, such as tensile strength, elongation at break and tear resistance. Aerosil-fumed silica serves to control the inﬂuence of temperature on mechanical properties. Aerosil-fumed silica is also used in paints and coatings to control rheological

Cohesion of Fractal Agglomerates

749

characteristics, as thixotroping agent, as an anti-settling agent and to help in the prevention of rust and corrosion. One of the most important applications of aerosil-fumed silica is its use as an active reinforcing ﬁller to improve the mechanical properties of silicone sealants. Aerosils are also employed as anticaking and ﬂow improvement additives. The fractal nature of pyrogenic silica has been investigated and established [58,59].

2.2.3. Ceramics The incorporation of porosity within a tailored structure gives porous ceramics many intrinsic properties such as high permeability, large surface area and good insulating characteristics. Porous ceramics have found a wide variety of applications as ﬁlters, membranes, sensors in biomedical and construction materials [60]. The microstructure combining porosity, particle and pore size distribution is a very important factor for many potential applications of porous ceramics. For instance, the increase in porosity of porous ceramic causes an increase in the permeability and the ideal combination of pore size and porosity could optimize the relationship of permeability and mechanical strength [61]. For some applications, the porosity has to be evacuated by sintering and new interfaces are thus generated. A fractal description of grain boundaries in a sintered powder metallurgical sample has been elaborated [62]. For catalytic applications, ceramics are developed with suitable porous network structures. The porosity requires optimization from the point of view of two factors. The component should have sufﬁcient strength to provide structural stability, and sufﬁcient porosity to offer efﬁcient catalytic activity. This requires suitable tailoring of the particle size distribution and sintering conditions of powder compacts [63]. Investigations relative to the sintering behaviour of the compacts made up of agglomerated powders determined that with the increase in sintering time and temperature their rate of sintering and densiﬁcation is slow compared to compacts of non-agglomerated powders. As a result, the agglomerated powders viewed as mass fractals are the ideal choice for fabricating porous solids suitable for various catalytic applications [64–66]. Many ceramics are produced commercially by compacting a powder and sintering. The mechanical and material properties of the ﬁnal product are strongly inﬂuenced by the spatial distribution of pores and particles, commonly denoted as the microstructure [67]. This has led to numerous investigations into the development of the microstructure during processing. Useful measures for characterizing the microstructure are the pore and particle size distributions. The development of the pore and particle size distributions during processing can be monitored by analysing samples taken at intermediate stages of the compaction

750

E. Pefferkorn

and sintering processes [63,68,69]. During compaction, the density increases, pores become progressively smaller, while the particle size is normally unaffected. In some cases, particles have been agglomerated prior to compaction to improve the strength of the green product. Agglomerates represent a second, relatively large class of particles, which are subject to compression and deformation during compaction. The microstructure of the product strongly inﬂuences the subsequent sintering process: pores either grow or shrink before reaching a stable size or disappearing completely. Similarly, particles shrink and disappear or grow into grains. Fractal cracks were determined to occur within dense material using Raman scattering. The parameters of such fractals reﬂect the continuity (connectivity) and homogeneity of the material at the nanostructural level [70].

2.3. Fractal aggregates in food processing Precipitation by polyelectrolytes is an important method in the processing of protein products since it provides high protein recovery at small polymer doses [71]. The ﬁrst stage of the recovery process is the formation of protein-polyelectrolyte complexes with intra-polymer and inter-polymer structures. The linear polymer appears as being decorated with globular proteins at this stage [72–74]. These complexes quickly aggregate to form primary particles of size ranging from 0.1 to 1 mm. Aggregation of these particles into large aggregates constitutes the ﬁnal process [73]. Casein proteins were shown to result in fractal aggregates when proteins, calcium ions and polyphosphate were mixed [75,76]. The rate of this aggregation process increases strongly with temperature and casein concentration. Aggregates were determined to have a self-similar structure with fractal dimension close to 2. The macroscopic properties of the basis fat crystal network control the quality characteristics of fat containing food products [77]. Attempts to correlate their macroscopic properties and the lipid composition and the polymorphic nature of the network were not successful. The level of structure of the fat crystal network that controls the macroscopic behaviour is its microstructure. The microstructures can vary from a diameter of 80 to above 120 mm. The microstructures assemble elements that are arranged in fractal geometries. It was concluded that the elastic constant of the network and the hardness of the fat crystal network might be derived from the images of the fractal network. When crystals stick together, they progressively form voluminous aggregates with a volume much greater than the total volume of the primary crystals. The aggregates may break up when the shear rate is increased. These forces are proportional to the cross section of the aggregate and therefore scale with the square of the aggregate radius. Besides

Cohesion of Fractal Agglomerates

751

break up of aggregates, shear stresses may also induce internal rearrangement of the aggregates, which may become more compact. If the aggregates have grown so far that a space-ﬁlling network has been formed, application of shear would cause break up of the network and internal rearrangement of the fragments [78]. Because of simultaneous crystallization and aggregation, crystals and aggregates may become sintered (growing together of crystals as a result of the formation of solid crystal bridges between adjacent crystals). The solid bridges formed can also be broken by the application of shear, and the resulting network fragments are attracted to each other by van der Waals forces. Fat crystal aggregates were assumed to be composed of very rigid non-fractal aggregates that are connected to each other by relatively weak links.

2.4. Fractal aggregates in agriculture and environment Fractal geometry provides the major contribution to improve our understanding of soils in incorporating the soil physical properties and the chemical and biochemical processes [79,80]. The key domains of soil physics are structure, porosity, permeability, strength and stability. All these characteristics are related with the spatial irregularity and heterogeneity of soils. Soil aggregates may be viewed as very heterogeneous in size, nature and cohesion. Tisdall and Oades distinguished among the following systems [81]: Aggregates larger than 2 mm. These aggregates consist of agglomerates held

together mainly by a ﬁne network of roots or by transient binding agents. A cross section of the agglomerate shows that it is porous and constituted of particles of 20–250 mm. Aggregates of size between 20 to 250 mm. These aggregates are stable to rapid wetting and are constituted of particles of 2–20 mm. They are very stable because they contain several types of binding agents whose effects are additive. Aggregates of size between 2 and 20 mm. These aggregates consist of particles smaller than 2 mm and are bonded together so strongly by persistent organic bonds that they are not disrupted by agricultural practices. Knowledge of the soil structure requires the complementary investigation on the solid and void size distributions within the system as well as that of the pore connectivity. Several studies were devoted to these objectives [82–85]. The relationship existing between the inter-particle cohesion within aggregates and agglomerates and the distribution of solids and voids is of major importance in the domain of the soil stability, namely for collapsible soils. Aggregates are the major component of the skeleton of such soils and the collapsibility is usually associated with bond break-up and closing of the inter-aggregate pores [86].

752

E. Pefferkorn

3. THE COHESION CONCEPT In many systems, organic binder was supplied to powders to allow the material to be safety handled and conveyed due to their sub-micron sizes. During stocking, handling and conveying from silos to manufactories, powders sustain attrition or break-up resulting in the production of new ﬁne particles. Several investigations have focused on such wet granulation processes that develop in wetting and nucleation, consolidation and growth and breakage and attrition. The challenge is to determine reliable methods to design formulations controlling efﬁcient granulation behaviours. Typically, the term granulation deﬁnes the granule agglomeration process carried out under agitation, in which the original particles can further be distinguished [87–90]. Efﬁcient powder mixing is essential to ‘‘uniform’’ binder dispersion that may be realized by conveying high powder ﬂux through the spray zone [91]. The critical binder volume fraction seems to be an additional important parameter [92].

3.1. Aggregate cohesion and additives The approach of granulation modelled by Iveson et al. [87] is a combination of three sets of processes: wetting and nucleation, consolidation and growth, attrition and breakage. The ﬁrst combination of wetting and nucleation mainly rests on the ‘‘quality’’ of the molecular interaction developed between the binder and the dry particles. The occurrences of optimal contact angle and spreading coefﬁcient may favour the large-scale reproducibility of the ﬁrst-step processes. The relative size of the binder droplet to primary particle size also conditions the development of the nucleation process. When binder and particles have similar sizes, the granulation resembles coalescence. When the binder has a large volume compared to the solid particles, the latter become sucked within the drop volume by an imbibition mechanism [93]. Since optimal binder dispersion infers uniform wetting, uneven binder distribution and wetting install a large size frequency of the growing binder/particles complexes with variable saturation degrees [94,95]. A relatively uninvestigated domain is relative to the supply of very small droplets of binder or binder solution to powders of larger size, when the relative drop size is small in comparison to the size of the powder. This aspect may concern the agglomeration of fractal aggregates (powders) when the binder delivery mode has to be especially controlled [32,96]. Granule consolidation rate and extent rest on the inter-related effects of capillary, viscous and frictional forces as well as on the particle size and energy of granule collisions. Granule growth rests on the combination of coalescence and breakage phenomena that ﬁnally cancel each other out to the beneﬁt of the formation of granules of average size and size frequency. Recently, an enriching

Cohesion of Fractal Agglomerates

753

comparison has been made between the rate of granule growth and the rate of aggregate growth in ﬂocculation. With the ﬂocculation approach two major functions were deﬁned: the collision frequency and the collision efﬁciency [97,98]. Different relationships for collision frequency were applied. Since all collisions do not lead to coalescence (or thickening) between the colliding granules, the degree of liquid saturation of granules in coalescence is expected to exert the same limiting roles than the degree of surface coating with destabilizing agents or the existence of energy gaps in colloid destabilization processes [99,100]. The application of these concepts to model granulation processes opened up new ways for revisiting their kinetic and mechanistic approaches as well as for initiating agglomerate cohesion studies on a more elementary basis [31,32,101].

3.2. Aggregate assemblies: scaling approach The investigation of Ehrburger and Jullien on aerosil silica systems [102] clearly illustrates the present concept of agglomerate cohesion. The authors derived the aggregate fractal dimension Df of aerosil silica systems from the pore size distribution curve [103]. The method was applied to aggregates generated on a lattice by the DLA mechanism [39] and the fractal dimension directly calculated by averaging the radius of gyration was estimated to be close to 1.75. According to the method developed in [103], one deﬁnes the minimum parallelepipedic box of volume Lx Ly Lz that contains the aggregate of N particles entirely. The box is then paved with cubes of increasing edges L, starting from L ¼ 1 (the lattice spacing) up to L ¼ max (Lx, Ly, Lz). One then deﬁnes as occupied cubes those containing at least one particle of the aggregate. One derives the cumulative volume V(L) of the holes as being the total volume of voids remaining inside the box after eliminating the occupied cubes. The volume V(L) is deﬁned by equation (3) and the fractal number Df derived from the slope of the representation of –dV(L)/dL vs. L was determined to be 1.75. V ðLÞ ¼ Lx Ly Lz N

ð3Þ

The same method was adapted to derive the fractal dimension of agglomerates of silica aggregates obtained by the secondary aggregation processes occurring during aggregate sedimentation. One may imagine that the deposition randomly induces some aggregate interpenetration and ends with the formation of new connections between the settled aggregates. One could imagine that all the connection possibilities were experienced in the presence of a great number of aggregates. This assumption is not valid since the method showed the deep of interpenetration to depend on the suspending medium. The difference in the fractal dimensions Df was attributed to the nature of the interaction existing between the silica surface and the liquid (water or undecane). Similar

754

E. Pefferkorn

observations were reported for carbon blacks [104]. This observation raises the question of the fractal dimension and the void volume and distribution for agglomerates of aggregates dispersed in a liquid, in a polymer or in air (in the powder state). Obviously, the extent of the aggregate interpenetration and the number of newly formed inter-aggregate connections are expected to control the cohesion of the resulting agglomerated system. Valuable characteristics of agglomerates formed by the interpenetration and sticking of two aggregates of mass i and j are as follows:

the the the the

connection frequency P(v, i, j), P total number of agglomerate conﬁgurations v Pðv; i; jÞ, pore frequency n(p), total porosity, VP and VPP.

These characteristics were determined for aggregates (clusters) and agglomerates resulting from the interpenetration of two clusters, each of the clusters being obtained using numerical simulations of diffusion- or RLA processes, as described in Sections 2.1.1 and 2.1.2. Each simulation run was stopped when the cluster of the desired mass was obtained and the spatial coordinates of all particles were then registered. At least ten clusters of identical mass were synthesized for statistical reasons to investigate clusters of various shapes. These hard clusters being strongly connected did not sustain further fragmentation or reconformation. The total volume VT of the cluster is determined taking into account the total volume of the individual adjacent boxes enclosing all the i particles belonging to both the inner and the peripheral open zones of the cluster. The aggregate volume corresponds to the total volume VI of the i constituting particles, the volume of each particle being equal to one. The porous volume VP is thus equal to the difference between the total volume VT attributed to the cluster and the cluster volume VI. This deﬁnition provides the same scaling law for the increase of the porous volume VP with the cluster mass i than does the application of the usual scaling laws of the fractal cluster for which VI is scaled by RDf , R being the radius of gyration and VT is scaled by R3. The resulting relationship shows VP to be scaled by i 3=Df for ic1. V P / R3 RDf / i 3=Df i / i 3=Df

i1

ð4Þ

Equation (4) accords with the variation of VP as a function of i derived from the analysis of the porosity of clusters of fractal dimension Df ¼ 1.8 and 2.0 obtained by numerical simulations of the diffusion- and RLA processes, respectively [39]. This methodology could be applied to agglomerates of two clusters of masses i and j. Two clusters were assembled so that all the possible non-overlapping conﬁgurations of the cluster agglomerate were experienced. To this end, one

Cohesion of Fractal Agglomerates

755

VPP

10000

VP

1000

100

10 100

1000 i+j

Fig. 2. Representation of the total pore volumes VP (open symbols) and VPP (ﬁlled symbols) as a function of the agglomerate mass (i+j). The pore volumes are averaged over all the non-overlapping conﬁgurations of agglomerates of clusters generated by the DLA (K, J) or RLA algorithms (’, &).

cluster is held ﬁxed and the second cluster is allowed to explore the six facets of the ﬁrst one. This movement is repeated ﬁve times so that at the end each of the six facets of the second cluster visited the six facets of the ﬁrst one. For each assembly sharing one or more faces of the extremely high number of agglomerate conﬁgurations obtained by assembling clusters of masses i and j, the pore volume VP of the agglomerate was determined using the similar method from the construction of adjacent boxes enclosing all the particles of the agglomerate. Obviously, the practical result corresponds to the average agglomerate pore volumes VP and VPP that correspond to the ﬁrst and second moments of the pore volume frequency. VPP provides an indication of the pore volume polydispersity since a greater variation of VPP with the agglomerate mass (i+j) indicates the prevalence of pores having larger volumes. Figure 2 shows the average pore volumes VP and VPP as a function of the agglomerate mass (i+j) for agglomerates obtained by assembling clusters generated by the diffusion- or RLA processes. The following relationships between VP, VPP and the agglomerate mass (i+j) were determined for agglomerate masses ranging between 50 and 800 by assemblage of clusters of even and uneven masses. For agglomerates of clusters generated employing the algorithm of the DLA or the RLA [39], the increase of the pore volumes is given by equations (5) and (6), respectively [106,107]. V P / ði þ jÞ1:62 ;

V PP / ði þ jÞ2:20

ð5Þ

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E. Pefferkorn

V P / ði þ jÞ1:70 ;

V PP / ði þ jÞ2:88

ð6Þ

For agglomerated DLA systems, the value 1.62 of the scaling law of the increase of VP with the agglomerate mass is slightly smaller than the value 1.67 determined for the parent DLA cluster. Agglomerated DLA systems tend to become more compact. Conversely, for agglomerated RLA systems, the value 1.70 of the scaling law of the increase of VP with the agglomerate mass is greater than the value 1.50 determined for the parent RLA cluster. Agglomerated RLA systems tend to become less compact. According to the increase of VPP with the agglomerate mass (Fig. 2), voids present within RLA agglomerated systems have larger volumes than those in DLA systems. Additionally, the exponents 2.88 and 2.20 show the portion of large volumes to increase more strongly for RLA agglomerated systems than for DLA ones. The agglomerate porosity VP/VT is represented as a function of the total agglomerate volume VI in Fig. 3 and the exponents of the scaling laws representative of the variations are 0.33 and 0.44 for DLA and RLA agglomerated systems, respectively. The porosity of DLA agglomerated systems is greater than that of the RLA ones despite the fact that the porosity of the latter ones increases more strongly with the agglomerate mass, at least in the domain of explored masses. Investigation of the pore volume frequency – the number n(p) of pores of volume p existing within agglomerated DLA and RLA systems – deserves a special interest since this characteristic is expected to inﬂuence the amount of binder additive required for establishment of the optimal degree of liquid saturation of 0.5

Porosity

0.4

0.3

0.2

0.1

0.0 0

200

400

600

800

1000

VI

Fig. 3. Representation of the agglomerate porosity VP/VT as a function of the volume VI occupied by the assembled particles of global mass (i+j). The solid lines represent ﬁts of the power laws of exponent 0.33 (J, DLA systems) and 0.44 (K, RLA systems).

Cohesion of Fractal Agglomerates

757

granules in wet granulation processes [107]. Extensive studies using DLA and RLA systems demonstrate the self-similar character of the pore volume frequency when using the reduced coordinates nðpÞ ðV 2PP =V P Þ and p/VPP as represented in Figs. 4 and 5, respectively by [105]. nðpÞV 2PP =V P / ðp=V PP ÞZ

ð7Þ

From equation (7), values of Z of 2.2 and 1.8 were derived from the slope of the reduced pore volume frequency at small values of the reduced pore volume represented in Figs. 4 and 5 corresponding to agglomerates of various masses generated from DLA and RLA systems, respectively. The decrease of the pore volume frequency with the pore volume indicates that pores of small volumes are always present at the maximal concentration even for agglomerates of extremely high porosity VPP. Equation (7) led to the conclusion that the global porous volume n(p) p attributed to pores of volume p decreases with the relationship p(1Z). The fact that the average porosity of the DLA agglomerate is greater than that of an RLA agglomerate of even mass (see Fig. 3) rests on this progressively decreasing volume frequency and on the fact that the number of small pores is greater for the former rather than the latter.

3.3. Cohesion against break-up rate The modelling of the wet granulation of powder and grains based on the concepts of particle size distribution in colloid aggregation can be extended to agglomerate 10000000

n(p)V2PP/VP

1000000

100000

10000

1000

100 0.0001

0.001

0.01

0.1

p/VPP

Fig. 4. Representation of the reduced void frequency nðpÞðV 2PP =V P Þ as a function of the reduced void volume p/VPP for agglomerates of masses equal to 300, 400, 500, 650 and 800 constituted by assembling cluster generated by DLA processes. Agglomerates are referenced by their average voids VPP and VP.

758

E. Pefferkorn 10000000

n(p)V2PP/VP

1000000

100000

10000

1000

100 0.0001

0.001

0.01

0.1

p/VPP

Fig. 5. Representation of the reduced void frequency nðpÞðV 2PP =V P Þ as a function of the reduced void volume p/VPP for agglomerates of masses equal to 149, 205, 415 and 580 constituted by assembling cluster generated by RLA processes. Agglomerates are referenced by their average voids VPP and VP.

and powder break-up resulting from collision processes. In this sense, attrition processes will not be taken into account for investigation of agglomerate cohesion. The fragmentation process means that aggregates become dispersed into single constitutive particles. However, this fully dispersed stage may not be reached for thermodynamic or kinetic reasons, and therefore at equilibrium or after a given period, fragments of smaller mass and size coexist with single particles. The unbreakable constitutive solid particle may be monodisperse or polydisperse in mass, size, and shape. In the present context aggregates are considered to be loosely assembled conglomerates of ﬁne particles bound together at their point of contact by weak van der Waals forces or by soft molecular bonding resulting from interaction with organic or inorganic binder additives. As for ﬂocculation processes, the granulation mechanism may be deﬁned on the basis of the shape of the aggregate mass distribution, and the relative variation as a function of time of the weight and number of average masses [34,101,107]. For ﬂocculation processes, controlled provoked fragmentation was found to occur in different ways. Agglomerates of lyophobic polystyrene latex particles with carboxylic acid surface groups instantaneously fragment when the density of the surface charge and the range of the electrical interaction are increased simultaneously [108]. Fragmentation does not lead to a fully dispersed system and the reduced mass distribution of the parent aggregates and fragments were found to be self-similar for aggregates generated by the RLA process [109].

Cohesion of Fractal Agglomerates

759

Fragmentation of such aggregates thus restores the suspension to the situation that existed earlier. This possibility depends on the system. Hydrophilic polystyrene latex particles with high density of various surface groups were determined to slowly aggregate in 1 mol/l NaCl suspension and to fragment in 0.05 mol/l NaCl, according to the range of the hydration forces in the two situations [110,111]. Within the fragmentation medium, instantaneous and delayed fragmentation ultimately led to a fully dispersed system, but the mass distributions determined during aggregation and fragmentation could not be correlated. Simultaneous granulation and granule fragmentation may coexist during mixing of powder and binder. When these two processes cancel each other out, a stationary situation arises where the mass distribution does not change with time. When the agglomerated particles sustain supplementary break-up forces, fragmentation sets in and the average masses S(t) and N(t) (equation (8)) of the fragments are smaller than the initial average masses of the parent agglomerates. The fragmentation process of the typical agglomerate Ay composed of y unbreakable assembled constituents is assumed to develop at random according to reaction (9) where the fragment Ai and Aj are composed of i and j unbreakable components. The rate constant of the fragmentation process B(i, j) is given by [112–116]. P 2 P n cðn; tÞ ncðn; tÞ SðtÞ ¼ Pn ; NðtÞ ¼ Pn ð8Þ ncðn; tÞ n n cðn; tÞ Bði;jÞ

Ay ! Ai þ Bj

ð9Þ

Bði; jÞ ¼ kði þ jÞa

ð10Þ

In the following part, we present elements of the recent theory of the rate of fragmentation [114]. The variation with time of the concentration c(n, t) of fragments composed of n unbreakable components at time t is given by @cðn; tÞ=@t ¼ aðnÞcðn; tÞ þ

1 X

cðy; tÞaðyÞf hnjyi

ð11Þ

y¼n

where a(n) is the rate of break-up of the aggregate of mass n and f hnjyi is the rate at which aggregates of mass n are produced from the break-up of aggregates of mass y. In the theory of Cheng and Redner homogeneous kernels of the forms (12) and (13) were taken into account [114] aðnÞ ¼ nl

ð12Þ

f hnjyi ¼ y 1 bhnjyi

ð13Þ

The rate b(n) of formation of fragments of mass n from aggregates of mass y is thus expressed by equation (14) and the treatment of equation (11) was carried

760

E. Pefferkorn

out on the basis of the reduced mass frequency given by equation (15) bðnÞ ¼ nn

ð14Þ

cðn; tÞS2 ðtÞ=N 1 / f ½n=SðtÞ

ð15Þ

N1 being the total number of single particles in the dispersed and agglomerated forms. Both a(n) and b(n) can be derived from the cluster mass distribution. The rate of fragmentation of aggregates of mass n, which is measured by the exponent l, may be calculated from the variation of j(n) in the range of large values of the reduced mass n/S(t) as expressed by equation (16). Clearly, the change in the concentration of aggregates of large masses should be highly affected by fragmentation. jðnÞ ¼ n2 expðanl Þ

ð16Þ

However, the rate of fragments of small masses can be derived from the slope of the function f [n/S(t)] (equation (15)) at small values of the reduced concentration. Clearly, the mass distribution of the smallest fragments is expected to be largely dictated by the rate b(n) of fragments of mass n resulting from the breakup of aggregates of mass n larger than y. Thus, the rate of formation of small fragments is measured by the value of the initial slope of the function f [n/S(t)] for n=SðtÞ 1. Therefore, the shape of the colloid mass distribution determines the mechanism and the rate of fragmentation. It must be noted that the method for derivation of the break-up rate only applies to granules composed of unbreakable grains stuck together with the aid of weak binder additives. With this limitation, the rate of break-up and the agglomerate cohesion are correlated.

4. BACKGROUND AND LITERATURE REVIEW The term ‘‘agglomerate strength’’ is not often well deﬁned but in general includes resistance to any kind of breakage during formation, handling and processing [117]. As the content of the book is dedicated to breakage, this paragraph only notices some experiments that were interpreted on the basis of numerical models.

4.1. From experimental observation towards modelling Fracture of solids always includes the creation of new surface area more or less at random that may be controlled by the stress characteristics. Additionally, the break-up force acting on a fragment depends only on the portion of the surface subtended by the fragment. The fragmentation of agglomerates is less complex

Cohesion of Fractal Agglomerates

761

in so far as the break-up only concerns a given number of bonds that must be severed to ﬁnalize the agglomerate rupture [24,30]. These concepts apply to break-up of agglomerates in simple shear ﬂow and the inﬂuence of structural heterogeneities is investigated by taking into account the mass of the agglomerate and its structure as measured by the fractal dimension. The irregular fracture model was elaborated to analyse the break-up of carbon black agglomerates [118]. Some information relative to the cohesion–structure correlation was derived from the behaviour of colloidal silica during uniaxial compaction [119]. Colloidal silica accords with the image of agglomerates composed of fractal blobs of constant structure that are connected by links of weaker cohesion. The pressure transmitted by colloidal silica samples during low-speed compaction is measured as a function of the volume fraction of the solid. Different behaviours were determined to occur for pyrogenic or precipitated silica. It appears that the mechanism of strengthening of precipitated silica is very different from that of pyrogenic silicas. For the ﬁrst, strengthening occurs by an increase of the size of the mechanical units, whereas for the latter, it probably occurs by breaking and restructuring processes. This could be attributed to the difference in the fractal dimension of the types of agglomerates. Pyrogenic silicas are tenuous and have a fractal dimension smaller than 2 whereas precipitated silicas exhibit a much higher mechanical strength and a fractal dimension larger than 2. Correlations could be established between the average high porosity, the presence of macro-pores and the resistance to break-up of catalyst carrier beads employed in the oil industry [120]. The behaviour of the beads under quasi-static compressive loading has been analysed and compared for samples with different macro-porosities to determine the inﬂuence of these macro-pores on the particle strength. The investigation related the bulk behaviour to the single-particle characteristics. Individual particle testing shows the Young modulus to decrease with an increase in the macro-porosity. The average crushing strength is maximal for the sample with the medium macro-porosity and minimal for the sample with the highest macro-porosity. As a result, the sample with the highest macro-porosity experiences most attrition. The other two samples sustained less breakage. The situation is, however, complex since the sample with the lowest macro-porosity is more resistant to breakage under low compressive loading and less resistant to breakage under higher compressive loading, compared with the sample of medium macro-porosity. The present aspect of agglomerate breakage also concerns de-aggregation processes of aggregated powders that may occur during ﬁne grinding. Fine grinding involves the breakage of bound aggregates rather than solid particles. The characteristics of the breakage were investigated by experimental studies of grinding kinetics [121]. The grinding rate and the product size distribution are determined by the frequency of the break-up events and the size distribution of

762

E. Pefferkorn

the fragments produced in each event. Breakage is expected to take the form of splitting of the aggregate into a few smaller pieces. Concomitantly, attrition and erosion of the periphery may be an important contributor to the size reduction process. As shown in equation (11), the rate of change in any size frequency is equal to the rate at which particles enter this size due to the breakage of larger size minus the rate at which material leaves the size by breakage into smaller sizes. The effects of the different breakage modes should be reﬂected in the form of breakage distribution. Bortzmeyer proposed a numerical model to interpret the attrition on the basis of single agglomerate that randomly sustains several successive impacts. The fragments were eliminated from the system and the total amount was determined as a function of time [27]. Bortzmeyer showed that the amount of ﬁnes does not account for the experimental determination of the amount of ﬁnes as a function of time. The discrepancy was attributed to the fact that the simulated erosion proceeds from destruction of a smooth surface and often produced single particles, while the test aggregates of small size came from erosion of surface asperities.

4.2. Determination of the rate of aggregate breakage Obviously, the kinetics of fragmentation should reﬂect the variation with time of the number N(t) and weight averaged S(t) masses deﬁned in equation (8) when the aggregate break-up is irreversible. When some fragments newly aggregate, the variations with time of N(t) and S(t) may not accord with the rate of break-up derived from equation (16). The mass frequency allows the average masses N(t) and S(t) to be calculated and, in the model of Cheng and Redner, their temporal variation is given by [114,122] SðtÞ NðtÞ t 1=l

ð17Þ

where the exponent l derived from equation (16) corresponds to the rate of fragmentation of large aggregates (equation (12)). Deviation from this irreversible fragmentation process may occur,which implies that the rate of variation of S(t) is different from that of N(t). The situation may be analysed taking into account the fragmentation model with detailed balance of Ernst and van Dongen, for which the aggregation prevails over fragmentation [123]. The model was shown to apply to the inverse situation where fragmentation prevails over aggregation [124] SðtÞ ¼ t 1=n

ð18Þ

where the exponent n derived from equation (14) corresponds to the rate of formation of fragment of small masses.

Cohesion of Fractal Agglomerates

763

Application of equations (12)–(18) to the experimental variations of the colloid mass frequency c(n, t) and to the variation with t of the aggregate average masses N(t) and S(t) leads to the deﬁnition of the fragmentation process as being partially reversible or irreversible. The break-up of clay particles agglomerated by adsorption of water-soluble polyelectrolytes, induced by increasing the range of the inter-particle charge–charge repulsive interactions, is irreversible since the sticking of new fragments is prohibited by the long-range interactions that are far beyond the range of van der Waals attractive forces. Conversely, when the fragmentation process of oxide particles agglomerated by the adsorption of polyelectrolytes resulting from modiﬁcation of the hydrophobic/hydrophilic balance characterizing the zone of mutual interactions of colliding fragments, the range of action of the repulsive forces exerted between hydrophilic moieties may be comparable to the range of action of the repulsive forces exerting between hydrophilic moieties. Therefore, the fragmentation may be reversible since some fragments may recombine under the effects of the attractive forces exerted between the hydrophobic moieties [125].

4.3. Recent models of fragmentation Investigation of the agglomerate break-up in the presence of hydrodynamic forces requires modelling of both the hydrodynamic forces FH acting on the agglomerate and the force FC responsible for the agglomerate cohesion [24]. Firstly, the hydrodynamic forces and the mechanical characteristics of the agglomerates (considered as non-permeable spheres) are brieﬂy resumed according to the presentation of Horwatt et al. Secondly, the fragmentation model based on the agglomerate cohesion resulting from the number of inter-aggregate links is shortly deﬁned [31,32].

4.3.1. The model of Horwatt et al. Figure 6 portrays the parent agglomerate and the fragment as two centroids of radius R and r centred at O and o, respectively; the fragment centroid being centred on the fracture vector at a distance (R–r) and C being the angle subtended by the fragment. The force binding the fragment to the parent agglomerate depends on the number v of bonds that must be severed prior agglomerate rupture. Thus, the cohesive force resisting rupture is given by FC ¼ Hv

ð19Þ

where H is the mean force of the doublet connection (particle–particle bond). Since the agglomerates are not perfectly spherical, they rotate through all possible orientations and occasionally will experience the maximal hydrodynamic

764

E. Pefferkorn

Fig. 6. Schematic representation of the parent agglomerate and the fragment of radii R and r centred at O and o. C describes the size of the of the fragment relative to the size of the parent agglomerate.

force FH during the course of the rotation. The hydrodynamic force FH acting on the fragment is a function of the portion of the cluster surface subtended by the fragment and independent of the internal structure of the fragment. Fragments within the agglomerate experience the maximal hydrodynamic force expressed by FH ¼

5 2 d 2 pR m g sin C 2

ð20Þ

d where m and g are the ﬂuid viscosity and the shear rate, respectively. Equation (20) expresses that the hydrodynamic force acting on any fragment depends only on the portion of the cluster surface subtended by the fragment. d At the critical rupture stress ðm g Þcrit for which FHZFC, rupture occurs for the values of the variable according to o 5n d F C ¼ Hv ðm g Þcrit pR2 sin2 C ð21Þ 2 This model clearly shows the respective inﬂuence of the hydrodynamic forces and the cohesive force resisting rupture of the intra-agglomerate bonds.

4.3.2. Fragmentation controlled by the number v of intra-agglomerate connections To accord the threshold conditions of the fragmentation deﬁned in equation (21) with the present model of fragmentation that includes the number v of connections as a critical parameter, we express the variable C as a function of the radius r of the fragment according to the schematic representation shown in Fig. 6. If the agglomerates are assumed to be composed of particles of

Cohesion of Fractal Agglomerates

765

diameter 1, equation (21) gives rise to 5 1 1 2 d pðm g Þcrit Hv 2 r R

ð22Þ

provided 0rrrR/2. From equation (22), one derives that application of the critd ical stress ðm g Þcrit to an agglomerate of radius R produces the fragment of radius r when less than v connections have to be severed. Agglomerates having more than v connections will not fragment. Inspection of equation (22) leads to the following comments. For a given v value, the value on the right-hand side of equation (22) decreases with increasing values of r and reaches the minimum for r ¼ R/2: the

fragmentation favours the formation of two fragments of even size; decreases with increasing value of R: the small agglomerate is more stable

against fragmentation than the large cluster. Equation (22) requires the determination of the number v of connections being established when the agglomerate is composed of two unbreakable aggregates. It is also assumed that the agglomerate cohesion may depend on the agglomerate mass (i+j), on the masses i and j of the two aggregates, and on the number v of connections. Additionally, the cluster porosity and the pore distribution may exert their role in the event of fragmentation. The present model thus assumes the agglomerate to fragment when the connection frequency P(v, i, j) is greater than a given constant value deﬁned as Pðv; i; jÞ ¼ ði=jÞði þ jÞv 1

ð23Þ

with io j [126]. Increasing v decreases the value of the right-hand side of equation (23), thus, allowing agglomerates of increasing mass (i+j) to resist fragmentation. Additionally, for a given v value, permanent sticking of aggregates of highly uneven masses is favoured to the detriment of aggregates of even masses. Finally, equation (23) accords with the priority order derived from equation (22).

5. THE COHESION OF FRACTAL AGGLOMERATES: AN ELEMENTARY MODEL Usually, powder presents a fractal structure or surface so that prediction of the total interaction energy between solids of irregular shapes is impossible. The present model of internal cohesion of agglomerates addresses assemblies of weakly agglomerated 3d-aggregates and the numerical study was elaborated to show to what extent the internal cohesion of a couple of stuck aggregates might

766

E. Pefferkorn

be simply determined by the number v of inter-aggregate connections and how v is correlated to the masses i and j of the aggregates or the agglomerate total mass (i+j). The investigation of the agglomerate cohesion employing agglomerates obtained by assemblage of 3d-aggregates on cubic lattices may serve to apprehend this special problem.

5.1. Aggregate formation Although off-lattice models are more realistic and may give important information concerning short-range structure, the lattice model may apply to systems for which ultimate contact between ﬂat solid surfaces is not required for weak assemblage of single grains. One may imagine that the presence of additives/ adhesives may stick together neighbouring primary particles belonging to different aggregates even when close contact between grains is not realized as it occurs for grain agglomerates strengthened by pendular liquid bridges [127]. Aggregates were generated using the algorithms of the DLA and RLA processes (see Section 2.1) with the usual constraints presented in equations (1) and (2). The parameters of the numerical study selected for realization of the initial aggregates are presented in Table 1. Each simulation run was stopped when the aggregate of desired mass i was obtained and the spatial coordinates of all constituents were then registered. At least ten aggregates of identical mass were synthesized for statistical reasons to take into account different spatial structures. These hard aggregates did not sustain fragmentation or reconformation and each aggregate constituted a single unbreakable grain.

5.2. Agglomerate formation Two aggregates of given mass were assembled so that all the possible nonoverlapping conﬁgurations of the agglomerate were experienced, one aggregate Table 1. Parameters of the numerical study

DLA

i

100

150

200

250

300

400

Di ¼ D1 ig

D1 g i P1 s

10 0.5 50 0.05 0.4

13 0.5 100 0.03 0.4

14 0.5 150 0.02 0.4

16 0.5 200 0.02 0.4

18 0.5 300 0.012 0.4

20 0.5

RLA Pij ¼ P11(ij)s

Cohesion of Fractal Agglomerates

767

being held ﬁxed and the second aggregate being allowed to rotate along its three axes. For each conﬁguration obtained by assembling aggregates of masses i and j, the set of assemblages P(v, i, j) constituted of v connections was calculated. Assemblies sharing one or more faces are taken into account and analysed. The P total number of sets v Pðv; i; jÞ was calculated for each couple of agglomerate P mass (i+j) as well the relative fraction v m Pðv; i; jÞ for the various values of m. The ratio deﬁned in equation (24) provides the relative portion F(m, i, j) of aggregates surviving stresses being able to disrupt agglomerates connected by less than m intra-agglomerate (or inter-aggregate) connections. Clearly for m ¼ 1, agglomerates of two aggregates only connected by a single connection will resist fragmentation and F(1, i, j) is thus equal to 1. P v m Pðv; i; jÞ Fðm; i; jÞ ¼ P ð24Þ v Pðv; i; jÞ The variable of interest is the mass of the initial clusters and agglomerates of systems of even and uneven masses were considered.

5.2.1. Number of con¢gurations P The total number, v Pðv; i; jÞ, of non-overlapping conﬁgurations resulting from the sticking together of two aggregates of even masses i or uneven masses i and j is represented in Fig. 7 as a function of the total mass (i+j) of the agglomerates. The lower and upper lines correspond to agglomerates obtained from the sticking of clusters generated by the algorithms of the RLA and DLA processes,

v

ΣP(v, i, j)

1000000

100000

10000 100

1000 i+j

Fig. 7. Representation of the total number of non-overlapping conﬁgurations obtained by all the possible assemblages of two aggregates of even (¢lled symbol) and uneven masses (open symbol) as a function of the mass (i+j) of agglomerates constituted by aggregates realized with the aid of algorithms of DLA (K, J) and RLA (’, &) processes.

768

E. Pefferkorn

respectively. The following relationship is true for agglomerates or aggregates of even (i ¼ j) or uneven masses i and j: X Pðv; i; jÞ / ði þ jÞd ð25Þ v d being 1.04 and 0.93 for DLA and RLA systems, respectively. The total number P of agglomerate conﬁgurations, v Pðv; i; jÞ, obtained for DLA systems is more than twice that obtained for RLA systems. Agglomerates of DLA aggregates offer two times more particles accessible to a mobile visiting aggregate than do the RLA systems. For a given type of aggregate, the number of complexions only depends on the total mass (i+j) of the agglomerates and does not depend of the masses i and j of the individual aggregates.

5.2.2. Connection frequency The numerical simulation provided the number v of connections concomitantly established for each agglomerate conﬁguration resulting from the sticking of two aggregates. The connection frequency P(v, i, j) was determined on the basis of P the total number v Pðv; i; jÞ of possible conﬁgurations. Figures 8 and 9 show two typical connection frequencies for agglomerates of mass (250+250) and (50+300) corresponding to sticking of aggregates generated by DLA and RLA processes, respectively. The maximal value of the ordinate and the slope of the variation of P(v, i, j) vs. v are functions of the agglomerate mass and the mode of aggregate synthesis. The following relationship applies to the frequency of connections: log Pðv; i; jÞ ¼ ð1 vÞ log Pða; i; jÞ

ð26Þ

1000000

P(v, 250, 250)

100000 10000 1000 100 10 1 1

3

6

9

12

15

18

v

Fig. 8. Representation of the number of non-overlapping conﬁgurations P(v, i, j) as a function of the number of connections v simultaneously established between two aggregates of equal masses of 250 generated with the aid of the DLA algorithm.

Cohesion of Fractal Agglomerates

769

100000

P (v, 50, 300)

10000 1000 100 10 1 1

3

6

9

12

15

18

v

Fig. 9. Representation of the number of non-overlapping conﬁgurations P(v, i, j) as a function of the number of connections v simultaneously established between two aggregates of different masses of 50 and 300, generated with the aid of the RLA algorithm.

where a is equal to 1 for DLA agglomerates and is very close to 1 for RLA agglomerates. Obviously, the connection frequency decreases exponentially with the connection number v that apparently did not exceed the value of 20.

5.2.3. Agglomerate cohesion The variable F(m, i, j) was calculated from equation (24) with the assumption that a given number m of connections may impede the agglomerate break-up and the portion of stable agglomerates is shown in Figs. 10 and 11 for DLA and RLA systems. In Figs. 10 and 11, it is shown that the agglomerate mass inﬂuences the cohesion term F(m, i, j) of small agglomerates but its inﬂuence vanishes above a given (i+j) value. For such agglomerates the best ﬁt of the variation of F(m) with m is given by the Johnson–Mehl equation (27) ln FðmÞ ¼ kmh

ð27Þ

where k and h are equal to 0.5570.01 and 1.034 or 0.3970.01 and 1.110 for agglomerates of DLA or RLA systems, respectively. The equation strictly applies for v43, while a deviation appears for lower values as shown in Fig. 12. Since the term [1F(m)] represents the fraction of agglomerates sustaining fragmentation when links of vrm connections become broken, the evolution with m of the amount of fragments may be similar to the evolution with time of the mass of fragments produced during the attrition test. Bortzmeyer determined the increase with time of the amount of fragments to be represented by the Johnson–Mehl equation [128,129]. In the experiment of Bortzmeyer fragments

770

E. Pefferkorn m 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

1

φ (m, i, j)

0.1

0.01

0.001

0.0001

0.00001 0

200

400

600

800

1000

i+j

Fig. 10. Representation of the internal cohesion F(m, i, j) of agglomerates constituted by assemblage of aggregates of even and uneven masses obtained with the aid of the DLA algorithm as a function of the agglomerate mass (i+j). F(m, i, j) represents the fraction of agglomerates surviving stresses being able to disrupt agglomerates connected by less than m inter-aggregate links (the corresponding values of m are indicated on the right side of the ﬁgure). m

1

1 2 3 4 5

0.1

6

φ (m, i, j)

7 8 9

0.01

10 11 12 13

0.001

14 15

0.0001 0

200

400 i+j

600

800

Fig. 11. Representation of the internal cohesion F(m, i, j) of agglomerates constituted by assemblage of aggregates of even and uneven masses obtained with the aid of the RLA algorithm as a function of the agglomerate mass (i+j). F(m, i, j) represents the fraction of agglomerates surviving stresses being able to disrupt agglomerates connected by less than m inter-aggregate links (the corresponding values of m are indicated on the right side of the ﬁgure).

Cohesion of Fractal Agglomerates

771

12

-ln[φ(m)]

8

4

0 0

4

8

12

16

20

m

Fig. 12. Johnson–Mehl plot of the internal cohesion F(m) of agglomerates formed by assemblage of aggregates generated with the aid of the DLA (&) and RLA (J) algorithms as a function of the number m of inter-aggregate connections.

came from the break-up of surface asperities that may behave like small fragments connected to aggregates of very large size. The characteristics of the links to be broken and the attrition time may be correlated in so far as the fragmentation of links of increasing connection number may become broken with increasing attrition time. With this reasonable assumption, the intuitive link between the cohesion of agglomerates and the attrition rate is highlighted. If one takes into account a stress fragmenting agglomerates of m connections, fragmentation of agglomerates of masses (2i), (2j) or (i+j) provides aggregates of masses i and j with the constraint that the concentrations of each class of broken agglomerates is independent of the agglomerate mass and corresponds to [1F(m)]. Agglomerates formed by assembly of RLA systems are characterized by a higher degree of cohesion than agglomerates of DLA systems: a greater amount of DLA agglomerates are broken under a given stress than RLA agglomerates. The higher brittleness of the DLA agglomerates may be further increased by their higher porosity as represented in Fig. 3.

5.3. Fragmentation of DLA and RLA agglomerated systems confronted to the cohesion model The DLA and RLA processes developing in unstable systems lead to aggregates (agglomerates) characterized by fractal dimensions equal to those of clusters generated by the numerical simulations [51,130–135]. It was thus interesting to analyse the fragmentation characteristics of these typical aggregates in the light of the present model, taking into account the model developed in Section 4.2. Spherical polystyrene latex particles of diameter near to 1 mm but of different

772

E. Pefferkorn

surface charge densities were selected as solid particles of mass 1. Owing to the presence of negatively charged sulphate surface groups, ﬁxing the ionic strength of the medium may control their stability characteristics in aqueous medium. When macromolecules are added to the latex dispersion, they exert a stabilizing effect at full coverage of the latex surface areas [136,137]. Addition of macromolecules to aggregated particles induces the fragmentation of the existing aggregates and, in the long term, leads to single stabilized particles. The polyvinylpyridine molecule may be viewed as a ﬂexible random coil whose maximal end-to-end distance is of the order of 2.5 mm. Immersing aggregates in concentrated polyvinylpyridine solutions induces the aggregate fragmentation and ultimately leads to complete dispersion of the latex particles [138,139].

5.3.1. Fragmentation of aggregates formed under conditions of di¡usionlimited aggregation The latex concentration was 10 g/l (2 1013 particles/l) and the aggregating medium was an aqueous solution of 1 mol/l NaCl at pH 3.0 and 201C, thus leading to fast aggregation of the suspended latex particles. After a given aggregation period, a sample of the suspension was transferred to an aqueous polyvinylpyridine solution at the same pH 3.0 and the system was homogenized by gently rotating the tank to establish conditions for perikinetic fragmentation. The fragmentation did not start immediately after immersion in the polymer solution since no difference in the aggregate mass frequencies could be determined in the aggregating and fragmenting media immediately before and after transfer. Polymer solutions at different concentrations were employed as fragmenting media and the polymer concentration was deﬁned in terms of C/d, C ¼ 3.06 g/l being the critical bulk concentration of non-overlapping coils, and d the degree of dilution of that medium. The variables are the average mass of the aggregates before fragmentation S(0) and the concentration C/d of the polyelectrolyte solution. Figure 13 shows typical results of aggregate fragmentation experiments, which express the variation with time of the weight S(t) and number N(t) of the average masses for immersion of aggregates in polyvinylpyridine solutions of concentration C/5, C/10 or C/30. The time lag before fragmentation is close to 2000 min whereas the variations of S(t) and N(t) as a function of t display different shapes and slopes. Two regimes of fragmentation were determined to occur, the ﬁrst being relatively low and the second being fast. The parameters of the fragmentation are given in Table 2 for the various experiments that were carried out with changing the parameter S(0) or the dilution d of the polymer solution. The rate of fragmentation of large aggregates (equation (12)) is derived from results shown in Fig. 14 that summarizes all the situations existing in the various

Cohesion of Fractal Agglomerates

773

N(t)

S(t)

100

10

1 100

1000

10000

100000

t (min)

Fig. 13. Polymer-induced fragmentation of aggregates generated in the 1 mol/l NaCl solution that favours the DLA of the latex particles. Representation of S(t) (open symbols) and N(t) (black symbols) as a function of the period of fragmentation in the polyvinylpyridine solution of concentration C*/5 (B, E); C*/10 (J, K) or C*/30 (&, ’).

Table 2. Parameters of the polymer-induced fragmentation of aggregates generated by diffusion-limited aggregation in 1 mol/l NaCl aqueous suspension

Dilution (d) 5 10 30 30 50

S(0)

l (equation (16))

l (equation (17))

50 40 100 30 55

First regime of break-up 0.56 9.2 0.56 5 0.56 1.9 0.56 5.3 0.56 1.76

l (equation (16))

l (equation (17))

Second regime of break-up 0.60 1.8 0.60 1.4 0.65 0.9 0.65 1.4 0.65 3

media at successive periods of fragmentation: l is equal to 0.56 and 0.6270.03 in the ﬁrst and second regimes of fragmentation, respectively. In addition, we note that in dilutions corresponding to d ¼ 5 to 30, N(t)ES(t) in the ﬁrst kinetic regime: the fragmentation strictly inverses the variations of the average masses that were determined for the DLA for which N(t) increased like S(t) too. Conversely, in the second regime, the decrease of S(t) and N(t) values indicates that the fragmentation merely resembles aggregate splitting with iﬃj for immersions in dilutions of d ¼ 10 or 30 since S(t) decreases faster than N(t) [108].

774

E. Pefferkorn

- ln(n2x c(n,t))

10

1 0

1

10

100

n/S(t)

Fig. 14. Polymer-induced fragmentation of aggregates generated in the 1 mol/l NaCl solution that favours the DLA of the latex particles. Representation of –ln(n2 c(n,t)) as a function of the reduced variable n/S(t) for experiments carried out in the polyvinylpyridine solution of concentration C*/5 (B); C*/10 (J) or C*/30 (&).

5.3.2. Fragmentation of aggregates formed under conditions of reactionlimited aggregation The latex concentration was 10 g/l (2 1013 particles/l) and the aggregating medium was 0.15 mol/l NaCl at pH 3.0 and 201C, thus leading to the RLA of the colloids. Fragmentation was induced by the transfer of small volumes of the suspension in the aqueous polyvinylpyridine solution at pH 3.0. Figure 15 shows the variation with time of S(t) and N(t) for systems fragmenting in polymer solutions of concentration C*/d with d ¼ 5, 10 or 30 and the characteristics of the variation of S(t) and N(t) are given in Table 3. The rate of variation of S(t) and N(t) and initial time lags seem to depend on the mass S(0) as shown in Table 3. A single value of exponent l was derived from results shown in Fig. 16 and the rate of aggregate fragmentation a(n) was determined to vary according to equation (12) with l ¼ 0.50. Immersion of aggregates in the polymer solutions did not induce fast phenomena since in all situations the average mass characteristics of the aggregates before and just after immersion are unchanged. After a certain delay, the mass distribution function c(n, t) vs. n becomes shifted towards smaller values of n. However, the mass distribution adopts a single shape expressing the self similarity of the mass distribution when the reduced variables c(n, t) S2(t) and n/S(t) are employed as expressed by equation (15). In addition fragmentation develops on the basis of a memory effect certainly related to ageing of inter-particle links since S(t) decreases like N2(t). In the aggregating medium of 0.15 mol/l NaCl, the average mass S(t) was determined to

Cohesion of Fractal Agglomerates

775

S(t)

100

N(t)

10

1 1000

10000 t (min)

100000

Fig. 15. Polymer-induced fragmentation of aggregates generated in the 0.15 mol/l NaCl solution that favours the RLA of the latex particles. Representation of S(t) (open symbols) and N(t) (black symbols) as a function of the period of fragmentation in the polyvinylpyridine solution of concentration C*/5 (B, E); C*/10 (J, K) or C*/30 (&, ’).

Table 3. Parameters of the polymer-induced fragmentation of aggregates generated by reaction-limited aggregation in 0.15 mol/l NaCl aqueous suspension. Single regime of fragmentation

Dilution (d)

Time lag (min)

S(0)

l (equation (16))

l (equation (17))

5 10 30 50

15,000 9000 5000 4500

20 18 40 45

0.50 0.50 0.50 0.50

1.38 1.32 1.07 1.09

increase according to N2(t) and the RLA model that favours the sticking of largest clusters. This means that the fragmentation of aggregates generated under conditions of the RLA process splits the aggregates Ay in two fragments Ai and Bj of relatively similar masses according to (9) [109]. For aggregates generated under perikinetic conditions, all possible conﬁgurations of the aggregates may be established as a result of the multiple binary collisions between aggregates of smaller mass. The experimental results relative to the fragmentation of aggregates indicate that aggregates generated by the RLA process fragment at a lower rate than those generated by the RLA process according to [140]. We believe that the small difference in the kinetic exponents l

776

E. Pefferkorn

-ln(n2x c(n,t))

10

1 0

1

10

100

n/S(t)

Fig. 16. Polymer-induced fragmentation of aggregates generated in the 0.15 mol/l NaCl solution that favours the RLA of the latex particles. Representation of –ln(n2 c(n, t)) as a function of the reduced variable n/S(t) for experiments carried out in the polyvinylpyridine solution of concentration C*/5 (B); C*/30 (&) or C*/50 (n).

of the ﬁrstly developed aggregate fragmentation regime (0.56 in the DLA process (see Table 2) and 0.50 in the reaction-limited process (see Table 3)) may be attributed to the small difference existing in the corresponding mean number of connections ov4 (2.05 and 2.20) determined by numerical simulation according to equation (28). ov4 does not depend on the agglomerate mass. P vPðv; i; jÞ hvi ¼ Pv ð28Þ v Pðv; i; jÞ Thus, we may conclude that the number of connections apparently is the factor that limits the agglomerate strength according to aðnÞ ¼ nl / 1=ov4

ð29Þ

The validity of the comparison between model and experiment rests on the important experimental results that fragmentation of real aggregates provides fragments that earlier existed as aggregates in the aggregating system. Agglomerate break-up ﬁrst applied to recently established connections between two aggregates. Progressively, older connections become disrupted and pre-existing situations were recovered. This is schematized by the two following signiﬁcant situations for which the aggregate mass distribution of fragments was compared to that of the parent aggregates. The reduced mass distribution function (equation (15)) of aggregates in the aggregating medium is equal to that of fragments in

Cohesion of Fractal Agglomerates

777

the fragmenting medium [108,109]. Fragmentation of aggregates generated under conditions of the DLA process delivers two fragments of highly uneven mass that really existed in the system earlier. Similarly, fragmentation of aggregates generated under conditions of the RLA process delivers two fragments of even masses that really existed in the system earlier too. It is thus obvious that due to this memory effect, aggregated systems of well-deﬁned characteristics experienced the fragmentation tests. Additionally, the rate of fragmentation is determined from the mass frequency of aggregates of high mass and not derived from the variation with time of the average masses N(t) or S(t) that were shown to depend on certain parameters of the system. This approach that produces a single scaling law a(n)pnl with a single value of the exponent l for identical systems is based on the self-similarity of the mass distributions that is theoretically assumed and experimentally demonstrated. For aggregates (in experiments) and agglomerates (in simulations) of systems generated under diffusion-limited or RLA processes, the differences existing between the average numbers of connections (v) determined from the numerical simulations and the rates of fragmentation determined experimentally is relatively small and cannot by itself serve to establish the validity of the model. Nevertheless, this correlation may serve to set the empirical relationship (29) between l and ov4. It is noticeable that the similar analysis applies to the fragmentation of aggregates generated in the presence and in the absence of shear [138].

6. FORWARD LOOK The usual explanation for slow aggregation is that the change from inter-particle collision to particle sticking requires the crossing of an energy barrier that is generally of electrostatic origin [100]. Such chemically controlled or RLA processes have been studied analytically, experimentally and numerically. In provoked, controlled fragmentation experiments, the resulting aggregates were found to be isotropically stable. As expected, the immersion of such aggregates in a medium of very low ionic strength did not induce aggregate fragmentation, which could only be triggered by an associated increase in the pH of the immersion phase [108,109]. The fragmentation gave rise to fragments of welldeﬁned average masses and mass distribution, while self-similarity could be demonstrated for the mass distribution curves of the aggregates and of the resulting fragments. Variation of the characteristics of the suspending liquid phase in these experiments provided clear evidence of the role of isotropic electrostatic forces in the break-up mechanism and thus in the internal stability of aggregates. The formation of inter-particle links in such RLA processes usually succeeded after a great number of attempts with a statistical chance of success. However, it could also result from the simultaneous establishment of multiple contact points,

778

E. Pefferkorn

so that inter-particle sticking succeeded when a given number of contacts could be established during the collision time. This last mechanism required the aggregates to be physically anisotropic and ageing of inter-particle links was in fact found to exert a major role in the structural characteristics of the resulting fragment suspension [109,141]. Despite the fact that aggregation (and fragmentation) of colloidal particles has been the subject of extended and various experimental, numerical and theoretical investigations, the aggregation controlled by the internal cohesion of the agglomerate potentially resulting from the sticking of two colliding aggregates has not been studied. This special aspect and the effect on the kinetics of aggregation and on the cluster mass frequency function were taken into account employing numerical investigation [126]. In our opinion, looking at the control of the internal cohesion on the basis of equation (23) of newly forming clusters may address various experimental situations, and especially the following two cases discussed below. The ﬁrst case corresponds to the aggregation of particles in the presence of hydrodynamic forces, which is very important from a practical standpoint. As pointed out by Mason [142] and Gregory [143], two particles approaching each other in a shear ﬁeld form a doublet, which can be either transient or permanent. In the former case, the doublet rotates through a certain angle before ﬂuid forces separate the particles. This image may be adapted to two colliding clusters. One may imagine that the break-up issue is relatively reduced when the encounter between the two clusters immediately induces a great number of inter-cluster connections. Conversely, when the two clusters are only connected by a single bond, the sticking certainly will be transient. Regarding the given conformation of the stuck clusters, the knowledge of the number of inter-cluster connections is thus of major interest for the determination of the evolution of the sticking towards rupture or permanent consolidation. The second case is relative to systems of marginal instability such as hairy or hydrated colloids and colloidal materials for which inter-colloid attraction forces are greatly reduced due to the small value of their Hamaker constant, such as lowdensity biological materials. One may imagine that permanent aggregation requires the formation of multiple inter-particle connections between the aggregates. Obviously, this supplementary constraint may complete the already existing conditions imposed to interpret the slow aggregation affecting colloids suspended in aqueous media of small ionic strength [100]. Similar conditions were expressed in the algorithm of the RLA process as expressed by equation (2).

6.1. Aggregate cohesion and aggregate growth Hydrated colloids were found to sustain very slow aggregation processes even in the presence of high electrolyte concentrations where the electrostatic

Cohesion of Fractal Agglomerates

779

100

N(t)

S(t)

1000

10

1 1

10

100 t (min)

1000

10000

Fig. 17. Colloidal aggregation. Representation as a function of time of the weight S(t) (m) and number N(t) (D) average masses of the aggregates obtained in 1 mol/l NaCl suspension at pH 3.0 (log–log scale). The initial particle concentration is 15 g/l, equivalent to 2.58 1013 particles/l.

interactions are effectively non-existent or of very short range [111]. The shell of these particles is highly water swollen and contains charged and neutral hydrophilic moieties with interconnecting water molecules [144]. Such systems conform to the classical theories of DLVO only if one replaces the value of the Hamaker constant generally accepted for polystyrene latex particles by a smaller value in the expression for the attractive forces between two approaching colloidal particles. For such systems, it may be expected that the sticking of particles and colloids is dependent of the structure stability being provided by the number of inter-aggregate connections. Figure 17 represents the variation as a function of time of the average masses N(t) and S(t) of aggregates of hydrated latex present in the aggregating medium of 1 mol/l NaCl aqueous suspension at pH 3.0. Three growth regimes are apparent. Aggregation is relatively slow over the ﬁrst 20 min; from 20 to 400 min, S(t) and N(t) increase with t according to equation (17). After 400, S(t) always increases strongly whereas the rate of increase becomes limited for N(t). Figure 18 representing the aggregate mass frequency according to equation (15) clearly demonstrates the existence of at least three different aggregate populations within the aggregating suspension. The mass distribution of population 1 (the smallest aggregates) is continuously decreasing: dispersed particles and small aggregates are present in the highest concentration and the aggregation process thus characterizes the reaction-limited process whereas the processes leading to aggregates of populations 2 and 3 are diffusion-limited.

780

E. Pefferkorn 10000

c(n,t) S2(t)/N1

1000

100

10

1

1.7

0.1 0.0001

0.001

0.01

0.1

1

10

n/S(t)

Fig. 18. Colloidal aggregation. Representation of the aggregate mass distribution (reduced concentration vs. reduced mass) at aggregation times t (min) of 3 (K), 50 (J), 125 (’), 595 (~) and 4410 (B). The curve refers to the full distribution of three populations and the line indicates the common initial slope at large aggregation times. N1 represents the latex particle number at time zero. The experiment was carried out in a mixture of D2O and H2O of density strictly equal to the density of the latex particles so that no aggregate settling occurred.

Fig. 19. Schematic representation of the aggregates belonging to population 1, 2 or 3.

Figure 19 proposes a schematic representation of the three aggregate populations as reported in [110]. The particles of the aggregates of population 1 are assumed to be linked only by single bonds and to be composed of linear particle segments oriented in all directions. Clearly, growth of this type of aggregate requires only the energy of doublet formation. The aggregates of population 2 are characterized by multiple inter-particle bonds and are composed of elements containing three or more closely packed particles. In this case, the probability of

Cohesion of Fractal Agglomerates

781

growth is greater as the agglomerate cohesion is proportional to the number of bonds formed simultaneously during approach of the aggregates and particles. Population 3 originates from the sticking of elements of types 1 and 2 and multiple intra-aggregate bonds therefore stabilizing these aggregates, possibly with readjusting rotations [145]. In contrast to aggregates of types 1 and 2, which contain only linear particle segments, the structure of type 3 aggregates contains rings of assembled particles. The stability against fragmentation may be expected to increase stepwise from population 1 to 3, since the fragmentation process requires the concomitant rupture of all inter-particle links. The formation of multiple populations during aggregation cannot result from the two usual modes of aggregation of lyophobic colloids. In the absence of energy barriers, since the rate of disappearance of colloidal

particles decreases with increasing size, at any point in time dispersed particles and aggregates of very small mass (size) disappear to the beneﬁt of the formation of aggregates of a given mean mass (size). This is in accordance with the single-bell-shaped mass distribution curves, which can be ﬁtted at any time point using one set of parameters [146]. In the presence of energy barriers, since the rate of disappearance of colloidal particles increases with increasing size, at any point in time dispersed particles and aggregates of very small size remain in the suspension while the concentration of aggregates of larger size slowly increases. The size distribution of the aggregates can be similarly described at any time point using a unique set of parameters [147]. This is no longer valid in the present case of hydrated colloidal particles immersed in a solution of high electrolyte concentration, where the attractive energy of van der Waals forces is strongly reduced and consequently relatively low forces drive the acceleration of the rate of approach of colliding particles or aggregates. In this situation, DLA may be expected to be highly reversible from kinetic and energetic points of view. As the formation of doublets requires interparticle sticking during the time of collision between two elementary particles, which is relatively short due to the high mobility of dispersed particles, doublet formation should be strongly reversible. However, the situation is different when we take into account the roles of (i) the simultaneous establishment of multiple inter-particle contacts and (ii) the simultaneous formation of multiple inter-aggregate bonds. (i) Aggregate formation resulting from the collision of dispersed particles (or very small aggregates) with large aggregates may be facilitated when a single particle is able to simultaneously establish two or three contact points with neighbouring particles belonging to an approaching aggregate (Fig. 19 (2)).

782

E. Pefferkorn

(ii) Aggregate formation resulting from the collision of large aggregates may be signiﬁcantly increased by the simultaneous establishment of multiple bonds between large numbers of particles belonging to the colliding aggregates (Fig. 19 (3)). The present aggregation model leading to three apparent populations was considered as a working hypothesis to be modiﬁed in the light of new numerical studies as shown in Section 6.2.

6.2. Aggregate growth in the presence of connection constraints In this numerical simulation of the RLA process with connection constraints expressed by equation (23), the part of the usual RLA process is determined by applying equation (2) with P11 ¼ 0.01 and s ¼ 0.4. This allows one to determine to what extent the connection constraint modiﬁes the kinetics of the aggregate growth and the aggregate mass frequency resulting from the application of the standard algorithm of the RLA process [126]. In the usual RLA process, the permanent sticking of the cluster is realized if the sticking probability Pij (equation (2)) is greater than the randomly selected number. Alternatively, the sticking fails. The exponent s and d of the scaling law of the total number of aggregate conﬁgurations (equation (25)) are connected since d ¼ 2s [148,149]. To take into account hydrodynamic effects in the RLA process, the following numerical procedure was implemented. The spatial conﬁguration of each agglomerate generated by sticking two randomly selected clusters of masses i and j was analysed, and the number v of inter-cluster connections was determined on the basis of the spatial coordinates of the particles. Then, the value of P(v, i, j) was determined and the permanent sticking was only allowed for conﬁgurations characterized by values of P(v, i, j) smaller than values of 2, 5, 20 or 50, according to the relationship (23). Figure 20 shows schematic 2d-conﬁgurations for agglomeration of two aggregates of masses 34 and 41. Conformation (a) is secured by seven inter-aggregate connections and thus resists fragmentation when the threshold value of P(v, i, j) was set to 10 in the simulation run. Conversely, assemblage generating conﬁgurations (b) is transient since the corresponding P(v, i, j) value derived from equation (23) is 15.5. All simulation runs with a given constraint P(v, i, j) were repeated ten times to reduce the statistical ﬂuctuations due to the relatively small number N1 ¼ 10,000 of particles at time zero. The kinetics of aggregation are derived from the stepwise growth of the aggregate average mass S(t) and N(t) (equation (8)) as a function of time. The time is incremented by N(t)/N1 when two clusters are selected, independently of the success or failure of the sticking procedure, N1 being the number of particles at

Cohesion of Fractal Agglomerates

783

Fig. 20. Representation of the assemblage of two 2d-aggregates of masses 34 and 41 developing non-overlapping conﬁgurations characterized by 7 (a) and 4 (b) connections. Each conﬁguration is deﬁned by its P(v, i, j) data.

100

C

N(t)

B

10

A

1 100

101

102 103 104 t (computer)

105

106

Fig. 21. Representation of the average mass N(t) of the computer generated clusters as a function of computer time for s ¼ 0.40 and A(v, i, j) o2, (&) and o5, (J).

t ¼ 0, before aggregation starts and N1/N(t) representing the number of particles and aggregates in the system at each time t. Figure 21 shows the variation of the number average mass N(t) as function of time. The two apparent domains of variation of N(t) successively result from the effect of the sticking probability P(i, j) with s ¼ 0.40 (A), corresponding to the

784

E. Pefferkorn 1000

-1.5

c(n,t) S2(t)/N1

100 10 1 0.1 0.01 0.001 0.01

0.1

1

10

n/S(t)

Fig. 22. Representation of the mass frequency – reduced concentration vs. reduced mass – of cluster generated with the constraints s ¼ 0.40 and P(v, i, j)oN.

usual RLA process that later becomes inﬂuenced by the effect of the connection constraint P(v, i, j). This latter effect increases markedly with time from (B) to (C). The supplementary severe constraint decreases the rate of cluster growth and ﬁnally limits the values of the average mass N(t) to 90 or 30 for A(v, i, j)o5 or 2, respectively. This variation of N(t) parallels the variation of N(t) shown in Fig. 17. The aggregate mass frequencies are shown in Fig. 22 (reduced coordinates, equation (15)) for the standard RLA process with Pij ¼ 0.01 and s ¼ 0.40, and in Figs. 23 and 24 (simple coordinates c(n, t) vs. n) for the standard process with the connection constraints P(v, i, j)r2 and 5, respectively. In Fig. 22, the curves display the usual characteristics: the mass frequency functions are self-similar and continuously decreasing with the initial slope equal to 1.5. Figures 23 and 24 represent typical mass frequency functions corresponding to the successive situations of domains of apparent RLA (A) and of prevailing connection constraints (B) and (C) with the development and the ﬁnal establishment of multiple populations of clusters when the connection constraint prevails. Its effect is marginal for P(v, i, j)o5 but well marked when o2. The effect of the connection constraints is clearly visible on 2d-aggregates generated by numerical simulation for the different values of the constraint P(v, i, j) equal to 2 or 5. Figure 25(a) and (b) portray two aggregates of 240 and 600 particles for P(v, i, j) equal to 2 and 5, respectively. To illustrate the heterogeneous structure of such aggregates, the simulation algorithm was adapted to generate bi-dimensional clusters. Figure 25(a) and (b) portrays the biggest clusters generated using values of A(v, i, j)o2 or o5, that were extracted when the average masses are stabilized or close to maximal values (domain C) in Fig. 21.

Cohesion of Fractal Agglomerates

785

10000

c(n, t)

1000

100

10

1 0

10

100

1000

n

Fig. 23. Representation of the mass frequency curves of clusters generated with the constraints deﬁned by s ¼ 0.40 and P(v, i, j) o2 in the growth domains of A, (J), B, (&) and C, (B). 10000

c(n,t)

1000

100

10

1 1

10

100

1000

n

Fig. 24. Representation of the mass frequency curves of clusters generated with the constraints deﬁned by s ¼ 0.40 and P(v, i, j) o5 in the growth domains of A, (J), B, (&) and C, (B).

For the threshold value 2, the cluster is compact nearly without voids and its fractal dimension is close to 2. When the value progressively increases, the cluster compactness decreases and ﬁnally the fractal dimension is close to 1.5, which is the value determined for planar clusters generated by the usual RLA

786

E. Pefferkorn

Fig. 25. Typical 2d-clusters generated by numerical simulation for the different values of A(v, i, j)o2 (a) (240 particles) or o5, (b) (600 particles).

process. Finally, aggregates generated with the connection constraint o5 display local anisotropy: compact islets are connected by less dense moieties. This nonhomogeneity allows the progress of cluster growth to be determined: the less dense moieties were ﬁrst generated while the compact islets appeared when the connection constraint is prevailing. Therefore, clusters of prevailing connection constraints are not real fractals and the relative portion of compact and diffuse zones may change the global fractal dimension when the usual analysis is applied to the cluster coordinates.

7. CONCLUSION In the light of the present model of agglomeration in the presence of connection constraints resulting from hydrodynamic stresses exerted on powders and grains while sticking, different agglomerate structures and morphologies may be built from the numerous manufacturing processes or syntheses that can be developed. One of the more obvious concerns the presence of strong anisotropies of the particle distribution and density within the agglomerate as shown in Fig. 25 (b). The cohesion model also shows the relative amount of agglomerates resisting a given stress to be independent of the agglomerate mass when the initial aggregates resulted from aggregation in the absence of shear. For the other systems generated in the presence of shear, the similar investigation remains to be done taking in mind that the main difference is relative to the fact that such systems may not be fractals as it has be determined.

Cohesion of Fractal Agglomerates

787

The numerical cohesion model based on the number of inter-aggregate connections being established on the sticking of two aggregates may be shown as being a very elementary model that simply illustrates some characteristics of real agglomerates such as the dependence of the porosity or the independence of the break-up characteristics on the agglomerate mass.

ACKNOWLEDGEMENTS The author warmly acknowledged S. Stoll and Y. Tatek for their great implication in the investigations relative to the aggregate fragmentation processes and in the numerical model of the aggregate cohesion.

Nomenclature

l n F(m, i, j) P P

v Pðv; i; jÞ v vPðv; i; jÞ

ov4 a(n) c(n,t) C Df Di fon9y4 i, j n (i+j) N1

exponent of the scaling law of fragmentation a(n). l also determines the variation of S(t) and N(t) with fragmentation time exponent of the rate of formation of fragments of small masses n from the fragmentation of aggregates of larger masses fraction of agglomerates resisting fragmentation when applied forces do not succeed to break-up inter-agglomerate links of vZm connections total number of agglomerate conﬁgurations for assemblage of two clusters of mass i and j taking into account all v values total number of connections corresponding to the total number of agglomerate conﬁgurations resulting from sticking of cluster of masses i and j mean value of the number of connections for an ensemble of agglomerate conﬁgurations of given characteristics rate of fragmentation of aggregates of mass n concentration of aggregates of mass n at time t of aggregation or fragmentation critical bulk concentration of non-overlapping coils of solute macromolecules (g/l) fractal dimension of aggregates mobility of the aggregate of mass i rate at which aggregates of mass n are produced from the breakup of cluster of mass y mass of the aggregate i or j n is used to deﬁne the mass of aggregates when the particular values of i and j are unknown (experimental situation) number of single particles in the system before initial aggregation or after complete fragmentation

788

N(p) N(t) Pij P(v, i, j)

S(t) v VP, VPP

E. Pefferkorn

number concentration of pores of size p within the agglomerate number averaged mass of aggregates or fragments at time t sticking probability of colliding aggregates of masses i and j total number of agglomerate conﬁgurations characterized by v connections for the assemblage of two aggregates of masses i and j weight averaged mass of aggregates or fragments at time t number of connections established when two stuck aggregates form an agglomerate of given conformation average values of the pore sizes within agglomerates

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CHAPTER 18

The Linear Breakage Equation: From Fundamental Issues to Numerical Solution Techniques Margaritis Kostoglou Division of Chemical Technology, Department of Chemistry, Aristotle University of Thessaloniki, Univ. Box 116, 54124 Thessaloniki,Greece Contents 1. Introduction 2. Problem formulation 2.1. The breakage equation 2.2. The breakage functions 2.2.1. Overview 2.2.2. Product and sum-type kernels 2.2.3. Erosion kernels 2.2.4. Sum of powers kernel 2.2.5. Discrete homogeneous kernels 3. Iterative and analytical solutions of the breakage equation 3.1. Iterative techniques 3.2. Analytical solutions 4. Self-similar solutions of the breakage equation 4.1. Formulation of the self-similarity problem 4.2. Closed form solutions of the self-similarity equation 4.2.1. Continuous kernels 4.2.2. Discrete kernels 4.2.3. Some features of the self-similar PSD 4.2.4. Is the self-similar PSD realizable? 5. Special cases of breakage 5.1. Limited breakage 5.1.1. Formulation of the steady-state problem 5.1.2. General behaviour – Asymptotic results 5.1.3. Analysis for the sum of powers kernel 5.1.4. Approach to the limiting steady state 5.2. Erosion 5.2.1. The erosion equation 5.2.2. Decomposition to generations – Analytical solution 5.2.3. Moments of the generations 5.2.4. Case study

794 796 796 797 797 799 800 801 801 802 802 804 806 806 808 808 809 811 812 813 813 813 815 818 819 820 820 822 824 825

Corresponding author. Tel.:+30-2310997767; fax:+30-2310997759;

E-mail addresses: [email protected], [email protected] (M. Kostoglou).

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12021-2

r 2007 Elsevier B.V. All rights reserved.

794 6. Methods of moments 7. Sectional methods 8. Current and future research topics on breakage equation References

M. Kostoglou 827 829 831 833

1. INTRODUCTION The linear breakage equation is a partial integrodifferential equation belonging in the more general class of population balances and describes the evolution of a particle size distribution (PSD) of a particle population undergoing breakage. The phenomenon of breakage is very important in basic science and engineering. It is encountered in literature under several names depending on the particular physical situation, e.g. crushing, milling, grinding, fracture, partition, disintegration, shattering, scission, fragmentation. The list of application areas is enormous. Apart from the area of size reduction of solids by mechanical means, which is described in detail in the present book, there are several other areas for which the breakage process is an essential aspect. Some of these areas, by no means an exhaustive list, are discussed below. During the combustion of coal the disappearance of solid bridges between elements of coal or soot particles leads to the so-called percolative fragmentation [1,2]. The polymer degradation using thermal, mechanical or radiation-induced means is used in polymer recycling and for structural characterization of polymers [3,4]. Attrition of the catalyst particles in ﬂuidized bed reactors is a major problem of chemical industry [5]. Particle breakage is also important for the transfer of coal particles in the form of a slurry (which determines the ﬂow properties) [6]. In biotechnology, the cell division process can be described as a spontaneous breakage process [7]. In industrial crystallization, crystal fragmentation is one of the phenomena determining the resulting crystal size distribution [8]. In fact, the so-called secondary nucleation is just a consequence of crystal fragmentation. The droplet sizes in a ﬂowing or stirred dispersion [9] or emulsion [10] are the result of a breakage cascade. The inﬂuence of bubble breakage in two-phase gas–liquid ﬂow (through its effect on the bubblesize distribution) is an issue that has emerged recently in the literature [11]. Applications in other scientiﬁc ﬁelds are also numerous ranging from meteorology (rain formation) [12] and astronomy (size distribution of asteroids) [13] to the fragmentation of volcanic ash [14]. Recently, it has been argued that the size distribution of the ice crystals in the Greenland ice sheet is the result of a breakage process [15,16]. Even everyday life problems such as car parking can be described in terms of the breakage equation [17]. Regarding more fundamental areas of physics, aspects of breakage are recognized in the atomic collision cascades, in energy cascades of turbulence and in the multivalley structure of the phase space of disordered systems [18].

The Linear Breakage Equation

795

Of course, the single breakage equation is not used isolated to describe quantitatively all the above phenomena. In some cases, other terms are added to the breakage term in the population balance equation. These terms, depending on the particular application, can include coagulation, continuous particle growth or dissolution and diffusion in the particle size domain. But even for these more complex cases there is a region of parametric space where the breakage equation alone is capable of describing the phenomenon. From the above, it is obvious why scientists from several disciplines have studied extensively the fundamental properties of the breakage equation. The linear breakage equation, as with the majority of linear problems in science, contains a very rich structure which is accessible with analytical tools. But although the study of the breakage equation began in the 1950s, there are still today many aspects of its structure that remain unrevealed. A previous detailed account of the subject was published in 1990 [19]. Since then, many new features of the breakage equation have been discovered. The scope and the ambition of the present work is to describe the knowledge on fundamentals of the linear breakage equation up to 2005. The structure of the present work is as follows: Firstly, the breakage equation is presented, then the properties having a breakage kernel are described and several breakage kernels are presented. It must be noted that this is not a review of breakage kernels; only kernels which will be used in what follows are presented. A general account of the iterative methods for the solution of the breakage equation follows. Despite the theoretical development of the subject, their main use has been the derivation of analytical solutions. Their application as an alternative to numerical methods is very limited. In the next section, all the known analytical solutions of the breakage equation are presented. These solutions in most cases refer to very simple breakage kernels whose main use has been the benchmarking of numerical codes. The fact that the breakage equation can admit constant shape (self-similar) large time asymptotic solutions for some categories of breakage functions was discovered a long time ago. Here, the large time behaviour of the breakage equation is examined in detail and all of the closed form self-similar solutions are presented. Another interesting mode of the breakage problem is that of limited breakage inspired from some practical applications where a critical particle size exists; a particle smaller than this does not break at all [20]. The limited breakage excludes the possibility of self-similar solution and leads to a static steady-state PSD (compare to dynamic steady-state PSD of selfsimilarity). The reason of the considerable development of the limited breakage theory is that the corresponding inverse problem (estimation of breakage kernel from PSD shape) is much simpler than that of self-similarity theory. Unfortunately, it is still not absolutely clear if limited breakage really does happen in nature or if it is a misinterpretation of incomplete experimental observations. However, the corresponding theoretical development is presented here. When the breakage event produces a ‘‘coarse’’ fragment of size close to the parent particle and a

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number of much smaller ‘‘ﬁne’’ particles, the process is called particle erosion [21], attrition [22], abrasion [23] or chain-end scission [4] depending on the physical situation. In that case, the solution techniques used for the conventional breakage problem are not appropriate. This special mode of breakage is analyzed in detail here and several analytical results are presented. To proceed from the fundamental issues to the requirement for numerical solution of the breakage equation for arbitrary breakage functions, two categories of solution methods are examined. The method of moments can be used for fast approximate solutions whereas the sectional method can result to very accurate solutions but it requires signiﬁcant computational effort. The discussion on numerical methods given here is neither an exhaustive review nor an application guide but contains only the basic information for the reader to get into the subject along with a suggestion of which method to use.

2. PROBLEM FORMULATION 2.1. The breakage equation The linear breakage process can be described in general by the following linear partial integrodifferential equation: Z 1 @f 0 ðx 0 ; tÞ ¼ p0 ðx 0 ; y 0 Þb0 ðy 0 Þf 0 ðy 0 ; tÞdy 0 b0 ðx 0 Þf 0 ðx 0 ; tÞ ð1Þ @t 0 x0 where t0 is the time, x0 the particle volume, f 0 (x0 , t) particle number density distribution, b0 (x0 ) breakage rate (literally speaking it is the breakage frequency but the term rate is used extensively in the literature) and p0 (x0 , y0 ) the probability distribution of particles of volume x0 resulting from the break-up of a particle of volume y0 . Let f 00 ðx 0 Þ ¼ f 0 ðx 0 ; 0Þ be the initial PSD. The total volume concentration, the total number concentration and the mean particle volume of the initial distribution are, respectively: Z 1 M¼ xf 00 ðxÞ dx ð2aÞ 0

Z

1

N0 ¼ 0

x0 ¼

xf 00 ðxÞ dx

ð2bÞ

M N0

ð2cÞ

The functions and variables already introduced can be expressed in dimensionless form as follows: x0 y0 x 0 f 0 ðx 0 ; tÞ x ¼ ; y ¼ ; t ¼ b0 ðx 0 Þt 0 ; f ðx; tÞ ¼ ð3Þ N0 x0 x0

The Linear Breakage Equation

797

bðxÞ ¼ b0 ðx 0 Þ=b0 ðx 0 Þ;

pðx; yÞ ¼ p0 ðx 0 ; y 0 Þ=x 0

and equation (1) can be written as Z 1 @f ðx; tÞ ¼ pðx; yÞbðyÞf ðy; tÞdy bðxÞf ðx; tÞ @t x The dimensionless moments of the PSD are deﬁned as Z 1 Mi ¼ x i f ðx; tÞ dx

ð4Þ

ð5Þ

0

The dispersivity s of the PSD is a measure of its extent and deﬁned as s ¼ lnðM 0 M 2 =M 21 Þ. An alternative way to describe the fragmentation process is through the use of the normalized cumulative volume PSD function FðxÞ ¼ Rx 0 yf ðyÞ dy (by deﬁnition F(N) ¼ 1). R x The cumulative volume fraction breakage kernel is deﬁned as Gðx; yÞ ¼ 1=y 0 zpðz; yÞ dz. Substituting the above in the breakage equation (4) one obtains the equivalent form: Z 1 @Fðx; tÞ ¼ bðyÞGðx; yÞdFðy; tÞ ð6Þ @t x In case of having a lower limit in the size of particles that can break (let us call this critical size Xm and its dimensionless counterpart xm ¼ Xm/x0), the dimensionless breakage equation takes the following form: Z 1 @f ðx; tÞ ¼ pðx; yÞbðyÞf ðy; tÞdy bðxÞf ðx; tÞ; x4x m ð7Þ @t x Z 1 @f ðx; tÞ ¼ pðx; yÞbðyÞf ðy; tÞdy; xox m ð8Þ @t xm

2.2. The breakage functions 2.2.1. Overview In general, the function p(x, y) should satisfy the following requirements: (i) Conservation of mass: Z

y

xpðx; yÞ dx ¼ y

ð9Þ

0

This equation presumes that the total volume of fragments resulting from the break-up of a particle of volume y, must be equal to y. Z k Z y y (ii) ð10Þ x pðx; yÞdx ðy xÞ pðx; yÞ dx where ko 2 0 yk

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This expression states the requirement that only breakage events take place with no rearrangement of mass allowed. It is an absolute condition based on the physical requirement that when breakage occurs such that a particle xZy/2 is formed, the volume contained within the smaller fragments (yx) must contribute to the total volume of the fragments smaller than (yx ). Further analysis concerning this condition can be found elsewhere [24]. For binary breakage (two fragments per parent particle), the above restriction is simpliﬁed being equivalent to a symmetry requirement in the sense p(x, y) ¼ p(yx, y). The above restriction on the form of breakage kernel is very important but it seems to have been overlooked in the literature on breakage; this may lead to physically unrealistic kernels being used to ﬁt experimental data. For example in [25] a kernel is employed (power-law kernel with a positive exponent) which violates condition (ii) and does not have any physical signiﬁcance, even though it is capable of ﬁtting the experimental data. In that case, a different breakage mechanism (and different breakage kernel) may be dominant. As another example, in [26] the asymptotic PSD is given for several kernels without physical meaning (i.e. power-law form with a positive exponent). (iii) The number of particles resulting from breakage of a single particle of volume y is given as Z y vðyÞ ¼ pðx; yÞ dx, ð11Þ 0

Evidently, the parameters of the breakage equation are the breakage kernel, p(x, y), and the breakage rate, b(x). In general, the form of these two functions depends on the physical process under consideration. Even for a particular physical process a variety of breakage functions has been proposed in the literature, sometimes quite different from each other [27]. The proposed functions belong to two major categories. The phenomenological breakage functions are based on physicochemical models of the breakage process and in general they do not include unknown (ﬁtting) parameters. Only the physicochemical parameters of the system under consideration are needed. On the other hand, the algorithmic breakage functions are just parameterized families of functions (usually one or two parameters), which are considered to include members close to the actual function. The unknown parameters must be determined by ﬁtting the parameterized functions to experimental PSDs. The ﬁtting procedure can be through the direct solution of the breakage equation (4) and further comparison of the theoretical PSDs to the experimental ones [28] or by the inverse problem methodology [29]. Another major classiﬁcation of breakage functions is made with respect to their homogeneity. The breakage frequency is homogeneous only if it has a power-law

The Linear Breakage Equation

799

form, i.e. b(x) ¼ xb. The breakage kernel is homogeneous if it has the form p(x, y) ¼ y(x/y)/y. This means that the normalized (with respect to the parent particle size) fragment size distribution does not depend on the parent particle size. The above properties (i), (ii) and (iii) of the breakage kernel may be transformed, respectively, to Z 1 zyðzÞdz ¼ 1 ð12Þ 0

Z

Z

k

1

zyðzÞdz 0

ð1 zÞyðzÞdz

for ko1=2

ð13Þ

1k

Z

1

yðzÞ dz

v¼

ð14Þ

0

The phenomenological breakage functions are usually very complicated [27,30,31] and it is not possible for them even to be approximated by simpler homogeneous functions. On the other hand, almost all the algorithmic breakage functions are homogeneous with the notable exception of the breakage rate used extensively in the solids comminution-grinding literature: bðxÞ ¼ ðkx b Þ=ð1 þ cx a Þ [32]. But even in this case a usual assumption is that the denominator is close to 1, justifying the use of a power-law rate [32]. The corresponding algorithmic breakage kernel used for the study of solids comminution-grinding is homogeneous with p(z) ¼ azc+bzd [33]. Algorithmic kernels have also been used extensively in the context of polymer degradation [3] as well as in the study of the general properties of the fragmentation equation in physics literature [19]. The ﬁrst generation of the algorithmic breakage kernels included a power-law kernel (permitting analytical manipulation of the breakage equation) and a normal distribution binary kernel (based on the central limit theorem) [34]. Hesketh et al. [28] developed the U-shaped binary kernel which is completely different from the previous kernels. Kostoglou et al. [35] generalized the normal and U-shape kernels by adding additional parameters.

2.2.2. Product and sum-type kernels Hill and Ng [36] developed two very general families of breakage kernels (i.e. product and sum kernels). Recently, Diemer and Olson [37] transformed the kernels of [36] to a simpler and more meaningful form and showed that almost every kernel can be written as a linear combination of the members of the above two families. The product-type kernel is yðzÞ ¼ v

zq1 ð1 zÞqðv1Þ1 Bðq; qðv 1ÞÞ

ð15Þ

800

M. Kostoglou

and the sum-type kernel is yðzÞ ¼

zq1 ð1 zÞv2 ð1 zÞqþv3 þ ðv 1Þ Bðq; v 1Þ Bð1; q þ v 2Þ

ð16Þ

where B is the beta function, v(y) ¼ v(Z2) is the number of fragments per breakage event and q40 is the parameter of the kernel. The product kernel shows a quite general behaviour ranging from fragmentation in equal pieces for q-N to an erosion-type behaviour for qo1. The term ‘‘erosion-type breakage’’ means that one of the fragments has a size close to that of the parent particle. The value q ¼ 1 corresponds to the random fragmentation in v fragments for both kernels. The shape of the above kernels for several combinations of parameters can be found in [37].

2.2.3. Erosion kernels The erosion-type kernel has the following general form [38]: yðzÞ ¼ P 1 ðzÞ for 0ozo1 ¼ 0 for 1 ozo2 ¼ P 2 ðzÞ for 1 2 ozo1

ð17Þ

with 2 1. Using the requirement in equation (13) results in the restriction e1re2 where the equality sign is for the binary breakage case. The parent particles retain their identity (and number) so Z 2 P 2 ð1 zÞdz ¼ 1 ð18aÞ 0

Z

1

P 1 ðzÞdz ¼ v 1

ð18bÞ

0

Substituting the kernel in the mass conservation equation (12), rearranging and using (18a) results in Z 1 Z 2 zP 1 ðzÞdz ¼ zP 2 ð1 zÞdz ð19Þ 0

0

A special category of erosion kernels is the binary erosion kernels such that e1 ¼ e2 ¼ e and P1(z) ¼ P2(1z); the simplest is the monodisperse erosion kernel P1(z) ¼ d(ze). The uniform erosion kernel is P1(z) ¼ 1/e. The power-law erosion kernel that assigns greater probability to smaller fragments has the form P1(z) ¼ (n+1)/e(1z/e)n. Obviously, the uniform kernel is a member of the family of the power kernels with exponent n ¼ 0. It must be noted that many kernels deﬁned in the entire interval [0, 1] can be cast in the form of the erosion kernel with a very good approximation. For example, the family of the sum-type kernels

The Linear Breakage Equation

801

(i.e. equation (16)) are practically zero everywhere except at the edges of the interval [0, 1], for high values of its exponent. This is also the case for the U-type kernel developed by Kostoglou et al. [35] for some values of its parameters.

2.2.4. Sum of powers kernel The sum of powers type kernel comprises a very important class of breakage kernels. The functional form of these kernels is yðzÞ ¼

n X

c i zk i

ð20Þ

i¼1

where kiA(2, N). Of course the coefﬁcients ci must be such as to conserve the P i total mass, that is ni¼1 k icþ2 ¼ 1. There are a variety of members of this family, from purely empirical kernels that closely ﬁt experimental data [33] to theoretical kernels that admit analytical solutions of the breakage equation (4) [39–41]. Also, the entire spectrum of product and sum type of kernels can be written in the above form (but for a large value of exponent q this is quite impractical since a large number of terms n are needed, so the above kernels are examined separately). In fact, (at least in principle) any kernel that is continuous in [0, 1] can be cast in this form provided that n is large enough.

2.2.5. Discrete homogeneous kernels The homogeneous breakage kernel is called discrete when the fragments are allowed to have only certain sizes [42]. Mathematically this means that the kernel can be written as a sum of Dirac functions, i.e. yðzÞ ¼

n X

ai dðz ci Þ

ð21Þ

i¼1

Pn The mass conservation requires that i ¼ 1 and the number of fragi¼1 ai cP ments resulting from a breakage event is v ¼ ni¼1 ai . The simplest discrete homogeneous kernels are the equal-size multiple breakage kernel: yðzÞ ¼ vdðz 1=vÞ

ð22Þ

and the general binary discrete kernel: yðzÞ ¼ dðz Þ þ dðz ð1 ÞÞ

ð23Þ

In the ﬁrst case, the parent particle is divided into v equal pieces and in the second case in two pieces of normalized size e and 1e, respectively (0oeo1/2). The second kernel in the case 1 belongs to the family of erosion kernels (binary erosion discrete kernel).

802

M. Kostoglou

3. ITERATIVE AND ANALYTICAL SOLUTIONS OF THE BREAKAGE EQUATION 3.1. Iterative techniques An important feature of equation (4) is that (due to its linearity) the solution for any initial PSD is obtained by appropriately superposing solutions for monodisperse initial distributions. The Green’s function with respect to particle size can be deﬁned as the solution of (4) for a monodisperse initial distribution with particles of size z and is denoted as fd(x, t; z). The solution for an arbitrary initial PSD is of the form Z 1 1 x ð24Þ f ðx; tÞ ¼ f 0 ðzÞ f d ; bðzÞt; z dz z z 0 By substituting fd(x, 0; z) ¼ d(xz) in the above equation, the following identity results as a conﬁrmation of its validity Z 1 f 0 ðxÞ ¼ f 0 ðzÞdðx zÞ dz ð25Þ 0

Equation (4) can be written in its Volterra integral form and solved using the method of successive substitutions, the convergence has been proven by Ramkrishna [43]. It can be shown that the above method leads to a solution of the form f ðx; tÞ ¼ f 0 ðxÞ þ

1 X

Ci ðxÞ

i¼0

ti i!

ð26Þ

At each iteration a new term is added to the above series. Another approach equivalent in nature, but more meaningful in physical content, has attracted the attention in recent years. The PSD is decomposed in generations as follows 1 X f ðx; tÞ ¼ f ðgÞ ðx; tÞ ð27Þ i¼0

where the generation g includes all the particles which have undergone g breakage events. This approach was introduced in [44] in the context of studying microbial and cell structures for which it arises naturally. Shortly thereafter, Lensu [45] generalized the method of generations as a mathematical tool for the study of fragmentation. He also stated that the generation index can be an additional internal variable and the breakage functions may be explicitly dependent on it in addition to the size dependence. From the mathematical point of view the method of generations is actually the method of successive substitutions but using an iteration scheme different than that leading to series (26). Now the key operator for the iterations is not the time derivative only but the term @f ðx; tÞ þ bðxÞf ðx; tÞ @t

The Linear Breakage Equation

803

Using techniques for the integration of a non-homogeneous linear differential equation, the method of successive substitutions takes the form f ð0Þ ðx; tÞ ¼ f 0 ðxÞebðxÞt f ðgþ1Þ ðx; tÞ ¼ ebðxÞt

Z x

1

Z

t

ebðxÞt pðx; yÞf ðgÞ ðy; tÞdydt

g ¼ 0; 1; 2

ð28Þ

0

The ﬁrst generation for the Green’s function fd(x, t; z) can be obtained explicitly to be bðzÞt f ð1Þ pðx; zÞ d ðx; t; zÞ ¼ e

1 eðbðxÞbðzÞÞt bðxÞ bðzÞ

ð29Þ

By expanding the exponential term in its Taylor series, the function for the ﬁrst P ti generation takes the form ebðzÞt 1 i¼1 Ai ðx; zÞ i!. Substituting it in the iterative scheme (28), it can be shown (by induction) that all the generations obey the above functional form. From equation (21) one can deduce that the Green’s function can be written as f d ðx; t; zÞ ¼ dðx zÞebðzÞt þ ebðzÞt

1 X i¼1

Ci ðx; zÞ

ti i!

ð30Þ

Substitution of the above series in equation (4), changing the summation indexes and equating terms of the same power of time leads to the following recursive formula for the functions Ci (i ¼ 1, 2, y, N) Z z Ciþ1 ðx; zÞ ¼ pðx; yÞCi ðy; zÞdy þ ½bðzÞ bðxÞCi ðx; zÞ ð31Þ x

C1 ¼ pðx; zÞ The above type of solution has been given for the case of binary breakage in [40] and generalized for general breakage in [38]. In both cases, the expansion of the form of equation (30) is introduced without explanation. It is important to notice that an expansion of the type of equation (30) is not possible for the case of an arbitrary (non-monodisperse) initial size distribution. This is due to the fact that an exponential in time term (i.e. as in equation (29)) cannot survive from the integration with respect to size in equation (22). This restriction does not exist for b(x) ¼ constant for which a solution of the form (30) can be found for arbitrary f0(x). So far, three series solutions of the equation (4) have been presented. The ﬁrst method is a simple power series in time. Its convergence properties are very poor. For a particular time, a large number of terms are needed to obtain a meaningful solution but as time tends to inﬁnity, the series tends to inﬁnity as well, for a ﬁnite number of terms. The second method is an expansion not with respect to time (as the ﬁrst one) but with respect to a new variable, the so-called generation. Each term of the expansion provides the exact size distribution of

804

M. Kostoglou

the corresponding generation for any value of time so the series tends to zero as time goes to inﬁnity for any number of terms. Of course as the breakage time increases, more terms are needed in equation (27). The major disadvantage of this method is of computational nature. The series of equation (27) is not separable with respect to the two variables x and t, as is equation (26). This means that a twodimensional recursive relation (28) is needed to determine additional terms in series (27). The third method combines the advantages of the other two. The solution is separable, with a very simple form of the time terms (product of powers with an exponential (30)). The recursive relation is just with respect to particle size (one-dimensional), equation (31). Furthermore, this method has the nice convergence properties of the second method. As time goes to inﬁnity, the series tends to zero for any number of terms due to the exponential. This method can be directly employed only for a monodisperse initial distribution but the superposition property enables its use for arbitrary f0(x). For the particular case of b(x) ¼ constant, the iterative scheme in equation (31) is simpliﬁed considerably. In this case the second and third iterative solution methods coincide and the term i of the series in equation (30) corresponds to the i-th generation of the particle population. For homogeneous breakage kernel and rate, the function 1z f d ðxz ; bðzÞt; zÞ is invariant with respect to the initial particle size z. This means that one can solve equation (4) for the particle size Green’s function with a particular value of z (usually the value z ¼ 1 is preferred) and then determine all the particle size Green’s functions for arbitrary value of z by using the above invariance (changing the scaling of the particle size, time and PSD). This is a very important property since equation (4) needs be solved for just one monodisperse initial distribution in order to ﬁnd the size distribution evolution for an arbitrary initial distribution, using the superposition principle.

3.2. Analytical solutions All the analytical solutions here will be given for the monodisperse initial distribution f0(x) ¼ d(x1) since the extension to the general initial distribution f0(x) through the superposition principle is straightforward. Some exact solutions for binary random kernel were known from the 1960s [46,47] and they were quoted and used extensively in the literature. For example, for b ¼ 1: f ðx; tÞ ¼ et dðx 1Þ þ ext ½2t þ t 2 ð1 xÞ

ð32aÞ

and for b ¼ 2: f ðx; tÞ ¼ et dðx 1Þ þ 2text

2

ð32bÞ

The Linear Breakage Equation

805

All these solutions are sub-cases of the very general analytical solution derived in [40] for the case of a power-law kernel: bþmþ2 ; 2; ð1 x b Þt f ðx; tÞ ¼ et dðx 1Þ þ ðm þ 2Þtx m M ð33Þ b where m ¼ (2v)/(v1) and M(a1, a2, a3) is the conﬂuent hypergeometric function of ﬁrst kind. The above general result can be simpliﬁed in terms of simpler functions (associated Laguerre polynomials, modiﬁed Bessel functions, elementary functions) for particular combinations of the parameters v and b. An account of these special cases can be found in [24]. Another kernel for which an analytical solution exists is a particular form of the two-term sum of powers þ2Þb kernel (i.e. pðzÞ ¼ c1 zk 1 þ c2 zk 2 ) with k1 ¼ b2, c1 ¼ c2 ¼ kðk22þ2b . The solution is: Z 1 b f ðx; tÞ ¼ et dðx 1Þ þ bðk 2 þ 2Þtx k 2 y bk 2 3 ey t dy ð34Þ x

Sub-cases of this particular kernel are the parabolic binary kernel (b ¼ 3, k2 ¼ 2) and the ternary random kernel (b ¼ 2, k2 ¼ 1) [40]. For the case of the equal size v-ary (v fragments) breakage kernel the PSD has the following discrete form f g ðx; tÞ ¼

1 X

ni dðx v i Þ

ð35aÞ

i¼0

The solution for ni’s can be written in the following form: ni ¼

i X

ci;j ev

jb

t

ð35bÞ

j¼0

where the coefﬁcients ci,j can be found from the recursive relation: 1þð1iÞb v v 1þb ci;j ¼ ¼ ci1;j ðj ¼ 0; 1; 2; . . . ; i; i ¼ 1; 2; 3; . . .Þ c i1;j v ib v jb 1 v ðijÞb ð35cÞ Application of the initial condition n0 ¼ 1, ni ¼ 0 (i ¼ 1, 2, y, N) leads to the relations c0;0 ¼ 1 ci;i ¼

i1 X j¼0

ci;j

ð35dÞ

ði40Þ

ð35eÞ

806

M. Kostoglou

From the above equations, the computation of all the coefﬁcients ci,j can be performed in the order shown in the following scheme: i¼0 1 2 3 4

j¼0 1 2 4 7 11

1

2

3

4

3 5 8 12

6 9 13

10 14

15

The above table refers to an i value up to 4 but its extension to larger values of i is trivial. Each cell of the table corresponds to a ci,j and the number in the cell is the order in which this particular ci,j will be computed. The off-diagonal elements are computed using equation (35c) whereas the diagonal elements using equation (35e). Actually, a set of coefﬁcients ci,j corresponds to a pair of values b, v. The particular case with b ¼ 1 was solved analytically in [48] by constructing an equation for the generating function and solving it employing the Laplace transform. Explicit results were shown only for an integral quantity corresponding to the viscosity of the dispersion.

4. SELF-SIMILAR SOLUTIONS OF THE BREAKAGE EQUATION 4.1. Formulation of the self-similarity problem The self-similar solutions of the breakage equation for the case of homogeneous breakage functions are presented here. An additional case for which selfbðxÞ similarity exists is for arbitrary b(x) and Gðx; yÞ ¼ g bðyÞ . This case admits different fragmentation patterns for particles of different size. Nevertheless, if one of the breakage functions is homogeneous the existence of self-similarity requires the other to be homogeneous too. The choice of the self-similarity variables for the breakage equation is not unique. The independent variable can be cast in self-similar form by scaling it with the inverse of volume speciﬁc area [49], mean particle diameter [50] or mean particle volume [51]. Other choices are z ¼ (1+t)xb and z ¼ txb (actually the two choices are equivalent since a large time asymptotic solution is sought). As regards the dependent variable, there are two distinct formulations of the selfsimilarity problem. The ﬁrst is the differential one based on the particle number density size distribution and is similar to that of [52] for coagulation. The second is the integral one based on the cumulative volume fraction. Only Peterson [25] has used exclusively the ﬁrst formulation. Most authors present both formulations but

The Linear Breakage Equation

807

use the second one since it appears to be more amenable to mathematical treatment. Assuming that a representative breakage kernel is of the form G(x, y) ¼ g((x/y)b), the use of new variables z ¼ xbt and c(z) ¼ F(x, t) eliminates the time dependence of equation (6) which is transformed as follows Z 1 zcz ðzÞ ¼ zcz ðzÞgðz=zÞ dz ð36Þ z

The solution of this equation leads to the self-similar size distribution. For convenience, one can use j(z) ¼ zcz(z) as the self-similar function, which depends on the breakage kernel through the expression: Z 1 jðzÞ ¼ jðzÞgðz=zÞ dz ð37Þ z

This is the starting point for ﬁnding self-similarity distributions. This equation is homogeneous which means that multiplying a solution with an arbitrary constant, a new one is obtained. To obtain speciﬁc results, the following total mass conservation constraint must be fulﬁlled. Z 1 jðzÞ dz ¼ 1 ð38Þ z 0 The alternative self-similarity distribution, similar to that for coagulation in [52], is M1 M0 jF ðZÞ ¼ 2 f x ð39Þ M1 M0 The equation for j is more compact and simpler than the corresponding equation for jF but on the other hand the function jF can be interpreted more easily from a physical standpoint. An important point is that jF can be directly constructed from an experimental PSD, which is not the case for j since the value of exponent b (a theoretical quantity) is needed. For these reasons, a procedure for computing jF from j is very useful. As a ﬁrst step, the relation between the particle number density distribution f(x, t) and the function j(z) is determined. Z b 1 @Fðx b tÞ 1 @ x t jðzÞ b f ðx; tÞ ¼ ¼ dz ¼ 2 jðx b tÞ ð40Þ x @x x @x 0 z x Expressing jF with respect to f(x, t) and using (40) leads to (taking into account that M1 ¼ 1) ! 1 Z b Zb ;t ¼ 2 j t ð41Þ jF ðZÞ ¼ 2 f Z M b0 M0 M0 Z

Z

1

1

f ðx; tÞ dx ¼

But; M 0 ¼ 0

0

b jðx b tÞ dx ¼ t 1=b x2

Z 0

1

jðzÞ dz z1þ1=b

ð42Þ

808

M. Kostoglou

Combining (41) and (42) leads to jF ðZÞ ¼

Z 1 b jðzÞ b j½Z ð dzÞb 1þ1=b Z2 z 0

The moments of the two self-similarity distributions are deﬁned as Z 1 Fi ¼ zi jðzÞ dz

ð43Þ

ð44Þ

0

Z

1

Zi jF ðZÞ dZ

FFi ¼

ð45Þ

0

and they are related to each other through FFi ¼ Fði1Þ 11=b Fi11

ð46Þ

b

From the relations (40) and (43) it is obvious that the function j is shifted towards high values of the independent variable with respect to the functions f, jF (due to the square of particle volume in the denominator). This must be taken into account in the numerical solution of (37) since seemingly insigniﬁcantly small values of j for small z can correspond to size regions with high particle number concentration. Rz We reserve the notation gðzÞ ¼ gðzb Þ ¼ 0 yyðyÞ dy for the homogeneous cumulative volume kernel corresponding to homogeneous number density kernel y. The kernel gðzÞ is physically more meaningful than g(z) which is used for mathematical convenience. The relation R 1 i between the moments R 1 i of the function g(z) and the kernel y (deﬁned as Gi ¼ 0 z gðzÞ dz and Yi ¼ 0 z yðzÞ dz, respectively) is as follows: Z 1 Z z1=b Z 1 Z 1 Gi ¼ zi yyðyÞ dy dz ¼ yyðyÞ zi dz dy 0

Z

ðiþ1Þb

yyðyÞ

¼ 0

0

0 1

1y iþ1

dy ¼ Z

yb

1 1 Yðiþ1Þbþ1 for ia 1 iþ1

ð47aÞ

1

y lnðyÞyðyÞ dy

G1 ¼ b

ð47bÞ

0

4.2. Closed form solutions of the self-similarity equation 4.2.1. Continuous kernels For certain simple forms of the breakage kernel, equation (37) can be solved analytically. Solutions for the power-law kernel (g(z) ¼ zm/b with m ¼ (2v)/(v1)) are available from the pioneering work of Fillipov [51] and have the form: jðzÞ ¼

zm=b ez Gðm=bÞ

ð48Þ

The Linear Breakage Equation

809

where G is the Gamma function. Peterson [25] solved the differential form of the self-similar breakage equation for this kernel. An important point is that he presented solutions for m42. The corresponding kernels satisﬁed the mass conservation requirement (12) but not the condition (13). A mathematically acceptable power-law kernel must have mr2; additionally, for practical purposes (ﬁnite number of fragments per breakage event) should be m41. The two-term sum of powers kernel is a generalization of the simple power law and the corresponding solution has been derived in [41]. The kernel and the corresponding solution are gðzÞ ¼ azb=b þ ð1 aÞzd=b jðzÞ ¼

ð49aÞ

Gðð1 aÞb=b þ ad=bÞ b=b z z e Uðð1 aÞðb dÞ=b; b=b d=b þ 1; zÞ ð49bÞ Gðb=bÞGðd=bÞ

where U is the conﬂuent hypergeometric function of the second kind which can be computed using mathematical routine libraries. Some solutions for special cases of the above kernel were already known (e.g. [53] for b ¼ 3, d ¼ 4, a ¼ 4). The power law–polynomial product kernel is a different generalization of the power-law kernel. The corresponding function g has the form: gðzÞ ¼

n X i¼0

ci zðgþ2þiÞ=b gþ2þi

ð50Þ

where ci (i ¼ 0, 1, y, n) and g are parameters of the kernel. For the case n ¼ 0 the above kernel degenerates to the power law. The n ¼ 1 case is just a sub-case of the general sum of powers kernel. The above kernel (50) has been introduced by Treat [54] who solved (using the Mellin transform) an equation equivalent to equation (37) for the self-similar (‘‘reduced’’ in his terminology) distribution in terms of G-Mejer functions. For the case n41 not covered by the other kernels, the analytical solution is too complicated to be of practical use and numerical solution is preferable. Nevertheless, Treat [54] used the exact solution to extract very interesting asymptotic results.

4.2.2. Discrete kernels The existence of a self-similarity solution for the case of binary equal-size breakage was reported in [55], where it was shown that the modiﬁed Gamma distribution used for approximating the PSD retains its shape after some time of breakage. Kostoglou [42] derived the explicit form of the self-similar distribution for the more general case of v equal-size fragments. The function g and the corresponding solution are gðzÞ ¼

0 1

z 2 ½0; v b Þ z 2 ½vb ; 1

ð51aÞ

810

M. Kostoglou

jðzÞ ¼

lnðv

b

Þ

1 X

!1 z

ik i

e

þ

1 X

i¼1

! v bi z

ki e

ð51bÞ

i¼1

Q where k i ¼ ð1Þi ij¼1 ðv bðj1Þ v b Þ1 The above self-similarity distribution is always unimodal and, in general, the location and the height of its maximum decrease as the parameter vb increases. As vb-1 the distribution shifts to higher values of x and for vb ¼ 1 (b ¼ 0) it does not exist, as it is well-known in the literature [56]. The number of terms needed in series (51b) for the computation of j(x) with a speciﬁed accuracy decreases as vb increases. For example, for four digit accuracy, six terms are needed at vb ¼ 2 and only three terms at vb ¼ 10. At the limit of large vb the self-similarity distribution takes the form jðxÞ ¼

ex ev lnðv b Þ

b

x

vb ! 1

as

ð52Þ

From the above relation the location of the maximum of j(x) can be computed analytically as x max ¼ lnðv b Þ=ðv b 1Þ

ð53Þ

Surprisingly enough, the approximate expression (53) is acceptable even for vb as small as 5, although it has been derived assuming vb goes to inﬁnity. Some basic features of j(x) can be found directly in a simple form using equations (63, 64). The total area, mean value and dispersivity of j(x) are: j0 ¼ 1= lnðv b Þ

ð54Þ

j ¼ ð1 v b Þ1

ð55Þ

sj ¼ ln

2 1 þ v b

ð56Þ

From the above relation it is obvious that the total area below j(x) decreases decreases monotonously as vb increases. The mean value of the distribution j also as vb increases and acquires an asymptotic value j ¼ 1. Finally, the dispersivity s of the self-similar distribution starts with a value close to zero for vb close to one and increases to an asymptotic value equal to ln(2) as vb goes to inﬁnity. Consequently, the general pattern is that of a narrow distribution with large mean value for vb close to 1, which shifts to smaller sizes and spreads out as vb increases. The form of the self-similarity function is reminiscent of a singular asymptotic expansion in which the outer expansion is the single exponential function and a boundary layer arises close to x ¼ 0 due to the boundary condition j(0) ¼ 0. The thickness of the boundary layer increases as the parameter vb decreases.

The Linear Breakage Equation

811

The number of terms required for the computation of j(x) for vb close to 1 is very large and the numerical errors make it impossible for vbo0.1 (double accuracy calculations). One way to tackle this problem is to replace the sum in equation (53b) with an integral and to ﬁnd j(x) in terms of ﬁnite integrals. The latter can be computed numerically. A much easier way is to approximate j(x) with a log-normal distribution by matching the ﬁrst three moments. The approximation is the following 1 2 x s=2 j0 japr ðxÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ ð57Þ e2sln j e 2psx where the parameters are given in equations (54–56). This approximation becomes better as vb decreases and is close to perfect (four digits accuracy) for vbo1.1. Therefore, the self-similarity distribution for vbo1.1 is given using jðxÞ ¼ japr ðxÞ. Kostoglou [42] also derived explicit self-similar distributions for the case of the general discrete binary kernel:

jðzÞ ¼

1 X 1 X

0

z 2 ½0; b Þ

gðzÞ ¼ 1

z 2 ½ ; ð1 Þb Þ z 2 ½ð1 Þb ; 1 b

!1 k i;j ½i lnð1 Þ þ j lnðÞ

i¼0 j¼0

1 X 1 X

ð58aÞ

k i;j e

x ð1Þib jb

ð58bÞ

i¼0 j¼0

where the coefﬁcients ki,j are obtained from the following recursive relation k 0;0 ¼ 1 k i;j ¼

1 1 ð1 Þib jb

h i ð1 Þði1Þbþ1 jb k i1;j þ ð1 Þib ðj1Þbþ1 k i;j1

ð59Þ

4.2.3. Some features of the self-similar PSD The asymptotic behaviour of the solution of equation (37) has been derived by several authors using different approaches. The small-z behaviour has been shown [29,54,56] to be jðzÞ / zg for gðzÞ / zg as z ! 0. The exponent g must be larger than one for a ﬁnite number of particles in the system. The small-z behaviour for different than the power-law form of g(z) (i.e. g(z) ¼ 0 for 0ozoz1) cannot be found using an asymptotic treatment. The large-z behaviour has been shown [54,56,57] to be 0

jðzÞ / zg ð1Þ ez where the prime denotes the derivative of the function.

ð60Þ

812

M. Kostoglou

If both sides of equation (37) are multiplied by zs and integrated with respect to z from 0 to N, it follows that Gs1 ¼

Fs Fs1

for s 0

ð61Þ

Taking into account that j1 ¼ 1 (volume conservation), the integer moments j(z) are given as Fi ¼

i Y 1 G j¼0 j1

for i ¼ 0; 1; 2; . . .

ð62Þ

Some basic features of the self-similarity such as the mean value distribution ¼ F1 =F0 and the dispersivity sj ¼ ln FF0 F2 2 can be found directly using equaj 1 tion (62): 1 G0 G0 sj ¼ ln G1 ¼ j

ð63Þ

ð64Þ

4.2.4. Is the self-similar PSD realizable? All the above analysis (closed form and numerical solution of the reduced equation (37)) is based on the implication that the function j corresponds to the asymptotic form of a PSD undergoing fragmentation. In practice, there is no evidence to ensure this. The crucial assumption for the derivation of equation (37) is that an asymptotic j exists (so transient terms can be omitted), so if it really exists it will be given from the solution of equation (37). On the other hand, it is not sure that the evolving PSD transformed to self-similar coordinates (let us denote it as j(x, t)) will reach the steady-state solution j(x). Alternatively the question is: Is j(x) a stable steady solution of the dynamic breakage equation capable to ‘‘attract’’ all the PSD trajectories independently from the initial condition? At present there is no formal answer to this question but an answer can be given based on the accumulated experience from numerical solutions of the dynamic breakage problem. It is interesting that despite the large number of works devoted to the derivation of exact forms of the self-similar PSD, no extensive checking to ensure if it really corresponds to the asymptotic solution of the dynamic problem has been performed. Recently, Mantzaris [58] solved numerically the breakage equation for several binary kernels and showed that in all cases a steady state is reached for the function jF(Z). But there is a notable exception: the binary equal-size kernel. In this case, Mantzaris [58] showed that

The Linear Breakage Equation

813

the standard deviation of the function jF(Z) oscillates periodically with respect to the transformed time tF ¼ ln(M1/M0). Based on the results of Mantzaris the relation between the function j(x) (i.e. the solution of the integral equation (37)) and the solution of the breakage equation (i.e. j(x, t)) transformed in self-similar coordinates are re-examined and the new picture is as follows: For any breakage kernel except the v-ary equal size one, the situation is what was thought by the scientiﬁc community for years, i.e. j(x, N) ¼ j(x). But for the equal-size breakage kernel the situation is different: the function j(x, t) in the limit t-N does not reach a steady state but oscillates periodically with respect to transformed time tF ¼ ln(M0/M1). The period of the oscillation is ln(v) and the amplitude of the oscillation depends on the difference between the initial distribution j(x, 0) and the function j(x) which is given in closed form in equation (51). For the particular case j(x, 0) ¼ j(x), the amplitude of oscillations is zero and the function j(x) fulﬁls the dynamic breakage equation at all times.

5. SPECIAL CASES OF BREAKAGE 5.1. Limited breakage 5.1.1. Formulation of the steady-state problem The steady-state distribution can be obtained from the solution of equations (7, 8) at the limit t ¼ N. Since, this method is quite impractical computationally, one may proceed in a different way. The following function is introduced [59]: Z 1 LðxÞ ¼ bðxÞ f ðx; tÞ dx ð65Þ 0

which represents the total number of particles with volume x, that suffer breakage during the entire process. This transformation essentially eliminates the breakage frequency. Integrating equations (7) and (8) from t ¼ 0 to N, one obtains: Z 1 f 0 ðxÞ ¼ pðx; yÞLðyÞ dy LðxÞ; x4x m ð66aÞ x

Z

1

f s ðxÞ ¼

pðx; yÞLðyÞ dy þ f 0 ðxÞ;

xox m

ð66bÞ

xm

where fs(x) is the dimensionless steady-state PSD. The function L(x) depends on p(x, y) and f0(x), whereas the steady-state size distribution fs(x) depends additionally on xm. The independence of the steady-state size distribution from breakage rate suggests that a steady state exists even for rates which lead the unlimited breakage problem to a shattering-type behaviour [60].

814

M. Kostoglou

To proceed one should modify the non-dimensionalization of the particle volume and of the steady-size distribution to render it independent of the initial distribution: x ¼

x x0 ¼ xm X m

2 0

0

X f ðx Þ ¼ m s ¼ x 2m f s ðxÞ f s ðxÞ M

ð67Þ

In the linear equations (66a, b), f0(x) can be interpreted as a kind of driving force. Indeed, considering an elementary form of the initial distribution such as x m Þ results. For an arbitrary initial distrif0(x) ¼ d(x1), a Green function f sg ðx; bution f0(x), application of the superposition principle leads to Z 1 x ¼ m Þ dy þ x 2m f 0 ðxx mÞ f s ðxÞ yf 0 ðyÞf sg ðx; ð68Þ y xm The second term in the right-hand side of the above equation represents the part of the initial distribution with sizes smaller than Xm, which remains untouched during the breakage process. In view of equation (68), it is not restrictive to assume that the initial distribution is monodisperse d(x1); any result obtained with the latter can be directly generalized for arbitrary initial distribution via equation (68). Using the monodisperse initial condition and the new function q(x) ¼ L(x)d(x1), equations (66a,b) are modiﬁed for a homogeneous breakage kernel as: Z 1 1 yðxÞ þ yðx=yÞqðyÞ dy qðxÞ ¼ 0 ð69aÞ x y ¼ x 2m f s ðxÞ

1 x mÞ y x m qðyÞ dy þ x 2m yðxx y xm y

Z

1

ð69bÞ

Equation (69) is a form of Volterra equation of the second kind, that can be solved using the well-known method given by Tricomi [61]; i.e. Z 1 qðxÞ ¼ yðxÞ þ Hðx; yÞyðyÞ dy ð70Þ x

Hðx; yÞ ¼

1 X

Anþ1 ðx; yÞ

n¼0

Z

y

Anþ1 ðx; yÞ ¼

Aðx; zÞ An ðz; yÞ dz

n ¼ 1; 2; 3; . . . ; 1

x

A1 ðx; yÞ ¼ Aðx; yÞ ¼

1 yðx=yÞ y

This solution is also impractical for computations; thus it is preferable to proceed in different ways in order to solve the equations (69a, b). It is observed that

The Linear Breakage Equation

815

with respect to the variable xm resembles the behaviour of the distribution f s ðxÞ the temporal evolution of a distribution. To demonstrate and exploit this analogy, the new time-like variable t ¼ ln(xm) is deﬁned and the alternative symbolism tÞ is used to denote the functional dependence on t. The initial condition f s ðx; 0Þ ¼ jðxÞ. As ‘‘time’’ t (for t ¼ 0), easily obtained from equation (69b), is f s ðx; increases the distribution ‘‘evolves’’.

5.1.2. General behaviour -- Asymptotic results The limit t-N: Nambiar et al. [62] solved numerically the dynamic equation and found the steady-state size distribution in the limit of large times. Kostoglou et al. [35] solved numerically equations (66a, b) to obtain directly the steady-state size distribution. Both works resulted in the conclusion that for xm51 the steady-state size distribution (called ‘‘limiting’’) is independent of the initial size distribution f0(x). Thus, there is a direct relationship between the steady-state size distribution and the breakage kernel. This relationship will be revealed in what follows. One is that can be computed from interested in the limiting steady-state distribution f sl ðxÞ equation (69b) in the limit xm-0. A particular solution to the above problem is given in [35] in the form of an inﬁnite series containing derivatives of the kernel of all orders. This solution has obvious restrictions but it is correct for the few kernels for which the series converges. A general solution is given here, valid for all cases where a steady state exists. Using the new variable of integration z ¼ ðx=yÞx m in equation (69b), the following relation is obtained: Z x 1 x 2 f sl ðxÞ ¼ lim x m yðzÞ x m dz ð71Þ x m !0 z 0 z To proceed with the above equation, the asymptotic behaviour of q(x) as x-0 must be known. It is interesting that only the asymptotic behaviour of q(x) is required for evaluating the limiting steady state and not the entire function deﬁned in the interval [0, 1]. To ﬁnd the asymptotic behaviour of q(x), the following procedure is adopted: Equation (69a) is multiplied by xs and then integrated with respect to x from x ¼ 0 to N. After some algebra the following relation results: QðsÞ ¼

YðsÞ 1 YðsÞ

ð72Þ

where Z

Z

1 s

QðsÞ ¼

x qðxÞ dx 0

1

x s yðxÞ dx

YðsÞ ¼ 0

The functions Q(s) and Y(s) are related to the Mellin transforms of the functions q(x) and j(x), respectively. Function Y(s) is a purely monotonic (decreasing) function of s for s40. It takes the values v and 1 at s ¼ 0 and 1, and tends to

816

M. Kostoglou

zero as s tends to inﬁnity. Function Q(s) is by deﬁnition (q(x)40) positive. According to equation (72), Q(s) takes positive values for s41 and diverges for s ¼ 1. For so1 it takes negative values with no physical meaning. The above behaviour clearly suggests that q(x) for x-0 behaves asymptotically like the power function x2. If A is a constant (dependent on j(x)) one may substitute for x-0, q(x) ¼ Ax2. Although this procedure is not mathematically precise, it is supported by the available exact solutions. Furthermore, similar considerations have been extensively used in the scaling theory of fragmentation [19]. The above statement is not valid for discontinuous kernels. The function q(x) is also discontinuous and in the limit of small x acquires a fractal-like structure with ever and ever more closely spaced discontinuities. Nevertheless, equation (72) also holds in this case but the asymptotic behaviour is observed only in an integral sense. For example, for the equal-size binary breakage (y(x) ¼ 2d(x1/2)) the asymptotic relation is not valid because there exist values of x arbitrarily close to zero where q(x) ¼ 0; it is valid, however, for the mean value of q(x) over ﬁnite regions of x. The above behaviour of q(x) suggests that a limiting steady state does not exist for discontinuous breakage kernels. In this case, the steadystate size distribution depends on the initial distribution. Substituting the asymptotic relation for q(x) in equation (71) and taking the limit, the following relation is obtained: A 1Þ ¼ f sl ðxÞ ¼ 2 f s ðx; x

Z

x

zyðzÞ dz

ð73Þ

0

The parameter R 1 A can be evaluated, from R the requirement of the total volume dx ¼ 1, as A ¼ ð 01 z lnðzÞyðzÞ dzÞ1 . The resulting equaconservation 0 x f s ðxÞ tion (73) is very important for the so-called inversion problem; i.e. from a measured it is very easy to determine the governing steady-state size distribution f s ðxÞ, breakage kernel from the relation: yðzÞ ¼

dðz2 f s ðzÞÞ dz zf s ð1Þ 1

ð74Þ

This inversion formula is similar to that for determining the bubble-size distribution from a measured chord length distribution. In both problems the result can be obtained from the ﬁrst derivative of an experimentally determined function. An extensive analysis for the inversion of such problems from noisy experimental data is given in [63]. Behaviour of q(x) for small x: It has been shown [35] that, for continuous kernels y(x), q(x)px2 in the limit x-0. This behaviour creates serious difﬁculties in the direct numerical solution of equation (69a). These difﬁculties can be easily overcome by taking into account the above behaviour to make the unknown function smooth; i.e. by multiplying with x2.

The Linear Breakage Equation

817

for small x: If, in the limit x-0, y(x) p xm (where m42 from Behaviour of f s ðxÞ mass conservation considerations) then a simple substitution into equation (73) / x m for x ! 0. This statement can be obviously extended for every leads to f s ðxÞ ﬁnite interval [0,a], where ao1, for which y(x) has a power-law form. Behaviour of q(x) for large x (x-1): We deﬁne the perturbation variable e ¼ 1x. It is interesting that the series solution (70) converges as en. Although this iterative solution can be used to ﬁnd q(x) as a power series of e, there is a more systematic way, as follows. Equation (69a) can be transformed into Z 1 1 1 y yð1 Þ þ qð1 yÞ dy qð1 Þ ¼ 0 1 y 0 1 y If y(x) is analytical in [0, 1] then it can be written as a MacLaurin series around 1 P i with respect to e. Assuming that qð1 Þ ¼ 1 i¼0 qi , carrying out the expansions and integrations and equating coefﬁcients of equal powers of e in the above equation, the coefﬁcients qi can be found as functions of the derivatives of y(x) at x ¼ 1. The above procedure is valid up to i ¼ n for kernels with continuous derivatives up to n-th order. The result for the zero and ﬁrst-order term of this expansion is qð1 Þ ¼ yð1Þ þ ðy2 ð1Þ y0 ð1ÞÞ

ð75Þ

tÞ for small t: If e ¼ 1xm ¼ 1et, equation (69b) can be Behaviour of f s ðx; transformed into

Z 1 Þ 1 xð1 f s ðxÞ ¼ ð1 Þ2 ÞÞ y qð1 yÞ dy þ ð1 Þ2 yðxð1 1 y 0 1 y For y(x) analytical in [0, 1], the terms of the above equation can be expanded in a power series of e. Using the series for q(x) mentioned above, after integration and collection of terms of the same order, a result of the following type is obtained ¼ f s ðxÞ

1 X

i f i ðxÞ

ð76aÞ

i¼0

are combinations of the derivatives yðjÞ ðxÞ and yðjÞ ð1Þ for where the functions f i ðxÞ joi. This procedure is valid up to i ¼ n for a kernel y(x) with derivatives continuous up to n-th order. The result for the zero, ﬁrst and second-order terms of the expansion is " ! 2 0 y ð1Þ y ð1Þ 3yð1Þ 0 ¼ yðxÞ þ ½yðxÞyð1Þ y ðxÞ þ yðxÞ f s ðxÞ þ1 2yðxÞ 2 2 2

yð1Þ 1 2 00 0 ðxÞ 2 ð76bÞ þxy þ 2 þ x y ðxÞ 2 2 where primes denote differentiation.

818

M. Kostoglou

5.1.3. Analysis for the sum of powers kernel A very efﬁcient method for the computation of the steady-state size distribution for kernels of this particular type will be given next. The kernel substituted into the equation (69b) for the steady state, after some algebra, leads to Z 1 n n X X k þ2 k i þ2 k i i f s ðxÞ ¼ ci x m y k i 1 qðyÞ dy x k i þ ci x m x xm

i¼1

i¼1

This result can be simpliﬁed as ¼ f s ðxÞ

n X

ci x k i ½x kmi þ2 ðM i þ 1Þ ¼

n X

i¼1

ci x k i F i

ð77Þ

i¼1

R1 where E i ¼ xm y k i 1 jðyÞ dy and F i ¼ x kmi þ2 ðE i þ 1Þ. It is obvious that the steady-state size distribution has exactly the same form as the kernel. Consequently, any singularity of the kernel at z ¼ 0 can be transferred analytically to the steady-state size distribution. This fact is very important for singular kernels (even practical kernels can be singular at z ¼ 0) for which the direct numerical solution for the steady-state size distribution is difﬁcult. Thus, the problem of computing the steady-state distribution is reduced to determining the weights Fi (i ¼ 1,2, y, n) which are functions of xm. To proceed in this way, the kernel is substituted in equation (69a). After some algebra (using the deﬁnition of Mi) the following result is obtained qðxÞ ¼

n X

ci x k i ðE i þ 1Þ

ð78Þ

i¼1

Taking into account that qðxÞ ¼ x k i þ1 dE i =dx, for i ¼ 1 to n, a system of differential equations results n X dM i ¼ cj x k i þk j 1 ðM j þ 1Þ; dx j¼1

i ¼ 1 to n

ð79Þ

Finally, by employing the relation between Ei and Fi, and the time-like variable t, the system of differential equations for the Fi’s is reduced to the following very simple form n dF i X ¼ cj F j ðk i þ 2ÞF i ; dt j¼1

i ¼ 1 to n

ð80Þ

For t ¼ 0 the steady-state distribution coincides with the kernel; thus the initial condition for the above system is Fi(0) ¼ 1 for i ¼ 1 to n. The weights of the limiting steady-state distribution Fi(N) can be obtained from the solution of the system of linear algebraic equations that result by setting the derivatives in equation (80) equal to zero. The solution to this system is Fi ¼ constant/(ki+2). The constant is speciﬁed to conserve the total mass of the system and the

The Linear Breakage Equation

819

ﬁnal result for the weights of the limiting steady-state size distribution is given as !1 n X 1 cj Fi ¼ ð81Þ k i þ 2 j¼1 ðk j þ 2Þ2 Exactly the same result is obtained by using equation (71) that provides directly the limiting steady-state size distribution. The system of linear differential equations with the given initial condition has the following solution [64] Fi ¼

n X

d ij elj t ¼

j¼1

n X

lj

d ij x m

where d ij ¼

n X

j¼1

w ij v jk

ð82Þ

k¼1

where i ¼ 1 to n and j ¼ 1 to n. In the above equations li is the eigenvalues, wij are the elements of the eigen column matrix and vij are the elements of the eigen row matrix, of a matrix A with elements aij ¼ ci(ki+2)dij where dij is the Kronecker delta. An alternative way for solutions of the system is by direct numerical integration. Advantages of the latter include its very easy implementation and the fact that a numerical integration reproduces all the steady states between the initial and ﬁnal value of xm. For the signiﬁcant case corresponding to n ¼ 2, a very simple solution results F1 ¼ 1 þ

ðx am 1Þ ðk 1 þ 2 c1 c2 Þ a

ð83aÞ

F2 ¼ 1 þ

ðx am 1Þ ðk 2 þ 2 c1 c2 Þ a

ð83bÞ

where a ¼ k1+k2+4c1c2.

5.1.4. Approach to the limiting steady state In the following, the general sum of powers kernel will be examined as regards the approach to the limiting steady state. As already discussed, the weights of the steady-state distribution are given by a relation of the form Fi ¼

n X

k d ik x l m

ð84Þ

k¼1

One of the eigenvalues lk of the matrix A must be zero (let us say l1 ¼ 0) to account for the limiting steady state. On physical grounds it is expected that the remaining eigenvalues would be real and negative. The approach to the steady state is determined by the smallest (in absolute value) eigenvalue. Summarizing, for the sum of powers kernel tÞ f sl ðxÞ / x bm f s ðx;

as t ! 1

ð85Þ

820

M. Kostoglou

where b ¼ b ¼ jl2 j and l2 is the smallest, in absolute value, non-zero eigenvalue of the matrix A. For example, let us explore the case of a binary product kernel (equation (15) with v ¼ 2). At ﬁrst, the matrix A is constructed and then a QR decomposition algorithm [65] is used to compute its eigenvalues. The exponent b is found to take the values 7, 7.5, 6.5, 5.5, 4.75, 4.15 and 3.66 for q ¼ 2, 3, 4, 5, 6, 7, 8, respectively. When q increases (kernel becoming steeper) the exponent b decreases. A generalization of this is the statement that as the kernel becomes steeper (the maximum derivative increasing) the approach to the limiting steady state is ‘‘delayed’’. In the limit of a kernel with inﬁnite steepness (discontinuous kernel) the exponent b tends to zero and, consequently, there is no limiting steady-state size distribution.

5.2. Erosion 5.2.1. The erosion equation The breakage equation (4) with the homogeneous erosion-breakage kernel given in equation (17) takes the form [38]: Z x=ð12 Þ @f ðx; tÞ 1 ¼ P 2 ðx=yÞbðyÞf ðy; tÞ dy bðxÞf ðx; tÞ @t y x Z 1 1 P 1 ðx=yÞbðyÞf ðy; tÞ dy þ ð86Þ y x=1 The ﬁrst integral is along a small region of sizes greater than x. A new integration variable s is used to denote the fractional deviation from x. The new variable is deﬁned by the equation y ¼ (1+s)x so that 0osoe2/(1e2). The Taylor series expansion around x of the product b(y)f(y, t) appearing in the integral is as follows: bðyÞf ðy; tÞ ¼ bðx þ sxÞf ðx þ sx; tÞ ¼ bðxÞf ðx; tÞ þ

1 X ðsxÞi @i bðxÞf ðx; tÞ i¼1

i!

@x i

ð87Þ

After substitution of the above series, the ﬁrst and second terms on the right-hand side of equation (86) take the form ðK 0 1ÞbðxÞf ðx; tÞ þ

1 X x i K i @i bðxÞf ðx; tÞ i¼1

i!

@x i

ð88Þ

where Z

2 =ð12 Þ

Ki ¼ 0

Z 2 si 1 zi P2 P ð1 zÞ dz ds ¼ iþ1 2 1þs 1þs 0 ð1 zÞ

ð89Þ

The Linear Breakage Equation

821

In order to obtain a formal expansion of the breakage equation, with respect to parameter e2, the following series expansion is used zi iþ1

ð1 zÞ

¼

1 X

aij zj

ð90Þ

j¼i

Q where aii ¼ 1 and aij ¼ ð1=ðj iÞ!Þ ji k¼1 ði þ kÞ for j4i Substitution of the above series in the expression for Ki results in Ki ¼

1 X

aij nj j2

ð91aÞ

j¼i

where nj are deﬁned as Z nj ¼ 2

1

zj P 2 ð1 z2 Þ dz

ð91bÞ

0

and represent some kind of dimensionless moments of the kernel P2. Employing the condition n0 ¼ 1 (using equation (18a)) which is an intrinsic property of the breakage kernel, it can be inferred that the nj’s are not functions of e2 and the series in equation (91a) is a formal perturbation expansion of the Ki’s for small e2. The series form of Ki is substituted in the breakage equation, the summation order of the double sum term is reversed and the terms with same order of e2 dependence are collected to give the perturbation expansion with respect to e2 " # j 1 X X @f ðx; tÞ x i @i bðxÞf ðx; tÞ j ¼ nj 2 a0j bðxÞf ðx; tÞ þ aij @t @x i i! j¼1 i¼1 Z 1 1 P 2 ðx=yÞbðyÞf ðy; tÞ dy þ ð92Þ y x=1 The terms in the brackets for j ¼ 1 and j ¼ 2 can be written in the form j ¼ 1;

j ¼ 2;

2

@xbðxÞf ðx; tÞ @ðxÞ

@xbðxÞf ðx; tÞ x 2 @2 bðxÞf ðx; tÞ bðxÞf ðx; tÞ þ @x @x 2 2

ð93aÞ

ð93bÞ

For moderate values of e2, more than one term of the expansion must be used, in which case the j ¼ 2 term (and higher-order terms) renders the new equation quite complicated. It is noted, however, that for this case (moderate e2) the original breakage equation can be handled easily by existing methods. The picture is quite different as e2 becomes small. In this case, the solution of the original equation becomes more difﬁcult. However, in the new equation it sufﬁces to retain only the ﬁrst term of the expansion that has a very simple form. It is interesting that the leading-order mass loss term (93a) is equivalent to that of

822

M. Kostoglou

erosion-breakage with fragments of ﬁxed sizes and breakage rate xb(x) [66]. This similarity does not hold for higher-order terms as one can ascertain by examining the j ¼ 2 term (93b) which is quite different for the two processes [67]. Finally, the erosive breakage equation for a homogeneous kernel takes the form Z 1 @f ðx; tÞ @½gxbðxÞf ðx; tÞ 1 ¼ þ P 1 ðx=yÞbðyÞf ðy; tÞ dy ð94Þ @t @x y x=1 where g ¼ n1e2 is the volume fraction that a parent particle loses per breakage event and the product gxb(x) is the mass erosion rate.

5.2.2. Decomposition to generations -- Analytical solution By inspection of the series solution of the original breakage equation one can infer that the i-th term of the series represents the size distribution of the particles that have undergone breakage i+1 times. The additional term is simply the size distribution of the remaining (unbroken) initial particles. For breakage kernels, which are not close to the uniform one, the various terms of the series can be widely differing functions resulting in a multimodal ﬁnal distribution. For this reason, it would be very useful to decompose the original equation to a hierarchy of equations that must be solved for the modes of the distribution. This approach has been exploited by Liou et al. [44] for the solution of the growth-breakage population balance describing the dynamics of microbial and cell cultures. The potential of the approach of Liou et al. to overcome problems encountered in the conventional solution methods of the population balance equation arising in biotechnology has been stressed by Villadsen [68]. In [45] the modes of the distribution are called generations and the development of the method is rather intuitive and based on biological system considerations and not on mathematical arguments. Furthermore, the direct application of the method of generations to equation (4) for an erosion kernel is useless because an enormous number of generations are needed to describe the evolution of the eroded particles while only a few generations are needed for the fragments. This leads to a reformulation of the method of generations so that after a breakage event only the small fragments are assigned a new generation index whereas the large fragment remains in the same generation. This approach is equivalent to applying the method of generations directly to equation (94) instead of equation (4). For convenience, the index i ¼ 1 represents the eroded initial particles, and the index i41 particles which are the ﬁne fragments of a cascade of i1 breakage events. In what follows the term parent particles is used for the generation with i ¼ 1 and the term fragments is used for the other generations.

The Linear Breakage Equation

823

The hierarchy of equations for the size distribution of the generations is: @f 1 ðx; tÞ @½gxbðxÞf 1 ðx; tÞ ¼ @t @x @f i ðx; tÞ @½gxbðxÞf i ðx; tÞ ¼ þ @t @x

Z

ð95aÞ

1

1 P 1 ðx=yÞbðyÞf i1 ðy; tÞ dy y x=1

for i ¼ 2 to N 1 ð95bÞ

@f N ðx; tÞ ¼ @t

Z

1

1 P 1 ðx=yÞbðyÞf N1 ðy; tÞ dy y x=1

ð95cÞ

where it has been assumed that no breakage occurs for particles of the N-th generation. The above equations must be solved successively with initial conditions f1(x, 0) ¼ f0(x) and fi(x, 0) ¼ 0 for i41. The complete size distribution is obtained by a simple superposition of the size distributions of the generations, as f ðx; tÞ ¼

N X

f i ðx; tÞ

ð96Þ

i¼1

The method of generations is considered useful for the general breakage problem, but it seems to be perfectly suited to the present case of erosive breakage. In the latter, the solution displays important features in several size classes so that an appropriate discretization for a successful numerical treatment seems unfeasible. On the other hand, the solutions of the generations equations are well-behaved functions amenable to conventional treatment. A particular advantage of the generation equations is that they can be solved analytically using the method of characteristics [69]. Taking into consideration the general solution of the growth population balance given by Williams [70] the following result is obtained: f 1 ðx; tÞ ¼

f i ðx; tÞ ¼

1 B1 ðBðxÞ þ tÞb½B1 ðBðxÞ þ tÞf 0 ðB1 ðBðxÞ þ tÞÞ xbðxÞ

1 xbðxÞ

Z

t

xðt 0 Þbðxðt 0 ÞÞ

Z

1 xðt 0 Þ=1

0

1 P 1 ðxðt 0 Þ=yÞbðyÞf i1 ðy; t 0 Þ dy dt 0 y

for i ¼ 2 to N 1

ð97bÞ

Z tZ f N ðx; tÞ ¼ 0

ð97aÞ

1

1 P 1 ðx=yÞbðyÞf N1 ðy; t 0 Þ dy dt 0 x=1 y

ð97cÞ

R where BðxÞ ¼ 1=ðgxbðxÞÞ dx, B1(z) is the solution to equation z ¼ B(x) and xðt 0 Þ ¼ B1 ðt t 0 þ BðxÞÞ.

824

M. Kostoglou

5.2.3. Moments of the generations Although the above analytical solution can be used for practical purposes, it is rather complicated. On the other hand, often the gross features and not the details of the distribution are of interest. This is the reason for extensively using the method of moments for solving population balances [71]. Additionally, the method of moments has been used in the physics literature to get a physical insight from the structure of the solution [56]. In order to proceed with the method of moments and obtain analytical solutions, the breakage rate is assumed to be of the power-law form b(x) ¼ xb. The k-th moment of the i-th generation is deﬁned as Z 1 M i;k ¼ x k f i ðx; tÞdx ð98Þ 0

The set of equations (95) are multiplied by xk and integrated with respect to x from 0 to N. The product rule is used for integrating the term with the derivative together with the fact that f(N,t) ¼ 0 due to physical considerations. As regards the double integral arising from the integral term, the order of integration is interchanged and the inner variable is scaled appropriately to permit the separation of the two integrals according to a standard procedure in breakage literature. The resulting system that describes the evolution of moments is (i ¼ 1, 2, y, N, k ¼ 0, 1, 2, y, N) dM i;k ¼ ð1 diN ÞgkM i;kþb þ ð1 dil ÞJ k M i1;kþb dt

ð99aÞ

where the Kronecker delta (dij is equal to 1 for i ¼ j and equal to 0 for i6¼j) is used to obtain a system in compact form, and Z 1 Ji ¼ zi P 1 ðzÞ dz ð99bÞ 0

ei1

This quantity is of order and it is in general different than the quantity ni e1i , except for the case i ¼ 1 where J1 ¼ g, and for the binary kernel where the two quantities are equal for every value of i. The above system is solved with initial conditions M1,k(0) ¼ Mk0 (the moments of the initial size distribution) and Mi,k(0) ¼ 0 for i41. For the case of constant breakage rate (exponent b ¼ 0), the system (99a) is closed and admits the following analytical solution gkt M i;k ¼ M k0 J i1 k e

M N;k ¼

M k0 J N1 k

t i1 ði 1Þ!

for i ¼ 1 to N 1

" # 2 1 ðN 2Þ egkt NX ðN 2Þ!t N2i ðN 2Þ! ðgkÞN1 gk i¼0 ðgkÞi ðN 2 iÞ!

ð100aÞ

ð100bÞ

The Linear Breakage Equation

825

5.2.4. Case study Constant breakage rate case: The particular case of constant breakage rate (b ¼ 0) is worthy of detailed study because it is amenable to a simpliﬁed treatment and its solution exhibits a very interesting behaviour. Using equation (97a) for b ¼ 0 results in f 1 ðx; tÞ ¼ egt f 0 ðxegt Þ

ð101Þ

irrespective of the type of breakage kernel. The monodisperse erosion-breakage kernel admits an analytical solution as follows. Substituting the monodisperse 0 kernel in equation (97b) and using the characteristic x(t0 ) ¼ xeg(tt ) yields the recursive relation (i ¼ 2 to N1): gðtt 0 Þ Z t 0 xe f i ðx; tÞ ¼ ; t 0 egðtt Þ dt 0 f i1 ð102Þ 0 Using f1 to start with, the recursive integration can be performed analytically to obtain t et t i1 e f 0 i1 f i ðx; tÞ ¼ i1 ð103Þ ði 1Þ! From the ﬁrst two moments of the above distributions (or, alternatively, using equation (100a)) one obtains t i1 ði 1Þ!

ð104aÞ

t i1 i1 t e ði 1Þ!

ð104bÞ

M i;0 ¼

M i;1 ¼

The well-known self-similarity transformation, originally proposed by Friedlander [52] for the solution of the coagulation equation, is here deﬁned at the generation level as M xM i;0 f i ðxÞ ¼ i;1 f ð105Þ M i;1 M 2i;0 ¼ f 0 ðxÞ which means that the shape Substitution into equation (103) gives f i ðxÞ of the size distribution for each generation is similar to that of the initial distribution. This type of self-similarity is quite different from the conventional one since it is not valid only in the large time limit but for all times, and it is not independent of the initial distribution but has exactly its shape. The present type of self-similarity has also been noticed in the study of the (equivalent with the i ¼ 1 equation of the present problem) polymer degradation with chain-end scission [66], a case for which it has been argued that no self-similarity solution exists [55]. The above (generation level) self-similarity solution is restricted to constant

826

M. Kostoglou

breakage rate (b ¼ 0) and monodisperse erosion kernel. Furthermore, it should not be confused with the global asymptotic self-similarity solution which exists for every homogeneous breakage kernel, including the homogeneous erosion kernel studied here. An interesting consequence of equation (103) is that, in the case of a monodisperse initial distribution, the size distribution of each generation is monodisperse as well. This is due to the fact that, as the particles of generation i ‘‘move’’ to smaller sizes, the particles of generation i1 ‘‘move’’ in such a way that (at every instance) the fragments they produce have the current size of the particles of generation i. Summarizing, the solution of equation (4) for constant breakage rate and monodisperse binary erosion kernel is t 1 X 1 t i1 e f ðx; tÞ ¼ et f ð106Þ 0 i1 ði 1Þ! i1 i¼1 For a more general than the binary monodisperse erosion kernel there is no analytical solution, although some features of the solution can be obtained using the moments equation (100a, b). Thus, the mean size of generation i is gt found to be x mi ¼ zi1 , where zm ¼ g=ðv 1Þ is the mean size of the fragm e ment size distribution. The dispersivity of generation i is s ¼ s0 þ ði 1Þsfragment where sfragment ¼ lnððv 1ÞJ 2 =gÞ may be called the dispersivity of the fragment size distribution. This means that the dispersivity of each generation tends to increase, compared to the dispersivity of the preceding one, by an amount equal to the dispersivity of the fragment size distribution. For the monodisperse kernel sfragment ¼ 0; thus, all the generations have the same dispersivity s0 as the initial distribution. Study of the N ¼ 2 case: To get an insight into the structure of the solution, some simple cases were examined, for N ¼ 2, power-breakage rate and monodisperse erosion kernel. The general solution for f1 (irrespective of N and erosion kernel) with power-breakage rate is f 1 ðx; tÞ ¼

ðx b gbtÞðbþ1Þ=b f 0 ½ðx b gbtÞ1=b x bþ1

for ba1

ð107Þ

where x b o1=gbt. It is noted that if this condition is not satisﬁed then f1(x, t) ¼ 0. This discontinuity originates from the fact that because of particle erosion the maximum particle size at time t is x ¼ ð1=gbtÞ1=b . For the limiting case b ¼ 0, the result has already been given in the analysis of the constant breakage rate case. For N ¼ 2 and monodisperse breakage kernel, equation (97c) takes the form Z t f 2 ðx; tÞ ¼ bðx=Þf 1 ðx=; t 0 Þ dt 0 ð108Þ 0

The Linear Breakage Equation

827

Substituting f1 from equation (107) and using a new variable of integration gives n 1=n Z 1 ððx=Þ ntÞ f 2 ðx; tÞ ¼ f 0 ðzÞ dz for na1 x x= or 1 f 2 ðx; tÞ ¼ x

Z

xet =

f 0 ðzÞ dz for n ¼ 1

ð109Þ

x=

6. METHODS OF MOMENTS The idea which led to the development of the method of moments is that in some cases the amount of information on the PSD given by a method based on discretization of the particle size domain, can be sacriﬁced in favour of the computational speed. For example, in a large plant simulator the grinding sub-model has to be as efﬁcient as possible even in the expense of having as only output, the total particle number concentration and the mean particle size. From the technical point of view the method of moments is a generalization of the methods of weighted residuals [72] having a trial function more general than a linear superposition of a basis function, i.e. f(x, t) ¼ F(x, t; c) where the function F has a known form and the vector c ¼ (c1, c2, y, cP) contains P unknown time-dependent parameters which can be found in the following way. Equation (4) is multiplied by the P power-law test functions x ai (i ¼ 1, 2, y, P) and then is integrated for x between 0 and N to give the system of equations (assuming an homogeneous fragmentation kernel): Z 1 dM ai ¼ ðJ ai 1Þ bðxÞf ðx; t; cÞ dx ð110Þ dt 0 Z 1 x ai f ðx; t; cÞdx ¼ M ai ð111Þ R1

0 ai

where J ai ¼ 0 x yðxÞ dx. The most widely used forms of the distribution F are pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ the log-normal distribution f ðx; tÞ ¼ ðc1 = 2pc2 xÞ exp½ð1=2c2 Þln2 ðx=c3 Þ and the Gamma distribution f ðx; tÞ ¼ ðc1 =Gðc2 þ 1ÞxÞðx=c3 Þc2 ex=c3 with (a1, a2, a3) ¼ (0, 1, 2) and x in the general case can be a power-law function of the particle volume [55,73]. For the case of a power-law fragmentation rate, b(x), the integrations in equations can be performed in closed form leading to a simple system of Ordinary Differential Equations (ODE’s) with respect to ai’s. If the b(x) does not have a power-law form, the integral in equation (110) must be computed numerically. The Hermite and Laguerre quadratures are ideally suited for the case of log-normal and Gamma distribution, respectively. Another candidate function

828

M. Kostoglou

for the representation of the PSD is the so-called modiﬁed Gamma distribution. This function has not been used for the solution of the breakage equation despite its close relation to the Rosin–Rammler distribution which is the basic tool for characterization of PSD’s resulting from fragmentation processes during size reduction of solids [74]. A systematic way to improve the log-normal method is the so-called interpolation between the moments method [73,75]. This method can be applied only for a power-law breakage rate using the following set of ai’s (a1, a2, y, aP) ¼ (0, 1, 2, y, P1). An explicit form of f is not assumed and each moment z appearing on the right-hand side can be found from the integer moments of the PSD by the following interpolation rule: " # QP1 P 1 X j¼0;jak ðz jÞ M z ¼ exp logðM k Þ QP1 ð112Þ k¼0 j¼0;jak ðk jÞ For P ¼ 3, the log-normal method is recovered while improved results can be found using P ¼ 4 and P ¼ 5. Larger values of P cannot improve the solution because the amount of information contained in the higher moments of the distribution is limited. According to the generalized method of moments, the PSD is approximated by a set of Dirac delta functions with unknown strength and location, i.e. (P is an even number) f ðx; tÞ ¼

P=2 X

w j dðx x j Þ

ð113Þ

j¼1

Substituting in equations (110) and (111) leads to the following system of Differential Algebraic Equations (DAEs). P=2

X dM ai ¼ ðJ ai 1Þ w j x aj i bðx j Þ dt j¼1 P=2 X

w j x aj i ¼ M ai

ð114Þ

ð115Þ

j¼1

This method is quite general and can be used for any fragmentation rate and kernel. It is developed for the solution of the aerosol growth equation (quadrature method of moments [76]) and aerosol coagulation equation (generalized approximation method [77]) independently. Kostoglou and Karabelas [73] used it for the solution of the fragmentation equation (generalized method of moments). Typical values of P are 4 [78] and 6 [79] and the best choice for ai’s seems to be (a1, a2, y, aP) ¼ (0, 1/3, 2/3, y, (P1)/3). The system (114–115) can be solved directly using an ODE–DAE solver ([73,78]) or using an ODE solver for (115)

The Linear Breakage Equation

829

simultaneously with special procedures from the theory of Gaussian integration to ﬁnd the weights wj and abscissas xj [79]. More recently, the so-called direct quadrature methods of moments was proposed [80]. In this case, a set of ODEs which can be solved directly for the unknowns (weights and abscissas) is derived.

7. SECTIONAL METHODS Very detailed studies on the application of the traditional general purpose numerical algorithms to the breakage problem can be found for ﬁnite differences, spectral and ﬁnite element methods ([81–83]). The conventional ﬁnite differences have been used extensively for the solution of breakage equations in bubble/ droplet applications where the form of breakage rate does not allow a large extent of breakage [84]. In case of solids fragmentation having a power-law breakage rate, the ﬁnite difference method is not appropriate. Some algorithms more speciﬁc to the breakage equation based mainly on collocation approaches have been developed over the last few years. B-splines and wavelets are used as basis functions in [85] and [86], respectively. A reﬁnement of the log-normal moments method using collocation with Hermitte polynomials as basis function on a grid moving according to the moments method is proposed in [87]. Finally, very recently the breakage equation has been transformed using self-similar coordinates and in this new form it can be solved using the Galerkin method with Legendre polynomials as basis functions [58]. In general, the above methods are very elaborate and very accurate (indispensable for fundamental studies) but not unconditionally stable. For example, a very sharp initial distribution (e.g. monodisperse) with a small number of basis functions may lead the algorithm to failure. For practical purposes, a different category of methods with unconditional stability and accuracy comparable with that of measurement techniques and also able to give more information (e.g. complete PSD) than the moments technique is needed. This category is the so-called sectional methods. According to the sectional method the particle volume coordinate is partitioned using the points vi (i ¼ 0, 1, 2, y, L). The particles with volume between vi1 and R v i vi belong to the i-class and their number concentration is denoted as Ni (i.e. v i1 f ðx; tÞ dx). The characteristic particle size for the class i is taken to be x i ¼ ðv i1 þ v i =2Þ. The direct sectional (ﬁnite volume) equivalent of equation (4) is (i ¼ 1, 2, y, L) L dN i X ¼ nij bj N j bi N i dt j¼i

bi ¼ bðx i Þ Z vi nij ¼ pðx; x j Þ dx v i1

ð116aÞ ð116bÞ ð116cÞ

830

M. Kostoglou

This system of ODEs can be solved analytically [88] and has been extensively used in the grinding literature as the fundamental equation and not a simpliﬁcation of the continuous form (4). Experimental values of Ni can be directly supplied from sieve analysis and nij can be found by ﬁtting the model (116) to the experimental Ni’s sequentially. The problem arises from the fact that the discretized form (116) does not conserve integral properties of the PSD and, in particular, total particle mass. From a physical point of view it is due to that any fragment has to be assigned to one of the discretized sizes so that it alters its mass. This problem was considered by Hill and Ng [89] who used two sets of unknown constants multiplied by both terms on the right-hand side of equation (116). These constants were computed from the requirement of internal consistency with respect to the total particle mass and total particle number. The term ‘‘internal consistency’’ of a discretization scheme with respect to a particular moment of the PSD refers to the ability of the discretized system to reproduce the discretized form of the evolution equation for the particular moment. Although this is a highly desired property (i.e. for the total particle mass, this is equivalent to the total mass conservation), it does not guarantee the exact computation of the moment. The procedure developed by Hill and Ng [89] requires complex analytical derivations that depend on the particular form of the fragmentation kernel and on the particular grid used for the discretization. The above authors made these derivations for three forms of the fragmentation kernel, two forms of the grid (equidistant and geometric) and exclusively for power-law fragmentation rate. The major drawback of their procedure is that it cannot be directly generalized for arbitrary parameters and implemented in a computer code. Vanni [90] improved the situation by replacing the requirement of internal consistency with respect to the total particle number, with a better handling of the second term on the right-hand side of equation (4) (death term). This new version (slightly less accurate than its predecessor) can be fully automated, i.e. computed numerically regardless of the fragmentation rate and kernel. A different approach for the development of a quite general sectional method with arbitrary fragmentation functions, arbitrary grid, exhibiting internal consistency with respect to two arbitrary moments has been followed by Kumar and Ramkrishna [91]. In that case the internal consistency is achieved by the proper sharing of fragments resulting from a fragmentation event among two of the discretized particle sizes. The coefﬁcients nij for the particular case of internal consistency with respect to total particle and mass are (written in a form somewhat more compact than in [91]) nij ¼ ð1 dij ÞZðx i ; x iþ1 ; x k Þ ð1 d1j ÞZðx i ; x i1 ; x k Þ Z

b

Zða; b; cÞ ¼ a

by pðy; cÞ dy ba

ð117Þ

The Linear Breakage Equation

831

Several improvements have been proposed for the above sectional approach. As an example, Attarakih et al. [92] very recently developed a method where the pivot (characteristic) size for each class is free to move between the boundaries of the class and in addition the grid moves as a whole to capture better the features of the PSD. The improved methods of this type can be implemented only through custom codes and cannot cast the problem to the form of system of ODEs directly solvable by commercial integrators. Conclusively, the suggestion here is to use the scheme of equations (116a, 117) with a geometric grid dense enough to capture the features of the required PSD.

8. CURRENT AND FUTURE RESEARCH TOPICS ON BREAKAGE EQUATION It seems that the single linear breakage equation has not many things still to reveal to us. Of course, many practical aspects such as the inverse problem are not solved yet but the required development is application speciﬁc; information about the nature of the kernel which can be derived based on ﬁrst principles is needed to achieve the inversion [93]. The combination of breakage with other mechanisms acting on the PSD (e.g. aggregation) has been studied in detail in literature but it has not been considered here since breakage is not the dominant mechanism. On the other hand, in case of breakage with dissolution or particle size diffusion the breakage is the dominant mechanism and the corresponding equations have to be studied thoroughly in view of the emerging applications for them. The recent needs for explaining, modelling and predicting size distributions of objects resulting from breakage processes are taking us beyond the linear breakage equation. The extensions to the breakage problem are divided into three categories: Generalized non-linear breakage, spatially distributed breakage and multidimensional breakage. Literally speaking, the last two categories are equivalent to each other as they refer to multiple dimensions but the difference is that the ﬁrst implies existence of spatial (external) variables with only one internal variable whereas the second refers to multiple internal variables. Of course, any combination of the above categories is possible. According to the generalized non-linear breakage, the fragmentation rate depends not only on the fragmented particle size, but also on the overall PSD. It was found that this generalization is necessary to ﬁt experimental data resulting from grinding processes [94]. A particular mode of the non-linear breakage is the so-called collisional breakage. In its more general form the mass between two collided particles is redistributed. This type of model can ﬁnd application to the recently emerged technology of nanoparticle production from emulsions [95] and to the particle coating processes [96]. In the simplest version of collisional breakage, the collided particles just break as a result of the collision event. The existence of self-similarity solutions to this

832

M. Kostoglou

problem has already been pointed out [97]. The extension of the breakage equation to spatial dimensions including phenomena like convection and diffusion has been studied and several analytical solutions were found [99]. These terms can be of phenomenological nature (e.g. modelling the actual mixing situation in grinding equipment [98]) or can refer to the actual processes. Breakage equations with more than one internal variable have been studied in the physics literature mainly with regard to large time asymptotic and scaling behaviour [100]. A recent extended analysis of the problem and the corresponding kernels makes possible the use of this equation as a modelling tool for practical breakage processes [101]. All of the above extensions of the breakage equation (which will be the new tools for the identiﬁcation and design of actual breakage processes) have to be studied in detail ﬁrst from the fundamental point of view (analytical, asymptotic and approximate solutions) and then based on this knowledge, speciﬁc numerical methods covering the whole range of weighting between accuracy, robustness and computational efﬁciency, must be developed.

Nomenclature

b(x) F f fd(x, t; z) f sl ðxÞ f s ðxÞ zÞ f sg ðx; Fi f0 fs G(x, y) g(z) Gi Ji L(x) M Mi Mi,k ni N0 p(x, y) P1(z), P2(z) q

dimensionless breakage frequency cumulative volume fraction function dimensionless particle number density function Green’s function for breakage equation limiting steady-state PSD in limited breakage normalized form of fs (equation (67)) Green’s function for steady-state PSD in limited breakage auxiliary parameters deﬁned in equation (77) initial condition for f steady-state form of f in case of limited breakage cumulative volume fraction breakage kernel transformed breakage kernel i-th moment of function g i-th moment of P1(z) (equation (99b)) auxiliary function deﬁned in equation (65) total particle volume concentration (volume fraction) i-th moment of function f k-th moment of the i-th generation of particles auxiliary variables deﬁned in equation (91b) initial total particle number concentration dimensionless breakage kernel functions for erosion kernel (equation (17)) parameter in breakage kernels given in equations (15,16)

The Linear Breakage Equation

Q(s) q(x) t v x x, y xm Xm x0

833

moment of order s of function q auxiliary function in limited breakage problem dimensionless time average fragments number per breakage event particle volume normalized by Xm dimensionless particle volumes dimensionless maximum particle stable volume maximum particle stable volume initial mean particle volume

Greek Characters e 1, e 2 parameters of erosion kernel (equation (17)) y homogeneous breakage kernel t time-like variable for limited breakage ( ¼ ln(xm)) Y(s) moment of order s of function y c(z) auxiliary self-similar PSD function j(z) self-similar PSD function (present approach) tF transformed time ( ¼ ln(M1/M0)) jF(z) self-similar PSD function (approach of [52]) FFi i-th moment of function jF Fi i-th moment of function j Prime denotes dimensional variables

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[65] J. Villadsen, M.L. Michelsen, Solution of Differential Equation Models by Polynomial Approximation, Prentice Hall, New York, 1978. [66] S. Hansen, J.M. Ottino, Powder Technol. 93 (1997) 177. [67] M. Kostoglou, Chem. Eng. Sci. 55 (2000) 2507. [68] J. Villadsen, J. Biotechnol. 71 (1999) 251. [69] L. Lapidus, G.F. Pinder, Numerical Solution of Partial Differential Equations in Science and Chemical Engineering, Wiley, New York, 1982. [70] M.M.R. Williams, J. Colloid Interf. Sci. 93 (1983) 252. [71] M.M.R. Williams, S.K. Loyalka, Aerosol Science: Theory and Practice, Pergamon Press, New York, 1991. [72] B.A. Finlayson, The Method of Weighted Residuals and Variational Principles, Academic Press, New York, 1972. [73] M. Kostoglou, A.J. Karabelas, Powder Technol. 127 (2002) 116. [74] G. Delagrammatikas, S. Tsimas, Chem. Eng. Comm. 191 (2004) 1362. [75] J.C. Barret, J.S. Jheeta, J. Aerosol Sci. 27 (1996) 1135. [76] R. McGraw, Aerosol Sci. Technol. 27 (1997) 255. [77] V.N. Piskunov, A.I. Golubev, J. Aerosol Sci. 33 (2002) 51. [78] M. Kostoglou, A.J. Karabelas, Powder Technol. 143–144 (2004) 280. [79] D.L. Marchisio, R.D. Vigil, R.O. Fox, J. Colloid Interf. Sci. 258 (2003) 322. [80] D.L. Marchisio, R.O. Fox, J. Aerosol Sci. 36 (2005) 43. [81] N.V. Mantzaris, P. Daoutidis, F. Srienc, Comput. Chem. Eng. 25 (2001) 1411. [82] N.V. Mantzaris, P. Daoutidis, F. Srienc, Comput. Chem. Eng. 25 (2001) 1441. [83] N.V. Mantzaris, P. Daoutidis, F. Srienc, Comput. Chem. Eng. 25 (2001) 1463. [84] L.M. Ribeiro, P.F.R. Regueiras, M.M.L. Guimaraes, C.M.N. Madureira, J.J.C. Cruz-Pinto, Comp. Chem. Eng. 19 (1995) 333. [85] R.C. Everson, D. Eyre, Q.P. Campbell, Comp. Chem. Eng. 21 (1997) 1433. [86] Y. Liu, M.O. Tade, Powder Technol. 139 (2004) 61. [87] R.A. Hamilton, J.S. Curtis, D. Ramkrishna, AIChE J. 49 (2003) 2328. [88] K.J. Reid, Chem. Eng. Sci. 20 (1965) 953. [89] P.J. Hill, K.M. Ng, AIChE J. 42 (1995) 1600. [90] M. Vanni, AIChE J. 45 (1999) 916. [91] S. Kumar, D. Ramkrishna, Chem. Eng. Sci. 51 (1996) 1311. [92] M.M. Attarakih, H.J. Bart, N.M. Faqir, Chem. Eng. Sci. 58 (2003) 1311. [93] M. Kostoglou, A.J. Karabelas, Chem. Eng. Sci. 60 (2005) 6584. [94] E. Bilgili, B. Scarlett, Chem. Eng. Technol. 28 (2005) 153. [95] R. Bandyopadhyaya, R. Kumar, K.S. Gandhi, Langmuir 16 (2000) 7139. [96] M. Alonso, F.J. Alguacil, AIChE J. 47 (2001) 1303. [97] M. Kostoglou, A.J. Karabelas, J. Phys. A: Math. Gen. 33 (2000) 1221. [98] C. Mihalyko, T. Blickle, B.G. Lacatos, Powder Technol. 97 (1998) 51. [99] M. Kostoglou, A.J. Karabelas, AIChE J. 50 (2004) 1746. [100] P. Singh, M.K. Hassan, Phys. Rev. E 53 (1996) 3134. [101] P.J. Hill, AIChE J. 50 (2004) 937.

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CHAPTER 19

Analysis of Agglomerate Breakage Mojtaba Ghadiri, Roberto Moreno-Atanasio, Ali Hassanpour and Simon Joseph Antony Institute of Particle Science and Engineering, University of Leeds, Leeds LS2 9JT, UK Contents 1. Introduction 2. Models of agglomerate strength and failure 2.1. Theoretical models 2.2. Phenomenological models and analyses 2.2.1. Weber number 2.2.2. Mechanistic analysis of the breakage of interparticle contacts 2.2.3. Chipping model 3. Distinct element method 3.1. Introduction 3.2. Agglomerate behaviour using distinct element method 3.2.1. Agglomerate damage 4. Agglomerate behaviour in a bed of particles subjected to shearing 4.1. Effect of size ratio on the breakage of agglomerate 4.1.1. Stress ratio 4.1.2. Damage ratio 4.2. Comparison with experiments 4.3. Relevance to granulation process References

837 839 840 841 841 843 845 846 846 847 849 862 862 863 865 866 869 870

1. INTRODUCTION Chemical, pharmaceutical and food industries amongst many others, use agglomerates either as intermediate or manufactured products. The mechanical strength of agglomerates under impact or shear deformation during handling and processing is of great interest to these industries for optimising product speciﬁcation and functionality. Agglomerates are formed by smaller particles, which have been brought together and joined to one another by a physical or chemical process [1]. Agglomerates can break during processing or transport making them less suitable for Corresponding author. Tel.: +44 113 343 2406; Fax: +44 113 343 2405; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12022-4

r 2007 Elsevier B.V. All rights reserved.

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their intended use due to formation of debris and hence quality degradation. Therefore, they need to have sufﬁcient mechanical strength to be functional but at the same time not to be so strong that they present further processing difﬁculties. This makes the study of the mechanical strength of agglomerates of great interest to industry and academia. In this chapter, we present the breakage characteristics of agglomerates subjected to two types of loading scenarios, namely (i) agglomerates impact against a wall and (ii) agglomerates immersed in a bed of particles subjected to rapid shear deformation. A full prediction of agglomerate strength and agglomerate breakage patterns has not yet been achieved although there is a large amount of work in the literature on agglomerate strength [2–26]. This difﬁculty arises from the degree of freedom and number of parameters that inﬂuence agglomerate structure and properties. Both experimental and computer simulation work suggest that agglomerates formed in the same way and impacted at the same velocity can still fail in different ways [10,12] and have shown different breakage patterns. Subero et al. [12] analysed experimentally the fragmentation pattern of agglomerates made of glass ballotini when an artiﬁcial porosity was created in the materials. In some of the impacts, the agglomerates suffered local damage in the region of the impact site only and no crack propagation occurred. Subero et al. [12] quantiﬁes the frequency of fragmentation of all the structures as a function of the agglomerate porosity. Mishra and Thornton [10] showed that for a certain range of porosities, agglomerates fragmented or showed local disintegration depending on the number of interparticle contacts. Furthermore, as shown by computer simulations, agglomerates with the same number of broken contacts can show different breakage patterns depending on the location of the broken contacts [16]. These evidences [10,12,16] suggest that the breakage pattern is strongly inﬂuenced by the path followed by the forces originated during impact. Therefore, in order to fully predict the mechanical strength and breakage pattern of agglomerates, it would be necessary to know the exact spatial distribution of particles and contacts within the agglomerates and then also to know the path of the force propagation. In systems made of many particles, the determination of the path of force propagation is a difﬁcult task and therefore macroscopic parameters such as packing fraction and coordination number need to be used. Obviously, the use of these parameters has the disadvantage that a large amount of information is lost, hence making it difﬁcult to predict the fragmentation patterns of agglomerates. The parameters that inﬂuence agglomerate strength can be classiﬁed into four types: single particle properties, interparticle interactions, agglomerate properties and external parameters, such as impact angle and impact velocity. The inﬂuence of some of these factors on the impact behaviour of agglomerates has previously been analysed [2–26], although a systematic study of the inﬂuence of these four types of parameter on the agglomerate strength does not exist yet. A systematic study would imply that, to clearly discern the inﬂuence of a particular

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property on the mechanical strength of agglomerates, the value of such a property could be varied without affecting the other properties. However, this is a very difﬁcult task from an experimental point of view. In experiments in order to change the value of one parameter, the material that is being used usually has to be changed with the consequent alteration of all the physical properties. However, this type of study can be easily carried out by computer simulations based on the Distinct Element Method (DEM) [27]. Signiﬁcant progress has been recently made in the study of the strength of granular materials using DEM. An improved understanding of the physical behaviour of agglomerates can be achieved by linking particle and bulk properties, for example [2,5,8,9,16,19,22], application of contact mechanics [27–32] and development of faster computational algorithms [33–36]. The use of DEM presents many advantages, the main one being the ability to investigate the inﬂuence of a speciﬁed property at particle scale on the breakage characteristics of agglomerates. Additionally, the possibility of probing the internal state of the systems such as the state of stress or the number of broken interparticle contacts within the agglomerate during mechanical loading, are features that cannot be easily diagnosed experimentally, and hence this makes DEM a powerful tool to study granular materials. The analysis of agglomerate breakage using DEM can address both microscopic and macroscopic points of view, for example, the number of broken interparticle contacts (microscopic) and the mass detached from the agglomerates and the agglomerate breakage pattern (macroscopic). Furthermore, DEM can be used to critically compare the predictions with those from theoretical models to identify critical factors affecting agglomerate behaviour. In addition, DEM can be combined with other models to provide a more fundamental approach to modelling, for example, in deﬁning the selection and breakage levels for population-balance modelling. Therefore, a review of various theoretical models is presented here, followed by a review of the results of impact damage analysis of agglomerates subjected to impact loading, as obtained by DEM, and wherever possible, a comparison with experimental results is presented.

2. MODELS OF AGGLOMERATE STRENGTH AND FAILURE Several attempts have been made to predict the breakage characteristics of agglomerates failure under different loading conditions by quantifying the stress required to break the agglomerates [37,38], by predicting the number of broken interparticle contacts [2,7,26] or even by quantifying the extent of breakage [13]. However, we can classify these models into two large groups: (1) purely theoretical and (2) phenomenological models.

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2.1. Theoretical models There are two widely cited theoretical models of agglomerate strength developed by Rumpf [37] and Kendall [38], respectively. Rumpf deﬁned the strength of agglomerates as the force required to break all contacts simultaneously on a prescribed failure plane. However, Kendall deﬁned agglomerate strength as the resistance of the agglomerate to crack propagation based on linear elastic fracture mechanics. Rumpf [37] estimated the agglomerate tensile strength, as the force per unit of area required to simultaneously break all contacts in the fracture plane. The ﬁnal expression shows a dependency on the porosity, e, diameter of the single particle, D and interparticle force, Fc, in the form 1 FC ð1Þ s ¼ 1:1 D2 In contrast, the model of Kendall is based on fracture mechanics principles and is therefore more rigorous than the model of Rumpf. Kendall [38] deﬁned agglomerate strength as the resistance against propagating a pre-existing ﬂaw. With this idea, he calculated the fracture toughness KC in a three-point bending test. The calculations of Kendall predict a value of tensile stress required to break the agglomerate in the form 5=6

s ¼ 15:6f4 GC G1=6 ðDcÞ1=2

ð2Þ

where f is the packing fraction (1e), G the interface energy, GC the fracture energy, (which is the measured value in experiments corresponding to the equilibrium value of G and includes all energy dissipated in the system by plastic deformation and damping, [38]), D the diameter of the particles and c the length of a pre-existing crack. In this model, the main inconvenience is the necessity to estimate the length of the ﬂaws in the material to be able to predict the agglomerate strength. The model of Kendall [38] is particularly suitable for describing brittle and semibrittle failure of agglomerates as it is consistent with the Grifﬁth criterion for crack propagation [39]. The model of Rumpf predicts the strength exclusively based on the interparticle bond strength and the average porosity of the agglomerate instead. This model may therefore be more suitable to describe the failure of ductile materials, which do not propagate any cracks. However, there is no detailed analysis in the literature that states which of the above models should be applied in a particular case. Moreover, Subero [11] showed that, as far as the dependency on the packing fraction is concerned, both models predict numerically similar values, albeit from very different functional relationships as given by these models. Furthermore, the concept of strength should be redeﬁned since agglomerates can suffer a size reduction in the form of detachment of small

Analysis of Agglomerate Breakage

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debris and not just by fragmentation as it was considered by Rumpf [37] and Kendall [38]. Therefore, it seems more appropriate to deﬁne the strength of agglomerates based on the type of damage incurred. The models of Kendall and Rumpf are neither able to predict the fracture pattern of agglomerates nor the fragment size distribution produced on impact, since agglomerates can develop different patterns of breakage depending on the impact velocity, (strain rate) and type of test [12,16,20,22]. Therefore, new theories need to be developed to predict the strength of agglomerates and this can best be done in terms of single-particle properties and packing conditions using DEM.

2.2. Phenomenological models and analyses Phenomenological models and analyses help us to extract the essential features of the phenomenon, for example, for the case of an impact process in agglomerates this could be to relate the degree of damage in the agglomerate to the impact velocity, taking into account the factors that inﬂuence agglomerate strength.

2.2.1. Weber number The ﬁrst of these phenomenological analyses is based on the use of the Weber number. Kafui and Thornton [2] analysed the effect of bond strength on agglomerate breakage using DEM and deﬁned the Weber number as the ratio of input energy to the average bond strength of an agglomerate. The expression of the Weber number for the case of impact of agglomerates is given by We ¼

r DV 2 G

ð3Þ

where r is the particle density, D the primary particle diameter, V the impact velocity and G the interface energy which is deﬁned by the Dupre´ equation (Israelachvili [40]) as G ¼ gA þ gB gAB

ð4Þ

where gA and gB are the surface energies of two particles made of different materials, A and B, in contact with each other and gAB is the interaction energy between them. For surfaces of the same material gAB is zero and therefore G ¼ 2g. Kafui and Thornton [2] analysed the effect of surface energy on the strength of 2-D regularly-packed agglomerates, having face centred cubic (fcc) and body centred cubic (bcc) structures, and they related the breakage of interparticle contacts to the Weber number. They expressed the damage in agglomerates in

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terms of the damage ratio, deﬁned as the ratio of broken contacts to the initial number of bonds. The damage ratio was related to the Weber number for a range of impact velocities and surface energies. They found that the curves corresponding to the values of surface energy between 0.1 and 1.0 J/m2 were reasonably uniﬁed when damage ratio was plotted as a function of Weber number. However, in a later work, Thornton et al. [6] obtained a better uniﬁcation by modifying the Weber number and deﬁning a lower limit of impact velocity, V0, below which no contact is broken. This modiﬁed Weber number, We0 , is given by We0 ¼

r DðV V 0 Þ2 G

ð5Þ

The above analysis was also carried out for a 2-D ordered packing with the surface energy in the range between 0.3 and 3.0 J/m2. Later, Subero et al. [7] carried out simulations using 3-D motion of particles and analysed the effect of the surface energy in the range 0.5–5.0 J/m2 in randomly packed agglomerates. They plotted the damage ratio as a function of Weber number, as shown in Fig. 1, for the range of surface energy between 0.5 and 5.0 J/m2, and found that their results were in agreement with the work of Thornton et al. [6], i.e. a good uniﬁcation of data for values of surface energies between 0.5 and 5.0 J/m2 was obtained. However, the surface energy values used in the above simulations were only varied by one order of magnitude. In a later work, Moreno-Atanasio et al. [26] varied the surface energy by two orders of magnitude and found that the use of the modiﬁed Weber number no longer uniﬁed the data adequately. They then proposed an alternative analysis based on the idea that the damage suffered by agglomerates during an impact event could be related to the incident kinetic

Fig. 1. Plot of damage ratio vs. Weber number for different values of surface energy [7].

Analysis of Agglomerate Breakage

843

energy and also to the physical and mechanical properties of the agglomerates. This model is described in the following section. The use of the modiﬁed Weber number is based on the assumption that the breakage of interparticle contacts is related to the ratio between the speciﬁc kinetic energy and the interface energy. However, no clear physical meaning was given to that relationship. In addition, the introduction of the minimum velocity under which no breakage of contacts was observed was purely empirical, since it ﬁtted better the simulation results but no physical interpretation was given to that new parameter or why it should appear in the expression of the Weber number.

2.2.2. Mechanistic analysis of the breakage of interparticle contacts Moreno-Atanasio and Ghadiri [26] analysed the breakage of interparticle contacts obtained from DEM simulations by using the Weber number. However, the analysis of their computer simulation results of impact breakage of agglomerates by using the Weber number was not successful. Figure 2 shows the damage ratio corresponding to the impact of four different agglomerates plotted as a function of We0 [26]. The data points correspond to the average damage ratio for the impact of four different agglomerates and the error bars correspond to the standard deviation of the data. When damage ratio was plotted vs. Weber number for surface energy values of 0.35, 3.5 and 35.0 J/m2 the different curves did not unify as reported in the work by other authors [2,12]. Therefore, a new model based on an energy balance was proposed to better explain the simulation results.

1.0 0.35 J/m2

Damage ratio

0.8

3.50 J/m2 35.0 J/m2

0.6 0.4 0.2 0.0 1E-6

1E-5

1E-4

1E-3 We'

0.01

0.1

1

Fig. 2. Relationship between damage ratio and modiﬁed Weber number, We0 . The data points correspond to the average damage ratio for the impact of four different agglomerates [26].

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844

The model development is based on the consideration that the work required to break interparticle contacts varies linearly with the incident kinetic energy [26]. Associated with each contact breakage is an amount of work to furnish the required surface energy and therefore the higher the incident energy the larger the number of broken contacts will be. The incident kinetic energy of agglomerates made of mono-size particles before impact, EK, is given by 1 E K ¼ N mV 2 2

ð6Þ

where N is the number of particles in the agglomerate, m the mass of a primary particle and V the impact velocity of the agglomerate. If the total number of broken contacts after impact is NB, the work for breaking these contacts, Wc, assuming that all contacts have the same contact area, Ac, is given by W c ¼ N B GAc

ð7Þ

Let us consider that the total work spent in breaking NB bonds is proportional by a factor k, to the incident kinetic energy of the agglomerate with k being the proportionality factor. 1 N B GAc ¼ kN mV 2 2

ð8Þ

Since the number of particles in the agglomerate, N, and the initial number of bonds, N0, are related through the coordination number [26], Z, damage ratio, DR, can be calculated as DR ¼

NB k mV 2 ¼ N 0 4Z GA

ð9Þ

If the contact area, A, between particles is estimated from the model of Johnson et al. [41] and the particle mass is substituted as a function of the particle density, an expression for damage ratio can be obtained in the form DR ¼ k

25=3

1 1 rD5=3 V 2 2=3 E 37=3 p2=3 Z ð1 n2 Þ2=3 G5=3

ð10Þ

Now considering the terms in equation (10), it is possible to deﬁne a new dimensionless number, D, as given by equation (11), incorporating particle density, particle diameter, elastic modulus and interface energy. D¼

rD5=3 E 2=3 V 2 G5=3

ð11Þ

Therefore the damage ratio, DR, is given by DR a We I 2=3 e

ð12Þ

Analysis of Agglomerate Breakage

845

where We is the Weber number and Ie the elastic adhesion index which is deﬁned as Ie ¼

ED G

ð13Þ

The hypothesis that the incident kinetic energy varies linearly with the work to break contacts has been shown to be true for agglomerates simulated by using DEM and containing between 500 and 10,000 particles and impact velocities between the elastic regime (o0.1 m/s) until the full disintegration of the agglomerates (5 m/s) [18,26]. In addition, the dependency of the damage ratio with the exponent 5/3 of the surface energy has also been tested by using computer simulations. Figure 3 shows the damage ratio plotted as a function of the new dimensionless group, D, for the impact of three different agglomerates with the values of surface energy of 0.35, 3.5 and 35 J/m2. Each data point are the average of four impacts and the error bars correspond to the standard deviation. The results can be compared with Fig. 2, where damage ratio was plotted vs. the modiﬁed Weber number [42]. It is clear that the curves corresponding to different values of surface energy become more uniﬁed when they are plotted using the new dimensionless group, as compared to the We0 .

2.2.3. Chipping model Ghadiri and Zhang [13] developed a model of attrition due to chipping for semibrittle materials based on the propagation of sub-surface lateral cracks. They

1.0 0.35 J/m2

Damage ratio

0.8

3.50 J/m2 35.0 J/m2

0.6 0.4 0.2 0.0 0.01

0.1

1

10 We

100

1000

1E4

(ED/Γ2/3)

Fig. 3. Relationship between damage ratio and new dimensionless group, D, for different values of surface energy. The data points correspond to the average damage ratio for the impact of four different agglomerates [26].

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846

estimated the extent of breakage, x, as the ratio of the volume of debris detached from the particles to the total volume and showed that this is given by x¼a

rV 2 lH K 2C

ð14Þ

where r is the density of the material, V the impact velocity, l the characteristic length of the system and a the proportionality factor which depends on particle shape and impact geometry and is determined experimentally. Ghadiri [43] showed that the extent of breakage expressed as the fractional loss per impact, x, is in fact related to the Weber number, We, since the relationship between fracture toughness KC, hardness, H and elastic modulus, E, is provided by the linear elastic fracture mechanics in the form K 2C ¼ 2EGð1 n2 Þ

ð15Þ

Therefore, the fractional mass loss per impact can be written in the following form, which is consistent with the phenomenological model of Moreno-Atanasio and Ghadiri for agglomerate breakage [26]. H x / We ð16Þ E The dependency of the extent of breakage on impact velocity and particle size was successfully veriﬁed by Zhang and Ghadiri [44] for MgO, NaCl and KCl. However, for agglomerates the power index of impact velocity is lower than two in some cases, but the reason for this is unclear. The knowledge of the power index of the impact velocity would give us an insight of the dissipation mechanism of agglomerates.

3. DISTINCT ELEMENT METHOD 3.1. Introduction The DEM, ﬁrst developed by Cundall [27], presents an alternative way to obtain an insight for particulate systems and provides fundamental information such as microscopic structure, interparticle forces, particle velocities, etc. Most importantly, this method makes it possible to relate the bulk mechanical behaviour of the assembly to individual particle properties. The DEM has been applied to systems of large number of particles subjected to different mechanical processes such as compression, impact, milling and shear [16,20,45,46]. A particular advantage of DEM is that a detailed examination of the micromechanics of the system, which determines the bond breakage and the internal microstructural information, can be made.

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The DEM works by cyclically updating the forces, accelerations, velocities and positions of the particles [27]. The interparticle force increments are calculated at all contacts from the relative velocities of the contacting particles using an incremental force-displacement law. The interparticle forces are updated and the new out-of-balance force and moment of each particle and new particle accelerations (both linear and rotational) are calculated using Newton’s second law. Numerical integration of the accelerations is then performed over the time step to give new particle velocities. Further, numerical integration provides displacement increments from which the new particle coordinates are obtained. Having obtained new positions and velocities for all the particles, the program repeats the cycle of updating contact forces and particle locations. Checks are incorporated to identify new contacts and contacts that no longer exist. In each cycle, every particle in the assembly is treated in the manner described above and the calculation cycle is repeated until the end of simulation [27]. The original code BALL was tested comparing the network of forces formed in a 2-D bed of particles with the results of photoelasticity experiments of De Joselin De Jong [42]. The comparison was purely qualitative but the good similarity between the network of forces in the simulation and the experiments was an indication of the success of the new method. However, the differences in force propagation and mechanical resistance between 2- and 3-D systems can be quite large. For this reason, the computer code BALL was modiﬁed to model a 3-D system and its name was changed to TRUBAL (meaning ‘‘true ball’’). Afterwards the code was further developed by Thornton and co-workers [28–31], where nonlinear contact deformations and adhesion were incorporated into the code and applied to the analysis of agglomerate strength. This code also incorporates the effects of friction and account energy dissipation by damping between particles. The analysis described in the following section is focused on the impact damage of agglomerates as carried out by computer simulations using DEM due to their technological importance and also the difﬁculty of using other approaches in analysing agglomerate breakage, e.g., continuum mechanics as discussed previously.

3.2. Agglomerate behaviour using distinct element method The impact process may be divided into three stages: the ﬁrst one is the compression of the agglomerate or loading stage, the second is the unloading of the agglomerate and the third is either the rebound or the deposition of the agglomerate on the target. Figure 4a and b shows the evolution of the force

M. Ghadiri et al.

848

0.01 m/s

0.5

0.05 m/s 0.1 m/s

0.4 Force (N)

0.5 m/s 2.0 m/s

0.3

4.0 m/s 0.2 0.1 0.0 0

10

(a)

20 Time (μs)

30

0.01 m/s 0.05 m/s 0.1 m/s 0.5 m/s 2.0 m/s 4.0 m/s

1.0 kinetic energy (Ekf /Eko)

40

0.8 0.6 0.4 0.2 0.0 0

(b)

10

20 Time (μs)

30

40

Fig. 4. (a) Evolution during impact of the force exerted on the target and (b) evolution of the kinetic energy during impact [18].

exerted by the agglomerate on the target (Fig. 4a) and the evolution of the kinetic energy during impact (Fig. 4b) for different values of the impact velocity. The force on the wall passes through a maximum after which the unloading stage of the agglomerate starts [6,8,18]. During the unloading stage, a part of the energy transmitted to the wall during loading is transferred back to the agglomerate, and this produces a decrease in the force exerted on the wall. The kinetic energy of the agglomerate is, then, recovered partially. The amount of energy recovered depends on the damage produced on the assembly and, therefore, on the impact velocity [6,18]. During the impact process, depending on the impact velocity, agglomerates can suffer damage which can be classiﬁed as microscopic or macroscopic. Microscopic damage is the breakage of contacts which is not necessarily linked

Analysis of Agglomerate Breakage

849

to macroscopic damage. Macroscopic damage is characterised by crack propagation and the detachment of fragments of different sizes from the agglomerate.

3.2.1. Agglomerate damage The study of agglomerate damage can be divided into two parts related to the breakage of contacts (microscopic) and to the fragmentation and detachment of particles of the agglomerate (macroscopic), respectively. The breakage of contacts can be studied using the damage ratio, which represents the number of broken bonds. However, this process does not reﬂect the formation of debris and fragments. The latter can be analysed by characterising the size distribution of the fragments [16,18,20,47].

3.2.1.1. Analysis of breakage of contacts Kafui and Thornton [2] expressed the extent of damage as the fraction of initial primary particle contacts that were broken as a result of impact. The number of interparticle bonds in agglomerate may be expected to reﬂect the bulk impact strength of the agglomerate and therefore the damage ratio would quantify the deterioration in the bulk strength of agglomerates as a result of an impact. For any agglomerate there is a threshold velocity, below which no damage, i.e. no bond breakage is observed and the agglomerate behaves in an elastic way. Above this minimum velocity, breakage of interparticle bonds is observed and it is usually near the impact site. This minimum velocity is intuitively related to the minimum energy required to break a single contact and this in turn is related to the surface energy. It is also expected that factors such as interparticle and particle-wall damping, friction coefﬁcient and local arrangement of particles around the impact site inﬂuence this minimum velocity. Thornton et al. [6] developed a correlation for the minimum velocity for a regular 2-D agglomerate with the surface energy values between 0.3 and 3.0 J/m2 as given by V 0 ¼ 0:0025 expð0:49GÞ

ð17Þ

In a later work, Kafui and Thornton [9] showed that the threshold velocity varies as a function of the interface energy G, according to the following form for a 3-D structured agglomerate with a range of surface energies between 0.2 and 4.0 J/m2. V 0 ¼ 0:17G1:5

ð18Þ

However, Moreno [18] found that the minimum velocity under which no damage was observed followed the relationship for values of surface energy between 0.35 and 35.0 J/m2 in the form V 0 ¼ 0:0095G0:81

ð19Þ

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Important differences between the three pieces of work are found which would make it possible to understand the origin of the different expressions. In the case of Thornton et al. [6], the agglomerates were 2-D structures; in the case of Kafui and Thornton [9] the agglomerate had a 3-D regular packing, whilst in the case of Moreno [18] the agglomerates had a random distribution of particles. In addition, in the ﬁrst two cases, only one agglomerate was used in the simulations, however in the latter case [18] four different agglomerates were tested to obtain a mean value of damage ratio for every impact velocity. In addition, three different values of surface energy spanning two orders of magnitude. Other differences such as porosity, or physical properties of the primary particles might be of importance to explain the differences between the three expressions. The Weber number and the modiﬁed Weber number as described previously in the previous section (equations (3) and (5)) has been used extensively to describe the agglomerate breakage. Kafui and Thornton [2] ﬁtted their damage ratio to a relationship in the form D / Web=2

ð20Þ

with the values of b of 0.131 and 0.250 for two-face centred cubic regular packings with 20 and 40 mm primary particle size but having the same agglomerate diameter. This clearly indicates that there are other factors such as the ratio of agglomerate size to particle diameter that are not considered when the Weber number is taken into account. Furthermore, using equation (20), Kafui and Thornton [2] compared the results of impact breakage of agglomerates with different packing fractions. The agglomerate with a lower packing fraction was found to be more resistant to damage as it could accommodate the transmitted energy more easily by microstructural plastic deformation. As the impact velocity is increased the damage ratio approaches unity asymptotically [7,18]. This clearly implies that a power law such as the one given in equation (20) is not suitable to ﬁt the full range of impact velocities. In addition, Subero et al. [7] found in their study that not only damage ratio but also the mass fraction of debris approaches an asymptotic value. Moreno and Ghadiri [24] proposed a new dimensionless group to describe the damage ratio since the Weber number was not useful in explaining the scaling of the damage ratio with surface energy for their simulations. The new dimensionless group also provides a dependency on the square of the impact velocity, and therefore on the incident kinetic energy of the agglomerate. However, this dependency does not hold true for high impact velocities. Despite extensive efforts made so far, there is still no mathematical relationship between the damage ratio and agglomerate properties for all impact test conditions (such as target mechanical properties) and physical and mechanical properties of the primary particles (such as elastic modulus, particle shape and density, for example).

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3.2.1.2. Fragmentation and breakage Microstructural damage or breakage of interparticle bonds leads to the detachment of particles from the agglomerate only when the sequence of broken bonds produces a cluster, which is completely detached from the agglomerate. The analysis of agglomerate breakage is usually carried out as a function of the impact velocity since the agglomerate behaviour varies from a purely elastic response where the agglomerate does not suffer from any type of damage to a complete disintegration into small pieces at high impact velocities. Figure 5a and b shows the normalised sizes of the two largest agglomerates as a function of the impact velocity for two different cases of surface energy (3.5 and 35.0 J/m2) [22]. In both cases, three regimes of fragmentation can be distinguished, which correspond to different values of impact velocity. However, the values of impact

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Fig. 5. Regimes of breakage for one agglomerates whose value of surface energy is (a) 3.5 J/m2 and (b) 35.0 J/m2 [22].

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velocities corresponding to these regimes depend on agglomerate properties. Therefore, when the terms low, medium or high impact velocities are used they are relative since for agglomerates with different properties these terms will correspond to different values of velocity. These regimes are Regime I (low impact velocity): small clusters are detached from the agglom-

erate and the size of the largest fragment shows a weak dependency on the impact velocity. Regime II (medium impact velocity): this is characterised by a fast decrease of the size of the residual fragment. Regime III (high impact velocity): the size of the largest fragment is much smaller than the initial agglomerate size and varies slowly with impact velocity. Regime I: low impact velocity: At low impact velocities, the agglomerate breakage is characterised by a slow decrease in agglomerate size. During this regime, the second largest cluster is less than 5% of the initial size of the agglomerate as shown in Fig. 5a and b, and the number of fragments detached from the agglomerate is very small as indicated by the large value of the largest cluster size. The small size of the second largest fragment after impact implies that the fragments detached from the agglomerate are really debris. This type of behaviour was ﬁrst described by Thornton et al. [9] and later reported as well by Subero et al. [7,11] and Moreno et al. [16,18,22]. Within this regime no large differences appear between the works reported by different authors [7,9,11,16,18,22]. Regime II: Intermediate impact velocities: Within this regime, the size of the residual cluster is very sensitive to the impact velocity (Fig. 5a and b). The size of the second largest cluster increases and passes through a maximum. The maximum in the curves corresponds to the fragmentation of the agglomerates into two fragments (Fig. 5a and b). This fragmentation is shown in Fig. 6 where the top and bottom views of four agglomerates impacted at velocities corresponding to the maximum of the second largest fragment are visualised. The four agglomerates are made of primary particles of 50 mm in radius, 31 GPa elastic modulus and 2000 kg/m3 particle density. The fragments are colour coded according to their sizes. This type of behaviour is in agreement with the work of Thornton et al. [9], who also observed the fracture of the agglomerate at relative intermediate velocities for 2-D agglomerates. However, neither Ning et al. [5] nor Subero et al. [7,11] observed fragmentation of agglomerates for any given value of impact velocity. In the case of Ning [5], the behaviour of weak lactose agglomerate was analysed by DEM. The agglomerates seemed to fail macroscopically in a ductile mode, i.e. extensive plastic deformation without any crack propagation as shown in Fig. 7. In addition, Subero et al. [7] simulated the impact breakage of agglomerates made of glass ballotini. They successfully quantiﬁed the increase in breakage of

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Fig. 6. Top view of four different agglomerates impacted at the velocities corresponding of the maximum of the second largest fragment as shown in Fig. 5a and b. Colour coding: light grey, largest fragment; red, second largest fragment; yellow, third largest fragment; green, clusters smaller than clusters in yellow and larger than 100 particles; cyan clusters between 4 and 100 particles; pink, doublets; blue, singlets [18].

Fig 7. Disintegration of weak lactose agglomerates Ning et al. [5].

the agglomerate with the impact velocity; however their simulated agglomerates did not fragment at all. Subero [7] argued that for an agglomerate to fracture, it is necessary for its structure to store sufﬁcient elastic strain energy required for crack propagation. It appears that the agglomerates of Subero [7] could not do this. However, neither Subero et al. [7] nor Ning [5] provided any explanation for the lack of crack propagation in their agglomerates.

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In a later work, Moreno and Ghadiri [18,22] showed that the effect of interface energy could explain the ﬁndings of Ning et al. [5] and partially explain the ﬁndings of Subero [7]. Their analysis of the effect of interface energy on the breakage of agglomerates [18,22] clearly showed that agglomerates with low surface energy (0.35 J/m2) do not fragment at any value of impact velocity and undergo extensive deformation accompanied by disintegration into small clusters. This behaviour was very similar to the ductile behaviour of continuum solids. In contrast, for agglomerates in which the particles were joined by a value of the surface energy of 3.5 J/m2 or higher, crack propagation was observed accompanied by the disintegration of a local region around the impact site into small clusters. This behaviour was more similar to the behaviour of semi-brittle continuum materials. Therefore, agglomerates seem to have a transition in their mode of failure when the surface energy is increased. The effect of surface energy can be clearly observed in Fig. 8 where the same agglomerate was impacted at values of impact velocities corresponding to the regime II and with different values of surface energy of the primary particles. Since Ning et al. [5] used a value of surface energy of 0.5 J/m2 the differences between the work of Ning et al. [5], and the work of Thornton et al. [9] and Moreno et al. [18,22] could be attributed to the effect of the surface energy. Subero et al. [7] used values of 0.5, 2.0 and 5.0 J/m2 and therefore the lack of fragmentation agglomerates reported in their work cannot be explained based on the

Fig. 8. Effect of the interface energy on the breakage pattern of agglomerates during regime II of breakage [22]. Colour coding: white, target; blue, singlets, pink, doublets; cyan, 4–100 particles; green, 4th largest fragment; yellow, 3rd largest fragment; red, 2nd largest fragment; grey, residual fragment.

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considerations of energy alone. Mishra and Thornton [10] analysed the failure pattern of agglomerates as a function of the packing fraction using agglomerates made of spheres with properties of glass ballotini. They found that for values of packing fraction less than 0.537 agglomerates did not fragment. At high values of packing fraction (0.602 or higher) their agglomerates fragmented into several pieces of comparable sizes. Finally, at the intermediate values of packing fraction (around 0.537) the agglomerates showed different behaviours when impacted in different orientations; in some orientations the agglomerates fragmented and in others they did not fragment and incurred local disintegration around the impact site. The agglomerates of Subero et al. [7] have a solid fraction of 0.565, which would place them in the intermediate regime. However, although a direct comparison is not feasible since the agglomerates were made of primary particles with different physical properties, in principle, the value of packing fraction could explain the lack of fragmentation of the agglomerates of Subero et al. [7]. Another factor that could inﬂuence the agglomerate strength and mechanical properties is agglomerate size. Moreno [18] found that small agglomerates made of 500 particles do not fragment for certain impact orientations and for any given value of impact velocity. In this case, the second largest cluster was always less than 10% of the initial agglomerate size. However, the largest agglomerate tested by Moreno [18] fragmented into 2, 3 or 4 pieces of similar sizes depending on the impact orientation. These differences appear exclusively due to agglomerate size since packing fraction, coordination number and physical properties of the primary particles were kept constant. Moreno et al. [16] analysed the oblique impact of agglomerates and found that the number of broken contacts decreases as the impact angle is reduced. However, the number of broken contacts in the agglomerate was roughly the same for all impact angles when the normal component of the impact velocity was kept constant. The tangential component of the impact velocity seems only to inﬂuence the location of the broken contacts and not their number as shown in Fig. 9. These observations have also been corroborated later by Behera et al. [48]. When the impact angle is 901, the broken contacts seem to be more uniformly spread through the agglomerate showing symmetry with respect to the perpendicular to the wall (Fig. 9). However, at 301 impact the broken contacts are more localised on one side of the agglomerates (the side from which the agglomerate moves to impact the wall). The pattern of breakage for 301, 451 and 901 is shown in Fig. 10. The number of broken contacts is approximately the same, however, the breakage pattern is completely different: At 451 the agglomerate fragments and at 301 no fragmentation is observed (Moreno et al. [16]). The experimental work of Samimi et al. [24] also concluded that the normal component of the impact is the main factor inﬂuencing the breakage of agglomerates within the chipping regime. However, when the agglomerate failure is in

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Fig. 9. Pattern of broken contacts at 301 and 901 impact angle. The damage ratio is 0.292 for both angles 301 and 901 [16].

Fig. 10. Inﬂuence of the impact angle on the breakage patterns of agglomerates.

the fragmentation regime, the tangential component of the impact velocity inﬂuences the breakage pattern. This can be explained by the ﬁndings of Moreno et al. [16] suggesting that a change of location of broken contacts produces a change in the failure pattern. However, the differences in the type of bonds

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and materials used by Samimi et al. [24] and Moreno et al. [16] require further analysis. So far, the breakage of agglomerates has been mainly related to the interface energy, packing fraction, agglomerate size and location of broken contacts. However, there is no systematic study on the effects of other parameters such as the elastic modulus or density. This is clearly needed to fully map out the mode of failure of agglomerates as a function of all parameters that inﬂuence agglomerate behaviour. Regime III: High impact velocities: In this regime, the input kinetic energy that is not dissipated is sufﬁcient to break most of the contacts of the agglomerate and the agglomerate behaviour is clearly characterised by the disintegration of the agglomerate into small fragments as suggested by Fig. 5a and b. However, there are differences in the mode of failure of agglomerates within regime III arising from different values of surface energy of the primary particles. Figure 11a and b shows the impact of an agglomerate with the same structure [18] but for two different values of surface energy (0.35 and 3.5 J/m2) impacted at 2.0 and 25 m/s, respectively. At the lowest value of the surface energy (0.35 J/m2) the agglomerate is shattered into small clusters and the largest fragment is deposited on the target (Fig. 11a). However, at higher surface energies (35.0 J/m2) the formation of clusters and large fragments away from the impact site is clearly observed as can be seen in Fig. 11b. Regime III of breakage occurs at much higher impact velocities for the case of 35.0 J/m2 than for the case of smaller surface energy (0.35 J/m2). The higher

Fig. 11. Agglomerate impact at different velocities of (a) 2.0 m/s and (b) 25 m/s during the third regime of breakage for two different values of surface energy (a) 0.35 J/m2 and (b) 35.0 J/m2. Colour coding: light grey, largest fragment; red, second largest fragment, yellow third largest fragment; green, fourth largest cluster; cyan clusters between 4 and 100 particles; pink, doublets; blue, singlets [22].

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residual kinetic energy leads to the fragments ﬂying away from the impact site (Fig. 11b). In addition, for the case of low surface energy, the agglomerates have disintegrated into small clusters made of a few particles. The volume of agglomerate disintegrated increases with impact velocity. However, for large values of surface energy the disintegrated agglomerates seem to be formed by fragments with straight contours suggesting that multiple fragmentation rather than disintegration is the origin of this structure.

3.2.1.3. Breakage patterns Some DEM and experimental studies have established qualitative relationships between some of the characteristics of agglomerates and their mode of breakage [10–12,16,18,22,49]. The features of the various work reported in the literature are summarised below. The experimental work of Subero and Ghadiri [12] for impact of agglomerates made of glass ballotini that related the impact velocity, porosity and breakage pattern in a systematic way for the ﬁrst time. It is interesting to compare the breakage pattern found experimentally by Subero and Ghadiri [12] with the breakage pattern obtained by DEM. The most important feature is that they found an increase in the frequency of fragmentation with impact velocity. The term ‘‘frequency’’ is used due to the fact that Subero and Ghadiri [12] found that some agglomerates did not fragment at all whilst others exhibited multiple fragmentations for the same range of impact velocities. Figure 12 shows the fragmentation histogram published by Subero and Ghadiri [12]. This feature could be further 0.9 U = 2 m/s

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Fig. 12. Frequency of fragmentation of agglomerates as a function of the artiﬁcial porosity [12].

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explained by the work of Mishra and Thornton [10] who showed that within a certain range of porosities agglomerates show large sensitivity to the impact site and they do not always fragment. The breakage patterns reported by Mishra and Thornton [10] based on DEM and the experimental results of Subero and Ghadiri [12] are relatively in good agreement, although a direct comparison is not feasible due to differences in particle properties and bond characteristics. These patterns can be summarised in four different types, which appear depending on the impact velocity and are plotted in Fig. 13. The ﬁrst is characterised by local damage at the impact site, the second by local damage and oblique fracture and the third by local damage and median cracks with or without secondary fragmentation and the fourth by multiple fragmentation. Some of the fragmentation patterns observed by Subero and Ghadiri [12] have also been reported by DEM simulations of Moreno and Ghadiri [18,22] for intermediate impact velocities, i.e. agglomerates fragmented only with local damage (Fig. 13a) or with local damage and detachment of side platelets Local damage only

(a) Impact site disintegrates into small debris (side view). Local damage + meridian fracture

(c) Median crack leading to fragmentation into large clusters (side view).

Local damage + oblique fracture

(b) Oblique cracks leading to the detachment of side fragments Multiple fragmentation

(d) Secondary fragmentation (top view)

Fig. 13. Experimental pattern of breakage of agglomerates by Subero and Ghadiri [12]. Patterns (a), (b) and (c) were also found by Moreno et al. [16,18,22].

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(Fig. 13b). When the impact velocity was increased, the presence of median cracks splitting the residual fragments into two large pieces was also observed (Fig. 13c). However, it was observed that the agglomerate size has an inﬂuence. Some of the smallest agglomerates tested by Moreno [18] made of 500 particles of 50 mm in size did not show fragmentation for certain impact orientations and for all impact velocities. When the agglomerates were made of 3000 particles the residual fragment divided into two large pieces. For agglomerates of 10,000 particles, the residual fragment divided into two, three or four pieces and they were always accompanied by local disintegration of the contact area and, very frequently, by the detachment of side platelets. These ﬁndings are summarised in Fig. 14. The increase in agglomerate size is associated with a tendency of dividing into a large number of pieces and there is an increase in the similarity between the mode of failure of continuum solids and agglomerates. It seems that the increase in agglomerate size with respect to their single-particle components shows a reduction in the importance of the discrete nature of the material. As more paths for the forces to propagate through the agglomerates are available and therefore the material behaves more as a continuum solid. This would explain, for example, the work of Schubert et al. [49] where the failure of cement agglomerates were simulated in 2-D. In this work, the authors observed experimentally and by DEM the same type of breakage pattern characterised by a pulverised cone of materials with median cracks and secondary fragmentation. This breakage pattern is schematically shown in Fig. 15. This type of pattern have a great similarity with

10000 particles

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Fig. 14. Schematical top view of the different breakage patterns found by Moreno [18]. Size effect on the fragmentation pattern of spherical agglomerates. The tendency to fragment into a larger number of fragments decreases with particle size.

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Radial cracks

Crush cone

Fig. 15. Fragmentation pattern found by Schubert et al. [49]. The pattern is characterised by a crushed cone of material with the rest of the agglomerate split by radial and median cracks [49].

the work of Arbiter [50] who showed that for sand cement agglomerates a compressed cone of material covered by a layer of debris is formed with radial cracks departing from the conical region. This similarity is also found in the work of Salman et al. [51] for continuum solid particles. The breakage pattern observed by Schubert et al. [49] has not been reported previously in any simulation work based on DEM. This is probably due to the high packing (70% volume fraction) of the agglomerates of Schubert et al. [49], which makes the agglomerates to have similar characteristics to a continuum solids than to any other agglomerates simulated by using DEM until now. With the development of computer speed, we are closer to being able to perform realistic comparisons between simulations and experiments. These comparisons have to be made not only at the agglomerate level (structure, porosity) but also at a single particle and bulk levels. This is going to require ﬁrst of all the use of realistic force–deformation relationships and the use of realistic bond characteristics (solid bridges, van der Waals, capillary or electrostatic forces) and also an adequate input of particle characteristics, such as shape and roughness. In conclusion, the results discussed above on the impact breakage characteristics of agglomerates under impact loading show that, factors such as surface energy, packing fraction, agglomerate size and impact angle can signiﬁcantly affect the mode of failure of agglomerates. The variation in the value of surface assigned at particle scale resulted in signiﬁcant changes on the mode of failure of agglomerates under impact loading – an increase in surface energy could change the mode of failure from ductile to semi-brittle failure [22]. Agglomerates showing high values of packing fraction (40.7) show modes of failure similar to those of solid particles [49]. For agglomerates with lower values of packing fraction, the mode of failure depends on both the impact orientation and the initial number of interparticle contacts [10]. When all other properties remain the same, the tendency of fragmentation of an agglomerate into a larger number of clusters increases with agglomerate size making the mode of failure of agglomerates, again, more similar to those of a solid particles [18]. When the impact angle is varied, whilst keeping the normal component of the impact velocity constant, the

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change in the tangential component of the impact velocity alters the mode of failure of agglomerates [16]. Nevertheless, despite the numerous amounts of research reported earlier reveal the inﬂuence of some of the single-particle properties on the impact breakage characteristics of agglomerates, further research is needed to study the inﬂuence of many other factors such as the particle density, the ratio of agglomerate diameter to single-particle diameter and the elastic modulus of primary particles.

4. AGGLOMERATE BEHAVIOUR IN A BED OF PARTICLES SUBJECTED TO SHEARING Agglomerates may deform and break within a bed of particles during shearing. This process commonly occurs in high shear granulators in process industries where the powder is under intense agitation and large forces are transmitted into the powder bed [52,53]. The existence of high level of stresses together with strong and rapid agitation would result in shorter processing time and provide the ability to produce granules with a higher strength. These features have made high shear mixer granulators attractive particularly in the pharmaceutical and detergent industries. It is widely recognised that, in addition to powder properties and binder characteristics, the properties of agglomerates formed by powder granulation strongly depend on the level of prevailing shear stresses as experienced by the powder bed inside the granulators, as deformation and breakage of agglomerates within granulators is predominantly affected by the level of shear stresses [54–56]. It is therefore necessary to understand the relationship between the properties of the agglomerates and nature and level of stresses that they experience. However, it is difﬁcult to measure or quantify the internal stresses experimentally within the bulk powder. As for the single agglomerates addressed in the previous studies, an appropriate approach for this purpose is the use of computer simulation technique by DEM.

4.1. Effect of size ratio on the breakage of agglomerate Experimental studies show that small particles within a bed under oedometric compression break more readily than large particles [57–59]. There is a cut-off limit of relative particle size, beyond which it is very difﬁcult to break the larger particles when they are surrounded by smaller particles and subjected to mechanical loading. Antony and Ghadiri [60] performed a detailed stress analysis of a large spherical inclusion inside a cubical periodic assembly subjected to quasi-static shear deformation. The average stress tensor of the inclusions was

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resolved into two components, viz. hydrostatic and deviatoric stresses. They showed that for particles having a size ratio greater than about 5, the nature of average stress on the inclusion was predominantly hydrostatic, as the ratio between the deviatoric and hydrostatic components was about 0.2 at the steadystate shearing regime. They suggested that the predominant nature of the hydrostatic (isotropic) stresses on large particles could retard their breakage, whilst the presence of strongly deviatoric (anisotropic) stresses on small particles would induce their breakage. However, the study considered the large inclusions as a single particle. In practice, e.g., in granulators, the large size inclusion tends to be in agglomerate form. Recently, Hassanpour et al. [61] have studied the deformation and breakage behaviour of an agglomerate in a bed of particles subjected to shear deformation using DEM. The agglomerate was prepared following the procedure of Moreno et al. [16] and hence its initial state of structure was isotropic and homogenous. Inclusions with several size ratios were generated and agglomerate behaviour was studied under shearing.

4.1.1. Stress ratio Hassanpour et al. [61] generated four different assemblies of non-adhesive particles with inclusion size ratios in the range 3–10 (the ratio between the diameters of agglomerate (D) and surrounding particles, d as shown in Fig. 16). The assemblies were then sheared at a constant normal pressure of 1 MPa (Fig. 17). The shear rate was about 300 s1. The stress tensor for the agglomerate was partitioned into the hydrostatic and deviatoric components [60] and the evolution of the stress ratio of the agglomerate (deviatoric/hydrostatic stresses) was

Fig. 16. The agglomerate within an assembly of primary particles [61].

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tracked down. The stress ratio is given by pﬃﬃﬃ 2ðsyy sxx Þ tD ¼ Stress ratio ¼ ðsxx þ syy þ szz Þ p

ð21Þ

where, tD is the deviatoric stress, p the hydrostatic stress and sxx, syy and szz correspond to the stresses on the agglomerate along directions shown in Fig. 17. As shown by Hassanpour et al. [61], the stress ratio with shear strain experienced by the agglomerate within assembly is different for cases of different size ratios (Fig. 18). The stress ratio attains a value of about 0.2 for the assembly with size ratio 10, which is the same as reported previously by Antony and Ghadiri [60] for a single large spherical inclusion in granular media subjected to slow shearing. This means that stresses on the agglomerate are predominantly hydrostatic

Stress ratio

Fig. 17. Sectional view of the shear box used in the simulation [61].

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Fig. 19. The maximum stress ratio [61] and the steady-state stress ratio [60] on the agglomerate within the assemblies with different size ratios.

(stress ratio less than 0.2). However, for the case of assemblies with lower size ratios the stress ratio is higher than that of size ratio 10. Hassanpour et al. [61] evaluated the maximum stress ratios during shearing as a function of the size ratio. The results are shown in Fig. 19. It can be seen that the stress ratio increases as the size ratio of the agglomerate is decreased, implying that the state of the stresses on the agglomerate becomes predominantly deviatoric, as the size ratio decreases. This agrees well with the results of Antony and Ghadiri [60], though the stress ratio was reported at a steady-state value. There is a transition at the size ratio of about 5, below which the stresses on the agglomerate are highly deviatoric (stress ratio 0.6).

4.1.2. Damage ratio Hassanpour et al. [61] observed that the agglomerate in a particle bed never fractured during the shear test when the size ratio of assembly was 10. However for the cases of size ratio smaller than 10, agglomerates underwent macroscopic breakage. They have calculated the damage ratio of agglomerate, at a particular time for assemblies with different size ratios (Fig. 20). It was observed that the damage ratio for the cases of size ratio 10 is lower than the other cases, and a value of 0.3 is obtained. It is interesting to note that despite some contact rearrangements within the agglomerate during shearing, the agglomerate does not loose its structural integrity at the end of the test. As it can be seen from Fig. 20, the damage ratio for the agglomerate with size ratio 7 is higher than that of size ratio 10, and the former experiences breakage when the damage ratio reaches 0.5 (corresponding to the shear strain of 0.13). For the case of assembly with size ratio 5, the damage ratio increases with shear

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Fig. 20. The evolution of stress ratio with strain during shearing [61].

strain to the value of 0.57, beyond which the agglomerate undergoes a macroscopic breakage (corresponding to the shear strain of 0.095). By increasing the shear strain the damage ratio increases to a maximum value of 0.74 and after this point the agglomerate disintegrates into smaller clusters (corresponding to the strain of 0.28). For the case of assembly with size ratio 3, a distinctive breakage occurs after the damage ratio reaches 0.49 (corresponding to the shear strain of 0.055). During shearing, the agglomerate damage ratio increases with shear strain to a maximum value of 0.75 and beyond this point the agglomerate disintegrates into smaller clusters similar to the case of assembly with size ratio 5. The agglomerates in the assemblies with different size ratios (when the ﬁrst sign of macroscopic breakage is observed during shearing) are plotted in Fig. 21. In this ﬁgure, different colours represent the clusters detached from the agglomerate. It can be seen from Fig. 21 that unlike the case of the assembly with size ratio 10, a distinctive level of breakage of the agglomerates is observed where their size ratio is equal or smaller than 7. Hassanpour et al. [61] report a stress ratio of about 0.4 that causes macroscopic breakage in the agglomerate for all assemblies with size ratios 3–7 (Figs. 18 and 20). This can be regarded as a failure criterion for agglomerates embedded in a bed subjected to shearing for the particular assembly under consideration.

4.2. Comparison with experiments Hassanpour et al. [61] carried out shear tests on real agglomerates made of calcium carbonate particles and polyethylene glycol (PEG) binder (applied as an

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0 0 .0 0. .00 018 0 0 0 16 0 .00 14 0. .00 12 0. 00 10 00 0 06 8

0.0016

0.0014 0.0018 0.0020 0.0022 0.0024 0.0026 0.0028 0.0030 0.0032

X Axis

Analysis of Agglomerate Breakage

0.0026 0.0024

X Axis

Assembly 1(size ratio 10)

Assembly 2 (size ratio 7) 0.0026

0.0024 0.0024 0.0022

Z Axis

Z Axis

0.0022 0.0020

0.0018

0.0018 0 0 .0 0 .0 02 0. .00 020 2 0 0 1 0 .0 01 8 0 .0 01 6 0 .0 01 4 0. .00 0102 00 0 06 8

YA

xis

0.0016 0.0014 0.0020

0.0020

0.0022

0.0024

0.0026

0.0028

0.0030

X Axis

Assembly 4 (size ratio 3)

Fig. 21. The agglomerate within an assembly with size ratios smaller than 10 after applying external stresses [61].

867

Assembly 3 (size ratio 5)

0.0032

0 02 8 YAx 0.0.001 16 4 is 0 .00 01 12 0 0 0.0 00 01 08 0. 0.0 .00 006 0 .0 0.0014 0 0.0018 0.0020 0.0022 0.0024 0.0026 0.0028 0.0030 0.0032 0.0034 X Axis 0.0016

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aqueous solution) to validate their simulation results. The granulation process was conducted in a Cyclomix high shear granulator, manufactured by Hosokawa Micron B.V. After the agglomeration process was completed, nearly spherical agglomerates were chosen under an optical microscope for shear experiments. Two different assemblies made of primary particles (calcium carbonate) with a size ratio of about 10 (Assembly A) and 3 (Assembly B) were prepared in a stainless-steel shear box. Here the two assemblies represent the two extreme cases of the simulation. The agglomerates were coloured to trace possible breakage. In each assembly four agglomerates were placed at different positions on the horizontal shear plane (mid level). Both the assemblies were loaded on the upper part of the box, which was then moved using a motor to shear the assemblies (similar to that of Fig. 17). After shearing, the assemblies were carefully examined to recover the agglomerate. The recovered agglomerate and the primary particles for both the assemblies are shown in Fig. 22. The agglomerates in Assembly A (Fig. 22a), where the size ratio was 10, have not suffered any breakage in agreement with those of the simulations. However in Assembly B (size ratio 3) some fragments are visible, while two agglomerates do not show any breakage (Fig. 22b). A possible explanation for the presence of undamaged agglomerates is their movement away from the shear band during normal compression. This could result in agglomerates being under less shear stress in a non-shearing region of the assembly. This, however, is not the case for the simulation as the agglomerate is exactly placed within the shear region of the assembly and its position monitored with time.

Fig. 22. The agglomerates within the Assembly A (size ratio 10) and Assembly B (size ratio 3) after shearing in a box shear cell [61].

Analysis of Agglomerate Breakage

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4.3. Relevance to granulation process An agglomerate breakage within a shearing bed of particles is clearly dependant on the size ratio. Hassanpour et al. [61] suggest that the stress ratio of 0.4 may be regarded as the failure criterion of an agglomerate within a shearing bed for the agglomerate considered. This criterion can be met for assemblies with size ratio equal or smaller than 7. However, the above failure criterion cannot be reached for the case of size ratios greater than about 7. This analysis would provide a direct application for the case of granulation process, where various stages, commonly known as nucleation, growth and breakage prevail [54,62–63]. The breakage of large agglomerates is a crucial stage before the completion of granulation processes, as it leads to the production of a desirable size distribution of agglomerates. The study by Hassanpour et al. [61] suggests that within a shearing mass inside granulators, the breakage of agglomerates having a size ratio larger than 7 would be difﬁcult due to the predominantly hydrostatic nature of stresses experienced by the agglomerates. This implies that larger agglomerates will have to be broken using an alternative process other than shearing the bed, e.g. using choppers or blades. However, it is worth pointing out that the study carried out by Hassanpour et al. [61] was limited to one single value of surface energy and the failure criterion depends on the magnitude of the surface energy [22,26,62]. In addition, spherical agglomerates were simulated, while agglomerates are not always in spherical form in high shear mixer granulators. Further, studies are in progress to examine the inﬂuence of surface energy and shape of agglomerate on the deformation behaviour.

Nomenclature Ac c D DR EK E Fc H Ie k Kc l m N NB

area of a contact (m2) crack length (m) particle diameter (m) damage ratio (–) incident kinetic energy of an agglomerate (J) elastic modulus (Pa) interparticle contact force (N) hardness (Pa) elastic adhesion index (–) proportionality constant in equation (8) (–) fracture toughness (Pa m1/2) characteristic length of a solid particle (m) particle mass (kg) number of particles in an agglomerate (–) number of broken contacts (–)

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N0 p V V0 Wc We We0 Z a b G GC g D e n r s sii tD f

initial number of bonds in an agglomerate (–) hydrostatic stress (Pa) particle velocity (m/s) velocity under which no contacts are broken in agglomerates (m/s) work for breaking one contact (J) Weber number (–) modiﬁed Weber number (–) coordination number (–) attrition propensity parameter (–) power law index in equation (20) (–) interface energy (J/m2) fracture energy (J/m2) surface energy (J/m2) dimensionless group in equation (11) (–) porosity (–) Poisson’s ratio (–) particle density (kg/m3) stress (Pa) components of the stress tensor (Pa) deviatoric stress (–) packing fraction (–)

REFERENCES [1] B.J. Ennis, J.D. Litster, in: R. Perry, D. Green (Eds.),Perry’s Chemical Engineers’ Handbook, 7th edition, McGRaw-Hill, New York, 2001, pp. 20–56 and 20–89. [2] K.D. Kafui and C. Thornton, Powders & Grains 93, the proceedings of the 2nd International Conference on Micromechanics of Granular Media, C. Thornton (Ed.), Balkema, Rotterdam, 1993, pp. 401–406. [3] A.D. Salman, D.A. Gorham, A. Verba, Wear 186–187 (1995) 92–98. [4] C. Thornton, D. Kafui, T. Ciomocos, IFPRI annual meeting, Urbana, US, 1995. [5] Z. Ning, R. Boereﬁjn, M. Ghadiri, C. Thornton, Adv. Powder Technol. 8 (1996) 15–37. [6] C. Thornton, K. Yin, M.J. Adams, J. Phys. D: Appl. Phys. 29 (1996) 424–435. [7] J. Subero, Z. Ning, M. Ghadiri, C. Thornton, Powder Technol. 105 (1999) 66–73. [8] C. Thornton, M.T. Ciomocos, M.J. Adams, Powder Technol. 105 (1999) 74–82. [9] K.D. Kafui, C. Thornton, Powder Technol. 109 (2000) 113–132. [10] B.K. Mishra, C. Thornton, Int. J. Miner. Process. 61 (2001) 225–239. [11] J. Subero, Ph.D. Dissertation. Impact Breakage of Agglomerates, Surrey University, 2001. [12] J. Subero, M. Ghadiri, Powder Technol. 120 (2001) 232–243. [13] M. Ghadiri, Z. Zhang, Chem. Eng. Sci. 57 (2002) 3659–3669. [14] K.D. Kafui, C. Thornton, M.J. Adams, Chem. Eng. Sci. 57 (2002) 2395–2410. [15] C. Thornton, M.T. Ciomocos, M.J. Adams, 10th Eur. Symp. Comminution, Heidelberg, Germany, 2002. [16] R. Moreno, S.J. Antony, M. Ghadiri, Powder Technol. 130 (2003) 132–137. [17] A.D. Salman, A.D. Gorham, A. Verba, Powder Technol. 130 (2003) 359–366. [18] R. Moreno, Ph.D. Thesis, University of Surrey. Guildford, UK, 2003.

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[19] D. Golchert, R. Moreno, M. Ghadiri, J. Litster, R. Williams, Adv. Powder Technol. 15 (2004) 447–457. [20] D. Golchert, R. Moreno, M. Ghadiri, J. Litster, Powder Technol. 143–144 (2004) 84–96. [21] M. Khanal, W. Schubert, J. Tomas, Granular Matter 5 (2004) 177–184. [22] R. Moreno, and M. Ghadiri, PARTECH 2004, Nuremberg. Germany, Paper 3.5, 2004. [23] M. Nishida, K. Tanaka, Y. Matsumoto, JSME Int. J. Series A 47 (3) (2004) 438–447. [24] A. Samimi, R. Moreno, M. Ghadiri, Powder Technol. 143–144 (2004) 97–109. [25] G.K. Reynolds, J.S. Fu, Y.S. Cheong, M.J. Hounslow, A.D. Salman, Chem. Eng. Sci. 60 (2005) 3969–3992. [26] R. Moreno-Atanasio, M. Ghadiri, Chem. Eng. Sci. 61 (2006) 2476–2481. [27] P.A. Cundall, O.D.L. Strack, Geotechnique 29 (1979) 47–65. [28] C. Thornton, C.W. Randall, Micromechanics of Granular Materials, Elsevier Science Publishers B.V., Amsterdam, 1988. [29] C. Thornton, J. Phys. D: Appl. Phys. 24 (1991) 1942–1946. [30] C. Thornton, K.K. Yin, Powder Technol. 65 (1991) 153–166. [31] C. Thornton, Z. Ning, Powder Technol. 99 (1998) 154–162. [32] F. A. Gilabert, A. M. Krivtsov, A. Castellanos, Meccanica, 41 (2006) 341–349. [33] A. Di Renzo, P. Di Maio, Chem. Eng. Sci. 60 (2005) 1303–1312. [34] H. Mio, A. Shimosaka, Y. Shirakawa, J. Hidaka, J. Chem. Eng. Japan 38 (2005) 969–975. [35] F.Y. Fraige, P. Langston, Adv. Powder Technol. 15 (2) (2004) 227–245. [36] D. Zhang, W.J. Witten, Powder Technol. 98 (1998) 223–230. [37] H. Rumpf, Proceedings of International Symposium on Agglomeration, W. A. Knepper (Ed.), Vol. 379, 1962. [38] K.J. Kendall, Powder Metall. 31 (1988) 28–31. [39] A.A. Grifﬁth, Philos. Trans. Roy. Soc. London A221 (1920) 163. [40] J.N. Israelachvili, Intermolecular and surface forces, Academic Press, London, Second Edition, 1985. [41] K.L. Johnson, K. Kendall, A.D. Roberts, Proc. R. Soc. London A. 324 (1971) 301–313. [42] G. De Josselin de Jong, A. Verrujit, Cah. Grpe fr. Etud. Rhe´ol. 2 (1969) 73–86. [43] M. Ghadiri, Powder Technology Handbook, K. Gotoh, H. Masuda, K. Higashitani (Eds.), 2nd edition, Marcel Dekker, New York, 1997, pp. 183–191. [44] Z. Zhang, M. Ghadiri, Chem. Eng. Sci. 57 (2002) 3671–3686. [45] C.C. Kwan, H. Mio, Y.Q. Chen, Y. Ding, F. Saito, D.G. Papadopoulos, A.C. Bentham, M. Ghadiri, Chem. Eng. Sci. 60 (5) (2005) 1441–1448. [46] R. Moreno-Atanasio, S.J. Antony, M. Ghadiri, Powder Technol. 158 (2005) 51–57. [47] M. Ghadiri, S. J. Antony, R. Moreno, Z. Ning, Development in Handling and Processing Technologies, W. Hoyle (Ed.), Royal Society of Chemistry, London, 2001, pp. 70–81. [48] B. Behera, F. Kun, S. McNamara, H.J. Herrmann, J. Phys.-Condens. Matter. 17 (24) (2005) S2439–S2456. [49] W. Schubert, M. Khanal, J. Tomas, Int. J. Miner. Process. 75 (2005) 41–52. [50] N. Arbiter, C.C. Harris, G.A. Stamboltzis, Trans. AIME 244 (1969) 118–133. [51] A.D. Salman, G. K Reynolds, J.S. Fu, Y.S. Cheong, C.A. Biggs, M.J. Adams, D.A. Gorham, J. Lukenics, M.J. Hounslow, Powder Technol. 143–144 (2004) 19–30. [52] P.C. Knight, T. Instone, J.M.K. Pearson, M.J. Hounslow, Powder Technol. 97 (1998) 246–257. [53] M. Landin, P. York, M.J. Cliff, R.C. Rowe, A.J. Wigmore, Int. J. Pharm. 133 (1996) 127–131. [54] B.J. Ennis, J. Li, G.I. Tardos, R. Pfeffer, Chem. Eng. Sci. 45 (1990) 3071–3088. [55] B.J. Ennis, G.I. Tardos, R. Pfeffer, Powder Technol. 65 (1991) 257–272.

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[56] G.I. Tardos, M.I. Khan, K.P. Hapgood, O.O. Ipadeola, J.N. Michaels, Powder Technol. 140 (3) (2004) 217–227. [57] O. Tsoungui, D. Vallet, J. Charmet, Powder Technol. 105 (1999) 190–198. [58] M.K. Gundepudi, B.V. Sankar, J.J. Mecholsky, D.C. Clupper, Powder Technol. 94 (1997) 153–161. [59] G.R. McDowell, M.D. Bolton, Geotechnique 48 (5) (1998) 667–679. [60] S.J. Antony, M. Ghadiri, J. Appl. Mechan. 68 (2001) 772–775. [61] A. Hassanpour, S.J. Antony, M. Ghadiri, Chem. Eng. Sci. 62 (2007) 935–942. [62] G.I. Tardos, M.I. Khan, P.R. Mort, Powder Technol. 94 (1997) 245–258. [63] J.D. Litster, K.P. Hapgood, J.N. Michaels, A. Sims, M. Roberts, S.K. Kameneni, T. Hsu, Powder Technol. 114 (2001) 32–39.

CHAPTER 20

Modelling of Mills and Milling Circuits Petya Toneva and Wolfgang Peukert Institute of ParticleTechnology, Friedrich-Alexander-University of Erlangen-Nuremberg, Cauerstrasse 4, D-91058 Erlangen,Germany Contents 1. Introduction 2. Examples of milling circuits 3. Modelling levels for milling circuits 3.1. The population balance model 3.2. The non-linear grinding kinetics 3.3. Modelling of the residence time distribution (RTD) in the case of continuous grinding 3.3.1. Combining of batch grinding data with RTD 3.3.2. Classical approaches from chemical reaction engineering 3.4. Modelling of milling circuits involving different mill types 3.4.1. High-speed hammer mills 3.4.2. Vertical spindle mills 3.4.3. Dry and wet ball mills 3.4.4. Air jet mills 3.4.5. Roller mills 4. Determination of model parameters 5. Flowsheeting for solid processes 5.1. General remarks 5.2. SolidSim – a new tool for ﬂowsheeting of particulate processes 5.3. Modelling of mill–classiﬁer circuits within SolidSim 5.3.1. Comminution 5.3.2. Classiﬁcation 5.3.3. Mill2classiﬁer circuits within SolidSim 5.4. The reference state method for modelling of mill–classiﬁer circuits 5.4.1. Classiﬁcation 5.4.2. Comminution 6. Discussion of the main inﬂuencing parameters 6.1. Inﬂuence of the mill operation mode on the product size distribution 6.2. How to produce narrow particle size distributions: the example of powder paint 6.2.1. Rotational frequency of the classiﬁer 6.2.2. Rotational frequency of the grinding disc

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Corresponding author. Tel.: +49-9131/85-29401; Fax: +49-9131/85-29402; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12023-6

r 2007 Elsevier B.V. All rights reserved.

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6.2.3. Air ﬂow rate and solids throughput 6.2.4. Design parameters References

906 906 910

1. INTRODUCTION Milling circuits are used in various industrial sectors in order to achieve desired product size distributions and product properties. The optimization and control of milling circuits is of high technological importance. Since size reduction is an expensive, high energy consuming process, the appropriate design of grinding systems is of primary importance for the minimization of the energy consumption. To provide cost and time savings as well improvement of productivity through optimization of a grinding circuit a mathematical simulation of the process can be applied. The simulation of milling circuits allows the control and optimization of existing milling systems as well as the design of new mill facilities. The following chapter gives a brief overview of the existing mathematical models for the simulation of grinding and classiﬁcation in mill–classiﬁer circuits, mainly for ball mills but also for other mill types. Examples of the simulation of closed circuits are divided according to the mill type and the level of sophistication introduced into the models.

2. EXAMPLES OF MILLING CIRCUITS In order to achieve the desired particle size distribution of a certain material the production chain can consist of several grinding stages connected with classiﬁcation units. The operation mode of the mills (batch, continuous, passage, cascade, pendulum) also inﬂuences the product size distributions. The classiﬁers can be integrated directly into the grinding machine and/or additional external classiﬁers can be integrated along the production chain. There are various types of interconnections between mills and classiﬁers. Figure 1 shows a schematic diagram of an open- and closed-circuit system. When a mill is operated without being directly inﬂuenced by a classiﬁer it is labelled to work in an open circuit

(a)

(b)

Fig. 1. Open (a) and closed (b) circuit systems.

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(Fig. 1a). In order to increase the grinding efﬁciency of a process, mills are connected with classiﬁers in a way that the classiﬁer returns coarse material to the mill feed. The simplest closed mill–classiﬁer circuit is that according to Fig. 1b. The produced material at the outlet of the circuit has a certain maximum size which is determined by the classiﬁer. In a closed mill–classiﬁer circuit all ﬂows other than the feed and the product are internal and are usually not quantiﬁed. Another possibility which is often applied for industrial production is that the circuits consist of a double recirculation system where ﬁne product from the ﬁrst classiﬁer is fed to the second one and tailings from the second classiﬁer are returned to the feed of the ﬁrst one. Advantages of changing open-circuit grinding mills to closed-circuit mills include efﬁciency, production of well-deﬁned particle size distributions and cost savings. It has been reported that for the same mean diameter of product, the power required for a closed-circuit system is about 5% smaller than that of the open-circuit system [1]. The removal of ﬁne particles by the classiﬁer results in less overgrinding in the closed circuit compared with the open circuit. A comparison of a mill–classiﬁer circuit consisting of a ring gap mill and a hydro classiﬁer with a wet grinding in passage operation has been carried out by Karbstein et al. [2]. A schematic diagram of the mill–classiﬁer circuit used for the investigations is shown in Fig. 2. With the mill–classiﬁer circuit much steeper distributions were achieved. The mean particle sizes from the closed circuit were smaller than those obtained from wet grinding alone. A number of mills, e.g. air classiﬁer mills, have a classiﬁer integrated in the mill chamber so that internal recirculation loops occur (Fig. 3). Various classiﬁer mills are integrated in an external recirculation loop including an external classiﬁer which removes undesired ﬁne particles. In this case especially narrow product size distributions are obtained. Typically applications include toner and powder paint production. In the former case jet mills are used whereas in the latter case hammer mills are employed. If the mill is operated with an external classiﬁer then the process model is built up by using two classiﬁers in series (Fig. 4). It has been

Fig. 2. Flowsheet of a wet mill–classiﬁer circuit.

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Fig. 3. Air classiﬁer mill ACM (Hosokawa Micron GmbH).

i i

i

i

i

1 i

2 i

1 i

2 i

Fig. 4. Two-classiﬁers-in-series model for the simulation of a grinding process with an internal and external recirculation loop.

reported that in wet ball mills material with particle sizes less than the grate opening are rejected back to the mill [3]. This phenomenon is also modelled assuming a hypothetical classiﬁer at the mill outlet. The parameters which have to be determined for a realistic process modelling are the grinding kinetics, the classiﬁcation and the residence time distribution. An important way to achieve steeper particle size distributions is to vary the mill operation mode, e.g. stirred media mills can be operated in circuit or passage mode. The former is used for batch or continuous grinding with corresponding storage tanks. The simplest case of a circuit operation mode with only one storage tank is shown in Fig. 5. The suspension circulates between the mill and a storage tank where the recycle is mixed with the feed. The product quality continuously increases with the time. This operation mode is characterized by wide residence time distributions (RTD) since the RTD of the mill is overlapped by that of the storage tank, which leads to broader size distributions of the product. One way to narrow the RTD is the passage operation mode. In this case the suspension is pumped with a high ﬂow rate from one storage tank through the mill to a product tank. One disadvantage of this operation mode is the high cleaning costs for the two tanks. In the cascade operation mode several mills are connected in series and the feed is pumped from one feed storage tank through the

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Fig. 5. A circuit operation mode of a stirred media mill with one storage tank (the ﬁgure is kindly provided by NETZSCH-Feinmahltechnik GmbH, Germany).

Fig. 6. Cascade and pendulum operation modes of stirred media mills (the ﬁgure is kindly provided by NETZSCH-Feinmahltechnik GmbH, Germany).

mills to another product tank (Fig. 6a). The milling result corresponds to that of n single passages. The cascade operation mode is characterized by a stepwise development of the product quality and requires high investment costs. Another more economical method to achieve steeper particle size distributions is the pendulum operation mode (Fig. 6b), where the suspension is pumped from one tank through the mill to a second one and then back to the ﬁrst one. In the passage operation mode both tanks are completely discharged during operation. One has to be aware that the pendulum operation mode requires high costs for the process control in comparison to the circuit operation mode.

3. MODELLING LEVELS FOR MILLING CIRCUITS Various models of mills and milling circuits are published in the literature with different precision and degree of sophistication. Mostly the grinding and

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classiﬁcation are investigated separately and then both unit operations are connected into a mathematical model by the recirculation feed. Therefore the modelling levels for the milling circuits depend on the degree of sophistication of the mill and the classiﬁer models introduced in the milling circuit model. The modelling of the grinding process is based on the population balance, i.e. the mass balance for discrete size intervals. The simulation of continuous grinding additionally requires the RTD in the milling circuit. Assuming that the mill is perfectly mixed in the radial direction (for example by rotating grinding parts integrated in the mill) but only partially mixed in axial direction, the population balance in the case of breakage can be expressed as i1 X dw i ðl; tÞ d2 w i ðl; tÞ dw i ðl; tÞ ¼ Si w i ðl; tÞ þ Sj bi;j w j ðl; tÞ þ Di ui 2 dt dl dl j¼1

ð1Þ

with Si speciﬁc breakage rate of particles of size class i, wi mass fraction of material in size interval i and bi,j breakage distribution parameter. The variable l denotes the space coordinate, t is the time. The left-hand side of the equation describes the accumulation. The ﬁrst and second terms on the right-hand side represent death and birth terms due to breakage, the third term describes the axial dispersion, characterized by the mixing coefﬁcient Di and the last term represents the convective transport of particles of size class i in the axial direction with velocity ui. The initial and boundary conditions for equation (1) are w i ðl; 0Þ ¼ f i ðlÞ w i ðl; tÞ ¼ ui w i ðl; tÞ Di dw i ðl; tÞ ¼0 dl

dw i ðl; tÞ dl

ð2Þ if l ¼ 0

if l ¼ L

ð3Þ ð4Þ

Equation (1) is the basis for the calculation of the breakage process. Taking into account the mill’s operational characteristics some of the terms are often neglected or further simpliﬁed. In the following subchapter the different levels of sophistication used for mill modelling are brieﬂy discussed.

3.1. The population balance model Assuming that the mill is perfectly mixed the third and fourth term from equation (1) can be neglected. In this case the most frequently used form of the population balance for the calculation of grinding processes is derived: i1 X dw i ðtÞ ¼ Si w i ðtÞ þ Sj bi;j w j ðtÞ dt j¼1

ð5Þ

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Over the years the most popular approach for the modelling of grinding is the assumption of ﬁrst-order (linear) breakage kinetics. This assumption has been proposed by Reid who derived an analytical solution of the batch grinding equation [4]. The linearity consideration implies that the speciﬁc rate of breakage for a certain size interval is time independent. The linearity of the speciﬁc breakage rate has been validated for many milling systems for short-time batch milling [5]. Nevertheless many authors have shown that the predictions based on the ﬁrstorder breakage kinetics deviate signiﬁcantly from the experimental ﬁndings with time progression of the grinding process (see Section 3.2). Fuerstenau et al. [6] expressed the batch grinding equation in terms of speciﬁc energy as independent variables instead of time. Detailed studies in three different scaled batch dry ball mills have shown that the feed size breakage rate functions fell on a single curve for various mill speeds, ball loadings and feeds when the inlet particle size distribution was plotted versus the speciﬁc energy input [7]. The proportionality of the speciﬁc rate of breakage to the power input to the mill allowed for the transformation of the population balance in terms of speciﬁc energy as follows: i1 X dw i ðEÞ þ SEj bij w j ðEÞ ¼ SEi w i ðEÞ dE

ð6Þ

j¼1

E is the speciﬁc energy input to the mill. SEi is the energy-normalized breakage rate parameter deﬁned as SEi ¼

Si P=W

ð7Þ

with P power input to the mill and W mill hold-up. The authors have transferred this method to the modelling of a high pressure roll mill taking into account the increasing of the energy dissipation with time due to interparticle friction and visco-plastic ﬂows [6]. The basic equation used for the modelling of grinding processes is the origin for various model extensions as for example the consideration of non-linear breakage kinetics.

3.2. The non-linear grinding kinetics The speciﬁc rate of breakage is usually determined in batch experiments by plotting the mass loss in the largest particle class as a function of time. The solution of the batch grinding equation for the largest class assuming a ﬁrst-order breakage rate is given by the following equation: w 1 ðx; tÞ ¼ w 1 ðx; 0Þ expðS1 tÞ

ð8Þ

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Fig. 7. First-order kinetics for different mass fractions of the top size interval.

This equation states that the breakage rate is proportional to the amount present to break. If the speciﬁc rate of breakage does not vary with time the plot has a constant slope (Fig. 7). If the breakage kinetics deviate from ﬁrst-order kinetics then they are termed non-linear breakage kinetics. Signiﬁcant deviations from the linear milling theory have been observed in experiments by different research groups for both dry and wet grinding [3,8–10]. The non-linearity of the grinding process is led back to the multi-particle interactions occurring in the mill, to the heterogeneity of the material or to fatigue effects with time progression [11]. When coarse and ﬁne particles are ground together in dry or wet ball mills as well as in roller mills the non-linear nature of the breakage process is explained by the multi-particle interactions between the coarse particles and the ﬁnes which surround the coarse ones. If the ground material is monodisperse, the speciﬁc rate of breakage remains constant until certain time t. Up to that time the coarse particle fraction prevails in the mill. With time progress of the grinding process the produced ﬁnes begin to affect the breakage rate. The coarse particles become surrounded by the ﬁnes and the ﬁnes absorb the supplied energy for impact. The stress concentration reduces because it is distributed over a large number of contact points due to the ﬁnes present. Hence the speciﬁc rate of breakage decreases as the ﬁnes in the mill increase. To describe the non-linear grinding kinetics a time-dependent speciﬁc rate of breakage has been proposed by Austin et al. [8,12] for the modelling of comminution. The speciﬁc breakage rate has been split into an acceleration–deceleration function k(t) and in an initial speciﬁc breakage rate S(0) as follows: S1 ðtÞ ¼ kðtÞ Sð0Þ

ð9Þ

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The non-linearity of the speciﬁc breakage rate has been characterized by Bilgili and Scarlett [11] in three types of functions taking into account experimental data published by different researchers (Fig. 8). Type I function occurs during milling of monodisperse particles as reported ﬁrst by Austin [13]. Type II and III breakage rate functions are characteristic for the breakage of a binary feed [11]. The signiﬁcant effect of the initial particle size distribution on the speciﬁc rate of breakage has been investigated by many authors using polydisperse feeds. [9–11,14,15]. In order to describe the non-linear breakage kinetics effects observed during comminution Bilgili and Scarlett [5] proposed a non-linear breakage model. They decomposed multiplicatively the speciﬁc breakage rate into apparent speciﬁc breakage rate function k(x) and a population-dependent part F[ ]. Z

1

SðxÞ ¼ kðxÞF

Pðx; yÞqðyÞdy

ð10Þ

0

The function F[ ] describes different types of non-linear kinetics. The function P(x,y) determines how signiﬁcantly the speciﬁc rate of breakage of particles of size x is affected by smaller particles of size yox. The population-dependent function describes the dependence of the breakage rate on the mass density of the surrounding particles. The modiﬁed expression for the speciﬁc rate of breakage has been implemented in the population balance model and applied to batch grinding [11]. Subsequently, it has been extended to open-circuit continuous grinding for the idealized cases of perfect mixing and plug ﬂow [5]. The parameter studies which have been carried out have shown the capability of the non-linear model to explain the experimentally observed deviations from the linear theory. Fuerstenau et al. [6,10] assumed a non-linear grinding kinetics in rod mill systems as one which order is less than unity. The system has been modelled as a linear combination of zero- and ﬁrst-order breakage kinetics. Non-ﬁrst-order kinetics have been reported by Cho and Austin [16] for wet media milling. The nonlinear effect has been considered in the population balance by the implementation of a two parameter ﬁtting speciﬁc breakage rate function of the type " l # t SðtÞ ¼ Sð0Þ 1 þ ; l41 ð11Þ c where S(0) is the constant speciﬁc rate of breakage, characteristic for short grinding times. The parameters c and l are the ﬁtting parameters. While discussing the time dependence of the speciﬁc rate of breakage it is appropriate to point out that processes such as fatigue or viscoelasticity are also time-dependent phenomena, which arise during comminution. These events can also arise into a time-dependent breakage rate.

882 P. Toneva and W. Peukert

Fig. 8. Characteristic deviations from the linear kinetic approach: Type I for breakage of monodisperse material; Type II and Type III for breakage of binary feed material.

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3.3. Modelling of the residence time distribution (RTD) in the case of continuous grinding In order to simulate the continuous grinding the RTD of the particles in a mill has to be taken into account. Several approaches for the consideration of the residence time in the continuous models for mills and mill–classiﬁer circuits have been reported.

3.3.1. Combining of batch grinding data with RTD All equations for batch grinding can be readily extended to continuous systems [13]. This procedure combines the knowledge of the particle size distribution Qi(t) in an identical mill operated in batch mode for time t with the RTD function E(t) of the continuous mill: Z 1 Qi ¼ Qi ðtÞEðtÞ dt ð12Þ 0

Qi is the particle size distribution from the continuous grinding. This approach is valid if the particle transport in the continuous mill is size independent, e.g. if all the particles in the mill move with the same RTD.

3.3.2. Classical approaches from chemical reaction engineering Other approaches are taken from classical chemical reaction engineering: the perfect mixing mill model, the plug-ﬂow mill model and the N-mixers in series approach. These models are introduced below.

3.3.2.1. The perfect mixing grinding approach Perfect mixing is an approximation which is often used for ball mills with a small diameter to length ratio [5,17,18], for high-speed hammer mills [19,20] and air jet mills. The perfect mixing mill model does avoid the problem of residence times in the mill. The residence time in the mill is assumed to be constant for every particle size and is determined from the mill hold-up W and the feed rate F in the mill as t ¼ W/F. The equation of an ideally mixed mill operating at steady state can be written as f i w i tSi w i þ t

i1 X

bi;j Sj w j ¼ 0

ð13Þ

j¼1

fi and wi represent the mass fraction of the material of size class i in the feed and the product, respectively. Si is the (apparent) speciﬁc rate of breakage of particles of size class i and bi,j is the fraction of breakage of progeny particles of size class j

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into fragments of size i. t is the mean residence time in the mill. One has to point out that in this case the size fraction of the material retained in the mill and the fraction leaving the mill is the same because of the assumed ideal mixing.

3.3.2.2. The plug-ﬂow model Another approximation used for modelling of continuous mills with a high length to diameter ratio is that the mill behaves as a plug ﬂow, e.g. there is no back mixing during comminution. For a mill operating at steady-state conditions the population balance for plug-ﬂow behaviour is written as Z 1 dqðl; xÞ ¼ SðyÞqðl; xÞ þ bðx; yÞSðyÞqðl; yÞ dy ð14Þ t dl 0 l is the axial position along the length axis. In this case the breakage depends not only on the particle size but on the actual position along the longitudinal axis. For solving of this integro-differential equation it is usually transferred to a sizediscrete form in order to provide a numerical solution. The size discretization allows easier computation and is more convenient for parameter estimation since the experimental data sets collected from grinding experiments are also in a sizediscrete form. The size-discrete form of equation (14) is as follows: i X dw i ðlÞ ¼ tSi w i ðlÞ þ t bij Sj w j ðlÞ dl j¼1

ð15Þ

Bilgili and Scarlett [5] have proposed a solution of the population balance for continuous grinding assuming a slowing-down of the dry grinding process by the selection function with a breakage function proposed by Klimpel and Austin [37] and a potential breakage rate constant. A comparison of numerical simulations performed by the authors for both ideal cases of plug ﬂow and ideally mixed mill have shown that the results differ in the larger particle size range. For one and the same distribution the product from the plug-ﬂow model is ﬁner than the product from the ideal mixing model. The difference has been explained by the different levels of mixing in both cases. In the plug-ﬂow mill the breakage rates change along the longitudinal axes, in contrast to that the breakage rate of the ideally mixed mill is determined at the mill outlet where the product is very ﬁne.

3.3.2.3. The N-mixers in series approach Another classical approach used mainly for the modelling of ball mills is the N-mixers in series model. The grinding process is modelled by N perfectly mixed cells. The number of cells N inﬂuences the RTD. With increasing number of cells the RTD gets narrower and approaches for large N the RTD of a system operating in plug-ﬂow regime. The N-mixers in series approach is usually applied for

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the modelling of the transport behaviour in stirred media mills. The breakage in that type of mill occurs along the entire length of the mill and the particles undergo a repeated breakage until they reach the mill outlet. For media mills the radial dispersion can be neglected and the mill can be simulated as a series of stirred cells of volume deﬁned by the distance between the stirrer discs [21]. Because of the intensive mixing in the grinding zone the number of cells is kept small (o5). The population balance equation is written as i1 X dðH n w ni Þ n n ¼ M n1 w n1 M w þ bi; j Sj H n w nj Si H n w ni i i dt j¼1

ð16Þ

M is the mass ﬂow rate and H is the hold-up. The superscripts denote the cell number. If the hold-up is constant for every cell then equation (16) can be written as i1 dw ni 1 1 n X ¼ n1 w n1 w þ bi; j Sj w nj Si w ni i t tn i dt j¼1

ð17Þ

Benzer [22] applied the N-mixers in series approach for the development of a back calculation model for an air swept ball mill. In order to describe the particle transport the mill has been divided into several compartments. The radial dispersion has been neglected and the mill simulated as a series of three perfectly mixed cells. The last cell has been assumed to operate in a closed circuit with a classiﬁer which represents the air sweeping in the mill. In order to consider the RTD in a stirred bead mill Dodds et al. [21] modelled the mill as series of mixed cells which volume has been deﬁned by the distance between the stirrer discs. The mill has been divided into ﬁve compartments and the radial classiﬁcation has been neglected. The inter-cell recycle factor has been used as an adjustable parameter in the model. The N-mixers in series model with a coupled back mixing is a more realistic approach used for the modelling of stirred bead mills.

3.4. Modelling of milling circuits involving different mill types In order to achieve a realistic modelling of mill–classiﬁer circuits the material behaviour in the mill as well as the efﬁciency of the classiﬁer have to be known. The grinding result depends on one side on the material properties related to comminution. These are mostly determined in single particle experiments. On the other side the product particle size distribution depends on the mill type. The mill type speciﬁes the stress mechanism and the stress intensity during comminution. Following mill models used for the calculation of the grinding in mills or/and milling circuits are brieﬂy introduced.

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3.4.1. High-speed hammer mills Because of their high rotational speed hammer mills are often modelled as fully mixed by the rotation action [19,23–26]. Fully mixed steady-state grinding with rapid ideal classiﬁcation through a discharge screen has been modelled by Jindal and Austin [26] for a batch operation and extended later on a continuous grinding and classiﬁcation with non-rapid product removal [23]. The non-rapid product removal implies that material which is ﬁne enough to pass the screen does not leave the mill completely, there exists a certain fraction which does not pass through the screen and remains in the mill. For the modelling of the grinding process the authors have assumed that the breakage law is ﬁrst order. The values of the speciﬁc rate of breakage, the breakage function and the variation of the speciﬁc rate of breakage with the mill hold-up have been determined pﬃﬃﬃ in batch tests. The particle size range for the calculation has been split into 2 intervals numbered 1 for the largest size [24]. In the present Chapter 1 is used to denote the largest particle size class and N for the smallest particle size interval. The modelling of the batch grinding by considering the presence of a death space inside a laboratory hammer mill have shown that the presence of a death space can lead to a considerable slowing-down of the breakage process even though the material’s breaking is according to ﬁrst-order breakage kinetics [27]. The order of the breakage kinetics may be inﬂuenced by this, i.e. ﬁrst-order breakage during batch tests may appear to be non-ﬁrst because of the existing death space. Austin [24] has proposed a concept for the modelling of grinding in a highspeed hammer mill taking into account the main physical parameters which have to be considered in the modelling of comminution. The fracture has been modelled assuming a speciﬁc impact energy distribution. The breakage function has been deduced from batch experiments. The hammer mill has been considered as fully mixed. Additionally, the damage accumulation has been taken into account. Austin could show that the effect of the damage accumulation is signiﬁcant and therefore cannot be neglected in simulating the grinding process. The experimental effort for the determination of the parameters needed for the detailed modelling is considerable.

3.4.2. Vertical spindle mills A mathematical model for a spindle mill with a built-in cone classiﬁer has been proposed by Austin et al. [28]. Because the rotating balls in the mill cause intensive mixing, the mill can be treated as fully mixed. The mill–classiﬁer circuit is built up according to Fig. 4. The ﬁrst classiﬁcation action arises from air expansion and the resulting falling back of large particles into the grinding zone; the second classiﬁer is the acting external classiﬁer. Both classiﬁcation effects can be

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and an overall treated as a single net classiﬁer with a net recirculation ratio C fracture of ﬁnes after classiﬁcation s i as follows: ¼ 1 ð1 þ CÞ 1 þ C0 C ð18Þ s i ¼ 1 ð1 þ s1;i Þð1 þ s2;i Þ

ð19Þ

Assuming the notiﬁcation according to Fig. 4 and taking into account the mass conservation, the mass balance leads to equation (20). gi þ ð1=FÞ i¼ ð1 þ CÞw

i1 P

j bi;j Sj W ð1 þ CÞw

j¼1

ðSi W =FÞ þ ð1 s1;i Þð1 s2;i Þ

ð20Þ

If s2,i ¼ 0 equation (20) reduces to the balance for a mill–classiﬁer circuit according to Fig. 1b. For the modelling of the vertical spindle mill the breakage probability Si and the breakage parameter bi,j have been determined in batch experiments [28]. Since the determination of the particle size distribution of the feed to the classiﬁer or of the recycle stream from the classiﬁer is difﬁcult, s2,i has been estimated from results of experiments on a similar classiﬁer. Sato et al. [29] applied the above model proposed by Austin to a ring-roller mill. The speciﬁc rate of breakage and the breakage distribution parameters have been again obtained from experiments in a batch roller mill. The classiﬁcation has been modelled by an empirical equation.

3.4.3. Dry and wet ball mills Ball mills with large diameter and short length (pancake type) are also modelled assuming ideal mixing. A few methods for the modelling of wet media milling, taking into account occurring internal classiﬁcation effects, have been proposed by Cho and Austin [16]. The fact that particles less than the gap opening remain in the mill has been considered by the implementation of a hypothetical exit classiﬁer in the model. The model is empirical and the calculation is done by a search routine for three model parameters. Since these parameters are of empirical nature, a parameter study is necessary in order to determine how these values change with changing operation conditions. The second method proposed by Cho and Austin is based on the non-equal ﬂow velocity. The mill is again assumed to be fully mixed in axial ﬂow direction. Additionally it has been considered that larger particles cannot follow the water stream because of their higher inertial forces and remain longer in the mill. The small particles are swept by the water ﬂow. Their mean residence time is the same as that of the water. The velocity of the larger particles ui is assumed by the authors to depend linear on the ﬂow velocity u0 according to

888

P. Toneva and W. Peukert

equation (21): ui ¼ k i u0

0 ki 1

ð21Þ

Austin proposed the following equation for the calculation of ki: ki ¼

1 1 þ ðx i =x k Þl

ð22Þ

where xk is the sieve size at which k ¼ 0.5. l is an adjustable parameter. Another way to model the different RTD of particles with different sizes is the assumption that the speciﬁc rate of breakage of larger particles is higher than that of smaller particles assuming a mean residence time for the calculations. For known values of fi, Si, ki, bi,j the mean residence time is varied until the desired product particle size distribution is obtained. If the RTD does not correspond to that of a fully mixed mill the mill can be modelled as such having an RTD corresponding to that of series of fully mixed reactors [30]. The mass ﬂow rate through each section is the same. For the modelling of dry grinding in tumbling mills Yildirim et al. [17] have proposed a model for tumbling mills in closed circuit using air separators for size classiﬁcation. The mill model used for the simulation assumes ﬁrst-order breakage kinetics. Minor modiﬁcations have been undertaken concerning the deﬁnition of the speciﬁc rate of breakage. S has been assumed to depend on the diameter d of the media used for grinding. The speciﬁc rate of breakage has been extended to a xi 1 Si ¼ A x 0 1 þ ðx i =mÞZ

ð23Þ

where A and m are functions of d. a and Z are taken as constants. x0 is a standard particle size of 1 mm. The wear law for the media has been assumed to be a Bond linear law. The same expression has also been applied by the authors for the modelling of wet ball mills [16]. A perfect mixing model for dry ball mills with a short length to diameter ratio has been proposed by Weedon [18]. The breakage function bij has been determined by single particle tests in a pendulum device. The author has assumed that each individual feed size has a speciﬁc breakage function. The data obtained from single particle breakage have been collected into a set of t-curves (family of lines that uses a normalization procedure to generate a characteristic t-parameter) and used to determine a matrix of appearance functions for each feed size, which is characteristic for the feed material and the mill.

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3.4.4. Air jet mills Dodds et al., Berthiaux and Dodds and Berthiaux et al. [21,31,32] have proposed the concept of the residual fractions for the modelling of grinding in ball and air jet mills (see also Chapter 8). They preferred a population balance formulation related to the cumulative oversize distribution R. i1 X dRi ¼ Si Ri ðtÞ þ ðSjþ1 Bi; jþ1 Sj Bi; j ÞRj ðtÞ dt j¼1

ð24Þ

The cumulative oversize distribution Ri(t) and the cumulative breakage distribution function Bi,j are related to wi and bi,j through Ri ðtÞ ¼

i1 X j¼1

w j ðtÞ and

Bi;j ¼

i X

bk;j

ð25Þ

k¼jþ1

In the case of continuous grinding the relation between the feed and product PSD depends on the grinding kinetics and on the residence time of the material in the mill. In general an RTD is taken into account. Assuming that the RTD does not depend on the particle size, Dodds et al. [21] proposed the following equation for continuous grinding, using Kapur’s solution [33] of the batch grinding equation: ! Z 1 p k X Rout ðxÞ t ¼ exp K ðkÞ ðxÞ EðtÞ dt ð26Þ Rin ðxÞ k! 0 k¼1 where E(t) denotes the RTD function. For the modelling of the air jet mill Dodds et al. divided the mill into three sections (Fig. 9): Perfect mixed grinding zone; Transport zone which is assumed to be in plug ﬂow; Classiﬁcation zone characterized by a grade efﬁciency curve.

Fig. 9. Model assuming the residence time distribution in an air jet mill [21].

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P. Toneva and W. Peukert

The three zones are characterized independently and then linked by the overall mass and population balance. The following equation describes the overall process [34]: " # Z x 1 I p ðxÞ E T Rin ðxÞ 1 GðxÞI p ðxÞ 1 1 GðxÞ ¼ þ RðxÞ dx RðxÞ ð1 GðxÞÞI p ðxÞ RðxÞ ð1 E T ÞI p ðxÞ I p ðxÞ 0 ð1 GðxÞÞ2 ð27Þ In this equation Ip includes both the batch grinding kinetics and the RTD. G(x) is the grade efﬁciency curve of the classiﬁer and ET is the total efﬁciency of the classiﬁer under the conditions of operation. ET and G(x) are determined by replacing the air jet section by an air inlet section. The RTD is considered by inclusion of the plug-ﬂow transport zone between the air jet and the classiﬁer. In continuous ﬂow systems particles of different sizes have different RTD in the mill and therefore are subjected to different degrees of grinding before reaching the mill outlet. This can be characterized by means of the RTD of the particles. Berthiaux et al. [32] modelled the RTD assuming the ﬂow through the mill as a bundle of ﬂow streams where each stream is independent and has no interaction with the others. Each stream is considered to be in plug ﬂow with residence time ts. The PSD at the outlet of the mill is then calculated as follows: Z 1 Rout ðxÞ ¼ Rðx; tÞEðt s Þ dt s ð28Þ 0

Gommeren et al. [35] proposed a detailed dynamic model for the process design in an air jet mill, taking into account the main unit operation taking place inside the mill. The modelling has been done into three modelling levels: Main model predicting the interactions between the mill and the classiﬁer; Flow simulation considering the particle motion and the particle–wall collisions; Model for the breakage behaviour of the single particle.

For the ﬁrst modelling level the closed mill circuit is divided into three zones as shown in Fig. 10. The population balance for zone 1 can be written as m1;i ðt þ DtÞ ¼ m1;i ðtÞ þ m2;i ðtÞP 21;i ðDtÞ þ

i1 X

bi;j Sj ðDtÞm1; j ðtÞ m1;i ðtÞP 12;i ðDtÞ Si ðDtÞm1;i ðtÞ

ð29Þ

j¼1

where m1,i is the mass of size fraction i in zone 1 and m2,i is the mass of size fraction i in zone 2. Similar to the classiﬁcation the particle transfer between the different zones in the mill is described by S-shaped functions (P21,i, P12,i) which give the probability that a particle with a size x is transferred from one zone to

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891

Fig. 10. Zones for the modelling of a closed jet mill–classiﬁer circuit proposed by Gommeren [35]: (1) outer zone of the mill chamber, (2) inner zone of the mill chamber, (3) classiﬁcation zone.

another. The probability functions used by Gommeren are empirical. The inﬂuence of the hold-up on the mill performance has been included in the transport functions by the time-dependent cut size and sharpness. The particle breakage has been described by the selection and the breakage functions. At the second level the ﬂow ﬁeld in the mill is modelled assuming that the particle concentration is low and the gas velocity distribution is not affected by the solid particles. The particle collisions have been taken into account assuming that binary collisions are dominant. For the modelling of the inter-particle collisions an approach initially proposed by Nanbu has been used [36].

3.4.5. Roller mills In roller mills the grinding results primarily by interparticle stresses generated within the particle bed. Fuerstenau et al. [6,10] proposed a model which is brieﬂy introduced. The population balance has been formulated in terms of cumulative energy input E considering the energy dissipation in the roll gap as follows: E

i1 S X dw i ðEÞ SE j ¼ iy w i ðEÞ þ y bij w j ðEÞ dE E E j¼1

ð30Þ

where y is an energy dissipation coefﬁcient. The shape of the breakage rate functions was the same as those found for rod and ball mills but the breakage rate functions were one order of magnitude higher then those determined for ball mills. Fuerstenau et al. [6] also proposed a model for the grinding kinetics of damaged particles considering the fatigue effect leading to non-linear grinding kinetics

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at a subsequent grinding stage. The authors modelled the strength distribution of the damaged particles by a modiﬁed gamma function.

4. DETERMINATION OF MODEL PARAMETERS The efﬁcient modelling of the grinding process (and in general of processes involving solids) is still not possible (and probably never will be) without collecting experimental data. The experimental effort for the determination of the model parameters can be considerable. There are few general approaches used in generating data for the steady-state modelling of mills and milling circuits: 1. Breakage functions and breakage probabilities are measured in single particle test devices using the same stress type as in the mill of interest (very well applicable for impact grinding) [19]. 2. Breakage rates and RTD are measured in laboratory batch tests and applied for the continuous steady-state modelling [29]. 3. Breakage and RTD are measured in a laboratory continuous mill of the same type and scaled-up to the mill of interest. 4. PSD and RTD are measured on the mill of interest and breakage characteristics are back calculated from them [37]. 5. Combined single particle, batch, laboratory test and real mill performance approach is used. Approach 1 is applied for the determination of material parameters related to comminution excluding the inﬂuence of the grinding device, e.g. the machine function. This approach has been used by Vogel and Peukert [19,20] for the modelling of closed hammer mill circuits. The modelling of the classiﬁer performance can be divided into three levels: 1. Ideal separation approach; 2. Efﬁciency curves based on theoretical S-shaped functions; 3. Efﬁciency curves measured on the classiﬁer of interest. The characterization of the two-phase ﬂow in mills and classiﬁers is still in its infancy. Very few systematic studies are known for both experiments and modelling of the ﬂow. The authors are currently studying for the ﬁrst time the one- and two-phase ﬂow through a classiﬁer mill by means of measurements by particle image velocimetry (PIV) and advanced computational ﬂuid dynamics (CFD) techniques.

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5. FLOWSHEETING FOR SOLID PROCESSES 5.1. General remarks The ﬂowsheet simulation is widely applied for the design and optimization of ﬂuid processes. Concerning the processes involving solids, the modelling is mainly concentrated on separate apparatus without taking into account the inﬂuence of the other unit operations along the production chain. Most solids processes consist of several apparatuses which are interconnected by the ﬂowing solids or ﬂuid/ solids streams. Often this is not only a sequential interconnection, recycling streams are a common addition. One very simple example for such a process is a sieve-crusher cycle, in which the product of the crusher is classiﬁed and the coarse material is recycled to the crusher. Flowsheet simulation of solids processes faces speciﬁc problems. The characterization of solids requires more parameters (e.g. size, porosity, humidity, shape) than a ﬂuid which may be characterized by a few parameters, i.e. temperature, pressure and concentration. The solids are characterized by distributed properties such as PSD, moisture content, etc. Distributed parameters require additional mathematical treatment, e.g. the fulﬁlment of population balances, which add new families of equations to the complex mathematics of ﬂowsheeting systems and thus need new or adapted solvers. The ﬂowsheet simulation permits to vary and optimize the arrangement of the devices along the production chain in order to achieve the desired product quality. In particle technology education the different apparatuses are commonly treated individually, neglecting their complex interaction. A newly developed ﬂowsheeting tool for solid processes SolidSim allows for a better understanding of the complex interactions between the unit operations along the production chain with respect to process optimization and product design.

5.2. SolidSim – a new tool for ﬂowsheeting of particulate processes SolidSim is a program tool which simulates solids processes consisting of several different processing steps including recycles. It also allows for a complete characterization of mixtures of particles with different particle size distributions and densities. The deﬁnition and correct treatment of attributes depending on other attributes, e.g. a moisture depending on the particles size, is supported. SolidSim is a simulation system especially designed for the simulation of processes including solids. It has been developed by a group of 11 institutes from seven different German universities. The program system is divided into three major parts: the simulation environment, the stream objects and the model library with the unit models. The simulation environment provides the basis of the

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system. It provides the graphical user interface, some basic functionality such as reporting, printing, saving and loading data, and it controls the calculation sequence. The material stream object provides the structures necessary to transport the information on the ﬂuids and solids from one module to the next one, and to provide the modules with all properties of the materials needed. The unit model simulates a single apparatus or unit operation. These basic elements are connected by different interfaces as deﬁned by the CAPE-OPEN-standard. Additionally, communication with other simulation packages is also possible using the CAPE-OPEN interfaces. A detailed ﬂowsheet simulated with SolidSim may be used as one module in an external CAPE-OPEN compliant simulation environment, using standard interfaces for the communication. Another method is to incorporate modules from other packages as unit models inside the SolidSim environment or to use external property packages inside SolidSim. A simpliﬁcation which is usually made in simulations is to deal often only with one particle attribute, for example when designing a classiﬁer it is assumed, that homogenous solids are classiﬁed, i.e. all particles have the same density, while in reality often mixtures with a distribution of the densities has to be treated. The stream structure of SoildSim allows the efﬁcient storage of and access to complex hierarchies of nested and dependent distributed solids properties [38]. In addition, the stream object provides the information to the unit models in a way that the model gets only the information needed for the model speciﬁc calculation, hiding all additional complexity. This is explained by means of a simple example: in industrial processes the grinding stock is often a mixture of a few materials. These have different densities, strength and moisture content. For the purposes of comminution the moisture content is not an input parameter for the simulation but the material stream object provides the moisture distribution further to a dryer taking into account the change into particle size resulting from comminution. Figure 11 shows the SolidSim ﬂowsheet for the processing of harbour sediment by the City of Hamburg in Germany [39]. In this process, contaminated material is separated into ‘‘clean’’ sand fractions and a highly contaminated silt fraction for land ﬁlling. Without going into detail, Fig. 11 is intended to show that SolidSim is already capable of simulating complex processes in solids processing. The industry partners involved in the ﬁrst tests of the still preliminary version of the tool (e.g. Dow Chemical) evaluated SolidSim as a promising tool for the efﬁcient handling of solids processes. Although the model library still requires extensions the positive industrial evaluation revealed the need for predictive models for solids processes. The existing program provides an excellent platform for further development. From a strategic point of view, this example shows that the existing product design paradigm still needs – and will need for a long time – model development on the level of unit operations. These models should aim at a clear separation of machine and material functions. Whereas material functions have been developed for impact grinding [40,41] where single particle impact is a

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Fig. 11. Flowsheet for processing harbour sediment in the city of Hamburg, Germany (METHA).

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reasonable assumption, the material function of particle ensembles are far from being understood (e.g. in a stirred media mill).

5.3. Modelling of mill– classiﬁer circuits within SolidSim The simulation of the comminution within SolidSim is divided into different levels of sophistication. On the one side a rough estimation of the grinding result can be achieved by implementing the short-cut models, including Rittinger’s, Bond’s or Kick’s law. These models provide as calculation result characteristic particle sizes, e.g. x80. By inserting an additional parameter, e.g. standard deviation, a PSD can be reproduced. On the other simulation level more detailed models for hammer mills and stirred media mills are implemented.

5.3.1. Comminution For the modelling of hammer mills at steady-state conditions a general model is implemented. The model is not bound to a speciﬁc mill geometry so that it is applicable for different mill designs. It also aims at a clear separation between material properties and machine speciﬁc features. Ideal mixing in the grinding zone is assumed. In this case the selection and breakage functions do not depend on the position coordinate. Furthermore it is assumed that all the particles are transported to the impact region, e.g. all particles are stressed. A quantitative approach for the characterization of the breakage probability has been used in the grinding model [19]. According to that approach the breakage probability of various materials of different size ranges can be represented by a single master curve not only for one impact but for more successive impacts k (Fig. 12). The

100

S/%

75

50

25

0

0

2 4 fmat k x (Wm,kin - Wm,min) /−

6

Fig. 12. Master curve of the breakage probability of various materials in the case of impact comminution.

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latter assumption is a reasonable approximation for a small number of impacts. For a large number of impacts a weakening of the particles may occur due to the formation and activation of internal ﬂaws and dislocations. Such a change in the internal morphology of the particles may lead to changing material parameters. For k successive impacts the equation for the breakage probability can be written as

S ¼ 1 exp f Mat xkðW m;kin W m;min Þ ð31Þ The parameter fMat is a material parameter, which comprises all material properties. Wm,min is the mass-speciﬁc energy threshold necessary for particle breakage and Wm,kin is the mass-speciﬁc kinetic impact energy. The particles from class i undergo within one comminution stage of duration dt k stress events with an energy Wm,kin,i. The change of the mass within one comminution stage can be written as: i1 X Dmi ¼ Si k i ; W m:kin;i mi þ bi; j k j ; W m:kin; j Sj k j ; W m:kin; j mj

ð32Þ

j¼1

In matrix form the population balance (PB) for all size classes is written as ~P ¼ m ~ F þ Dm ~ ¼ ½jjIjj jjSjj þ jjbjj jjSjj m ~ F ¼ jjZjj m ~F m

ð33Þ

jjZjj represents the comminution matrix which summarizes the data of the grinding kinetics of all particle classes. If the stress frequency and the impact energies as well as the breakage probability and the mass transfer coefﬁcients are known, the population balance for one comminution step can be solved by this simple matrix multiplication. Some of the information necessary for the calculation is hard to determine experimentally. The stress frequency and the stress intensity for example cannot be measured directly, they depend on the mill geometry and the multi-phase ﬂow inside the mill. Assuming the ﬂow surrounding the impact element is a potential ﬂow and the form of the impact elements being cylindrical, the impact probability of different sized particles can be calculated. The impact probability is deﬁned as the fraction of particles which cannot follow the ﬂuid ﬂow and become impacted on the cylindrical element. CFD calculations (for particles with density 2700 kg/m3) have shown that to a ﬁrst approximation the impact probability of particles larger than 10 mm is above 90%. Furthermore the impact velocity of the particles can be assumed as equal to the inﬂow velocity. Under these conditions k and Wm,kin are equal for all particle sizes. In this case the breakage probability and the mass transfer coefﬁcient depend only on the particle size and on the stress conditions. Since there is no experimental data about the impact velocity and the impact angle inside the mill, it has been assumed that the impact velocity is equal to the rotor velocity. There are two equivalent ways to calculate the comminution result. If the analytical functions for Sj(k,Wm,kin) and bij(k,Wm,kin) are explicitly known they can be

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implemented in equation (33) and the population balance can be solved by multiplication of the feed PSD with the comminution matrix for the corresponding stress intensity. If this is not the case, a breakage probability and a breakage function determined by single particle experiments can be implemented in equation (33). The result after n stressing events arises after multiplying the comminution matrix n-times with the feed PSD. Assuming that there is no change in the particle properties during comminution, e.g. neglecting the fatigue effect, the population balance for n stressing events can be simpliﬁed to ~ P ¼ jjZjjn m ~F m

ð34Þ

The population balance in its matrix form is often applied to model the grinding process. With respect to the ﬂowsheeting simulation of solid processes this form of the population balance allows short calculation times.

5.3.2. Classi¢cation The modelling of the classiﬁcation process within SolidSim is also divided in short-cut models and detailed models. The classiﬁcation process is often described by separation functions. The simplest case which can be implemented is the ideal separation. If a sieve is used as a classifying unit there are also separation functions implemented which include parameters describing the sharpness of classiﬁcation. If the classiﬁcation takes place by a sifter there are two modelling levels available. The ﬁrst level is based on the type of classiﬁcation used. There are short-cut models for sifters working in counter-current or crosscurrent ﬂows and such using centrifugal forces for the classiﬁcation, e.g. impeller wheel classiﬁers. The more detailed model for the calculation of the classiﬁcation process is based on an approach proposed by Husemann et al. [42,43]. This model includes the effect of the design parameters on the separation action as well as agglomeration effects occurring during classiﬁcation.

5.3.3. Mill2classi¢er circuits within SolidSim In SolidSim a mill–classiﬁer circuit can be built up as shown in Fig. 13. The grinding and the classiﬁcation are modelled separately and connected in a circuit by the recirculation loop. The mixer is added to mix the feed with the material rejected from the classiﬁer. The classiﬁer can be modelled as a sieve with an ideal sharp classiﬁcation or by other simple one- or two-parameter separation functions. Another possibility for the classiﬁcation is to choose a sifter as a classiﬁer (case acc. to Fig. 13). There are short-cut models available but also more sophisticated models which take into account certain agglomeration effects during classiﬁcation.

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Fig. 13. Build-up of a mill–classiﬁer circuit into SolidSim.

The parameters for the breakage probability necessary for the modelling are available from single particle tests. For the modelling of the breakage function the most common types of functions proposed by Austin [44], Vogel [19] and Kerlin [45] can be chosen. Concerning the impact velocity an average impact velocity can be considered or an impact velocity distribution function proposed by Kerlin [45] can be taken into account. This is a Maxwell-distribution function.

5.4. The reference state method for modelling of mill– classiﬁer circuits Another method for the modelling of tumbling mills and the corresponding mill–classiﬁer circuits at steady-state conditions is the reference state method, proposed by Espig et al. [46,47]. The method uses characteristic curves for the description of the grinding and classiﬁcation processes. The derivation of the characteristic curves is based on physical and/or empirical dependences resulting from the real operation state in a technical plant. A valuation and simulation software solution (PMP, Grainsoft GmbH) has been developed by the authors enabling the process engineer to determine the optimal working point for the milling process. For this purpose process data are collected, evaluated and

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interpreted including operating experience. The method is designed to estimate further experimental conditions, e.g. for scale-up. The quantiﬁcation of the mill models is carried out by the user, passing through several model levels, taking into account physical dependencies and empirical equations developed from experiments over the years. In the following sub-sections this procedure is explained for both classiﬁcation and milling.

5.4.1. Classi¢cation The modelling of the classiﬁcation process is carried out on three subsequent levels. On the ﬁrst level the classiﬁcation at a certain operating state is described by the separation efﬁciency curve. It can be explicitly calculated if two mass ﬂows and two PSDs from the three streams in the classiﬁcation process are known. The separation efﬁciency curve can also be determined if the three PSDs (inlet, coarse, ﬁnes) are known. The disadvantage of the separation curve is that it changes its form with varying operational conditions. If the classiﬁer efﬁciency is not sufﬁcient, a certain amount of material is not classiﬁed at all. As a result the separation curve resembles a ﬁshhook. Therefore a transformation of the separation curve is undertaken in the second modelling level, in order to achieve a standardized curve for various operational conditions. The transformation is described as follows: T ðx Þ ¼

T ðx Þ T 0 T ðx=x s Þ T 0 ¼ 1 T0 1 T0

ð35Þ

The separation size xs as well as the unclassiﬁed fraction T0 depend on the operational conditions and are important parameters for the characterization of the classiﬁcation process. These are called state variables and the transformed curve T is known as standardized curve. If the standardized curve is invariant for varying operational conditions (which is proved by sensitivity analysis) the third modelling level is set up. At this level the operational state is characterized by the main inﬂuencing parameters. The correlations have an empirical or physical nature. The inﬂuencing parameters can be related to the feed as well as to the operational conditions. One simple approach is that the classiﬁcation depends on _ C , rotational three operation parameters, e.g. mass ﬂow at the classiﬁer inlet m speed of the classiﬁer n and air volume ﬂow rate V_ . The dependence of the state variables xs and T0 on the inﬂuence variables is described by a power law as follows: a _ aC1 na2 V_ 3 x s ¼ a0 m

ð36Þ

b _ bC1 nb2 V_ 3 T 0 ¼ b0 m

ð37Þ

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5.4.2. Comminution For the comminution the N-cell model is used (basically applied for ball mills). As mentioned previously (in Section 3.3.2.3 ) the model is suitable for the description of the axial mixed transport of the material inside the mill. The transport through the mill takes place with constant velocity. Assuming a ﬁrst-order rate of breakage with respect to the residue and a two-parameter RTD the product cumulative size distribution QP is described as SðxÞ t N QP ðxÞ ¼ 1 ð1 QF ðxÞÞ 1 þ N

ð38Þ

t denotes the mean RTD and the mixing is considered by the number of cells N. It has been proven that this equation is a good approximation to the solution of the population balance equation. Taking into account the invariance condition W SðxÞ ¼ const.

ð39Þ

the continuous comminution process with feed F can be represented by the following characteristic curve, which is denoted by the authors as energy characteristic curve [46]: P eff W ðxÞ ¼ Fz

"

1 QP ðxÞ 1 QF ðxÞ

1=N

#1 1

ð40Þ

This curve can be considered invariant with respect to the operational parameters and can be easily determined from one steady operation mode. For this purpose the mass ﬂow as well as the PSD at the mill inlet and outlet have to be known. The effective energy input for comminution can be measured or estimated from actual mill parameters. The number of cells N in the model is estimated from the ratio between mill length and mill diameter. Various investigations have shown that the characteristic curves for different operational conditions are slightly shifted. In this case the shift of the curves is modelled by additional parameters. This can be for example the ratio between the inlet particle size and the ball diameter. In [48] it has been demonstrated that the energy efﬁciency curve is successfully applied for the prediction of the grinding result in different tumbling mills running under various operation modes. Another method to model complex technological systems as well as separate process steps is the theory of Markov chains. The objective of such investigations is the estimation of conditions, under which the mill–classiﬁer circuit has stable steady-state characteristics. A formulation for the transient calculation of the behaviour in a circuit has been proposed by Mizonov et al. [49].

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6. DISCUSSION OF THE MAIN INFLUENCING PARAMETERS There is a large number of parameters which inﬂuence the product size distribution. On one side these are material properties, on the other side there are mill characteristics which have to be taken into account. The product particle size distribution results from the interaction between the material and the mill. A way of choosing the proper milling device in order to achieve the desired product properties has been described by Mu¨ller and Polke [50] for impact rotor mills and stirred bead mills. To achieve the desired ﬁnes of a certain material, the material properties have to be taken into account. According to the hardness of the material and its plastic behaviour the mill type is assigned. The mill type deﬁnes the stress mechanism used for comminution. For materials tending to plastic deformation the mill has to be additionally cooled during comminution. The mill type also depends on the desired product ﬁnes, for instance in dry grinding particles up to several micrometres can be produced. For grinding in the nanometre range wet grinding in stirred media mills with additional stabilization stages is applied (see Chapter 13). The attrition occurring during comminution is also an important cost factor which has to be taken into account for the speciﬁcation of the mill type. Hammer mills for example are preferably used for grinding of soft materials because of the high attrition effects on the strike elements. For hard materials and smaller particle size distributions air jet mills are used.

6.1. Inﬂuence of the mill operation mode on the product size distribution The mill operation mode has a large inﬂuence on the product size distribution (the variety of operation modes is described in Section 2). In closed circuit operation narrow particle size distributions can be achieved at long grinding times. The same result can be achieved after a few passages using passage operation. Figure 14 shows the grinding results of limestone produced in a circuit and a passage operation mode at otherwise equal conditions. It is evident that the particle size distribution produced in a circuit mode is broader compared to the passage operation mode. The difference between the two operation modes is considerable especially at a lower number of passages. However, in the case of circuit operation narrow particle size distributions can also be achieved if the throughput is increased (Fig. 15). A comparison between the grinding results in a closed mill–classiﬁer circuit with a passage operation mode of the mill is shown for an inorganic pigment in Fig. 16. The ﬂowsheet of the mill–classiﬁer circuit, which is built up by a ring gap mill and a hydro classiﬁer, is shown in Fig. 2. One can see that in the mill–classiﬁer circuit

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903

Fig. 14. Width of the particle size distributions of limestone as a function of the speciﬁc energy input produced in circuit and passage operation mode at otherwise equal conditions [2].

Fig. 15. Inﬂuence of the throughput on the width of the particle size distribution at a circuit operation of a stirred disc mill [2].

much steeper and ﬁner particle size distributions are obtained in comparison to wet grinding alone. One has to take into account that the ﬁnes of the PSD in the case of circuit operation are strongly dependent on the separation efﬁciency of the classiﬁer.

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Fig. 16. Particle size distributions of an inorganic pigment. Comparison of a mill–classiﬁer circuit with a wet grinding in passage operation [2].

100 90 80

cum. % passing

70 60 50 40 30 20 10 0 1.0

5

10

50 100 particle size x / µm

500

1000

Fig. 17. Calculated particle size distributions in a dry ball mill at different mill operation modes based on the same energy efﬁciency curve [48].

Espig and Reinsch [48] have calculated the resulting product size distribution at various operation modes (Fig. 17) using the same energy efﬁciency curve (see Section 5.4.2). The results show that the coarsest particle size distribution at the mill discharge is produced in an open-circuit mode, following the batch process.

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The ﬁnest and steepest particle size distribution is obtained in a closed-circuit mode. The sharpness of the product particle size distribution depends in this case on the separation efﬁciency of the classiﬁer used.

6.2. How to produce narrow particle size distributions: the example of powder paint This sub-section handles brieﬂy the criteria for choosing the right mill and the possibilities of optimization of the mill operation mode in order to achieve narrow size distributions. As an example the comminution of powder paint is chosen. Powder paints are ﬁlled polymers in dispersed form. Their production often includes comminution and classifying steps. It is required that the particle size is tuned to a well-deﬁned, usually very narrow size distribution. The maximum particle size of powder coatings generally lies between 40 and 100 mm. Because the particles tend to produce unwanted agglomerates, the minimum of ﬁnes is restricted to 5–10 mm. In order to produce powder paints in the desired particle size range, ﬁrstly the most suitable grinding mechanism (impact, compression or cutting) should be chosen. Compression is not suitable due to plastic ﬂowing of the polymer matrix of the powder paint. Cutting does not provide the necessary ﬁneness. Therefore the only possible stress mechanism is impact. Secondly, the impact mill suitable for the grinding of power paint has to be speciﬁed. Unsuitable are mills providing a high thermal stress (e.g. mills with a narrow grinding slot). The powder paints are reactive systems which tend to stick at higher temperatures. The high temperature sensitivity of the material requires cooling of the system during grinding. Suitable are hammer mills with a high air throughput (necessary for the particle transport to the classiﬁer) which acts as cooling agent. High air throughputs can also be achieved in air jet mills but one has to take into account the high energy consumption of this mill type. An advantage of the latter is that abrasion does not play a role. Since the maximum particle size (40–100 mm) can also be achieved in a hammer mill, this mill type would be the more proﬁtable solution. As already mentioned, the maximum particle size in the case of powder paint production is limited. Therefore, a classiﬁcation step has to be incorporated into the process. The classiﬁcation can be realized by combining the mill operation with either an internal or external sieve or with an internal or external sifter. The sieve is applicable for particle sizes larger than 300–500 mm, therefore for the production of powder paint a classiﬁer is needed. Air classiﬁer mills combine in a compact construction both high air throughputs (necessary for cooling) with an internal classiﬁer. Since powder paint is susceptible to dust explosions, a pressure-shock resistant construction is needed (safety engineering). Once the mill

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has been chosen, the process and operation parameters of the hammer mill need to be speciﬁed.

6.2.1. Rotational frequency of the classi¢er The classiﬁer speed inﬂuences the maximum particle size produced and the RTD of the particles within the mill. With increasing classiﬁer speed the particle size distribution shifts to the ﬁner particle size range. At the same time the amount of coarse material rejected to the mill increases, i.e. the mean residence time of the particles increases.

6.2.2. Rotational frequency of the grinding disc Since the energy input increases with increasing speed of the grinding disc, the breakage probability of the particles increases and the amount of ﬁnes produced after one breakage event on the hammer also increases. The average particle size as well as the maximum particle size of the product size distribution decrease. For the production of ﬁnes less stressing events are required. The operational conditions of the mill need to be adjusted in such a way that the RTD of the material in the mill is minimized. One has to take into account that if the energy supply is not sufﬁcient for crack generation, e.g. if the amount of produced ﬁnes is too small relative to the adjusted cut of the classiﬁer, the mill hold-up increases and the mill becomes clogged.

6.2.3. Air £ow rate and solids throughput The increasing of the air ﬂow rate leads to coarser particle size distributions but improves the cooling of the system. An increasing of the throughput shifts the average particle size to the coarser size range (attention: this trend is not necessarily valid for the width of the distribution). The inﬂuence of the throughput on the average particle size produced gets more signiﬁcant at higher rotational speeds of the grinding disc.

6.2.4. Design parameters Next to the proper selection of the operational parameters the design parameters also inﬂuence the product size distribution. The height of the classiﬁer and the geometry of the classiﬁer blades crucially affect the separation efﬁciency of the classiﬁer. The geometry and the number of impact tools as well as the geometry of the internal chamber wall (e.g. linear or shroud ring) also inﬂuence the amount of ﬁnes produced.

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In order to produce narrow size distributions by comminution, following factors have to be taken into account:

Mechanisms of particle stressing (choice of machine); Stressing intensity (e.g. impact velocity); Impact frequency (RTD); Material properties; Mass concentration; Volumetric ﬂow rate; Rotational frequency.

The ﬁrst three factors have been already discussed above. The inﬂuence of the material properties on the comminution result can be determined in single particle experiments. Galk et al. [51] have shown that results of single particle experiments can be transferred to technically relevant impact mills. The comparison between the single particle tests and the grinding experiments obtained under realistic conditions indicate that single particle tests may be a reliable tool for the characterization of the quality and efﬁciency of real grinding processes. Owing to the idealized impact conditions in the case of single particle impact the particle size distribution which is obtained is the narrowest possible one that can be achieved by impact comminution. It can be interpreted as the physical limit of the grinding process. Narrower distributions in technical grinding can only be achieved if either the mechanical material properties are changed or the produced unwanted ﬁnes are removed by an additional classiﬁcation step. The latter is often chosen for the optimization of the process. For powder production in air classiﬁer mills, cyclones, cyclone classiﬁers or impeller wheel classiﬁers are connected to the mill. The product quality and the yield can be improved signiﬁcantly with the help of an efﬁcient classifying technique [51]. Not only the technical improvement but also the process control contributes to the production of narrow particle size distributions. The exact compliance of the process parameters is therefore of main importance. The particle size distribution is very sensitive to changes into the air and solids ﬂows. In-line particle size analyses have shown that feed ﬂuctuations (caused e.g. by ﬂuctuations in the mass ﬂow of the feed) lead to additional coarsening of the maximum particle size and therefore to large ﬂuctuations in the product quality. However, the changes in the average particle size remain negligible. In order to detect accruing ﬂuctuations by change in the throughput and to ensure narrow product size distributions during the process ﬂow, in-line or on-line measurements of the particle size distribution are required. Additional steps which can be incorporated into the process in order to achieve narrow size distributions include control of the upper size limit by a classiﬁer and/or even additional classiﬁers necessary for the reduction of the ﬁnes fraction.

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Nomenclature A bi,j ||b|| Bi,j c C C0 C Di E E E(t) ET fi f 0i fMat F F[ ] gi H Ip ||I|| k k(t) k(x) ki ki K(k) l mi ~F m ~P m _C m M n n n N pi P

constant in equation (23) (1/s) breakage distribution parameter (–) matrix of the breakage distribution parameter (–) cumulative breakage distribution function (–) ﬁtting parameter in equation (11) recirculation ratio (–) recirculation ratio (–) net recirculation ratio of two classiﬁers (–) axial dispersion coefﬁcient of size class i (m2/s) rescaled speciﬁc energy (J/kg) speciﬁc energy input (J/kg) residence time distribution function (1/s) total efﬁciency of the classiﬁer (–) fraction of mill feed for size class i (–) fraction of feed which for size class i (–) material parameter (kg/(J m)) feed rate to the mill (kg/s) dimensionless non-linear function (–) fraction of circuit feed for size class i (–) cell hold-up (kg) batch grinding kinetics coupled with residence time distribution (–) unit matrix (–) exponent in equation (26) acceleration–deceleration parameter in equation (9) (–) apparent speciﬁc breakage rate function for particles of size x (1/s) constant for particles of size i in equation (20) number of stressing events of size class i (–) Kapur function related to size class i (sk) axial position along the length axis (m) mass of size fraction i (kg) feed vector of the mass distribution in the interval classes (kg) product vector of the mass distribution in the interval classes (kg) mass ﬂow rate of the material at the classiﬁer inlet (kg/s) mass ﬂow rate (kg/s) cell number in equations (15) and (16) (–) number of stressing events in equation (34) (–) rotations per minute of a classiﬁer in equations (36) and (37) (1/min) cell number (–) mass fraction of product of size class i (–) power input to the mill (W)

Modelling of Mills and Milling Circuits

Peff P(x,y)

P12,i, P21,i

q(l,x) QF QP r1,i, r2,i Ri R0 Rin(x) Rout(x) s1,i, s2,i s i S ||S|| S(0) Si Sini S1 SEi t ts T0 T u0 ui V_ wi w1(x,0) w1(x,t) W Wm,kin Wm,min W y y

909

effective power input to the mill (W) dimensionless weighting function expressing the contribution of the generic size y to the disappearance rate of the particles of size x (–) S-shaped function representing the probability of transfer of particles of size class i from zone 1 to zone 2 and from zone 2 to zone 1, respectively (–) mass fraction density (1/m) cumulative particle size distribution of the feed material (–) cumulative particle size distribution of the product material (–) coarse fraction of size class i from classiﬁer 1 and 2, respectively (–) cumulative size distribution larger than size class i (–) cumulative size fraction retained above size class i (–) cumulative oversize fraction with respect to particle size x at mill inlet (–) cumulative oversize fraction with respect to particle size x at mill outlet (–) weight fraction of size class i for classiﬁer 1 and 2, respectively, which is recycled back with the coarse stream (–) overall fraction of ﬁnes after classiﬁcation (–) speciﬁc rate of breakage (1/s), breakage probability (–) matrix of the breakage probability (–) initial speciﬁc breakage rate (1/s) speciﬁc breakage rate of particles of size class i (1/s), breakage probability of particles of size class i (–) initial speciﬁc breakage rate parameter (1/s) speciﬁc breakage rate for the coarsest size class (1/s) energy-normalized breakage rate parameter for size class i (kg/J) time (s) residence time of a plug-ﬂow stream (s) not classiﬁed fraction in a separation process (–) standardized separation curve (–) mean axial ﬂow velocity of water (m/s) mean axial ﬂow velocity of particles of size class (m/s) air volume ﬂow rate (m3/h) mass fraction of material in size class i at the mill outlet (–) mass fraction in the coarsest size class at time t ¼ 0 (–) mass fraction in the coarsest size class at time t (–) mass of hold-up in the mill (kg) mass-speciﬁc kinetic impact energy (J/kg) mass-speciﬁc threshold energy for particle breakage (J/kg) energy efﬁciency curve (–) daughter particle size (m) energy dissipation coefﬁcient in equation (30) (–)

910

xi xk xs ||Z||

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particle of size class i (m) particle sieve size at which k ¼ 0.5 (m) separation size (m) comminution matrix (–)

Greek Symbols l m Z G(x) t

ﬁtting parameter in equation (11) (–) constant in equation (23) (–) constant in equation (23) (–) grade efﬁciency curve of the classiﬁer (–) mean residence time in the mill (s)

REFERENCES [1] A. Kobayashi, H. Nagasaka, K. Iizuka, H. Yoshida, Separat. Puriﬁcation Technol. 36 (2) (2004) 157–165. [2] H. Karbstein, F. Mu¨ller, R. Polke, Aufbereitungs-Technik 36 (10) (1995) 464–473. [3] L.G. Austin, C.A. Barahona, N.P. Weymont, K. Suryanarayanan, Powder Technol. 47 (3) (1986) 265–283. [4] K.J. Reid, Chem. Eng. Sci. 20 (1965) 953–963. [5] E. Bilgili, B. Scarlett, Chem. Eng. Technol. 28 (2) (2005) 153–159. [6] D.W. Fuerstenau, P.C. Kapur, A. De, KONA 21 (2003) 121–132. [7] S. Malghan, D.W. Fuerstenau, Zerkleinern, in Dechema-Monographien, Verlag Chemie., 1976, pp. 613–630. [8] L.G. Austin, P. Bagga, Powder Technol. 28 (1) (1981) 83–90. [9] D.W. Fuerstenau, A.-Z.M. Abouzeid, Int. J. Mineral Process. 31 (1991) 151–162. [10] D. Fuerstenau, A. De, P. Kapur, Int. J. Mineral Process. 74 (2004) 317–327. [11] E. Bilgili, B. Scarlett, Powder Technol. 153 (1) (2005) 59–71. [12] L.G. Austin, J. Shah, J. Wang, E. Gallagher, P.T. Luckie, Powder Technol. 29 (2) (1981) 263–275. [13] L.G. Austin, Powder Technol. 5 (1971/1972) 1–17. [14] R.K. Rajamani, D. Guo, Int. J. Mineral Process. 34 (1–2) (1992) 103–118. [15] R. Verma, R.K. Rajamani, Powder Technol. 84 (2) (1995) 127–137. [16] H. Cho, L. Austin, Powder Technol. 143–144 (2004) 204–214. [17] K. Yildirim, H. Cho, L. Austin, Powder Technol. 105 (1–3) (1999) 210–221. [18] D. Weedon, Minerals Eng. 14 (10) (2001) 1225–1236. [19] L. Vogel, W. Peukert, KONA 21 (2003) 109–120. [20] L. Vogel, W. Peukert, Chem. Eng. Sci. 60 (18) (2005) 5164–5176. [21] J. Dodds, C. Frances, P. Guigon, A. Thomas, KONA 13 (1995) 113–124. [22] H. Benzer, Powder Technol. 150 (3) (2005) 145–154. [23] L.G. Austin, V.K. Jindal, C. Gotsis, Powder Technol. 22 (1979) 199–204. [24] L.G. Austin, Powder Technol. 143 (4) (2004) 240–252. [25] H. Schallnus, J. Schwedes, Dissertation, TU Braunschweig, 1987. [26] V.K. Jindal, L.G. Austin, Powder Technol. 14 (1971/1972) 35–39. [27] C. Gotsis, L.G. Austin, Powder Technol. 41 (1985) 91–98. [28] L.G. Austin, P.T. Luckie, K. Shoji, Powder Technol. 33 (1) (1982) 113–125. [29] K. Sato, N. Meguri, K. Shoji, H. Kanemoto, T. Hasegawa, T. Maruyama, Powder Technol. 86 (3) (1996) 275–283. [30] H. Cho, L.G. Austin, Powder Technol. 122 (2002) 96–100.

Modelling of Mills and Milling Circuits [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51]

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H. Berthiaux, J. Dodds, Powder Technol. 106 (1999) 78–87. H. Berthiaux, C. Chiron, J. Dodds, Powder Technol. 106 (1999) 88–97. P.C. Kapur, P.K. Agrawal, Chem. Eng. Sci. 25 (6) (1970) 1111–1113. H. Berthiaux, C. Varinot, J. Dodds, in 5th World Congress Chem. Eng., San Diego, CA, USA, July 14–18, (1996) 220–225. H.J.G. Gommeren, D.A. Heitzmann, J.A.C. Moolenaar, B. Scarlett, Powder Technol. 108 (2000) 147–154. R. Illner, H. Neunzert, Transport Theory Stat. Phys. 16 (2–3) (1987) 141–155. R.R. Klimpel, L.G. Austin, Powder Technol. 38 (1) (1984) 77–91. E.U. Hartge, M. Pogodda, C. Reimers, D. Schwier, G. Gruhn, J. Werther, KONA 24 (2006) 146–158. E.U. Hartge, M. Pogodda, C. Reimers, D. Schwier, G. Gruhn, J. Werther, Aufbereitungs-Technik 47 (1–2) (2006) 1–10. C. Subero-Couroyer, M. Ghadiri, N. Brunard, F. Kolenda, Powder Technol. 160 (2) (2005) 67–80. L. Vogel, W. Peukert, Powder Technol. 129 (2003) 101–110. K. Husemann, R. Wolf, R. Herrmann, B. Hoffmann, Aufbereitungs-Technik 35 (8) (1994) 393–403. K. Husemann, Aufbereitungs-Technik 31 (7) (1990) 359–366. L.G. Austin, Powder Technol. 106 (1999) 71–77. H.P. Kerlin, Dissertation Technische Universita¨t Braunschweig, 1980. D. Espig, V. Reinsch, Int. J. Mineral Process. 44–45 (1996) 249–259. D. Espig, H. Schmidt, ZKG Int. 50 (10) (1997) 564–570. D. Espig, V. Reinsch, in 10th European Symp. Comminution, Heidelberg, Germany, 2002. V. Mizonov et al., Chem. Eng. Process. 44 (2) (2005) 267–272. F. Mu¨ller, R.F. Polke, Powder Technol. 105 (1999) 2–13. J. Galk, W. Peukert, J. Krahnen, Powder Technol. 105 (1–3) (1999) 186–189.

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Part IV: Applications

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CHAPTER 21

Particle Strength in an Industrial Environment Gabrie M.H. Meestersa,b, a

DSM Food Specialties, P.O. box 1, NL 2600 MA Delft,The Netherlands Delft University of Technology, Julianalaan 136, NL 2628 BL Delft,The Netherlands

b

Contents 1. Introduction 2. Particle breakage and wear 3. Simulating particle strength by numerical modelling 4. Fatigue by repeated loading 5. Fundamental multiple granule breakage tests 6. Experimental studies on multi particle breakage 7. Relation between single- and multi-particle 7.1. Prediction of breakage in pneumatic transport with repeated impact testing 7.2. Differences between the repeated impact tester and pneumatic transport system References

915 916 923 924 926 926 928 929 932 938

1. INTRODUCTION Formulation is the science to form blends of active ingredients with auxiliary compounds that can be transformed by e.g. granulation techniques for solid formulations. The structure is made such that it is suitable for the application ﬁeld for which it is designed. Much formulation work has been done for decades in the ﬁeld of pharmacy. Powder dosage recipes can be transformed in this way into tablets for easy administration. Patients swallow tablets much quicker and with a better mouth feel than powder, which ﬁrst has to be dissolved or dispersed in water. Special polymer coatings can target the release of the active component in a tablet, either in a speciﬁc place (the bowl) or give a controlled or delayed release. Industrial customers desire better ﬂowability and metering properties, low dust formation during handling, better product stability, etc. Consumers of solid products set high demands for convenient use of products and therefore demand for Corresponding author. Tel.: +31(0)15 2792434; Fax: +31(0)15 2792219; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12024-8

r 2007 Published by Elsevier B.V.

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formulation technology. Mixing cocoa with sugar and milk is nowadays much easier than in the past, because products for instant cocoa and chocolate drinks have been developed. Sugar does come in many different forms depending on the use. Large crystals are used in pies, cubes or crystalline sugar for coffee and tea, powder sugar for pancakes, etc. In many industrial processes, granules, agglomerates, crystals or any other solid are made with the desired particle properties. Although produced according to speciﬁcations, many of these products no longer conform to speciﬁcations after handling and transportation by e.g. pneumatic transport line, during ﬁlling and dosing or after transportation by rail or road. This is most of the time caused by breakage of particles, since they lose some mass due to all kinds of wear. Particle strength technology relies highly on experience. Only limited fundamental understanding of granule strength has been obtained and hardly any fundamental prediction for mechanical behaviour of particles is available.

2. PARTICLE BREAKAGE AND WEAR Breakage of particles in industrial processes has been described in the literature with many different terms. The main problem in distinguishing and using these terms has been the lack of understanding as to what has actually happened to the particles in the test or process equipment. Well-designed tests will be able to isolate a different cause of damage to the granules. Unfortunately, most tests for assessing attrition tendency reported in the literature simply generate damage to the granules, and this damage is only related to the test process, not to a mechanism. Ennis and Sunshine [1] described the urge to understand breakage mechanisms, as breakage tests applied to predict strength problems in process equipment often generate breakage mechanisms that differ from the one dominant in the real process. When comparing different products on particle strength, it is important that all products are subjected to the same mechanisms. Differences in test mechanisms easily result in completely wrong conclusions. To avoid this, more fundamental understanding of breakage mechanisms is essential. Austin [2] used fracture and abrasion to investigate the mechanisms in autogenous grinding of rocks. Considering his ideas for breaking particles, the following terms will be used to qualify the type of damage to the particles: If the force is associated with a normal impact damage to the granules, the breakage process will be called attrition, or if these normal forces break the granules into fragments directly at the ﬁrst impact, it will be called fracture; here usually a higher force is exerted to the particles. Figure 1 shows a typical result of an attrition process.

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Fig. 1. Example of damage caused by normal forces; this type of damage is attrition.

If, however, the impact velocity has a signiﬁcant tangential component, the impact damage shows a different effect: it results in more spherical, polished particles. If the impact occurs at low velocities, impact damage only occurs very locally, resulting in a smooth granule that gradually becomes smaller. This we call abrasion. If the impacts have a large velocity (higher force is exerted), the granule loses large areas of its surface and this results in signiﬁcant damage to the granules. This we then call chipping. Table 1 summarises the different damage processes (Fig. 2 and 3). Hutchings [3] deﬁned attrition to include both large-scale fracture of particles and the ﬁner-scale removal of material by wear. Very often it is seen that it takes many events before damage to the granules can be observed. During these events cracks propagate and ﬁnally lead to damage. This damage progression is called fatigue. The ﬁrst step in understanding the behaviour of industrial agglomerates is to review the theory of an extremely simple model of these particles: the perfectly elastic, homogeneous sphere. Analysis of this model system is available in literature, but hardly any applicability for the layered, non-homogeneous, non-spherical and not perfectly elastic granule is known. Assuming a ﬂat interface between two spheres, Herz [4] expressed the contact pressure at the interface as qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2 3F a r p sy ¼ ð1Þ 2p a3 and he deduced that at the centre of the sphere (r ¼ 0) the pressure equals sy ¼

3F 2pa2

and

sx;z ¼

5 3F 12 pa2

ð2Þ

in which sy is the contact pressure at the interface between both spheres, s x and sz the stress perpendicular to the direction of the contact force, a the radius of

918

Table 1. Damage characterisation related to the loading conditions of a granule

Direction

Magnitude Normal

Tangential

Low force ‘‘Wear’’

High force ‘‘Fracture’’

G.M.H. Meesters

Particle Strength in an Industrial Environment

919

Fig. 2. Comparison of granules before and after damage due to tangential impacts: by abrasion the granule surface becomes extensively polished and any sharp edges disappear.

Fig. 3. Two attrition mechanisms: erosion (left photo), if only the outside layer is damaged and fatigue (right photo) if the damage extends to the inside of the granule and initially is hardly measurable, but increases signiﬁcantly in time.

contact, rp the radius of the sphere and F the total contact force between the spheres. The area of contact rapidly decreases in size. For a big 1 cm glass sphere, the radius of contact is 1% of the particle diameter. For a 1 mm glass bead this contact area only amounts 0.1% of its radius. Theoretical treatment of plastic behaviour of perfect spheres is for example given by Hutchings [3]. He described impact times and contact radii for small metal spheres and investigated the effects of particle size of both purely elastic and purely plastic particle impacts. He described contact stresses between

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elastic spheres striking a plain target that deforms either purely elastically or purely plastically. He showed that the maximum stresses are independent of size of the sphere. He deduced for a purely elastic impact the maximum local pressure Pe that occurs inside the particle, the elastic contact time te and the contact radius ae, that relate to each other according to 1280 1 4 2 Pe ¼ rv 243p4 f ðEÞ

2=5

5p t e ¼ 1:47 4

r2=5

r v 1=5

ðf ðEÞÞ2=5

ae ¼ ½1:25 prf ðEÞ1=5 rv 1=5

ð3Þ ð4Þ

FðEÞ ¼ 1 n21 =E 1 þ 1 n22 =E 2 where r equals particle density, v the particle velocity, f is a function of geometry only (and constant for spheres), E Young’s modulus, r the particle radius and n the Poisson ratio. According to elastic theory, an ever-smaller area of contact will lead to an everhigher stress at the contact area between two spheres. For very small particles this stress becomes too high at a molecular level, and particles will deform due to an additional mechanism: local plastic deformation. Hess and Scho¨nert [5] discussed the typical behaviour of plastic yielding for small particles and identiﬁed the transition regime from brittle breakage to plastic yielding. They studied local force maxima in force deformation curves and identiﬁed the appearance of brittle breakage by a sharp decrease in measured compression force and any second or later peaks in the compression force as extended breakage of the debris. Absence of the brittle breakage phase was identiﬁed as plastic deformation behaviour. As the shapes of the compression curves showed a large scatter, they characterised these curves individually by a curve characteristic number. If a particle deforms plastically during an impact, the maximum local pressure, the contact time and contact area are shown to be [3] Pe ¼ Y te ¼

ae ¼ 2

p r p ð2r=3PÞ2=5 2

1=2

r pv

1=2

2r 3P

1=4 ð5Þ

where Y is the yield stress of the material and P the local compression pressure. Finally, Hutchings argued that the average strain introduced into a metal during

Particle Strength in an Industrial Environment

921

either elastic or plastic loading is e ¼ 0.2 a/rp, (only dependent on the ratio of contact radius a to sphere radius rp) and used contact time and average strain to calculate average strain rate upon impact. Timothy and Pearson [6] discussed the contact pressure distribution occurring when plastic deforming spheres are compressed. Kendall [7] fundamentally stated a brittle ductile transition limit depending on particle size, limiting grinding possibilities of granules. In comminution it is very important to understand whether a granule fails under certain loading conditions in a brittle way, and thus creates (many) fragments, or in a plastic way. Kendall [7] showed that it is possible to deduce a size below which only plastic deformation takes place. As plastic deformation does no longer reduce particle size, this has strong implications in the grinding industry. Puttinck [8] focused on plastic deformation of polymer surfaces. He investigated the transition in compression behaviour from elastic to plastic deformations. He discussed indention experiments and argued that particle size effects dominate all ductile–brittle transitions. The total strain energy in a granule increases with its volume (with the third power of particle size), while the energy released by crack propagation depends on the created crack surface (square of particle size). So for a given material, particle and test geometry there must be a critical size of the tested particles. This size is according to x 0C ¼

E

G

Y 2 f ðx 0 =l c Þ

ð6Þ

where E is Young’s modulus, G the fracture surface energy, Y the yield stress in uniaxial test, x0 the particle size, lc the length of the cracks present and f a function that is a characteristic of each test. Puttinck suggested that the combination (EG/Y) identiﬁes the materials toughness and is the material property that describes the crack propagation phenomenon. Moreover, he suggested that particle size effects also would have an effect on fatigue phenomena. Ahuja [9] tested nearly perfect spheres (1 mm diameter) of polycarbonate (example of ductile material) and polystyrene (example of a brittle material) for compression behaviour. Experimental breakage stress was compared to the theoretical breakage stress of bulk polymer material and a good agreement was found for these small particles, indicating that in the absence of micro-cracks the strength of particles is well understood. Weichert [10] calculated particle breakage energy from particle compression experiments. He explained the observed distribution in fragment size, using Grifﬁth theory. Hardly any particle is perfect, and therefore the effects of cracks or homogeneities often have to be taken into account. Using Weibull’s distribution theory, brittle breakage can be described successfully, assuming crack initiation from the inside. However, sometimes breakage is

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G.M.H. Meesters

also initiated from the surface. Statistics can nevertheless be applied to the whole fragmentation process according to Weichert [10]. He argued that from an energy release rate analysis it follows that G¼

s2 Lc f ðgeometry; crackvelocityÞ E

ð7Þ

where s is the stress at a far point away from the crack, Lc the crack length, E Young’s modulus and f a function depending on particle geometry and material crack velocity only. From his bifurcation analysis it follows that fragmentation cumulative distributions can be described as Q ¼ Q(dp, Wm), where dp is the size of the fragments and Wm the elastic energy applied to the particles during cracking. This important result means that once for a single size class of particles exactly the fragmentation behaviour is known, the results can be translated to other experiments. Kanda et al. [11] investigated the effect of crack numbers in a glass test piece and clearly observed a decrease in bending strength of the glass with increasing number of cracks. Stevenson and Hutchings [12] described the indention fracture of brittle materials and spheres. They related critical stress intensity of the brittle materials to the Vickers indenter and the indention pressure. Kanda et al. [13] used Hertzian theory to directly calculate the speciﬁc fracture energy Efrac/Mp (in which Efrac is the fracture energy and Mp the particle mass). 2=3 " #5=3 E frac 1 1 n2 Pc ¼ 4:99 prp Mp E d 2p

ð8Þ

where Pc is the compression breakage force of the particle, dp the particle’s diameter, E Young’s modulus, n Poisson’s’ ratio and rp the particle’s density. Furthermore, he argued that strength is a structure sensitive property and discussed the use of Weibull coefﬁcients to quantify product uniformity. Meyers [14] discussed micro structural effects of dynamic failure. Dynamic failure can occur by tension, shear or compression. It is inﬂuenced by grain size, presence of a second type of grains, texture, impurities, crystallographic structure, pre-strain, inter granular phases and voids. From the above literature it is concluded that when granules contain microcracks and have different sizes, it is very difﬁcult to compare results. Therefore, only experimental results of particles of approximately equal size should be compared. Furthermore, micro-cracks have a major inﬂuence on granule strength measurement. For industrial granules, however, it is extremely difﬁcult to quantify these.

Particle Strength in an Industrial Environment

923

3. SIMULATING PARTICLE STRENGTH BY NUMERICAL MODELLING With increasing availability of computing power, the results obtained from computer modelling become more impressive. Simulations have been reported for agglomerates that are built from spherical and realistic particles with glue inbetween and granules that have even been created by a computer agglomeration process. This chapter will review results obtained from computer simulations, in order to verify whether it is likely that industrial agglomerates can be studied using these programs. Thornton [15] calculated the internal stresses of model agglomerates in compression tests. Advanced computer graphics help to obtain insight in the force ﬁeld distribution inside a granule. Later, Thornton [16] discussed the fundamentals of impacting elastic particles with a wall. His fundamental research, which also included the adhesion force, provided the basis for extended computer modelling work on single particle breakage. Shipway and Hutchings [17] compared numerical solutions of stress distributions inside impacting or compressed granules with experimental results of nearly ideal particles. They showed that material tensile strength can be deduced from sphere compression experiments: s ¼ 0:7

F p d 2p

ð9Þ

where s is the maximum (horizontal) tensile strength inside the particle, F the vertical loading force and dp the particle diameter. This shows that the compression breakage force is directly related to material tensile strength. Kafui and Thornton [18] presented numerical results of normal particle wall impacts for face cubic centred agglomerates. The computer program ﬁrst prepared special agglomerates out of perfect spheres and binder materials. Afterwards, it calculated their behaviour during a collision with a solid wall. The program analysed all the fragments that were created during the particle-wall impact, in order to obtain insight into the breakage mechanisms. Miller and Wadsworth [19] discussed computer calculations on layered particles as proposed for a new nuclear reactor design. These particles consisted of a core, protected by a buffer layer and three coating layers. Finite element computer simulations have been used to understand the mechanical behaviour of these layered, nearly spherical particles. Lian et al. [20] showed that the behaviour of non-compressible Newtonian ﬂuid in-between two colliding spheres could be approximated using simple equations. Their simpliﬁed equations are very suitable for using in computer programs describing the principles of agglomeration.

924

G.M.H. Meesters

Thornton et al. [16] performed computer impact simulations of agglomerates comprising over 1000 primary particles in a two-dimensional array. They could recognise three regimes of impact: shattering, semi-brittle and elastic. Below a certain impact velocity, they observed no damage to the granules. One may conclude that computer analysis has advanced a lot towards realistic simulations of stressed agglomerates. However, the current simulation techniques only study nearly ideal systems (such as agglomeration of sand or glass beads). Industrial agglomerates are still far too complex to analyse in this way. Multi particle breakage (such as that occurring in ﬂuid beds, conﬁned compression tests or high shear mixers) can in principle be translated into many single particle breakage processes. Unfortunately, the dynamic problem of granular movements (a two-phase ﬂow problem) is already difﬁcult to analyse. Nevertheless, some promising steps made have been mentioned in literature. Potapov and Campbell [21,33] studied impact-induced particle breakage by making use of computer-glued agglomerates. They studied the effects of impact velocity and the effect of a possible shock wave. Apart from the ratio of impact energy over breakage energy of the agglomerate, also the ratio of impact velocity over sound velocity was shown to be important. The shape of the fragments was related to the velocity ratio.

4. FATIGUE BY REPEATED LOADING Repeated loading or multiple impacts often much better describe the real industrial events which particles encounter. In a pure brittle solid, a crack opening is in principle reversible. According to Lawn and Wilshaw [22], during unloading of a specimen, cracks can close reversibly when only little plastic deformation has occurred to the specimen. However, crack healing is not often observed, as air molecules rapidly contaminate the interface. Cracks can accumulate in a material when stressed repeatedly and the accumulation of micro cracks can result in reduced material strength (fatigue). Peters et al. [23] investigated size reduction processes on coal particles. They investigated both comminution processes and single particle breakage. According to Peters et al. coal comminution should be analysed by the number of breakage cycles, whereas single particle breakage should be investigated at a microstructure level. They tried to model coal breakage by combining three functions: loading probability, breakage probability and fragmentation size. In compression tests the force of breakage is strongly related to fragment size and, to a lesser extent, to the size of the parent particle; however, the number of fragments is hard to predict. With increasing number of fragments, they ﬁnd less inﬂuence of the parent particle size. When a single particle compression test is continued after primary breakage of the particle, the test starts measuring breakage of the

Particle Strength in an Industrial Environment

925

fragments. Therefore, results at the continuing tests are much more difﬁcult to interpret. Yashima et al. [24] described fatigue testing on spherical particles with (semi-) static compression tests with compression cycle frequencies up to 100 Hz. To investigate fatigue effects on granules they developed suitable test systems. A relatively slow test system uses a magneto motive system, which compresses granules with a very long cycle time (0.01 Hz). A fast test system uses an electrodynamic resonance system (100 Hz) and realises a large number of compression cycles in every experiment. In their experimental results, they showed clearly that fatigue strength can be signiﬁcantly less than strength as determined from direct particle compression tests. They observed an endurance limit: the apparent force a granule can withstand indeﬁnitely. For sodium glass beads this limit is of the order of a million stress cycles. The fatigue breakage strength is of the order of one ﬁfth compared to the mean compression strength. They described their fatigue load deformation curves as the sum of a series of hysteresis curves. They deﬁned fatigue fracture energy as N 1 X E¼ E pm þ E n ð10Þ m¼1

where E is the total work done on the particle, Epm the contribution of individual hysterics loops and En the work done during the ﬁnal compression that breaks the granule. This is under the assumption that fracture occurs at the Nth compression. They also observed that fragments from the fatigue experiments become coarser when fatigue has occurred at a lower stress ratio. They calculated the stress ratio for each compression experiment relative to the average compression breakage stress. Goder and Kalman [25] discussed fatigue characteristics of particles by repeatedly loading a bed of particles in a conﬁned cylinder and quantifying the damage to the granules. They characterised fatigue curves by plotting the percentage of broken particles (undersize) as a function of compression stress and number of repeated cycles and showed that granule breakage can occur at much lower forces due to fatigue. Kalman et al. [26] showed that for large tablets a relation can be found between particle wear, particle compression and crush tests. Often, wear experiments are the most convenient for particle strength measurement, because they are fast and yield reproducible results. They also investigated the effect of ageing of compacts under different humidities and assessed fatigue characteristics. This literature shows that fatigue effects in particle strength have been studied in detail and can be extremely important. Fatigue breakage can occur at less than 20% of the force needed to break the particle in a single experiment. An endurance limit of the order of one million collisions is observed for (single particle)

926

G.M.H. Meesters

glass bead model systems. This limit has only been investigated in single particle compression tests; this research extends investigations onto the fatigue effects for particle impact experiments. Results above, concerning the effect of repeated compression, have been studied with a quite high compression speed and with brittle particles. Beekman [27–29] modiﬁed a stress strain set up to measure fatigue and repeated stress–strain relations with industrial products.

5. FUNDAMENTAL MULTIPLE GRANULE BREAKAGE TESTS De Silva [30] discussed prevention, reduction and measurement of attrition and wear in process equipment. Due to the large number of variables involved, general recommendations are impossible. As industrial processes often show complicated behaviour, a fundamental understanding of these particle attrition tests is therefore only scarcely present in literature. For attrition tests to be useful, it is essential that they simulate the conditions in the process equipment for which information is required. Paramanathan and Bridgwater [31,32] described application of a special annular attrition cell. Their test deﬁned both stress and strain on the granules in a very uniform and accurate way. During the test operation a very well deﬁned boundary layer is formed, which generates very uniform shear conditions. Potapov and Campbell [21] used computer simulation to simulate the annular shear cell. Their results were in agreement with the Gwyn description for attrition rate. Chouteau et al. [34] discussed the development of a test set-up in which a jet of air is used to accelerate granules against a target. Solids were impacted with the target multiple times to investigate the effects of multiple impacts. Shipway and Hutchings [17] argued that in order to assess attrition, a minimum impact velocity should be deﬁned. With most impact test set-ups this impact velocity cannot even be measured in principle, because both particle impact velocity and particle impact orientation need to be controlled accurately to do so. The literature shows that only few attrition tests exist that have a very transparent mode of operation. Beekman [27,29,35], Pitchumani [36–39] and van Verkoeijen [40,41] described a repeated impact tester which is correlated to industrial attrition which occurs in pneumatic transport lines.

6. EXPERIMENTAL STUDIES ON MULTI PARTICLE BREAKAGE Bembrose and Bridgwater [42] have given a thorough review on attrition and attrition test methods. They described the sources of attrition and discussed the

Particle Strength in an Industrial Environment

927

majority of available test methods. They distinguished them into single- and multiple particle tests. Single particle tests give fundamental information, but their results are often difﬁcult to relate to particle attrition. Multiple particle tests often give statistically reliable results, but are more empirical in nature. Multiple particle tests offer the advantage of testing a great number of particles simultaneously. If in the test all particles experience approximately the same treatment, much more accurate results are obtained. Fundamental understanding of these particle breakage processes are however difﬁcult to understand, although several attempts have shown good results [43]. Single particle tests often yield inaccurate results and are often very difﬁcult to perform in a statistically reliable way, however, they result in direct strength information about the particles. Ghadiri et al. [44] have investigated attrition of fcc-catalysts (ﬂuid bed cracking catalyst) in a ﬂuid bed set-up and they compared attrition results with single particle impact tests. They quantiﬁed attrition by sieving, using a sieve one size smaller than the lower limit of the original particle size distribution and deﬁned the attrition rate as the fractional loss of mass of the mother particles per unit time. Jet angle results were studied, to understand the attrition causes. Dessacle et al. [45] reported on attrition evaluation for catalysts in circulating ﬂuid beds, by using the jet-cup attrition test. Static charging of particles was avoided by ﬂuidising with humid air. They showed examples of purely eroding particles and of fragmenting particles. The ﬁrst type showed a rapid decay in attrition rate, whereas the second type showed a nearly constant attrition loss. For comparing attrition behaviour in ﬂuidised beds, therefore, long test periods of several hours were required to result in reliable estimates for fragmentation sensitive granules. Ponomovera et al. [46] reported two attrition tests for cracking catalysts. They used an instrument in which inner and outer drums rotate at velocities of 100 and 1440 rpm, respectively. The granules are contained within these moving drums and show signs of extensive abrasion and attrition. Furthermore, Ponomovera and Kontorovich used a special vibro-saw, which vibrated a box containing some granules with amplitude of 6 mm. Results were reported (as a percentage of abrasion) as a function of test period (typically up to 30 min). In their experiments they showed that addition of a special lubricant powder could reduce impact damage to the catalyst particles. An alternative to the rotating drums, that still obtains controlled particle–particle collisions to generate attrition, is shaking an amount of granules in a box. Deitz [47] described a vibrating box test, in which a bed of 50 ml of coal granules is vibrated at 60 Hz and accelerations up to 7 g (vibration amplitude of about 0.5 mm). He found that attrition rate decreases to an asymptotic value on repeated testing for successive periods. A problem when using ﬂuid bed breakage tests is that the granules in the bed can get statically charged. If this charging is allowed to build up, it can disturb the

928

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bed and test operation, as granules can become adhered to the wall. The effect of static charging can be reduced by providing humid air in the ﬂuid bed or by a better granule surface conductivity, as this promotes draining of the charge on the particles [48]. Especially in process equipment, the interactions between granules can result in complex mechanical effects on particle characteristics, though the principles of granule–granule interactions are easily explained [50]. Briscoe [49] reviewed pneumatic conveying equipment and proposed measures to reduce attrition problems when transporting weak particles. Depending on small differences in equipment set-up, large differences can be obtained when granule attrition rates in different ﬂuid bed tests are compared. Patel et al. [51] argued that these differences in attrition rates observed in literature might well stem from different gas distributors, as these are not at all standardised (not even among ﬂuid bed attrition tests). Furthermore, he showed that for a constant gas velocity, a shallow ﬂuid bed causes more attrition due to the lower pressure drop over the bed. Taylor [52] reported attrition experiments in which results from a standard attrition test apparatus are compared to a negative pressure pneumatic conveying model system. Results reported showed very good agreement between the attrition test results and the attrition observed in the pneumatic conveying set-up. Furthermore, a very smooth relation is demonstrated between the particle attrition and the average particle velocity.

7. RELATION BETWEEN SINGLE- AND MULTI-PARTICLE Austin [53] has established relations between particle compression strength and speciﬁc fracture energy for grinding. Cleaver et al. [54] reported an extensive study on impact attrition of sodium carbonate crystals. In their attrition study, they related percentage of attrition loss to particle density, impact velocity, length, critical stress intensity factor and a constraint factor. In the range of impact velocities that is interesting for pneumatic conveying (roughly 10–40 m/s), they identiﬁed 20 m/s as a transition velocity below which only chipping occurs. Above 20 m/s the sodium carbonate crystals fragment if they hit the wall side-on. High-speed cameras were used to demonstrate these mechanisms. When performing repeated impacts with their crystals, they only observed high attrition rates during the initial impacts but found a constant attrition rate after several impacts, independent of the crystal size. The initial high attrition rates were attributed to the presence of poly-crystalline and damaged particles, thus granules with local weak spots. Johnsson and Ennis [55] tried to relate particle material properties to the attrition phenomena occurring during particle processing. Results from

Particle Strength in an Industrial Environment

929

single-particle micro-indention tests were related to attrition results obtained in a model pneumatic conveying loop. Gahn and Mersmann [56] reported an extensive study on attrition of salt crystals in crystallisers. He related attrition behaviour, which is a source for extended nucleation in a crystallisation process, to particle size, hardness, Young’s modulus, density, particle impact velocity and surface fracture energy. Gahn and Mersmann [56] gave theoretical prediction of attrition rates and experimental veriﬁcation for crystals in a crystalliser. They deduced a minimum impact energy, below which no attrition should occur: W min 64

m3 G3 K 3r H 5

ð11Þ

where Wmin is the minimum impact energy required to observe attrition damage for a crystal, m the crystal shear modulus, G the crack propagation energy, Kr an efﬁciency constant for the collision and H the crystal hardness.

7.1. Prediction of breakage in pneumatic transport with repeated impact testing Particle breakage during pneumatic conveying has been extensively studied before, and efforts have been made to simulate pneumatic conveying using computer simulations. Chapelle and Abou-Chakra et al. [57] have developed a tool kit for describing degradation in dilute phase pneumatic conveying. They used a centrifugal accelerator that gives both normal and tangential velocity components to particles. The angle at which the particles impact can be preset and studied. From the accelerator experiments a breakage matrix is constructed and applied to the pneumatic conveying rig. They claim their model is especially good in predicting the number of ﬁnes. Vogel and Peukert [58] constructed a model for impact breakage, based on Weibull probability statistics and a dimensional analysis approach. From experiments with their single particle impact device, a material function parameter can be determined. This dimensionless parameter can be regarded as a particle property. Frye and Peukert [59,60] applied the previous work to pneumatic conveying and added a computational ﬂuid dynamics (CFD) simulation in order to accurately determine particle velocity during pneumatic conveying. From their CFD simulations they found large tangential velocity components and therefore added a shear tester for simulating degradation due to tangential velocity components. Salman et al. [61] looked at the fragmentation of particles during pneumatic conveying; they found no fragmentation in horizontal pipelines. The fragmentation is strongly dependent on impact angle, the closer the angle to 901 the bigger the probability of fragmentation. Their simulations show that the ﬁrst impact in the bend is at the highest angle; hence, the ﬁrst

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collision causes the major fragmentation. High-speed video recordings in a pilot scale pneumatic conveying rig using glass tubing conﬁrm this [38]. The ﬁrst impact of the particles is at high normal velocity; second and third impacts have much smaller normal velocity components. The most common way to assess the mechanical strength of an irregularly shaped granular material is to mimic conditions that happen in the process equipment. Many test set-ups are just scaled-down versions of the proposed process equipment (small ﬂuid beds, small screw conveyers or small transport loops). A disadvantage of the scale-down approach is that the particle–particle interaction processes in small process equipment can be very complex and can be dependant on the size and geometry of the test equipment. More fundamental information about particle strength is obtained by test methods that focus on the mechanical behaviour of only a single particle. For example, by impacting different types of granules with the same velocity with the same target, and monitoring the sustained damage, a very straightforward test for impact strength assessment is obtained. Though the conditions in such a test set-up can be much better controlled than in multiple particle tests, the results are still difﬁcult to interpret, because the particles are only rarely spherical and homogeneous and the extent of damage strongly depends on the particle orientation upon impact. Particle impact tests often falsely indicate that a certain particle can withstand some impact velocity against a target, because the impact took place at one of its stronger areas. This part will focus on a test method that will improve understanding of the damage caused by collision processes by ensuring that particles will collide randomly with a target an extremely large number of times with a very well deﬁned impact velocity [27,29,36–39,62,63]. By doing so, the impact velocity can be tuned to cause only damage to the particles if they are hit on their weakest edge. Therefore, the damage conditions in the test have become nearly uniform for all impact angles, and the associated damage therefore is related to the weakest parts of the granules only. Closely associated with this is that the large number of impacts signiﬁcantly improve statistical reliability, so that even for very few particles (down to 20 individual granules) reasonable accurate strength investigations can be performed. Beekman et al. [27,29,35,36,62] designed and developed a desktop tester to asses attrition behaviour of small quantities of granules (Fig. 4). Pitchumani et al. [37–39] improved the design to the crank slider repeated impact tester (RIT) (Fig. 5). The question asked in the research of van Laarhoven [63] was, can attrition to granules in the repeated impact tester be compared to attrition during dilute phase pneumatic conveying? If this is possible, the RIT can determine much faster the attrition behaviour of granules than when a pilot scale pneumatic conveying rig is used. A typical RIT measurement takes in the order of half an hour

Particle Strength in an Industrial Environment small mass for tuning 0

931 L=15 cm

2A=4cm second arm only present for stability

2A'=4mm

mechanical amplitude control

50 Herz sieve shaker

Fig. 4. Repeated impact tester as developed by Beekman [27,29,35,36,62].

1 Guide shaft 2 Particle box 3 Linear ball splines 4 Crank shaft 5 Flywheel

Fig. 5. The crank slider repeated impact tester as developed by Pitchumani [37–39].

per sample ,whereas the pilot scale pneumatic conveying rig takes in the order of half a day per sample. The RIT allows for isolation of the normal velocity component, giving the possibility to test the inﬂuence of attrition during pneumatic conveying. Attrition seen during pneumatic conveying can be correlated to the attrition measured during repeated impact testing. This gives a tool able to quickly test newly developed granules on resistance to attrition, for example during pneumatic conveying, or particle–particle interaction in ﬂuidised beds, and even catalyst degradation and damage during charging and discharging hoppers. For each system different conditions need to be set in the RIT.

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7.2. Differences between the repeated impact tester and pneumatic transport system Typical impact velocities of the tested granules in the RIT can be set and vary between 1 and 5 m/s. During transport of granules in the dilute phase pneumatic transport system (PTS), typical air velocities are 20–40 m/s. Typical number of bends in a conveying system are in the order of 6–10 bends. A test in the repeated impact tester has typically 500–10,000 impacts. In order to correlate the two systems, a few assumptions are made. The granule velocity in the PTS is assumed to be 0.7 times the air velocity. Previously Frye and Peukert [64] found the particle velocity to be 5–25 m/s at air velocities of 40 m/s. They used a CFD package and extended it to calculate the particle impact velocity accurately. The simpliﬁcation is made that the highest attrition rate occurs at the highest particle velocity. Therefore, 0.7 times the air velocity is taken as particle velocity. Salman et al. [61] found that particles in straight tubing hardly collide and the ﬁrst impact in a bend causes major damage. Therefore, the assumption is made that a granule impacts once in each bend; hence, the number of bends is equal to the number of collisions in the PTS. The attrition a granule undergoes is related to the mass speciﬁc kinetic impact energy (Wm,tot) of the granule at the moment of collision. The total attrition a granule undergoes is then related to the kinetic energy of each collision, multiplied by the number of collisions (k). Because during pneumatic conveying the air pressure is controlled, the superﬁcial air velocity vair is calculated over the oriﬁce. The particle impact velocity vpart is simpliﬁed to the value of 0.7 vair, which results in W m;tot ¼ 12v 2part k ¼ 12ð0:7v air Þ2 k

ð12Þ

This yields the total kinetic impact energy (Wm,tot). The mass loss of the granules is the measure for attrition. In order to compare low speed RIT with the PTS measurements, a plot of the total kinetic impact energy versus the mass loss is constructed and analysed. Because the attrition mechanism removes surface asperities the granules are expected to become more spherical. In order to quantify the sphericity, a shape factor is determined. Bouwman et al. [65,66] summarised and evaluated existing shape factors. A shape factor is able to discriminate between the sphericity of initial granules and the granules after attrition. The test described by Beekman [27,29,36,62] realises a large number of particle-target impacts in a very short time period by shaking one or more granules in a box. When the displacement, box geometry and cycle time are well designed, this set-up can assure that any particle in the box will collide with at least two walls of the box every cycle. As during each vibration cycle the particles typically

Particle Strength in an Industrial Environment

933

Fig. 6. Sugar crystals treated in the RIT. Numbers in the pictures reveal the number of normal impacts.

Fig. 7. PuroxS particles (sodium benzoate granules from DSM Special Products BV, The Netherlands) which show a rounding off during repeated impact. Numbers in the picture show the number of normal impacts which the particles have experienced.

Fig. 8. Shape of PuroxS granules during repeated impact testing, as a function of the number of normal impacts [63].

travel twice the box height and four times the vibration span, the particles in the box achieve high impacts at high vibration frequency only when the box has a large height and the vibration has a wide span. The test must be able to achieve impact velocities that are relevant for testing agglomerate strength. As the granules are expected to break or get damaged at

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Fig. 9. Particle size distribution of PuroxS during attrition experiments in the repeated impact tester. Legend shows which psd corresponds with the number of collisions [63].

Fig. 10. Mass loss during dilute phase pneumatic conveying of PuroxS as a function of the number of bends the particles went through [63].

Particle Strength in an Industrial Environment

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Fig. 11. Particle size distributions of PuroxS particles as a function of the number of bends in a pneumatic conveying system, for 20 (a), 30 (b) and 40 (c) m/s air velocity [63].

936

G.M.H. Meesters

Fig. 12. Mass loss as a function of the total kinetic energy (Wm,tot) transferred to the particles in the repeated impact tester and the pneumatic conveying system compared (for PuroxS).

their weakest places, the test impact velocity can be signiﬁcantly less than those used in single particle impact experiments that are often reported in literature. The changing of the shape of granules is shown in the next two ﬁgures. In Fig. 6 sugar crystals loose their rectangular shape due to the many normal impacts (attrition) and do not break. Figure 7 shows granules which are already round but are smoothed and more rounded off. Figure 8 shows the results in a graph. From Fig. 9 it can be clearly seen that the size distribution of the particles does not change in shape but gradually shifts to smaller sizes. This indicates that the particles loose their mass gradually from the surface and do not break. These PuroxS particles were also put through a PTS (pilot scale) consisting of stainless steal tubing of a total length of 27 m containing six bends. The solids content was kept below 5 wt% (a typical dilute system). The particles were transported with three different superﬁcial air velocities of 20, 30 and 40 m/s. Figure 10 shows the mass loss in the PTS after passing the samples through the system several times. Figure 11 shows the evolution of the particle size distributions as a function of the number of bends, for the three air velocities applied. In order to compare the results from the RIT and the PTS mass fraction which remained as a function of the total kinetic energy (Wm,tot) transferred to the

Particle Strength in an Industrial Environment

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particle during the impacts in the RIT and the PTS were calculated, which is given in Fig. 12. It is clearly shown that the RIT can predict up to about 7 kJ/kg of energy transferred what the attrition behaviour of the PuroxS particles in the pneumatic conveying line will be. If one considers that 7 kJ/kg corresponds to 76, 34 and 19 bend in the PTS for 20, 30 and 40 m/s respectively, the results can easily be used to optimise the particles for a better performance in the PTS using the repeated impact tester, since normally there are anywhere between 2 and 10 bends per conveying set-up. These kinds of tests and test systems still have a lot of drawbacks and imperfections, but prove to be very useful during product optimisation. Nomenclature

e G n m r rp sy sx sz s a dp E Efrac Epm En F f H G k Kr lc Pc Mp N Pe PTS RIT

average strain introduced (–) fracture surface or crack propagation energy (N/m2) Poisson’s ratio (–) crystal shear modulus (N/m2) particle density, (kg/m3) particle density (kg/m3) contact pressure at the interface between both spheres (N/m2) stress perpendicular to the direction of the contact force (N/m) stress perpendicular to the direction of the contact force (N/m) tensile strength (N/m) radius of contact (m) particle diameter (m) Young’s modulus (N/m2) fracture energy (J) energy contribution of hysteresis loop (J) energy applied in normal direction (J) force. (N) function (–) crystal hardness (N/m2) particle breakage energy (J) number of collisions (–) efﬁciency constant for collision (–) length of the cracks (m) compression breakage force (N) particle mass (kg) number (–) maximum local pressure (N/m2) pneumatic transport system (–) repeated impact tester (–)

938

Q rp te v vair vpart Wm Wm,tot Y x

G.M.H. Meesters

fragmentation cumulative distribution (%/m) radius of the sphere or particle (m) elastic contact time (s) particle velocity (m/s) superﬁcial air velocity (m/s) particle impact velocity (m/s) elastic energy applied to the particles during cracking (J) mass speciﬁc kinetic impact energy at moment of collision (J) yield stress of the material (N/m) particle diameter, size of fragment (m)

REFERENCES [1] B.J. Ennis, G. Sunshine, Symp. on Attrition and Wear in Powder Technol., Utrecht, The Netherlands, 1992. [2] Austin, J. Powder Technol. 46 (1986) 81–87. [3] I.M. Hutchings, J. Phys. D. Appl. Phys. 10 (1977) 396–406. [4] H. Herz, J. Reine Angew Math. 92 (1881) 156. [5] W. Hess, K.I. Scho¨nert, Chem. E Symp. Series 63 (1980) . [6] S.P. Timothy, J.M. Pearson, I.M. Hutchings, Int. J. Mech. Sci. 29 (10/11) 713–720. [7] K. Kendall, Proc. Roy. Soc. London, A-series 361 (1705) (1979) 245–263. [8] K.E. Puttinck, J. Phys. D-Appl. Phys. 11 (4) (1978) 596–603. [9] S.K. Ahuja, J. Powder Technol. 16 (1975) 16–22. [10] R. Weichert, Int. J. Miner. Process. 22 (1988) 1–8. [11] Y. Kanda, M. Kikuchi, C. Endo, T.J. Hohma, Powder Technol. 52 (1987) 167–170. [12] A.N.J. Stevenson, I.M. Hutchings, J. Mater. Sci. Lett. 15 (1996) 688–690. [13] Y. Kanda, S. Sano, S. Yashima, J. Powder Technol. 48 (1986) 263–267. [14] M. Meyers, J. Phys. III 4 (September) (1994) . [15] C. Thornton, Acta. Mech. 64 (1986) 45–61. [16] C. Thornton, K.K. Yin, M.J. Adamas, J. Phys. D Appl. Phys. 29 (1996) 424–435. [17] P.H. Shipway, I.M. Hutchings, Philos. Mag. A 67 (6) (1993) 1405–1421. [18] D. Kafui, C. Thornton, First Part. Technol. Forum, Denver, USA, 1994. [19] G.K. Miller, D.C. Wadsworth, Nucl. Technol. 110 (June) (1995) 396–406. [20] G. Lian, M.J. Adamas, C. Thornton, J. Fluid Mech. 311 (1996) 141–152. [21] A.V. Potapov, C.S. Campbell, First Part. Technol. Forum, Denver, 1994. [22] B.R. Lawn, T.R. Wilshaw, Fracture of Brittle Solids, Cambridge University Press, Cambridge, 1975. [23] A.W.P.G. Peters, M. Hagg, J.I. van Brakel, Chem. Symp. Series 69 (1979) 1–19. [24] S. Yashima, H. Horita, F. Saito, T. Minahara, Sci. Rep. Res. Inst., Tohoku Univ. 31 (2) (1983) 236–253 [25] D. Goder, H. Kalman, Part. Part. Sys. Charact. 15 (1998) 150–155. [26] Kalman et al., Powder Technol. 104 (1998) 214–220. [27] W.J. Beekman, Measurement of the mechanical strength of granules and agglomerates, Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 2000. [28] W.J. Beekman, G.M.H. Meesters, B. Scarlett, Compression test method and apparatus for determining granule strength, Genencor International, Palo Alto, USA, US 2000 712034, EP 1337832A, 2000. [29] W.J. Beekman, G.M.H. Meesters, T. Becker et al., Powder Technol. 130 (2003) 367–376.

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[30] S.R. De Silva, Symp. on Attrition and Wear in Powder Technol., Utrecht, The Netherlands, 1992. [31] B.K. Paramanathan, J. Bridgwater, Chem. Eng. Sci. 38 (2) (1983) 197–206. [32] B.K. Paramanathan, J. Bridgwater, Chem. Eng. Sci. 38 (2) (1983) 207–224. [33] A.V. Potapov, C.S. Campbell, J. Powder Technol. 81 (1994) 207–216. [34] N. Chouteau, P. Guigon, J.F. Large, AICHE Symp. series, 1996. [35] M. Gebert, R. Pitchumani, W.J. Beekman, G.M.H. Meesters, B. Scarlett, Repeated impact apparatus and method for characterising granule strength, Genencor International, Palo Alto, USA, Patent Application WO 02/04923 A2, US20000611631, 2002. [36] R. Pitchumani, S. Arce Strien, G.M.H. Meesters et al., Powder Technol. 140 (2004) 240–247. [37] R. Pitchumani, G.M.H. Meesters, B. Scarlett, Powder Technol. 130 (2003) 421–427. [38] R. Pitchumani, Breakage characteristics of particles and granules, Ph.D. Thesis, Delft University of Technology, Delft, The Netherlands, 2003. [39] R. Pitchumani, N. Gupta, G.M.H. Meesters, B. Scarlett, Part. Part. Sys. Charact. 20 (2003) 1–4. [40] D. Verkoeijen., G.M.H. Meesters, P.H.W. Vercoulen, B. Scarlett, Powder Technol. [41] D. Verkoeijen, The measurement and modelling of granulation processes – a turbo, Ph.D. Thesis, Delft University of Technology, The Netherlands, 2001. [42] C.R. Bembrose, J. Bridgwater, Powder Technol. 49 (1987) 97–126. [43] Y.C. Ray, T.S. Jiang, J. Powder Technol. 49 (1987) 193–206. [44] M. Ghadiri, J.A.S. Cleaver, V.G. Tuponogov, J. Werther, Powder Technol. 80 (1994) 175–178. [45] G. Dessacle, F. Kolenda, J.D. Raymond, First Part. Technol. Forum, Denver, USA, 1994. [46] T.P. Ponomovera, S.I. Kontorovich, E.D. Shchukin, Kinetics Catal. 21 (1980) 97–101. [47] V.R. Deitz, Proc. 15th DOE Nucl. Air Cleaning Conf., Dept. of Energy, Washington D.C., 1979, 379–393. [48] D. Boland, D. Geldart, J. Powder Technol. 5 (1971) 289–294. [49] B.J. Briscoe, Powder Technol. 88 (1996) 255–259. [50] O. Molerus, J. Powder Technol. 88 (1996) 309–321. [51] K. Patel, A.W. Nienow, I.P. Milne, J. Powder Technol. 49 (1986) 257–261. [52] H. Taylor, Proc. World Congress on Part. Technol. 3, 1997. [53] Austin, J. Powder Technol. 53 (1987) 145–150. [54] J.A.S. Cleaver, M. Ghadiri, N. Rolfe, J. Powder Technol. 76 (1993) 15–22. [55] L. Johnsson, B. Ennis, First Part. Technol. Forum, Denver, USA, 1994. [56] C. Gahn, A. Mersmann, J. Powder Technol. 85 (1995) 71–81. [57] H. Abou-Chakra, P. Chapelle, U. Tuzun et al., Part. Part. Syst. Charact. 21 (1) (2004) 39–46. [58] L. Vogel, W. Peukert, Powder Technol. 129 (2003) 101–110. [59] L. Frye, W. Peukert, V. Burk, K. Sommer, Analysis of bulk solids attrition: recent results and open questions, Proc. PARTEC, Nuremberg, Germany, 2004. [60] L. Frye, W. Peukert, Chem. Eng. Process. 44 (2005) 176–186. [61] A.D. Salman, N.L.J. Hounslow, A. Verba, Powder Technol. 126 (2002) 109–115. [62] B. Scarlett, W.J. Beekman, G.M.H. Meesters, R. Pitchumani, Particles-Their Strength and Weaknesses, Key Eng. Mater. 230–232 (2002) 203–212. [63] B. Laarhoven-van, S.H. Schaafsma, G.M.H. Meesters, Chem. Eng. Sci., 2006, submitted. [64] L. Frye, W. Peukert, Powder Technol. 143–144 (2004) 308–320. [65] A. Bouwman, J.C. Bosma, P. Vonk, J. Wesselingh, H.W. Frijlink, Powder Technol. 146 (2004) 66–72. [66] A.M. Bouwman, Form, formation and deformation, Ph.D. Thesis, Groningen University, The Netherlands, 2005.

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CHAPTER 22

The strength of pharmaceutical tablets Iosif Csaba Sinka,a,,1 Kendal George Pitta,2 and Alan Charles Francis Cocksb a

Merck Sharp and Dohme Ltd, Hoddesdon, Herts EN11 9BU, UK Department of Engineering Science, University of Oxford,Oxford OX13PJ, UK

b

Contents 1. Introduction 2. Tablet compaction 3. Measurement of tablet strength 4. Effect of process parameters on tablet strength 4.1. Punch velocity and dwell time effects 4.2. Effect of pre-compression 4.3. Compression position in die 5. Compaction simulators and intelligent industrialisation 5.1. Compaction simulators 5.2. Effect of punch velocity and dwell time 5.3. Tablet formulation: discrimination of bonding and compression effects at the molecular level 6. The effect of tablet structure on strength 6.1. Characterisation of tablet microstructure 6.2. Effect of density distribution on the strength and friability of capsule-shaped tablets 6.3. Effect of density distribution on the strength, friability and failure mode of round curved-faced tablets 7. Summary and conclusions References

941 943 945 948 949 951 951 952 952 954 955 956 957 957 962 967 969

1. INTRODUCTION About 80% of all medication is administered as tablets. Tablets are produced by die compaction, which consists of ﬁlling a die with powder, compression between two rigid punches and ejection from the die. The resulting tablet is subject to bulk handling and other post-compaction operations during which it must maintain Corresponding author. Tel:+44 116 2522555; Fax: +44 116 2522525; E-mail: [email protected] 1

Current address: Department of Engineering, University of Leicester, University Road, Leicester LE1 7RH, UK. 2 Current afﬁliation: GlaxoSmithKline, Ware, UK.

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12025-X

r 2007 Elsevier B.V. All rights reserved.

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mechanical integrity. At the same time, tablets must be capable of disintegrating or dissolving in the body to achieve the required bioavailability characteristics after administration. These properties depend on the formulation of the powder or granules and the choice of process parameters during manufacturing. One of the simplest ways of formulating a pharmaceutical powder is by mixing the active ingredients, excipients and lubricants to form a uniform powder blend. However, a granulation step is often necessary to achieve content uniformity, acceptable ﬂow properties, avoid particle segregation, assure physical and chemical stability and improve compactibility. Compactibility is deﬁned as the ability of the granules or powder to form strong compacts. This chapter discusses issues related to tablet strength. The production cycle of high-speed rotary presses is described in Section 2. The process parameters are deﬁned and the kinematics of punch motion is examined together with the force–displacement response of the powder. The diametrical compression test, which represents the main strength characterisation test in the industry, is described in Section 3. The strength of the tablets is dependent on the state of densiﬁcation of the powder, which is given by the applied force. The strength of ﬂat and curved-faced tablets is discussed and other characterisation methods, such as bending tests are presented. The effect of primary manufacturing process parameters on tablet strength are discussed in Section 4. The kinematics of punch motion is dictated by press speed and the geometric characteristics of press and tooling. The punch velocity and dwell time are examined in more detail for a range of production presses. The role of pre-compression in high-speed tabletting is discussed together with other process parameters. Section 5 describes the use of compaction simulators in pharmaceutical formulation design and process development. Section 6 examines issues related to tablet strength from a more fundamental point of view. Experimental techniques used to characterise compact microstructure are presented. The density distribution in more complex, capsule-shaped tablets is examined using X-ray computed tomography (CT). The strength, friability, erosion and disintegration behaviour of capsule-shaped tablets is discussed in detail in relation to the tablet structure. The effect of friction between powder and tooling on the density distribution in round, curved-faced tablets is examined. Identical tablets (same weight, thickness and material) were compressed using different die wall lubrication conditions, which results in opposite density distribution patterns. The use of ﬁnite element analysis is introduced as means of understanding and predicting tablet microstructure. A constitutive model for powder compaction is described and calibrated. The friction coefﬁcient between powder and die wall is measured using a die instrumented with radial stress sensors. The effect of density distribution on tablet strength, friability and failure mode is discussed with special reference to formulation design and process development.

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2. TABLET COMPACTION Tablets are mass produced using high-speed rotary presses. A rotary press consists of a turret (or die table) with a number of tool stations, which consists of upper punch – die – lower punch assemblies. A typical die table is presented diagrammatically in Fig. 1. As the die table is rotated, each station passes successively through the die ﬁll frame, then between pre-compression and main compression rollers (where the granules are compressed to form a tablet) and ejection cam, where the tablet is ejected from the die. The powder feed mechanism consists of a mass ﬂow hopper attached to a feed frame. The feed frame usually contains a number of paddle wheels, which transfer the powder from the hopper outlet to the die opening. The powder ﬂows into the die under the effect of gravity and is assisted by the paddle wheels (force feed), downward motion of the lower punch (suction ﬁll), weight adjustment mechanism (the powder is partially ejected), metering wheel (to improve weight uniformity) and the centrifugal forces and vibrations in the system. A more detailed description of the die ﬁll mechanisms on rotary presses using force feed frames is given elsewhere [1]. Powder compression usually occurs in two stages, under the pre-compression and main compression rollers. The role of pre-compression is to reduce the porosity in two stages as the entrapped air in high-speed tabletting is a known cause of defects such as cracks and laminations [2]. After compression, the top punch is removed and the tablet is ejected by the lower punch as it travels through

8

7

7

4

1

6 6

1

9 10

2

11 3 4

6

5

7 (a)

8

2

(b)

Fig. 1. Rotary press production cycle (a) top view and (b) unfolded view; 1 – die table, 2 – ﬁll cam, 3 – feed wheel, 4 – die ﬁll area, 5 – metering wheel, 6 – pre-compression roller, 7 – main compression roller, 8 – ejection cam, 9 – upper punch, 10 – die, 11 – lower punch.

I.C. Sinka et al.

944 compaction (top punch)

punch head

ejection

5

Displacement, mm

Vertical displacement Land

40

0.4 0.2

30

0 0

0.7 0.8

20

-5

Force, kN

Roll radius head radius

50

10

10

-10 lower punch displ. punch tip (a)

-15 (b)

0 0

0.2 dwell

0.4 0.6 Time, sec

0.8

Fig. 2. Compression kinematics (a) detail of punch head – compression roller interaction during compression and (b) punch displacement proﬁles and forces for a lubricated placebo formulation. The reference point for displacement is the top of the die table.

the ejection cam. The punch travel between die ﬁll, compression rollers and ejection cam is guided using cam mechanisms. Figure 2 illustrates the compaction process in more detail. Compression occurs when the punch head makes contact with the compression roller. The kinematics of punch motion is determined by: 1. geometric characteristics such as the radius of the die table, the radius of the compression rollers and the geometry of the punch head. The punch head is characterised by a radius, and a ﬂat section, which is termed as land, Fig. 2a. 2. Processing conditions: turret speed (rpm). The resulting vertical displacement of the punch in contact with the roller can be determined analytically [3]. At maximum punch displacement, the punch remains stationary with respect to die while the roller is in contact with the ﬂat portion of the punch head. This time interval is referred to as the dwell time (Fig. 2). Figure 2b presents the force and displacement response of a placebo powder formulation subject to a compression schedule where the upper punch displacement mimics the punch head – compression roller interaction and the lower punch is stationary. Figure 2b illustrates that during dwell the force response does not present a ﬂat plateau, which is due to the time dependent elasto-plastic behaviour of the powder that interacts with the mechanical frame loaded elastically. The compression event is fast; in this case the compaction force has risen to over 40 kN in less than 10 ms. A Loading schedule where the lower punch is stationary allows measuring the friction coefﬁcient between powder and die wall. For the lubricated placebo

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formulation presented in Fig. 2b the upper and lower punch forces are almost overlapping, indicating reduced friction between powder and die wall, which results in low ejection forces. The material response and friction behaviour are discussed in detail in Section 6.

3. MEASUREMENT OF TABLET STRENGTH In industrial practice, the most commonly applied strength measurement for a compact is the diametrical compression test. The test was developed independently at the same time by Carneiro and Barcellos [4] in Brazil and by Akazawa [5] in Japan and is referred to as the ‘‘Brazilian’’ or ‘‘indirect’’ tensile test as a tensile fracture is induced in a disc-shaped material by compressive loading across the diameter. The test is simple and easy to perform and has been widely used to determine the tensile strength of a variety of brittle materials such as concrete [6], coal briquettes [7], gypsum [8] and lactose tablets [9]. An analytical solution exists for the stress state in a thin elastic disc subject to loading across the diameter [10]. The tensile stress st is st ¼

2P pDt

ð1Þ

In practice it is important to determine the tensile strength as a function of compaction pressure, which is the principal process variable affecting strength. The tensile strength – compaction pressure proﬁle of aspirin tablets is presented in Fig. 3. The tensile strength increases as the compaction pressure is increased

Tensile Strength (MPa)

2.5 3.75 mm

2

2.5 mm 1.5 1.25 mm 1 0.5 0 0

100 200 300 Compaction pressure (MPa)

400

Fig. 3. Tensile strength – compaction stress proﬁle for aspirin tablets.

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Simple tensile Failure

“Triple cleft” Tensile failure

Compressive Failure

Fig. 4. Failure modes in diametrical compression test.

and this trend is observed for most pharmaceutical materials. High pressures are, however, not always desirable. After a certain threshold defects such as cracks and lamination start appearing in most materials, thus the strength is reduced. Also, the porosity is decreased at high pressures, which can impede on ﬂuid ingress, thus altering the bioavailability behaviour of the tablet. The labels in Fig. 3 indicate tablet thickness, illustrating that the results are dependent on tablet geometry. By deﬁnition, a material property is independent of test specimen size. Some underlying assumptions of the test method are presented below. The remaining of the chapter examines the factors contributing to tablet strength at increasing levels of complexity. The Brazilian method is applicable to materials that exhibit brittle failure in tension. The failure modes of tablets were described by Mitchell [11] as presented in Fig. 4. For some brittle materials padding is required between the sample and the load to assure adequate load distribution [9] and promote failure in tension [12] as opposed to crushing at the interface between loading platen and tablet. Another fracture mode, the triple cleft, was also identiﬁed as failure in tension [12]. High compressive stresses around the loaded part will result in non-tensile failure due to shattering and cracking in the loading region. Hence, the validity of the diametrical compression test under a given set of conditions to determine a tensile strength from a fracture load can be assessed by examining the specimen fragments after failure [8]. The elastic stress analysis (equation (1)) is based on the assumption that the material is linear elastic. The effect of elasto-plastic material behaviour on the diametrical compression response was studied by Procopio et al. [13] in more detail. Nevertheless, equation (1) remains the basis of determining the strength of tablets in industrial practice. Pharmaceutical tablets commonly have curved faces. Two popular shapes, round and elongated (the so-called ‘‘capsule-shape’’) are illustrated in Fig. 5. In practice, the strength of both round and complex shape tablets is also measured by diametrical compression across the diameter or the longest axis. The break force does not take into account the dimensions, shape or mode of failure of the tablet. A relationship for determining the tensile strength of round, convex-faced tablets was obtained experimentally by Pitt et al. [14] for shaped

Strength of Pharmaceutical Tablets

947

D

b

W

a

A

W t

t

(a)

(b)

l

Fig. 5. Geometric characteristics of common tablets (a) round and (b) elongated (capsule) shape.

Tensile strength (MPa)

0.8 0.7 0.6 0.5 0.4 0.3 0

0.5 1 1.5 Face curvature ratio

2

Fig. 6. Tensile strength of convex-face aspirin tablets.

homogenous gypsum castings.

st ¼

10P pD2

1 t t W 2:84 0:126 þ 3:15 þ 0:01 D W D

ð2Þ

Figure 6 presents tensile strength data for curved-faced aspirin tablets determined using equation (2). The optimum face curvature to produce the mechanically strongest tablets is observed at a ratio of face curvature to diameter of ca. 0.7. Tablets at the extremes, for example, ﬂat face (R/D ¼ 0), and deep convex tablets (D/R ¼ 1.47) are inherently weaker. Similar punch-shape affects have also been reported by Newton et al. [15] for microcrystalline cellulose tablets. Cantilever methods for strength testing have been adapted for elongated tablets. For this purpose beam compacts of rectangular section are formed and subjected to three- or four-point bending, where the stress distribution in the specimen is

I.C. Sinka et al.

948

non-uniform, varying from zero at the neutral axis to a maximum at the outer edge surface. The modulus of rupture is calculated from the load at fracture [16]. A drawback with the three-point bending method is that there is a large contribution from shear stresses to the failure force [17] and as such the application of this method to several geometries is inappropriate. In addition, beam testing accentuates the effects of surface conditions on the measured strength and the test can give results considerably different from the true tensile strength [7]. The formulae are summarised by Stanley [17]. For a deﬁned geometry, it is possible to determine the second moment of area (and bending stress) rigorously. An approximate relationship was presented by Stanley and Newton [18] for the tensile strength elongated (capsule-shaped) tablets subject to three-point bending. 3 Pl W þ 2a st ﬃ 2 d 2 6A þ bd

ð3Þ

A four-point bending test produces a region of constant bending moment between two inner loading points [17]. However, due to the relatively small size of pharmaceutical tablets three-point bending is the more common choice. The size of the compact can be quite small. Hancock et al. [19,20] investigated the use of threepoint bending for the characterisation of very small powder compacts of approximately 20 mg. Elastic properties such as Young’s modulus or Poisson’s ratio of the material can be estimated from the linear regions of stress–strain curves. Furthermore, some indication of material properties such as ductility or brittleness can be determined from the graphs of displacement vs. force [20]. The testing methods for determining the strength and elastic properties of a material are well developed and form international standards. These behaviours are presented in detail in materials engineering textbooks [21,22]. Strain rate sensitivity whilst conducting the tensile test should also be considered. Increasing the load rate of concrete cylinders has been reported [11] to result in higher observed tensile strengths. Rees et al. [23] recorded similar observations for lactose tablets and concluded that discrepancies in tensile strength values determined using different testing instruments might be partially attributable to differences in the loading rate applied by the testing machines.

4. EFFECT OF PROCESS PARAMETERS ON TABLET STRENGTH The mechanical and bioavailability properties of tablets are dependent not only on the composition of the powder blend but also on the process parameters during compaction, which are deﬁned in Section 2. The dominant process parameter that

Strength of Pharmaceutical Tablets

949

determines tablet strength is the compression pressure. As the compression pressure is increased, the tensile strength of the material increases. For example, the tensile strength–compaction pressure proﬁle of aspirin tablets is illustrated in Fig. 3. Modern high-speed rotary presses have a large number (i.e. tens) of other adjustable process parameters. The most basic machine settings are: Press (turret) speed. The productivity of a press is proportional with the press

speed and the number of tooling stations. The press speed affects all of the processes during tablet compaction. – Die ﬁll: suction effect, centrifugal forces and powder ﬂowability. – Pre-compression and main compression: For a given press and tooling the punch displacement proﬁle are ﬁxed. The turret speed therefore inﬂuences the strain rate, dwell time and material behaviour. – Ejection: friction is dependent on velocity and other factors as discussed in Section 6. Pre-compression and main compression force. These parameters are adjusted independently by the vertical positioning of the rollers with respect to the die. The pre-compression and main compression forces are independent on turret speed; however, there is some interaction for strain rate-dependent powders. Die feed frame speed. The feed frame is usually driven by a separate motor, which can be adjusted. Compression position in the die. After die ﬁll the lower punch is usually moved downwards. The top surface of the metered powder is below the die table before it is contacted by the top punch. The effect of the process parameters on tablet strength is discussed below.

4.1. Punch velocity and dwell time effects Punch speed and dwell time are two parameters commonly used to help rank different compression events. These two parameters are dependent on press speed and vary from one compressing machine to another as functions of the size of the die table and compression rollers. Armstrong [24] determined punch speed and dwell time for a range of tablet presses, Table 1. We note that the punch speed is not constant during compression, Fig. 2a,b. When in contact with the roller the punch accelerates to a maximum speed, which is then reduced to zero when the roller makes contact with the ﬂat portion of the punch head during dwell. Therefore, any study of punch speed effect must use a consistent deﬁnition of the speed. The effect of compression speed and dwell time on tablet strength is different for each powder formulation. Conversely, the strength of tablets compressed

I.C. Sinka et al.

950 Table 1. Tablet press compaction parameters (after Armstrong [24])

Machine type

Productivity per die (tablets/min)

Punch speed (mm/s)

Dwell time (ms)

F Press (single station) B Press Manesty Express Manesty Nova Press

85 44 100 100

139 163 416 720

0 10.8 3.9 2.14

Fig. 7. Impact of machine proﬁle on the tensile strength (a) microcrystalline cellulose formulation and (b) lactose formulation.

Table 2. Effect of increasing punch velocity on the tensile strength of plane-face aspirin tablets (150 MPa compaction pressure)

Punch velocity (mm/s)

Tensile strength (MPa)

0.008 0.08 0.33 30 500

1.8 1.81 1.76 1.00 0.6

from the same material depends on the type of press used. Microcrystalline cellulose is considered a plastic material in the sense that the densiﬁcation mechanism involves plastic deformation at the contact areas between particles. Lactose, however is a brittle material. Figure 7 illustrates that the properties of microcrystalline cellulose tablets are affected by the pressing conditions while lactose is relatively insensitive to the machine settings and compression proﬁles. The impact of increasing the compaction speed can be seen in the Table 2 for aspirin. As the material strength is primarily dependent on its densiﬁcation

Strength of Pharmaceutical Tablets

951

(or compaction pressure), tablets were compressed to the same pressure. Table 2 illustrates that at low punch velocities (i.e. electro-mechanical testing machines), the tensile strength of the aspirin tablets remains constant, whereas when the velocity is increased (i.e. high-speed rotary presses), the tensile strength is reduced.

4.2. Effect of pre-compression Rotary tabletting machines are equipped with pre-compression rollers, which can be of the same size or smaller than the main compression roller. The purpose of these rollers is to apply a small force (pre-compression) to the powder prior to the main roller applying the main compaction force. Pre-compression can be useful in overcoming tabletting problems such as capping and lamination because it partially expels air from the powder bed and prolongs the compression interval for timedependent materials. Pre-compression typically becomes effective at a pressures greater that 10% of the main compaction pressure. Table 3 illustrates the effect of pre-compression for a lactose-based formulation.

4.3. Compression position in die The position within a die where compression occurs can have an impact on tablet strength, as illustrated in Fig. 8 for curved-faced tablets. The lower down the tablet is compressed the lower the tensile strength. These tabletting problems are often attributed to either air entrapment or increased ejection sliding distance. Cracks and laminations are induced in the tablets at high compaction pressures and as a result the tensile strength is reduced as illustrated in Fig. 8. It can be noted that for ﬂat-faced aspirin tablets (Fig. 3), the strength increases monotonically and capping is not observed. The behaviour in Fig. 8 illustrates that the tablet shape has in this case a notable effect of on tablet behaviour. Table 3. Effect of pre-compression on lactose containing tablet formulation (convex-face tablets)

Pre-compression (MPa)

Main compression (MPa)

Observation

0 0 8 12 12 20 20

120 210 160 160 220 140 160

Capping Lamination Surface cracking Surface cracking Slight-surface defect Satisfactory compact Satisfactory compact

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952

Tensile Strength (MPa)

6 5 4 3 Tablets delaminated

2 1 0 0

50 100 150 200 Lower Compaction Pressure (MPa)

Fig. 8. Impact of die compression position on tensile strength of convex-faced aspirin tablets.

5. COMPACTION SIMULATORS AND INTELLIGENT INDUSTRIALISATION A fundamental understanding of the mechanisms of compaction can assist the formulation of powders. Experimental characterisation is however necessary because behaviours such as compactibility and compressibility are dependent on the processing conditions as illustrated in Section 4. Experimental data are generated using either small-scale production presses with additional instrumentation or specially designed instrumented presses. The instrumented presses vary in sophistication from general purpose material testing machines to high-speed hydraulic systems, which in the pharmaceutical industry are referred to as compaction simulators.

5.1. Compaction simulators The amount of drug substance available for the early stages of formulation design and process development is usually limited. In such situations it is not possible to validate the manufacturing process of solid dosage forms using full-scale rotary presses, which require large quantities of powder to ﬁll the feed frame and operate the press at steady-state conditions. However, compaction simulators can be used to evaluate the performance of a small amount of drug formulation. A compaction simulator is illustrated in Fig. 9a. It consists of a loading frame, die table and two independent servo-hydraulic systems controlled by a computer, which operate the upper and lower punches as highlighted in Fig. 9b.

Strength of Pharmaceutical Tablets

953

Upper crosshead

Loading frame column

Lower crosshead (a)

Upper accumulator, valve, actuator, load cell, punch holder, punch and LVDT assembly Die table Lower accumulator, valve, actuator, load cell, punch holder, punch and LVDT assembly

(b)

Fig. 9. Compaction simulator (a) general view and (b) schematic diagram.

Running separate machine trials can be expensive in terms of both time and material. Compaction simulators can evaluate the effect of different machine settings on the properties of the tablets using a minimal quantity of powder material. Compaction simulators are precision instruments that generate accurate results. The displacement proﬁles of the punches can be programmed such that they mimic the compression schedule of any single station or rotary press running at a given speed. Proﬁles can be obtained in a number of ways which include: machine drawings; kinematics calculations [3]; high-speed photography; radiotelemetry or the use of instrumented punches. The punch penetrations are set corresponding to rotary presses at various levels of pre-compression and main compression forces. In addition, the compaction simulators can accommodate punch and die sets from rotary presses; therefore, it is possible to evaluate the performance of tablets having various shapes. The latest (the so-called third generation) compaction simulators available today allow more detailed investigation of the die ﬁll phenomena by independent programming of various parameters of the feeding process. A compaction simulator experiment typically consists of programming displacement proﬁles for the punches, as exempliﬁed in Fig. 2b. In addition to the displacements and forces applied by the upper and lower punches, the following parameters are measured: Radial stress at the die wall: the axial stress applied by the top punch is in-

creased during compaction and is transmitted to the bottom punch and the die wall. Measurement of die wall stress is necessary to characterise the behaviour of material during compaction and to measure the friction coefﬁcient between powder and die wall using methodologies described elsewhere [25]. The

I.C. Sinka et al.

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compaction simulator can be ﬁtted with dies instrumented with radial pressure sensors. The radial pressure data interpretation is discussed in Section 6. The take-off force necessary to push the tablet off the lower punch after ejection, which can be measured by instrumenting the push-off arm with a load cell. All subsequent interpretation is derived from the six parameters above. The data can be used to assist formulation design and process development as well as setting up rotary presses. The disadvantages of using compaction simulators are related to high capital costs and the requirement for skilled labour for operation and interpretation of the results.

5.2. Effect of punch velocity and dwell time As described in Section 4.1, for a given rotary tablet press, the punch speed and dwell time are interrelated and depend upon the press speed. As the speed of rotation of the tablet press is increased, the punch velocity is increased and the dwell time is reduced. Using a compaction simulator the punch displacement proﬁles can be prescribed arbitrarily; therefore, it is possible to decouple the impact of process parameters on the tensile strength of tablets. Further to the aspirin example presented in Section 4, the effect of speed and dwell time was explored using a compaction simulator by running the proﬁles with either the dwell time kept constant with the punch velocity being changed (Table 4) or varying the dwell time and keeping the punch velocity constant (Table 5). In both Table 4. Effect of punch velocity on the strength of aspirin. Compaction pressure: 150 MPa. Dwell time: 260 ms (constant)

Velocity (mm/s)

Tensile strength (MPa)

50 500

4.2 4.0

Table 5. Effect of dwell time on the strength of aspirin. Compaction pressure: 150 MPa. Punch velocity: 50 mm/s (constant)

Dwell time (ms)

Tensile strength (MPa)

35 160 260 600 3,800

1.3 3.2 4.0 7.0 7.4

Strength of Pharmaceutical Tablets

955

studies, the compression force was set same to allow comparison of the results. The data in Tables 4 and 5 illustrate that for this material the tablet strength is more sensitive to dwell time than to punch velocity. As discussed in Section 4 the punch velocity is variable during compression, while the punch head is in contact with the compression rollers. The need for a consistent deﬁnition of speed was highlighted for obtaining comparable results. Compaction simulators can be programmed to execute linear displacement proﬁles, i.e. constant punch velocity. While such experiments do not mimic the material behaviour on a rotary press, the effect of strain rate on the material behaviour during compression can be evaluated more rigorously.

5.3. Tablet formulation: discrimination of bonding and compression effects at the molecular level Most tablets contain ingredients in addition to the active pharmaceutical ingredient (API). These additional ingredients are termed excipients. Excipients are needed both to bulk out the actives, to facilitate the compression or to modify the biopharmaceutical properties of the tablet. Examples of excipients are:

Fillers to bulk up the tablet to an acceptable size for the patient to handle. Binders to assist in granule and compact formation. Disintegrants to assist the tablet break down in the ﬂuids of the body. Wetting agents, particularly to assist with the dissolution of hydrophobic API. Lubricants to reduce the friction between powder and die. Anti-adherents to prevent sticking to tool faces. Glidants to assist with powder ﬂow into the die on the tablet press. Anti-oxidants or metal chelating agents to help stabilise chemically the API. In addition the ﬁllers can incorporate polymers, which can modify the release of the API from the tablet, for example to give controlled release of the medicament over a number of hours.

All these excipients can interact to affect the compression properties of the ﬁnal tablet. The accuracy and precision of compaction simulators allows discrimination of bonding and breakage effects at particle level. This can help explain and guide formulation efforts. For example polymorphism of the bulk drug can be a concern in pharmaceutical compression. This is because the most stable form of a polymorph would be expected not to form bonds as strong as the less thermodynamically stable form. Direct compaction of two polymorphs of a bulk drug (illustrated in Fig. 10a) shows that the Form A, the more thermodynamically stable form, has

I.C. Sinka et al.

956 5.00

4.00 3.00 2.00 Form B

1.00 0.00 100

(a)

Form A

Tensile Strength (MPa)

Tensile Strength (MPa)

5.00

4.00 3.00 2.00 1.00 0.00

150

200

250

300

0

350

Average Compaction Pressure (MPa)

(b)

100

200

300

400

Lower Compaction Pressure (MPa)

Fig. 10. Compaction behaviour of different polymorphs (a) bulk drug of two polymorphs and (b) bulk drug of two polymorphs after dry granulation.

a lower tensile strength than Form B. However, it is of interest to note that after formulation of the bulk drug as a tablet, these differences in tensile strength can be removed, with the ﬁnal formulated products having similar tensile strengths (Fig. 10b). A similar phenomenon has been reported by Joiris et al. [26] for polymorphs of paracetamol. Fundamental studies on tensile strength and breakage can be readily run on compaction simulators to delineate the impact of different compression conditions. However, these complex interactions may be difﬁcult to translate to the industrial setting particularly for multi-component mixtures or for composite systems. Hence in the interest of intelligent industrialisation, it is sometimes easier to run a given formulation against a bank of tablet machine proﬁles to predict the net effect in a production setting.

6. THE EFFECT OF TABLET STRUCTURE ON STRENGTH In this section, we explore the relationship between tablet structure and tablet properties. Experimental characterisation methods of the density distribution in tablets are reviewed. The strength and friability behaviour of capsule-shaped tablets is characterised experimentally and the results are discussed together with density distribution maps obtained using X-ray CT. The effect of processing parameters on material microstructure, which determines the ﬁnal properties of the tablets, is investigated by examining the role of the friction interaction between powder and tooling. We introduce the use of ﬁnite element analysis to predict density distributions in round, curved-faced tablets. The results are validated by density maps using the indentation mapping method. The effect of tablet structure on strength, friability, abrasion and fracture mode are discussed.

Strength of Pharmaceutical Tablets

957

6.1. Characterisation of tablet microstructure Density distribution measurements have been conducted on powder compacts since the early 1900s [27] and included techniques based on differential machining, hardness tests or on X-ray pictures of lead grids in compacts. The more modern techniques [28] include X-ray CT, nuclear magnetic resonance imaging (NMRI) and acoustic wave velocity measurements. These techniques have been applied to pharmaceutical tablets [25,29–33].

6.2. Effect of density distribution on the strength and friability of capsule-shaped tablets The density distribution in capsule-shaped tablets made of microcrystalline cellulose was determined using X-ray CT [31]. The tablets were compressed using a manual press under controlled die wall frictional conditions. In subsequent work, a series of tablets was manufactured from the same material using the same tooling on a single-station press. Experimental tests on the hardness, friability, abrasion and disintegration of the tablets were carried out. The results are discussed with reference to the density distribution tomographs as follows. Figure 11 presents the density distributions in a capsule shape tablet. The top face of the tablet presents a break-line, while letters are de-bossed to the lower face of the tablet for identiﬁcation purposes.

Fig. 11. Density distribution in tablets using X-ray CT [31].

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958

The tablet presents high-density regions around the edge and low-density regions near the ﬂanks of the break-line and around the lettering. High-density regions are associated with higher strength and can be beneﬁcial around the edges to reduce breakage. However, high-density regions also require longer time to disintegrate when in contact with ﬂuids. Low-density regions imply low local strength of the material, which may result in elegance problems during bulk handling operations of the tablets. These aspects are examined below. A series of tablets were compressed using the same tooling as the tablet in Fig. 11 on a Manesty F3 single-station tablet press running at 71 rpm. The press was instrumented with a data acquisition system to monitor the forces applied by the punches. The compression conditions are presented in Table 6. The average tablet weight was 0.192 g. The compression force (measured at the lower punch) and the average relative density of the tablets were changed by modifying the eccentric settings from batch to batch. The weights and thicknesses were measured using a Metler Toledo Balance and a Mitutoyo Thickness gauge PK-505, respectively, and used to determine the relative density of tablets presented in Table 6. The break force of the tablets across the long axis was measured using a Schleuniger hardness tester. The failure mode corresponds to tensile failure as indicated in Fig. 12. Some ﬂattening at the contacts was observed, especially for tablets compressed to lower relative densities. A number of 6 tablets were tested at each average relative density level (i.e. from each batch in Table 6). The break force of the tablets is illustrated in Fig. 12 as a function of relative density. Given the tablet shape, the break force values were not converted into strength properties, i.e. using equations (1,2). The break force increases exponentially as the density is increased. The material (microscrystalline cellulose) was not lubricated, yielding relatively high break force values. The disintegration was measured using an Erweka ZT 4 disintegration tester using the method described in the US and European pharmacopoeias. The disintegration time as a function of average tablet relative density is illustrated in Table 6. Compression of capsule-shaped tablets

Batch

Eccentric setting

Average lower force (N)

Average RD

ECC27 ECC28 ECC29 ECC30 ECC31 ECC32 ECC33

27 28 29 30 31 32 33

844 930 1054 1291 1730 2358 2621

0.422 0.453 0.493 0.528 0.575 0.627 0.652

Strength of Pharmaceutical Tablets

959

200 180

Break force, N

160 140 120 100 80 60 40 20 0 0.2

0.3

0.4 0.5 Relative Density

0.6

0.7

Fig. 12. Strength of capsule shape tablets. The failure mode is also illustrated. 30

Disintegration time, min

25 20 15 10 5 0 0.2

0.3

0.4 0.5 0.6 Average Relative Density

0.7

Fig. 13. Disintegration of capsule shaped tablets: disintegration time as a function of average relative density. The high-density regions that disintegrate slowly in the disintegration bath are also illustrated.

Fig. 13. Although microcrystalline cellulose can be used in tablet formulations to aid disintegration, at higher relative densities the disintegration time of the singlecomponent material increases exponentially. A visual observation of the disintegration process revealed that during the ﬁrst few seconds in the disintegration bath, the tablets break up in large blocks and the small parts disintegrate rapidly. Some larger parts (as indicated in Fig. 13 after 1 min) remain and their size is slowly reduced. Characteristic is the presence of horseshoe-shaped clusters (Fig. 13b), which originate from the ends of the

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960

capsules where the density is high, see X–Z plane in Fig. 11. These regions disintegrate very slowly, sometimes tens of minutes after the rest of the tablet had disintegrated. The friability and erosion of the tablets was tested using an Erweka tablet friability tester. Friability and abrasion tests are designed to determine the behaviour of the tablets resulting from wear due to shocks, abrasion and attrition. The friability test is carried out in a cylinder with a curved paddle, which is used to demonstrate the resistance to shocks. A cylinder with 12 blades is used in the abrasion tests to examine abrasion and attrition behaviour. These are tests standard and were carried out according to Pharmaeuropa speciﬁcations (Pharmaeuropa Vol. 5, No.3, September 1993, pp. 282–283). Each batch was tested using the standard method for friability and abrasion testing (25 rpm, 100 revolutions). The results are presented in Fig. 14a. It can be observed that the weight loss in the abrasion drum is larger than the weight loss in the friability drum. For tablets made of unlubricated microcrystalline cellulose, which have high strength, the standard tests indicate relatively low abrasion. Therefore, the tests were extended to 200, 300, 500, 1000 and 2000 revolutions for each batch and the weight was measured at each stage. The results are presented in Fig. 14b and illustrate a strong dependence of weight loss with the average density of the tablets. The friability and abrasion results are, however, relatively similar for batches of tablets compacted to the same average relative density. Further to the quantitative results presented in Fig. 14, it is instructive to examine the appearance of the tablets from a more qualitative point of view. Random tablets were selected from friability and abrasion testing from Batch ECC32 (Table 6) after 100 and 2000 revolutions, respectively. The results are presented in Fig. 15 and indicate that tablets compressed to relatively high

Friability

100 revolutions

ECC27F ECC32F ECC30A

2.5

50

2

40

1.5 1 0.5

Weight loss, %

Weight loss, %

3

Abrasion

(a)

EC C 27 EC C 28 EC C 29 EC C 30 EC C 31 EC C 32 EC C 33

0

ECC29F ECC27A ECC32A

ECC30F ECC28A ECC33A

ECC31F ECC29A

30 20 10 0

(b)

ECC28F ECC33F ECC31A

0

500 1000 1500 No. of revolutions

2000

Fig. 14. Weight loss during friability and erosion capsule-shaped tablets (a) standard test method and (b) extended number of revolutions. Labels indicate eccentric settings (Table 6) and friability (F) or abrasion (A) tests, respectively.

Strength of Pharmaceutical Tablets

961

Fig. 15. Friability and erosion of high-density tablets (a) friability and (b) abrasion.

Fig. 16. Friability and erosion of low-density tablets after 100 revolutions (a) friability and (b) abrasion.

density do not present visual erosion. The lower faces of the tablets, which present the letters MSD de-bossed, are not very clear even for the control tablets; this type of elegance issue is commonly observed for underlubricated powder materials (no lubricant was used in this example). Batch ECC28, which was compressed to a low average density (Table 6) presents erosions of the letters on the lower face and in the regions adjacent to the break line on the upper face as illustrated in Fig. 16 after 100 revolutions. These regions correspond to low-density distributions presented in Fig. 11. By 1000 revolutions, the features on the faces of the tablets erode almost completely as illustrated in Fig. 17. However, we note that the band area is still relatively intact, in spite of being an obvious edge prone to erosion. This is attributed to the fact that the local RD is very high in this area. The edges of the break lines are still visible, partly because these areas are recessed into the tablet, partly because the RD is very high at the band area just under the break line.

962

I.C. Sinka et al.

Fig. 17. Friability and erosion of low-density tablets after 1000 revolutions (a) friability and (b) abrasion.

In conclusion, friability and erosion are associated with low-density areas in tablets. For the situation examined the erosion test results in more weight loss than the friability test. The erosion progresses as follows: (1) characters at the bottom, (2) picking at the top of the tablet adjacent to the break line, (3) erosion of top cups, (4) erosion of bottom cup and (5) progressive overall erosion. The severity decreases when the average relative density of the tablet is increased. It is also interesting to note that the upper and lower faces do not erode in a symmetrical manner (Fig. 17). This can be attributed to either die ﬁll or pressing conditions on the single-station press, which are different from the manual compression of tablets examined using X-ray CT in Fig. 11.

6.3. Effect of density distribution on the strength, friability and failure mode of round curved-faced tablets The effect of friction between powder and tooling on the density distribution in tablets is characterised experimentally using the indentation hardness method. Finite element analysis of tablet compaction is carried out to further understand the origin and the controlling factors of density variations. The density distributions predicted by the model are veriﬁed experimentally. The effect of microstructure on strength, failure mode, friability and abrasion are discussed below. Curved-faced tablets made of microcrystalline cellulose were compressed in a 25 mm diameter die using concave punches with a radius of 19.82 mm. The die was ﬁlled by hand with attention to obtaining a uniform initial packing. The lower punch was maintained stationary during compression. To examine the effect of friction on the density distribution, two identical tablets are produced (i.e. having

Strength of Pharmaceutical Tablets

963

Fig. 18. Relative density distribution in curved faced tablets. Experimental density maps (a) high friction, (b) low friction; numerical modelling for (c) high friction (d) low friction.

same weight, height) under different die wall lubrication conditions. First, the die and punches were cleaned (i.e. degreased) before compaction, giving a highfriction coefﬁcient between powder and tooling. Second, the die and punches were pre-lubricated by compressing pure magnesium stearate (a common pharmaceutical lubricant), which results in deposition of a thin lubricant layer at the die wall thus giving a reduced friction coefﬁcient. The density distribution in the two tablets was characterised experimentally by mapping the hardness in a vertical cross-section of the tablet using a method presented elsewhere [25]. Tablets compressed in a clean die present high-density regions around the outer periphery, Fig. 18a. High friction prevents relative sliding of the powder with respect to the tooling into the cup area of the top punch. The situation can be described as compressing columns of powder to different height, which result in the high-density region around the outer edge. During compression complex powder movement occurs in the die. For the low-friction case, relative sliding of the powder with respect to the punch face induces an opposite density distribution trend (Fig. 18b) from the high-friction case (Fig. 18a). The results suggest that it is possible to engineer different microstructures in two identical tablets (same height, weight, materials, etc.) by controlling the friction coefﬁcient between powder and die wall. Tablet compression is an engineering problem that can be examined using ﬁnite element analysis. Finite element analysis is a numerical method that allows for the solution of the equations of equilibrium, equations of compatibility and the constitutive law [34–37]. The material is a mechanical continuum, which is described by the compatibility equations that relate displacements to strains.

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The behaviour of the powder during compaction is described by a constitutive model. One of the simplest constitutive models is Hooke’s law for linear elastic materials which states that the stresses are proportional to the strains and the proportionality constant is termed as Young’s modulus. In a continuum model for powders, the particle–particle interactions are incorporated in a macroscopic constitutive model. The constitutive models used to capture the behaviour of powders during compaction are, however, more complicated than Hooke’s law [38]. We adopt the Drucker–Prager cap model [39] which was developed for granular and porous materials and was used for pharmaceutical tablet compaction modelling [25,40–44]. The model consists of a shear failure line and a cap surface. The shear failure line captures that powder compacts have different strength in tension and compression, which is characteristic to granular and porous materials. The shear failure line is described by cohesion and internal friction angle, which are determined using simple experiments on powder compacts as illustrated in Fig. 19a.

e

(5)

(6)

(4) (3)

(5a)

(5b)

(2) (1) (5c)

Effective Stress, MPa

150 125 100 0.90

50

(c)

10 9 8 7 6 5 4 3 2 1 0 0.2

m

-25 (b)

0 0

25 50 75 100 125 150 Hydrostatic Stress, MPa

0.35 Poisson's Ratio

Young's Modulus, GPa

(a)

(5d)

0.85 0.80 0.75

25

b

R b

0.93

75

0.4 0.6 0.8 Relative Density

0.3 0.25 0.2 0.15 0.1 0.05 0 0.2

1 (d)

0.4

0.6

0.8

1

Relative Density

Fig. 19. Drucker–Prager cap model and elastic parameters for microcrystalline cellulose (a) calibration procedures: 1 – uniaxial tension, 2 – pure shear, 3 – diametrical compression, 4 – uniaxial compression, 5 – triaxial testing, using the following procedures: (i) classic triaxial tests at constant cell pressure, (ii) simulated closed die compaction, (iii) radial loading in stress space and (iv) hydrostatic compression, (b) families of yield surfaces as function of relative density, indicated by the labels, (c) Young’s modulus as a function of relative density, and (d) Poisson’s ratio as a function of relative density.

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The cap surface captures the densiﬁcation behaviour. The surface is described by an ellipse, which is characterised by its size and shape that can be determined using a die instrumented with radial stress sensors. The calibration procedure is described elsewhere [25] and is based on the stress conditions corresponding to closed die compaction and on the normality rule. The instrumented die experiment ﬁxes one point on the yield surface. To determine additional points triaxial testing can be used [45]. It is important to note that the four plastic parameters of the model (i.e. cohesion, internal friction angle, size and shape of the ellipse) are not constant but change during the densiﬁcation. The evolution of the yield surface should therefore be described as a function of the state of the material. Figure 19b presents yield surfaces for microcrystalline cellulose where the material parameters are expressed using relative density as state variable. In addition to the plastic parameters of the model which are described by the yield surface, the elastic properties (Young’s modulus and Poisson’s ratio), are also determined as functions of relative density as presented in Fig. 19c,d, respectively [25]. The friction between powder and die wall is described by Coulomb’s law. The powder compaction literature indicates that the coefﬁcient of friction depends on factors such as contact pressure, local powder velocity, sliding distance, temperature and wall roughness [46–51]. For powder pressing application, the friction coefﬁcient can be measured using sliding piece devices [48] or dies instrumented with radial stress sensors, the latter technique being adopted here. The data analysis is based on the Janssen–Walker method of differential slices [52], and the friction coefﬁcient can be determined using a procedure presented by Sinka et al. [40] D sB sT z=H sT m¼ ln 4H szrr sB sB

ð4Þ

The friction coefﬁcient is dependent on the contact pressure as presented in Fig. 20b, which presents data for microcrystalline cellulose for clean and prelubricated die wall conditions. The coefﬁcient of friction increases as the punch velocity is increased during high-speed tabletting. The effect of punch velocity was examined using a compaction simulator and the effect of the velocity dependent friction coefﬁcient on the density variations was discussed elsewhere in more detail for ﬂat-faced tablets [40]. In addition to the constitutive model and friction, to conduct ﬁnite element analysis, knowledge of the following input factors is also necessary: tablet shape, sequence of punch motions and initial conditions that describe the state of the powder after die ﬁll (i.e. initial density distribution). In this analysis, the tablet geometry corresponds to the size and punch curvatures used in the indentation hardness mapping experiments. During compression the lower punch was

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σT

1

H

σz+dσz τz

dz

σrad

Coefficient of Friction

0.9

z

σz

0.8 0.7 0.6 clean die

0.5 0.4 0.3 0.2

lubricated die

0.1 0

(a)

σB

0 (b)

15

30 45 60 75 Contact Pressure, MPa

90

Fig. 20. Friction coefﬁcient between powder and die wall (a) data analysis and (b) friction coefﬁcient as a function of contact pressure for clean and pre-lubricated die wall conditions.

maintained stationary. The powder is assumed to form a uniform aggregate in the die before compaction. Numerical analysis has been carried out to predict the density distribution in tablets after compaction for both clean and pre-lubricated die wall conditions. The results are presented in Fig. 18c,d. It can be observed that the model predictions are in good agreement with the experimental data presented in Fig. 18a,b. In the following, we examine the effect of density distribution on tablet hardness, failure mode, friability and abrasion. Series of tablets of same weight were compacted to different relative densities using both clean and pre-lubricated tool conditions. The strength was measured using diametrical compression and the results are presented in Fig. 21a. It can be observed that for the same average density the break force is not unique. The breakage behaviour of a tablet under diametrical compression is discussed elsewhere in more detail [42]. Tablets compacted in pre-lubricated dies have a low-density area around the edge in contact with the loading platen. This results in contact ﬂattening, which can be observed visually as well as by examining the force–displacement curves when forces below the fracture load are applied. Of particular interest is the fracture mode, illustrated in Fig. 21b. All tablets compressed in a clean die (which have high density around the outside edge) split diametrically into two parts, similar to the failure of a brittle disc under point loading across the diameter. All tablets manufactured in a pre-lubricated die, which have a low density around the outside edge, delaminate. Quantitative results of the friability and abrasion of tablets compacted in clean and pre-lubricated dies were presented elsewhere [42]. The general conclusions are similar to the behaviour of capsule-shape tablets discussed in

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lubricated

unlubricated

Break Force, N

200 die 180 lubricated 160 die 140 120 100 80 60 40 20 0 0.35 0.40 0.45 0.5 0.55 0.60 0.65 (a) Relative Density (Average)

(b)

Fig. 21. Effect of density distribution in tablets on (a) strength and (b) failure mode of tablet compacted in clean and pre-lubricated tooling.

Section 6.2: low-density regions erode preferentially. For tablets compressed in clean dies erosion and friability occurs at the top of the curved faces, while the edges remain intact. Tablets compressed in pre-lubricated dies present stronger cups, while the edges erode gradually or smaller bits are detached from these regions. It can be concluded that the density distribution in tablets is a result of the combined effect of the contributing factors (material, friction, geometry, pressing schedule and initial conditions). By performing parametric studies, it is possible to identify material properties for the powder composition or processing parameters that yield quality tablets by design, thus reducing trial and error. The aim is not necessarily a uniform microstructure but one that confers desired ﬁnal tablet properties. Thus, numerical analysis can be used in formulation design, process development, tablet image and tool design. In terms of the failure mode it is important to note that the Drucker–Prager cap model described above, where the material behaviour is assumed isotropic, is not able to differentiate between the breakage modes illustrated in Fig. 21b. Although further work is needed to predict failure, the model presented in this sections offers insight into the tabletting process that help explain some of the unexpected features of diametrical compression behaviour.

7. SUMMARY AND CONCLUSIONS The strength of tablets is dependent on the composition of powders or granules and the parameters of the compression process. The process parameters are deﬁned based on the operating cycle of high-speed production presses.

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Tablet strength is primarily dependent on the compression state of the material, which is created by the compression pressure. The strength measurement techniques used in industrial practice are reviewed. The effect of process parameters such as speed, pre-compression, or pressing position in the die has been discussed using experimental data and it was shown that their relative effect is dependent on the powder or granule material. The use of compaction simulators for formulation and process design is discussed. Modern compaction simulators allow separate analysis of compression parameters speciﬁc to production presses as well as various die ﬁll mechanisms. From a mechanistic point of view, the material behaviour is described using a constitutive model. Experimental procedures used to calibrate a modiﬁed Drucker–Prager cap model where the elastic and plastic parameters of the model are expressed as a function of the state of the material are presented. The friction interaction between powder and die wall was measured using a die instrumented with radial stress sensors. A state-of-the-art pharmaceutical powder compaction model is presented. Understanding powder compaction requires knowledge of the following:

powder or granule behaviour during compression, i.e. constitutive model, friction interaction between powder and tooling, tablet shape, sequence of punch motions, and initial condition of the powder after die ﬁll.

These factors cannot be viewed in isolation. The internal structure of the tablet is determined by their combined effect. Therefore, generalisations are not recommended and each situation, that is powder material, friction and processing parameters should be considered individually. The effect of friction between powder and die wall is discussed in more detail to illustrate that different tablet microstructures can be engineered by manipulating the die wall lubrication conditions. The prediction of the model is compared with experimental data for round curvedfaced tablets. The effect of density distribution on tablet strength, friability, erosion, failure mode or disintegration is illustrated for round- and capsule-shape tablets. High-speed tablet compaction is a complex process and requires rigorous analysis. The contributing factors (material behaviour, friction, geometry, pressing sequence and initial conditions) can be studied experimentally and described mathematically. The use of numerical analysis allows sensitivity ranking of the material and process parameters within the operating space of a practical situation and can, therefore, contribute to improving product quality by design instead of trial and error. The experimental and numerical approaches presented in this chapter can be used aid formulation design and process development. A more fundamental understanding of particle–particle interactions, the evolution

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of anisotropy in powder compacts and the powder-air interaction during high-speed tabletting is, however, needed to improve the predictive capability of these tools.

List of symbols st szrr sT sB m P D t W l A a b H z

tensile strength (MPa) radial stress (MPa) axial stress (upper punch) (MPa) axial stress (lower punch) (MPa) coefﬁcient of friction between powder and die wall applied force (N) compact (or die) diameter (mm) compact thickness (mm) central cylinder thickness (see also Fig. 5) (mm) distance between supports (mm) area of curved segment (mm2) height of curved segment (cup depth) (mm) width of the tablet (mm) specimen height (mm) height of radial stress sensor (mm)

REFERENCES [1] I.C. Sinka, L.C.R. Schneider, A.C.F. Cocks, Int. J. Pharm. 280 (1–2) (2004) 27–38. [2] K.G. Pitt, I.C. Sinka, in M. Hounslow, A. Salman (Eds.), Handbook on Granulation, Elsevier, Amsterdam, 2007, pp. 735–778. [3] E.G. Rippie, D.W. Danielson, J. Pharm. Sci. 70 (1981) 476–482. [4] F.F.L. Carneiro, A. Barcellos, RILEM Bull. 18 (1953) 99–107. [5] T. Akazawa, RILEM Bull. 16 (1953) 11–23. [6] P.J.F. Wright, Mag. Concrete Res. 7 (1955) 87–96. [7] R. Berenbaum, I. Brodie, Br. J. Appl. Phys. 10 (1959) 281–287. [8] E. Addinall, P. Hackett, Civil Eng. Pub. Works Rev. 59 (1964) 1250–1253. [9] J.T. Fell, J.M. Newton, J. Pharm. Sci. 59 (1970) 688–691. [10] S.P. Timoshenko, J.N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 1970. [11] N.B. Mitchell, Mater. Res. Stand. 1 (1961) 780–788. [12] A. Rudnick, A.R. Hunter, F.C. Holden, Mater. Res. Stand. 1 (1963) 283–288. [13] A. Procopio, A. Zavaliangos, J.C. Cunningham, J. Mater. Sci. 38 (2003) 3629–3639. [14] K.G. Pitt, J.M. Newton, R. Richardson, P. Stanley, J. Pharm. Pharmacol. 41 (1989) 289–292. [15] J.M. Newton, I. Haririan, F. Podczeck, Powder Technol. 107 (2000) 79–83. [16] J.P. Den Hartog, Adv. Strength Mater., MaGraw-Hill, New York, 1952. [17] P. Stanley, Int. J. Pharm. 227 (2001) 27–38. [18] P. Stanley, J.M. Newton, J. Pharm. Pharmacol. 32 (1980) 852–854. [19] B.C. Hancock, S.D. Clas, K. Christensen, Int. J. Pharm. 209 (2000) 27–35. [20] B.C. Hancock, C.R. Dalton, S. Clas, Int. J. Pharm. 228 (2001) 139–145.

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[21] F. Ashby Michel, R.H. Jones David, Engineering Materials 1: An Introduction to properties, Applications and Design, 3rd edition, Elsevier Butterworh-Heinemann, Amsterdam, 2005. [22] F. Ashby Michael, R.H. Jones David, Engineering Materials 2: An Introduction to Properties, Applications and Design, 3rd edition, Elsevier Butterworth-Heinemann, Amsterdam, 2005. [23] J.E. Rees, J.A. Hersey, E.T. Cole, J. Pharm. Pharmacol. 22 (Suppl.) (1970) 64S–69S. [24] N.A. Armstrong, Int. J. Pharm. 49 (1989) 1–13. [25] I.C. Sinka, J.C. Cunningham, A. Zavaliangos, Powder Technol. 133 (2003) 33–43. [26] E. Joiris, P. Di Martino, C. Berneron, A.M. Guyot-Hermann, J.G. Guyot, Pharm. Res. 15 (1998) 1122–1130. [27] D. Train, Trans. Inst. Chem. Eng. 35 (1957) 258–266. [28] J.J. Lannutti, MRS Bull. 22 (1997) 38–44. [29] G. Nebgen, D. Gross, V. Lehmann, F. Mu¨ller, J. Pharm. Sci. 84 (3) (1995) 283–291. [30] B. Eiliazadeh, K. Pitt, B. Briscoe, Int. J. Solids Struct. 41 (21) (2004) 5967–5977. [31] I.C. Sinka, S.F. Burch, J.H. Tweed, J.C. Cunningham, Int. J. Pharm. 271 (2004) 215–224. [32] C.-Y. Wu, O.M. Ruddy, A.C. Bentham, B.C. Hancock, S.M. Best, J.A. Elliott, Powder Technol. 152 (2005) 107–117. [33] A. Djemai, I.C. Sinka, Nuclear magnetic resonance imaging of density distributions in tablets, Int. J. Pharm. 319 (1–2) (2006) 55–62. [34] T.J.R. Hughes, The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, Dover Pubns., Mineola, NY, 2000. [35] K.-J. Bathe, Finite Element Procedures in Engineering Analysis, Prentice-Hall, Englewood Cliffs, London, 1982. [36] R.D. Cook, Finite Element Modeling for Stress Analysis, Wiley, Chichester, 1995. [37] T. Belytschko, Nonlinear Finite Elements for Continua and Structures, Wiley, Chichester, 1999. [38] J.R.L. Trasorras, R. Parameswaran, A.C.F. Cocks, ASM Handbook, Vol. 7, Powder Metal Technologies and Applications, ASM International, Materials Park, OH 440730002, 1998, pp. 326–342. [39] D.C. Drucker, W. Prager, Quart. Appl. Math. 10 (1952) 157–175. [40] I. C. Sinka, J. C.Cunningham, A. Zavaliangos, Proc. PM2TEC 2001 International Conference on Powder Metallurgy and Particulate Materials, New Orleans, LA, USA, 13–17 May, 2001, Part 1, pp. 46–60. [41] A. Michrafy, D. Ringenbacher, P. Tchoreloff, Powder Technol. 127 (2002) 257–266. [42] I.C. Sinka, J.C. Cunningham, A. Zavaliangos, J. Pharm. Sci. 93 (8) (2004) 2040–2052. [43] C. -Y. Wu, O.M. Ruddy, A.C. Bentham, B.C. Hancock, S.M. Best, J.A. Elliott, Powder Technol. 152 (1–3) (2005) 107–117. [44] I. C. Sinka, A. C. F. Cocks, in P. R. Brewin, O. Coube, P. Doremus, J. H. Tweed (Eds.), Modelling Powder Compaction, Springer, 2007, pp. 225–244. [45] I.C. Sinka, A.C.F. Cocks, C.J. Morrison, A. Lightfoot, Powder Metall. 43 (3) (2000) 253–262. [46] R.F. Mallender, C.J. Dangerﬁeld, D.S. Coleman, Powder Metall. 17 (34) (1974) 288–301. [47] E. Ernst, D. Barnekow, PM ‘94 World Congress on Powder Metallurgy, Les Editions de Physique, 1994, Vol. 1, pp. 673–676. [48] E. Pavier, P. Doremus, International Workshop on Modelling of Metal Powder Forming Processes, Grenoble, France, 21–23 July, 1997, pp. 335–344. [49] P. Mosbah, D. Bouvard, E. Ouedraogo, P. Stutz, Powder Metall. 40 (4) (1997) 269–277. [50] B. Wikman, H. A. Haggblad, M. Oldenburg, International Workshop on Modelling of Metal Powder Forming Processes, Grenoble, France, 21–23 July, 1997, pp. 149–157. [51] S. Roure, D. Bouvard, P. Doremus, E. Pavier, Powder Metall. 42 (2) (1999) 164–170. [52] R.M. Nedderman, Statics and Kinematics of Granular Materials, Cambridge University Press, Cambridge, 1992.

CHAPTER 23

Crystal Growth and Dissolution with Breakage: Distribution Kinetics Modelling Giridhar Madrasa, and Benjamin J. McCoyb a

Department of Chemical Engineering, Indian Institute of Science, Bangalore 560 012, India Department of Chemical Engineering and Materials Science, University of California, Davis, CA 95616, USA

b

Contents 1. Introduction 2. Experimental background 3. Distribution kinetics 4. Results and discussion 5. Conclusions References

971 974 978 982 985 986

Abstract The growth and dissolution (monomer addition and dissociation) of crystals in the presence of breakage is described by a reversible distribution kinetics model based on population balance equations (PBEs). Breakage (or fragmentation) is the process by which the parent particles are broken into smaller entities of signiﬁcant size. Denucleation, wherein particles smaller than the stable nucleus size dissolve instantaneously, is also included in the model. A numerical solution for the PBEs shows the evolution to a steady state crystal size. Various parameters, such as the breakage coefﬁcient, breakage kernel, and the interfacial energy (through the Gibbs–Thomson effect), inﬂuence the crystal growth– dissolution process, and are quantiﬁed. It is shown that when crystal growth occurs with breakage, the number concentration and the average size of the crystals attain steady state even in the absence of aggregation. This steady state value of the crystal size obeys a power-law relationship with the breakage parameter.

1. INTRODUCTION The population balance approach to particulate dynamics has been used extensively to provide a quantitative description of a wide range of phenomena. The primary concept is that the distribution function for particle or molecular properties

Corresponding author. Tel.: +91 80 293 2321; Fax: +91 80 360 0683; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12026-1

r 2007 Elsevier B.V. All rights reserved.

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varies with position in space and time. The distribution function for a continuous property x is deﬁned as c(x,r,t) dx, representing the number density (or volume concentration) of particles at time t and position r with properties in the range from x to x+dx. Though this concept is related to equilibrium statistical mechanics, understanding the dynamics of the distribution requires a non-equilibrium approach. Such approaches have been previously developed through Smoluchowski’s theory of diffusion-inﬂuenced coagulation [1] and the kinetic theory of gases via the Boltzmann equation [2]. Many problems involving the time evolution of particulates, such as those in crystallization, cannot be solved by conventional conservation equations and need to be solved by the proposed population balance approach [3–7]. Population balance equations (PBEs) can be applied to numerous problems. Generically referred to as distribution kinetics and dynamics, this method is a powerful tool for solving problems with particle-mass distributions [3,5] or molecular-weight distributions [8,9]. The basic equations are similar for particles and macromolecules, except for rate coefﬁcient expressions. During recent years, we have solved PBEs for crystal growth [10], Ostwald ripening [11–13], polymorph separation, and aggregation [14]. We have applied PBEs to various polymer reactions [15,16], including chain and step polymerization, and random [17], mid[18] and end-chain scission [19]. We have also extended this method to various other problems, such as crystallization of polymers [20,21], ﬂuid mixing [22], reaction and precipitation [23], granular mixing [24], granular densiﬁcation [25], oscillator synchronization [26], and polymorph transformation [27]. We believe that a multitude of natural and industrial phenomena can be analyzed as a combination of four major processes, namely aggregation, breakage, monomer addition, and monomer removal. For example, particles can become larger because of aggregation or monomer addition or a combination of both processes. Similarly, particles can become smaller due to monomer removal or breakage or a combination of both. Polymer degradation by end-chain scission occurs by monomer removal and chain polymerization occurs by monomer addition. On the other hand, polymer degradation by random chain scission and step-growth polymerization occurs by breakage and aggregation, respectively. The four combined processes in crystals have been hypothesized as a way to explain observed relaxation behaviour of glasses [28,29]. In many crystallization processes, growth and aggregation, along with its reverse (breakage), occur simultaneously. Aggregation during crystallization [30,31] results not only in the enhancement of particle size but also determines product characteristics such as mechanical strength, size distribution, or crystal shape [32]. Using size-independent kernels, several investigators have discussed modelling crystallization with aggregation in studies of alumina trihydrate [33,34], calcium oxalate particles [35], and barium sulphate [36] crystals.

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Previous investigations of crystallization modelling have not explicitly accounted for breakage, though the concepts of breakage have been used in population balance modelling for other applications such as polymerization processes [37,38] and particle aggregation [39–41]. The introduction of a breakage term in the development of crystallization kinetics [32] accounts for the existence of an observed maximum agglomerate size [34,42,43] and the decrease in the agglomeration kinetics observed at high seed masses [32]. The decrease in agglomeration rate is usually modelled by including a factor that stops the agglomeration process at long times. For example, these models [34,44,45] include a function of the supersaturation in the agglomeration kernel such that the agglomeration rate continuously decreases, reaching zero as supersaturation decreases to unity. Recent experimental data [42,43] indicate that a steady state occurs even at constant supersaturation, suggesting that the decrease in the agglomeration rate might be explained by a steady state attained because of a balance between growth and breakage. Recently, crystallization has been modelled [46,47] with a size-independent agglomeration kernel and a breakage kernel of the form, kxnU The value of n depends on various factors, including the mechanism of breakage. For the cases n ¼ 0 or 1, analytical solutions or exact similarity solutions [39] can be obtained, respectively. The case of n ¼ 1/3 is applicable when the breakage rate is proportional to the radius of the particle [46,48]. Breakage can also occur by attrition in crystallizers [49,50] due to shocks caused by contact of the particles with the stirrer [5,51]. The breakage equations have also been widely used in polymer engineering [8], particle breakage [39], liquid–liquid dispersions [52], attrition of carbon [53], and grinding of coal [54,55], alumina [56], and quartz [57]. Breakage is often random, resulting in products being distributed uniformly over all sizes. For those particles smaller than the stable nucleus size and hence unstable, estimates of their dissolution (or denucleation) rate indicate that denucleation is nearly instantaneous [12]. A consequence of neglecting the important denucleation process is that particles would fragment to molecular size, an unrealistic behaviour for crystallization with either homogeneous or heterogeneous nucleation. Unstable crystals vanish spontaneously giving up their mass to solution resulting in deposition on larger particles. This process continues with the decline of the particle number density and increase of the average particle mass. The variance initially increases as the distribution broadens by growth of large crystals and then decreases continuously until eventually vanishing. The theoretical ﬁnal state after an inﬁnite time is a single large particle in equilibrium with the solution. Thus, crystal growth and denucleation always lead to Ostwald ripening in absence of breakage. Breakage, however, interrupts the asymptotic time dependence usually associated with Ostwald ripening. The balancing of growth, denucleation, and breakage allows a steady-state crystal-size distribution

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to be reached. This is in contrast to the asymptotically decreasing number of particles and increasing average size, as with Ostwald ripening. Such difﬁculties do not occur for crystal growth with aggregation in the absence of breakage, because particles will grow as long as the supersaturation is higher than unity and will aggregate until a single particle is realized. This observation is consistent with previous modelling studies of crystal growth with aggregation and no breakage [14] and crystal growth in the absence of breakage or aggregation [10], wherein the supersaturation decreases and the crystal-size distributions show an asymptotic evolution to a single particle. In this work, we show how a combination of particle growth (and dissolution) with breakage can be formulated to describe the time evolution of crystal-size distributions. We show that an important parameter is the interfacial energy, which inﬂuences the stability of nuclei as well as equilibrium conditions and hence driving forces for growth or dissolution. We show that the number concentration and the average size of the crystals attain steady state, which has been observed experimentally. It is shown that this steady state is achieved even in the absence of aggregation and depends on the breakage parameter.

2. EXPERIMENTAL BACKGROUND Several types of equipment are available for investigating crystallization processes and products. In chemical engineering, the development of crystallizers has been in the direction of improving continuous processes. The advantages of continuous operation compared to batch operation are higher throughput, better control of optimal conditions, and smaller crystallizer size. These advantages must be balanced against speciﬁc drawbacks like the slow attainment of steady state and potential instability of operation. The classiﬁcation of crystallizers is often based on how supersaturation is initiated (by evaporation, cooling, saltingout, etc.). Among continuous processes [58,59], the Mixed Suspension Mixed Product Removal (MSMPR), draft tube bafﬂe (DTB), and Krystal crystallizers and variations of these are in common use. Many of these can be operated either as cooling or evaporative crystallizers and are predominantly intended for rapid crystallization of inorganic products. For certain applications, like crystallization of sugar where kinetics are slow in medium to high viscosity systems, tower-type vertical crystallizers are used. Compared to other conﬁgurations, this crystallizer is an application of the plug-ﬂow concept and results in a narrow crystal-size distribution [60]. The issue of fragmentation in crystallization is of great interest as it inﬂuences crystal size in several ways. During crystallization processes, a supersaturated solution nucleates much more readily when crystals of the solute are already

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present or deliberately added. The term secondary nucleation or heterogeneous nucleation is used for this particular pattern wherein the nucleation results from the presence of crystals in the supersaturated solution. There have been several comprehensive reviews on heterogeneous nucleation [61–67]. These theories comprise two main categories; in the ﬁrst, the origin of the heterogeneous nuclei is attributed to the parent crystal. The second origin of heterogeneous nucleation is attributed to the solute and includes impurity concentration and nucleation due to ﬂuid shear. Mechanical breakage during crystallization can occur by collisional or turbulent ﬂuid break-up. Several mechanisms exist for the attrition process. The most important are the result of crystal collisions with other crystals or with surfaces (impeller, wall) of the crystallizer [63,64,67]. The production rate of the attrition fragments depends on the ﬂuid dynamics of the suspension, on the mechanical properties of the solid material, and on the physico–chemical properties of the liquid. The probability of a fragment denucleating increases with decreasing size. As the solubility of a particle increases with decreasing size (causing Ostwald ripening) for a given solution concentration, the supersaturation will decrease with increasing crystal size. Despite the large number of studies on crystallization, experimental studies on the breakage that occur during crystallization are few. In typical experiments [68–70], the growth and breakage kinetics are investigated in a crystallizer at different temperatures and stirrer speeds. The crystallizers are usually equipped with a draft tube and four bafﬂes [69,70] that are aligned perpendicular to each other to ensure effective turbulent mixing. The turbulent ﬂuid motion of the suspension is usually achieved by a four-bladed [71] or a six-bladed [72] disk turbine impeller. The impeller speed is carefully controlled and the effect of the turbulent mixing on the crystallization is quantitatively observed by conversion of the impeller speed to the turbulent dissipation energy and the turbulent shear force. Although both particle collision and turbulent ﬂuid shear are promoted by an increase in the mixing intensity, the crystal size is maximized at a certain agitation speed due to a dynamic equilibrium between particle aggregation and breakage [72]. The effect of crystal-impeller and crystal-vessel collisions on the breakage rate was studied [71]. While the geometry and the position of the impeller had a small effect on secondary nucleation rate, the distance of the impeller from the bottom of the crystallizer inﬂuenced both the crystal-impeller and crystalbottom collisions. The number of impeller blades had only a minor effect on the crystal-impeller collisions but it had quite a signiﬁcant effect on crystal-bottom collisions. In all these experiments, the evolution of crystal-size distribution was monitored by a Coulter counter and/or a laser light scattering (Malvern Mastersizer) device and the image analysis was carried out using a scanning electron microscope (SEM). An impact instrument [73] was developed to test the attrition propensity of single crystals. The principle was to accelerate a single particle in a jet of saturated

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solution circulating in a vertical tube. The jet of solution carrying the particle penetrates into a vessel full of saturated solution in which a steel or glass target has been immersed. The particle is impacted on the target and collected at the bottom of the chamber containing the target. The velocity of the impact was between 2 and 10 m s–1 and a high-speed camera recorded the impact. This allowed the inﬂuence of the impact on the crystallization kinetics of sodium chloride to be determined. In another study [74], the precipitation kinetics of calcium oxalate CaC2O4 was investigated at the laboratory scale with a DTB precipitation reactor at different residence times, feed concentrations, stirrer speeds and with feed point locations inside and outside of the draft tube. Increasing linearly with power input, the measured disruption rates accounted for breakage due to particle splitting and attrition. It was inferred that both decreased agglomeration efﬁciency and increased breakage led to a decrease in the number of agglomerates observed at higher stirrer speeds. The experimental setup consisted of a polyethylene draft tube and four bafﬂes. The contents were stirred with a three-blade marine-type propeller pumping the suspension upward in the annulus and downward inside the draft tube. The feed tubes and withdrawal tube could be positioned exactly in the reactor and scaled up with high accuracy maintaining the supersaturation proﬁle in the system. The particle size distribution was determined with a Coulter counter. To avoid secondary changes of the precipitate during the particle sizing, a suitable electrolyte (saturated solution of calcium oxalate with 3 wt% sodium chloride) was used that fulﬁls the conductivity criteria necessary to detect particles as they pass through the oriﬁce and does not dissolve particles. Feed rates, temperatures of the vessel and the jacket, pH, and calcium ion concentration were monitored online throughout the experiment. As measured by the calcium ion concentration in the liquid phase corresponding to the supersaturation in the reactor, steady state was attained after nearly 10 mean residence times. Then samples were taken and analyzed in the Coulter counter. Samples were also withdrawn to investigate morphology of the crystals under the light microscope with a video camera module CCD and the SEM. Most of the experimental techniques require sampling from the suspension, which alters the quality of the crystal population. Additional errors result from the subsequent sizing or counting methods, which underestimate the number of ﬁnest particles in the system. These problems can possibly be ameliorated with an in situ sensor that measures in-line crystal-size distributions. One such experimental technique [75] is based on determining the turbidity spectra, from which size distributions can be calculated. Polychromatic light is passed to the sensor by optical ﬁbres. After the light has crossed the sensor window, scattering by ﬁne particles results in the attenuation of its intensity. The transmitted light is again passed via optical ﬁbres to a photodiode array spectrophotometer,

Crystal Growth and Dissolution with Breakage

977

which delivers the turbidity spectrum of the suspension. With this procedure the fragmentation of potassium sulphate crystals was investigated in a roundbottomed vessel equipped with bafﬂes and draft tube with an impeller to stir the reactor contents. The particles released in the medium by fragmentation covered a large-size distribution, which was monitored by the sensor described above. Additional particle size determinations were performed ex situ after sampling using a Coulter light scattering apparatus, where the sizing principle is based upon Fraunhoffer scattering for diameters greater than 2 mm, and upon Mie diffusion for diameters below this value. Finally, samples removed from the solution and dried were observed with a SEM to study the size, shape, and surface state of the crystals. The mass of the seeding sample and the stirring rate determined the mechanism of fragmentation, which was primarily attributed to turbulent ﬂuid crystal interactions. A more extensive use of in-process analytical techniques like optical turbidimetry, Attenuated Total Reﬂectance Fourier Transform Infrared (ATR FTIR) spectroscopy, ultrasonic spectroscopy (USS), and X-ray diffraction (XRD) to give direct access to the processing parameters needed for the optimization of crystallization processes has been developed [76,77]. With this facility, dynamic measurements of the onset of crystallization, supersaturation of the mother liquor and crystal-size distribution of the particles produced during the isothermal nucleation, growth, and breakage of monosodium glutamate monohydrate were obtained. The onset of crystallization was determined by the use of optical turbidity measured with a turbidometric ﬁberoptic reﬂectance immersion probe. ATR FTIR measurements within the batch reactor were carried out with an immersion probe equipped with a ZnSe conical internal reﬂection element. Through measurement of the solute concentration, the supersaturation was determined. USS, which involves measurement of the attenuation of transmitted acoustic waves through solid/liquid suspensions, allows the crystal size distribution (0.1–1000 mm) and solid content to be examined in situ during crystallization processes. An on-line USS spectrometer provided solution samples drawn directly from the crystallizer for size analysis. Characterization of crystallographic form was carried out by an online XRD system. With this multitechnique online experimental setup, the simultaneous and dynamic measurements of crystallization onset and supersaturation of the mother liquor together with the crystal size distribution and crystallographic form of the product crystals was obtained. The nucleation and growth processes taking place within a batch reactor were measured. It was possible to obtain an estimate of the critical cluster sizes, follow the time evolution of the crystal size distribution, assess the inﬂuence of particle breakage, and derive growth kinetic parameters [77]. Despite a large number of experimental investigations on crystal growth and kinetics, the determination of crystal-size distribution in the presence of breakage is not straightforward and is still a developing area.

978

G. Madras and B.J. McCoy

3. DISTRIBUTION KINETICS The equations for reversible growth and aggregation-breakage processes for crystals are based on distribution kinetics [10,11] for a well-mixed system; hence, the r dependence is suppressed here. Spatial dependences could be addressed by a compartment model, where different ﬂuid conditions would be assumed for the interacting regions (compartments) in the vessel. We let x and xm represent the masses of crystals and monomer, respectively. For a spherical particle the particle mass x is related to the crystal mass density rc and particle radius r by x ¼ (4/3)pr3rc. A general representation of reversible addition of monomer, M(xm), to a crystal, C(x), is k g ðxÞ

CðxÞ þ Mðx m Þ Ð Cðx þ x m Þ k d ðxÞ

ð1Þ

with rate coefﬁcients, kg and kd, for growth and dissolution, respectively. The crystals may aggregate with rate coefﬁcient ka or fragment with breakage rate coefﬁcient kb, k a ðxÞ

CðxÞ þ Cðx 0 Þ Ð Cðx þ x 0 Þ k b ðxÞ

ð2Þ

In general, all the four rate coefﬁcients are functions of x, and these expressions have a signiﬁcant inﬂuence on the system dynamics. As the primary aim is to examine the inﬂuence of breakage, in this work we ignore aggregation. Governing equations. The PBEs [10,11,14] that govern the distribution of the crystals, c(x,t), and of the monomer, m(x,t) ¼ m(0)(t)d(xxm), are based on mass conservation for the processes represented by equations (1) and (2): Z 1 Z x @cðx; tÞ ¼ k g ðxÞcðx; tÞ mðx 0 ; tÞ dx 0 þ k g ðx x 0 Þcðx x 0 ; tÞmðx 0 ; tÞ dx 0 @t 0 0 Z 1 0 0 k d ðxÞcðx; tÞ þ k d ðx Þcðx ; tÞdðx ðx 0 x m ÞÞ dx 0 x Z 1 k b ðxÞcðx; tÞ þ k b ðx 0 Þcðx 0 ; tÞOðx; x 0 Þ dx 0 Idðx x Þ ð3Þ x

and @mðx; tÞ ¼ mðx; tÞ @t

Z

1

kgðx 0 Þcðx 0 ; tÞ dx 0 þ

0

þ Idðx x Þ

Z

1

kd ðx 0 Þcðx 0 ; tÞðxxmÞ dx 0 x

x xm

ð4Þ

Denucleation of crystals of mass x* at rate I is represented as a loss term in equation (3). With the opposite sign, I would refer to nucleation, which is not considered here. The mass of denucleated crystals adds to solute in equation (4), where the nuclei, each of mass x*, are composed of monomers of mass xm. The

Crystal Growth and Dissolution with Breakage

979

expression O(x,x0 ) appearing in equation (3) is the stoichiometric kernel representing the probability of breakage of x into x0 . Breakage of C(x) can occur into multiple unequal sized crystals, C(x0 ), C(x00 ),y,C(xx0 x00 y). According to Diemer and Olson [78], the following breakage kernels with qZ0 can be formulated for breakage into unequal fragments: Power-law product, Oðx; x 0 Þ ¼ NGðNqÞ=½GðqÞ=GðqN qÞðx=x 0 Þq1 ð1 x=x 0 ÞqðN1Þ1 =x 0

ð5aÞ

Power-law sum, Oðx; x 0 Þ ¼ N½N 1 ðx=x 0 Þq1 ð1 x=x 0 ÞN2 =Bðq; N 1Þ þ ð1 1=NÞð1 x=x 0 ÞqþN3 =Bð1; q þ N 2Þx 0

ð5bÞ

in terms of the number of daughters, N, and the beta function, B(a,b) ¼ G(a)G(b)/ G(a+b). Equations (5a) and (5b) are extensions of earlier work by Hill and Ng [79], and reduce to most known breakage kernels [18]. The random binary breakage kernel is O(x,x0 ) ¼ 2/x0 , which holds if N ¼ 2 and q ¼ 1. When q becomes inﬁnite, the kernel for breakage into N equal-sized fragments is O(x,x0 ) ¼ Nd(xx0 /N). The common breakage kernels encountered in crystallization have been summarized [80]. When breakage produces two fragments, the distribution function is composed of two delta functions centred at the length values of the fragments. If, for example, breakage results in the formation of two equal fragments, the two delta functions overlap. However, in case of erosion, two delta functions, one centred at the monomer and the other at the difference is formed. Formation of two fragments with different mass ratios also occurs when the distribution functions are positioned according to mass fractions. Another breakage kernel is the parabolic distribution function, which has been formulated [81] in terms of a constant that can vary from 0 to 3. In this work, we consider only the random breakage kernel. For growth and dissolution in equations (3) and (4), Dirac delta distributions of breakage products are formulated as monomer removal kernels, d(x(x0 xm)) and d(xxm). When the monomer distribution, m(x,t) ¼ m(0)(t)d(xxm), is substituted into equation (3), the integration yields for the growth and dissolution terms, k g ðxÞcðxÞmð0Þ þ k g ðx x m Þcðx x m Þmð0Þ k d ðxÞcðx; tÞ þ k d ðx þ x m Þcðx þ x m Þ ¼ x m @½ðk g ðxÞmð0Þ þ k d ðxÞÞcðxÞ=@x þ ðx 2m =2Þ@2 ½ðk g ðxÞmð0Þ þ k d ðxÞÞcðxÞ=@x 2 þ

ð6Þ

where we have applied Taylor series expansions and cancelled identical terms of opposite sign. Equation (6) has the form of growth and dispersed growth terms customarily written for crystallization kinetics [5]. The growth rate coefﬁcient, xm[kg(x)m(0)+kd(x)], ensures that an equilibrium state will eventually be

980

G. Madras and B.J. McCoy

reached. The dispersed growth coefﬁcient, (x2m/2)[kg(x)m(0)+kd(x)], increases with increased monomer concentration m(0), as expected. Growth dispersion is therefore a consequence of distribution kinetics embodied in the PBEs. Moments. Integral forms of the rate expressions in the population balances lend themselves to calculations by moments, deﬁned as integrals over x, Z 1 cðnÞ ðtÞ ¼ cðx; tÞx n dx ð7Þ 0

The zeroth moment, c(0)(t), is the time-dependent molar concentration, and the ﬁrst moment, c(1)(t), is mass concentration (mass/volume) of crystals. The average crystal (particle) mass is the ratio, cavg ¼ cð1Þ =cð0Þ

ð8Þ

Measures of polydispersity deriving from the second moment are variance, cvar ¼ cð2Þ =cð0Þ =cðavgÞ

2

ð9Þ

and polydispersity index, cpd ¼ cð2Þ cð0Þ =cð1Þ

2

ð10Þ

Rate coe⁄cients. Diffusion- and surface-controlled growth can be expressed in terms of particle mass x, k g ðxÞ ¼ kg x l

ð11Þ

where l ¼ 1/3 represents diffusion-controlled growth [13]. When growth is limited by monomer deposition and dissociation at the particle surface [82], the rate coefﬁcient is proportional to the particle surface area so that l ¼ 2/3. If the deposition is independent of particle size, then l ¼ 0. Other values of l may be realistic for other rate processes [83,84]. In the current study we consider only l ¼ 0 but the main conclusions are not different for other values of l. Several kernels for the dependence of breakage rate on size are based on ﬂow-induced normal stresses for solid particles [85] and drop and bubble breakup in turbulent dispersions [86,87] and polymer degradation [18,19]. The breakage rate coefﬁcient typically increases with increased particle mass, kb(x) ¼ kbxn or exp(kbxa). The size independent case is represented by n ¼ 0 while the linear dependence of rate on size is given by n ¼ 1. When the rate of breakage increases with size, larger particles fragment at a higher rate than smaller particles. For particles undergoing breakage in crystallization, the possibility exists of shattering which is similar to encountering a phase transition analogous to gelation [80]. Shattering is characterized with an increase in the rate of breakage with the decrease of particle size (no0). In this case, mass is no longer conserved in the particulate phase and a new phase of inﬁnitesimally small particles is formed. Avoiding the possibility [88] that no0, we have used n ¼ 0, 1/3, 2/3 and 1 in the current work.

Crystal Growth and Dissolution with Breakage

981

Microscopic reversibility (detailed balance) [89,90] follows from the equilibrium condition in equation (6) and allows the calculation of kd(x) given an expression for kg(x), k d ðxÞ ¼ mð0Þ eq k g ðxÞ

ð12Þ

To establish an expression for the dissolution rate coefﬁcient, we substitute the Gibbs–Thomson equation, equation (11), into equation (12) to get

where kd ¼

kgm(0) N

k d ðxÞ ¼ kd x l exp½o=ðx=x m Þ1=3

ð13Þ

o ¼ ð4px 2m =3r2c Þ1=3 2s=k B T

ð14Þ

and

Equation (14) deﬁnes o, the ratio of monomer interfacial energy to thermal energy, which plays a key role in controlling nucleation, growth, and ripening (denucleation). Substituting the rate coefﬁcient expressions yields @cðx; tÞ=@t ¼ kg x l cðx; tÞmð0Þ ðtÞ þ kg ðx x m Þl cðx x m ; tÞmð0Þ ðtÞ kd x l exp½oðx=x m Þ1=3 cðx; tÞ þ kd ðx þ x m Þl exp½oððx þ x m Þ=x m Þ1=3 cðx þ x m ; tÞ Z 1 0 kb x n cðx; tÞ þ 2b x n cðx 0 ; tÞ dx 0 =x 0 Idðx x Þ

ð15Þ

x

Dimensionless equations. The governing PBEs can be made dimensionless to minimize the number of parameters and to simplify the numerical analysis. Dil1 viding equation (15) by m(0)2 allows deﬁnition of the scaled quantities, N kgxm ðnÞ l ð0Þ ð0Þ ð0Þ n ¼ cðnÞ =ðmð0Þ x ¼ x=x m ; y ¼ tkg mð0Þ 1 x m ; S ¼ m =m1 ; C ¼ cx m =m1 ; C 1 xmÞ l ð0Þ ln J ¼ I=ðkg mð0Þ2 1 x m Þ; cb ¼ kb =ðkg m1 x m Þ

ð16Þ

Note that x is the number of monomers in a particle. Substituting the new variables into equation (4) yields dS ðyÞ=dy ¼ ½SðyÞ þ eOa CðlÞ þ Jx

ð17Þ

Because equation (17) is a moment equation, Oa is evaluated at the averagesized particle, Oa ¼ oX(Cavg)1/3. The mass balance is always satisﬁed, thus, ð1Þ SðyÞ ¼ S0 þ Cð1Þ 0 C ðyÞ

ð18Þ

This is substituted into equation (15) to give the dimensionless equation for C, ð1Þ l l @Cðx; yÞ=@y ¼ ½S0 þ Cð1Þ 0 C ðyÞ½x Cðx; yÞ þ ðx 1Þ Cðx 1; yÞ

xl expðox1=3 ÞCðx; yÞ þ ðx þ 1Þl expðoðx þ 1Þ1=3 ÞCðx þ 1; yÞ Z 1 0 cb xn Cðx; yÞ þ 2cb x n Cðx0 Þ dx0 =x0 Jdðx x Þ ð19Þ x

982

G. Madras and B.J. McCoy

The critical nucleus size and Gibbs–Thomson factor have the scaled forms [13] x ¼ ðo= ln SÞ3 and OðxÞ ¼ o=x1=3

ð20Þ

The scaled moment equations are determined by applying the operation R1 n 0 ½ x dx, " # " # 1 1 X X ðn jÞCðlþnjÞ þ eOa CðlþnÞ þ ðn jÞð1Þj CðlþnjÞ dCðnÞ =dy ¼ S CðlþnÞ þ j¼0

cb C

ðnþnÞ

þ 2cb C

j¼0 ðnþnÞ

=ðn þ 1Þ x

n

ð21Þ

It follows that the crystal moment equations for n ¼ 0, 1, and 2 are dCð0Þ =dy ¼ cb CðnÞ

ð22Þ

dCð1Þ =dy ¼ ðS eOa ÞCðlÞ Jxn

ð23Þ

dCð2Þ =dy ¼ ðS þ eOa Þ½2Cðlþ1Þ þ CðlÞ ð1=3Þcb Cðnþ2Þ Jx

2

ð24Þ

Equations (17) and (23) together, independently of the value l, satisfy the mass balance, d(S+C(1))/dy ¼ 0, which is the differential of equation (18). Equilibrium conditions for the zero-th moments are determined by setting J and the time derivatives equal to zero.

4. RESULTS AND DISCUSSION The integro-differential difference equation (19) was solved by a Runge–Kutta technique with an adaptive time step. All results reported here are for a growth coefﬁcient independent of mass, thus l ¼ 0. The distribution C(x,y) was evaluated at each time step sequentially. Because C(x,y) lies in the semi-inﬁnite domain, it was converted to a bounded range (0,1) by the mapping function, xx* ¼ (Cavgx*)y/(1y) with 0ryr1. The grid for this mapping is ﬁne in the range of prevalent sizes and coarse at very high and very low sizes. This ensures that y varies from 0 to 1 when x varies from x* to N. The variable y is centred at 0.5, and the distribution is centred around x ¼ Cavg, which requires fewer intervals. Denucleation was accounted at each time step by eliminating particles less than the critical nucleus mass and allowing the ensuing soluble mass to increase the supersaturation according to equation (17). This affords a check of mass conservation (equation (18)) at every step. Denucleation ensures that the distribution is zero for particles smaller than the critical nucleus, i.e., for xrx*. In equation (19), the breakage integral was evaluated at each time step by Simpson’s rule. Because the distribution is zero when xrx*, and the breakage integration is from x* or x (whichever is greater) to N. The mass variable (x) was divided into 10,000 intervals and the adaptive time (y) step varied from 0.001 to 0.1.

Crystal Growth and Dissolution with Breakage 4 2 1 0.8 0.6

5x10-4 10

0.4

-4

ψb = 0 1

10 θ

100

ψb = 0

(b)

5 x10-3 10-3

10-4 5x10-4

100 Cavg

C(0)

200

10-2

(a)

0.2 0.1

983

80 10-3

60

5 x10-3 40

1000

0.1

10-2 1

10 θ

100

1000

Steady state value of Cavg

200 (c) 100 90 80 70 60 50 40 30 10-4 10-3 Breakage parameter, ψb

10-2

Fig. 1. Effect of breakage parameter cb on the time evolution of (a) particle number concentration C(0); and (b) particle average mass Cavg. (c) Shows the variation of steady state particle average mass with the breakage parameter.

The values of o can be directly calculated from the fundamental parameters given by equation (14), which for vapour–liquid systems [91,92] range from 2 (methanol at 350 K) to 33 (mercury at 290 K). For solids, o would be smaller than these values and will be around o ¼ 1. Thus, unless otherwise speciﬁed, the parameters used in the calculations are So ¼ 10, o ¼ 5, C(0) o ¼ 1, l ¼ 0, n ¼ 1 avg avg with an initial delta distribution, Co(x) ¼ C(0) ¼ 100. o d(xCo ) and Co Calculations involving growth, denucleation, and breakage show that the supersaturation, number, average size, and polydispersity of crystals reach a steady state. The evolution to the steady state for the crystal number and average size is shown in Figs. 1a and 1b, respectively. This evolution to steady state can be understood by observing the scaled moment equation (22), dC(0)/dy ¼ cbC(n) J. It has been experimentally observed [31,32,34] that the agglomeration process stops when the particles reach a particular size. Conventional models based on a size-independent agglomeration kernel without breakage will not predict a steady state because the size would continuously increase with time. However, the current model shows that the average crystal size reaches a steady state even when aggregation does not occur. Steady state, dC(0)/dy ¼ 0, implies that

984

G. Madras and B.J. McCoy

the rate of denucleation is equal to cbC(n), and thus depends on the value of cb, as illustrated in Figs. 1a and 1b. In the absence of breakage (cb ¼ 0), Ostwald ripening dictates that the crystal number concentration would continue to decrease asymptotically [11] until only a single crystal remains in equilibrium (Fig. 1a). We have previously shown [47] that breakage starting from different initial conditions due to seeding and the effect of changing the initial crystal-size distribution from the delta to the exponential distribution is minor for the evolution of the crystal number concentration and crystal size. Further, the number concentration and average size at steady state are independent of the initial condition. Figure 1c shows that the steady state value of the average size, Cavg s , depends on the breakage parameter, cb, as a power law, Cavg ¼ 7.337c0.347 and thus a s b log–log plot is linear. The so-called inverse problem is to determine parameters or kernel expressions that best describe experimentally-observed behaviour. Rationally choosing such expressions requires an understanding of their mathematical behaviour. This result is extremely useful because the parameter cb can be back calculated based on the experimental steady state value of average size. Figures 2a and 2b show the effect of n on the time evolution of crystal number concentration and crystal-size distribution for cb ¼ 102 and 103, respectively. As discussed earlier, n ¼ 0, 1/3, 1, respectively, represent the cases when the breakage rate is independent of size, dependent on crystal radius, and linearly dependent on crystal mass. As expected, the parameter n greatly affects the evolution of the crystals but the inﬂuence decreases as cb becomes smaller. Figures 3a and 3b show the evolution of the particle average size for different o for cb ¼ 104 and 105, respectively. Unlike the effect of n, the inﬂuence of o does not decrease as cb becomes smaller. For larger o, the rate of decrease of the number concentration and the increase of average size is faster because the rate of denucleation is faster at higher o. This is consistent with experimental data that indicates that the average size of metals grows more slowly than that of liquids. 300 200

400

0 (a)

300

0

20 0.1

2/3 ν =1

ψb = 10-2 1

10 θ

100

1/3

200 -3

ψb = 10

Cavg

Cavg

1/3 100 90 80 70 60 50 40 30

(b)

2/3 100 90 80 70 60 50 0.1

ν =1

1

10

100

θ

Fig. 2. Effect of n on the time evolution of the particle average mass Cavg for (a) cb ¼ 103; and (b) cb ¼ 104.

Crystal Growth and Dissolution with Breakage

985 200

200

2

100 90 80 70 60

ω =5

150

1

2 ψb = 10-5

1 ψb = 10-4

50 40 30 0.1

(b)

ω =5

Cavg

Cavg

(a)

1

10 θ

100

1000

100 95 90 0.1

1

10

100

1000

θ

Fig. 3. Effect of o on the time evolution of the particle average mass Cavg for (a) cb ¼ 104; and (b) cb ¼ 105.

5. CONCLUSIONS An aim of the current work was to construct a simple and effective distribution kinetics model for crystal growth and dissolution accompanied by breakage. The model incorporates all relevant thermodynamic and kinetic inﬂuences, yet is amenable to straightforward computations. Interfacial energy effects, essential to this goal, enter via the Gibbs–Thomson equation for particle size and solubility, as well as through the critical nucleus size, which governs denucleation. Thermodynamic principles dictate that smaller crystals are more soluble than larger crystals and that crystals less than the critical nucleus size are unstable and vanish instantaneously. The quantitative model is based on population balances to describe the growth of crystals with breakage. The model allows assessment of various parametric inﬂuences, such as the initial particle size distribution, breakage kernel, aggregation rate, and interfacial energy leading to dissolution of smaller particles and denucleation of subcritical crystals. When growth occurs in the absence of breakage, the supersaturation decreases and Ostwald ripening causes an asymptotic evolution to a single particle. However, when crystal growth occurs with breakage, the number concentration and the average size of the crystals attain steady state. Growth dispersion, a natural consequence of population dynamics, is affected by the breakage rate and inﬂuences the time required to reach steady state. According to the present model, the steady state value of the crystal size has a power dependence on the breakage parameter. These results provide a simple way for the inverse problem to determine the breakage parameter based on experimental data. These computational results, based on a discretization of the PBE, provide new insights into the nature of the steady states that have been observed experimentally.

986

G. Madras and B.J. McCoy

Nomenclature c C I J kB kg kd kb m(0)(t) m(0) N r r* S x xm x*

crystal-size distribution cxm/m(0) N , dimensionless crystal-size distribution denucleation (or nucleation) rate l I/(kgm(0)2 N xm ), dimensionless denucleation rate Boltzmann’s constant rate coefﬁcient for growth of crystal rate coefﬁcient for dissolution of crystal rate coefﬁcient for breakage (fragmentation) molar concentration of solute as a function of time molar concentration of solute in equilibrium with plane crystal surface radius of the crystal radius of critical nucleus m(0)/m(0) N , supersaturation crystal mass monomer mass mass of critical nucleus

Greek symbols y o cb l n r rc x

l tkg m(0) N xm, dimensionless time Gibbs–Thomson ratio of interfacial to thermal energy for a crystal of monomer size ln kb/(kgm(0) N xm ), dimensionless breakage parameter exponent on mass in the growth rate coefﬁcient expression, equation (11) exponent on mass in the breakage rate coefﬁcient expression r/r*, dimensionless crystal radius crystal mass density x/xm, dimensionless crystal mass

Subscripts and superscripts (n) 0

superscript indicating n-th mass moment of crystal-size distribution subscript indicating initial condition.

REFERENCES [1] R.M. Ziff, in: F. Family, D.P. Landau (Eds.) Proceedings of the International Topical Conference on the Kinetics of Aggregation and Gelation, University of Georgia, NorthHolland, Amsterdam, 1984, pp. 191–199. [2] S. Chapman, T.G. Cowling, The Mathematical Theory of Non-Uniform Gases, 3rd edition, Cambridge University Press, London, 1970. [3] H.M. Hulburt, S. Katz, Chem. Eng. Sci. 19 (1964) 555. [4] A.D. Randolph, Can. J. Chem. Eng. 42 (1964) 280. [5] A.D. Randolph, M.A. Larson, Theory of Particle Processes, Academic Press, New York, 1971.

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CHAPTER 24

Liberation of Valuables Embedded in Particle Compounds and Solid Waste Wolfgang Schubert and Ju¨rgen Tomas Otto-von-Guericke-University Magdeburg, Universitaºtsplatz 2, 39106 Magdeburg,Germany Contents 1. Amount of particle compounds and solid waste 2. Waste comminution in the recycling industry 2.1. Processing objectives 2.2. Theoretical aspects of liberation of valuables 3. Classiﬁcation of waste into brittle, rubber-elastic and ductile 4. Crushing of solid waste and particle compounds 4.1. Stressing modes 4.2. Size reduction by compaction 4.3. Equipment for comminution 4.3.1. Machines used for rubber-elastic and ductile materials 4.3.2. Machines used for brittle materials 5. Simulation of breakage and liberation of particle compounds using the discrete element method 5.1. Motivations for applying the discrete element method 5.2. Description of the simulation method 5.3. DEM model and its calibration 5.4. Crack patterns, particle size distributions and liberation degrees 5.5. Crushing chambers of machines 6. Conclusions References

989 991 991 991 993 995 995 996 997 997 1003 1004 1004 1006 1007 1010 1013 1016 1017

1. AMOUNT OF PARTICLE COMPOUNDS AND SOLID WASTE Recent decades have seen continually growing waste amounts. More than 1.8 billion tonnes of waste are produced in the European Union (EU) each year; that’s 3.8 tonnes per person during the year 2000 in the EU-15 (including 15 countries) and 5 tonnes per person in Central and Eastern Europe [1]. The total waste

Corresponding author. Tel.: +49 391 6718783; Fax: +49 391 6711160; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12027-3

r 2007 Elsevier B.V. All rights reserved.

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Not declared 12%

Construction and demolition 31%

Municipal waste 14% Industrial waste 15%

Mining and quarrying 24%

Fig. 1. Total waste generation divided up into sectors in Western Europe in 2002 [1].

generation in the EU-15 rose by nearly 13% within 10 years between 1990 and 2000. Half of this waste came from the manufacturing industry, construction and demolition activities (Fig. 1). Recycling of glass and paper had not been increased sufﬁciently quickly to reduce the overall disposal volumes of these waste streams. Despite rising recycling rates, landﬁlling remained the most common treatment for waste [2]. Additionally, the amount of waste annually generated in the EU is increasing at a higher rate than ever before according to the enquiry of waste generation [2]. Therefore, the EU 6th Environment Action Programme of the European Community identiﬁes waste prevention, management and recycling as one of the top priorities, aiming at a signiﬁcant overall reduction in the volumes of waste generated. Considering these facts and the increasing raw material prices in the global market, recovery of valuables from waste by recycling is of fundamental importance to our society. Valuable components in solid waste are mostly bonded to other components in the form of compounds, composites or assemblies. Comminution, i.e. crushing, grinding or shredding, is one of the operations that is able to liberate the valuable components from the others so that these components can be separated by physical processes afterwards. In this chapter, waste is classiﬁed into two main groups: particle compounds and solid waste. Particle compounds can be found in the sector of construction and demolition (Fig. 1). They cover about 73%, which is equivalent to 22% of the total waste amount [3], and occur in the form of concrete and its modiﬁcations such as reinforced concrete, clay and asphaltic concrete. Solid waste is scrap and municipal waste, consisting of steel and non-ferrous scrap, electronic and cable scrap, composites and plastics, wood and paper, with a proportion of more than 20% of the total amount of waste [3].

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2. WASTE COMMINUTION IN THE RECYCLING INDUSTRY 2.1. Processing objectives The objectives of comminution in the recycling industry are similar to those in mineral, chemical, pharmaceutical and food processing industries. These goals are to liberate the valuable component locked in the particles, to produce particles of the desired size and shape, to increase the surface of the particles and to generate activated particle surfaces [4]. In detail, crushing and shredding of waste and scrap are carried out [5]: to liberate components that form compounds, composites or assemblies and

make them available for following operations. It is the principal objective of waste comminution when separation processes reclaiming the valuables come downstream. to produce particles or fragments of a deﬁned size distribution that meets consumer requirements or is favourable for operations following comminution. This product can be characterized by – increased bulk density, – better handling and ﬂowability, – accelerated microbial degradation. to liberate contaminants so that subsequent operations can isolate them, to de-ﬁbre composites, to de-compact packed waste.

2.2. Theoretical aspects of liberation of valuables More than 60 years ago Gaudin [6] deﬁned terms that characterize concisely the liberation of valuables out of minerals. His approach can be fully adopted by waste recycling. During reading, the terms ‘‘ore’’ and ‘‘mineral’’ should be replaced by ‘‘waste’’ and ‘‘component’’ respectively: Particles of ore can, of course, consist of a single mineral in which case they are termed free particles, or they may consist of two or more minerals in which case they are locked particles. Locked particles are binary, ternary, quarternary, etc., as they contain two, three, four, or more minerals. The degree of liberation m of a certain mineral or phase is the percentage of that mineral or phase occurring as free particles in relation to the total of that mineral occurring in the free and locked forms. Conversely, the degree of locking of a mineral is the percentage occurring in locked particles in relation to the total occurring in the free and locked forms [6].

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Fig. 2. Size reduction of amorphous and polycrystalline solids.

The liberation degree is the ratio of values that are obtained by weighing (mass based) or counting (number based): m3 ¼

mfree ; mfree þ mlocked

m0 ¼

nfree nfree þ nlocked

ð1Þ

The corresponding locking degrees are y 3 ¼ 1 m3 ;

y0 ¼ 1 m 0

ð2Þ

The partial liberation degree is used when particles containing worthless components are considered as liberated. The percentage of the valuable component is often added to the symbol, e.g. m3,50. Figure 2 supports Gaudin’s explanations; he continues: ‘‘The very fact that a large clump is reduced in size to small particles does not result in rupturing the bond between adjacent dissimilar minerals; but it restricts the occurrence of locking to a relatively small portion of the original lump, and this produces a result practically equivalent to freeing. However, if the physical properties of the adjacent minerals are sufﬁciently dissimilar, or if the bond between them is notably weaker than either of them, fracture may take place preferentially at the boundary. In that case, comminution results in true freeing of minerals in addition to a restriction in the occurrence of locking. It can, therefore, be said that liberation is increased by comminution by two means, liberation by size reduction and liberation by detachment [6].’’ Liberation by size reduction can be explained by a lattice that is placed on a particle, which consists of two components of equal abundance (Fig. 3a). When we assume that breakage occurs only along this lattice and not preferentially along the grain boundaries, then the liberation degree calculated after crushing depends only on this lattice. If we could adjust the mesh size of the lattice e.g. by a discharge screen in a crusher, we would observe that with narrower lattice, seen in Fig. 3b, the liberation degree of the product rises.

Liberation of Valuables

(a)

993

(b)

Fig. 3. Two differing lattices over a particle consisting of two components of equal abundance.

The liberation degree of a two-component material can be determined by separation of either the valuable or the worthless component of the crushed material. Three techniques are customary: hand sorting, using a visual characteristic of the component for separation, e.g.

a grey tone, gravity separation, using a heavy liquid in which one component ﬂoats. The

liberation degree is calculated from the obtained density distribution [7], dissolution of a component by using chemicals, e.g. hydrochloric acid. The

liberation degree correlates with the mass dissolved [8]. These techniques can be adapted for multi-component systems as well.

3. CLASSIFICATION OF WASTE INTO BRITTLE, RUBBER-ELASTIC AND DUCTILE Materials can be characterized by their material behaviour as elastic, elastic–plastic or elastic–viscous. However, materials as they exist in reality, e.g. compounds and composites, exhibit mostly mixed behaviour such as elastic–plastic–viscous. Thus, real materials have to be classiﬁed by their fracture behaviour, i.e. their stable or instable crack propagation during stressing, into non-brittle and brittle. Non-brittle can be divided further into rubber-elastic and ductile (Table 1). Stress–strain curves characterize the deformation of materials, which leads to a similar classiﬁcation. These curves are recorded when a specimen is compressed by stress. On the left hand side in Fig. 4, the fractional shortening of the specimen parallel to the applied compression is the strain e. Brittle materials behave elastically nearly until the breakage point, marked B, whereas ductile materials are

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Table 1. Classiﬁcation of materials found in waste and scrap according to the deformation behaviour of these materials [5] Strength (Hardness) Fracture behaviour

Deformation behaviour

brittle rubber-elastic ductile

elastic - brittle-elastic - rubber-elastic elastic-plastic

plastics, elastomers

elastic-viscous

wood paper, cardboard, biogenous waste, plastics, textile ﬁbres

low (soft)

medium (medium hard)

high (hard) concrete, glass, cast iron steel, wrought alloys of nonferrous metals

Added by the authors.

B

ductile Y

Stress σ

Stress

a

b

Strain

Strain ε

permanent recoverable deformation

Fig. 4. Typical stress–strain curves. On the left side, each B represents the breakage point of the corresponding material [9]. On the right, elastic–viscous material behaviour at high (a) and low stressing velocities (b) [10].

elastic up to the yield point, marked with Y, but then they deform plastically before fracturing. The ability to undergo large permanent deformation before fracture is called ductility. If the applied stress is removed while a ductile material is in the plastic range, part of the strain is recoverable, but there remains permanent deformation [9]. Solid waste and particle compounds are comminuted by cutters and hammers, which often apply such a high stress to the material that it becomes heated. One can say that most of the energy consumed by a crushing machine – about 70–80% – dissipates in heat by deformation and friction.

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Stressing velocity and temperature can inﬂuence considerably the deformation behaviour of the material being crushed. So, some additional statements have to be made. Materials of purely elastic behaviour are not inﬂuenced by temperature and stressing velocity; materials of elastic–plastic behaviour depend on temperature, but not on stressing velocity, and materials of elastic–viscous behaviour depend on both temperature and stressing velocity. An increase in velocity leads here to higher stresses during compression at low temperatures (Fig. 4, right side) [10]. Table 1 presents components of waste, scrap and particle compounds and classiﬁes them based on their deformation behaviour. These components occur in mixtures, assemblies, compounds and composites. In assemblies, components are joined together by bolting, clamping or welding. Examples are modules resulting from cars, household appliances or electronics. Dismantling, i.e. disassembling of components manually, is often the ﬁrst operation to start the liberation of valuables for recycling. Compounds and composites are solid materials that are composed of two or more substances, which have different physical characteristics. Each substance retains its identity while contributing desirable properties to the compound or composite. The substances are closely bonded together by interlocking (hook-like), macroscopic force bonds (friction) or microscopic physical and chemical bonds (adhesion effects, covalent ionic bonds). Often one substance acts as matrix. In particle compounds, at least one substance appears as discrete particles and another one as matrix. The substances in composites exist in the form of layers, ﬁlms, coatings or ﬁbres. Most compounds and composites are anisotropic in their nature, i.e. they exhibit properties with different values when measured in different directions. A typical example of a particle compound is concrete, which is made of hardened cement paste (the matrix) and aggregates (gravel, crushed stones) embedded in this matrix. Fibreglass-reinforced plastic and multi-layer packaging materials for liquids are well-known composites.

4. CRUSHING OF SOLID WASTE AND PARTICLE COMPOUNDS 4.1. Stressing modes Rumpf [11] deﬁned different modes of stressing of brittle particles at one solid surface or between two solid surfaces as impact, double impact, compression, attrition and shear (see Fig. 5). However, the fracture behaviour of rubber-elastic and ductile materials differs considerably from that of brittle materials. Schubert [10] therefore suggests

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Fig. 5. Stressing modes in general: (a) impact, (b) double impact, (c) compression, (d) attrition and (e) shear [11].

Fig. 6. Stressing modes occurring in cutters and shredders: (a) cutting, (b) shearing, (c) tearing, and (d) bending [10]. F is force, M torque, w wall thickness and s gap width.

cutting, shearing, tearing and bending stresses being typical to cause fractures in large to medium-sized non-brittle fragments (Fig. 6). Cutting is characterized by a force transmission into the material along the sharp edge of a knife (force per line length), the wedge angle of which is usuallyo601 (Fig. 6a). Due to tension, the failure zone propagates near the knife-edge. The fracture is the result of shear and tensile stresses; the latter is produced by a splitting effect of the knife. The counter-block acts as a supportive device like an anvil. When pressure is put on a large surface by two knives being in the same plane, shearing occurs (Fig. 6b). An increasing shear stress induces a gliding of the material within this shear plane. So, both knives, the wedge angle of which ranges from 801 to 901, are actively involved in the process. High shear stresses require here a relatively small gap so0.2w. With a wider gap, the pieces are so sharply bent that the fast rotating crushing tool can hit holes in it. In this way, tensile stress becomes dominant and develops tear stress (Fig. 6c). In comparison to cutting and shearing, tearing requires a wider gap between both tools, normally a multiple of the wall thickness of the feed plates. Tearing becomes more effective when the tool can already engage into some holes at the beginning, seen in Fig. 6c. In some applications, e.g. crushing of rails (Fig. 6d), bending stress is utilized [10,12].

4.2. Size reduction by compaction Until now it is supposed that size reduction only happens by breaking large pieces into fragments. However, ductile materials in extreme shapes, e.g. spirals, plates

Liberation of Valuables

997 Not capable of compacting

high yield strength, low fracture strain, low tear resistance Strong tendency to brittle fracture

Cast iron*

Wrought alloy AlMgCu1 Wrought alloy AlMg3

Al 99.5 Cu 99.99 Zn 99.99 More capable of compacting low yield strength high fracture strain low strain hardening high tear resistance

Brass CuZn15

Steel St14 O3

Brass CuZn37

Strong tendency to ductile fracture

Alloy steel Less capable of compacting high yield strength high fracture strain high strain hardening high tear resistance

Fig. 7. Compaction and fracturing vs. material properties of scrap metal [13,14]. * added by the authors.

or tangles, provide one more mechanism of size reduction: compaction. Compaction reduces the volume of pore space and makes a dense mass from the processed material. Examinations found that ductile metals with high fracture strain and low yield stress compact easily (Fig. 7) [13,14]. These malleable materials can be widely detected in their pure forms in tins, cans, tubes, or in mixed forms in composites such as wires, cables and multi-layer packaging. During stressing in mills, they often twist and wind around themselves into bodies of dense structure and round contour. Their particular quality can be utilized by a separation process downstream.

4.3. Equipment for comminution 4.3.1. Machines used for rubber-elastic and ductile materials Equipment for comminution can be classiﬁed according to the way in which forces are applied, shown in Fig. 8. Distinguishing features are the stressing mode, the stressing velocity provided by rotary or translatory tools, and the gap width between the moved and ﬁxed tools. The feed materials are coarse to medium-sized pieces of rubber-elastic and ductile fracture behaviour. In this section, the feed materials are named pieces whereas the crushed products are called fragments.

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Fig. 8. Machines for rubber-elastic and ductile materials classiﬁed according to stressing mode, stressing velocity and gap width [10].

Following the scheme in Fig. 7, the machines are introduced from left to right; ﬁrst the translatory shears. They are represented by guillotine and alligator shears, which are applied mainly to shredding of steel, non-ferrous-metal scrap [15] and large-area scrap textiles [16]. Rotor shears not only comminute large to medium-sized bulk waste, such as furniture, mattresses, carpets, bicycles, but also waste wood, old tyres, bio waste and old ﬁles [17]. According to their main stressing zone (Table 2), rotor shears are classiﬁed into axial-gap rotor shears (Fig. 9) and radial-gap ones (Fig. 10). The name of that category of machines indicates the dominant stressing mode: shearing. Concerning the axial-gap rotor shears, the axial gaps of both the single- and the multi-rotor shears are relatively small. The radial gap of the radial single-rotor shear is also small but here it causes an additional cutting stress in the pieces at the beginning of crushing. The radial gap of the axial-gap multi-rotor shear is much larger; and therefore, the ﬁrst cracks in the pieces crushed by this machine are initiated by tearing stresses. Besides crushing, fragment deformation and friction attrition also consume a large amount of energy in multi-rotor shears, and let the speciﬁc stressing energy rise considerably. Depending on the machine construction, around 50–90% of the consumed energy is effectively wasted [18]. A cascaded arrangement of rotors improves the feeding conditions of axial-gap rotor shears (Fig. 9c) and makes the rejection possible for pieces that cannot be comminuted. The many technical difﬁculties with waste comminution often require special crushing tools, which many machine suppliers provide [17]. For example,

Axial-gap rotor shears

Radial-gap rotor shears

Number of rotors

Multi-rotor/double-rotor Single-rotor

Single-rotor/double-rotor Single-rotor

Circumferential speed

o 0.5 m/s

0.5–5 m/s

5–20 m/s and more

801–901 front e. Shearing

Radial gap sR 901 front edge Shearing

o 0.5 m/s

Liberation of Valuables

Table 2. Classiﬁcation of rotor shears; w is the wall thickness of the feed material [12]

Schematic illustration (top view)

Main stressing zone Wedge angle Stressing mode

Axial gap sA sAo0.2w 901 side edge Shearing

901 side edge Shearing

999

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W. Schubert and J. Tomas

Fig. 9. Axial-gap rotor shears, schematic: (a) single-rotor shear (version Alpine), (b) double-rotor shear (version SID) and (c) quadruple-rotor shear (version Untha) [12].

Fig. 10. Radial-gap rotor shears, schematic: (a) indexable knife shear with swivel arm (version Lindner), (b) indexable knife shear with pusher (version Holzmag), (c) block knife shear (version MeWa) [12].

low-speed radial-gap rotor shears produce effectively small fragments, but extreme shapes of waste pieces and their low packing density can require extra devices such as hydraulically operated swivel arms or pushers, which compact the feed material and press it into the crushing chamber (Fig. 10). Fragments crushed by a radial-gap single-rotor shear are smaller than those crushed by an axial multi-rotor type. This can be attributed to the great number of rotor knives, the deﬁned comminution geometry and the usage of discharge grates. It should be remarked that rotor shears have been investigated in detail by Woldt [19]. The block knife shear in Fig. 10c, equipped with ﬂywheel and robust block knives, crushes thick-walled metal pieces, small electro-motors and transformers. Its rectangular shaped knives rotate at circumferential speeds of between 5 and 20 m/s. During cutting rotary knives interact with ﬁxed stator knives. The revolving knives are mounted on an open or closed rotor. So, the rotor of a low-speed rotor cutter, seen in Fig. 11, is either a closed ﬂat rotor (b) or an open one (a), which is made up of staggered discs. These low-speed cutters are used to cut plastic

Liberation of Valuables

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Fig. 11. Low-speed rotor cutters, schematic: (a) with a disc rotor for plastic waste (version Rapid), (b) with a ﬂat rotor for waste wood (version Pallmann) [22].

Fig. 12. Low-speed rotor shredders, schematic: (a) version Neuenhauser, (b) version M&J, (c) shredder version for scrap aluminium and cars [10].

waste and waste wood with low noise and low dust emission, an advantage over high-speed cutting mills. For voluminous materials, such as ﬁlms, foams and barrels, open rotors are suitable; whereas closed rotors are more effective in cutting of compact materials like high molecular weight polymers or wood [20,21]. The low-speed rotor shredder generates tensile stress in the feed material perpendicular to the rotor axis by means of toothed elements mounted on the rotor (Fig. 12). This tensile stress causes tearing, which is the main stressing mechanism occurring in this machine. Depending on its properties, material is compacted energy-intensively in the stressing zone. In case of blockage, which happens when the material is compacted ﬁrmly and closely, the direction of the rotation of the rotor can be reversed. Rotor shredders are widely used for pre-comminution of scrap aluminium and cars. They shred not only different types of waste, such as bulk, household, wood, paper, cardboard and metal chips, but also moist and sticky mineral bulk materials. If solidly constructed and sufﬁciently equipped with robust teeth, rotor shredders can comminute cohesive, rock-bearing soils as well as asphalt and building rubble for recycling purpose. Different models of bale looseners and sack openers can be added to this class of shredders [17].

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Fig. 13. High-speed rotor shredders shown as swing-hammer shredders, schematic: (a) to (c) horizontal shaft shredders, (d) to (f) vertical shaft shredders; (1) rotor with swing hammers, (2) anvil, (3) discharge grate, mesh or gap, (4) device for removing uncrushable pieces; (A) feed, (P) product [23].

High-speed rotor shredders apply tensile or tear stresses in conjunction with bending and torsion. Their crushing tools are either ﬁrmly ﬁxed to the rotor, such as pins, cams, toothed slats and knives, or moveably mounted on the rotor, such as swing hammers. Swing-hammer shredders, which are the most commonly used high-speed rotor shredders, can be subdivided into horizontally and vertically mounted rotor shredders (Fig. 13). These machines can crush pieces to sizes of a deﬁned fragment size distribution, liberate valuables out of wastes, or only reduce the size of the pieces to increase the bulk density. Examples of liberation include scrap steel, such as cars and mixed scrap [24,25], aluminium, waste incineration scrap, lead batteries, and electro-technical scrap from industry as well as from household. Applications to increase the bulk density are the crushing of scrap sheet metal, metal chippings [26], waste wood, used paper and household waste. Different designs of the comminution zone, particularly the upper and lower sections of the machine casing, have an effect mainly on the comminution rate. However, if the circumferential speed of the rotor is reduced, the energy necessary for compaction prior to comminution increases. For this reason, swinghammer shredders should always operate at high circumferential speeds [27]. The swing-hammer shredders with horizontally mounted rotors derive directly from the traditional hammer crushers used in mineral processing. Nevertheless,

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they have been widely adapted to the shredding of waste. For example, the versions (a) and (b) in Fig. 13 were developed speciﬁcally for the comminution of scrap cars. In (a), the material circulation above the rotor is homogenized by a buffer or feed zone. This also allows for the rejection of uncrushable pieces through a ﬂap. Variant (b) was designed to reduce the residence time of material that is still in the crushing chamber although sufﬁciently ﬁnely comminuted. The ultimate aims are higher throughput and lower wear [18]. A special swing-hammer shredder called Kondirator (Fig. 13c) comminutes steel scrap containing hard and unbreakable pieces of steel without any damage to the machine. In case of blockage of unbreakable pieces in the gap between rotor and grate, the direction of rotation of the rotor can be reversed. Additionally, the anvil and grate are redesigned so that blockages of the rotor can be avoided. Swing-hammer shredders with vertically mounted rotors have no discharge grates. In the versions shown in Fig. 13d–13f, the distance between the impacting tools and the wall of the machine casing represents the gap width that deﬁnes the output particle size. Impacting tools are ring hammers (d), extended hammers (e), or chain-like tools (f) [23,28]. Detailed studies of swing-hammer shredders are available in [29,30] dealing with horizontally mounted rotors, and in [31] dealing with vertically mounted rotors.

4.3.2. Machines used for brittle materials In principle, the same equipment and processing technologies as those used in mineral processing can be applied to crushing of brittle particle compounds like building rubble [34]. The most common machines are jaw crushers, impact roller crushers and impact crushers (Fig. 14) [35]. However, building rubble differs from minerals with regard to properties that can suddenly change during crushing such as the feed particle size, moisture content, contamination content and the amount of iron pieces derived from steel-reinforced concrete. This requires modiﬁcations of the used machinery and processing such as automatic adjustments of gap width and rotor speed for impact crushers or horizontally arranged jaw crushers.

Fig. 14. Crushers used for recycling of building rubble: (a) eccentric jaw crusher [4], (b) impact double-roller crusher (version O&K) [32], (c) impact crusher (version Hazemag) [33].

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Table 3. Distinguishing features of crushers used for recycling of building rubble [35] Jaw crusher

Impact roller crusher Impact crusher

Stressing mode

Compression

Characteristic kinetic parameter Reduction ratio Strength of the feed material Moisture content Product size d80 is determined by Wear costs

Stroke number 275–400 min1 6 500 MPa

Impact, double impact Roller tip speed around 20 m/s 7 200 MPa

o 5% 0–150 mm gap width Low

o 15% o 8% 0–250 mm gap width, 0–80 mm gap width, roller tip speed rotor tip speed Intermediate High

Impact Rotor tip speed 28–42 m/s 20 300 MPa

Table 3 gives a comparison between the most common crushers. The size reduction ratio is here deﬁned as the feed opening related to the gap width in the crusher.

5. SIMULATION OF BREAKAGE AND LIBERATION OF PARTICLE COMPOUNDS USING THE DISCRETE ELEMENT METHOD 5.1. Motivations for applying the discrete element method Basically, project engineers need a lot of experience for designing crushers or comminution plants. Statements about the expected particle size distribution of the product or the energy consumption during crushing can only be made by referring to already existing plants or by doing experiments. Although experimental results can be used beneﬁcially for the design of an industrial machine, the machine planned has to be of the same type, crushing the same material as the one on which the experiments had been carried out. Additionally, the number of these experiments rises rapidly when complex structured materials such as compounds and composites are examined. In this context, the alternative approach of using computer simulations can be an attractive way for the engineer to receive information about the liberation degree of the crushed material and its particle size distribution depending on the energy consumption and mass ﬂow rate. In the past a large number of detailed investigations into the phenomena of particle breakage were carried out. Quartz and limestone particles were tested by impact and compression experiments, in which single particles or particles in a particle bed were stressed [36,37]. These examinations show that it is difﬁcult to predict the particle size distribution of the product by a generalized ‘‘crushing law’’. Reasons for this are the many simpliﬁcations that have to be made in the

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modelling to get any applicable result, so the usage of elastic spheres of homogeneous and isotropic structure instead of real irregular particles, or the usage of the Hertz-theory for the description of the material behaviour instead of a theory for elastic–plastic and viscous–elastic material behaviour. Additionally, the behaviour of particles changes drastically under stress, especially when the ﬁrst cracks occur and later when the generated fragments interact with each other. This has not been taken into account in any modelling done to date. Despite these simpliﬁcations, powerful rules for designing crushers and mills have been developed. For example, the well-known Bond model calculates the mean particle size of the product from the feed particle size and the speciﬁc energy consumption called work index [38]. It is commonly used for the scale-up and design of ball mills. Analytical models are used for designing high-pressure roller mills, impact mills and stirred media mills; see Scho¨nert [39], Vogel and Peukert [40] and Kwade [41], respectively. Nevertheless, all these rules and analytical models suffer the handicap that their validity is restricted to a particular material and to a particular stressing condition determined by the crushing apparatus. There are obviously too many variables which control the ultimate outcome of a crushing process that a generalized ‘‘crushing law’’ could not be formulated. This is the point where the discrete element method (DEM) intervenes and takes a new approach to the problem. This method can be applied to more realistic inhomogeneous and anisotropic materials such as particle com-

pounds of various sizes and shapes such as fragments and agglomerates, either two- or three-dimensional; materials that have not only macroscopically ‘‘basic’’ material behaviour like elastic or brittle, but also complicated properties resulting from a superposition of different microscopic element behaviour like elastic–plastic or visco–elastic; composites consisting of components that differ in their mechanical properties such as the Young’s modulus or compressive strength. Furthermore, the dynamics of a crushing event can be simulated by this method, i.e. the generation and propagation of cracks and stressing paths and their development into a fracture pattern, caused by forces that change over time and location. The stressing conditions can herein vary widely from impact over shearing to compression. The results can be characterized by quantities derived from process engineering like energy consumption, particle size distribution or liberation degree. The next sections give a brief survey of the DEM and explain this method with examples.

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5.2. Description of the simulation method The DEM was developed by Cundall and Strack [42] to predict the failure behaviour of rock and soils in geomechanics. More recently it has proved a versatile numerical tool, particularly suitable for the simulation of granular and particulate systems. In general, the DEM is based on the concept that individual basic elements – in most cases distinct rigid spheres [43] – are considered to be individual and connected only along their boundaries by physically based interaction laws. These laws between the spheres, also called contact laws, can differ widely depending on the components being modelled. After generation, the spheres are placed in a random pattern similar to a web, contacting each other. They make up the DEM model, which can be a differently shaped body, fragment or agglomerate. Micro-properties, such as density, stiffness, Coulomb friction and bond strength, are assigned to the spheres and their contacts. They control the macro-properties, i.e. the macroscopic material behaviour of the model, such as elasticity, plasticity, failure behaviour or compressive strength. Selected micro-properties can be assigned to certain spheres to model a speciﬁc component of a compound. The number of spheres determines the resolution of the model, i.e. how ﬁne and precise a model works, but also the time needed for computing. Walls give the arrangement of the spheres’ position and orientation in space. They are the boundaries in the form of lines and circles (two-dimensional) or planes and cylinders (three-dimensional). These wall elements can be ﬁxed or moved by external forces. Using walls, stressing of the mechanical system of bonded spheres can be applied by modelled obstacles, containers, machinery tools and crushing chambers. As mentioned above, the DEM is capable of simulating a dynamic phenomenon like breakage. So, all the forces involved in particle deformation and crack generation are computed during simulation, and with them accelerations, velocities and positions of all the spheres and walls. The velocities of the walls, which may be zero, and the optional external forces serve as boundary conditions in a system of differential equations of motion. For the numerical calculations, the DEM uses a time-stepping algorithm, in which the balance of forces and moments (Newton’s and Euler’s laws of motion) are applied to each sphere, a force–displacement law to each contact, and an update of position to each wall. This is done after each timestep (Fig. 15): At the start of each timestep, the set of contacts is updated from the known particle and wall positions. The force-displacement law is then applied to each contact to update the contact forces based on the relative motion between the two entities at the contact and the contact constitutive model. Next, the law of motion is applied to each particle to update its velocity and position based on the resultant force and moment arising from the contact forces and any body

Liberation of Valuables Initial Conditions sphere positions and velocities at start

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Calculation of contact forces

Force-Displacement Law applied to each contact e.g. the linear spring model : normal force: Fn = kn Un shear force: Fs = ks Us

Boundary Conditions external forces acting on boundaries

Force Balance and Moment Balance applied to each sphere translational motion: ∑ Fi,j = mj xi,j j

rotational motion: i=1,2 for 2D

∑ Mi,j = Jj ωi,j j

i=1,2,3 for 3D

Update of the positions of spheres, walls and contacts

_ and x€ stand for force, mass, overlap, stiffFig. 15. Calculation cycle. F, m, U, k, M, J, o ness, torque, rotational moment of inertia, angular velocity and acceleration respectively.

forces acting on the particle. Also, the wall positions are updated based on the speciﬁed wall velocities (see Cundall et al. [43]).

5.3. DEM model and its calibration DEM models must be oriented and adjusted to their originals, i.e. to the real objects. This is done with the help of experiments in a calibration procedure, in which the DEM model is adjusted to the real material properties, i.e. the results of both experiments and simulations are compared and as a consequence of it the DEM model is changed until its behaviour meets that of the real object. To understand better the DEM and its calibration procedure, the building material ‘‘concrete’’ is given here as an example of a real object. It is a typical particle compound material consisting of many different components, from which the following two main components are chosen for the model [44]: aggregates, such as granite or basalt gravels and sand, sized between 0.1 and

16 mm, hardened cement paste as a matrix, in which the aggregates are embedded.

Concrete is widely applied to construction and after many years of usage it returns again as building waste for recycling purposes. Related to the breakage patterns, concrete behaves as if it is a more or less brittle material, see Table 1, but from the perspective of continuum mechanics it is a material of combined elastic–plastic–viscous behaviour. The concrete used in the experiments is of

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Table 4. Properties of the real concrete ball [44] and the DEM-model [45] Characteristics

Hardened cement paste

Properties of the real concrete, B35, ball of 150 mm in diameter Shape Matrix forming the ball Dimension of the components Crystals in the mm range Solid density, mass fraction 1790 kg/m3, 24% Tensile strength of concrete 4 N/mm2

Aggregates Irregular particles 0.1–16 mm in diameter 2570 kg/m3, 76%

Micro-properties of the 2D-DEM model, circle of 150 mm in diameter Number and shape 2283 spheres 120 spheres Diameter of the spheres 2.3 mm 2–16 mm Solid density, mass fraction 1790 kg/m3, 18% 2570 kg/m3, 82% Tensile strength of each bond 6.5 N/mm2

type B35, which speciﬁes its compressive strength and tensile strength of at least 35 and 4 N/mm2, respectively (Table 4). To determine the crushing behaviour of single pieces of concrete, impact experiments are carried out using an air cannon, crushing balls of 150 mm in diameter [44]. After accelerating by pressed air in a tube, the concrete ball crashes into a target at a velocity controlled by the air pressure. The generation of cracks on the surface of the ball is caught on a high-speed digital camera. These impact experiments can check the simulations and deliver experimental data for model calibration. One of the real concrete balls serves as the master for the DEM model. The DEM model is created by commercial software called Particle Flow Code [43]. A cross-section of the real concrete ball is the basis for the two-dimensional DEM model, which uses more than 2,000 spheres. The spheres are arranged in a plane, contacting each other (Fig. 16). The contacts are viscously damped and furnished with solid bridge bonds, which are characterized by stiffness and strength. Coulomb friction is assigned to the spheres, the uniform 2.3 mm of which represent the hardened cement paste, i.e. the matrix, with its mass fraction of 18%. The aggregates are represented by the coarser spheres, 2–16 mm in size and of Gaussian distribution. The density of each component, the hardened cement paste and the aggregates, is assigned to the corresponding spheres (Table 4). The calibration follows an iteration algorithm that starts by entering the initial micro-properties of the spheres and contacts (Fig. 17). These properties should be in agreement with the original concrete material. Then the crushing event is simulated as an impact of a ball against a wall similar to the real experiment that uses the air cannon. The results of this simulation are compared with those obtained by the experiment. If both results agree, the calibration ends with a working DEM model adjusted to a particular stressing condition. Otherwise, the simulation has to be repeated with varied model micro-properties. Note that the results of both simulation and experiment are expressed as

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1009 Aggregate, d=2-16 mm

Hardened cement paste, d=2.3 mm (a)

y

z

(b)

x

Fig. 16. The two-dimensional DEM model of the concrete ball (a) in the x–y-plane, and (b) in the x–z-plane [45].

quantities of process engineering such as particle size distribution and liberation degree [45]. The calibration procedure leads to a DEM model that should ﬁt sufﬁciently well the real concrete ball. The interesting question is now how good the DEM model copies the breakage behaviour of the real concrete ball.

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W. Schubert and J. Tomas Input of the initial micro-properties of the spheres and contacts, which control the macro-properties of the model. Variation of the microproperties. DEM-simulation of impact. Determination of crack patterns, particle size distributions and liberation degrees.

Pool of experimental data: digital movies, particle size distributions and liberation degrees.

Do results of simulation match those of experiment?

No

Yes Usage of the calibrated model for new DEM-simulations, in which other stressing conditions can be investigated.

Fig. 17. The calibration procedure, schematic.

5.4. Crack patterns, particle size distributions and liberation degrees Experiments and simulations deliver the desired crack patterns in the concrete ball. The photographs in Fig. 19 display the fragments of the concrete ball reassembled after the impact test. On the left panel, the corresponding simulations are given as images. Note that the simulation results are not only given numerically in numbers, like positions and velocities of the spheres and walls, but also graphically in the form of images. In these images, spheres that are in contact and bonded are covered by a virtual skin called cluster. If the maximum stress applied to a bond exceeds its corresponding bond strength, the bond breaks during simulation. If enough bonds are broken, the cluster breaks and a crack pattern emerges, containing smaller clusters, single spheres and the paths between them. In Fig. 19, it can be seen that the simulated crack patterns are similar to those produced by the experiments. This good result can be additionally supported by the fact that crack patterns experimentally obtained from brittle materials follow speciﬁc rules, which were thoroughly investigated by Rumpf [46] and Scho¨nert [47]. They found that after contacting the impact plate, the material near the contact point undergoes a mainly elastic–plastic deformation, which rapidly

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Fig. 18. Crack pattern in a brittle sphere impacting against a wall [44].

extends to a larger contact region. Shortly after contacting, the so-called cone of ﬁnes is formed from the material that breaks and collapses unselectively inside this cone by large tensile and shear stresses. Owing to its high velocity, the cone is almost immediately pressed into the concrete ball. Between the cone surface and the surrounding material, sliding and shearing planes arise, containing shear stresses t and compressive stresses s (Fig. 18). Finally, ring tensile stresses sj emerge and cause the concrete ball to break. The fragments produced by this violent burst are shaped like segments of an orange. The theoretical velocity of the elastic wave propagation, which is responsible for the energy transfer to the crack propagation front, ranges from 1200 to 6700 m/s [44]. It is worth noting here that these crack patterns can be found not only in the photographs of the experiments but also in the images of the simulations. The zones in which the cone of ﬁnes and the remaining cone are situated, and the secondary cracks can be clearly identiﬁed (Fig. 19). These patterns have not been found in any previous simulations [48,49]. The size of the cone of ﬁnes, which appears as a wedge in 2D simulations, increases with rising impact velocities. This was expected in the experiments and so observed in the simulations. These correlations indicate that breakage phenomena can be simulated appropriately by the DEM. The next step is to evaluate quantitatively the simulation. For process engineers, the particle size distribution and liberation degree are suitable process parameters to characterize the progress of comminution; but these evaluation methods are not available in the commercial DEM software PFC [43]. Therefore, its results have to be converted into quantities like the particle size distribution in the following way. After simulation of the impact the produced clusters are scanned. Their largest

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Zone of the cone of fines

Zone of the remaining cone v = 25 m/s

Secondary crack

Meridian crack

v = 35 m/s

Fig. 19. Crack patterns in a concrete ball after impact at different stressing velocities. The left panel shows images of the simulated impact, here ﬁve milliseconds after touching the wall; the right panel shows photos of the crushed balls, taken after the experiments [45].

x- and y-dimensions are the Feret’s diameters [50], which are taken to group the clusters in size classes. To these classes, the masses of the corresponding clusters are assigned. The size classes and their masses are the basis for the particle size distribution ﬁnally calculated. The liberation degree is determined likewise by spheres representing the valuable component. The mass of these spheres in each cluster is related to the total mass of the cluster [51,52]. Figures 20 and 21 show the wide range in which the results of simulations correspond to those of experiments.

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100 Experiments v = 40-50 m/s

Cumulative distribution in %

Liberation of Valuables

80 60 40 20

Simulation v=45 m/s

0 0

50 100 Particle size in mm

150

Fig. 20. Particle size distributions of the crushed product obtained by experiments and simulations [45].

Liberation degree in %

70 60 Experiment

50 40

Simulation

30 Exponential fit (Simulation)

20 10 0 10

30 50 Stressing velocity in m/s

Fig. 21. Liberation degrees of the aggregate component obtained by experiments and simulations [45].

5.5. Crushing chambers of machines Recently the process chambers of different crushers and mills were modelled with the help of the DEM. The process dynamics in ball mills, autogenous and semi-autogenous mills have been investigated most commonly [53–55], whereas simulations of impact crushers are still rare [56]. The DEM modelling of a crushing machine focuses on its essential part: the crushing chamber with its tools, i.e. the space and the surfaces where comminution takes place. Parts of the machine that are not involved in the process, such as the frame or driving unit, are left out. The crushing chamber is a full-scaled geometrical model designed often by professional CAD software allowing fast changes to the construction elements. Figure 22 shows two crusher models containing here only one piece of feed material. In the case of three-dimensional simulations, the observer can see the sequence of events from different perspectives under different magniﬁcations.

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feed chute

baffle plates fixed jaw (transparent) rotor

swinging jaw impact bar closed opening

Fig. 22. Three-dimensional DEM models of process chambers of an impact crusher and Blake jaw crusher [57].

Fig. 23. Time sequence of events in which a concrete cylinder is crushed in an impact crusher at a rotor tip speed of 20 m/s, obtained by a 3D-DEM-simulation [58].

The crushing chamber of an impact crusher is marked by the rotor and bafﬂe plates, against which in our example, a concrete cylinder impacts (Figs. 23 and 24). Different stressing conditions can be set by various rotor velocities, bafﬂe plate shapes and bafﬂe plate positions. Provided that the feed material is well-calibrated, simulations can predict changes in product quality when stressing conditions alter. As seen in Fig. 25b, the particle size distributions of the simulations are obtained in expected order: the higher the rotor tip speed, the ﬁner the product. In the following example, a large number of impact crusher tests on concrete material were done to search the best conditions for aggregate recycling [59,60]. When the obtained experimental data were judged appropriate for DEM

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Fig. 24. Fragments ‘‘produced’’ within the gap between bafﬂe plate and rotor [58].

0.8

Experiment 25m/s Experiment 27m/s Experiment 29m/s Experiment 31m/s Experiment 33m/s Experiment 35m/s Simulation 30m/s

0.6 0.4 0.2 0 0

(a)

Particle size distribution

Particle size distribution

1

10

20 30 40 50 Particle size in mm

1 0.8 0.6 0.4

Simulation 20m/s Simulation 30m/s Simulation 40m/s

0.2 0 0

60 (b)

10

20 30 40 50 Particle size in mm

60

Fig. 25. Particle size distributions of the impact crusher product received from both experiments and simulations [58].

modelling, the simulations started, using the same settings as in the crusher experiments:

Feeding rate of the concrete pieces: 4.5–13.3 t/h, Smallest gap width: 20–35 mm, Rotor tip speed: 25–38 m/s, Test specimens: concrete balls and cylinders of type B35.

In the crusher experiments, the particles trajectories were recorded by a highspeed video camera and the energy consumption was measured by a torque meter and a speed gauge. The crushed product is characterized by the liberation degrees and the particle size distributions. The latter is shown in Fig. 25a together with the size distributions derived from the simulations.

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The simulated particle size distribution lies slightly below the experimental ones. This is caused by the model accuracy, which is determined by the number of the spheres used in the model and their sizes. In our example, the smallest spheres in the modelled cylinder are 6.4 mm in diameter. Spheres smaller in size and therefore larger in number not only increase the accuracy of the model, but also result in increased CPU processing time. Thus, a compromise between accuracy of the model and the computing time always has to be made.

6. CONCLUSIONS Comminution is a basic requirement for liberation of valuables, which are embedded in waste. Waste consists of components that have mainly non-brittle, i.e. rubber-elastic or ductile fracture behaviour. For this material, modern comminution equipment has been developed to a large extent in the last decade to meet the requirements of the applicants. A description of this machinery including its classiﬁcation is given. The DEM shows its great potential for modelling particle compounds and simulating impact events. The simulation of impact events is only one of many applications of DEM in process engineering. The DEM is now in the phase of reproducing mechanical processes. Changes in product properties by varying stressing conditions can be predicted qualitatively, and to some extent also quantitatively. Professional software such as MathCAD, Excel and MatLAB is linked to the DEM allowing a proper evaluation of the simulations. In the future it is expected that the DEM will expand its performance and capability so that it can support process engineers in process design and adjustment, process optimization, development of new equipment.

Nomenclature

mfree mlocked nfree nlocked sA sR w m3

mass of liberated particles (kg) mass of locked particles (kg) number of liberated particles number of locked particles axial gap (mm) radial gap (mm) wall thickness (mm) liberation degree, mass based

Liberation of Valuables

m0 y3 y0

1017

liberation degree, number based locking degree, mass based locking degree, number based

REFERENCES [1] Data based on joint Eurostat/OECD Questionnaire of 2002/2003, www.europa.eu.int/ comm/eurostat. [2] European Environment Outlook Rep. (EEA), Rep. No. 4/2005, www.eea.eu.int, European Environment Agency, Copenhagen, Denmark, 2005. [3] European Environment Information and Observation Network (EIONET), www.waste.eionet.eu.int/wastebase. [4] H. Schubert (Ed.), Handbuch der Mechanischen Verfahrenstechnik, Bd. 1, 1. Auﬂ. WILEY-VCH Verlag, Weinheim, 2003. [5] G. Schubert, Aufbereitungstechnik 43 (2002) 6–23. [6] A.M. Gaudin, Principles of Mineral Dressing, McGraw Hill Book Co., New York, 1939, pp. 70–71. [7] J. Tomas, J. Friedrichs, Erzmetall 50 (1997) 562–571; p. 9. [8] L. Kiss, K. Scho¨nert, Aufbereitungstechnik 30 (1980) 223–230. [9] R. S. Carmichael, Rock, Encyclopædia Britannica, Encyclopædia Britannica, London, 2006. [10] G. Schubert, S. Bernotat, Int. J. Miner. Process. 74 (2004) 19–30. [11] H. Rumpf, Chem. Ing. Tech. 37 (3) (1965) 187–202. [12] D. Woldt, G. Schubert, H.-G. Ja¨ckel, Int. J. Miner. Process. 74 (2004) 405–415. [13] G. Schubert, Stand und Entwicklungstendenzen bei der Sortierung von Schrotten und Abfa¨llen, Freiberger Forschungshefte A 850 TU Bergakademie Freiberg, 1999, pp. 1–35. [14] D. Schubert, Zerkleinerung der Metalle in einer diskontinuierlich arbeitenden Hammermu¨hle, Master Thesis, TU Bergakademie Freiberg, 1998. [15] G. Schubert, Aufbereitung metallischer Sekunda¨rrohstoffe, 1. Auﬂ., Dt. Verlag fu¨r Grundstofﬁndustrie, Leipzig, 1984. [16] D. GroXkreutz, Die Gestaltung des Schneidwerkzeuges beim ziehenden Schneiden textiler Fla¨chengebilde, Ph.D. Thesis, University Kaiserslautern, Shaker Verlag Aachen, 1996. [17] H.-G. Ja¨ckel, G. Schubert, Chem. Ing. Tech. 69 (1997) 640–648. [18] G. Schubert, Aufbereitungstechnik 43 (2002) 13. [19] D. Woldt, Zerkleinerung nicht-spro¨der Stoffe in Rotorscheren und -reiXern, Freiberger Forschungshefte A 887 der TU Bergakademie Freiberg, 2005. [20] W. Michaeli, M. Bittner, L. Wolters, Stofﬂiches Kunststoff-Recycling, Carl Hanser Verlag, Mu¨chen, Wien, 1993. [21] V. Hess, Schneidmu¨hlen, H. Pahl (Eds.), Zerkleinerungstechnik, Fachbuchverlag/ Verlag TU¨V Rheinland, Leipzig, 1993, pp. 236–248. [22] D. Woldt, G. Schubert, H.-G. Ja¨ckel, Size reduction by means of low-speed rotary shears and cutters, 10th European Symp. on Comminution, Heidelberg, 2002. [23] S. Sander, G. Schubert, Chem. Eng. Technol. 26 (2003) 409–415. [24] A.A. Nijkerk, W.L. Dalminj, Handbook of Recycling Techniques, Nijkerk Consutancy, Laseurlaan, 2001. [25] G. Schubert, Neue Hu¨tte 25 (6) (1980) 201–206. [26] K. Go¨pel, G. Schubert, Neue Hu¨tte 37 (3) (1992) 93–97. [27] S. Sander, G. Schubert, H.-G. Ja¨ckel, Int. J. Miner. Process. 74 (2004) 385–393. [28] J. Kirchner, G. Timmel, G. Schubert, Powder Technol. 105 (1999) 274–281.

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[29] J. Kirchner, Mikroprozesse und EinﬂuXgro¨Xen bei der Zerkleinerung der Schrotte und Metalle in Shreddern mit horizontal angeordnetem Rotor, Freiberger Forschungshefte A 860 TU Bergakademie Freiberg, 2000. [30] St. Sander, Grundlagen der Zerkleinerung in HammerreiXern, Freiberger Forschungshefte A 871 TU Bergakademie Freiberg 2003. [31] G. Timmel, Zerkleinerung von Metallen in Shreddern mit vertikalem Rotor, Freiberger Forschungshefte A 862 TU Bergakademie Freiberg, 2001. [32] D. Leininger, Aufbereitungstechnik 4 (1988) 197–202. [33] P. Guntermann, Marktu¨bersicht 2000, Zement-Kalk-Gips 53 (2000) 3. [34] J. Hanisch, H.G. Ja¨ckel, M. Eibs, Aufbereitungstechnik 32 (1) (1991) 10–17. [35] J. Hanisch, Aufbereitungstechnik 35 (8) (1994) 423–432. [36] K. Scho¨nert, Aufbereitungstechnik 32 (9) (1991) 487–494. [37] H. Schubert, Aufbereitungstechnik 34 (10) (1993) 495–505. [38] F.C. Bond, Min. Eng. 4 (5) (1952) 484–494. [39] K. Scho¨nert, U. Sander, Powder Technol. 122 (2002) 136–144. [40] L. Vogel, W. Peukert, Chem. Eng. Sci. 60 (2005) 5164–5176. [41] A. Kwade, Powder Technol. 105 (1999) 14–20. [42] P.A. Cundall, O.D.L. Strack, Geotechnique 29 (1979) 47–65. [43] P.A. Cundall, O. Potynondy, H. Konietzky, Particle Flow Code 2D/3D, Version. 3.1, Itasca Consulting Group Inc., Minneapolis, MN, USA, 2005. [44] J. Tomas, M. Schreier, T. Gro¨ger, S. Ehlers, Powder Technol. 105 (1999) 39–51. [45] W. Schubert, M. Khanal, J. Tomas, Int. J. Miner. Process. 75 (1–2) (2005) 41–52. [46] H. Rumpf, Aufbereitungstechnik 14 (2) (1973) 59–71. [47] K. Scho¨nert, Powder Technol. 143–144 (2004) 2–18. [48] C. Thornton, M.T. Ciomocos, M.J. Adams, Powder Technol. 105 (1999) 74–82. [49] A.V. Potapov, C.S. Champbell, Powder Technol. 120 (2001) 164–174. [50] R.H. Perry, D.W. Green, Perry’s Chemical Engineers’ Handbook, 7th edition, The McGraw-Hill Companies Inc., US, 1999, pp. 20–28. [51] M. Khanal, W. Schubert, J. Tomas, Granul. Matter 5 (4) (2004) 177–184. [52] M. Khanal, Simulation of crushing dynamics of an aggregate-matrix composite by compression and impact stressings, Ph.D. Thesis, docupoint Verlag, Magdeburg, 2005. [53] D. Morton, S. Dunstull, Miner. Eng. 17 (2004) 1199–1207. [54] R.D. Morrison, P.W. Cleary, Miner. Eng. 17 (2004) 1117–1124. [55] B.K. Mishra, C. Murty, Powder Technol. 115 (2001) 290–297. [56] N. Djodjevic, F.N. Shi, R.D. Morrison, Miner. Eng. 16 (2003) 983–991. [57] W. Schubert, H. Jeschke, M. Khanal, J. Tomas, DEM-simulation of mineral processing machines, Proc. of the 4th colloquium Sorting, Berlin, 2005. [58] H. Jeschke, W. Poppy, W. Schubert, Concrete comminution in impact crushers– experiments and simulations, Aufbereitungstechnik 47 (6) (2006) 4–21. [59] H. Jeschke, W. Poppy, Aufbereitungstechnik 45 (5) (2004) 25–32. [60] H. Jeschke, Massenstromzerkleinerung von Beton, Ph.D. Thesis, Reihe 3, Nr. 852, VDI-Verlag, Du¨sseldorf, 2006.

CHAPTER 25

Attrition in Fluidised Beds Renee Boereﬁjn,a, Mojtaba Ghadirib and Piero Salatinoc a

PURAC Biochem b.v., P.O. Box 21, 4200 AA Gorinchem,The Netherlands Institute of Particle Science and Engineering, University of Leeds, Leeds LS2 9JT, UK c Dipartimento di Ingegneria Chimica, UniversitaØ degli Studi di Napoli Federico II, P. le V.Tecchio 80, 80125 Napoli, Italy

b

Contents 1. Introduction 2. Process-dependent attrition 3. Main hydrodynamic effects 3.1. Simple ﬂuidised bed systems – elutriation and size distribution 3.2. Jet effects 3.3. Terminator and cyclone effects 4. Attrition in wet systems 5. Attrition in reacting systems 6. Standard tests and characterisations 7. Conclusions Acknowledgements References

1019 1020 1025 1025 1029 1032 1032 1036 1041 1049 1049 1051

1. INTRODUCTION The aim of this chapter is to address principal causes and hydrodynamics resulting in particle attrition in major ﬂuidised bed applications, as there are the nature of particle breakage phenomena, as related to the location of attrition

(e.g. bulk of the ﬂuidised bed, jetting region, terminator) and bed hydrodynamics; the inﬂuence of liquid addition in wet systems, e.g. in ﬂuidised bed granulators; the mutual inﬂuence between attrition and chemical reactions, as experienced

by fuel and sorbent particles in ﬂuidised bed combustors/gasiﬁers. The present chapter contains a comprehensive overview of attrition studies in systems with homogeneous distribution and jet distribution of the ﬂuidising gas, Corresponding author. Tel.: +31-183-695-641; mob: +31-646-444-773; Fax: +31-183-695-607; E-mail: r.boereﬁ[email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12028-5

r 2007 Elsevier B.V. All rights reserved.

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R. Boereﬁjn et al.

but does not aim at an in-depth survey of these ﬂuidised bed attrition studies. A recent review on attrition of ﬂuidised particles is provided by Werther and Reppenhagen [1], and the reader is referred to this review for speciﬁc treatment of most of these attrition studies. The background of this chapter is represented by a longstanding research activity on ﬂuidised bed attrition of two groups along two distinguished lines of research, which have in themselves a common root: particle breakage in energy systems, i.e. processes intended either for the generation of fuel or for the generation of heat and electricity. Common features of these processes are the large scale of the reactors and the limited accessibility to the reaction environment to which individual particles are exposed in full-scale equipment. In order to obtain an idea of the general interest in this ﬁeld, a search on key words ‘‘attrition resistant’’ and ‘‘sorbent’’ yields no less than 73 PCT (US and WO) patent ﬁlings, whereas replacing ‘‘sorbent’’ by ‘‘catalyst’’ results in 180 ﬁlings. Ghadiri et al. [2–4] at Surrey and Leeds have studied attrition of catalysts used in the oil industry. The research group of Salatino and co-workers at Naples (e.g. [5,6]) has studied attrition of carbon, sorbents and ash in combustion/gasiﬁcation systems. Distinct approaches have resulted from each line of research: Ghadiri and co-workers have characterised the single particle behaviour and have combined this with process models, e.g. a model of ﬂuidised jet hydrodynamics derived by the same group at Naples University [7,8], to predict the in-process behaviour of materials as a function of operating conditions and equipment design parameters. This approach is suitable for systems where the material being studied is the main or sole (solid) material being processed in the system, as is the case with ﬂuid catalytic cracking (FCC). The Naples group has studied mainly reactive materials, e.g. carbon and lime, which are added to a system which consists for the largest part of practically inert material, such as sand. In order to predict the effect of operating conditions and equipment design parameters on material and process efﬁciency, small scale test facilities were developed to match the conditions of the real processes as closely as possible, taking into account the mutual interaction between attrition and the course of chemical reactions. This chapter provides an overview of the state of our understanding on these two topics.

2. PROCESS-DEPENDENT ATTRITION Attrition is commonly deﬁned as undesired particle breakage, sometimes as erosive wear, i.e. a superﬁcial size reduction process in a sheared environment. Fluidisation is the term used to denote the ﬂuid-like behaviour of powders induced by the ﬂow of an interstitial ﬂuid (gas or liquid). Fluidised beds are used for a

Attrition in Fluidised Beds

1021

variety of processes and the conditions to which powders are subjected vary as much. Food/pharma systems are commonly operated under slight vacuum, in order to prevent contamination, and at temperatures up to 2001C, with superﬁcial gas velocities up to 2 orders of magnitude above the minimum ﬂuidisation level (o2 m s1), and powders commonly contain an organic component. In petrochemical and energy production processes, inorganic powders may be used as catalysts, and conditions may be stretched to 10001C and transport velocities up to 100 m s1. Let us turn to the processes at hand to identify the occurrence and modes of attrition as related to the relevant hydrodynamics. Figure 1 shows different types of arrangements used for ﬂuidised bed processes, namely a catalytic cracking unit, a combustor, a granulator, and a mixer. Table 1 lists some typical features of such systems, relevant to attrition.

Standpipe

Cyclone

(b)

Riser

(a)

Solids Feed

Return Valve

Heat Exchanger

Air Supply

(c)

Liquid Supply

(d) FLUIDISED POWDER BED

NOZZLES SPRAY

powder bed length SPRAY-ON SECTION

Bag House

powder bed height

air distributor plate

1 SPRAY-ON 2 SECTION

CONDITIONING 3 SECTION

SC

PIC

Belt Feeder

powder bed width Plenum

Fig. 1. Typical layout of different ﬂuid bed processes: (a) ﬂuid catalytic cracking unit; (b) circulating ﬂuidised bed combustor; (c) ﬂuidised bed granulator; (d) continuous ﬂuidised bed mixer.

R. Boereﬁjn et al.

1022 Table 1. Basic features of some ﬂuidised bed processes Catalytic cracking

Combustion

Granulation

Mixing

Type of bed Range of superﬁcial gas velocities Distributors

Circulating 410 m s1

Circulating ﬃ5 m s1

Captive o2 m s1

Captive o2 m s1

Perforated plates

Nozzles

Gill plates

Separation Termination

Cyclones Sometimes

Cyclone Usually

Gill plates, with orientation, or perforated plates Bag House NA

Bag House NA

Let us consider as an example a typical cracking unit operation, as illustrated in Fig. 1a, with a solids load of about 500 tonnes. The oil feed is introduced into the riser and joins the ﬂow of regenerated catalyst up into the reactor. The endothermic cracking reactions already take place in the high velocity (about 20 m s1) riser, with a solids ﬂux of about 50 tonnes per minute. In the design in Fig. 1a, the reactor is a dilute phase ﬂuidised bed. Many other designs of FCC units include a fast ﬂuidisation reactor riser, featuring a terminator (Fig. 1b), where particles may impact at velocities of about 20 m s1 to be subsequently deﬂected into the cyclones [9]. After the reactor, the particles move back to the regenerator, where the coke deposit on the particle surface is burnt off. The regeneration process thus generates the heat of reaction for the cracking process. While moving from the reactor to the regenerator, the particles may pass through cyclones and, depending on the design of the unit, may pass through a grid before entering the dense ﬂuidised bed of the regenerator. Before reaching this grid, the particles may encounter an impact plate, intended to enhance particle distribution in the area below the grid. Typical particle velocities here are about 25 m s1, and velocities are about 50 m s1 at the grid holes of a few centimetre in size, spaced at a triangular pitch of about 0.5 m. Apart from suffering mechanical stresses, the particles also participate in chemical reactions and undergo a certain degree of surface sintering. An average daily amount of 4–8 tonnes of ﬁnes may be lost mainly due to attrition in a unit of this type. The above ﬂuid catalytic cracking process description exempliﬁes that a single process may contain a wide variety of attrition mechanisms, as there are, e.g. impact and rapid shearing, causing surface abrasion and fragmentation, and these are always a material response to a sequence of process conditions. Therefore, guidance is needed on how closely conditions of tests, which aim to predict the material behaviour in a process, need to match the actual conditions in that process, and this depends strongly on the material composition and its process history. Several authors have addressed the design of catalysts for FCC [10], FischerTropsch [11] and methylamine production ﬂuidised bed processes [12].

Attrition in Fluidised Beds

1023

Fig. 2. Surface abrasion (left) and fragmentation (right).

The circulating ﬂuidised bed combustion (CFBC) system of Fig. 1b consists of a riser, terminated with a blind T, followed by a cyclone separator and a return standpipe. This system will be described in more detail in Section 5 below. The granulation system of Fig. 1c consists of a simple rectangular ﬂuidised bed operated in plug ﬂow by a directional air ﬂow induced by the gill plate distributor towards the over/underﬂow weir, with exhaust ﬁlter separators. This system will be discussed in Section 4. A ﬂuidised bed blending system in its simplest form is shown in Fig. 1d, and is the basis for discussion in Section 3. Generally, breakage mechanisms may be distinguished between two extremes: surface abrasion and fragmentation or disintegration (Fig. 2). A spouting bed test according to Forsythe and Hertwig [13], as explained in Section 6 below, may be used to compare fresh and used FCC, which exhibit a special combination of these mechanisms [14], involving the rupture of the neck between burrs (Fig. 3a,b), which then leave a crater-like impression. Time effects are signiﬁcant, and difﬁcult to account for in-process and material design and testing. In their landmark paper on catalysts designed for longevity, Contractor et al. [10] showed that it can take over 100 days of continuous running for a good catalyst to reach equilibrium. Other examples of ﬂuidised bed processes in which particle attrition may be signiﬁcant are drying (see e.g. [15]), combustion (see e.g. [5]) and desulphurisation (see e.g. [6,16–18]). Attrition occurs through a number of distinct pathways that determine the performance of ﬂuidised bed combustors (FBC). Chirone et al. [5] have comprehensively surveyed attrition of carbon particles during ﬂuidised bed combustion. They highlighted the complex interplay of chemical reactions (devolatilisation, heterogeneous combustion of fuel char) with mechanical processes (abrasion, impact) in determining the fate of a fuel particle in an FBC and, in turn, the combustor performance. Attrition phenomena determine to a large extent:

the carbon loading establishing at steady state in the bed, the axial proﬁles of solids concentration along the riser, the levels of unburnt solid carbon at the exhaust, the role of char in the formation/destruction chemistry of gaseous pollutants, e.g. nitrogen oxides [19] and precursors of dioxins/furans [20].

1024

R. Boereﬁjn et al.

Fig. 3. (a) Fresh 75–90 mm FCC debris. (b) Used 75–90 mm FCC debris.

The study, after initially focusing on coals and low-volatile fuels, has recently been extended to biomass and waste-derived fuels, highlighting speciﬁc features of these fuels [21–22]. Cammarota et al. [23] have broadened the scope of the analysis to embody the fate of fuel ash beyond fuel particles burn-off (Fig. 4). They have shown that the properties and extent of primary ash particles (PAPs) released along burn-off and their subsequent attrition signiﬁcantly inﬂuence the inventory and particle size distribution of bed solids in the FBC. These features, in turn, are key parameters to control ﬂuidised bed heat transfer, control of bed temperature uniformity, balance between bottom and ﬂy ash, and emission of particulates at the exhaust. Using experimental and methodological tools similar to those employed in the characterisation of fuel attrition, the Naples group have carried out in-depth investigations of attrition of calcium based sorbents (limestone, dolomite) used for in situ desulphurisation in ﬂuidised bed combustion [6,24] in a mechanistic way. In particular, the interplay between chemical reactions (calcination, sulphation) and mechanical processes, which determine attrition, have been clariﬁed with reference to the prevailing mechanism of particle breakage: primary fragmentation

Attrition in Fluidised Beds

1025

3

1 Raw Fuel

PAPSD

Bottom Ash COMBUSTION

DEVOLATILIZATION

Coarse Char

2 4

Fine Char

5

Circulating Ash 6

Fly Ash and Unburnt Carbon

Fig. 4. The fate of a fuel particles and the primary ash particle size distribution (PAPSD) [23].

during calcination, abrasive wear, impact loading. It is noteworthy that in situ desulphurisation by calcium-based sorbents during ﬂuidised bed combustion represents one ﬁeld in which particle attrition may be regarded as a desirable rather than undesired process. In fact, better exploitation of the sorbent by disclosure of the unreacted core of exhausted sorbent particles to the reactive environment might be effectively accomplished by intentionally promoting in-bed attrition of the sorbent. Altogether, the research work on attrition of reactive solids underlines the need for realistic testing and additional efforts on validation of observations from both ‘‘cold’’ and ‘‘hot’’ models, which in some way may misrepresent the phenomena in the ‘‘real’’ process.

3. MAIN HYDRODYNAMIC EFFECTS 3.1. Simple ﬂuidised bed systems – elutriation and size distribution In the context of this chapter, simple bed systems are considered 2-phase, gas–solids systems without chemical reactions other than ad/absorption, such as (de-)hydration. In practice, this would apply to ﬂuidised beds used for conditioning (cooling and/or drying), drying, and mixing, either in batch or continuous mode, with or without ﬁnes recycle. Speciﬁc consideration will be given to jet effects. Fluidisation relies on an evenly distributed air supply. In general, this is provided by a gill or perforated plate, whereas in smaller equipment, where the investment of a high power compressor or blower is no object, sintered plates may be used. Table 2 shows a range of studies into attrition in ﬂuidised beds using perforated

1026

Table 2. Overview of studies of ﬂuidised bed attrition based on the superﬁcial or excess gas velocity Authors Blinichev et al. [25]

Dimension of attrition rate

Model equation Ra ¼

0:21d or =d p 18f 2:6 90js u5:8 s rf 10 1:3 1:3 d 1:9 p sbr rp

Merrick and Highley [26]

Ra ¼ C M b ðus umf Þ

Lin et al. [27]

Ra ¼ C expð0:162ðus umf ÞÞ

Donsı` et al. [28]

Ra ¼ Cðus umf Þ

t0:6

Wc dp

Sishtla et al. [29]

Rbed ¼ 1:6 103 ðus umf Þ0:66 ðHGIÞ0:05

Seville et al. [30]

Ra ¼ Cðus umf Þ

Material

Time of operation (h)

Range of (us -- umf) (m s1)

Range of uo (m s1)

0–11

20–240

Background ﬂuidisation

Attrition debris criterion

1–8 mm holes in perforated plate

None

Elutriate

Grid type

s1

3–5 mm NaCl, 3–5 mm nitrofosk, 3–5 mm silica gel

kg s1

Three types of coal 100 1.59 and 3.18 mm diameter 5 mixtures of 133–354 mm Silica sand and 10–113 mm char 0.4–3.0 mm coal in Not indicated 0.2–1.0 mm sand

0.67–2.67

Not indicated

Not indicated

0.67–2.67 m s1

Elutriate (smaller than 63 mm)

0.1–0.32

up to 53.5

492 holes 0.24 cm wide in triangular pitch

None

Elutriate

0.55–1.3

82–135

76 upward pipe holes, 1.5 mm wide

None

Elutriate and bed inventory

s1

500–841 mm char

0.3–0.6

67–134

None

Elutriate and bed inventory (smaller than 500 mm)

kg s1

1.18–2.80 mm Up to 0.25 sand agglomerates

0.15–1.15

74–111

Six bubble caps with three oriﬁces, 0.24 cm wide, 301 downward 139 holes, 1.5 mm wide on 12 mm triangular pitch

None

Elutriate

s1

kg s1

1

14

R. Boereﬁjn et al.

Attrition in Fluidised Beds

1027

plates with and without background ﬂuidisation, in which the superﬁcial or the excess gas velocity, the difference between the superﬁcial and minimum ﬂuidisation gas velocities, was taken as the main hydrodynamic parameter. A linear relation between gas velocity and attrition rate is indicated in Table 2 by Merrick and Highley [26], Donsı` et al. [28], and Seville et al. [30]. The exponential correlation of Lin et al. [27] appears to be an exception to the general form, but replotting their data shows that there is no need for an exponential correlation, and that an equally good ﬁt is also achieved with a linear regression. Also Blinichev et al. [25], and Sishtla et al. [29] appear to present exceptions, the former reporting a non-linear relation with an exponent of 5.8, the latter 0.66. Blinichev et al. [25] used a material similar to the NaCl for which Ghadiri et al. [2] report an exponent of 5.1 for jetting attrition (Table 3 below). From the descriptions it appears that the jets are dominating the attrition phenomena, but this still does not explain this high exponent value presented, nor the remark by Sishtla et al. [29] that their result is in broad agreement with that of Blinichev et al. [25]. The deviation in exponent value of Sishtla et al. [29] can be explained by the fact that their data show an initial rapid, non-linear rise, due to initial rounding off and equilibration, after which a linear dependence of attrition on time follows. The hydrodynamic regimes in drying and mixing systems tend to be in the turbulent regime, with the superﬁcial gas velocity in excess of 20 times the minimum ﬂuidisation gas velocity for the mean particle size and density, to avoid the risk of de-ﬂuidisation in the presence of a coarse fraction. This fact turns the mixing system into an effective elutriator or air classiﬁer, which in some cases can be useful for inﬂuencing the performance of the product, e.g. in dustiness or dispersion into a solution. It does, however, affect the bulk density negatively, and may result in a higher volume dosage for the same weight, i.e. higher packaging cost, not to speak of the added cost of handling the ﬁnes recycle stream. Heat (and mass) transfer at high excess gas velocity decreases gradually, mainly due to the increased voidage in the bed, and the decrease of particle cluster size [31]. Elutriation has been extensively studied, and most recently an overview of available correlations was provided by Stojkovski and Kostic [32]. They found that the correlation provided by Geldart et al. [33] was providing the best ﬁt to literature data and their experiments, and also proposed a simple alternative correlation. Interestingly, all correlations are based on Froude and Reynolds numbers applied to the superﬁcial gas velocity, many also include dimensionless groups of ratios of particle to ﬂuid density and terminal to superﬁcial gas velocity. None would therefore appear to apply to beds whose hydrodynamic behaviour is governed by the penetration of jets, or which operate with spouts or turbulent regions. In such cases, at least for captive beds, the transport disengagement height [34] needs to be observed and sufﬁcient freeboard length (and width) allowed.

1028

Table 3. Some distributor conﬁgurations

Simple Jet (perforation)

Side-entry Jet

Ghadiri and Boereﬁjn [41]

Ayazi-Shamlou et al. [42]

Maximum Penetration Jet

Shrouded Jet – upward

Shrouded Jet – downward

Zenz [43] and Parker et al. [44]

Zenz [43] and Seville et al. [30]

Closed arrow: jet air supply; open arrow: background ﬂuidisation air supply. R. Boereﬁjn et al.

Attrition in Fluidised Beds

1029

Attrition typically translates into a broadening of the particle size distribution. The effect of changes in the particle size distribution on the entrainment or elutriation of particles has been the subject of some controversy in the literature. Adding ﬁnes to a ﬂuidised bed may result in a decrease of entrainment, as ﬁnes may adhere to coarse particles and reduce or delay bubbling [35]. In turbulent beds, on the other hand, addition of ﬁnes may cause the carry-over of coarse particles, despite their free fall velocities being in excess of the superﬁcial gas velocity [36]. Rodrı´ guez et al. [37,38] have performed recent studies on the relation between attrition and elutriation for Geldart group A and C powders, e.g. iron oxides, and report little observable difference for these materials. The review of Stojkovski and Kostic [32] indicates that even though all correlations may be applicable only to a small range of materials and conditions, all are based on some form of dimensional analysis for the same material properties and process conditions, and have as main prediction quantity the so-called rate constant, which is in fact not the rate of change of the amount of ﬁnes, as the name suggests, but rather the ratio between ﬁnes concentrations in bed and freeboard. All correlations are based on ﬁxed conditions and ignore the transient nature of the bed inventory – noteworthy.

3.2. Jet effects In industrial processes with high temperatures and/or large solid ﬂuxes, causing thermal degradation or erosion, perforated plates may provide more robust air supply compared to meshes or gill plates, though this may come at the price of less even distribution. Perforated plates may be equipped with caps to increase the surface area over which they are active [8], or may be covered by a wire mesh [39]. Any design modiﬁcation that breaks up the jet will have a positive effect on the attrition rate, except when particles are entrained into the jet before its path is broken. In order to verify all jet holes are active, the design procedure of Fakhimi and Harrison [40] may be used. Massimilla [8] provides an overview of modes of gas discharge in ﬂuidised beds of particles, and distinguishes a number of physical geometries. Table 3 shows some geometries and where references with attrition studies are also available. The ﬁnal two types are indicated by Zenz [43] to have minimal attrition, whereas the side-entry jet, especially when it is opposite another, i.e. impinging jet, is useful for jet milling [42]. Table 4 shows a range of attrition studies where the oriﬁce gas velocity was taken as the main hydrodynamic parameter. Concentrating on the main design parameters, the oriﬁce gas velocity and diameter in Table 4, it is clearly shown that most correlations are of the type Ra / un d hor

ð1Þ

1030

Table 4. Overview of studies of ﬂuidised bed attrition based on the oriﬁce gas velocity Authors Chen et al. [45]

Model equation

f ðd p ; jso Þ ¼

Ra ¼ Crf u3o d 0:55 for uor3.6 m s1; or Ra ¼ Crf u2o d 0:55 for uo43.6 m s1 or

Sishtla et al. [29]

2:15 Ra ¼ 2:8 1012 ðr1:56 u3:12 f o ÞðHGIÞ

Werther and Xi [47]

Ra ¼ Crf d 2or u3o

Ghadiri et al. [2,4,48]

Ra ¼ Cuno d hor with n ¼ and h ¼

Material

Time of operation (h)

Range of (usumf) (m s1)

Range of uo (m s1)

Grid type

Background ﬂuidisation

Attrition debris criterion

Variable

Elutriate (smaller than 75 mm)

s1

115–274 mm siderite iron ore and 210 mm lignite char

Up to 24

0.2–0.5

Up to 300

1.47–3.18 mm single jet in porous plate

kg s1

200 mm FCC

12–80

Not indicated

33–303

s1

0.97–4.00 mm mullite aluminasilicates 500–841 mm char

8–12

0.45–8.0

0.5–30

Variable Downward pipe holes 0.8–19 mm wide 3.6–15.5 cm None single tapered jet

14

0.3–0.6

67–134

106 mm spent FCC, 125 mm fresh FCC 425–600 mm NaCl

Up to 260

0.2

25–100

10–22

0–0.85

25–125

jso þ ð1 jso Þð1 d p =d op Þk

pd 2 pﬃﬃﬃﬃ Ra ¼ Cðuo rf Þ2:5 z or 4

Zenz and Kelleher [34] Kono [46]

rf Qðbuo Þ2 f ðd p ; jso Þ with Wd p rp h i ð1 jso Þ 1 ð1 d p =d op Þk

Ra ¼ C

Dimension of attrition rate

5:1 for NaCl 3:31 for FCC 0:4421:11 for NaCl 0:620:76 for FCC

s1

kg s1 h1

90–106 mm FCC

Six bubble caps with three oriﬁces, 0.24 cm wide, 301 downward 0.5–2.0 mm single jet in porous plate 73, 110 and 175 3.0 mm holes in triangular pitch

Elutriate

Elutriate (smaller than 88 mm)

None

Elutriate and bed inventory (smaller than 500 mm)

Variable

Elutriate (smaller than 23–35 mm)

None

NaCl: all smaller than 355 mm; FCC: all smaller than 75 mm

R. Boereﬁjn et al.

Attrition in Fluidised Beds

1031

with values of n ranging from 2 to 5.1, and of h from 0 to 1.11, as reported by different authors. Notwithstanding this basic similarity, each of these correlations has been obtained for different materials, experimental set-up and operating conditions, making it difﬁcult to compare them. Ghadiri et al. [4] coupled a model of single particle attrition [49] with a hydrodynamic model of ﬂuidised bed jets [7,8]. Two basic assumptions are made The jet region contribution is dominant in the sum of attrition processes. Breakage occurs by impact of particles entrained freely into the jet.

Assumption 1 has been validated by Ghadiri et al. [48]. Assumption 2 was validated by Boereﬁjn [50] for oriﬁce-to-particle size ratios above about 10 for small particles (FCC). For ratios smaller than 10, the jet may close and a rapidshear abrasion may dominate attrition. Similar observations were made by Stein et al. [39] for large particles (glass) for oriﬁce-to-particle size ratios above 2. Vaccaro [51] notes that below oriﬁce-to-particle size ratios of 7.5 the jet half angle does not depend on the Froude number based on oriﬁce gas velocity and either particle size or oriﬁce diameter. A single particle impact attrition test device, presented in Section 6 below, can be used to develop a relation between the impact attrition rate and the impact velocity ui: Ri / um i

ð2Þ

where m is usually about 2 for crystalline materials. The impact attrition index m is sensitive to internal and surface defects and particle structure, e.g. the presence of polycrystals, work-hardening, and fatigue [3,48,49,52]. A hydrodynamic model of ﬂuidised bed jets [7,8] may be used to compute the solids entrainment rate Ws and the particle velocities up as a function of the oriﬁce gas velocity and diameter. This yields an overall expression for the dependence of the attrition rate on oriﬁce gas velocity and diameter as follows: l m k Ra / W s Ri / uko um / uokþlm ð3Þ p / uo uo and s t m sþtm Ra / ðW s =W g ÞRi / d sor um p / d or ðd or Þ / d or

ð4Þ

where Wg is the gas mass ﬂow rate and l, t are the exponents for the correlations between the particle velocity and oriﬁce gas velocity at constant oriﬁce diameter and the oriﬁce diameter at constant oriﬁce gas velocity, respectively, as computed by the model. Ranges for the above exponents are shown in Table 5 [41]. The combination of single particle impact tests and hydrodynamic modelling allows prediction of values for n and h to within 10% [41]. It should be noted that

R. Boereﬁjn et al.

1032 Table 5. Ranges of hydrodynamic exponents

Exponent

Min

Max

h k l m s t

0.44 1.3 0.9 2.3 0 0.1

1.1 1.9 0.9 2.6 0.2 0.5

due to the submerged nature of the jet, particle velocities in the jet rarely exceed 10 m s1, even at oriﬁce gas velocities in excess of 60 m s1.

3.3. Terminator and cyclone effects The end of a riser, called a terminator, plays an important role in circulating ﬂuidised bed attrition. Terminators can be designed for longevity, or to reduce attrition. A blind tee (Fig. 1b) is often employed, and a bed of powder may accumulate in there to soften the impact. Alternatively, a straight elbow can be used, or a large radius bend, which typically wears out at the 151 angle [53]. Zenz and co-workers [34,43] have investigated this extensively and recommend using the blind tee. Sesti-Osseo and Donsı` [54] have proposed a novel geometry, integrating a cyclone-type separator into the terminator, but do not report on longterm wear of equipment or materials. Recently, Werther and Reppenhagen [55] have quantiﬁed cyclone losses and proposed a model for this. Solids abrasion in their cyclone model is a linear function of the ratio of cyclone inlet gas velocity squared to the square root of the solids concentration.

4. ATTRITION IN WET SYSTEMS In addition to combustion and cracking processes, particle formation processes such as ﬂuidised bed granulation also involve some particle breakage. Fluidised bed granulation is a common production process for powders in the agricultural (e.g. fertilisers, feed), pharmaceutical, food, and ﬁne chemicals (e.g. detergents) industry. No less than three reviews in the last 15 years signify the interest in ﬂuidised bed granulation [56–58]. Fluidised bed granulation processes tend to differ by industry, as shown in Table 6. In the food and pharma industry, granulation may involve spraying a solution or dispersion onto a carrier, which may consist of the same material.

Attrition in Fluidised Beds

1033

Table 6. Types of ﬂuidised bed granulation system by industry (MT denotes metric tonnes)

Industry

Bottom/top spray

Continuous/batch

Scale

Food/pharma Detergents Agriculture

Top Top Bottom

Batch Continuous Continuous

0.1–10 MT 1–30 MTph 1–10 MTph

Drying is an essential part of the granulation process, which makes it a relatively slow process. In the detergents industry, apart from granulation by spraying pure liquids and solutions onto carriers, reactive granulation is common, where a liquid acid (mixture) may be sprayed onto an alkaline carrier, and heat of reaction promotes drying, but does pose the requirement of a conditioning (cooling) step, which may result in breakage and generation of ﬁnes. Typically, in ﬂuidised bed granulation kinetics, a distinction is made between layering growth and coalescence. Coalescence is actually a rare phenomenon in industrial practice, reserved for those cases which satisfy all of three conditions: 1. Agglomerates contain sufﬁcient liquid content to merge. 2. The liquid is viscous enough to stabilise the agglomerates. 3. Bed ﬂuidisation is vigorous enough to prevent wet quenching. The ﬁrst and third criteria are captured in a so-called Akkermans or Flux Number [59,60], deﬁned as the ratio of a spray ﬂux and the ﬂuidisation momentum. The second criterion is captured in a Stokes criterion [61], comparing particle inertia and liquid viscosity, which is complemented by the criterion to avoid agglomeration and remain in a coating regime [62]. Helpful to satisfy the ﬁrst criterion is the ratio of droplet to particle size, which should be close to or above unity [63]. Top spray systems are ﬂexible with respect to growth kinetics, bottom spray systems always operate in layering mode. Extensive work by Hounslow and co-workers [64] has shown that the equipartition of kinetic energy principle (EKE) applies to net growth of granules in most ﬂuidised bed granulation systems. This principle assumes that the collision speed of a primary particle or agglomerate thereof scales with its momentum, i.e. the product of speed (squared) and mass remains constant irrespective of size and density. Fluidised bed granulation processes have until recently been characterised mainly by the net growth kinetics, where no distinction is made between instantaneous growth and breakage, which occur in parallel. Only recently an attempt has been made to quantify instantaneous breakage rates [65], by performing granulation experiments in which the spray was switched off after a certain time. The granule size distribution mean was shown to increase steadily during

R. Boereﬁjn et al.

1034

spray-on, but after spray-on would decrease sharply and then gradually reduce to an asymptotic value. The standard deviation of the granule size distribution was shown to remain unchanged during the breakage period, and even to remain similar to that in the ﬁnal stages of the spray-on period. Population balance modelling has been applied successfully to describe growth kinetics and has resulted in quantitative prediction of growth as a function of operating conditions and material properties by relating the growth kernel to the Akkermans or Flux Number [66,67]. Biggs et al. [65] showed that the same growth kinetics applied to the spray-on phase in their experiments, and then quantiﬁed a separate breakage kernel for the phase after spray-on. The point in time when the spray-on is switched off for experiments with the same spray-on rate can be designated by a liquid-to-solid ratio (L/S). The growth kernel is that prescribed by the EKE theory [64]: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 1=3 1=3 b0 ¼ d i þ d j ð5Þ þ di dj with di and dj the sizes of primary particles participating in the growth process, whereas the breakage process is assumed to progress according to: b1 et=t

ð6Þ

where the exponent accounts for the asymptotic limit to size reduction observed, which could be related to a survival-of-the-ﬁttest selectivity. Table 7 shows the resulting values for the growth (b0) and breakage (b1) kernels. Numerically, values for b0 and b1 appear to be very close indeed, indicating that the instantaneous net growth process is only marginally stronger than (i.e. overcoming) the breakage process during spray-on. Figure 5 shows the resulting ﬁt between model and experimental data for the volume weighted mean diameter, D4, 3. Population balances are number-based, and therefore take into account the reduction of the number of collisions as agglomerates grow and the number of particles in the system decreases. The steady increase of the value of t as L/S increases may indicate that as the growth process progresses, the agglomerates that survive become inherently stronger as well, in spite of their larger size. A Table 7. Growth and breakage kernels and time constants (experimental details as in caption to Fig. 5)

L/S (wt/wt)

Spray off (s)

0.05 0.10 0.20

450 900 1500

b0 (m5/2 s1) 12

2.1770.06 10

b1 (m5/2 s1)

t (s)

1.8070.14 1012 4.7871.14 1012 3.6970.37 1012

276730 297784 421754

Attrition in Fluidised Beds

1035

Fig. 5. Model predictions and experimental data of subsequent growth (open symbols) and breakage (closed symbols) phases. 90–150 mm Glass ballotini and polyethyleneglycol MW4000 binder, sprayed on at 4.8 kg h1 at 801C, bed temperature 401C, diameter 0.4 m.

value of t of 200 s results in a size reduction of nearly 80%, whereas a value of 400 s results in a size reduction of about 50% after 10 min, which is a common conditioning time span in industrial applications – signiﬁcant ﬁnes generation, product loss, bad ﬂowability and dustiness may be the consequences. In their review on breakage in granulation, Reynolds et al. [68] comment on the above approach that, though useful, it is fundamentally ﬂawed, as breakage is a ﬁrst order process and growth is second order. The success of the net-growth approach [66], i.e. explicitly ignoring breakage altogether and using the EKE growth kernels to quantify net growth rates indicates that such simpliﬁcation is often both convenient and justiﬁable. Also White et al. [69] use a simple netseeding term and ignore attrition in their population balance model of a silicon particle production ﬂuidised bed process, which provides satisfactory results in comparison with the experimental data. Tan et al. [70] use population balance modelling to compare three different breakage modes during granulation, which they deﬁne as attrition, fragmentation, and binary breakage. Attrition is deﬁned as a surface erosion process, resulting in a slight size reduction of granules and some ﬁne debris. Fragmentation is complete shattering into primary particles, and binary breakage describes the breakage of granules into parts of similar size. The modelling of so-called breakage-only indicates that only binary breakage is approaching the breakage rates of the experiments. This is somewhat curious as the growth model assumes a layering

R. Boereﬁjn et al.

1036

growth, and not coalescence. It remains difﬁcult to mechanistically distinguish unsuccessful growth from attrition.

5. ATTRITION IN REACTING SYSTEMS Most of the research activity on attrition in reacting systems has regarded attrition of fuel and/or sorbent during ﬂuidised bed combustion and gasiﬁcation. Salatino and Chirone [71], extending previous work by Chirone et al. [5], classify attrition of fuel particles during ﬂuidised bed combustion (FBC) as follows: Primary fragmentation: the breakage of fuel particles associated with internal

overpressure due to volatile matter release during devolatilisation. Primary fragmentation may yield coarse (non-elutriable) and ﬁne (elutriable) particles depending on the extent of fragmentation. Extensive testing of different carbon materials (coals, biomass- and waste-derived fuels) has shown that primary fragmentation of fuel particles may be extensive (generation of more than 1000 fragments per single mother fuel particle have been observed during ﬂuidised bed combustion of some anthracites) as related to internal overpressures building up within the particle during late devolatilisation. The mechanistic framework of primary fragmentation developed at Naples has received conﬁrmation by other groups [72]. Primary fragmentation can be characterized in terms of multiplication factors and fragmentation probabilities for any given fuel and particle size with fairly established experimental procedures. Attrition by surface abrasion: generation of char ﬁnes by rubbing of char particles with bed solids and with reactor walls and internals. Attrition by surface abrasion generates ﬁne char particles whose size is such that they are readily elutriated. No doubt, the most remarkable feature of attrition by surface abrasion of char under reactive conditions is that it is enhanced by orders of magnitude when compared with attrition under non-reactive conditions. Indeed, internal burning, either extended to the entire char particle or conﬁned in a cortical region, increases particle porosity and, correspondingly, brings about a loss of mechanical resistance. Internal burning and surface wear suffered by char particles in a ﬂuidised environment are combined into the so-called ‘‘combustion-assisted attrition’’ mechanism. Detailed single particle combustion-attrition models have been developed [73,74]. However for practical design purposes the use of equation E c ¼ CðU U mf Þ

Wc d

ð7Þ

may be preferred [75], in view of its simplicity. The attrition constant C depends on the fuel type and on oxidising conditions and can be characterised by purposely developed experimental protocols.

Attrition in Fluidised Beds

1037

Fragmentation by loss of particle connectivity: fragmentation of ‘‘percolative’’ na-

ture due to enlargement and overlapping of pores or to the burn-out of solid bridges responsible for the connectedness of the carbon particle. Both coarse (non-elutriable) and ﬁne (elutriable) particles may be generated by this kind of fragmentation, which, notably, does not require any external force to make the particle fall apart. The relative importance of fragmentation by loss of particle connectivity versus attrition by surface wear was addressed by Salatino et al. [21]. Fragmentation by impact breakage: breakage of char particles induced by impact loading. It can lead to either chipping or fragmentation depending on impact velocity and particle failure mode. This mechanism is comparatively less extensively characterised, partly because it is closely related to detailed design of reactor components like the distributor and the terminator. A mechanistic framework for the attrition of sorbent particles, used for in situ desulphurisation in FBCs, has been developed by the Naples group similarly to that developed for fuel particles. In atmospheric FBCs, under overall oxidising conditions, limestone-based sorbents undergo the following reactions: CaCO3ðsÞ ! CaOðsÞ þ CO2ðgÞ

ð1Þ

CaOðsÞ þ SO2ðgÞ þ 12O2ðgÞ ! CaSO4ðsÞ

ð2Þ

Sorbent particles ﬁrst undergo calcination (reaction 1) yielding porous calcium oxide, which eventually reacts with SO2 following reaction 2. The latter reaction yields compact calcium sulphate as the product. Calcination has a much shorter time scale than sulphation, so that for all practical purposes the two reactions can be considered in series. Sorbent sulphation has been extensively investigated in the past. Sulphation most typically conforms to the core-shell pattern (the reaction front in the sorbent particle divides the porous unreacted CaO inner core from the dense reacted CaSO4-rich outer shell), but ‘‘network’’ and ‘‘uniform’’ sulphation patterns have also been occasionally observed. Sulphation of the unreacted CaO core is hindered by the onset of strong diffusional resistance to SO2 transport across the CaSO4-rich outer layer. Degrees of calcium conversion seldom exceed 30–40%, and overstoichiometric Ca/S feed molar ratios (Z2–3) are required to obtain sulphur capture efﬁciencies larger than 90%. Substantial changes in the particle size distribution of sorbents can be brought about by particle attrition/fragmentation during calcination/sulphation in FBCs. Scala et al. [6,24] have established a classiﬁcation of sorbent attrition phenomena during ﬂuidised bed calcination and sulphation based on the relevant breakage mechanism and on the size of generated fragments (Fig. 6). According to this

1038

R. Boereﬁjn et al.

Fig. 6. Attrition mechanisms in calcination and sulphation (from Ref. [6]).

classiﬁcation, that partly echoes that established for fuel particles, the following mechanisms are identiﬁed: Primary fragmentation Attrition by surface abrasion Secondary fragmentation.

Primary fragmentation occurs immediately after the injection of the particles in the bed, as a consequence of thermal stresses due to rapid heating of the particles and of internal overpressure due to carbon dioxide emission. This may result in the generation of coarse as well as ﬁne fragments. The damage induced by overpressures is dependent on the texture and the mechanical properties of

Attrition in Fluidised Beds

1039

the sorbent, and can be characterised in terms of alterations to the original particle size distribution of the native sorbent by fairly well established experimental procedures. Attrition by surface abrasion generates ﬁne particles that are quickly elutriated. This breakage process is essentially related to the occurrence of surface wear as the emulsion phase of the ﬂuidised bed is sheared by the passage of bubbles. Attrition by surface abrasion of sorbents under reactive conditions displays a complex phenomenology, which results from conﬂicting effects of particle calcination and sulphation [76]. The increase of sorbent particle porosity induced by calcination brings about a dramatic loss of mechanical resistance of the sorbent particle at ﬁrst, and a pronounced increase of sorbent attrition rate. This phenomenon takes place over a time scale of tens of seconds to minutes, i.e. the time scale of particle calcination. Subsequently, and over a much longer time scale (several hours), the progress of sulphation brings about the build up of a less porous and much harder sulphur-rich outer shell enclosing an unreacted core. As a consequence of this surface hardening of the particle, the attrition rate decays to values that may be several orders of magnitude smaller than that of the lime (Fig. 7). Secondary fragmentation generates coarser fragments that are typically retained within the reactor for relatively long residence times (i.e. long enough to give rise to extensive conversion). Secondary fragmentation is generally the consequence of damage by impact loading of sorbents, a process that has been recently investigated by Scala et al. [24,77] (Fig. 8). It has been shown that the progress of calcination and sulphation signiﬁcantly affects the impact fragmentation behaviour, much like resistance to surface wear. In particular, the raw sorbent is the most resistant and the calcined one is the less resistant to impact loading. Raw and fully sulphated limestones exhibit fragmentation patterns typical of brittle or semi-brittle materials. Lime particles undergo attrition by impact loading according to a failure mode involving ‘‘plastic’’ deformation and disintegration into polydisperse

Fig. 7. Elutriation during calcination and sulphation of limestone.

1040

R. Boereﬁjn et al.

Fig. 8. Crack formation in simultaneously calcined and sulphated limestone (from Ref. [24]).

fragments. An interesting aspect of the phenomenology is related to the composite (core-shell) nature of fully sulphated sorbents: the properties of the outer sulphated shell dominate impact damage at small impact velocity, the unreacted core takes part in the fragmentation process at higher impact velocities. Altogether, the relevance of sorbent attrition to the performance of FBC fuelled with high-sulphur fuels has been addressed [17,78] in the light of the multiple and sometimes conﬂicting consequences of particle breakage. On one hand sorbent attrition is responsible for the generation of ﬁne particles of elutriable size whose residence time in the combustor can be much shorter than the time required to achieve full sulphation. This feature would limit sorbent exploitation. On the other hand, attrition could promote the breakage of the outer sulphated shell and the exposure of the unconverted core to the reactive environment, with positive implications on sorbent exploitation. The analysis of attrition under reactive conditions suggests that different chemical reactions and particle breakage modes combine with each other so as to give rise to multiple pathways to particle attrition. It cannot be claimed that a single attrition testing method can provide all the information required for proper design or optimised operation of an FBC. Assessment of the co-operative effect of the different attrition pathways can only be accomplished with a combination of reliable data from down-scaled test facilities (e.g. Fig. 9) and of integrated, multilayer modelling approaches based on single particle models and global models. Salatino and Chirone [71] analysed the various sections of the CFBC loop with the aim of underlining those which are potential sources of attrition. An updated version of the result of their analysis is reported in Table 8. The classiﬁcation of sorbent attrition processes developed at Naples is now a generally recognised framework for analysing sorbent particle behaviour in ﬂuidised bed desulphurisation processes [79].

Attrition in Fluidised Beds

1041

Fig. 9. Down-scaled ﬂuidised bed combustor for attrition characterisation at Naples [6]. Table 8. Locations and mechanisms of attrition in a circulating ﬂuidised bed combustor Attrition mechanism Primary fragmentation Combustor section

Riser

Dense bed – jetting region Dense bed – emulsion phase Dilute region Terminator

Cyclone Standpipe, EHE

Surface wear

Impact

++

++

++

+ +

+

: absent; +: minor relevance; ++: dominant.

6. STANDARD TESTS AND CHARACTERISATIONS Several attrition test methods exist, some based on ﬂuidisation technology, some intended to mimic a process employing this. The use of small ﬂuidised bed jet devices for testing the material resilience became a matter of importance with the advent of the ﬂuid catalytic cracking process. Forsythe and Hertwig [13] were the

R. Boereﬁjn et al.

1042

ﬁrst to employ a single oriﬁce jet device (Fig. 10), followed by Gwyn [80] with his triple oriﬁce jet device (Fig. 11). The Forsythe and Hertwig [13] test has become the basis of ISO 5937, introduced 1980, withdrawn 2002, identical to BS5688 Part 26, speciﬁcally intended for use by the detergent industry for boric acid and borates, but used more

Connection replaced by Dust Collection Filter during Test

Glass Tube Flow Meter

Air Supply

Fig. 10. Typical layout of ﬂuidised bed jet test device as used in ISO 5937 [81], based on Forsythe and Hertwig [13].

Dust Collector Perforated Plate (dashed lines to guide the eye) Fines Disengager Syringe (Water Supply)

Air Supply

Mass Flow Controller

PI Perforated Plate

Fig. 11. Typical layout of a test device as used in ASTM D5757-95 [83], following Gwyn [80].

Attrition in Fluidised Beds

1043

widely for detergent ingredients as well [81]. It is commonly accepted that the oriﬁce jet tests are useful to mimic the following processes: Circulating ﬂuidised beds, used in combustion, catalytic cracking, sorption, etc. Dense phase pneumatic conveying [82].

The Gwyn [80] device has been adopted in ASTM D5757-95 [83], which prescribes calculation of the Air Jet Index (AJI): m1 m0 þ ms m00 AJI ¼ 100 ð8Þ ms with m1 the mass of ﬁnes collected at 1 h, m0 the mass of ﬁnes at the start, ms the starting mass, and m00 the mass of ﬁnes collected after 5 h. Cairati et al. [84] have shown that molybdate catalyst particles do not reach steady state equilibrium within 1 h in the Forsythe and Hertwig test, and Ghadiri and Boereﬁjn [41] show that NaCl particles only reach this after 5 h. Recently, the so-called Davison Jet-Cup test (Fig. 12) has become of interest [85]. This test employs a small cylinder with a small tangential jet at the bottom through which gas is supplied. Compared to the ﬂuidised bed oriﬁce jet tests, this test is more suited to mimic cyclone attrition. Zhao et al. [86] found that the breakage mechanism in ﬂuidised bed jets is abrasion, whereas in the Jet-Cup test it is a combination of abrasion and fragmentation, the latter being dominant by far. In general they found a 1:1 correlation between the amounts of material broken in both tests, which is in itself a curious ﬁnding given the difference in breakage mechanism. In only one case did they ﬁnd that a material very prone to fragmentation showed a very different amount of breakage in the two tests. The Jet-Cup test is being developed into an ASTM Standard under number WK282 by ASTM Subcommittee D32.02. The Jet-Cup test allows for calculation

Dust Collector

Flow Meter Air Supply Water Bath

PI Sample Holder

Fig. 12. Jet-Cup schematic.

R. Boereﬁjn et al.

1044

of the Davison Index (DI): 0c 1 100 þ H f G Bm C DI ¼ @ s A 100 100 G

ð9Þ

with c the mass fraction of ﬁnes, ms the starting mass, Hf the mass fraction of ﬁnes left in the sample, and G the mass of ﬁnes in the starting material. As an alternative to DI, another index termed WD has been proposed [85]: Cum: wt% fines ¼ WD t þ C

ð10Þ

or the slope of a time plot of ﬁnes generated, with C the intercept, similar to G/m in DI and to the constant used originally by Gwyn [80]. There are two test methods for speciﬁc use with enzymes in the detergent industry: a single oriﬁce ﬂuidised bed jet test called the vertical elutriation test, similar to

the ISO 5937 device shown in Fig. 10, and the Heubach Type III (Fig. 13), a ﬂuidised bed, in which four steel balls, 20 mm

in diameter are moved around by a motor-driven impeller to cause attrition of the granules [87]. York et al. [88] have modiﬁed the Heubach Type III to their purpose by using tungsten balls and increasing the impeller speed to 75 rpm. The comparison of the shearing/crushing action in the Heubach Type III with any actual process has to date not been carried out. A different type of Heubach test, Heubach Dust meter Type I (Fig. 14), is in use in the ﬂour, pigment, and paint ﬁller industry [89], namely a rotating drum through which a moderate airﬂow is run to remove ﬁnes below 30 mm. It is unclear how the Heubach Type III operated in stressed condition with the balls can reliably simulate process conditions in the industry. However, the other tests, Heubach

Fig. 13. Heubach Type III (left) and internals (right).

Attrition in Fluidised Beds

1045

Fig. 14. Heubach Type I.

Types I and III without balls, and the vertical elutriation test, have as their purpose not the estimation of attrition in a process, but rather the liberation of ﬁnes below 150 mm from ﬂour and enzymes, as these are commonly allergenic or sensitising. Heitbrink [90] provides an analysis of Heubach Type III performance as a function of operating conditions and material properties, also in comparison to another dustiness test, called Midwest Research Institute (MRI) test. Hamelmann and Schmidt [91,92] provide an overview of dustiness tests. In the absence of tests, which directly model the critical process steps with sufﬁcient sensitivity and accuracy, industry has to resort to running new materials under strict monitoring conditions through a plant whilst collecting samples and then using these tests to measure the amount of dust generated. As mentioned in Section 3, single particle impact tests may be used to complement understanding and prediction of ﬂuidised bed jet attrition [93]. This test method, developed by Ghadiri and co-workers at Surrey/Leeds [49], employs an air eductor and a target, as shown in Fig. 15. It consists of a funnel-shaped inlet section guiding the particles into an eductor tube, which ends in a collection chamber, where particles impact on a rigid horizontal target plate made of sapphire. The tube does not need to be very long, nor is positive air supply required. The particle velocity before impact is measured with dual light diodes or with a laser Doppler velocimeter. The impact product is analysed gravimetrically. As a criterion for attrition, the mass of debris passing through a sieve, with a size of two BS410 sieve sizes below the lower limit of the original size, is chosen. In the low range of impact velocities, where mainly surface damage (chipping) occurs, the ﬁnes produced will be far smaller than the original mother particles. Cleaver et al. [94] have shown that the attrition results are in this case insensitive to the speciﬁc criterion applied, as long as the sieve size, used for the separation of debris from the

R. Boereﬁjn et al.

1046

Fig. 15. Single particle impact rig.

Relative Frequency (-)

Mother Particles Coarse Product

Fine Product

Particle Size (Arb. Units)

Fig. 16. Particle size distribution for chipping attrition.

Attrition in Fluidised Beds

1047

mother particles, lies in between the particle size distributions of the ﬁne product and coarse product, as shown schematically in Fig. 16. The attrition rate is then simply deﬁned as the ratio of the mass of ﬁne product to the initial sample mass of mother particles: Ri ¼

M fineproduct M motherparticles

ð11Þ

An alternative design, speciﬁcally for fragmentation tests on sorbents, such as limestone, is shown in Fig. 17. This device has been used successfully to correlate

Fig. 17. Fragmentation impact rig developed by Salatino and co-workers [77].

Fig. 18. Scaled-up Jet-Cup (courtesy of PSRI Chicago).

1048

Table 9. Overview of common test methods and relevant parameters Oriﬁce size (mm)

Oriﬁce number

Cut-off size (mm)

Sample size (g)

Test duration

25 Pneumatic conveying (dense phase); attrition in 25 35 circulating ﬂuidised beds

14

0.375

1

150

50

1h

7 10

125 20

50 50

10 min Mass fraction below 150 mm 1 and 5 h 1 h: ðm1 m0 =ms Þ 100; 5 h: AJI

Quality control

24

0.4 1 0.381, 10 mm 3 triangular pitch 1.0 1

150

60

40 min

Weight increase on ﬁlter

1 hr

DI or WD

Source/standard

Purpose

Relevance

Fluidised bed jet tests

Forsythe and Hertwig [13] ISO 5937 [81] ASTM D5757-95 [83] and Gwyn [80] AISE vertical elutriation test [95]

Catalyst attrition Borates attrition Catalyst attrition

Dust liberation

Tube diameter (mm)

Air ﬂow rate (l min1)

Test method

25

Attrition measurement

Jet-Cup

ASTM WK282

Catalyst attrition

Cyclones

NA

21

1.5

1

20

5

Heubach Type I

DIN 55992-1 [89]

Dust liberation

Dustiness

NA

20

NA

NA

30

50

5 min

Weight increase on ﬁlter

Heubach Type III

AISE attrition test Dust liberation/ [95] attrition

Quality control

70

20

NA

NA

50

20

10 min

Weight increase on ﬁlter

Single particle impact test

Ghadiri et al. [2,3,48]

Termination, regeneration impact

NA

Impact attrition

2 BS410 4th root NA of two sieve sizes below lower size limit of starting material (single sieve cut)

Mass fraction of debris

R. Boereﬁjn et al.

Attrition in Fluidised Beds

1049

observed fragmentation levels in a ﬂuidised bed combustion process, and shows promise of predictive capability [77]. It should be noted that in all these tests, dust collection is tedious and critically dependent on the layout and even the materials of construction of the equipment, as bends and the use of non-conducting material may result in signiﬁcant build-up and material loss. For this reason, PSRI Chicago has scaled-up the Jet-Cup test to several inches in diameter, so it can be used with sizeable amounts of material, and also in conditions relevant to the process it mimics (Fig. 18). Table 9 lists dimensions and operating conditions for the above tests. Commonly, test protocols include procedures for preconditioning (temperature, moisture, dust content of ﬁlters) of equipment and sometimes also materials.

7. CONCLUSIONS Attrition in ﬂuidised beds occurs in the bulk of the bed by shearing and in the

distributor region by a combination of impact and rapid shearing. Test methods are available to evaluate attrition of particle in ﬂuidised beds.

Great caution should be used in careful selection of appropriate test methods representing the actual process conditions. Some modelling capability is available but only limited to mechanical attrition. Once the system is reactive, the whole approach for characterisation becomes empirical, and needs to be carried out on carefully scaled-down processes. The role of material properties is not well understood or taken into account. Single particle impact testing provides an indirect method of evaluating the attrition propensity, and once coupled with the ﬂuidised bed hydrodynamics, goes some way in providing a predictive tool. More fundamental understanding of attrition in ﬂuidised beds requires a more rigorous modelling, coupled with reliable kinetic data on the rate of particle attrition. The application of distinct element method may provide this capability in the long term. In the short term the use of population balance models, coupled with proven reliable breakage kernels may provide helpful insights for controlling the attrition process in ﬂuidised beds.

ACKNOWLEDGEMENTS The authors wish to acknowledge support from their respective industrial partners, as there are ENEL, Alstom, EDF, Pﬁzer, Shell, Unilever, the EPSRC, as well as their colleagues for enthusiastic collaboration: Riccardo Chirone, Fabrizio Scala, Fabio Montagnaro, Antonio Cammarota, Massimo Urciuolo, Jamie Cleaver, Craig Bentham, Dimitris Papadopoulos, Chi-Chi Kwan, the workshop at the University of

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Surrey. Finally, permission of PSRI Chicago for publishing photographs and comments, as well as their kind hospitality to RB are gratefully acknowledged.

Nomenclature

AJI c C D4, 3 DI d dor dp d 0p f G h H Hf HGI k Kc l m m0 m00 m1 ms M Mb n Q Ra Rbed Ri s t ui umf uo up us W

Air Jet Index () weight fraction ﬁnes in Jet-Cup test () Constant (see Tables 3 and 4, equation (7)) Mass weighted mean diameter (m) Davidson Index () linear particle dimension (m) distributor oriﬁce size (m) mean particle size (m) initial mean particle size (m) distributor free area () mass of ﬁnes in starting material of Jet-Cup test () power index () hardness (Pa) weight fraction of ﬁnes after Jet-Cup test () Hardgrove Grindability Index (kg) power index () fracture toughness (N m3/2) power index () power index () mass of ﬁnes at start of ASTM D5757-95 test (kg) mass of ﬁnes after 5 h of ASTM D5757-95 test (kg) mass of ﬁnes after 1 h of ASTM D5757-95 test (kg) total mass at start of ASTM D5757-95/Jet-Cup test (kg) mass (kg) total bed weight (kg) power index () volumetric oriﬁce gas ﬂow rate (m3 s1) ﬂuidised bed jet attrition () ﬂuidised bed bulk attrition () single particle impact attrition () power index () power index () particle impact velocity (m s1) minimum ﬂuidisation velocity (m s1] oriﬁce gas velocity (m s1) particle velocity in the jet (m s1) superﬁcial gas velocity (m s1) residue bed weight (kg)

Attrition in Fluidised Beds

Wc Ws Wg WD z b b0 b1 rf rp sbr t js jso

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carbon loading (kg) solids entrainment rate (kg s1) gas mass ﬂow rate (kg s1) Weeks and Dumbill [85] constant () number of grid jet holes () correction factor () growth rate constant (m5/2 s1) breakage rate constant (m5/2 s1) ﬂuid density (kg m3) solids density (kg m3) crushing strength (Pa) processing time, breakage time constant (s) particle sphericity () initial particle sphericity ()

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CHAPTER 26

A Mechanistic Description of Granule Deformation and Breakage Yuen Sin Cheong,a,c Chirangano Mangwandi,a Jinsheng Fu,a Michael J. Adams,b Michael J. Hounslowa and Agba D. Salmana, a

Department of Chemical and Process Engineering,University of She⁄eld, Newcastle Street, She⁄eld, S13JD, UK b Centre for Formulation Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK c Presently at Department of Materials Science and Metallurgy, University of Cambridge, Pembroke Street,Cambridge,CB2 3QZ, UK

Contents 1. Introduction 2. Origin of granule strength 2.1. Autoadhesion 2.2. Wettability and surface energy of solids 2.3. Adhesion models 2.3.1. Non-deformable solids 2.3.2. Deformable solids 2.4. Friction models 2.4.1. Static friction (adhesive peeling) 2.4.2. Sliding friction 2.5. Liquid bridges 2.5.1. Static capillary force 2.5.2. Viscous junctions 2.6. Solid bridges 3. Macroscopic granule strength 3.1. Micromechanical descriptions of granule strength 3.1.1. Ensemble elastic modulus 3.1.2. Rumpf’s theory of granule strength 3.1.3. Kendall’s theory of granule strength 3.2. Measurement of granule strength 3.2.1. Diametric compression experiments 3.2.2. Impact experiments 3.2.3. Multi-granule testing 3.2.4. Quantiﬁcation of breakage propensity

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Corresponding author. Tel: +44-114-222-7560; Fax: +44-114-222-7501; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12029-7

r 2007 Elsevier B.V. All rights reserved.

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3.3. Discrete element method (DEM) simulations 1099 3.3.1. Diametric compression simulations 1099 3.3.2. Impact simulations 1100 4. Practical considerations 1102 4.1. Understanding wet granulation mechanisms 1102 4.1.1. Effect of impact velocity 1105 4.1.2. Effect of primary particle size 1106 4.1.3. Effect of binder content, binder viscosity and binder surface tension 1107 4.1.4. Remarks 1109 4.2. Targeted performance 1109 5. Concluding remarks 1113 References 1116

1. INTRODUCTION Particulate materials exist ubiquitously in both nature and industries as assemblies of discrete solid grains making intimate contact with each other where the interstitial matrix may be occupied by air or a liquid. At the gigantic scale, 10% of the earth’s surface is covered by natural particle assemblies such as sand dunes in deserts, soil sediments and pebble beaches [1]. In contrast, particle assemblies at small length scales include daily consumer products such as individual coffee or detergent granules and pharmaceutical tablets. Such particulate systems not only differ in their macroscopic size but their constituent particles also span several orders of magnitude in size. These seemingly simple systems have tremendous impact on human affairs and the global economy. In industry, the amount of particulate solids or powders being processed in the world is enormous and only second to water [2]. Approximately 10 billion metric tonnes of particulate products are manufactured worldwide every year utilising nearly 10% of the total energy produced [1]. Most of these products range from low-cost raw agricultural grains and construction aggregates to high value-added particles manufactured through sophisticated processes in the pharmaceutical and chemical industries. As pointed out by Iveson et al. [3], particulate solids account for approximately 80% of the ﬁnal and intermediate products in the chemical industries in the United States. The consequent annual revenues generated by these industries amount to US$1 trillion. In addition, the ceramics, electronics and biological sectors are driving rapidly the development of advanced particulate materials at the nanometre scale. The behaviour of particle ensembles is rich and differs considerably from that of ordinary solids, liquids and gases [4]. One may recall that the measurement of time once relied upon the ability of sand to ﬂow freely through an hourglass in a liquid-like manner. In spite of this similarity to the liquid state, sand piles up as a heap at an angle of repose due to friction, instead of spreading across a surface to form a thin layer as seen with liquids [2]. For cohesive powders such as

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cornﬂour, ﬂowability is hindered by the adhesive and frictional forces operating between the particles, unlike the free-ﬂowing sand in an hourglass. In a pack of breakfast cereal, larger nuts are usually found to accumulate at the top of a bed of smaller grains. This anomaly is called the ‘‘Brazilian nut’’ effect, which is counter-intuitive to the expectation that larger and heavier nuts should settle to the bottom of the cereal pack [5]. Despite engineers having long standing interest in the static deformation and ﬂow of particle assemblies, the underlying principles of these phenomena are still not well understood and a consensus has not been reached [2]. As a result, the design of operations and equipment for particle processing is often inefﬁcient and failure is sometimes unexpected. For instance, chemical engineers may have to deal with the blockage of silos and hoppers caused by arch formation and poor solid ﬂowability during discharge of ﬁne powders. One of the usual practices is to hammer the silo or hopper wall to encourage ﬂow but the problem is only temporarily remedied [1]. Furthermore, capillary condensation of moisture induces the formation of liquid bridges between hydrophilic particles in a humid and warm environment. Soluble materials dissolved in the contacting liquid bridges may subsequently recrystallise to form solid bonds that lead to caking of materials. Strong massive caked solids may eventually cause silos to collapse and an additional milling operation is required to crush the cake to recover the products [6]. Moreover, size segregation is inevitable when particles of different sizes are mixed mechanically, for example, in pharmaceutical industries. Consequently, preferential accumulation of materials according to particle size will occur in a similar way to the ‘‘Brazilian nut’’ phenomenon. Adversely, this results in the inhomogenous distribution of ingredients in the ﬁnal products leading to a large number of rejects. Another aspect is that ﬁne particles generated from comminution and conveying disperse in air as dust. If the particles are triboelectrically charged they may detonate causing devastation of workplace and loss of human lives. In addition, dust inhalation has an adverse health effect since some materials, which are harmless in the non-particulate solid state, become toxic after being ﬁnely comminuted [7]. In tackling the above problems, it has been proven useful to aggregate ﬁne particles to form larger entities known as granules [3]. This can be achieved through a wet granulation process where ﬁne particles are agitated with a liquid binder in a mechanical mixer. The binder may solidify upon a separate drying process to form solid bridges holding the particles together. When the addition of binder is undesirable, granulation can be facilitated using methods such as roller compaction and pressure swing ﬂuidised bed granulation [8]. These size enlargement techniques are widely used in various sectors ranging from the pharmaceutical and detergent industries to ceramic and mineral processing. Generally, pharmaceutical and detergent products are sold to the end users as granules while ceramic and mineral powders are granulated as intermediate ‘‘green bodies’’ for further processing. When ﬁne powders are agglomerated to

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form larger masses, the ratio of the interparticle attractive potentials to the gravitational force is reduced signiﬁcantly. Hence, the cohesive forces, which are dominant for ﬁne particles inhibiting particle movement, can be overcome by the granule weight resulting in improved ﬂowability. This leads to more accurate metering and easier material handling as the blockage of storage silos and conveying pipes is reduced. Ingredients that tend to segregate can be bound together to obtain product with uniform compositions as desired in pharmaceutical industries. Furthermore, the loss of precious materials and the release of hazardous substances through dust emission can be minimised with a reduced risk of dust explosion. Another advantage of size enlargement is that granules may dissolve more effectively as they tend to disperse readily compared to their constituent particles that are likely to ‘‘clump’’ when in contact with a ﬂuid. In a powder processing plant, granules are subjected to various external mechanical stresses during the different manufacturing routes such as the pharmaceutical production scheme depicted schematically in Fig. 1. Granule strength is the key issue in determining the efﬁciency of the processes highlighted in grey. For example, collisions between granules or the impact of granules with the mixer wall and impeller are observed in high shear mixers causing attrition and breakage of the granules [3,10]. There is evidence suggesting that granule breakage is useful in promoting a more uniform distribution of the binder within a given batch of granules [11]. After high shear granulation, the granules may be discharged onto a vibratory conveyor belt and transported to a storage silo. During storage, granules in the lower part of the silo experience substantial static compressive forces that arise from the weight of the granules accumulating above. Hence, once granules are formed, they should be sufﬁciently strong to resist these handling and storage damages to avoid product degradation and debris generation.

Crystalliser

Liquid formulation & filling lines

Filter

Blending

Dissolution

Dryer

High shear granulation

Tabletting

Mill

Granule conveying

Storage

Fig. 1. Flowchart showing typical granule processing routes (highlighted in grey) in pharmaceutical industries. (After Barrett et al. [9], with permission.)

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Some drugs in the pharmaceutical industries are designed to be administered directly to the human body through inhalation [12–14]. This is useful in delivering drugs that may cause adverse side effects with the appropriate dosage to the target site only and is commonly used in pulmonary disease treatment. Hence, it is desirable to manufacture these drugs in a granular state, which can be disintegrated and dispersed readily to form an aerosol while having sufﬁcient strength to survive the mechanical stresses during handling and transportation. Moreover, for example, detergent granules manufactured in the form of end products should be sufﬁciently weak to disintegrate and disperse readily for rapid dissolution [15]. It is plausible to improve the dissolution properties by making granules with a high air volume fraction or porosity but to the detriment of the strength. In contrast, it is essential to achieve complete crushing of granules in a compaction process to ensure that tablets with a homogenous density distribution and appropriate strength are produced. The presence of strong granules as inclusions may act as defects in the tablets. Consequently, cracks are likely to be initiated from these ﬂaw sites causing the tablets to fracture. Similarly, a substantial reduction in the strength of ceramic structures is expected if granules, which act as the intermediate green bodies, are not eliminated completely during the forming process [16]. Thus, granule strength should be tailored carefully to seek the balance between the resistance to handling damages and end-use requirements. Typically, the minimum strength of a granule is expected to range from 1 to 10 MPa based on the tensile strength measured for dry compacted pharmaceutical granules handled in actual processing plants [17]. Granule strength is also important in granulation processes. The binder is typically a solution or a melt and causes the feed particles to granulate due to the formation of capillary and viscous liquid bridges [3]. If the granules are weak they will be broken down by the mechanical action of the granulator but, if they are too strong, very large granules will be formed. Thus, it is important to control the strength of the forming granules to achieve a product that has the required granule size distribution. This chapter attempts to reiterate that granule strength stems from the interparticle adhesion and friction forces, which is central to the control of granule strength; the term adhesion is used broadly here to include any type of interaction that binds the primary particles together including solid bonds caused by the presence of a binder. Therefore, the engineering of granule strength can be achieved qualitatively by manipulating the interfacial properties of particle surfaces and the interfacial and bulk properties of the binder, if present. This approach is of great importance in pharmaceutical industries that has stringent time limits to release a product to the market. Furthermore, strict legislation and ﬁnancial constraints make trial and error experiments, which require large amounts of pharmaceutical materials, impractical. Further demand for this correlation is clear after the launch of the good manufacturing practice (GMP) and process

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analytical technology (PAT) initiatives. These initiatives require the food and drug industries to implement control strategies on their manufacturing processes, aiming for effective and efﬁcient production with reduced wastage of materials [18]. The successful prediction of granule strength from the microscopic particle interactions is believed to allow these industries to incorporate appropriate process control to fulﬁll the objectives of the GMP and PAT initiatives.

2. ORIGIN OF GRANULE STRENGTH As mentioned in the Introduction, various size enlargement processes, either wet or dry, with different degrees of agitation are available to produce granules with complex packing structures (distribution of contacts) and bonding mechanisms. Despite this, granules can be treated as an assembly of discrete particles where external perturbations are transmitted via the interparticle contacts in the normal and tangential directions. This section contains a brief account of existing theories and techniques employed to describe the interparticle interactions including autoadhesion, friction, liquid bridges and solid bonding. These microscopic interactions control the macroscopic elasticity and strength of granules based on the packing of the constituent particles, as will be described in Section 3.1.

2.1. Autoadhesion Despite the existence of molecular adhesion, its effect is not apparent if the separation between the surfaces of two distinct solids is large compared to the equilibrium separation between atoms, z0 (typically a few angstroms). The parameter z0 is the distance at which the attractive and repulsive forces between neighbouring atoms equilibrate. It has been observed that two approaching solid surfaces jump into contact when the separation is reduced to a dimension close to z0 [19]. This is due to the fact that the surfaces are now within the range of action for molecular attractive forces to operate thus, pulling the surfaces together. This phenomenon is more profound if the surfaces are clean and smooth. Furthermore, the attractive forces become comparable to the weight of the contacting solids, as the body size is made smaller. The adhesion force scales with the ﬁrst power of the solid size as discussed in Section 2.3. Comparing the relative magnitudes of adhesion and gravity forces acting on a solid while reducing the body size, the gravity force (weight) diminishes more rapidly than the adhesion force, since it depends on the third power of size [20]. Following the above arguments, it is not surprising that molecular adhesion is the main mechanism responsible for the structural integrity of a binderless granule composed of ﬁne constituent particles. During a granulation process, the

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constituent particles are compacted by mechanical agitation to form granules with a close-packed structure. As a result, the separations between the particles are reduced to within the range of action for molecular adhesion to operate. The adhesion is further enhanced if the particles are of sub-micrometer size. However, it should be noted that the surfaces of real particles are seldom smooth. Hence, the real contact between the constituent particles within a single granule is commonly due to the pairwise contacts between asperities of the particles. Molecular adhesion can be interpreted in terms of energy since work is required to pull apart the contacting solid surfaces. As illustrated in Fig. 2, an interface between the solid surfaces is formed when contact is established. The energy corresponding to the formation of a unit area of this contact is called the interface energy, denoted g12. An attempt to separate the contacting surfaces involves not only the elimination of the solid–solid interface but also the creation of two new solid surfaces. The amount of reversible work required to separate a unit area of the surfaces from contact to inﬁnity is deﬁned as the work of adhesion, WA, which can be written as the Dupre´ equation shown below [21]. W A ¼ g1 þ g2 g12

ð1Þ

where g1 and g2 are the surface energies of two dissimilar solids 1 and 2, respectively. The surface energy of a solid is the energy expended to form a unit area of the solid. For normal ambient conditions, adsorption of foreign vapour onto the solid surface is inevitable. Hence, the solid surface energy measured in air is much smaller than that obtained in vacuum. For two identical solids in contact, each with a surface energy of gS in vacuum, equation (1) reduces to W A ¼ 2gS

ð2Þ

since g1 ¼ g2 ¼ gS and g12E0 [22]. The determination of the work of adhesion according to equations (1) and (2) is difﬁcult by direct measurement except for a few special cases when atomically smooth interfaces may be formed, e.g. mica and some elastomers. However, a reasonable estimate of these energy

Solid 1, 1 Solid 1, 1 CONTACT

−12

SEPARATION Interface, 12

WA

Solid 2, 2 Solid 2, 2

Fig. 2. Schematic diagram illustrating the total energy change (work of adhesion, WA) associated with the separation of a contact interface.

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parameters can be made based on the wetting behaviour of the solid [23], as described later.

2.2. Wettability and surface energy of solids The wettability of a solid refers to the tendency of a liquid drop to spread on its surface. For a given solid surface, only limited spreading will occur for a drop of a partially wetting liquid so that a ﬁnite contact angle, yE, is formed between the liquid–vapour interface and the solid surface, as illustrated in Fig. 3. At equilibrium, the energy of this solid–liquid–vapour system can be described using the well-established Young’s equation (see [24]) gSV ¼ gLV cos yE þ gSL

ð3Þ

where gSL is the solid–liquid interface energy while gSV and gLV are the solid surface energy and liquid surface tension, respectively. There is a reduction in the solid surface energy caused by the adsorption of vapour on a solid surface. When this is taken into account by including the spreading pressure, pe ¼ gSgSV equation (3) becomes [25] gS ¼ gLV cos yE þ gSL þ pe

ð4Þ

gS being the surface energy of solid in vacuum. It appears from equation (4) that the surface energy of a solid, which determines the work of adhesion between two contacting solids, can be measured from contact angle experiments. Unfortunately, these experiments suffer from a drawback that gSL and pe are not known a priori and independent measurements of these parameters are not reliable. Instead of measuring all the variables in the Young’s equation, Fox and Zisman [26] suggested that gS for a given solid can be estimated from the contact angles of a homologous series of liquids on the same solid surface. They observed that a straight line can be ﬁtted through a Fox–Zisman plot of cosyE as a function of gLV for the series of liquids. A measure of the solid surface energy is given by the critical surface tension gC obtained by extrapolating the straight line to cosyE ¼ 1. This parameter signiﬁes that liquids with a surface tension that is less than gC spread instantaneously over the solid surface. By inspection of equation (4), the Liquid drop

E Solid surface

Fig. 3. A typical drop proﬁle of a non-wetting liquid resting on a solid surface.

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implications of this empirical approach are that pe is negligible and gSL vanishes as cosyE approaches unity. Despite the lack of evidence for these assumptions, this method has been proven to provide satisfactory estimates of the surface energy of low-energy solids such as hydrocarbon surfaces with certain pure liquids interacting via dispersion forces only [27]. For other systems where simultaneous dispersion and non-dispersion forces (polar interactions such as hydrogen bonding) exist, the total surface energy of a single phase gT can be decomposed into energies due to dispersive gD and nondispersive gP interactions [28]. gT ¼ gD þ g P

ð5Þ

According to Wu [29], the total solid–liquid interface energy for a system comprising both dispersive and non-dispersive interactions can be described by gTSL ¼ gTS þ gTLV

D 4gD 4gPS gPLV S gLV D gD gPS þ gPLV S þ gLV

ð6Þ

Thus, substitution of equation (6) into equation (4) and assuming pe 0 gives gTLV ð1 þ cos yE Þ ¼

D 4gD 4gPS gPLV S gLV þ D gD gPS þ gPLV S þ gLV

ð7Þ

Since the dispersive and non-dispersive components for the surface tensions are known, it can be seen from equation (7) that the total surface energy of a solid can be estimated by measuring the contact angle of two non-wetting liquids on the solid surface. Such an investigation was undertaken by El-Shimi and Goddard [27] where methylene iodide with mainly dispersion forces and water having both dispersion and non-dispersion forces were used as the probe liquids on lowenergy hydrocarbon surfaces and human skin. Close agreement was found between the solid surface energies calculated from equation (7) and the critical surface tensions extracted using the Fox–Zisman method, which was shown to be applicable for low-energy surfaces.

2.3. Adhesion models The work of adhesion can be inferred by considering the total force required to separate two contacting surfaces, which is known as the pull-off force. As reviewed by Tabor [22], the theoretical calculations of the pull-o¡ force can be performed based on the principles set forth in the proceeding sections. It is essential to note that the contacting solids are treated as continua with molecularly smooth surfaces in these theoretical considerations. In relation to particulate systems, discrete particles are approximated as solid spheres in these theories. However, the results are still applicable for rough grains since the

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contact geometry between a pair of asperities approximately coincides with that of two spherical surfaces, i.e. curved surfaces each deﬁned by a radius of curvature [30]. The adhesion effect arising from surface attraction is commonly known as autoadhesion.

2.3.1. Non-deformable solids The mutual attractive forces between individual atoms at sufﬁcient distances apart are mainly electrostatic in nature. These attractive forces are commonly referred to as dispersion forces and were shown to be proportional to the inverse seventh power of the distance between the two atoms by London (see [21]). Using pairwise addition of the dispersion forces between all the atoms in two non-deformable (hard and rigid) solid spheres, Bradley [31] derived the following expression for the pull-off force, PC, exerted between macroscopic spheres: P C ¼ 2pR W A

ð8Þ

with the effective radius, R ¼ (R1R2)/(R1+R2) where R1 and R2 are the radii of the contacting spheres; note that R is equal to the radius of a sphere in contact with a ﬂat surface. Experimentally, for example, the value of PC required to separate silica spheres of different sizes has been found to vary with the effective radius as a straight line passing through the origin [31]. The ﬁtting of equation (8) to the experimental data yielded a work of adhesion of 33.8 mJ m2. This value is smaller than that of 50 mJ m2 measured by Kendall et al. [32] but comparable to the value of 36 mJ m2 obtained by Heim et al. [33]. The discrepancy in different experiments may be attributed to the difference in the surface conditions of the silica spheres, such as the roughness and contamination. The rigid sphere assumption of Bradley [31] implies that the solids were in point contact with no distortion of the contact region. Real elastic spheres tend to ﬂatten under the action of the surface attractive forces resulting in a circular contact area. This phenomenon was observed experimentally when rubber spheres were placed in contact under small or no externally applied force [19]. Hence, the elastic deformation of solids must be accounted for to provide a more realistic model for the adhesion between solid particles.

2.3.2. Deformable solids The development of the adhesion models describing the effect of surface adhesion forces on the elastic deformation of solid particles was based on the well-established Hertz theory [34]. Therefore, it is necessary to summarise this theory for two contacting elastic spheres loaded externally in the absence of surface forces.

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2.3.2.1. Hertz contact model The contact problem between two smooth and frictionless elastic spheres was studied by Hertz [34] where the deformations of the spheres were restricted to the contact region as justiﬁed by the high local stresses involved. According to Hertz [34], the radius of the contact circle, a, formed when two such spheres are pressed together, varies with an externally applied force of, P, as follows: a3 ¼

3R P 4E

ð9Þ

where 1 n21 1 n22 1 ¼ þ E E1 E2

ð10Þ

such that E1, E2, n1 and n2 represent the Young’s moduli and Poisson’s ratios of the spheres, respectively. The distribution of the compressive pressure within the contact is such that it attains a maximum at the centre of the contact circle and decays to zero at the periphery as sketched in Fig. 4. According to this deformed proﬁle, the displacement of the centres of the spheres towards each other, d, given by d¼

a2 R

ð11Þ

It follows from equations (9) and (11) that the load–displacement relationship takes the following non-linear form: P¼

4 1=2 3=2 E R d 3

ð12Þ

The Hertz analysis is a good approximation for large bodies when the externally applied force is large compared to the surface forces. Nevertheless, it is Hertzian contact pressure distribution

H (r, )

z0 a r

Fig. 4. The Hertzian contact proﬁles of two elastic spheres in contact (adapted from Tabor [22]). Note that the equilibrium separation z0 is exaggerated for clarity.

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commonly observed that the contact area is larger than the Hertz prediction as the external force approaches zero. This is a consequence of the dominant contribution from the surface forces to the elastic deformation of the spheres at low external forces. Hence, the elastic contact deformation of lightly loaded solid spheres can be related to the work of adhesion, WA required to pull the spheres apart. The determination of WA is dependent on the exact shape of the deformed surface near the contact region. There are two main results based on two different assumed deformation proﬁles.

2.3.2.2. Derjaguin, Muller and Toporov (DMT) adhesion model In the study of Derjaguin and co-workers [35], the deformed surfaces outside the contact of two adhering spheres were postulated to follow the Hertzian proﬁle as ﬁrst proposed by Derjaguin [36], even though the surface attraction cause a larger contact area than the Hertz prediction. In principle, the elastic reaction force of the deformed spheres is balanced by the surface attractive forces and any externally applied force. The total surface force not only consists of the contact forces within the enlarged contact area but also the noncontact forces exerted in the annulus region surrounding the contact. Thus, the total surface adhesion energy WS0 can be written as the sum of the molecular energies Wcontact and Wnon-contact due to the contact and non-contact forces deﬁned above. W 0S ¼ W contact þ W non-contact

ð13Þ

To calculate the total surface force pulling the spheres together, one can consider the rate of change of W 0S by varying the displacement between the centres of the spheres, d. Separation of the spheres occurs as d-0 and hence the pulloff force PC corresponds to the PS at point contact dW 0S PC ¼ ¼ 2pR W A ð14Þ dd d!0 which resembles the result of Bradley [31] in equation (8), even after taking elastic deformation into account. Perhaps, this result is not surprising as Derjaguin et al. [35] emphasised that the DMT analysis was intended for small and hard particles with large elastic moduli.

2.3.2.3. Johnson, Kendall and Roberts (JKR) adhesion model In contrast to the DMT analysis, Johnson et al. [19] suggested that the stress distribution within the contact area must be altered from the Hertzian distribution by the surface attraction and that all the attractive forces act within the contact area. The contact stress distribution can be found by the superposition of the compressive Hertzian distribution for an enlarged contact area and a

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JKR contact pressure distribution

z0 a

Fig. 5. Figure showing the JKR contact proﬁles for two elastic spheres in contact in the presence of surface attraction (after Johnson et al. [19]).

tensile stress distribution similar to that generated by a rigid punch pressing onto an elastic plane [37]. This hypothesis gives rise to an inﬁnite tensile stress at the boundary of the contact circle causing the contact surfaces to meet perpendicularly at just outside the contact area forming a neck (see Fig. 5). However, slight peeling of the sphere surfaces near the edge of the contact area must occur in reality such that the surface attraction is maintained in equilibrium by a ﬁnite tensile stress as argued by Johnson [37]. Consider two solid spheres brought into contact from inﬁnity by a small external force, a surface adhesion energy, WS, is released due to the formation of the enlarged circular contact interface. To ensure equilibrium, the total energy of the system, WT, must achieve a minimum where the elastic energy stored in the deformed spheres, WE, balances the expended potential energy of the external force, WM, and that dissipated in surface adhesion energy, WS. Therefore, WT ¼ WE WM WS

ð15Þ

On this basis, the expression relating the enlarged contact radius, a, to the external force, P, is given by [19] qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3R a3 ¼ P þ 3pR W þ 6pR W A P þ ð3pR W A Þ2 ð16Þ A 4E It can be seen that this expression reduces to the Hertz equation (equation (9)) if surface attractions are neglected for which WA ¼ 0. The total normal force acting over the contact area, PJKR is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð17Þ P JKR ¼ P þ 3pR W A þ 6pR W A P þ ð3pR W A Þ2

Y.S. Cheong et al.

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If an external tensile force (i.e. negative P) is applied to separate the spheres from contact to inﬁnity, the pull-off force PC expressed below is obtained such that a real solution to equation (16) exists [19] P C ¼ 1:5pR W A

ð18Þ

t is worth noting that the JKR pull-off force is 25% smaller in magnitude than that of the DMT analysis (equation (14)) as a result of the modiﬁed contact stress distribution. Furthermore, detachment occurs at a ﬁnite contact radius, unlike the gradual diminution of the contact area to a point contact predicted by the DMT. If the peeling process of the contact interface is viewed as a Mode I fracture (crack opening), there is an analogy to Grifﬁth’s criterion [38] for the fracture of brittle solids to describe the detachment of the spheres. Hence, the JKR pull-off condition corresponding to the sudden separation of the spheres in a ‘‘jump-wise’’ manner is expected once the rate of release of the stored elastic energy in the contact exceeds the rate of gain in the surface adhesion energy (see [39]). To verify the JKR analysis, smooth and soft elastomer spheres were used since it was argued that dust particles or surface asperities may be pushed into the bulk of the elastomer due to the small elastic modulus, thus permitting good intimate contact [19]. The variation of the contact radius of the elastomer spheres with the applied force was found to agree remarkably well with equation (16) when the work of adhesion was taken to be 71 mJ m2. This implied that the surface energy of each elastomer surface was approximately 35 mJ m2 according to equation (2). The work of adhesion was reduced by an order of magnitude when the elastomer contact was wetted with water. These energy values were proved to be convincing when inserted into Young’s equation resulting in a contact angle of 641 for water on the elastomer contact. This value was consistent with the measured value of 661 for a water drop resting on the same elastomer surface. Furthermore, the JKR theory was supported by experiments where soft gelatine spheres of different sizes were pressed onto Perspex ﬂat [19]. When a tensile force is applied to separate elastomer spheres, it has been observed that large uncertainty in the contact radius measurement was inevitable near the pull-off condition as the attainment of contact equilibrium was prolonged due to the viscoelastic behaviour of elastomers [19]. In spite of this, the dependency of the contact radius on the applied force agreed reasonably with the JKR theory within the limits of experimental uncertainties. In addition, the contact interface was observed to detach in an unstable way. Apart from the original experimental veriﬁcation, the JKR theory was realized experimentally for adhesive contact between a wide range of materials in different contact geometries. For example, these include the crossed cylinder conﬁgurations of the polymer ﬁbre monoﬁlaments studied by Briscoe and Kremnitzer [40]

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and the mica on mica contact investigated by Homola et al. [41] using the surface force apparatus (SFA). Recently, the contact between a 140 nm cantilever tip and a mica surface was shown to exhibit JKR behaviour in the atomic force microscopy (AFM) study of Carpick et al. [42]. The theory was shown to break down when surface roughnesses comparable to atomic diameter were considered in the molecular dynamic simulation of Luan and Robbins [43].

2.3.2.4. Limits of DMT and JKR Despite different deformation proﬁles being assumed, the DMT and JKR analyses were demonstrated to yield the same dependency of the pull-off force on the particle radius [22]. Thus, it is difﬁcult to ascertain the selection of the appropriate model to describe the adhesion of elastic particles, apart from using nature of the detachment of the particles as a criterion. A formal approach follows the suggestion of Tabor [22] to identify the limits of application of the DMT and JKR theories using the following dimensionless parameter !1=3 R W 2A z¼ ð19Þ E 2 z30 Essentially, this parameter may be interpreted as the ratio of the neck formed around the contact circle to the range of action of the surface forces characterised by the equilibrium separation, z0. As Tabor [22] pointed out, the neck for two large adhering particles with small elastic moduli like elastomers is large compared to z0. In this regard, the surface forces outside the contact area diminishes rapid with separation between surfaces and can be neglected in accord with the JKR theory. In the other extreme, the DMT is applicable to describe the adhesion between small and hard particles where the neck size is small to permit the action of surface forces outside the contact area. A form of dimensionless group similar to z but with a prefactor of 2.92 was derived by Muller et al. [44]. It was shown that the pull-off force resembles the DMT prediction for zo1, whereas a transition to the JKR pull-off occurs at z45. This transition is supported by the numerical computation performed by Greenwood [45].

2.4. Friction models The adhesion of solid particles summarised in the preceding section represents only the normal interaction between solid surfaces. When a contact interface is subjected to tangential loading, relative motion is resisted by a friction force operating in the direction opposing the tangential force. This action of the friction force is dissipative in nature and may be regarded as the work per

Y.S. Cheong et al.

1070

unit sliding distance. The description of the friction force, F, acting between two non-adhering macroscopic bodies is based on the Amonton’s law m¼

F P

ð20Þ

where P is the external force applied normally to the system and m the coefﬁcient of friction. Sliding between two surfaces is not possible unless a limiting value of the frictional force is reached, i.e. FomsP, where ms is the coefﬁcient of static friction. Once sliding occurs, the relationship in equation (20) still applies to describe sliding friction although the value of the dynamic value of m may be different from the static value. Numerous observations have shown that the frictional force is independent of the geometric area of contact and the velocity at which two contacting bodies slide over each other. Nevertheless, it is clear that equation (20) is not applicable for all cases in practice since friction still exists when the external load is reduced to zero or even made negative, especially in the context of cohesive particles (see, for example, Gao et al. [46]). In describing the frictional contacts of cohesive powders, the commonly used approach is Coulomb’s law in the following form: F P ¼ tc þ m A A

ð21Þ

where A is the cross-sectional area of the powder assembly under consideration and tc is the cohesive shear strength of the powder while other symbols are as deﬁned in equation (20). It can be seen that friction is now ﬁnite at zero applied force due to the contribution from the attractive forces operating between ﬁne particles characterised by the parameter tc. However, it was reported by Kendall [47] that the frictional force for ﬁne powders is incompatible with Coulomb’s law where a signiﬁcant reduction of friction force was noticed at high applied forces. Furthermore, smaller particles are usually associated with friction coefﬁcients larger than that measured for macroscopic blocks of the same material [47]. According to the argument of Kendall [47], the inadequacy of the Coulomb’s law is attributed to the assumption that the externally applied force and the surface attraction between ﬁne particles are independent of each other. Thus, it is essential to consider the interdependence between friction and adhesion. A special feature of particles is that they may form point contacts even when their surfaces are rough. This arises because contact may occur at single asperities due to the acute radii of particles. The friction of most particles corresponds to the adhesion mechanism in which temporary junctions, formed by short-range attractive forces such as van der Waals interactions, are intermittently ruptured [48]. Such an interfacial process is common because most particles are relatively stiff and elastic so that bulk dissipation processes are negligible. The interfacial frictional force is related to the real area of contact, AR,

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by the following relationship: F ¼ ti A R

ð22Þ

where ti is the interfacial shear stress required to rupture the adhesive junctions. It should be noted that this does not imply that there is an adhesive force between the particles at a zero applied load since the real area of contact is then also zero according to this relationship. Since the real area of contact increases as the 2/3 power of the normal load according to the Hertz equation (9), the frictional force will also vary in this way provided that the surfaces of the particles are clean. In practice, most particles are contaminated with organic materials that exhibit a pressure dependence of the interfacial shear strength so that the load index is in the range of 2/3 to unity [49].

2.4.1. Static friction (adhesive peeling) The effect of a tangential force, T (i.e. equal and opposite to friction force) on the behaviour of an adhesive contact between two elastic spheres pressed together by an external force was investigated by Savkoor and Briggs [39] by extending the JKR adhesion model. According to their theoretical formulation, the contact radius a between the spheres can be related to the external normal force P and the friction force F as follows: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ3 2 3R 4 T 2E5 a3 ¼ ð23Þ 6pW A P þ 9W 2A p2 R2 P þ 3pR W A þ 4E 4G where 1 2 n1 2 n2 þ ¼ G G1 G2

ð24Þ

Comparing equations (16) and (23) with the normal force kept constant, it can be seen that the contact radius diminishes with increasing tangential force leading to a ‘‘peeling’’ mechanism. This ‘‘peeling’’ process proceeds in a stable manner and will be complete once T reaches a critical value, TC, thus rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ G T C ¼ 2 ð6pW A P þ 9W 2A p2 R2 Þ ð25Þ E and the contact radius diminishes to a3C ¼

3 R ðP þ 3pR W A Þ 4 E

ð26Þ

The contact radius measured in the experiments of Savkoor and Briggs [39], where a rubber hemisphere was pressed onto a glass slide at different normal loads, exhibited clear reduction under a monotonically increasing tangential

1072

Y.S. Cheong et al.

force. In addition, it was concluded that the data were in reasonable agreement with equation (23) within the limits of experimental error. The different processes that may occur at the end of contact peeling will be considered in the following subsection.

2.4.2. Sliding friction In describing the sliding friction between small cohesive particles, it was pointed out by Kendall [47] that the external force and the surface attraction is interrelated such that the effective load acting over the contact interface PJKR is given by equation (17). Consequently, a modiﬁed form of Coulomb’s law is obtained based on Amonton’s law as follows: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ T ¼ m P þ 3pR W A þ 6pR W A P þ ð3pR W A Þ2 ð27Þ Qualitatively, this result supports the dependence of friction on particle size observed for ﬁne powders and the anomaly where friction decreases at high applied force as indicated by the variation in the gradient of equation (27). In the sliding experiments carried out by Briscoe and Kremnitzer [40] using adhesive polyethylene–terephthalate monoﬁlaments, the measured tangential force as a function of normal applied force was found to follow a non-linear trend prescribed by equation (27). Despite this, ﬁtting of the theory to the experimental data required a work of adhesion, which was inconsistent with the measured surface energy of the polymer [47]. Presumably, this can be explained by the failure of the model to account for contact peeling prior to the onset of sliding. Alternatively, the data may be explained by a power law relationship as mentioned previously such that the normal load is the sum of the applied and adhesive values according to DMT theory [50]. Experimental evidence for a reduction in the contact size emerged from the recent AFM study of Carpick et al. [42] in accordance with the analysis of Johnson [30]. A more rigorous analysis was performed by Thornton [51] to examine the conditions at the end of the contact peeling (TZTC), in which the contact interface between the adhering elastic spheres was ﬁrst assumed to peel followed by a smooth transition to complete sliding. This assumption coincides with the conclusion of Johnson [30] by treating the contact interface, in the presence of normal and tangential forces, as a crack using the principles of fracture mechanics. Recalling the resultant normal stress distribution for the JKR adhesion model, there is a central portion of the contact area of radius a1 which is subjected to compressive stresses. This compressive region is surrounded by an annulus of tensile stresses rising to an inﬁnite value at the contact periphery (see [52]). It was argued that peeling of the contact area was only necessary until the contact

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radius is reduced to [51] a31 ¼

3R P1 4E

ð28Þ

where P 1 ¼ P JKR

P JKR P 1 3P JKR

3=2 ð29Þ

with PJKR given by equation (17) and hence, T ¼ mP 1

ð30Þ

Expression (30) was found to follow the whole range of the data obtained in the sliding experiments of Briscoe and Kremnitzer [40] satisfactorily including the results at negative normal loads. It should be noted that the work of adhesion measured experimentally was used in the analyses of Thornton [51] in contrast to the inconsistent values assumed in the work of Kendall [47]. Hence, it is reasonable to accept equation (30) in preference to equation (27) based on the comparison between the two models with the experiments of Briscoe and Kremnitzer [40], although equally close agreement is obtained with the power law model discussed above.

2.5. Liquid bridges When a powder mass is stored in humid environment, a common phenomenon is capillary condensation of water vapour causing caking due to the formation of liquid bridges at the interparticle contacts. During wet granulation, however, liquid binder is deliberately added to a powder mass to induce size enlargement. The resulting wet granules can exist in several different levels of liquid saturation. The granules are in the pendular state at low liquid saturation where the primary particles are held together by discrete liquid bridges with the interstitial space between the particles ﬁlled by air. Further increases in liquid saturation caused by either continuous binder addition or granule consolidation leads to the capillary state, where the entire interstitial space is occupied by the binder (see [3]). An intermediate state between these saturation levels is known as the funicular state. In these granular states, there are additional contributions to the interparticle forces from the static capillary action and dynamic viscous effect of the liquid binder. This section will focus on the theoretical description of the forces generated due to pendular liquid bridges connecting a pair of spherical bodies only. The theoretical analyses of the funicular and capillary states are considerably more complex and 2D results are given by Urso et al. [53].

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2.5.1. Static capillary force For a pair of stationary spheres with a small separation distance, the formation of a pendular liquid bridge connecting the spheres requires that the Gibbs free energy of the system attains a minimum value. Under such circumstances, the surface proﬁle of the bridge at a given solid–liquid contact angle has a mean curvature, xm, which can be deﬁned by two principle radii of curvature in orthogonal directions r1 and r2, respectively. Conventionally, a positive radius of curvature is drawn with its centre lying in the liquid mass. The mean curvature of the bridge surface must remain constant so that the surface tension of the liquid–vapour interface is balanced by the hydrostatic pressure difference between the interior and exterior of the liquid mass DP, as observed by Plateau [54]. This relationship is described by the Laplace–Young equation as follows: 1 1 DP ¼ gLV xm ¼ gLV þ ð31Þ r1 r2 At equilibrium, the forces at any point along the axial direction of a liquid bridge are equal otherwise there would be ﬂow of the liquid. For a gravity-free analysis, it follows that the liquid bridge force between the spheres can be computed at any axial position of the bridge by summing the contributions from the axial component of surface tension and the axial force due to the hydrostatic pressure difference, DP. For small liquid bridges, the distortion of the bridge proﬁle due to gravity can be neglected [55]. When considering a large pendular bridge with its axial direction oriented along the gravity ﬁeld, the buoyancy acting on the portions of the spheres submerged in the liquid should be accounted for in estimating the bridge force. In the context of wet granular materials, the constituent particles may be treated as rigid bodies so that the particle geometry is undistorted under the action of the liquid bridge forces. The hydrostatic pressure ﬁeld across a cross-section of a liquid bridge can be evaluated analytically by solving equation (31) numerically as demonstrated, for

rA

rC R

rN

ϕ β

A

2S

Fig. 6. The geometry of the toroidal approximation with particles of radius R, separated by a distance 2S, connected by a liquid bridge of half-ﬁlling angle b and contact angle j [55].

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example, by Erle et al. [56] and Orr et al. [57]. Nevertheless, such procedures are computationally inefﬁcient and impractical for applications such as granular dynamics computer simulations. To simplify the problem, one can adopt a toroidal approximation ﬁrst proposed by Haines [58], in which the meridional bridge proﬁle is taken as a circular arc as shown in Fig. 6. Following geometric arguments, it can be shown that [59] ðr N r A Þ ¼

A þ R sin b tanðp=2 b jÞ

ð32Þ

n o1=2 r A ¼ ð1Þn A2 þ ½R sin b ðr N r A Þ2

ð33Þ

r C ¼ R sin b

ð34Þ

n¼

0; ðb þ jÞ4p=2

1; ðb þ jÞop=2

ð35Þ

where A ¼ S+R(1cosb) such that b is the half-ﬁlling angle. In practice, one may express the half-ﬁlling angle in terms of the volume of the liquid bridge since the amount of liquid added to a powder mass can be measured reliably (see [60,61]). It may be seen that the local curvature of the toroidal proﬁle varies from ð1=r A þ 1=r N Þ1 at the neck to ð1=r A þ 1=r C Þ1 at the solid–liquid interface reﬂecting that the mean bridge curvature is not constant and hence errors are introduced in the estimated bridge forces. The accuracy of this approximation will be discussed later. Neglecting the effects of gravity, Fisher [62] proposed the gorge method to estimate the total attractive force Pgorge acting on a pair of equal spheres at the neck of a toroidal liquid bridge (see Fig. 6) P gorge ¼ 2pr N gLV pr 2N DP

ð36Þ

On the right hand side of equation (36), the ﬁrst term is the positive (attractive) contribution from the axial component of surface tension whereas the second term is the hydrostatic component, which can be positive for DPo0 depending on the bridge geometry. However, Adams and Perchard [63] argued that it is more physically appropriate to employ the boundary method to calculate the attractive force at the solid–liquid interface where forces are transmitted to the spheres, so equation (36) was modiﬁed accordingly. Both methods predict that there is a rapid decrease in the attractive bridge force with increasing separation between the spheres as indicated by the dimensionless plots in Fig. 7. Furthermore, the bridge force attains a maximum at an intermediate half-ﬁlling angle and this angle increases with the separation. The exceptional monotonic decline of the bridge force with increasing half-ﬁlling angle at zero separation is a consequence of the large pressure deﬁcit and a similar explanation applies as b

Y.S. Cheong et al.

1076 1.0

5° 10°

β = 1°

0.8

20°

0.6 F*

40°

0.4

0.2 'Gorge' 'Boundary' 0.0 10-6

10-5

10-4

(a)

10-3

10-2

10-1

100

101

S* 1.0

S* = 0

'Gorge' 'Boundary'

S * = 0.001

0.8

S * = 0.01

F*

0.6

S * = 0.1 0.4

0.2

0.0 0 (b)

10

20

30 β (°)

40

50

60

Fig. 7. The variation in the dimensionless attractive force (F ¼ F/2pRgLV) between equal spheres as a function of (a) the dimensionless half-separation distance (S ¼ S/R) with constant half-ﬁlling angle and (b) the half-ﬁlling angle with constant dimensionless halfseparation distance, evaluated using the toroidal approximation for zero contact angle by both ‘gorge’ and ‘boundary’ methods.

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diminishes (see Fig. 7(b)). Clearly, the ‘‘boundary method’’ yields a larger bridge force than that calculated using the ‘‘gorge method’’ and the predictions deviate from each other as b increases. In the theoretical work of Lian et al. [64], the gorge method was found to provide a more accurate total bridge force than the boundary method, which was within 10% of that given by exact numerical solution of the Laplace–Young equation. They further demonstrated that the accuracy of the toroidal approximation could be improved by introducing simple scaling coefﬁcients. Recently, another closeform approximation for the total bridge force as a function of a scaled dimensionless half-separation distance was developed by Willett et al. [55]. Several criteria are available to specify the critical separation distance at which a pendular bridge ruptures. Erle et al. [56] pointed out that bridge rupture is expected at a ﬁnite rather than a zero neck radius. According to the hypothesis of De Bisschop and Rigole [65], bridge rupture occurs when the half-ﬁlling angle is reduced to a critical value. However, this hypothesis was shown to be incorrect by Mazzone et al. [66], who identiﬁed that the bridge ruptures at a critical separation distance beyond which no stable liquid bridge can be formed since no solution to the Laplace–Young equation exists. On this basis, Lian et al. [64] found that this critical separation distance is given by the following empirical expression: 2Sc ¼ ð1 þ 0:5jÞV 1=3

ð37Þ

One of the most comprehensive experimental measurements of the forceseparation curves for liquid bridges were performed by Mason and Clark [67] using 30 mm polythene hemispheres connected by liquid bridges of different volumes. The gravitational distortion of the bridge proﬁle and buoyancy effect were minimised by immersing the hemispheres in water, which has the same density as the bridge liquid (a mixture of di-n-butyl phthalate and liquid parafﬁn). The force-separation curves resulted from numerical solution of the Laplace–Young equation and the gorge toroidal approximation were shown to exhibit comparable agreement with most experimental data of Mason and Clark (see [64]). However, deviation of the calculated bridge forces from the measured forces was observed at small separation and large bridge volume. Similar discrepancy at small separation, where the measured forces were lower than expected, was observed by Erle et al. [56] and Mazzone et al. [66]. However, this phenomenon was not observed in the experimental studies of Willett et al. [55] with unequally sized sapphire spheres having a zero solid–liquid contact angle. Moreover, the rupture distances measured by Mason and Clark were observed to follow equation (37) reasonably well as shown by Lian et al. [64]. Fairbrother and Simons [68] quoted a 10% deviation of the measured rupture distances from the predictions of equation (37) for equal spheres of glass ballotini connected by liquids having various solid–liquid contact angles.

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For large liquid volumes and large spheres, the effect of gravity can no longer be neglected and this was considered in detail by Mazzone et al. [66] and Adams et al. [69]. Essentially, the liquid proﬁle bridging a pair of equal spheres with the axial direction oriented along the gravity ﬁeld (rotating Fig. 6 by 901) was shown to become asymmetric and thus the mean curvature is not constant. A strong gravitational effect is also manifested by a draining phenomenon where there is continuous reduction in the upper half-ﬁlling angle with increasing separation distance [69]. As a result, there is a difference in the forces acting on the upper and lower spheres, which is equal to the weight of the liquid bridge due to buoyancy. In addition, a bridge ruptures at a smaller separation compared to the gravity free case. When the surfaces of particles are neither smooth and nor chemically homogenous, this leads to wetting hysteresis in which the contact angle is greater than or less than the equilibrium contact angle when the liquid is advanced or retracted, respectively. A characteristic of this phenomenon is that the contact line will remain stationary (i.e. pinning) if the contact angle is intermediate between the advancing and receding contact angles. Once either of these limits is attained, slippage of the contact line will occur. The consequences are that the total liquid bridge force increases when the spheres are separated so that the liquid is retracted with the contact angle being intermediate between the advancing and receding contact angles. Experiments with liquid bridges between spheres have shown that the capillary forces decrease when the receding contact angle is reached allowing the contact line to slip [70]. On reduction of the separation distance, pinning of contact line ﬁrst occurred followed by slippage once the advancing contact angle was reached. This implies an extended bridge rupture distance and that capillary interaction can be dissipative rather than conservative as generally assumed due to the hysteresis in the force–separation curve. Wetting hysteresis is believed to be the contributory factor causing a lower than expected bridge force at near zero separation [70].

2.5.2. Viscous junctions In the above analysis of capillary liquid bridge forces, the particles are assumed to be stationary. The viscous effect of the liquid, however, may be accounted for by the lubrication solution between rigid spherical particles when there is relative normal motion between their centres and provided that the ratio of the separation distance and their radii is sufﬁciently small [71]. Consider the case depicted in Fig. 8, the upper sphere approaches the stationary lower sphere at a normal velocity, nn, causing symmetric radial ﬂow. The parameter nn can be taken as the relative normal approach velocity if both spheres are in motion and would be negative if the spheres were separating. Considering the general case of a power

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z

vn

B h(r)

Liquid bridge

h0

r

Fig. 8. Schematic diagram showing the squeeze ﬂow between two rigid spheres with the cylindrical coordinate system adopted [72].

law liquid, the shear stress is given by the following relationship: q @v r trz ¼ m0 @z

ð38Þ

where m0 is the ﬂow consistency, q the power law index and vr the radial component of the velocity ﬁeld. The continuity equation based on mass conservation in the integral form may be written as [72] Z h2 2pr v r dz þ pr 2 v n ¼ 0 ð39Þ h1

where the gap may be represented by the parabolic approximation h1 ðrÞ ¼ h01

r2 2R1

and

h2 ðrÞ ¼ h02 þ

r2 2R2

ð40Þ

so that hðrÞ ¼ h1 þ h2 ¼ h0 þ

r2 2R

ð41Þ

Note that h01 and h02 are half-separation distances at r ¼ 0 for the lower and upper spheres, respectively. The pressure gradient in the liquid dp/dr is given by the lubrication approximation as follows: dp @trz ¼ dr @z

ð42Þ

Substitution of equation (38) into (42) with the boundary conditions @vr/@z ¼ 0 at z ¼ 0 (zero shear rate at the mid-plane) and vr ¼ 0 at z ¼ h1(r) ¼ h2(r) (no slip boundary condition) yields the radial velocity distribution. # " q 1 @p 1=q h2 h1 ðqþ1Þ=q hðrÞ ðqþ1Þ=q vr ¼ z ð43Þ q þ 1 m0 @r 2 2

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Combining equations (39) and (43) and integrating between p ¼ pr to patm and r ¼ r to B results in the radial pressure distribution p(r) with respect to the atmospheric pressure patm: Z 2q þ 1 q q B rq pðrÞ ¼ 2m0 vn dr ð44Þ 2qþ1 q r ðhðrÞÞ The total viscous force in the normal direction can be obtained by the surface integral of equation (44) Z 2q þ 1 q q B r qþ2 P vn ¼ 2pm0 vn dr ð45Þ 2qþ1 q 0 ðhðrÞÞ For a Newtonian ﬂuid where q ¼ 1 and m0 ¼ Z is the ﬂuid viscosity, equation (45) reduces to [71] P vn ¼ 6pZðR Þ2

vn h0

ð46Þ

which is consistent with the analysis of Frankel and Acrivos [73]. Numerical solutions and two closed-form approximations for different values of q and B were deduced by Lian et al. [72]. For the force developed in tangential viscous interactions, Pvt, Lian et al. [74] proposed the use of the following asymptotic analytical solution derived by Goldman et al. [75] who studied the motion of a rigid sphere parallel to a rigid wall bounding a semi-inﬁnite viscous ﬂuid 8 R P vt ¼ ln þ 0:9588 6pZRv t ð47Þ 15 h0 vt being the tangential relative velocity of the spheres and R ¼ R for the case of two spheres. They argued that the viscous resistance between two spheres connected by a small liquid bridge is similar to that between a sphere and ﬂat surface at small separations where the ﬂuid resistance in the central inner region between the sphere and the ﬂat dominates.

2.6. Solid bridges It was pointed out that a network of liquid bridges can form within a powder mass or granules as a result of capillary condensation. There are cases where the constituent particles and other contaminants may react chemically with or dissolve in the liquid bridges. Following evaporation of the liquid, the dissolved materials may recrystallise or precipitate to form solid bridges between the constituent particles and thus causing caking. In granulation, the drying of binders in the form of a solution will cause the formation of solid bridges. The solidiﬁcation

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may result in bridges with complex morphologies and microstructures depending on the type of solution binder employed [76]. The development of the interparticle force between a pair of cylindrical rods of 5 mm diameter connected by a solidifying liquid bridge was monitored by Tardos and Gupta [77]. The measured attractive forces during the solidiﬁcation of pure polymer bridges were two orders of magnitude larger than the initial liquid bridge weight, despite the occurrence of several sharp drops in the measured force indicating the formation of internal voids and cracks during shrinkage of the bridges. Furthermore, it was probable that larger voids were created in larger bridges during solidiﬁcation leading to lower bridge strengths compared with those associated with smaller bridges. In contrast, the crystallisation of liquid bridges containing saturated salt solutions was shown to generate repulsive forces, which were several hundred times the initial bridge weight [77]. Microscopic inspection of the fracture surfaces of granules bound together by these solid bridges revealed that fracture through the solid bridge (cohesive failure) and detachment of the bridge from the particle surfaces (adhesive failure) were both possible. Similar fracture mechanisms emerged from the impact experiments of Subero and Ghadiri [78] using porous glass ballotini granules composed of solidiﬁed polymer binder. The theoretical computation of solid bridge binding forces is not well documented in the literature but the case of cohesive failure was considered by Bika et al. [76]. Although shrinkage of liquid bridges is expected during solidiﬁcation, it was conjectured in their analysis that the bridge proﬁle is preserved with a welldeﬁned neck similar to that shown in Fig. 6. Assuming that bridge fracture occurs across the narrowest cross-section or neck, the normal rupture force Psb is related to the bridge strength ssb [76] P sb ¼ pr 2sb ssb while the bridge neck size rsb is given by " #c r sb CS V b ¼ 2b R rp R3

ð48Þ

ð49Þ

where R is the constituent particle radius, rp the particle density, Vb the initial liquid bridge volume, CS the total concentration of solid dissolved in the liquid, b and c are numerical constants tabulated in ref. [76]. For bridge volumes larger than 0.1 ml, the experiments of Pepin et al. showed that the constants b and c could be approximated as 0.42 and 1/3. To determine the bridge strength, Bika et al. [76] expressed ssb in terms of the macroscopic crushing strength of granules sf using the Rumpf’s theory (see Section 3.1.2) as follows: " #2c 2 1 g C S V b ssb ð50Þ sf ¼ pb g rp R3

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Y.S. Cheong et al.

eg being the porosity of the granules. ssb can be evaluated empirically by fracturing macroscopic solid bridges formed between two tablets under three point bend [76]. It was found that the measured values of ssb for the macroscopic bridges agreed reasonably well with those inferred from equation (50) using two model systems of lactose and mannitol granulated with different liquids having different solubilities of the powders. In addition, they pointed out that bridge strength comparable to the tensile strength of the pure polymer binder could be achieved if the constituent particles have low solubility in the binder. However, if foreign substances such as dissolved base powder and surfactant were present, the bridge strength was dependent on the compatibility of the dissolved materials with the liquid binder.

3. MACROSCOPIC GRANULE STRENGTH The strength of a material, on a macroscopic scale, can be interpreted as the resistance to permanent deformation and fracture. To characterise the stiffness, an elastic modulus can be deﬁned as the gradient of the stress–strain response. Typically, for particles and granules, they would be approximated as spheres and force-displacement data from diametric compression would be ﬁtted to the Hertz equation (12). When fracture occurs, it is common to attribute material strength to the maximum stress corresponding to the initiation of crack growth. The value of the stress depends on the intrinsic toughness of the material and the size of the ﬂaw at which fracture starts. For large specimens, it is possible to apply fracture mechanics procedures to accurately measure the toughness (e.g. [79]). However, for single particles and granules it is usually possible to only deﬁne a nominal fracture stress both because of the geometry and because the breakdown mechanisms are likely to be complex, e.g. fragmentation involving multiplecrack pathways. The stiffness and strength of materials are strongly dependent on the evolving surface and bulk stress ﬁelds. For a homogenous elastic sphere (or particle) in contact with an external body, the classical theories of Hertz–Huber [80] and Lurje [81] can be superposed to describe the overall stress distribution within the sphere [82]. Recently, Shipway and Hutchings [83] derived numerical solutions for the elastic stress ﬁelds developed in spheres under uniaxial compression and free impact against a platen. If plastic deformation is initiated, the resulting stress ﬁeld is expected to depart dramatically from the elastic case. It was suggested that Prandtl’s solution [84] might provide a reasonable description of material deformation in a sphere experiencing plastic deformation [85]. Catastrophic failure of solid particles will take place once the maximum allowable stress in the material is exceeded. The failure modes can be classiﬁed into

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three categories, viz brittle, semi-brittle and ductile failures depending on the extent of plastic deformation experienced by the material during fracture. Brittle failure occurs without signiﬁcant plastic deformation whereas substantial plastic deformation can be found in material failing in a ductile manner. An intermediate case where brittle fracture occurs at the boundaries of a small plastically deformed region at the crack tip is termed semi-brittle failure [78]. The above theories provide accurate descriptions of the stress ﬁelds in homogenous non-porous solid particles where local stresses can be transmitted throughout the entire volume of material. Attempts have been made to extend continuum theories of elasticity and plasticity to predict the mechanical response of granular materials to external loads [86,87]. Nevertheless, the predictions have not always been successful as some results are contradicted by experimental observations [88]. Numerous studies have conﬁrmed that heterogeneous stress propagation occurs in particle assemblies such that forces are transmitted through discrete chains of particles [89–91]. The dispute on the limit of continuum mechanics to describe the behaviour of a granular system was resolved recently by the numerical studies of Goldenberg and Goldhirsch [92]. It was demonstrated that the stress response of a granular system resembles that of a continuum solid provided the region of deformation is small compared to the volume of the entire system. In addition, the continuum characteristics are enhanced by increasing levels of friction and disorder of the system. The similarity in the deformation characteristics between granules and continuum solids were noted in the review of Bika et al. [17], despite the inherent disordered structure of granules. From a microscopic perspective, the mechanical properties of an ensemble are dictated by the bonding between the constituent particles. The interparticle bonds in the region of load application may be ruptured causing the particles to shear apart before the load can be transmitted throughout the medium in contrast to a homogenous elastic systems [93]. Thus, it is clear that generally the strength of a granular medium is invariably governed by the interparticle bonding mechanisms rather than the strength of individual constituent particles. Furthermore, the load transmission in a granular body is affected by the internal particle packing. To account for these features, the micromechanical models detailed in the succeeding sub-sections were proposed to estimate the strength of granular materials.

3.1. Micromechanical descriptions of granule strength 3.1.1. Ensemble elastic modulus The elasticity of a particle assembly, which is characterised by the ensemble elastic modulus, is of practical signiﬁcance. For instance, this parameter is the decisive factor in controlling the dimensional change of compacted tablets as a

Y.S. Cheong et al.

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consequence of the residual stresses induced during compaction. Moreover, it determines the amount of elastic energy stored in moist granules during collisions encountered in a granulation process. Successful coalescence is promoted when there is insufﬁcient recovery of elastic energy to overcome the bonding developed at the interface between colliding granules [94]. In addition, Kendall and coworkers [32] discussed the possibility of inferring the surface energy of solids from the elasticity of powder compacts. Their work forms a basis for relating the ensemble elastic modulus to the packing and properties of the constituent particles, which is summarised below. The normal contact stiffness, kn, of two autoadhesive spherical particles each with a radius R and identical properties under a small external normal load of P is deﬁned as [32] kn ¼

P PR ¼ 2 d a0

ð51Þ

where a0 is given by "

9pW A R2 ð1 n2 Þ a0 ¼ 4E

#1=3 ð52Þ

which is the JKR contact radius at zero external load. Clearly, the contact stiffness deﬁnition implies the Hertzian force–displacement law instead of the JKR relationship. This inconsistency will be examined later. If autoadhesive particles are stacked vertically together to form a single-particle chain or string with a nominal cross-sectional area of (2R)2 supporting a small tensile load at both ends, the ensemble elastic modulus, E 0 can then be expressed as [32] E0 ¼

nominal stress P=ð2RÞ2 P E ¼ ¼ ¼ a0 strain dð2RÞ 2ð1 n2 ÞR d=2R

Thus, combining equations (51), (52) and (53) leads to " #1=3 9pW A E 2 0 E ¼ 32Rð1 n2 Þ2

ð53Þ

ð54Þ

Since a 3-D ordered cubic array of particles comprises a square packing of these particle chains, it can be shown that the ensemble elastic modulus also follows equation (54), assuming equal contribution of each particle chain to stress transmission. A similar analysis was performed on three other regular packings having denser packing than the simple cubic array, viz, cubic-tetrahedral, tetragonal– sphenoidal and hexagonal arrays. In these arrays, the nominal cross-sectional area is reduced as the particles are packed closer leading to an increase in the coordination number, i.e. additional interparticle contacts are established. Hence, it is necessary to resolve the extra contact stiffnesses in the direction of the load

Mechanistic Description of Granule Deformation and Breakage

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application. It follows that there is a rapid rise of the ensemble elastic modulus with the fourth power of the solid fraction of the assemblies such that the following scaling applies [32]: " #1=3 9pW A E 2 0 4 E ¼ 13:3ð1 g Þ ð55Þ 32Rð1 n2 Þ2 In this equation, the air volume fraction of a dry autoadhesive ensemble is deﬁned as the porosity, eg so that the solid fraction f ¼ 1eg. The foregoing analysis was investigated by Kendall et al. [32] using three-point bend tests of beam shaped powder compacts manufactured from submicron zirconia, titania, alumina and silica powders. To produce beam compacts of different solid fraction, each powder was initially mixed with a different amount of aqueous polyvinyl alcohol solutions, which were subsequently removed by heating. It was demonstrated that the measured compact elastic modulus ﬁtted closely to the fourth power dependence on solid fraction. Conformity of the fourth power scaling was also reported in the studies of Trappe et al. [95] and Yanagida et al. [96] using weakly attractive colloidal particles and powders with a broad size distribution, respectively. Hence, it can be concluded that this scaling is applicable to powder compacts with random packing. This correlation was postulated to be the consequence of the f2 dependences of the solid fraction on the density of particle packing and coordination number of each particle [32]. However, the problem associated with equation (55) is that the surface energies of the powders used in the experiments of Kendall et al. [32] were underpredicted with the elastic moduli measured in the bend tests, except for the case of titania. The low surface energy values were explained in terms of surface contamination where surface adhesion was reduced considerably. Nevertheless, Thornton [97] disagreed with attributing surface contamination as the only reason for the low-energy values and pointed out the inconsistency of equation (51) with the JKR force–displacement law. The normal contact stiffness when corrected can be expressed as [52] pﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ E 3 P1 3 PC kn ¼ a0 pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ ð56Þ ð1 n2 Þ 3 P1 PC with P1 and PC deﬁned in equations (17) and (18), respectively. Under zero external load, the normal contact stiffness can be simpliﬁed to k n ¼ 0:6

E a0 ð1 n2 Þ

ð57Þ

which means the ensemble elastic modulus of a simple cubic packing is overpredicted by a factor of 1.67 using equation (55) [97]. Hence, there is an underestimation of the work of adhesion by a factor of 9.26.

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Y.S. Cheong et al.

Further experiments of Abdel-Ghani et al. [98] on bend tests of glass ballotini powder compacts highlighted another inadequacy in the model of Kendall et al. [32], which assumed a uniform transmission of the applied stress throughout the entire volume of the ensemble. In their analysis, the work of adhesion for the glass spheres was measured separately and inserted into equation (55). It was found that the ensemble elastic modulus was overestimated by several orders of magnitude compared to the bend test results for the glass compacts. To resolve this issue, Adams et al. [91] proposed that the external load acting on the ensemble is carried via equally spaced discrete particle chains with a spacing factor of s. Combining this approach and Thornton’s [97] analysis, they arrived at the following prediction of the ensemble elastic modulus of a simple cubic packing of autoadhesive particles: " #1=3 2:4 9pW A E 2 0 E ¼ 2 ð58Þ s 32Rð1 n2 Þ2 Although s is not known a priori, discrete element simulation data suggested that the typical value of this parameter is of the order of several particle diameters [90]. The prediction of equation (58) was still an order of magnitude larger than the elastic modulus measured by bending glass ballotini beam compacts [98]. This discrepancy could be interpreted more appropriately in terms of foreign molecules contaminating the glass particle surfaces which themselves may also be rough [91].

3.1.2. Rumpf’s theory of granule strength The deﬁnition of strength for particulate media is not universal but determined by the loading conﬁgurations and the failure mechanisms. A granule usually fails in a tensile mode where separation of contacting particles occurs across some failure planes forming macroscopic fragments. Hence, it is appropriate to deﬁne granule strength in terms of the maximum allowable tensile stress acting perpendicularly to the cross-sectional area of a failure plane. In Rumpf’s [99] pioneering theoretical treatment of granule tensile strength, tensile forces were assumed to transmit through a granule body via the bonds between the constituent particles. Different mechanisms of bonding were considered including autoadhesion, liquid bridges and solid binder. To simplify the problem, a granule was assumed to comprise a random distribution of monodisperse spheres where each interparticle bond on the failure plane contributed equally to sustain an external tensile force. The consequent effect is that the failure surfaces separate simultaneously, once the critical tensile stress is exceeded. Summing the interparticle tensile forces across the entire failure plane, the following theoretical relationship for the ultimate tensile strength of granule, sf was

Mechanistic Description of Granule Deformation and Breakage

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developed [99] 1 g Fb sf ¼ 1:1 g ð2RÞ2

ð59Þ

eg, R and Fb being the air volume fraction of the granule (granule porosity), radius of the constituent particles and binding force, respectively. Nevertheless, the constituent particles of real granules are often poly-disperse and nonspherical. In respect to this problem, it was proposed that R in the foregoing should correspond to the mean radius, for instance the surface-volume mean radius of the constituent particles [15]. For granules bound together by surface forces, the above expression can be written in terms of the Hamaker constant, A, and the gap between the constituent particle surfaces, z 1 g A sf ¼ 1:1 ð60Þ 48z2 R g This analysis correlates the macroscopic granule strength to microscale parameters, viz the constituent particle properties and their packing density characterised by the granule porosity. There is evidence that equation (59) provides a satisfactory prediction of tensile strength for wet granules when the binding force due to liquid bridge is used in place of Fb as demonstrated experimentally by Rumpf [99], Schubert [100] and Hartley et al. [101] and more recently by the computer simulation of Gro¨ger et al. [102]. However, Pierrat and Caram [103] compared Rumpf’s theory with published experimental data and revealed that the theory overestimates the strength of various types of wet granules in the pendular state, where the liquid content was too low to ﬁll the interstitial spaces. It is possible that these unﬁlled pores possess the stress ampliﬁcation effect similar to a crack. Consequently, these granules might fracture in a brittle manner by rapid crack propagation instead of the simultaneous rupture postulated by Rumpf [99]. Moist granules with pores saturated with liquid are likely to deform plastically resulting in a more uniform stress ﬁeld and hence it is arguable that Rumpf’s theory is applicable. This transition in failure mode was identiﬁed in the experimental investigation of Fu et al. [104] on the impact breakage of wet granules containing different amount of liquid binder.

3.1.3. Kendall’s theory of granule strength Most dry granules and powder compacts fail by brittle fracture where unstable cracks propagate rapidly through the assemblies. Furthermore, granules from identical batches exhibit a distribution of tensile strengths, which are sensitive to the distribution of defects and the critical defect size within the granules. These typical characteristics of brittle fracture were encountered in the studies of

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Kendall et al. [105] on ceramic compacts, Adams et al. [106] on polymer bound sand compacts, Abdel-Ghani et al. [98] on autoadhesive glass assemblies and recently Antonyuk et al. [107] on dry granules. The description of the tensile strength for these particle assemblies using Rumpf’s theory is obviously inadequate as the pronounced effect of defect size was ignored. This is because the stress ﬁeld in a cracked body is highly non-uniform particularly in the immediate vicinity of the defect tip. Hence, the calculation based on the uniform stress distribution assumed by Rumpf [99] is anticipated to overestimate the granule strength since granules are weakened by the high local stresses generated due to the presence of defects and inclusions. A more realistic theoretical tensile strength of a granule can be derived through the summation of the energy required to rupture each interparticle junctions located on the fracture plane [105]. This is in contrast to the force summation employed by Rumpf [99]. Assuming monosized spheres, a fracture energy uf to separate two adhering particles with a contact radius of a0 is expended to create two new solid surfaces. Nonetheless, there is also a simultaneous release of the elastic energy ue due to recovery of the material from deformation under the action of the surface forces at zero external force. Hence, the net fracture energy can be written as [105] uf ¼ pa20 W A ue

ð61Þ

Adopting the JKR adhesion model, the elastic energy with no externally applied force is ue ¼

1=3 2=3 1 4 3ð1 n2 Þ ð3pW A RÞ5=3 15 2R 2E

ð62Þ

whereas a0 is given by "

9pW A R2 ð1 n2 Þ a0 ¼ 4E

#1=3 ð63Þ

so that " uf ¼ 4:736

p5 W 5A R4 E2

#1=3 2 2

ð1 n Þ

ð64Þ

For a simple cubic packing of particles, the number of contacts per unit area across a cleavage plane, i.e. the contact density, (2R)2, whence the total fracture energy Ucubic associated with this plane is given by " #1=3 p5 W 5A 2 2 U cubic ¼ 0:074 2 2 ð1 n Þ ð65Þ E R

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Similar to the case of ensemble elastic modulus, Kendall et al. [105] concluded that the total fracture energy increases rapidly with the fourth power of the solid fraction of regular assemblies. On this basis, the ensemble fracture energy, Uf, can be derived as [105] " #1=3 5 2 2 4 W A ð1 n Þ U f ¼ 59:7ð1 Þ ð66Þ 4E 2 R2 This fourth power scaling was supported experimentally by fracturing beamshaped powder compacts with different solid fractions manufactured from titanium dioxide and aluminium oxide in a three-point bend conﬁguration [105]. However, the values of WA ﬁtted to the experimental data were two orders of magnitude larger than the equilibrium work of adhesion for both powders. This is because energy loss caused by plastic deformation ahead of the crack tip is inevitable when this quantity is measured in fracture experiments [98,108]. Despite the discrete nature of a particle assembly, it can be treated as a continuum body provided the constituent particles are much smaller than the macroscopic size of the assemblage [105]. Applying continuum fracture mechanics, rapid crack propagation in a tensile mode (Mode I) occurs at a critical defect size present in the assembly, c, which can be related to a critical stress intensity factor, KC [109]. pﬃﬃﬃ K C / sf c ð67Þ where Uf ¼

K 2C E0

ð68Þ

Rearranging equations (67) and (68), yields the critical tensile stress for fracture [105] 0 1=2 E Uf sf / ð69Þ c This continuum approach is justiﬁed when the fracture stress obtained by fracturing beam compacts of titania containing through edge notches was observed to decay with c according to equation (69) in three-point bend tests [105]. Notches of different length were deliberately introduced to the edge of the compacts to test the sensitivity of strength to defect size. Although this theory is successful in the case of artiﬁcially cracked compacts, questions were raised about the size of the natural defects. As Kendall [108] demonstrated, the natural defect size present in an unnotched sample was several orders of magnitude larger than the size of the constituent particles. Further analysis revealed that the inclusion of small agglomerated powders in the compact constitutes such defects [16]. This has also been observed by other workers [109].

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Recently, Coury and Aguiar [110] provided an interesting comparison between Rumpf’s and Kendall’s theories by measuring the strength of ﬁltration cake and tumbling drum granules. Their studies revealed that neither Rumpf’s theory nor Kendall’s approach could represent the interparticle interactions within granules found in different applications. Rumpf’s theory was shown to describe adequately the process of ﬁltration cake removal because the cake (granular layer) was formed by slow and uniform deposition of powder. This resulted in layers with low adhesion and uniform porosity, which could be separated simultaneously without crack propagation. However, the granules agglomerated in tumbling drum were observed to be ruptured by propagating cracks under diametrical compression. Hence, Kendall’s approach provided better prediction for the granule strength. This was because tumbling drum granules possessed irregular particle packing and cracks that could be nucleated at voids surrounded by strong interparticle links leading to their subsequent propagation through the granular structures. Therefore, rough estimates of granule strength may be facilitated using the appropriate theories depending on the breakage mechanism in the application concerned. The underlying mechanisms for granule breakage can be revealed by single-granule fracture studies. Some of the experimental investigations and computer simulations of single-granule crushing are summarised in the following sections, with a particular emphasis on spherical dry granules.

3.2. Measurement of granule strength 3.2.1. Diametric compression experiments Quasi-static diametric compression is a common technique for studying the crushing strength of single granules [17,110–112]. It provides a means of measuring the indirect tensile strength of a granule, i.e. the tensile hoop stress required to split the granule apart. This is an appropriate method since real applications usually involve spherical granules rather than beams or bars as studied by Muller et al. [109] and Kendall [108], for example. Furthermore, granule material parameters and the energy utilised to fracture a granule can be estimated from the load–displacement (load–deformation) curve. However, it has been argued that the granule crushing strength measured in diametric compression must be interpreted with great care, as it is only representative for highly brittle and isotropic materials [17]. The deformation characteristics of a dry granule before fracture are strongly dependent on the bonding mechanisms and the defect distribution, which are inﬂuenced by the granulation method. For instance, Sheng et al. [113] showed that polymer bound alumina granules having complex internal structures exhibited a mixture of brittle and ductile behaviour. A variation in crushing strength was reported by Kendall and Weihs [114] and Samimi et al. [115] for granules prepared

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under nominally identical conditions. These observations may be attributed to the susceptibility of granule strength to the structural heterogeneity of granules such as the critical size of the internal voids or inclusions. Antonyuk et al. [107] demonstrated that the deformation behaviour of granules can also be a function of compression speed and granule size. Generally, it is possible to classify the deformation of a granule by inspecting the load–displacement curve obtained by compressing it between two parallel platens. Typically, for dry granules they follow a characteristic elasto-plastic deformation where there exists a non-linear Hertzian regime at small displacements. After a yield limit, where plastic deformation is initiated, the load appears to increase linearly with the displacement of the platens according to the following relationship [116]: P ¼ 2pRHd

ð70Þ

where H is the hardness of the granules which varies from 1.1 sy at ﬁrst yield to 3 sy at full plastic deformation, sy being the uniaxial yield stress of the granules. For the diametric compression of granules where two contact zones exist, the relative approach between the centre of the granule and either platen d is half of the total displacement of the moving platen. Such deformation behaviour was observed in the compression studies of binderless granules produced by spray-drying [114] and ﬂuidised bed granulation [13]. Similarly, granules bound with solid binder such as detergent [115] and zeolite granules [107] were shown to deform elasto-plastically. When the elastic strain is negligible, a granule deforms plastically giving a linear load–displacement relation as exhibited by binderless limestone granules [93] and sodium benzoate granules [107]. Fracture of granules was identiﬁed as corresponding to an abrupt reduction in the platen force. In cases where substantial elastic strain is stored prior to fracture, granules may be disrupted through unstable crack propagation, possibly at sonic speed as suggested by Antonyuk et al. [107]. In contrast, the crack propagation for plastically deformed granules is more stable. Sometimes, partial fracture may occur without total disruption of the granules. This was observed in the compression studies of Sheng et al. [113] on polymer bound granules for which the measured forces ﬂuctuated during deformation. It has been proved useful to treat a granule as a continuum and infer the material parameters from the load–displacement data using the contact mechanics theory outlined in Section 2.3.2.1. The constituent particles of the granule must be small compared to the macroscopic granule size for the continuum approach to be justiﬁed [108]. This was pursued by Kendall and Weihs [114] to determine the elastic modulus and yield strength of spray-dried granules composed of submicron zirconia particles. Since no binding agent was used, the granules were postulated to be bound together by van der Waals attraction. However, the agreement of the inferred elastic modulus with that calculated using a micromechanical model based on autoadhesive particles

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(see Section 3.1.1) was not satisfactory. It was argued that the initial loading of the granules might be a consequence of the irreversible ﬂattening the protruding granule surface rather than the apparent elastic deformation shown in the compression data [114]. More recently, a similar treatment was performed to calculate the material parameters from diametric compression data of polymerbound alumina granules [113], detergent granules [115] and industrial granules [107]. Nonetheless, comparisons with results from micromechanical modelling were not possible due to the complex interactions between solid binder bridges and the constituent particles. Kapur and Fuerstenau [93] investigated the quasi-static behaviour of dry binderless granules by compressing limestone pellets of 8–20 mm in diameter. At fracture, a cone of material was generated at opposing poles of a spherical pellet. These poles corresponded to the contact points between the pellet and the upper and lower plates of the compression device. The general fracture pattern was splitting along a vertical plane creating two hemispherical halves. They suggested that fracture is initiated once the separation between the interparticle bonds along a potential fracture plane exceeded a critical value. Since it was assumed that no preferential fracture plane existed, they proposed that all the interparticle bonds were subjected to the same horizontal tensile force. By equating the total compressive work experienced by the cones at the poles until fracture to the strain energy stored in the thin disk volume encompassing the fracture plane, the crushing strength was related empirically to the pellet porosity and limestone powder surface area. This relationship reﬂected the divergence of the strength of dry, porous pellet from that of homogenous, elastic body. A later quasi-static compression test conducted by Arbiter et al. [117] using large sand–cement spheres (up to 120 mm in diameter) yielded fracture patterns similar to that obtained by Kapur and Fuerstenau [93]. Inspection of the contact area at a load slightly less than that corresponding to the fracture value revealed that minute cracks were densely distributed along the periphery of the contact area. The periphery was indicated by the sharp change in radius of curvature, which was expected to be heavily stressed. The breakage efﬁciency for diametrical compression to produce fragments of a speciﬁc size was deduced and was compared with that for free-fall impact. It was found that the energy input necessary to initiate fracture in free-fall impact was twice that required by quasistatic compression. Moreover, their work indicated that slow compression and low-velocity impact induced geometrically similar stress ﬁelds in spherical sand–cement spheres. The formation of cone was ﬁrst identiﬁed by Newitt and Conway-Jones [118] when wet sand granules fractured under compressive loading. They pointed out that the surfaces of the conical volume of materials coincided with the slip planes in the direction of the principal shear stresses. As a result of the indentation of the cone into the granule body, circumferential tensile hoop stresses were induced,

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which led to the propagation of meridian cracks across the granule. The signiﬁcance of platen friction in cone formation was also emphasised. Referring to the experiments of Arbiter et al. [117], the maximum tensile stresses generated at the perimeter of the contact area might result in the minute cracks observed (see [107]). Apart from strength testing, diametrical compression has been conducted as a complimentary test to visualise the sequence of crack formation in fertiliser granules under impact loading [119]. In this investigation, the load application was continued after fracture to examine the subsequent crack formation. Primary fracture into two hemispheres was observed followed by secondary fracture of the hemispheres into quadrants and segments. Consequently, Salman and co-workers [119] concluded that fertiliser quadrants collected from low-velocity impacts against a platen were the consequence of secondary fracture preceded by primary fracture.

3.2.2. Impact experiments Impact experiments can be used to simulate the dynamic conditions in a granulator and granule handling equipment where granule–granule and granule–wall collisions are involved. Nevertheless, only limited information can be extracted from impact experiments due to the short impact duration. The failure patterns for various types of granules are documented in the comprehensive review of Salman et al. [120], which are summarised in Figs. 9 and 10. In addition to post-impact product examination, several investigators have used high-speed imaging techniques to capture the evolution of an impact event. Arbiter et al. [117] performed a detailed experimental study of the fracture process for large sand–cement spheres (70–120 mm) released from increasing heights above a massive ﬂat steel target. It was observed that cracks dividing the spheres always propagated from the contact region. Furthermore, meridian plane fractures were found to be the dominating failure mechanism at low dropping heights. As the height was increased, there was not only an increase in meridian fracture planes

Fig. 9. Example of the typical impact fracture of the three generic types of granules under moderate impact conditions. These granules are between 4 and 5 mm in diameter [121].

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Fig. 10. Failure forms of solid, wet and binderless granules from the side view. They change with increasing impact velocity from left to right. (After Reynolds et al. [121], with permission.) (a) solid granule; (b) wet granule; and (c) binderless granule.

but oblique fracture planes also started to develop. Eventually, the spheres failed by multiple oblique fractures at the greatest height of fall investigated. Fracture was found to initiate from a conical surface where its base corresponded to the contact area of the sphere at the instant of failure. As this cone was indented into the specimen, the sphere was subjected to tensile hoop stress resulting in meridian crack propagation from the periphery of the contact area. Oblique cracks observed at greater released heights were shown to coincide with the trajectories of maximum compression stress developed when an elastic sphere was in contact with a platen under static loading. Photoelastic experiments were carried out by loading a resin disk with a ﬂat platen under constant loads to simulate the stress distribution within the sand–cement spheres under free-fall impact. From the photoelastic simulation, the stresses were found to concentrate at the corner of the ﬂat portion of the disk as indicated by the maximum fringe order. Thus, Arbiter et al. [117] explained that the failure along the conical surface might start from the periphery of the contact area, as this was the region of high stress concentration. A similar explanation was offered by Salman et al. [119] to interpret the failure patterns observed by impacting smaller fertiliser granules (3.2–7.2 mm) against a rigid massive target. At low impact velocities, meridian plane failure dominated the fracture of the granules into hemispheres or quadrants. However, they concluded that meridian plane cracks were absent at high impact velocities as indicated by the largest surviving fragment, which showed rotational symmetry. In addition, they also found a crushed cone around the contact region in contrast to

Mechanistic Description of Granule Deformation and Breakage

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the undamaged cones generated in the free-fall impact tests of Arbiter and coworkers [117]. At the highest impact velocity employed, disintegration occurred leaving a cone of compacted powder on the target surface. A 2-parameter Weibull equation was applied to characterise the probability of breakage for granules impacted at a speciﬁc impact angle and impact velocity. For the impact of these fertiliser granules at various angles, Maxim et al. [122] deﬁned the critical impact failure velocity as the velocity to cause 63.5% of the granules in a batch of nominally identical samples to fracture. Adopting a quasi-static approach, they assumed that the maximum force experienced by the granules before fracture was given by the Hertz equation (12). Integration of equation (12) in dimensionless form allowed the normal component of the critical impact failure velocity, Vf to be expressed in terms of the fracture force and the material properties of the granules [122]. A similar relationship was derived by Knight et al. [123] for the impact of steel spheres against glass surfaces, in which the kinetic energy of an impacting steel sphere was assumed to be converted into the strain energy stored in the sphere at the instant of maximum compression. Maxim et al. [122] demonstrated that the fracture force could be estimated by compressing the granules diametrically up to fracture and the values of Vf calculated on this basis were in reasonable agreement with those measured in impact experiments. In the high-speed imaging investigation of Subero and Ghadiri [78], different breakage behaviour was exhibited when 30 mm granules with a polymer binder containing artiﬁcial macro-voids were impacted against a target plate. It was interesting to note that a cone was not generated at the impact site in any of their experiments. Local disintegration prevailed at low impact velocities and, consequently, particles adjacent to the impact site detached from the main structure of the granule forming singlets, doublets and triplets. Moreover, there was an increase in the extent of disintegration at higher impact velocities. This was considered to be the effect of macro-voids as the presence of these voids at the impact location hindered the transmission of stresses through the granular structure. However, fracture combined with local disintegration might occur when the impact velocity was raised depending on the internal structure of the granule. In this case, signiﬁcant stresses could be transmitted to the bulk of the granules. Thus, cracks were available to split them into several large clusters by side chipping or a combination of meridian and oblique crack fractures. It was found that if fracture took place, the accompanying local disintegration was less severe compared to the cases for which fragmentation was not observed. Generally, it was concluded that the fragmentation of a granule could be promoted by increasing the impact velocity or macro-void size and number. In the impact crushing of 150 mm concrete spheres, fractures along meridian planes with the fragmentation of the conical region at the impact site into ﬁnes were reported by Tomas et al. [124] in contrast to the intact cone observed by Arbiter et al. [117]. This was explained as the result of plastic deformation at the

1096

Y.S. Cheong et al.

impact site causing local crushing. Furthermore, meridian cracks were initiated along the periphery of the contact area of the spheres due to the development of tensile stresses, which eventually propagated and divided the spheres into segments. However, it was suggested that a circular crack circumscribing the contact area would have formed if the spheres experienced elastic deformation at the impact site. The convergence of the circular crack as it propagated into the spheres might lead to the formation of a coherent cone. Besides these macroscopic failure modes, the failure mechanism of interparticle bonds was examined for glass ballotini granules bound with a polymer using scanning electron microscopy in the work of Subero and Ghadiri [78]. Two different interparticle bond failure mechanisms were identiﬁed namely adhesive and cohesive breakages. Adhesive breakage refers to the separation of contacts at the interface between the binder and primary particle resulting in a smooth fracture surface. Internal failure across the binder bridge with rough and irregular fracture surfaces is termed cohesive breakage. However, a correlation between the bond fracture toughness and the macroscopic granule impact breakage behaviour was not reported. A completely different failure pattern was described by Boereﬁjn and co-workers [12] when studying the impact breakage of weak lactose granules. Their results indicated that the disintegration of these granules was due to the bulk plastic deformation induced in the agglomerates. However, embrittlement of the lactose granules was found in a humid environment. In the presence of moisture, the transformation of amorphous lactose to the monohydrate state was initiated, which was postulated to cause the formation of brittle solid bridges. Extensive experimental investigations were carried out by Samimi et al. [125] to examine the inﬂuence of the structural characteristics of detergent granules on the impact breakage propensity. Since the granules contained a viscous binder, humidiﬁcation and storage temperature were found to modify the bonding mechanism between the constituent particles, thus, giving different extents of breakage. In addition, the manufacturing route was demonstrated to be important in determining the granule strength by tailoring the degree of consolidation of the granules.

3.2.3. Multi-granule testing It is clear that mechanical testing of single granules is informative in revealing the complete stress–strain behaviour and the associated failure mechanisms. However, the disadvantage of such tests is that a large number of experiments are necessary to obtain statistically reliable granule strength. A robotic tester was developed by Pitchumani et al. [126] to automate single-granule compression tests. However, from the standpoint of monitoring the degradation of granular

Mechanistic Description of Granule Deformation and Breakage

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product in handling equipment, it may be sufﬁcient to express the strength of a bulk granule sample in terms of an average value. In this case, strength measurements can be performed by stressing multiple granules simultaneously. To avoid tedious single-granule crushing tests, Adams et al. [127] proposed a granule strength measurement using a uniaxial conﬁned compression of a bed of granules. Based on the Mohr–Coulomb failure criterion, the shear strength of a single granule was related to a constant of proportionality determined from the linear portion of the pressure–volume compaction curve of the granule bed at large strain. The inferred failure force was of the same order of magnitude as the crushing load measured separately by fracturing single-polymer-bound silica granules under diametric compression. Besides the application of a slow compression velocity, multiple granules are commonly subjected to vibratory motions to assess the breakage propensity. For example, Beekman et al. [128] investigated repeated impacts (fatigue failure) of a group of enzyme granules enclosed in containers attached to a sieve shaker. In their experiments, the enzymes granules were weakened by the fatigue loading resulting in either attrition or gross fragmentation. In the study of Utsumi et al. [129], the strength of fragile ferric oxide granules was examined by agitating the granules on a sieve of a known size using a shaker capable of imposing rotating and vibrating motions. The extent of breakage was expressed as the sum of abrasion rate of the granule bed on the sieve surface and the wear rate of the granules within the bed.

3.2.4. Quanti¢cation of breakage propensity In single-granule fracture experiments, the analysis of failure patterns and the fracture surfaces of the resulting fragments provides insights into the failure mechanisms. In addition, the extent of breakage can be assessed by examining the size distribution of the fragments and fragment size distributions serve as the breakage function for population-balance modelling of a rate process. Nevertheless, there are not standard guidelines available to determine the size of a fragment. This is because most fractured products are irregular in shape and the fragment size depends on the orientation of the fragment under inspection [130,131]. For example, the probability of an elongated fragment passing through a certain sieve size is determined by the orientation of the longest axis of the fragment. Therefore, it is a common practice to represent the size of a fragment as a linear dimension at the particular orientation during inspection. This is usually expressed in terms of equivalent diameter of a sphere, such as equivalent volume diameter [130]. After deﬁning the size of fragments, the distribution of fragment sizes over a range of sizes can be expressed in several forms of empirical relationships. In the

Y.S. Cheong et al.

1098

comminution and fracture of brittle solids, the commonly employed relationship is the Rosin–Rammler equation. It is deﬁned by two empirical constants n and xc namely the width of the fragment size distribution and mode size, respectively [132]. Its cumulative form can be written as follows: n x Q ¼ 1 exp ð71Þ xc where Q is the cumulative probability of a fragment having a size smaller than x. This distribution function is useful in describing skewed-size distributions of comminuted products as pointed out by Djamarani and Clark [133]. The theoretical analysis of Gilvarry and Bergstrom [134] suggested that the parameter xc can also be interpreted as the spacing between the defects within a specimen. This proposal was based on the fact that the fracture of brittle solids is a consequence of crack initiation at a defect site. In addition, Cheong et al. [135] pointed out that n could be used to indicate the fracture form based on their investigation of the fracture of glass spheres impacting a target at different velocities and impact angles. The cumulative fragment size distribution can also be plotted on a Gates–Gaudin–Schuhmann double-logarithmic plot [117,131]. Two distinct straight lines with different slopes distinguishing the debris from coarse fragments are usually observed. The complete size distribution of fragments is necessary to describe the extent of breakage if fragmentation occurs. In the case of the attrition and the wear of granules, ﬁne debris is generated leaving a large intact residue. The analysis of the size of individual debris is difﬁcult since ﬁne particles adhere strongly to each other. According to Boereﬁjn et al. [12], the breakage of weak lactose granule under impact can be quantiﬁed as the proportion of the mass lost by the feed granules after impact. The large residues were separated from the debris by a sieve of a certain size smaller that the feed granules. Attributing all handling loss to the residues, Boereﬁjn et al. [12] deﬁned the lower limit of the breakage extent as x ¼

Md 100% Mf

ð72Þ

while the upper limit, where all handling loss was attributed to the debris, was taken as xþ ¼

Mf Mr 100% Mf

ð73Þ

with Md, Mr and Mf being the debris mass, residue mass and feed granule mass, respectively. This breakage parameter was shown to be directly proportional to the impact kinetic energy since the data scale with the square of the impact velocity.

Mechanistic Description of Granule Deformation and Breakage

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3.3. Discrete element method (DEM) simulations Recently, computer simulation has been a robust tool to study the deformation and breakage of particulate systems. This is because the same particle conﬁguration can be tested repeatedly and information about different parameters at any instant of time can be retrieved for further analysis. Furthermore, computer simulation offers the advantage of revealing information, such as different energy dissipations or load transmission paths, which are not accessible through physical experiments. The DEM is a suitable tool for studying the macroscopic response of a particulate system, which depends on the discrete behaviour of its constituent primary particles. This is in contrast to ﬁnite element simulations where the average local stress response of the material is described using constitutive relationships. The study of granule breakage mechanisms under external forces is one of the examples of the application of DEM simulation. The evolution of granule damage is modelled as a dynamic process by tracing the motion of the constituent particles throughout the impact event using Newton’s law of motion. The resulting particle motion is determined by the interaction at the interparticle contacts in addition to external forces such as gravity. The simulation is advanced over a large number of small-time steps and the particle motion is updated continually. This methodology was initially proposed by Cundall and Strack [136]. Attempts to study granule degradation were initiated at Aston University in the UK by incorporating well-established contact mechanics interaction laws into the methodology of Cundall and Strack [136]. The simulations were ﬁrst carried out in 2D [137] and later extended to 3D by Ciomocos [138]. They are now capable of simulating the interactions between elastic, spherical, frictional and autoadhesive particles. The earlier version of the code by Thornton and Yin [52] considered only elastic deformation at autoadhesive interparticle contacts. Plastic yield was accounted for in the subsequent version developed by Thornton and Ning [139]. A slightly different approach was adopted by Potapov and Campbell [140] to represent an elastic solid by ‘‘gluing’’ polyhedral elements together. The glue at the interface between two elements could withstand certain tensile stresses before fracture occurred. Using this modiﬁed technique, correlation between the breakage patterns of an elastic solid and different fracture mechanisms was established.

3.3.1. Diametric compression simulations There are only limited DEM simulations for the diametric compression of granules, presumably because substantial computing time is required owing to the low compression speeds employed. In the study of Thornton et al. [141], distinctive breakage behaviour was observed depending on the porosity and thus the

1100

Y.S. Cheong et al.

packing density of the granule being investigated. They concluded that the fracture of a granule across a well-deﬁned surface is only possible if the granule structure is sufﬁciently dense and rigid, i.e. high density of interparticle contacts. This allows the compressive loads to be transmitted along a discrete path into the granule body from the compression platens. Consequently, sliding between the particle clusters adjacent to this force transmission path occurs leading to the ultimate fracture. This may explain the phenomenon in which dense granules always fracture whereas loosely packed granules fail by progressive displacement of the constituent particles from the contact zones. In addition, two highly fragmented zones were formed at the contact zones when a dense granule fractured along a primary meridian plane in the diametric compression test simulated by Thornton et al. [141]. Similar breakage behaviour was also noted by Khanal et al. [142] in their DEM simulations of diametric crushing of concrete spheres. This is consistent with the cone formation and diametric fracture reported in the slow compressions of wet granules [118] and dry limestone pellets [93], described previously. Finite element computation suggested that the diametric crack should propagate from the peripheries of both contact zones of the granule, where a high tensile stress existed [142]. Recent advancement in X-ray microtomography has enabled Golchert et al. [143] to incorporate real granule structures into DEM simulation to study the effects of granule structure and shape under compressive loading. Their results demonstrated that a spherical granule was more resistant to damage compared to a granule of irregular shape. Again, the breakage behaviour was determined by the probability of forming a strong force transmission path.

3.3.2. Impact simulations Besides diametric compression simulations, the Aston code has been used to investigate granule impact breakage against a plane surface [131,144–147]. It appears that the resulting breakage patterns are very similar to those generated in diametric compression tests, except that there is only one stress application point in impact. The recent review of Mishra and Thornton [148] reported that there are ﬁve factors governing the breakage behaviour of granules under impact. They are the impact velocity, bond strength (interface energy), granule porosity, coordination number of the constituent particles and the local structural arrangement of particles near the impact region. Investigating the combined effects of impact velocity and porosity, signiﬁcant breakage was found to occur when the impact velocity exceeded a certain threshold value. Once breakage took place, dense granules always fractured while loose ones disintegrated. Those with intermediate porosities exhibited mixed-mode failure where both fracture and disintegration occurred. Furthermore, they compared the breakage behaviours

Mechanistic Description of Granule Deformation and Breakage

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between similar granules, one with a greater particle contact density than the other. The denser granule fractured in contrast to disintegration shown by the granule with the lower contact density. It was postulated that signiﬁcant stresses were transmitted through the bulk of the granule with the higher contact density storing sufﬁcient elastic energy for fracture. They also suggested that different breakage patterns could be obtained when different parts of a given granule were subjected to impact. This was due to the differences in the local particle arrangement near the impact location. Similar observations emerged from the DEM simulations of Moreno et al. [149] where a granule was impacted at various oblique angles on to the target plane. Moreover, the component of the impact velocity normal to the target plane was found to be the main cause of granule damage under oblique impact. Using the same code, Kafui and Thornton [150] simulated the collision between a pair of similar granules to understand the fragmentation process due to the impact arrangement. In DEM simulations, the extent of breakage can be characterised by the size distribution of the fractured products in a similar manner to that performed in physical experiments. Depending of the intensity of the applied stresses, the fragments can be several clusters of constituent particles that remain intact after a breakage event and some single constituent particles. The size of a cluster can be deﬁned using an equivalent diameter of a solid sphere having the same volume as the cluster [131,147]. Subero et al. [147] attempted an experimental veriﬁcation of their DEM simulations of granule–wall collisions by constructing physical granules using glass beads bound together with a polymer binder. Using the deﬁnition of fragment size deﬁned above, it was found that the fragment size distribution resulting from the simulation was comparable to the experimental results in the complement region, i.e. the fragment size range smaller than 50% of the original granule diameter. However, it should be noted that disintegration of granules was observed in the simulation of Subero et al. [147] rather than the fracture behaviour shown by their physical granules. In a study of the impact breakage of weak lactose granules, Boereﬁjn et al. [12] calculated the lower and upper limits of the granule breakage in physical experiments using equations (72) and (73), respectively. Their parallel DEM results indicated that the actual extent of breakage was within these limits conﬁrming the robustness of the DEM methodology. In another numerical study, Kafui and Thornton [131] reported that the exponent of the power law relationship describing the fragment size in the complement region was a function of the interparticle bond strength. The damage ratio was ﬁrst proposed by Thornton et al. [144] as an alternative measure of the extent of breakage of granules in DEM simulations. This ratio is deﬁned as the proportion of interparticle bonds broken within a granule due to a breakage or deformation event compared with the initial state of the granule. Further work showed that the damage ratio is proportional to a dimensionless group known as the Weber number, Wb [144,147], which takes the following

Y.S. Cheong et al.

1102

form: Wb ¼

2rp V 2i R WA

ð74Þ

where r, R, WA and Vi are the density and radius of the constituent particles, the work of adhesion required to separate an interparticle contact and the impact velocity of the granules, respectively. However, it is uncertain if these parameters should be ascribed to the properties of the bulk granules or those of the individual constituent particles [131]. Furthermore, an asymptotic limit of efﬁciency was identiﬁed beyond which no further breakage is possible with additional energy input [142,147].

4. PRACTICAL CONSIDERATIONS 4.1. Understanding wet granulation mechanisms In a practical context, it is desirable to assess the inﬂuence on the kinetics and growth mechanisms of a wet granulation process due to the modiﬁcation of an existing formulation or the use of a new combination of particle and binder systems. Although satisfactory quantitative predictions are not available currently, it has long been established that granule deformability has a strong effect on granule growth behaviour [151–153]. In most wet granulation processes especially in a mixer granulator, granules collide between each other and the equipment walls. These impact events may result in either rebound or coalescence and sometimes fracture of the granules, which controls the balance between granule growth and size reduction. In a collision, granules that deform easily will form a large area of contact as the impact kinetic energy is dissipated forming a stronger bond that is more likely to outlive ensuing collisions and thus resulting in successful granule coalescence. Conversely, strong granules, which are less deformable, form a weaker bond that will be easily broken apart by subsequent collisions and thus limiting granule growth. The following discussion of granule strength is restricted to that of wet granules where the interstitial spaces between the primary particles are occupied by liquid and air. The collision between a pair of granules is difﬁcult to investigate directly, so a more controlled granule–wall impact conﬁguration is often used as a step towards understanding granule impact and rebound behaviour. The deformability of a granule impacting a plane surface can be expressed in terms of the contact ratio, i.e. the ratio of the deformed area between the granule and the surface to the original cross-sectional area of the undeformed granule. This contact ratio can be related to the dynamic strength of the granule such as the dynamic yield stress [153]. Given the rate-dependent viscous dissipation of the liquid binder within the

Mechanistic Description of Granule Deformation and Breakage

1103

granule during impact, the dynamic strength is sensitive to strain rate when a critical value is exceeded [154–156]. With respect to granule coalescence, the coefﬁcient of restitution can be a useful parameter for predicting the aggregation efﬁciency that is the proportion of granule and/or primary particle collisions that result in an aggregation event. The coefﬁcient of restitution is the ratio of the rebound and impact velocities. For example, a value of zero would imply that the colliding granules would stick together rather than rebound. The breakage of granules can also be expressed in terms of the critical impact velocity to cause crack initiation during impact [104]. The amount of deformation that is experienced by granules in a granulator can be expressed in terms of the Stokes deformation number, Stdef as follows [3]: St def ¼

rV 2rel 2Y

ð75Þ

where Y is the dynamic yield stress, r the granule density and Vrel the relative velocity between the impacting granules. Iveson et al. [154] demonstrated that the mechanical strength of granules is strain rate dependent when a critical compression speed was exceeded. Arguably, it is more realistic to measure the dynamic yield strength under the impact conditions expected in the granulator of interest. The Stokes deformation number is also used to demarcate different regimes of granule growth (see Fig. 11). Granules that do not deform easily during a collision have a small value of Stdef and will only begin to coalesce and grow if liquid binder is expelled to their surfaces due to consolidation to bind the granules together; such systems are termed induction growth systems [3].

"Dry" FreeFlowing Powder

0.1

"Crumb"

Slurry/ Over-Wet mass Steady Growth Increasing Growth Rate

Increasing Deformation Number, Stdef = gUc2/2Yg

Nucleation Only

f(Stv)

Induction Decreasing Induction Time

0

100% Maximum Pore Saturation, smax = ws(1-min)/| min

Fig. 11. Proposed modiﬁed regime map. The nucleation-to-steady growth boundary and steady-growth-to-induction-growth boundaries are functions of Stdef and there is no distinct rapid growth regime [3].

Y.S. Cheong et al.

1104

Binder Viscosity (mPa.s)

10,000 Storkes Number Analysis 1,000 Granules Formed 100

Paste Formed

10

1 0

50 100 150 200 250 Median Size (microns) of the Constituent Particles

Fig. 12. Binder viscosity vs. median particle size showing regions in which granules did and did not form for the agglomeration of glass ballotini with silicone oils in a high shear mixer. Line shows prediction of equation (76) [157].

However, granules with intermediate values of Stdef deform easily during a collision to form a larger contact area leading to steady growth. However, granule breakage is induced at greater values of Stdef. Keningley et al. [157] also developed a strain criterion which describes whether a wet granule will break or survive in high shear granulation, depending on the amount of strain resulted from the compression during the impact. Assuming that granule deformation depends on the pressure loss through the ﬂowing viscous ﬂuid between particles upon impact, the collisional kinetic energy can be equated to the plastic deformation energy of the granule to obtain the following: 2m ¼

3g rD uo d 3;2 1 540 1 2g m

ð76Þ

where em is the maximum compressive strain, eg the granule porosity, rD the granule density, uo the granule impact velocity, d3,2 the sauter mean diameter, m the binder viscosity. They mentioned that the granules will break when the maximum strain, em, exceeds 0.1. This model allows a plausible interpretation of the effect of binder viscosity and primary particle size on the ability to form granules during high shear granulation (Fig. 12) and the predicted limit is shown to be reasonably consistent with the experimental data. Experimental studies have indicated that the impact deformation and breakage of wet granules are governed by various process and formulation-related factors, as described in the following subsection.

Mechanistic Description of Granule Deformation and Breakage

1105

Restitution coefficient

0.25 S=0.115

0.2

S=0.135 S=0.165

0.15 0.1 0.05 0 0

5

10 15 Impact velocity, m/s

20

Fig. 13. The variation of the coefﬁcient of restitution with impact velocity for calcite granules with different binder contents. (After Fu et al. [156], with permission.)

4.1.1. E¡ect of impact velocity In a granulation process, the velocity of granules is dictated by the type of granulator and the operating conditions. The variation of the coefﬁcient of restitution (granule deformation) with impact velocity was investigated by Fu et al. [104] for spherical calcite granules with various amount of liquid polyethylene glycol (PEG 400) colliding with a plane surface. According to the typical proﬁles depicted in Fig. 13, the coefﬁcient of restitution curves attained maxima at some intermediate impact velocities with sticking at smaller and greater values of two critical velocities. The lower limit arises because the surfaces of the granules were covered by a layer of liquid binder. The release of the elastic strain energy stored during impact is insufﬁcient to rupture the liquid junctions formed between the granule and the target. It was argued that the rising parts of the curves were a consequence of the elastic recovery of the granules being sufﬁcient to overcome the viscous dissipation due to the liquid junctions, since the stored elastic energy increases with velocity in the elastic regime [104]. At high impact velocities, the coefﬁcient of restitution was observed to decay approximately with an inverse power of impact velocity as a result of plastic deformation. It is worth noting that such decay was more rapid than that expected from contact mechanics calculations assuming that the granules are treated as homogenous elasto-plastic materials [104]. No improvement in the predictions was found by accounting for ﬁnite plastic deformation and strain hardening effects in the contact mechanics formulations [158]. Arguably, the contact deformation of the

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Y.S. Cheong et al.

Fig. 14. Dynamic yield stress as a function surface mean particle size for glass ballotini with water and glycerol binders. (Adapted from Iveson and Litster [153], with permission.)

granules at high impact velocities exceeded the limits allowable in quasi-static contact mechanics. Despite the discrepancy associated with wet granules, the coefﬁcient of restitution of granules held together by solid bonds was consistent with the prediction of elasto-plastic contact mechanics with strain hardening [158].

4.1.2. E¡ect of primary particle size Fu et al. [156] showed that the coefﬁcient of restitution decreases with increasing primary particle size for the granule types that were investigated. Although it was acknowledged that the distribution of primary particle sizes can complicate experimental results, this effect was concluded to be due to an increase in interparticle contacts, and hence the density of interparticle forces. Similarly Iveson and Litster [153] found a decrease in the dynamic yield stress of agglomerates with an increase in primary particle size, when water was used as a binder (see Fig. 14). They proposed that decreasing particle size decreases the average pore size between particles and increases the volume density of interparticle contacts. This increases both the capillary and the interparticle frictional forces and thus explains why the yield stress increases when water is used as the binder compared with glycerol. However, they also considered why there was not a signiﬁcant inﬂuence of particle size on the dynamic yield stress when a more viscous binder (glycerol) was used as the binder. Their arguments were based on lubrication theory, which predicts that the viscous force to dominate as interparticle spacing decreases, thus increasing the yield stress.

Mechanistic Description of Granule Deformation and Breakage

1107

Fig. 15. Relationship between tensile strength, st, and wet granule saturation, S. (After Schubert [100], with permission.)

4.1.3. E¡ect of binder content, binder viscosity and binder surface tension Schubert [100] investigated the relationship between tensile strength and the saturation of wet granules. As shown in Fig. 15, he argued that it would depend on the granule state as described in [159]. In the ﬁgure, Sp corresponds to the liquid saturation at which there is a transition between the pendular and funicular states, and Sc denotes the lower limit of the capillary state. In the pendular state, a binder forms discrete lens-like rings (liquid bridges) at the points of contacts between particles, leaving air as a continuous medium. The capillary state describes the completely saturated granule and the funicular state is intermediate between these two states. The tensile strength is expected to increase monotonically with increasing saturation in the funicular state (SpoSoSc) as both bridge bonding and bonding caused by regions ﬁlled with liquid contribute to the tensile strength. In addition, the tensile strength is then expected to decrease at high levels of saturation, as the material becomes a paste. Schubert [100] found reasonably close agreement between experimental observations and this hypothesis. However, this is a problem of considerable complexity given that there are not 3-D analytical solutions available for the forces developed in the funicular state and that it is necessary to account for the inﬂuence of the capillary forces on the friction at interparticle contacts, in addition to the capillary and viscous forces generated by deforming the liquid junctions as will be discussed below. Iveson and Litster [153] examined the dynamic yield stress of cylindrical pellets made from either 19 or 31 mm ballotini with water, glycerol and surfactant binders. They found that with glycerol as the binder, increasing the binder viscosity from 0.001 (for water) to 1 Pa s greatly decreased the amount of pellet deformation as shown in Fig. 16. They attributed this observation to an increase in viscous dissipation. However, they also observed that the effect of binder content is complex. With water as a binder, at low moisture contents, increasing the amount increased the yield stress. However, at greater moisture contents, this resulted in a reduction in the yield stress, whereas with glycerol they found that increasing

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Y.S. Cheong et al.

Fig. 16. The dynamic yield stress as a function of binder content for pellets made from two different sized glass ballotini with water, glycerol (Gly.) or NDBS surfactant solutions. (Adapted from Iveson and Litster [153], with permission.)

the amount always caused the yield stress to increase monotonically. They argued that this complex behaviour is a result of a balance between the interparticle friction, capillary and viscous forces that all resist granule deformation. For example, they suggested that increasing the binder content can reduce interparticle friction by lubrication, whereas capillary forces are increased up to the saturation point. In addition, the viscous forces were considered to increase with increasing binder as more would be squeezed between the primary particles. They suggested that for low viscosity binders, an increase in binder content will increase the capillary forces and hence the yield stress. However lubrication effects should eventually dominate, resulting in a decrease in strength at higher binder contents. For binders with a higher viscosity, the viscous forces should dominate, and hence an increase in binder content will increase the agglomerate yield stress (up to the point at which the agglomerate becomes a slurry). Recently, Fu et al. [156] found that increasing the binder content of wet calcite granules resulted in an increase in the kinetic energy dissipated on impact and hence leading to a reduction in the coefﬁcient of restitution (see Fig. 13). By diluting glycerol with water, calcite granules containing the same amount of liquid binder but with different viscosities were prepared. As shown in Fig. 17, the coefﬁcient of restitution of these granules was found to decrease monotonically with increasing binder viscosity. This was attributed to the effective lubrication by the more viscous binders and so reducing the interparticle friction. However, it should be noted that the magnitude of the friction does not directly relate to the energy dissipation. Actually, high interparticle friction stabilises an assembly resulting in less energy dissipation [160].

Mechanistic Description of Granule Deformation and Breakage

1109

0.25

Restitution coefficient

0.2

0.15

0.1

0.05

0 0

500 1000 Binder viscocity, mPa s

1500

Fig. 17. Relationship between binder viscosity on the restitution coefﬁcient at an impact velocity of 5.86 m s1 made from Durcal 15 (calcite) and a binder ratio of 0.15 [156].

4.1.4. Remarks Despite the promising predictions of the growth regime maps depicted in Figs. 11 and 12, the effects of binder properties on the impact deformation of wet granules are strongly dependent on the systems being investigated. Iveson and Litster [161] commented that even a qualitative prediction of the impact deformation of wet granules and the resulting growth behaviour required a knowledge of the relative contributions of the interparticle friction, capillary and viscous forces. To resolve this problem, it is possible to employ the discrete particle computer simulations of the type described by Lian et al. [74] where the interparticle interactions are parameterised by the particle properties and interfacial interaction between the particles and liquid binder. However, this work was limited to the pendular state for which closed-form interactions laws are available.

4.2. Targeted performance For many granulated products, it is necessary to re-disperse the constituent particles in liquid media for applications involving, for instance, coffee or detergent granules. When two contacting particles without a binding agent being present are immersed in liquid, an additional force known as the solvation force co-exists with the van der Waals and possibly electrostatic forces acting between the particle surfaces. It has been found that this solvation force is repulsive for hydrophilic surfaces, whereas attractive solvation forces operate between hydrophobic surfaces in the case of water [21]. Readers are referred to the comprehensive review of Israelachvili [162] for further information on the

Y.S. Cheong et al. 80

100

70

90 Contact angle (°)

Surface tension (mN/m)

1110

60 50 40 30 20 Pure IPA

10

70 60 50 40 30

0 0

(a)

80

2

4 6 8 10 12 IPA concentration (% v/v)

14

16

35

(b)

40

45 50 55 60 65 Surface tension (mN/m)

70

75

Fig. 18. The variations of (a) the surface tension of the aqueous IPA solutions and (b) the contact angle of the solutions on a polystyrene substrate with increasing concentration of IPA [163].

molecular forces acting on particles in liquids. It follows that the dissociation of granules occurs in a liquid if the solvation forces acting on the constituent particles are repulsive. Hence, the dissociation process can be correlated to the interfacial interactions between the liquid medium and the particle surface. In the experimental work of Adams et al. [163], a reduction in granule fracture strength was observed when binderless granules composed of autoadhesive polystyrene particles were compressed diametrically in liquids of different surface tension (see Fig. 19(a) for results). The liquids were prepared by mixing different amounts of isopropanol (IPA) with pure water to vary the surface tension from ca. 71 to 40 mN m1 as shown in Fig. 18(a). The interfacial interaction between the liquids and the polystyrene particles was determined by measuring the contact angles of the liquids on a compact of the polystyrene particles. It may be seen from Fig. 18(b) that the contact angle decreased progressively with increasing concentration of isopropanol in the aqueous alcohol solutions. The contact angle for pure water on the polystyrene surface was found to be approximately 931 as would be expected for a hydrophobic surface. A reduction in the contact angle implied an increase in the spreading tendency of the aqueous alcohol solutions at higher concentration of isopropanol due to stronger solid–liquid interaction. According to the proposal of Ottewill and Vincent [164], it is believed that the isopropanol molecules adsorb onto polystyrene surfaces with the alkyl chains (CH3CH2CH2–) directed towards the particle surfaces at low isopropanol concentrations. This is because dispersion interactions are favoured between the polystyrene surfaces and alkyl chains, which are both hydrophobic. Under such circumstances, the hydroxyl groups of the alcohol molecules are exposed to the solution phase offering a hydrophilic surface to establish hydrogen bonding with water molecules. This implies that the aqueous isopropanol solutions were effectively interacting on a surface with increasing hydrophilicity capable of polar interactions, as the concentration of isopropanol was increased. Hence, spreading of the solutions was promoted as manifested by a reduction in the contact angle.

Mechanistic Description of Granule Deformation and Breakage

1111 Isopropanol molecule

Fracture stress (kPa)

40 Curved surface of a particle 30 A 20

B

10 Curved surface of a particle

0 40

(a)

REPULSIVE FORCES

45

50

55

60

65

70

75

Water molecule

(b)

Surface tension (mN/m)

Fig. 19. (a) The variation of the fracture strength of polystyrene granules compressed diametrically in aqueous IPA solutions of reducing surface tension (adapted from Adam et al. [163]) and (b) a schematic diagram illustrating the origin of the repulsive solvation forces (adapted from Israelachvili and McGuiggan [165]).

It may be noticed from Fig. 19(a) that there is a weak inﬂuence on the fracture strength of the liquids with surface tensions greater than approximately 42 mN m1. A profound effect is evident when the liquid surface tensions were reduced to less than 42 mN m1. The granules disintegrated almost instantaneously into clusters of constituent particles without the input of mechanical energy when immersed in 15% v/v aqueous isopropanol solution with a surface tension of 39 mN m1. In a ﬂuid environment, the autoadhesion between the constituent particles of the granules is characterised by the solid–liquid interface energy gSL instead of the solid surface energy of polystyrene gS. Hence, the work of adhesion for two adhering solid surfaces in a liquid medium WA0 is given by [19,166,167] W 0A ¼ 2gSL

ð77Þ

and rearrangement of equation (4) gives gSL ¼ gS gLV cos yE pe

ð78Þ

The above study [163] was complicated by the fact that adsorption of the isopropanol molecules on the polystyrene particle surfaces is possible, as discussed with respect to the contact angle measurements. Consequently, it is not justiﬁable to ignore pe to obtain a quantitative variation of the interparticle adhesion, i.e. gSL with the surface tension of the aqueous isopropanol solutions based on equation (78). Qualitatively, the fracture strength remains relatively constant when the granules were immersed in liquids with surface tensions greater than about 42 mN m1. Hence, this may imply that there is insigniﬁcant variation of gSL over this range of liquid surface tensions. Two effects of the presence of ﬂuid molecules on the attenuation of the autoadhesion between solid surfaces were discussed by Kendall [20]. The ﬁrst is relevant to the adsorption of ﬂuid molecules on the solid surfaces due to

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spreading. The consequent result is that a thin ﬁlm of ‘‘contaminant’’ is deposited, which prevents direct molecular contact between the solid causing a reduction in adhesion between the surfaces. This is consistent with the interpretation given above that immersion of the polystyrene granules in the aqueous isopropanol solutions results in adsorption of solvent molecules at the interparticle contacts. In addition to the adsorption of solvent molecules, consideration should be given to the second effect due to solvation forces resulted from the ﬂuid molecules. For a polystyrene granule immersed in aqueous isopropanol solution, one may consider an interparticle contact as sketched in Fig. 19(b). As discussed above, it is assumed that there was preferential adsorption of the alkyl chains of the isopropanol molecules onto the hydrophobic polystyrene particle surfaces. Hence, a hydrophilic layer was created as the hydroxyl groups of the isopropanol molecules were exposed to the solution phase. As a result, it was possible for water molecules to bind to these hydroxyl sites through hydrogen bonding to form an ordered layer of liquid structure. It is this layer of molecules from which the repulsive nature of water solvation forces arises [165]. The formation of new hydrogen bonds between the water molecules and the hydroxyl groups implies a disruption of the strong hydrogen bonding network of water to enable reorientation of the water molecules in region ‘‘A’’ (see Fig. 19(b)). Such a process is energetically unfavourable and might lead to antiparallel alignment of the water molecules in region ‘‘B’’, further away from the hydroxyl groups generating a repulsive force as shown in Fig. 19(b) [165]. Furthermore, the curved surfaces of the polystyrene particles might cause a non-uniform distribution of the number density of water molecules. The repulsive forces might be enhanced in certain regions where the number density was high. Following this hypothesis, strong hydration of the granules was manifested by the sudden rupture of the granules in 15% v/v isopropanol as the polystyrene particles were pushed apart due to the repulsive forces. In more dilute isopropanol solutions, these repulsive hydration forces could attenuate the adhesive forces between the polystyrene particles and thus reducing the granule strength. Conversely, there would not be a disruption of the hydrogen bond network when a granule was immersed in pure water since no hydrogen bonds could be established between water molecules and hydrophobic polystyrene surfaces. Hence, the attenuation in the interparticle adhesion in pure water is more appropriately interpreted as a consequence of water adsorption on the polystyrene particles through dispersion forces. This section demonstrates that the dissociation of granules is strongly dependent not only on the interaction of the particles but also the effect of the surrounding medium. For targeted performances, it may not be always possible to modify the medium and hence, the interfacial properties of the particle surfaces should be tailored to achieve the desired response of granules. Some sophisticated technologies such as reversibly wetting surfaces [168] and stimulus responsive materials [169] offer exciting potential where dispersion of granules can be triggered

Mechanistic Description of Granule Deformation and Breakage

1113

by a wide range of stimuli including pH, temperature and electric ﬁeld. The effectiveness of particle surface modiﬁcation can be accessed qualitatively using simple experiments, such as contact angle measurement.

5. CONCLUDING REMARKS The macroscopic strength of granules is a function of interparticle bonding mechanisms and the packing of the constituent particles. Three main bonding mechanisms namely autoadhesion, liquid bridges and solid bonding are reviewed with a particular emphasis on theoretical descriptions based on the properties of the constituent particles and the binder. With the advent of microscale characterisation techniques such as AFM, lateral force microscopy and nano-indentation, these theoretical treatments may be examined more quantitatively. Surprisingly, these continuum theoretical formulations were found to provide a good approximation for interactions between nanoscale contacts (see [42]). The spatial distribution of primary particles is accessible using X-ray microtomography down to the resolution of approximately several micrometers per pixel with a typical desktop tomographer [170]. However, this technique is restricted to particle and binder systems having clear contrast in the X-ray absorption. Analytical micromechanical models have highlighted the importance of fracture mechanics concepts in correlating the constituent particle properties and particle packing to the macroscopic granule strength. Furthermore, a realistic prediction of granule strength requires the non-uniform stress transmission within granules to be properly accounted for given the complexity of the rupture process. The alternative approach is the use of discrete computer simulations to examine the deformation and fracture characteristics of granules. It has been established that an external force is transmitted along many load-bearing particle chains leaving a large number of redundant interparticle contacts within a granule, which has provided a basis for some theoretical micromechanical modelling. According to DEM simulations, granules are likely to fracture through interparticle sliding across shear weakened planes created along the discrete force transmission paths within the granule dissipating energy as frictional work. It is possible to correlate granule strength parameters such as the dynamic yield stress or the coefﬁcient of restitution to the granule growth behaviour. Practically, the measurement of granule deformation under dynamic conditions provides a means of accessing the outcome of granulation for a combination of formulation and granulator. It should be pointed out that it is difﬁcult to perform experimental parametric studies on the deformation characteristic of wet granules by varying one parameter without affecting other properties of the particle-binder

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system. Perhaps, discrete particle simulation similar to that carried out by Lian et al. [74] should be pursued further by incorporating real structures of granules into DEM computations. Moreover, the appropriate control of the interfacial interaction between a liquid media and the constituent particles of granules enables granule breakage to be tailored. Hence, sophisticated surface treatments that modify such interactions deserve further investigation given the increasing demand in controlled dispersion applications. Nomenclature

A AR a a0 a1 aC c CS D3,2 E E E0 F Fb G G h0 H kn KC Mf, Md, Mr n P P1 PC Pgorge PJKR Psb Pvn

cross-sectional area of a powder assembly (m2) real area of contact (m) contact radius (m) Johnson, Kendall and Roberts (JKR) contact radius at a zero externally applied load (m) contact radius at the end of contact peeling [51] (m) critical contact radius when stable contact peeling terminates (m) critical defect size in a particle assembly (m) concentration of solid dissolved in a liquid bridge (g cm3) Sauter mean particle diameter (m) Young’s modulus (Pa) effective Young’s modulus (Pa) ensemble modulus (Pa) friction force (N) binding force between two contacting particles [99] (N) shear modulus (Pa) effective shear modulus (Pa) minimum separation distance between two spheres (m) hardness (Pa) normal contact stiffness (N m1) critical stress intensity factor (Pa m1/2) masses of feed granules, debris and residue, respectively (kg) width of fragment size distribution (–) normal applied load (N) normal force when interparticle sliding commences [51] (N) pull-off force (N) capillary force due to gorge method (N) effective JKR normal load (N) normal rupture force of a solid bridge (bond) (N) normal viscous force (N)

Mechanistic Description of Granule Deformation and Breakage

Pvt q Q r rN rsb R R s S Sc T TC u0 ue uf vr vn vrel vt V Vb Vf Vi W WA WA0 Wb Ws0 WT, WE, WM, WS

x xc Y z z0 b

1115

tangential viscous force (N) power law index of equation (38) cumulative fragment size distribution (–) local radius of curvature of a pendular liquid bridge proﬁle (m) liquid bridge neck radius (m) neck radius of a solid bridge (m) radius of a sphere or particle (m) effective radius of two spheres or particles (m) spacing factor (–) half-separation distance between a pair of spheres (m) critical half-separation for liquid bridge rupture (m) tangential applied force (N) critical tangential force when stable contact peeling terminates (N) impact velocity (ms1) elastic energy recovered during interparticle contact separation (J) fracture energy associated with interparticle contact separation (J) radial velocity of a ﬂuid under squeeze ﬂow (ms1) normal relative velocity between two spheres (ms1) relative velocity between two spheres (ms1) tangential relative velocity between two spheres (ms1) liquid bridge volume (m3) initial liquid bridge volume (m3) normal component of the critical impact failure velocity (ms1) normal impact velocity (ms1) work done or energy (J) thermodynamic work of adhesion (J m2) work of adhesion in a liquid medium (J m2) Weber number (–) total surface adhesion energy due to Derjaguin, Muller and Toporov (DMT) theory (J) total energy of a pair of spheres or particles in contact, elastic energy stored in the system, mechanical potential energy of the system and surface adhesion energy of the system, respectively (J) fragment size (m) mode size of fragments (m) dynamic yield strength (Pa) distance between two particle surfaces (A˚) equilibrium separation (A˚) half-ﬁlling angle (1)

1116

d eg em g g T, g D , g P g12 gSV, gLV, gSL gS Z j m m0 mS n pe yE rp sf ssb sy tc ti trz x+ , x xm z

Y.S. Cheong et al.

relative displacement between the centres of two contacting spheres (m) granule porosity (–) maximum compressive strain (–) surface free energy (J m2) total, dispersive and non-dispersive surface energies, respectively (J m2) interface energy between two phases (J m2) interface energies between the solid–vapour, liquid– vapour and solid–liquid interfaces, respectively (J m2) surface energy of solid in vacuum (J m2) viscosity (Pa s) contact angle (1) coefﬁcient of friction (–) ﬂow consistency coefﬁcient of static friction (–) Poisson’s ratio (–) spreading pressure (J m2) equilibrium contact angle (1) particle density (kg m3) fracture stress (Pa) fracture strength of a solid bridge (Pa) uniaxial yield stress of a granule (Pa) cohesive shear strength of powder due to Coulomb’s law (Pa) interfacial shear stress (Pa) shear stress of a power law ﬂuid (Pa) upper and lower limits of breakage extent, respectively (%) mean curvature of a pendular liquid bridge proﬁle (m1) Tabor dimensionless group (–)

REFERENCES [1] J. Duran, Sands, Powders and Grains: An Introduction to the Physics of Granular Materials, Springer, London, 1999. [2] P.G. de Gennes, Rev. Mod. Phys. 71 (1999) S374–S382. [3] S.M. Iveson, J.D. Litster, K. Hapgood, B.J. Ennis, Powder Technol. 117 (2001) 3–39. [4] H.M. Jaeger, S.R. Nagel, R.P. Behringer, Rev. Mod. Phys. 68 (1996) 1259–1273. [5] J.C. Williams, Powder Technol. 15 (1976) 245–251. [6] H. Purutyan, P.B. H, G.I. Tardos, Chem. Eng. Prog. 101 (2005) 22–28. [7] B.J. Ennis, J. Green, R. Davies, Chem. Eng. Progress 90 (1994) 32–43. [8] M. Horio, Powder Technol. 1–3 (2003) 1–7. [9] P. Barrett, B. Smith, J. Worlitschek, V. Bracken, B. O’Sullivan, D. O’Grady, Org. Process Res. Dev. 9 (2005) 348–355.

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CHAPTER 27

Descriptive Classiﬁcation: Failure Modes of Particles by Compression Ian Gabbott,a Vishal Chouk,a Martin J. Pitt,a David A. Gorhamb and Agba D. Salmana, a

Chemical and Process Engineering, University of She⁄eld, She⁄eld S13JD, UK b Faculty of Technology,The Open University, UK

Contents 1. Introduction 1.1. Complementary nature of impact and compression tests 1.2. Compression tests 2. Experimental setup 3. Results 3.1. Glass 3.2. Aluminium oxide 3.3. Fertilizer 3.4. Polymethylmethacrylate 3.4.1. Form (1) 3.4.2. Form (2) 3.4.3. Form (3) 3.4.4. Form (4) 3.5. Calcium carbonate granules 3.5.1. Primary particle size 3.5.2. Binder content 3.5.3. Speed of compression 3.6. Wet granules 3.7. Binderless granules 4. Summary References

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1. INTRODUCTION Consider a sphere forced into contact with a rigid plane and initially ﬂattened as shown in Fig. 1a. The force may come from impact or compression. There will be a circular area of contact and stress from the force distributed within the sphere. If there are no discontinuities, this stress will be distributed within the area of contact and Corresponding author. Tel.: +44 114 222 7560; Fax +44 114 222 7501; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12030-3

r 2007 Elsevier B.V. All rights reserved.

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Sphere in forced contact with rigid plane (a)

between two planes (b)

Fig. 1. Sphere in forced contact (a) with rigid plane and (b) between two planes.

reach down in the form of a cone. If compressed between two planes, there will be a corresponding contact area and stress zone on the other side as shown in Fig. 1b. The fundamental difference between impact and compression is the rate at which deformation (i.e. strain) can occur. Thus, lower velocity impact may have the same effect as compression. If, however, the particle has time-dependent characteristics of elasticity or ﬂow, then the speed of application of the force determines if it acts as a brittle body or otherwise. Looking at Fig. 1b, it can be seen that the compressive load will force material in towards the centre of the sphere along the axis of compression. Consequently, there will tend to be a movement of material out from the centre at a plane normal to this. In simple terms, a tennis ball when squashed will bulge out around the middle. This gives rise to tensile rather than compressive stresses, and it is these which typically cause major failure of the sphere. The forces on the sphere are of course matched by equal and opposite forces in the plane, and if the plane is the weaker material it will crack. However, in this chapter we will only consider the failure of the spherical body. If the plane is not rigid, but itself deforms, then the area of contact will increase, spreading the load on the sphere. Thus, Fig. 1a can be approximated by the situation of a sphere compressed between a rigid plane and a deformable one so that sphere failure largely comes from contact with the rigid plane. For a perfectly elastic homogeneous isotropic sphere, it may be shown mathematically that the greatest tensile stress occurs round the edge of the contact circle, and thus a ring-shaped crack is often observed. As this derives from the theory of elastic bodies in contact by the great physicist Heinrich Hertz [1], it is referred to as Hertzian cracking. Further tensile stresses occur within a cone below the contact area, so crazing1 or fragmentation may be observed within this 1 An event caused by very localised yielding at highly stressed regions associated with scratches and ﬂaws in a material which sometimes occurs prior to fracture, most commonly in amorphous, brittle polymers such as polystyrene and polymethylmethacrylate (PMMA). Crazing differs from cracking in that it can absorb fracture energy and even help to increase the fracture toughness of the material.

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region. (Hertz himself observed the complementary situation of ring and cone cracks on a glass plate being pressed with a hard sphere.) For such an ideal particle, we should therefore be able to calculate from the stress tensors and material tensile strength the plane of failure and the force required. However, most particles have imperfections and microcracks causing failure before this point and in less certain directions. Even an elastic particle may be anisotropic in its response to stress – this is true of some minerals, notably sedimentary ones such as limestone or shale. There are two other non-elastic modes of deformation which may occur. First, the particle may be plastic, which effectively means that strain energy is not stored elastically, but is dissipated in ﬂow of material to give permanent change of shape. For some materials, a force applied slowly will give plastic deformation, but the same force applied quickly can give brittle failure. In both elastic and plastic deformation, the sphere’s volume will be largely conserved, meaning that the diameter increases in the plane normal to the axis of compression. In simple terms, both a tennis ball (elastic) and a ball of clay (plastic) will bulge out when compressed. Second, the particle may be to some extent compactable – that is to say capable of densiﬁcation. If the particle is an agglomerate of smaller particles with space between them, then these particles may become packed more closely. In this case, the diameter does not increase. A third situation exists when a spheroid is made up of more than one material, and its behaviour depends on the combined effects. A good mathematical description of the stress ﬁelds involved in these situations is given in a recent paper by Antonyuk et al. [2], who also identiﬁed the breakage phases by means of a force–displacement diagram. However, once a sphere is partially fragmented (at least into two portions), then further loading will cause failure of the fragments in a way which is due to their individual stress ﬁelds and cannot be predicted from the stress ﬁeld of the complete sphere. It should be noted that failure can occur either during the period when a load is applied or when the load is released. In the latter case, the elastic recovery of the material may open microcracks which were held closed during compression. In this chapter, the results of compression tests are described for spheroidal materials demonstrating the different types of behaviour:

Glass which is essentially elastic with brittle failure from microcracks PMMA which is elasto-plastic Aluminium oxide which is compactable Granules which display more complex behaviour because of the interaction of the binder and the constituent particles.

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1.1. Complementary nature of impact and compression tests As mentioned above, the results of impact and compression tests are sometimes similar, sometimes not. According to Beekman et al. [3], ‘‘The major difference between impact and compression tests is that the rate of strain is inevitably high during an impact, but can be controlled during compression. In some aspects though, impact tests and compression tests are complementary. The impact test assesses better attrition and the compression tests the fracture. The impact test is more convenient whereas the compression is more accurate. The impact test chooses random points over the whole surface of the particle and the compression test follows the history of two particular points. The impact test measures the fracture behaviour as a function of the impact velocity and the compression test measures the applied stress directly’’.

1.2. Compression tests A review of uniaxial compression and a discussion on the validity of tensile stress measurement is given by Darvell [4]. These are commonly applied to rock or concrete samples to determine crushing strength, or tensile strength (by splitting a cylinder, sometimes known as the Brazilian test). Various types of materials have been studied in compression tests. Jaeger [5] used sandstone, marble and limestone and found that failure is a result of the extension of cracks along diametral planes. Static compression of two glass spheres between two rigid ﬂat platens has been studied by Gilvarry and Bergstorm [6], Kschinka et al. [7], Shipway and Hutchings [8], and Salman and Gorham [9]. Soil aggregates have been studied by Hadas and Wolf [10] and ceramics by Wong et al. [11]. Quartz sand has been studied by Breval et al. [12] and rubber by Tatara [13]. Agglomerates of various materials have been studied by Antonyuk et al. [2], Beekman et al. [3], Schubert [14], Shinohara and Capes [15], Golchert et al. [16], Thornton et al. [17], Tunon and Alderborn [18], and Sheng et al. [19]. Hiramatsu and Oka [20] derived expressions for the stress distribution in sphere subjected to uni-axial compression. These equations were modiﬁed by Shipway and Hutchings [8] in their study on compression of spheres under uniaxial loading using lead glass spheres. They noted that under compression the glass spheres tended to fracture into wedge-shaped segments, and the fracture initiated at surface of the sphere at a critical value of tensile stress. They also noted that the properties of platens greatly inﬂuenced the fracture strength of the material. Scho¨nert [21] observed the existence of inelastic effects in contact region of glass spheres before breakage takes place. These inelastic effects could be seen as the indentation of cone-shaped volume into the sphere causing

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hoop stress which completely changes the stress ﬁeld. Scho¨nert [21] reported that as the size of a particle decreases, both the particle strength and amount of inelastic deformation will increase. Failure of PMMA particles in compression was studied by Chen et al. [22] and Salman and Gorham [23]. The former [22] observed that in the case of quasistatic compressive stresses, there is clear dependence of peak strength and strain rate. The so-called Hertzian crack, which occurs with elastically deforming material, usually starts with a ring crack. For an impacted or compressed sphere the development of a ring crack around the contact area has been reported by Salman et al. [24], Verral [25], and Kienzler and Schmitt [26], while the plastic deformation associated with conical fragmentation of the material under the contact area has been reported by Rumpf [27], Santurbano and Fairhurst [28], Shipway [8] and Salman et al. [29]. Rumpf [27] related the type of fracture to the size of the sphere. Scho¨nert [21] related it to the particle material and Shipway [8] related the type of fracture with the relative size of the contact area and the type of deformation between the particle and the target or anvil. Different initiation mechanisms have been reported: Hiramatsu and Oka [20] suggest that the compression stresses would be responsible for the crushing near the contact area and the circumferential tensile stresses responsible for the fracture. Rumpf [30] stated ‘‘elastic tensile stresses are always required to initiate a brittle fracture’’. Scho¨nert [21] and Kienzler and Schmitt [26] reported that for elastic spheres, the longitudinal surface stress is responsible for initiating the crack whereas the latitudinal tensile stress is responsible for the plastic case. Shipway [8] reported that for both glass and sapphire spheres (0.7 mm) as elastically deforming targets, the internal shear stress is responsible for the initiation, but the fracture propagation is by the action of circumferential tensile stresses. Antonyuk et al. [2] concluded that where elastic behaviour was dominant, the process starts with concentric ring cracks and high-speed crack propagation within the body. However, if the particle undergoes mainly plastic deformation, then ring tensile stresses are achieved which lead to meridian crack formation by relatively low-velocity crack enlargement. Discrete Element Method (DEM) simulations carried out by Golchert et al. [16] suggest that granule shape, and therefore structure, are important in determining the type and extent of breakage, particularly during processing, rather than the effects of particulate or binder properties. Further, they suggest that the ‘‘network of contacts inside a granular media has a more deﬁnitive effect on breakage behaviour than that of the shape of the object with a load placed upon it between two parallel plates’’. They also argue that since most granules contain primary particles of size range between 1% and 10% of the total granule diameter, existing breakage theorems such as those according to Rumpf [30] and Kendall [31]

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are unsuitable since they assume much smaller primary powder particles. Instead, granule strength is dependent on the weakest link in the system, the strength of the binder bridge [32]. This chapter reports on an extensive series of quasi-static compressive tests to elucidate the failure modes of spheres of aluminium oxide, PMMA and glass, and granules of fertilizer and calcium carbonate. They exhibit distinctive modes of failures that are representative of a much wider range of particles, for example, polystyrene, nylon, hard wood, sapphire and tungsten carbide, aluminium, copper and steel. The results obtained in the quasi-static compression are found to be similar to low-velocity failure modes of these materials obtained in earlier impact studies of these particles [24]. It seems that in impact, surface cracks play a major role, while in slow compression internal cracks may come into play. Similarity between the failure modes of low-velocity impact and slow compression suggest that in low velocity also internal cracks play a major role in the initiation of the failure.

2. EXPERIMENTAL SETUP The compression of spherical particles was carried out using various testing machines; more speciﬁc details of which are given in each section. For particles weaker than glass, the ﬂat surfaces could be made of glass to allow direct observation. However, for glass spheres it was necessary to use ceramic or metal platens. Aluminium oxide (Al2O3) was found to be suitable. The procedure was therefore to apply a load and release it, then observe the results. This was then repeated with a fresh sphere and an increased load. Thus, the effects up to total failure were recorded. In a separate series of experiments, calcium carbonate granules were progressively compressed between steel platens and the process recorded using a Redlake high-speed camera. For spheres of aluminium oxide, fertilizer and PMMA, two glass anvils were used with a microscope for recording the deformation process. A prism was also used underneath the specimen to observe the deformation and crack development. A high-resolution video camera together with U-matic video recording was used to record the failure process for later analysis. For aluminium oxide, each individual particle was compressed to a certain load and then unloaded for examination. Several loads were applied, with the loading range chosen carefully to show the main features of the failure process. The loading range covers from no permanent deformation to catastrophic failure. The video camera was used to record the change in contact area during loading and unloading, together with the crack propagation sequences. (Fig. 2)

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1127 light

glass platen

microscope

glass platen prism Camera

Fig. 2. Video recording of particle failure via prism.

3. RESULTS 3.1. Glass A number of low strain rate compression tests on soda-lime glass spheres between 1 and 12.7 mm diameter were carried out in an Instron machine. The general behaviour is shown diagramatically in Fig. 3. If loading was stopped before catastrophic failure of the sphere occurred, then Hertzian cone cracks were usually found at both ends of the sphere. As in the case of impact [9,24] and indentation [33], chips of material were removed by the growth of lateral cracks probably during the unloading phase. If loading was continued, then the most common form of catastrophic failure occurred by the propagation of meridian plane cracks, which usually divide the specimen into two or more similarly sized segments. An important point to note is that these fragments typically contain sections of the Hertzian cone crack cutting into one or both ends. Often the direction of the cone crack appears to turn towards the axis through an angle approximately 901, an exaggerated form of the change different from the impact failure. These Hertzian cracks do not appear to have lead directly to the catastrophic failure of the sphere, contrary to what has been previously suggested as a mechanism [21]. For weaker specimens (i.e. those failing at a low load) the sphere breaks into segments. For spheres breaking at higher loads, the segments are broken into smaller fragments and are accompanied by a signiﬁcant quantity of ﬁnely comminuted material. A typical example from this higher load region is shown in Fig. 4, which is from a 12.7 mm sphere. The distinctive outline of the fragment shows where material originally along the axis between the loading points, especially near the ends, has been lost. The boundary of this lost region is often part of the Hertzian

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Fig. 3. Various stages in failure of glass spheres under compression: (a) Hertzian ring cracks at both ends with partial extension of cracks C1 near surface; (b) Surface cracks extend so that small portions separate as chips C1 and C2. This probably occurs during the unloading phase by elastic recovery extending the cracks and joining by lateral weaknesses L1. Ring crack R1 penetrates conically. See Fig. 6; (c) The crack along R1 extends to give cone failure under the loading point and splitting of the sphere along one or two planes along the direction of the force. Other fractures within the body are visible. (d) (e) (f) show the typical distribution of increasing fractures within the particle.

crack, leading to a characteristic proﬁle of the ends of most fragments. The direction of the markings on the fracture surface visible in Fig. 4 suggests that the meridian plane crack initiated from a sub-surface origin close to one of the contact areas. Each fragment from a single sphere indicates fracture initiation from the same end. In addition to these large fragments, one or two cones of material from contact regions are often found after these compression tests, especially when the

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Fig. 4. Segment from a compression test of a 12.7 mm glass sphere.

Fig. 5. Compacted cone from a 12.7 mm glass sphere. Base diameter is 2.9 mm.

specimen fails at a high load. A typical example from a 12.7 mm sphere is shown in Fig. 5, and is very similar in shape to those found by Shipway and Hutchings [8] with much smaller spheres. The material in this piece is white in appearance and under the microscope shows evidence of intense fragmentation and compaction. In some specimens the curved surface of the cone or its entire bulk is of very low strength and crumbles to the touch. The original contact surface is ﬂat and often ﬁssured with circumferential and other cracks. Similar features are found in compression tests for all the sizes of sphere tested. However, the average load for catastrophic failure varied from 9500 N for

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1130 L C2

R1 C1

Fig. 6. Hertzian cone crack system in glass.

Fig. 7. The crushing of an aluminium oxide sphere.

the 12.7 mm sphere down to about 200 N for the 1 mm size. Some larger sized specimens failing at the highest loads can end up as fragments which are too small to be recognisable in form. This is the only exception we have found to the characteristic failure pattern along meridian planes in compression tests. Figure 6 shows how smaller chips are released by compression and release. Outside the internal cone R1 a longer chip C1 can be released, and smaller lateral cracks such as L can give rise to smaller chips such as C2.

3.2. Aluminium oxide Figure 7 shows the progressive failure of a single aluminium oxide sphere. In the initial stages of compression, ﬂats were formed at each end of the sphere in contact with glass platens. The ﬂattening of the contact area in the form of circular crushing as shown in Fig. 7a, which is associated with no change in diameter outside the contact area, suggests that densiﬁcation occurred. Eventually, as in normal impact done in previous studies [24], catastrophic failure by cracks along meridian planes occurred. Figures. 7b and 7c show the typical formation of halves and quadrants around a central pulverized cone. Figure 7d shows further crushing of quadrants. Figure 8 shows diagrammatically and photographically the principal failure modes of different particles, all by meridian plane fractures. These were generally halves or quadrants but an occasional particle formed three roughly equal segments as shown in Fig. 8d.

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(a)

(b)

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(c)

(d)

Fig. 8. The various modes of failure of aluminium oxide.

Fig. 9. Shape of aluminium oxide particle ‘‘triple cleft’’.

More complex failure patterns can be observed, but it appears that most of these were a result of secondary damage occurring during the continued loading of the primary fragments after the initial fracturing had taken place. Figure 9 shows a shape analogous to the triple cleft fracture often found in diametrically

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compressed cylinder tests. However, the primary features of failure were very similar to examples of normal impact. Figure 10 summarises in plan and vertical section the modes of failure. The contact points are initially crushed; then a meridian crack starts from the centre, soon after bisecting the sphere. Further compression causes more crushing and a greater area of contact. The hemispheres then themselves split, with the ‘‘triple cleft’’ shape indicating the stress pattern. (In about 9% of cases the sphere splits into approximately three equal segments rather than quarters.) Records of the failure stresses under static compression are summarised in Table 1. The test was terminated when the particle lost contact by the sections falling away. In a minority of cases this was two pieces at an average loading of 320 N. In most cases, the particle remained in contact as the force was increased until further fragmentation occurred. However, in a further minority of cases there was no fracture until the particle fell into three pieces, which required the greatest force. It is postulated that for most particles a signiﬁcant internal weakness effectively deﬁned a meridian plane where failure occurred. Further compression of hemispheres causes symmetrical failure. However, in particles with a sufﬁciently homogeneous centre, there is no preferred meridian plane and the sphere fails

radial crack crushing

crushing

median crack (1) crushing

median crushing crack (2) median crack (1)

crushing

median crack (3)

median crack triple cleft (1)

triple cleft (2)

Fig. 10. The various stages in the compression of aluminium oxide granules: upper set is the plan view, lower set is the side view. Table 1. Results of average load and relative frequency of observation of the four primary failure forms in compression tests. A total of 65 specimens are represented here

Form of failure

Average load

Proportion

(a) Two hemispheres (b) One hemisphere, two quadrants (c) Four quadrants (d) Three equal segments

320 N 350 N 400 N 425 N

9% 43% 39% 9%

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in threefold symmetry at a higher force. It should be noted that the two failure forms involving three primary fragments (Fig. 8b and 8d) are quite distinct and easily distinguishable from each other in both impact and static compression tests.

3.3. Fertilizer Similar fracture patterns are observed in case of fertilizer granules, the composition and more speciﬁc details of which are given by Salman et al. [34]. The normal sequence of events for the granule is plastic deformation along with disintegration on both the ends, then splitting into two planes ﬁrst (Fig. 11a and 11b) and then continued loading of one or both hemispheres to produce patterns as shown in Fig. 11c and 11d. The three-way symmetry of Fig. 8d is occasionally found in compression tests, but one was not observed during its formation. This may be explained by supposing that very few of this sort of granules were sufﬁciently free of ﬂaws to show failure in the strong mode. It was also interesting to note that much of the compression damage under the contact area was produced as loading of the remaining hemispherical and quadrant fragments continued after meridian plane crack formation had divided the granule. Change in the lateral diameter during compressive loading suggests that no densiﬁcation occurred during plastic deformation.

3.4. Polymethylmethacrylate The process of compression of one sphere is shown as frames from a video in Fig. 12. Several damage forms were identiﬁed, and their schematic diagrams are shown in Fig. 13.

R

R

(a)

(b)

R

(c)

median crack

R median crack

(d)

Fig. 11. Sequence of failure modes of fertilizer particle during compression.

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Fig. 12. Frames from video showing the expanding contact area and crack development in a PMMA sphere adapted from Ref. [23].

Fig. 13. Failure mode of PMMA particles shown schematically Adapted from Ref. [23]: upper set is the plan view, lower set is the side view.

3.4.1. Form (1) When the sphere is compressed initially, ﬂat areas are formed at each end of the sphere in contact with the glass platens. By increasing the load, a gradual increase in the contact areas was observed (Fig. 12(i) and Fig. 13a). Upon removing the load, both ends of the sphere were observed to have permanent ﬂat deformation. Radial crazing could be seen very clearly around the contact area (Fig. 14 and 15). As the contact area increases, some of this crazing will fall within it. Due to the compression pressure, some of the cracks within each of these crazing zones begin to heal once they are within the contact area. The compressed spheres do not exhibit any piling up of material by the side of the compression zone. Fig. 17c shows the cone after the material is broken into half.

3.4.2. Form (2) At a higher load, radial cracks could be seen around the contact area (Fig. 13g). These cracks were observed to have developed during the unloading process. These cracks extend radially about the contact surface and internally around the

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Fig. 14. Photograph showing radial crazing in PMMA spheres.

Fig. 15. Photograph showing angel wing structure.

compressed zone underneath the contact area. Two or three cracks could have developed at this stage. By subsequently sectioning the sphere, these radial cracks were seen to have initiated from the upper end of the crack close to the contact area. Generally, these cracks initiate from one side of the sphere. In addition, the radial crack is found to be very similar to the impact damage form B as shown earlier by Salman et al. [24], shown in Fig. 16.

3.4.3. Form (3) Upon further loading, median cracks would develop. By close observation of fragment surface, it was determined that these cracks initiated from the centre of

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Fig. 16. Photograph of impact damage form B, adapted from Ref. [23].

the sphere and then developed outward (Fig. 17b and 17c). These were always bigger in size than radial ones. Again, we found that this crack could develop under both loading and unloading conditions. These cracks lead to the splitting of the sphere into two hemispheres (Fig. 12(iii)). Further loading will lead to the splitting of one or both hemispheres into two segments (Fig. 12(iv)) or a number of segments depending on the number of the median cracks.

3.4.4. Form (4) After the splitting of the sphere by median crack, continued loading on the fragments will lead to development of a crack between the two compressed sides of the sphere. A crack similar to the triple cleft crack was produced (Fig. 13i and 13j). It was observed to have maximum of four clefts, mainly related to the number of segments in the fragments. By increasing the load further, the segments would be fragmented, which demonstrates signiﬁcant plastic deformation. Compression tests have also been carried out on individual complete quadrants from aluminium oxide spheres and PMMA, which results in a very similar proﬁle to those illustrated in Fig. 17. This suggests that most of the failure along the central core of the compression specimen occurs as a ﬁnal stage due to continuous loading of one or more divided segments.

3.5. Calcium carbonate granules Compression tests performed on calcium carbonate granules formed by high shear granulation using a Zwick/Roell materials tester and Red Lake high-speed

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Fig. 17. PMMA fracture surfaces: (a) a median crack surface from failure form h; (b) centre of the sphere where the median crack is initiated; (c) the cone after the sphere has been broken into half.

video camera showed a range of different failure characteristics. The granules discussed in this section were all made by melt granulation in a high shear mixer at 400 rpm with a solid polymeric binder (polyethylene glycol, average molecular weight 1500 Da). Unless stated otherwise, the mass median primary particle size is 40 mm and the granulation time is 10 min. Typically, such granules undergo elastic deformation, followed by plastic deformation and eventually fail by meridian crack propagation. Figure 18 shows the range of failure modes of calcium carbonate granules that have been carefully removed post failure and examined under a microscope. Failure is most commonly catastrophic failure along meridian planes, denoted by a sharp drop in the load–displacement curve, plotted live by the materials tester (Fig. 19). These granules typically split into halves, triple segments or quadrants. The ﬂattened area in the centre of the granules indicates the level of plastic deformation that has occurred during compression, prior to catastrophic failure. Pitchumani et al. [36] reported similar behaviour for spray-dried sodium benzoate granules in the size range 1.2–1.9 mm, which showed irreversible plastic deformation after unloading that could not be described by Hertzian theory. Despite this inelastic behaviour, they argue that the breakage can be classed as

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1 mm

1 mm

1 mm

Fig. 18. Calcium carbonate granules having a solid polymeric binder typically fail by catastrophic failure along meridian planes. Granules commonly split into halves, triple segments or quadrants. 8 Macroscopic fracture

Load, N

6

4

Region A

Region B

2

0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Displacement, mm

Fig. 19. Load–displacement behaviour for a single solidiﬁed calcium carbonate/PEG 1500 granule of diameter 1.3 mm under uniaxial compression. Failure (macroscopic fracture, the mechanics of which are discussed in more detail by Bika et al. [35]) occurs at 6.4 N, having been compressed by just over 0.1 mm. The region after fracture (region B) represents the compression of broken granule fragments and their rearrangement under the moving piston, resulting in a ﬂuctuating force.

semi-brittle because of the brittle fracture of the granules, denoted by a sharp drop in the force, similar to that in Fig. 19. They also suggest that ‘‘local inhomogeneities inside the particles promote microcracking and result eventually in the growth of a median macroscopic crack that splits the particles into two fragments’’.

3.5.1. Primary particle size Figure 20 and Fig. 21 show 2.5 mm calcium carbonate granules having a mass mean primary particle size of approximately 60 mm and were granulated for 30 min. During compression between two stainless-steel platens, the granules

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Fig. 20. Photograph showing meridian crack propagation in a calcium carbonate granule.

Fig. 21. Photograph showing cone residue left on the platen.

were seen to display a change in the horizontal diameter similar to fertilizer granules, followed by meridian crack propagation. In some cases, there was also a cone-shaped residue left on the platen. Figure 22 shows a granule similar in size and made under the same conditions but having much smaller primary particles, of mass median size 2 mm. These granules also displayed plastic deformation followed by meridian crack propagation; however, there was no observed change in the lateral diameter of the granules during compression, and no cone-shaped residue was left on the platen.

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Fig. 22. Meridian crack propagation in granules made with small primary particles. Table 2. Properties of granules having different binder contents

Average single granule failure PEG wt% load, N (S.D.)

Water content – before drying (%)

PEG content – Porosity – after after drying (%) drying (%)

15 35 100

8.62 6.77 0.02

2.1 4.8 12.2

2.50 (0.56) 5.16 (1.14) 9.77 (2.28)

7.8 7.5 1.8

3.5.2. Binder content To investigate the effect of binder content on failure mode, granules with different binder contents were made by granulating using polyethylene glycol and water binder solutions. The binder content could be altered by changing the PEG concentration (PEG wt%) in the solution, since the more volatile component (water) was removed after the granulation by drying in an oven. The resulting properties of these granules are given in Table 2. A comparison of these granules using a high-speed camera to record the failure pattern reveals that the granule binder content has little or no effect on the failure mode. Each of the granules shown in Fig. 23 reveals catastrophic failure by vertical meridian crack propagation, splitting the granule into two halves. There also appears to be little difference in the particle deformation and contact area prior to failure.

3.5.3. Speed of compression During the compression testing of calcium carbonate granules with a solid polymeric binder, it was noted that the failure mode changes with the speed of

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Fig. 23. Compression of dry granules containing different polyethylene glycol binder contents (Table 2) reveals no signiﬁcant difference in failure mode. Each of the granules fails by meridian crack propagation, splitting the granule into two halves.

compression. While the failure mode remains by meridian crack propagation, the number of large cracks along meridian planes increases with compression speed. The amount of plastic deformation of the granule prior to catastrophic failure also increases. This can be clearly seen in Fig. 24, in which the ﬂattened area seen on the plan view of the granules increases in area, supported by side-on images showing that the width of the granules compressed at higher speeds is signiﬁcantly lower.

3.6. Wet granules The strength of liquid-bound granules and agglomerates depends on three forces, namely, interparticle friction, capillary and surface tension forces in the liquid between the particles, and viscous forces in the liquid between the particles [37]. Figure 25 shows photographs of silica glass ballotini (mass mean size 75 mm) pellets bound with a range of different viscosity silicone oils after a total applied compressive strain of 0.4 at a range of strain rates [38]. Iveson and Page [38] reported two observed modes of failure. At low speeds, the pellets were found to display brittle behaviour and fail by large-scale cracking. These specimens were so badly damaged that they fell apart and could not be removed in one piece. Conversely, at high-compression speeds, pellets bound with 1.0 and 60 Pa s silicone oil displayed no visible cracking (Fig. 25b and 25c). The authors suggest that the pellets instead failed by uniform plastic ﬂow and also add that they could easily be removed in one piece after the test. For pellets made with 1.0 and 60 Pa s silicone oil, the failure modes are reported to have changed from brittle to plastic with increasing deformation velocity. It can also be concluded that the average velocity at which the transition from brittle to plastic occurs decreases with increasing liquid viscosity, suggesting that the brittle-to-plastic transition depends on the relative importance of the viscous forces. Iveson and Page [38] also note that the effects described were

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Fig. 24. Calcium carbonate granules compressed at increasing crosshead speeds show both enhanced levels of plastic deformation and number of large meridian cracks at the point of catastrophic failure.

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Fig. 25. Pellets having experienced 10 mm of deformation at a range of speeds and binder viscosities. The initial pellet dimensions were 25 mm height 20 mm diameter. The diameter of the base platen is 38 mm for scale [38].

independent of platen lubrication, particle size and, to some extent, particle shape. In a separate but related work Iveson et al. [39] also reported that when the crosshead of the load frame was stopped, the reported force returned quickly to zero, indicating that little elastic energy was stored in the deformed pellet and that most of the deformation was therefore plastic (Fig. 26). Granules bound with PEG 400, which have a viscosity of 0.12 Pa s show some similarities to the results observed by Iveson and Page under comparable conditions (Fig. 25), in that they eventually fail by large-scale cracking, although it is difﬁcult to tell from their work precisely at which point the brittle to plastic transition occurs. There also appears to be signiﬁcant deformation prior to failure, something not seen for dry granules (Figs. 20–23). Figure 27 shows the compression of water-bound glass ballotini pellets at the pellet-to-pellet boundary. Iveson and Page [40] discovered that pellets with

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Fig. 26. Compression of 1.2 mm calcium carbonate granules bound with PEG 400 at 1 mm/s showing extensive deformation before catastrophic failure.

Fig. 27. Compressed glass ballotini/water pellets showing (a) deformation in the bond zone and (b) deformation in the bulk of the weakest pellet [40].

uniform properties deform (by ‘‘barrelling’’ along the planes of slip) most at the contact area where the two pellets meet (Fig. 27a) but that pellets with high internal porosities or ﬂaws would tend to deform at their weakest part, which was usually close to the middle of one of the pellets (Fig. 27b). They also noted that for water-bound pellets there was little elastic recovery of the bond.

3.7. Binderless granules Submicron polystyrene particles with a mean diameter of 516 nm were granulated with water in a small-scale high-speed food processor (Bosch MCM 5380) with

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modiﬁed impeller for 1 h at 300 rpm [41]. The resultant granules were of high sphericity and in the sieved size range of 3.35–4.0 mm. The ﬁnal binderless granules were produced by drying in an oven at 601C to eliminate the water from the wet masses. Completion of the drying process was determined by no further

Fig. 28. High-speed camera images showing the primary fracture followed by the secondary breakage of a binderless polystyrene granule compressed diametrically at 0.1 mm/ min. Note that the images are numbered to indicate the sequence as the time interval between each image shown is different [41].

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change in mass. Further, the granules were then degassed in a vacuum oven at 251C for 1 h and then conditioned in a 0% relative humidity environment, again at 251C for a minimum of six weeks. At strains beyond the elastic limit the granules experienced irreversible deformation as a result of densiﬁcation at the contact zone. The elastic limit was deﬁned as the point at which the data deviated from the straight line passing through the origin, corresponding to the strain at which the data deviated from the Hertz equation [42]. Granule failure, corresponding to a sharp drop in the load–displacement curve, was captured using a high-speed video camera (Fig. 28). Failure is ﬁrstly by meridian crack propagation, resulting in the splitting of the granule into two hemispheres. Secondary fracture results in each hemisphere splitting into further fragments of varying size and shape. A small cone-shaped fragment of granular material was found adhering to each compression platen after the fractured granule was removed. Since the exact details of crack propagation could not be recorded using a high-speed camera operating at 6000 fps, it was concluded that the fracture process is fast and unstable: a characteristic of brittle failure, consistent with the abrupt reduction in load at fracture displacement. However, the breakage behaviour of these granules is more accurately classed as semi-brittle due to limited but irreversible deformation preceding fracture [43]. This irreversible deformation may result in rearrangement of the constituent particles of the granule near the regions adjacent to the contacting platens.

4. SUMMARY Most of the glass spheres tested ﬁrst display Hertzian cone cracking and under further compression fail catastrophically by the propagation of meridian plane cracks. Glass spheres generally show a cone of inelastic deformation under each contact area, and the meridian plane cracks initiate below this cone. Due to symmetric loading they are driven right through the sphere. The signiﬁcant plastic deformation found in PMMA in compression can be explained by its high-strain rate dependence. For PMMA a close similarity to impact tests conducted by Salman et al. [29] can be obtained if the compression is asymmetric, with soft curve platen on one side and a hard ﬂat one on the other. In this case, an approximately conical region was found at the end loaded by ﬂat platen and as was found in the impact case [29] the meridian crack was initiated from the tip of the cone. For aluminium oxide plastic deformation occurs without any change in the material in the contact area which suggests that densiﬁcation may have occurred. This material shows a combination of small-scale cracking and displacement of

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grains in a conical region under the contact area with some degree of localised ﬂow processes that together can be referred to as compression failure. This conical region of compression failure causes a hoop stress and meridian plane cracks to form. Similar fracture patterns are observed in case of fertilizer granules. The normal sequence of events for fertilizer granules is plastic deformation along with disintegration at both ends, followed by splitting into two planes and then continued loading of one or both hemispheres resulting in failure by cracks along meridian planes. The calcium carbonate granules fail by ﬂattening of the contact area. This ﬂattening may form a compacted cone followed by meridian cracks. It is thought that during compression the energy generated in the compression is dissipated when the conical volume is indented into the granule structure generating a wedging action, thus inducing tensile hoop stress along oblique and meridian planes leading to fracture of granules. Internal tensile stress may be responsible for the lateral increase in diameter. The failure pattern remains more or less similar for roughly spherical granules. However, the loading at which failure occurs will depend on factors such as type and amount of binder, primary particle size and manufacturing process conditions. There have also been a large number of careful experiments on annealed spheres, hemispheres and quadrants which gives clear evidence of a different mode of failure. Meridian plane cracks form ﬁrst, with subsequent fragments, including the characteristically shaped region of crushing along the axis that was also seen by others, occurring as secondary failures. If cone cracks occur, they are contained within segments, and do not directly contribute to fragmentation processes. For wet granules there were two observed modes of failure, namely brittle and plastic. At low speeds, the pellets were found to display brittle behaviour and fail by large-scale cracking. Conversely, at high-compression speeds, pellets instead failed by uniform plastic ﬂow. For binderless granules, at strains beyond the elastic limit they experienced irreversible deformation as a result of densiﬁcation at the contact zone. Failure of these granules is ﬁrst by meridian crack propagation, resulting in the splitting of the granule into two hemispheres. Secondary fracture results in each hemisphere splitting into further fragments of varying size and shape. A small cone-shaped fragment of granular material was found adhering to each compression platen after the fractured granule was removed.

REFERENCES [1] H. Hertz, J. fur die Reine und Angewandte Math, 92 (1881) 157–171. (English translation: Miscellaneous Papers, Macmillan, 1896, pp. 146–162). [2] S. Antonyuk, J. Tomas, S. Heinrich, L. Morl, Chem. Eng. Sci. 60 (14) (2005) 4031–4044. [3] W.J. Beekman, G.M.H. Meesters, T. Becker, A. Gaertner, M. Gebert, B. Scarlett, Powder Technol. 130 (1–3) (2003) 367–376.

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CHAPTER 28

A New Concept for Addressing Bulk Solids Attrition in Pneumatic Conveying Lars Frye Bayer Technology Services GmbH, PT-PT-ESP, 51368 Leverkusen,Germany Contents 1. Introduction 2. Fundamentals 2.1. Components of pneumatic conveying installations 2.2. A systematic deﬁnition of the term attrition 2.3. Attrition research 2.3.1. Process scale 2.3.2. Single-particle scale 2.3.3. Material scale 3. A new concept for addressing attrition in pneumatic conveying 4. Determination of the process function 4.1. Dilute phase conveying 4.1.1. Approach by Euler-Lagrange 4.1.2. Geometry of the computational domain 4.1.3. Determination of impact conditions from numerical simulations 4.1.4. Cumulative number distributions of the stress conditions 4.2. Dense phase conveying 4.2.1. Stress mode in plug ﬂow conveying 4.2.2. Stress intensity from measurements of the pressure exerted by plugs 4.2.3. Stress intensity from analysis of Discrete Element Methods (DEM) 4.3. Conclusions from the determination of the process function 5. Determination of the material function 5.1. Test material 5.2. Sample preparation and determination of attrition rate 5.3. Single-particle experiments 5.3.1. Experimental setup and parameters 5.3.2. Discussion of experimental results 5.3.3. Experimental validation of process function for dilute phase conveying

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Corresponding author. Tel.: +49 214-30-61928; Fax: +49 214-30-96-61928; E-mail: [email protected]

Handbook of Powder Technology, Volume 12 ISSN 0167-3785, DOI: 10.1016/S0167-3785(07)12031-5

r 2007 Elsevier B.V. All rights reserved.

1150 5.4. Continuum mechanical material properties 5.4.1. Application of existing attrition models on results obtained in single-particle experiments 5.4.2. Hardness and fracture mechanical properties 5.4.3. Thermo-mechanical properties 5.4.4. Experiments on the process scale 5.5. Qualitative model of attrition in pneumatic conveying 6. Bridging the gap between academic research and industrial needs 7. Summary References

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1. INTRODUCTION Pneumatic conveying has long been applied successfully in industry to transport bulk solids. Due to advantages like ﬂexibility in routing, dust-free transport or low maintenance costs, the number of bulk solids that are pneumatically conveyed has constantly increased ever since conveying was ﬁrst applied on an industrial scale. On the other hand, one of the unresolved problems associated with pneumatic conveying is the undesired product degradation due to attrition. The main reason for this is that neither the stresses the particles experience during conveying nor the material properties of the conveyed bulk solids relevant to attrition are known. The same holds for the mechanisms leading to attrition. The fact that attrition signiﬁcantly affects product quality, which in times of the rapidly developing product engineering cannot be tolerated, has brought research attention back to this problem. After brieﬂy discussing the fundamentals of pneumatic conveying (Section 2.1), and a systematic deﬁnition of the term attrition (Section 2.2), the state-of-the-art of attrition research is presented in Section 2.3. Based on the conclusions drawn from this, in Section 3 a new concept which introduces a material and a process function to attrition research is presented. Sections 4 and 5 describe how the process and material functions were determined. With the results obtained, a qualitative attrition model, which is presented in Section 5.5, was developed. In Section 6, the results of academic research presented are discussed from an industrial point of view. Speciﬁc requirements of industry with respect to attrition research are presented and a way to combine academic research and industrial expectations to achieve signiﬁcant progress in attrition research is outlined.

2. FUNDAMENTALS Pneumatic conveying was ﬁrst used on an industrial scale at the end of the 19th century to transport crops [1,2]. As the list of 300 products given by Klinzing et al. [1] underlines, the bandwidth of bulk solids that have been successfully conveyed has constantly increased ever since.

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The success of pneumatic conveying results from its advantages over other transport systems, e.g. mechanical conveyors. In pneumatic conveying, the products are transported in a closed system, which minimizes the danger of product contamination. By using nitrogen, even explosive bulk solids can be safely conveyed. Besides that, pneumatic conveying installations offer a high ﬂexibility in transport route. If necessary, the piping can be adapted to spatial constraints. By employing diverter valves, products can be picked up from as well as transported to different parts of a plant. And ﬁnally, pneumatic conveying systems possess hardly any rotating components, which results in low maintenance costs and a high degree of automatization. On the other hand, a clear disadvantage of pneumatic conveying systems is the energy consumption, which is signiﬁcantly higher than for mechanical conveyors. Furthermore, depending on the application, the conveying distance is limited to somewhere between 1000 and 3000 m. Subject to bulk solids properties and isometry of the conveying lines, pipeline wear as well as attrition of the conveyed particles will occur. These effects can be minimized by a suitable conveying design, to completely eliminate them however is impossible.

2.1. Components of pneumatic conveying installations Pneumatic conveying systems always consist of the same basic components. For establishing the gas ﬂow inside the pipes, a differential pressure generation unit is needed. For this purpose, according to Pahl [3] different types of fans, blowers or compressors are used. If the pressure generation unit is located downstream of particle addition as shown in Fig. 2, the pneumatic conveying system is called a positive pressure system. In case the pressure generation unit is located upstream of particle addition, i.e. behind the separation/storage section, a negative pressure system is obtained. As Wirth [4] points out, the maximum technically achievable pressure difference in negative pressure systems is Dp ¼ 0.6 bar. Such systems are thus limited to short conveying distances and low bulk solids load ratios1 mo10. With positive pressure systems, longer distances can be covered. In mining for example, distances of up to 3000 m are realized at pressure differences of Dp ¼ 4 y 6 bar. One of the most demanding tasks in pneumatic conveying is the technical realization of particle addition as well as the adjacent particle acceleration. Here, it has to be differentiated between negative and positive pressure systems. Besides that, it has to be decided whether the particles have to be metered into the conveying pipe. Based on these technical requirements, numerous technical solutions like vacuum nozzles, venturi feeders, ﬂuid-solid pumps, rotary valves or 1

The bulk solids load ratio m is the ratio of bulk solids mass ﬂow and gas mass ﬂow.

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blow tanks have been developed. Details on their design and functionality can be found in Klinzing et al. [1], Pahl et al. [3] or Siegel [5]. For all conveying systems, the conveying zone consists of piping which is connected with pipe bends, diverter valves, etc. Usually, the pipes are either positioned horizontal or vertical. Inclined pipes are avoided wherever possible. Due to the change in conveying direction in pipe bends, these are subject to increased wear. Besides that, pipe bends have the highest bulk solids attrition rates per unit length in any conveying system. Consequently, with respect to wear and attrition, they are the critical points in every conveying zone. As Fig. 1 shows, in addition to conventional bends (a), different designs like the blinded tee (b) or the vortex drum bend (c) have been developed to minimize wear and attrition. More details on this have for example been given by Hilgraf [6]. Finally, at the end of a pneumatic conveying system, the particles have to be separated from the gas ﬂow. In case of large particles, e.g. granular plastics as depicted in Fig. 1, a silo with a downstream ﬁlter might be sufﬁcient for separation. Besides that, cyclones are employed to separate particles and conveying gas. Details on the design and proper choice of separation units can be found in Refs. [1,5]. Besides the construction of a conveying system, the particle movement inside the conveying pipes is decisive for the amount of particle attrition that is generated. Depending on the process parameters, i.e. conveying gas velocity and bulk solids load ratio as well as particle respectively material properties, different modes of conveying develop. These are shown in Fig. 2. With decreasing gas velocity u or increasing bulk solids load ratio m, the modes of conveying are

Fig. 1. Basic components of pneumatic conveying systems.

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Fig. 2. Modes of conveying for horizontal pneumatic conveying as a function of conveying gas velocity u and bulk solid load ratio m.

passed through from top to bottom. Modes with low bulk solids load ratios belong to the so-called dilute phase conveying regime, while those with high-load ratios are summarized under the term dense phase conveying. In homogenous ﬂow, the particles are almost evenly distributed over the pipe cross-section. Here, the gas velocity is high, while the bulk solids load ratio is low. If the gas velocity is decreased or the load ratio is increased, the energy is no longer sufﬁcient to keep all particles suspended in the gas stream. As a consequence, in horizontal conveying a particle layer forms at the bottom of the pipe. This leads to a reduction in free cross-section and thus to an increase in gas velocity. Consequently, the particles above the layer are still conveyed in homogenous ﬂow, while the particles in the layer move much slower or do not move at all. In this case, a so-called stationary layer has formed. A further decrease in gas velocity or increase in load ratio leads to dune ﬂow. The dunes form and move through impulse transfer from particles which are still suspended in the gas ﬂow and impact on the particle layer. The dunes can transform into plugs, which ﬁll the complete pipe cross-section. Finally, if the whole pipe is ﬁlled with particles and the bulk solid possesses good air retention capabilities, a moving bed-type ﬂow can be established. In this case, the particles are ﬂuidized and move ﬂuid-like through the conveying pipes. As Wirth [4] has shown, the modes of conveying which have been discussed for horizontal conveying are found in vertical conveying as well. Weber [7] developed a state diagram of pneumatic conveying, which correlates the pressure drop per unit length in a conveying system with the conveying gas velocity for different bulk solids load ratios and thus allows the identiﬁcation of ﬂow modes as a function of process parameters. Since bulk solids properties have a signiﬁcant

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inﬂuence on the conveying characteristics of the material, Weber’s state diagram is not universally applicable, but has to be determined individually for each bulk solid that is to be conveyed.

2.2. A systematic deﬁnition of the term attrition As the discussion of the different modes of conveying in the previous section has shown, the transport of particles and gas through the conveying pipes takes place in a complex multiphase ﬂow. In the course of this transport interparticle as well as particle-wall contacts result in stress events, which lead to undesired particle attrition as well as pipeline wear. At ﬁrst, product degradation due to attrition was tolerable since the products conveyed were mainly crop, coal or ore. But the increase in number of bulk solids that are conveyed as well as higher demands on product quality in the course of the rapidly developing product engineering directed research focus back onto attrition formation. The negative effects of attrition are manifold. Attrition can lead to changes in the particle size distribution of bulk solids, in their speciﬁc surface, their density or ﬂowability [8,9], which in the worst case leads to plugged conveying pipes and consequently to the failure of the conveying installation. Besides that, attrition might lead to increased capital costs, i.e. if additional separation units have to be installed to separate attrition fragments from the excess air. A fundamental problem of attrition research is that different deﬁnitions of the term attrition exist in literature. Thus, depending on the point of view, different physical phenomena are subsumed under the term attrition. A systematic deﬁnition of the term attrition that is introduced in the following paragraphs will show that this does not necessarily have to be a contradiction. Besides that, this deﬁnition will be used to structure the discussion of published research on attrition and furthermore is regarded essential to obtain universally applicable research results. In general, the ﬁnes that occur in a bulk solid as a consequence of pneumatic conveying are named attrition. These ﬁnes result from material removal of the bulk solid particles. Depending on the physical scale on which attrition is regarded, different mechanisms are responsible for attrition formation. To obtain a coherent, systematic deﬁnition of attrition, the multiscale approach developed by Peukert [10] for comminution is transferred to attrition processes. Here, advantage is taken of the fact that comminution and attrition are connatural processes which at ﬁrst sight only differ such that product changes due to comminution are desired, while they are undesired as a consequence of attrition. On the process scale, attrition can be deﬁned as the undesired damage of disperse bulk solids as a consequence of mechanical stresses that lead to the

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formation of fragments. The stresses the particles experience are deﬁned by stress mode, stress intensity and the number of stress events. These so-called stress conditions induce a material-speciﬁc response, which leads to the occurrence of attrition. On the process scale, a clear separation of stress conditions and material inﬂuence is thereby impossible. Such a separation is permitted by looking at the single-particle scale. On this, particles are characterized by an integral particle strength as well as their size and form. If single particles are stressed in a well-deﬁned manner, a stress state is induced in the particle. In case these stresses exceed the particle strength, the material fails. In contrast to comminution, where the induced stress state extends throughout the whole particle, the stresses leading to attrition are usually locally conﬁned. Consequently, it can be differentiated between breakage and attrition on the single-particle scale. Breakage covers the whole particle which completely disintegrates into fragments. In attrition on the other hand, only part of the particle is stressed. Through this, attrition fragments are formed at corners or edges, the mother particles however remain largely intact. The already mentioned integral particle strength is a result of distribution and individual strength of imperfections in the particle volume. The strength thus seems suitable for the characterization of particle breakage. For attrition on the other hand, it does not appear to be the right property since in this case the stressed zones are located close to the particle surface. Whether breakage or attrition occurs on the single-particle scale mainly depends on stress intensity and the particle resistance to this stress. It can thus happen that a product degradation identiﬁed as attrition on the process scale is in fact caused by breakage on the single-particle scale. On the material scale, breakage as well as attrition fragments are formed by cracks that propagate through the particles. On this scale, the material is characterized by continuum mechanical properties like modulus of elasticity, tensile strength or fracture toughness. To reliably measure these properties, stress conditions that posses exactly deﬁned stress and deformation states are required. The above material properties are not dominated by imperfections. Consequently, they appear suitable for describing the attrition phenomena deﬁned in the discussion of the single-particle scale, since the particles are – as stated before – only partially stressed. As a consequence, attrition formation is not expected to be dominated by imperfections, and is likely to be described by continuum mechanical properties. On the molecular scale, both attrition and breakage lead to the failure of chemical bonds or intermolecular interactions. On this scale, a differentiation between both phenomena is no longer possible. A detailed analysis of this scale is beyond the scope of this chapter. Consequently, only the above three scales will be analysed.

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2.3. Attrition research An analysis of the work that has been published in context with attrition shows that many researchers only focus on one of the four scales of attrition, i.e. process, single particle, material or molecular scale. An investigation of attrition phenomena across the different scales is however missing. This has lead to numerous different approaches to attrition, especially on the process and single-particle scales. The reason for this has been strikingly named in a publication by the British Materials Handling Board [8]. There, it has been stated that a simple model experiment is best suited to describe and quantify attrition formation, if it is of elemental nature. These experiments are usually based on clearly deﬁned stress conditions, which however might vary signiﬁcantly in intensity and number of stress events from the industrial pneumatic conveying process. As such, a transfer of the results obtained in the experiments to the industrial process cannot be always ensured. Findings from experiments, which in their setup are closely related to industrial processes, usually cannot be generalized even if they provide good results for a certain industrial application. This is due to the fact that in such experiments, the stress conditions, i.e. mode, intensity and number are distributed and furthermore largely unknown. In particular, the previous paragraph clariﬁes that the isolated view of one of the physical scales of attrition is not sufﬁcient to obtain universally applicable results. Moreover, the development of a physical model of attrition will only be possible if attrition formation is understood on all four scales. In the following discussion of the published scientiﬁc work on attrition, the previously deﬁned physical scales of attrition will be used for structuring.

2.3.1. Process scale An extensive study on the process scale was carried out by Wypych and Arnold [11], who solely focused on the optimization of process and plant parameters to reduce attrition. Bends with a special geometry, e.g. vortex elbow, gamma bend or blinded-tee bend are presented, which reduce attrition compared to common long- or short-radius bends. In these, the bulk solid does not directly impact the bend wall anymore. Instead, the impacts take place on a stationary product layer. Besides that, the potential for optimization of particle addition, pipe diameter and modes of conveying with respect to attrition is discussed. Kalman [12] comes to similar conclusions with respect to possible optimizations. However, in contrast to Wypych and Arnold [11], he considers taking advantage of attrition as a unit operation. As such, he proposes to convey a bulk solid that is to be comminuted in a subsequent process step under conditions that promote attrition. The aim is to reduce the energy consumption in comminution. Furthermore, he investigates the effect of different bends as well as of process parameters on attrition.

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Konami et al. [13] determined the inﬂuence of the amount of ﬁnes on attrition. For this purpose, plug ﬂow conveying experiments were carried out in a pilot plant installation. These authors show that a higher amount of ﬁnes leads to lower attrition rates, which is explained with a damping effect of the ﬁne particles. Apart from that, they observed decreasing attrition rates with an increasing number of passes of the bulk solid through the conveying system. This was explained with an increasing roundness of the particles. Finally, they found that the pipes were blocked when a certain amount of ﬁnes was exceeded. Both effects are commonly known from industrial conveying systems. In Ref. [14], Coppinger et al. compare attrition tests carried out for dilute and dense phase conveying with experiments in a ﬂuidized bed, with a vibrating sieve which contains four steel balls as well as with compression tests. This comparison leads to qualitative relations between the measured attrition rates. Due to the undeﬁned and from test method to test method varying stress conditions, a generalization of the results is not possible. This is also the case for the approach given in Ref. [15] to use a vibrating sieve for the characterization of different bulk solids. As a result of such experiments, differences in attrition rate can be detected, but the transfer to bulk solids handling unit operations appears questionable due to the complex and undeﬁned stress conditions on the sieve. The approach followed by Neil and Bridgwater [16,17] appears more promising. They conducted experiments in a shear cell, in which the ﬁlling height of the bulk solid was chosen in such a way that an attrition zone stretching across ﬁve to six particle diameters was formed. The experiments were conducted under deﬁned normal stresses and shear rates. Afterwards, the mass of attrition was determined as a function of stress conditions by sieving. Through this experimental procedure, the particles were stressed under conditions relevant for numerous technical applications. The authors show that the ratio of mass of attrition and initial mass of particles yA can be calculated according to equation (1), which is an extension of an approach developed by Gwyn [18]. Herein, A, f and b are the empirical parameters, s is the normal stress and sc the stress at which single particles of the investigated materials failed in compression tests. The parameter m goes back to Gwyn and reﬂects the change in attrition rate with time. st f yA ¼ A sc

!b with

m ¼ bf

ð1Þ

The term in brackets in equation (1) is a measure for the fraction of total energy that leads to attrition. The parameter f was identiﬁed as an attrition parameter, which is constant for a given material. b quantiﬁes the velocity of material degradation. In Ref. [17], Neil and Bridgwater also consider results from experiments in a ﬂuidized bed and a screw pugmill and show that attrition formation in these two devices can also be described with the above approach. Furthermore, it was found

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that m was constant regardless of the unit operation. In the eyes of the authors, this raises hope that the presented approach is suitable to identify material properties to characterize the attrition behaviour, which are independent of the unit operation. The studies presented exemplify that it is difﬁcult to obtain universally applicable results from attrition experiments on the process scale. In most cases, it is not sufﬁcient to simply adapt the stress conditions in model experiments to those of the process of interest. They also have to be as deﬁned as possible. As Neil and Bridgwater [16,17] have shown, this might lead to results that can be generalized. However, the deﬁned stress conditions, which are usually realized in single-particle experiments, have to reﬂect those of the industrial processes. How this can be achieved will be discussed in the following section.

2.3.2. Single-particle scale Rumpf [19] introduced a differentiation between the stress modes in mills, i.e. impact/friction, compression/shear and non-mechanical stresses, which are the basis for many experimental studies on the single-particle scale. Since the stress modes in comminution and attrition are identical, similar approaches have been followed in attrition research. However in this case, the stress intensities and usually also the number of stress events is much lower than in comminution. Most of the studies can be attributed to the stress modes I (compressive stress) and II (impact) as deﬁned by Rumpf [19]. Besides that new approaches from tribology exist, where single particles are stressed by sliding friction. These studies were analysed since sliding friction is one of the stress modes possibly responsible for attrition in plug ﬂow conveying. Extensive studies to analyse attrition formation due to particle-wall impacts were carried out by the research group of Ghadiri. In Ref. [20], it was shown by using high-speed imaging that the mechanisms leading to attrition of sodium carbonate monohydrate crystals vary as a function of impact velocity. Below a threshold velocity, no attrition was observed. With increasing velocity, Chipping occurred, which is characterized by attrition fragments shaped like small plates which are formed by uniﬁcation of cracks located close to the surface. At high velocities, particle breakage, so-called Fragmentation is observed. Here, cracks propagate through the whole particle which is completely destroyed (see also Ref. [21]). Furthermore, a strong inﬂuence of particle orientation on attrition induced by impacts was found by Cleaver et al. [20]. As expected, corners and edges are most sensitive to attrition. The above change in attrition mechanisms was also found for extruded polymethylmethacrylate particles by Papadopoulos and Ghadiri [22]. In contrast to the crystals, plastic deformation at the contact interface was found for the PMMA particles at low velocities, which did not lead to any detectable amount of attrition. Experiments on the single-particle scale have the advantage over experiments on the process scale in that due to the deﬁned stress conditions a correlation of

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experimental results and intrinsic material properties is possible. In 1992, Ghadiri and Zhang developed an attrition model, which has been described in Refs. [23,24]. This model is only valid for materials that show a semi-brittle material behaviour, which is characterized by local plastic ﬂow at the contact points, i.e. corners and edges, where the yield stress is exceeded. The model is based on ﬁndings from fracture as well as indentation fracture mechanics. It is assumed that attrition formed due to impacts can be approximated by quasi-static indentation of the bulk solid with a hard indenter. In this context, it was taken advantage of the fact that the crack morphology caused by impact stresses resembles that caused by stressing a surface with an indenter [25,26]. Another assumption is that attrition fragments due to chipping are only formed through lateral cracks, which are located close to the surface. Furthermore, the contact parameters needed for modelling are approximated from hardness measurements (cf. Refs. [23,24]). As a result, the amount of attrition given as the ratio of volume of attrition Va and initial particle volume V can be calculated according to equation (2) from the particle density rp, the particle diameter xp, the hardness H, the fracture toughness Kc and the impact velocity of the particle vp,i. a is a ﬁtting parameter. rp x p H 2 Va ¼a v p;i V K 2c

ð2Þ

With this model, these authors achieve a separation of process and material parameters for normal impacts. However, it is assumed that the number of stress events does not have any inﬂuence on the attrited volume. Fatigue due to repeated impacts is thus disregarded. It could however be implemented by introducing hardness as a function of the number of stress events [24]. The transfer of the above model to pneumatic conveying has been given by Taylor [27]. He shows that equation (2) is suitable to qualitatively reproduce the attrition observed in pneumatic conveying. However, the fact that no adaptation of the stress conditions to those in the pneumatic conveying systems was carried out prevented a possible quantitative description. A similar correlation to that given in equation (2) was developed by Gahn [28] to model attrition that occurs in suspension crystallizers. As Ghadiri and Zhang, Gahn models the impacts between crystals and the stirrer blades as normal impacts. By analysing the stress state in the crystal under simplifying assumptions – mainly that the hypothesis by Rittinger is valid – Gahn develops a correlation between the kinetic impact energy Ei,kin and the size a of the plastic zone in the crystal for the contact of a hard indenter with a soft (crystal) surface as well as for the contact of a soft indenter, i.e. crystal corner with a hard surface. p E i;kin ¼ H V a3 8

ð3Þ

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Equation (3) serves as a foundation for the derived correlation to model attrition, which is given in equation (4). Like a, a is a ﬁtting parameter. 4=3

2=3

rp x p H V 8=3 Va ¼ a v V G bmax p;i

ð4Þ

As Gross and Seelig [29] have shown, the product of shear modulus G and speciﬁc fracture plane energy bmax is equal to the square of fracture toughness Kc, if linear elastic material behaviour is assumed. Consequently, with only slight deviations in the exponents, the material inﬂuence is reﬂected identically in the models of Gahn [28] and Ghadiri and Zhang [23]. This is not surprising, since both models are based on hardness measurements. The research group of Salman [30,31] also conducted numerous single-particle experiments. Due to the low toughness of the granular fertilizer particles, the stress intensities chosen by the authors lead to particle breakage. In the singleparticle experiments, the inﬂuence of impact velocity, impact angle and particle size on attrition was investigated. In parallel, pictures of particle breakage during impact were taken to identify the breakage mechanisms as a function of the varying stress conditions. The decision whether a particle was broken or not was based on a visual inspection. For each impact angle respectively particle diameter, the resulting number distributions of intact particles obtained for the different impact velocities were approximated by a Weibull function (cf. equation (5)). Herein, np,int is the number of intact particles while c and m are the ﬁtting parameters of the Weibull function. v m

np;int ¼ 100e

p;i

c

ð5Þ

The parameter m is independent of impact angle and particle size. The parameter c on the other hand is a function of those two quantities. In combination with a two-dimensional ﬂuid dynamics model, the particle trajectories in a pipe bend of a pneumatic conveying system were calculated. With the stress conditions that were determined from these, the number of intact particles was calculated according to equation (5). The comparison with experimental result showed a good agreement, and the assumption underlying the calculation that fatigue does not play a role in attrition of the investigated particles was validated by the experimental results. The Weibull function is also used by Vogel [32] to describe the impact comminution of particles. By combining dimensional analysis and fracture mechanical approaches, Vogel develops a correlation for calculating the breakage probability PB. In equation (6), the inﬂuence of material properties and process parameters is considered separately. If the breakage probability is interpreted as the failure of the weakest link in a chain [29,32], the chain length can be set equal to the particle diameter xp. The external stress can be calculated from an energy

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balance. Wm,kin is the mass-speciﬁc kinetic energy at the moment of impact, whereas Wm,min is an energy threshold below which no comminution takes place. This parameter was introduced by Vogel based on experimental results. Finally, fMat is the reciprocal value of the material resistance against the external stress, while k is the number of stress events, i.e. impacts. P B ¼ 1 ef Mat xp k ðW m;kin W m;min Þ

ð6Þ

With the approach of equation (6), Vogel succeeds in quantitatively describing the comminution result obtained in different mills. The required parameters fMat and Wm,min were determined in impact experiments on the single-particle scale. However, Vogel points out that these material properties which reﬂect the material-speciﬁc reaction on comminution are particle properties. Since parameters like particle shape as well as number, form and distribution of dislocations respectively imperfections are also of inﬂuence on the comminution result, comminution cannot be solely described by continuum mechanical properties. Whether the results can be used to describe attrition has to be questioned since the stress intensities occurring in attrition in most cases lie below Wm,min. Consequently, it is expected that these processes are not dominated by imperfections and dislocations, but can be described by continuum mechanical properties only. Besides the studies concerned with the impact of particles, single-particle compression tests were also used to try and describe breakage and attrition behaviour. In such experiments, the stress intensities can usually be adjusted more accurately than in impact experiments. Compression tests are regarded as a suitable experimental setup since as Gildemeister [33] and StieX [34] demonstrated experimentally as well as theoretically for PMMA particles, the stress state induced in the particles during compression resembles that in impact experiments. These ﬁndings have been validated more recently by results presented by Shipway and Hutchings [35,36]. Goder et al. [37] examined, whether compression tests can be employed to determine the resistance of particles against attrition respectively breakage. Although these tests were conducted with particle collectives, they were classiﬁed into the single-particle scale since the stress conditions are deﬁned and the absolute particle numbers are still small compared to industrial scale pneumatic conveying installations. In the experiments, the particle failure due to repeated stressing was investigated. The results were visualized in fatigue diagrams, which depict the fraction of destroyed particles as a function of compression force and number of stress cycles respectively events. Through the experiments, it was demonstrated that fatigue can be important in bulk solids handling. Beekman et al. [38] also use compression tests to characterize the failure behaviour of particles. However, in contrast to Goder et al. [37], single particles are stressed since through this, distributions in particle strength can be resolved. These authors show that structural differences in the investigated granules are

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reﬂected in the results of the compression tests. Besides that, the signiﬁcance of fatigue is stressed. It is thus not sufﬁcient to measure the initial particle strength, this value has rather to be determined as a function of number of stress events, since repeated stresses are the normal case in bulk solids handling. The results that will be discussed in the following paragraphs stem from a comparatively young research discipline called tribology. According to Jost [39], the term tribology was introduced in 1966 by the Department (Ministry) of Education and Science. It was derived from Greek tribos ¼ rubbing. The main focus of tribology lies on research and technology of contacting surfaces in relative motion. Tribology thus covers friction including lubricants and the corresponding interactions at the contact interfaces of solids as well as solids and liquids or gases. The scope of tribological research is the optimization of technical solutions by reducing energy and material losses due to friction and wear. If the general deﬁnition of tribology is used, attrition can be regarded as a tribological process. This is however limited to the stress mode sliding friction, which presumably dominates attrition formation in plug ﬂow conveying. The transfer of tribological knowledge to attrition processes could therefore help to achieve progress in the research on attrition in dense phase conveying. What happens at the contact interface during sliding friction is still largely unknown. As Krim [40] asserts, most of the knowledge about friction still goes back to the friction laws established by da Vinci, Amontons and Coulomb. According to these, the friction force, which is directed opposite to the motion of two surfaces in sliding contact is proportional to the normal force both surfaces are pressed against each other. Furthermore, the friction force is assumed to be independent of the contact area and the sliding velocity. Up to now, it was disproved that friction is primarily caused by the surface roughness of the contacting surfaces. The assumption that friction always leads to wear and attrition was also identiﬁed as being wrong [40]. The same holds for the adhesion of molecules. It is rather assumed that friction is caused by oscillations of the atomic grids of the surfaces, which are induced by the relative motion of the surfaces [40,41]. Part of the mechanical energy needed for maintaining the relative motion between the contacting surfaces is dissipated by these so-called phonons ﬁrst by conversion into sound and then into heat. As a result, the macroscopic friction laws cannot be applied on an atomic scale. Here, a generalized deﬁnition of friction laws is needed, which has been described in Refs. [40,41]. An important conclusion drawn from this is that the atomic friction force is directly proportional to the microscopic contact area, which is determined by the surface asperities in contact. This area usually differs signiﬁcantly from the macroscopic contact area. It is beyond dispute that friction is responsible for attrition in a tribological system. By reproducing tribological stresses in the so-called tribometers, it was possible to identify different attrition mechanisms. According to Czichos and Habig [42], these are Surface Disruption, Abrasion, Adhesion and Tribochemical

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Reactions. The latter however do not play any role in the attrition of polymers. Attrition due to surface disruption is caused by repeated stressing of a surface by the asperities of the contacting surface. Due to the elastic stresses that are induced, the formation, accumulation or growth of cracks close to the surface is initiated, which ultimately leads to the formation of attrition fragments, which in shape resemble small plates. A theoretical explanation for the occurrence of these fragments was given by Suh [43] in his delamination theory. Attrition due to abrasion always occurs when one of the contact materials is much harder than the other one. The surface asperities of the harder material penetrate into the softer surface and lead to material removal. For this, different mechanisms of material removal are responsible [42,44,45]. Micro-ploughing leads to the formation of furrows in the softer surface, which are created by the surface asperities of the harder material. A repeated stressing of the soft surface ﬁnally leads to attrition formation due to fatigue. In micro-cutting, a chip is formed ahead of the asperities ploughing through the material. In addition, if the materials are brittle, micro-fragmentation can occur. Finally, in adhesion the high local contact pressures at the asperities lead to contact forces exceeding the strength of the atomic bonds of one of the contact partners, which results in a material transfer. Since for a long time, tribological studies were mainly focused on metallic contacts as observed in bearings, the above mechanisms were at ﬁrst validated for metals. But as Uetz and Wiedemeyer [46] as well as Yamaguchi [47] have shown, the same mechanisms can also be observed in tribological systems in which one or even both of the contact partners are polymers. The discussion also exempliﬁes the complexity of the processes leading to attrition in sliding contacts. Despite numerous approaches, so far this has prevented the successful simulation of attrition formation. A detailed discussion of the various approaches was given in [48] as well as by Meng and Ludema [49]. Like the previously deﬁned physical scales of attrition (cf. Section 2.2), the results of tribological studies discussed above show, how experiments on the single-particle scale can serve to identify attrition mechanisms. The results obtained were veriﬁed by employing high-resolution methods like atomic force microscope (AFM) or scanning tunnelling microscopy (STM), which are usually applied in the ﬁeld of micro- and nano-tribology [44]. This clearly illustrates that a deeper understanding of attrition can only be obtained, if it is analysed on different scales, i.e. process, single particle and material scales.

2.3.3. Material scale In contrast to the studies presented so far, research on the material scale is usually of a theoretical nature. This mainly arises from the fact that the processes leading to attrition can hardly be measured on the material scale. Consequently, models

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were developed to try and correlate continuum and fracture mechanical material properties with the results obtained in experiments on the single-particle scale. Potapov and Campbell [50] model the crack propagation in two-dimensional discs, which are stressed by impacts. These discs are built from stiff triangles which are connected by elastic strings characterized by a normal (Kn) and a tangential (Kt) stiffness. Through this, the discs are modelled as ﬂexible. For initiation of crack propagation, a boundary tensile stress st,cr is deﬁned, which leads to failure of the strings. The energy release rate of this failure is set equal to the energy stored elastically in the string. By dimensional analysis, Potapov and Campbell identiﬁed three parameters and studied their respective inﬂuence on attrition. These parameters were deﬁned with regard to relationships that are relevant in comminution. The ﬁrst one, given in equation (7) is the ratio of kinetic energy Ekin and the minimum energy Ecrack,min required to initiate a crack that propagates through the whole particle. The higher this ratio, the higher is the awaited particle destruction. E kin E crack;min

¼

p rp x p K n 4 s2t;cr

ð7Þ

The second parameter is the ratio of impact velocity vp,i and the speed of sound c of elastic waves in the particle. This goes back to Charles [51], who found experimentally that a single impact with a high velocity leads to a different breakage pattern than a single impact with the same total energy at a lower velocity. The last parameter is the Poisson ratio n. The simulation results show that the Poisson ratio does not have any inﬂuence on the breakage pattern. With increasing vp,i/c however, an increasing asymmetry of the fragments, which is reﬂected in a higher aspect ratio,2 was observed. These ﬁndings are in line with experimental results by Charles [51]. According to Potapov and Campbell, the inﬂuence of the dimensionless kinetic energy on the size distribution of the fragments as given by equation (7) corresponds qualitatively with experimental results. During the ﬁrst phase of contact, only cracks close to the surface are observed, which do not propagate into the particle but to the particle surface. This could be an indication that the simulations are capable of modelling the chipping mechanism identiﬁed by Ghadiri (cf. Ref. [23]). In a later study, Potapov and Campbell [52] investigated the attrition behaviour of particles under shear stresses. A comparison of the results with the ﬁndings by Neil and Bridgwater [16], showed a good agreement. Yet another result of the simulations is that the attrition rate is proportional to the total energy applied by the external stresses. Thus the material parameter introduced by Neil and Bridgwater, which in the experiments was determined to be 0.26of r1, has to take on the value f ¼ 1. It is however doubtful that the conclusion drawn by Potapov 2

The aspect ratio is the ratio of the largest length and the largest width of the fragment.

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and Campbell, i.e. that the attrition rate only depends on the energy brought into the system and thus is independent of attrition mechanism, stress mode or plastic ﬂow, really holds in this generalized form. Thornton et al. [53] utilized discrete element methods (DEM) to simulate the breakage behaviour of agglomerates under impact stresses. The agglomerates, which contain imperfections and dislocations are formed from elastic elements circular in shape, which only touch in a single contact point. Friction and adhesion due to surface energy are accounted for in the two-dimensional simulations. In agreement with the results obtained by Potapov and Campbell [50], it was found that for impact velocities vp,i ¼ 1 m/s a compression wave originates at the contact point and propagates through the agglomerates. At the front of the wave, only small regions experiencing tensile stresses exist, while the majority of the contact points between the discs the agglomerates are formed of experience compression stresses. This leads to a plastic deformation of the agglomerates. At the rear end of the wave, tensile stresses occur which destroy the contacts in the plastic zone and lead to the formation of small fragments. The plastic zone is much smaller for lower impact velocities. For these, cracks propagate from the contact point to the agglomerate surface and the material behaviour characterized as semi-brittle by Ghadiri and Zhang [23] is found. Below a boundary velocity, which increases exponentially with the strength at the contact points, the agglomerates show no breakage at all. The extension of the DEM simulations to three dimensions is introduced by Kafui and Thornton in [54]. Due to the higher complexity, no imperfections and dislocations were implemented in the agglomerate structure. The main ﬁnding from these simulations was that an impact velocity exists for each contact point strength, which leads to a minimum number of breakage planes. For lower velocities, only parts of these planes are generated, while no additional planes occur at higher velocities. Here, only the fragments are further destroyed. Based on the simulations by the group of Thornton [53,54], Moreno et al. [55] studied the inﬂuence of impact angle on the breakage behaviour of agglomerates. It was shown that the normal component of the impact velocity is the dominating factor of inﬂuence. If it is kept constant, the number of bonds that break in an agglomerate also remains constant regardless of impact angle. The changing tangential component of the impact velocity however changes the breakage pattern. Consequently, although the number of broken bonds remains identical, for a constant normal impact velocity component, more attrition fragments are observed under an impact angle a ¼ 451 than for normal impacts. Although Moreno et al. do not compare their simulation results with experimental results, the ﬁndings are in-line with experimental ﬁndings by Ruppel [56]. More recent results published by Samimi et al. [57] conﬁrm that normal impacts do not necessarily lead to the highest amount of agglomerate breakage, but that in some cases, oblique impacts have a more signiﬁcant effect on breakage.

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Han et al. [58] determined the stress conditions that a particle experiences in a geometrically simple conveying setup by using a combination of DEM and computational ﬂuid dynamics (CFD). These results were then used to theoretically predict the attrition rates with help of the attrition model developed by Ghadiri and Zhang [23]. Han et al. do not only employ the chipping model, but also incorporate a model for particle fragmentation originally derived by Papadopoulos [59]. The impact velocity was used as a criterion to distinguish between chipping and fragmentation. A comparison of the model results with experimental results obtained by Bell et al. [60] in a pneumatic conveying system of identical setup as used in the simulations, only showed an acceptable agreement. Han et al. hold the assumptions made in the simulations responsible for the discrepancies. However, they did not consider that the breakage model used was derived for normal impacts, which are an exception in pneumatic conveying systems. Consequently, they disregarded the inﬂuence of impact angle, which as discussed above is assumed to be responsible for at least part of the observed differences. A study combining DEM simulations and impact experiments on the singleparticle scale has been carried out by Antonyuk et al. [61]. These authors conducted single-particle impact experiments with different types of agglomerates. By applying high speed as well as SEM imaging, the breakage behaviour of the granules was investigated. It was shown in the experiments that the breakage behaviour depends on both macroscopic and microscopic agglomerate structure. For approximation of the breakage probability, the Weibull function was applied. Subsequently, a DEM model was developed to simulate the particle-wall impacts. With this two-dimensional model, the same breakage patterns as observed in the experiments were obtained.

3. A NEW CONCEPT FOR ADDRESSING ATTRITION IN PNEUMATIC CONVEYING The above discussion clariﬁes two things. First, in attrition research, often only partial aspects are considered, which has lead to numerous experimental and theoretical approaches. These usually do not provide universally applicable results. Second, at present the inﬂuence of bulk solids particle and material properties on pneumatic conveying characteristics as well as on attrition is hardly understood. Accordingly, the design of pneumatic conveying systems is still based on time consuming and costly pilot plant or even industrial scale tests. As a consequence, an approach to describe attrition processes was developed, which considers the multiscale nature of attrition formation (cf. Refs. [62,63]). This approach goes back to the work of Rumpf [64,65], Krekel and Polke [66] and Peukert and Vogel [67]. Rumpf was the ﬁrst who analysed comminution processes by distinguishing between machine and material parameters. He

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stated that the main parameters affecting comminution results are stress mode, stress intensity and number of stress events as well as particle size and shape and material composition. This approach in which a mill is not regarded as a black box, but where the stress conditions of different mills are used as characteristic features to distinguish between them was later also applied by Krekel and Polke [66]. They introduced the term process function which describes the combined effect of process or machine parameters and material properties on the chemical and physical product properties. Peukert and Vogel [67] divided the process function into a machine and a material function. The machine function describes the connection between design and machine-speciﬁc parameters and the driving potential for the engineering operation, i.e. the stress conditions in a mill. The material function summarizes the reaction of the materials used on this driving potential. This concept was transferred to the problem of attrition in pneumatic conveying. Since no single part of machinery can be directly associated with this process, the machine function was renamed the process or conveying function in order to obtain an unambiguous terminology. The process function thus differs from what Krekel and Polke [66] have deﬁned. The term material function is used as introduced by Peukert and Vogel [67] (More details can be found in Ref. [32]). Consequently, attrition formation in pneumatic conveying is interpreted as the result of a process function describing the stress conditions the particles are subjected to in the conveying pipelines, and a material function summarizing the material-speciﬁc response to the process function in terms of intrinsic material properties. This approach is schematically shown in Fig. 3, where in the left part the common practice of designing a pneumatic conveying system is shown, which is based on an integral view of attrition processes. In the multiscale approach on the other hand, it is differentiated between the inﬂuence of process parameters and material properties on attrition.

Fig. 3. Description of attrition with a process and a material function.

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The process function of course depends on the mode of conveying. For dilute phase conveying attrition will be mainly caused by particle-wall and inter-particle impacts, whereas in plug ﬂow conveying friction is assumed to be the dominating stress mode. Stress intensities as well as the number of stress events will vary greatly likewise. For the material function, it is as yet unknown which properties can be used to at least qualitatively predict the material reaction to the stress conditions. Both functions are of course not fully independent, but can be separated sufﬁciently for the investigation of attrition processes.

4. DETERMINATION OF THE PROCESS FUNCTION To successfully further the understanding of attrition in pneumatic conveying, it is essential to determine the stress conditions the particles are subjected to during conveying. These largely depend on the mode of conveying since, as the discussion in Section 2.1 has shown, the particle movement inside the pipes signiﬁcantly changes with the mode of conveying. If the different modes of conveying are analysed, two fundamentally different stress modes occur in homogeneous dilute phase ﬂow on the one hand and plug ﬂow conveying on the other. While particles in dilute phase ﬂows are mainly stressed by oblique interparticle or particle-wall impacts, sliding friction is the stress mode, the particles forming the shell of a plug are subjected to in plug ﬂow conveying [68]. In all other modes of conveying, these two fundamental stress modes coexist with varying stress intensities and numbers of stress events, thus leading to a varying relative contribution to overall attrition. As a consequence of these results, homogeneous dilute phase ﬂow and plug ﬂow conveying were chosen as reference modes of conveying for the determination of the process function.

4.1. Dilute phase conveying 4.1.1. Approach by Euler-Lagrange To determine the stress conditions in dilute phase conveying, a commercial CFD code – CFX-Tascﬂow by AEA Technology – was used. For the calculation of the particle trajectories, the approach by Euler-Lagrange was employed. Since more detailed descriptions can be found for example in Ebert [69] or Huber and Sommerfeld [70] only a brief discussion of the calculation procedure is given here. The ﬂuid phase is modelled as a continuum by solving the conservation equations for mass, momentum and energy. Since the conveying process is modelled as isothermal, the conservation of energy can be disregarded in this context. To incorporate turbulence, the scalar variables in the above equations are expressed in terms of mean and ﬂuctuating components, through Reynolds-Stress

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averaging. This procedure leads to the so-called RANS (Reynolds-AveragedNavier-Stokes) equations. The resulting system of equations is not closed, thus further assumptions have to be made. This is done by applying a turbulence model. In the present case, the model used was the k-e turbulence model by Launder and Spalding [71]. It assumes that turbulence can be characterized by two quantities: a characteristic eddy life time and a characteristic eddy length scale. Both quantities can be calculated from the turbulent kinetic energy k and the energy dissipation rate e (the amount of kinetic energy per mass and time converted into internal energy of the ﬂuid by viscous action), which in turn can be deduced from the RANS-equations. In contrast to the ﬂuid phase, the particles of the disperse phase are modelled separately (Lagrange). The trajectories the particles follow through the conveying pipes are calculated by considering the forces acting on them. The particles are assumed to be spherical and inert. Particle-particle interactions are neglected and no particle source terms were included in the turbulence equations. Therefore, turbulence is not modulated by the discrete phase. Furthermore, lift forces have been neglected. The particle motion is thus calculated by applying the Basset-BoussinesqOseen (BBO) equation (see Ref. [72] for details). Since in general the density of bulk solids is approximately three orders of magnitude larger than that of the continuous phase, the terms for virtual mass, pressure gradient and Basset forces, which are included in the original BBO equation, were disregarded in the calculations. To verify whether the neglect of particle interactions is justiﬁed, the ratio of spacing between two particles Sp/p and particle diameter xp is calculated according to Crowe et al. [72]. In equation (8), rp is the particle density and cm,p denominates the mass concentration of the disperse phase. Its value is thus identical to the bulk solids load ratio m. Sp=p p rp þ cm;p rf ¼ 6 cm;p rp xp

ð8Þ

If the minimum value of Sp/p/xp is calculated for the bulk solids under investigation, (Sp/p /xp)min ¼ 7.4 is obtained. This value is only slightly lower than the boundary value of 10 given by Crowe et al. [72]. It can thus be concluded that the particles are largely independent and that the disregard of particle interactions is a reasonable ﬁrst approximation. The calculation of the particle trajectories follows the principle of two-way coupling. First, the ﬂuid ﬂow ﬁeld is calculated without the presence of particles. Once a converged solution is obtained, particles are injected into the computational domain and the particle trajectories as well as the particle source terms are calculated. Afterwards a new ﬂuid ﬂow ﬁeld in consideration of the particle source terms is calculated. This process is repeated until a converged solution is obtained.

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4.1.2. Geometry of the computational domain For the determination of the stress conditions, a pipe-bend model possessing the geometry shown in Fig. 4 was used. This setup was chosen since pipe bends generate the highest attrition rates per unit length. The 901 pipe bend has a diameter D ¼ 80 mm and a ratio of radius of curvature rB to diameter D of rB/D ¼ 5. Preceding the bend is a straight pipe with a length le ¼ 2400 mm. This pipe was added for practical reasons since an experimental setup of the same geometry was constructed in parallel. The particle inlet is at the left-hand side of the pipe. For the numerical simulations the particles enter the domain evenly distributed across the inlet surface with the velocity up,0, which is perpendicular to the inlet surface. In the attrition experiments they are introduced along the pipe axis. The gas velocity at the inlet is u. For means of reduced calculation time, the computational model was created as a symmetric half model. This geometry was transformed into a computer readable grid structure, which was used for the calculations. Particle-wall collisions were modelled as fully elastic with a coefﬁcient of restitution e ¼ 1, since no experimental values of e were found in literature for the bulk solids used.

4.1.3. Determination of impact conditions from numerical simulations With the calculation procedure described so far, it is possible to calculate particle trajectories for given boundary conditions, but no information about the (distribution of) stress conditions for particle-wall impacts can be obtained directly from Tascﬂow. Therefore, an external Fortran routine was developed to extract the stress conditions from the numerical simulations. The ﬁrst step is to determine where the particle-wall collisions take place. For this purpose a large number of particle trajectories (2025) were calculated and the respective particle positions in terms of their x-, y- and z-coordinates (cf. coordinate system in Fig. 4) as well as the corresponding up, vp and wp velocity components at each integration timestep are saved to so-called result ﬁles. With help of the Fortran routine, the particle positions and velocities are evaluated and the change Ddp/W in the smallest distance between the particles and the pipe wall is monitored. This change is deﬁned in such a way that it is negative when the particle approaches the wall and positive if the particle recedes from the wall. A particle-wall impact thus has taken place, when the sign of Ddp/W changes from negative to positive and furthermore when the change in the angle a as deﬁned in Fig. 4 lies above a threshold value Da40.1. This second criterion has to be fulﬁlled as well since, as the trajectory of particle 2 in Fig. 4 shows, not every change in sign of Ddp/W is necessarily caused by a particle-wall impact. An overview of the calculation sequence to determine the particle-wall impacts is given in Fig. 5.

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Fig. 4. Geometry and dimensions of the pipe-bend setup used for CFD simulations.

Fig. 5. Calculation sequence to determine particle-wall impact points from particle trajectories.

Since Tascﬂow returns the ‘‘after-crash’’ values of the particle velocity at the point of minimum distance between a particle and the wall, the correct particle position and velocity before impact are the values for the timestep (n2). Due to the fact that a particle-wall impact always takes place between two integration timesteps, the exact position of impact is not known. It is therefore extrapolated from the particle position directly before impact as outlined in Fig. 6. By means of a transformation of the original coordinate system given in Fig. 4, the x-y-z- as well as the x0 -y0 -z0 -coordinate systems are obtained. For the determination of the stress conditions, it is feasible to use cylindrical coordinates.

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Fig. 6. Extrapolation of point of impact based on particle position right before impact.

The surface vector in the bend S B , which describes all points on the bend surface, can be calculated according to equation (9). 0 1 r B cos j1 þ D2 cos j1 cos j2 B C S B ¼ @ r B sin j1 þ D2 sin j1 cos j2 A ð9Þ D 2 sin j2

The transformation the angles j1 and j2 the surface vector S B normal vector N B of equation (10)).

of the original coordinate system allows the calculation of from the global coordinates x, y and z. With these angles, can be calculated, which is in turn necessary to obtain the the tangential plane to the particle-wall impact point (cf.

NB ¼

@S B @S B @j1 @j2

ð10Þ

If it is assumed that a particle impacts the wall with the velocity of timestep (n2), the impact angle aW between the particle velocity vector vp and the tangential plane to the impact point can be calculated according to equation (11). Herein, an is the angle between the normal vector N B of the tangential plane and the particle velocity vector vp . 0 1 0 1 un2 p n2 N v C B B p C n2 B B v n2 C ð11Þ ; v ¼ aW ¼ 90 an with an ¼ arccos@ A p p @ A jN B jvn2 p w n2 p As shown in equations(12) and (13), with knowledge of aW the particle velocity vector vp can be divided into a component normal to the wall vp,n and one parallel to it vp,t. v p;n ¼ vp cos aW ð12Þ v p;t ¼ vp sin aW

ð13Þ

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The particle coordinates and velocities at the moment of wall impact are determined and saved with assistance of the previously discussed Fortran routine. In parallel, the total number of particle-wall impacts Np/W is computed. Subsequently, the maximum and minimum values of aW, vp,n, and vp,t are determined for the ﬁrst particle-wall impact in the bend. From these values, based on a ﬁxed number of intervals a speciﬁc interval size is generated for each quantity. All values are ﬁled into these intervals, which leads to cumulative number distributions of the stress conditions. The same procedure is also applied to obtain a distribution for the total number of impacts Np/W. A full account of the calculation of the impact points as well as a validation of the simulation results and a study of different parameters of inﬂuence as obtained by dimensional analysis can be found in Frye [73]. The following section is focused on the results obtained from the determination of the process function for dilute phase conveying.

4.1.4. Cumulative number distributions of the stress conditions In Fig. 7, the stress conditions, i.e. wall impact angle aW, normal velocity component vp,n and tangential velocity component vp,t are given for polypropylene (PP 1040 N). These were determined for the ﬁrst particle-wall impact inside the pipe bend. Both gas entrance velocity u0 and particle entrance velocity up,0 (perpendicular to entrance surface) were set to 41 m/s. The bulk solids load ratio was m ¼ 1, the gravity vector g was set positive in negative y-direction (cf. Fig. 4). Q0 denominates the cumulative number distribution of the property given on the abscissa. If the cumulative number distribution of the impact angle is analysed, it becomes obvious that the maximum impact angles aW,max ¼ 351 are unexpectedly small. This leads to low values of vp,n and correspondingly higher values of vp,t.

Fig. 7. Cumulative number distributions Q0 of (a) wall impact angle aW, as well as (b) normal (vp,n) and tangential (vp,t) component of particle velocity upon impact for PP 1040 N.

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Fig. 8. Visualization of impact points for PP 1040 N at gas entrance velocity u0 ¼ 41 m/s and particle entrance velocity up,0 ¼ 41 m/s.

To rule out errors in the processing of the data from the simulations, the impact points were visualized. As Fig. 8 shows, the determined locations are in fact particle-wall impact points. The results given in Fig. 8 also provide an explanation for the low-wall impact angles. As it can be seen, the position of the tangential plane of the impact points is signiﬁcantly inﬂuenced by pipe curvature. This results in decreasing impact angles for particles which do not impact the pipe wall close to the symmetry plane. Thus, both bend curvature and pipe curvature are of inﬂuence on the wall impact angles. Since the results for the stress conditions were conﬁrmed in further simulations for other polypropylenes as well as for polyethylenes, polymethylmethacrylates and polystyrenes, it can be concluded that – contrary to what is commonly assumed – attrition in dilute phase conveying of polymers is partly, probably even predominantly caused by sliding friction during particle-wall contacts. This assumption is supported by results published by Mills and Mason [74] and Klinzing et al. [1]. In both works, it is reported that pipeline wear of brittle wall materials during pneumatic conveying of abrasive products increases with increasing impact angle and reaches a maximum for normal impacts. For pipes made of ductile materials however, wear increases with impact angle until the maximum wear is reached at an impact angle of about 201. Subsequently, wear decreases with increasing impact angle. For normal impacts, it is only 10% of the maximum value observed at an impact angle of 201. A change in attrition mechanism is held responsible for this. While wear of brittle materials is caused by crack propagation from the contact point at which the material is stressed, ductile materials deform plastically. Here, wear is caused by micro-cutting. If these ﬁndings are transferred to pneumatic conveying of polymeric materials, it can be concluded that in dependence of material behaviour and due to

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the stress conditions shown in Fig. 7, and in particular due to the high-tangential component of the impact velocity, different mechanisms contribute to attrition. Details on this will be given in the discussion of the material function. In further simulations, the particle entrance velocity up,0 was varied, since in pneumatic conveying, the particles usually possess lower velocities than the conveying gas due to inertia effects. Besides that, the particle velocity component vp,0 oriented in positive y-direction, i.e. in parallel with the entrance surface was included in a simulation. This way, particles possessing a transverse velocity as a result of wall impacts in a straight pipe were simulated. The stress conditions derived from these simulations are given in Fig. 9. As expected, the cumulative number distributions differ due to the changes in boundary conditions. The maximum impact angle however remains unchanged. For vp,0 ¼ 0 m/s both components of impact velocity decrease with decreasing particle velocity up,0 in x-direction. Furthermore, the distributions become narrower. An explanation for this can be deduced from an analysis of the impact points, which with the exception for the parameters up,0 ¼ 41 m/s, vp,0 ¼ 0 m/s have been visualized in Fig. 10. While the particles entering the pipe with up,0 ¼ 41 m/s only slightly change their path upon entry into the pipe bend and more or less directly proceed to the outer pipe wall (cf. Fig. 8), the particles with up,0 ¼ 30 m/s are signiﬁcantly affected by the gas ﬂow. Their trajectories follow to a certain extend the trajectories of the gas ﬂow, which results in later impacts on the pipe wall and thus in larger impact angles. At the same time, vp,n and vp,t decrease as a consequence

Fig. 9. Cumulative number distributions Q0 of (a) wall impact angle aW, as well as (b) normal (vp,n) and tangential (vp,t) component of particle velocity upon impact as a function of particle entrance velocities up,0 and vp,0 for PP 1040 N.

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Fig. 10. Visualization of impact points for PP 1040 N at gas entrance velocity u0 ¼ 41 m/s and (a) up,0 ¼ 30 m/s, (b) up,0 ¼ 20 m/s as well as (c) up,0 ¼ 41 m/s and vp,0 ¼ 10 m/s.

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of the lower particle entrance velocity. If up,0 is reduced to 20 m/s, the particles gather in the lower pipe section. Responsible for this are the gravity force which causes the particles to sink in the straight pipe section as well as a few particlewall impacts in the bottom part of the straight pipe. This results in narrow distributions of the stress conditions. If besides up,0 a second particle velocity component vp,0 ¼ 10 m/s opposing the direction of gravity is superposed at the entrance, an opposite effect occurs. Through numerous particle-wall impacts, the particles are widely distributed across the pipe cross-section and particle-wall impacts take place in a large section of the pipe bend. Consequently, the distributions of the stress conditions are quite wide, although they still lie within the boundaries of those distributions calculated without a superposed vp,0. Concluding, it can be said that the simulations carried out to determine the process function for dilute phase conveying provide physically interpretable results for the stress conditions, which, as has been shown in more detail by Frye [73], can be regarded qualitatively as well as quantitatively correct. The most important result is that the particle-wall impacts take place under lower angles than usually assumed (aW,av ¼ 221). Because of the resulting low normal impact velocity component of vp,n,av ¼ 16 m/s and the large tangential impact velocity component of vp,t,av ¼ 36 y 41 m/s, it has to be concluded that not only impact stresses contribute to attrition. In fact, based on the results obtained, it is evident that depending on the material behaviour sliding friction signiﬁcantly contributes to attrition in dilute phase pneumatic conveying. The number of stress events was not analysed in detail in this study, since it was assumed that the ﬁrst particle-wall impact in the pipe bend is responsible for most of the observed attrition. In this case, the number of stress events is one. If the number of stress events is evaluated for a pipe bend with rB /D ¼ 5 under the assumption of fully elastic particle-wall impacts, i.e. e ¼ 1, on average each particle experiences 2 y 3 wall impacts in the pipe bend [73].

4.2. Dense phase conveying The determination of the process function for dense phase or plug ﬂow conveying is more complex than for dilute phase conveying, since the particle numbers are very high. As a consequence, the possibilities to model the particle motion are much more restricted. Consequently, important quantities like the wall normal and wall shear stresses exerted by the plugs had to be measured. In connection with the analysis of DEM-results, the process function for plug ﬂow conveying can be determined. In a ﬁrst step however, the stress mode the particles forming a plug are subjected to is discussed.

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4.2.1. Stress mode in plug £ow conveying To identify the stress mode in plug ﬂow conveying, the dynamics of plug movement are analysed. Here, it is distinguished between vertical upwards and horizontal conveying. In vertical conveying, plugs are formed if the gas velocity lies below the saltation velocity of the particles. Sinking particles lead to a local increase of the bulk solids load ratio. As a result, the free surface decreases, the gas velocity increases until it ﬁnally exceeds the saltation velocity of the particles and the particles are transported as a plug [75]. In horizontal conveying especially during the transport of cohesionless bulk solids like polymers, a stationary layer is a prerequisite for the formation of plugs. The force exerted by the stationary layer induces an axial compression of the particles which leads to an increased wall friction. Through this, the plug is stabilized [76]. In both cases, plug transport is a quasi-stationary process. In vertical conveying, the plugs constantly lose particles at their lower end, which are picked up by succeeding plugs. In horizontal conveying, the plugs pick up particles from the stationary layer at their front, while they lose particles to the layer at their rear end. The forces responsible for the formation of stable plugs are shown schematically in Fig. 11 for vertical upward and horizontal plug ﬂow conveying. Plug transport is effectuated by the pressure force Fp,f that is exerted by the ﬂuid on the plug. According to Legel and Schwedes [77], this force results on the one hand from the particle resistance and on the other from the pressure drop of the channels present in the plug. In vertical conveying, the particle weight Fg and the friction force between the particles and the wall Ffr, act opposite to Fp,f. The friction force originates from the axial compression of the plug. In vertical

Fig. 11. Balance of forces on a plug in (a) vertical upward and (b) horizontal conveying as well as (c) balance of the acting pressures and tensions on a differential plug element.

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conveying, the friction force is evenly distributed along the pipe circumference, while it is asymmetric in horizontal conveying due to the particle weight. This is expressed through the additional term Ffr,g. Furthermore, in horizontal conveying the force Fl exerted by the stationary layer acts against plug movement. Besides that, the layer exerts a side pressure force Fl,sp, which is present on both sides of the plug and can thus be disregarded (cf. Refs. [76,77]). If a differential plug element is regarded, the same balance for pressure p, axial compression sx as well as wall normal and tangential stresses sW and tW is obtained for both horizontal and vertical conveying. The axial compression originates from the particle weight in vertical conveying and from Fl in horizontal conveying. The wall normal and tangential stresses mainly result from the axial compression. In vertical conveying, the wall normal stress is evenly distributed along the pipe circumference, while a distribution is obtained in horizontal conveying due to the acting particle weight [78]. With respect to the stresses the particles are subjected to, it can be concluded that the particles experience different compressions in axial and radial direction due to the acting sx and sW. This might lead to attrition for brittle bulk solids. For polymers however, this mechanism is of minor importance. Due to the acting forces and the resulting axial compression of the plugs, it can be assumed that the particles forming the core of the plug, i.e. grey area in Fig. 11(b), are ﬁxed relative to each other. This assumption is corroborated by experimental results obtained by Guiney et al. [79], Mi and Wypych [80] and Va´squez et al. [81]. Consequently, the highest particle stresses are apt to occur between those particles forming the shell of the plug and the pipe wall, since this is where the highest relative velocities occur. Under these circumstances, the stress mode is sliding friction and the stress intensity is determined by the acting wall normal stress and the plug velocity.

4.2.2. Stress intensity from measurements of the pressure exerted by plugs The analysis of experimental results is based on three different studies in which the wall normal pressure exerted by moving plugs was determined experimentally. Mi and Wypych [78,80] used a sensor, which simultaneously measures the total pressure at the bottom of the pipe as well as the pressure pf of the conveying gas. From these values, the pressure a plug exerts on the pipe wall pW respectively the wall normal stress sW can be calculated. For horizontal plug ﬂow conveying of a polymer with a median diameter of xp ¼ 3.12 mm and a density rp ¼ 865 kg/m3 in a pipe with a diameter D ¼ 105 mm, sW ¼ 510 Pa was obtained. If it is assumed that the maximum number of particles np is lined up along the pipe circumference, the average contact force Fc,p/W,av between an individual particle and the wall can be calculated from the measured wall normal stress sW

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and the circumferential area of a pipe segment Acirc which is one particle diameter xp wide (equations (14) and (15)). X F c;p=W;i ¼ sW Acirc with Acirc ¼ pDx p ð14Þ i

P F c;p=W;av ¼

F c;p=W;i

i

np

with

np ¼

Acirc p 2 4 xp

ð15Þ

The calculation of the contact pressure between an individual particle and the pipe wall is based on the equations for elastic contact derived by Hertz [82]. The radius a of the contact circle between a spherical particle and a planar wall is calculated as follows: !1=3 3F c;p=W;av 12 x p 1 1 n2W 1 n2p a¼ with þ ð16Þ ¼ EW Ep E~ 4E~ For the calculation, the common values for cast steel, i.e. EW ¼ 169 GPa for Young’s modulus and nW ¼ 0.3 for Poisson ratio are used as wall material properties [83]. As particle properties, values valid for polypropylene, i.e. Ep ¼ 1.5 GPa and np ¼ 0.37 were used [84]. This leads to a radius a ¼ 14 mm. With equation (17), the contact pressure between an individual particle and the pipe wall was calculated to pc,p/W ¼ 6.5 MPa. pc;p=W ¼

F c;p=W;av pa2

ð17Þ

Due to the assumptions made for the calculation and the fact that the wall and particle properties had to be estimated, the above value of pc,p/W can only be regarded as an estimate of the magnitude of the stress intensity. If for example the estimated value for Young’s modulus of the bulk solid differs 5% from the actual value, an error of 3.3% is made in the calculation of pc,p/W. Besides that, it was assumed that the maximum number of particles is lined up along the circumference. This is however unlikely, and leads to an underestimation of pc,p/W. On the other hand, polymers are viscoelastic. Consequently, plastic deformations are present at the contact area, which are not included in Hertz theory. The contact pressure is thus overestimated by applying Hertz theory. Finally, deviations from a spherical particle shape would also lead to lower contact pressures pc,p/W. To validate the above value of pc,p/W, two more experimental studies were analysed in the same manner. Wall normal stresses were also measured by Va´squez et al. [81]. For spherical polyester particles with xp ¼ 2.5 mm, the wall pressure in a horizontal aluminium pipe of D ¼ 50 mm in diameter was determined to sW ¼ 290 Pa. From that, the contact pressure pc,p/W was calculated for an aluminium pipe with EW ¼ 69 GPa, nW ¼ 0.33 [83] and for spherical polyester

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particles with Ep ¼ 3 GPa and np ¼ 0.32 [84,85]. The resulting contact pressure is pc,p/W ¼ 6.7 MPa, which is in excellent agreement with the previously calculated one. Finally, the same calculation procedure was applied to a study carried out by Niederreiter and Sommer [86], who investigated vertical upwards conveying of polypropylene particles with xp ¼ 3 mm in an acrylic glass pipe with D ¼ 50 mm. Thus, EW ¼ 3 GPa and nW ¼ 0.32 were used for the calculations. For the particles, the same values for modulus and Poisson ratio were used as in the ﬁrst analysis. From the measured normal wall stress sW ¼ 1800 Pa, the contact pressure pc,p/W ¼ 7.5 MPa was calculated. This value once again is in good agreement with the two previously calculated ones. Thus, the magnitude of the stress intensity can be regarded as correct.

4.2.3. Stress intensity from analysis of Discrete Element Methods (DEM) Increased computer power has permitted the simulation of dense phase ﬂows for simple geometrical setups and assemblies of up to several tens of thousands of particles. Presently, the main problem in the simulation is to adequately model the particle-particle interactions, which presents a great challenge due to the numerous contact points and large particle numbers in dense assemblies. A method that has been applied largely and successfully to model the particle motion in dense systems is the DEM, originally developed by Cundall and Strack [87]. It has been successfully applied for modelling dense phase ﬂows [88–90], which is why this method is used here for the determination of the stress intensity from theoretical data. As already stated, the main problem in simulating dense particle systems is the correct simulation of inter-particle forces. This is difﬁcult since each particle not only interacts with other particles in direct contact but through these contacting particles also inﬂuences particles which are further away. To solve this problem, Cundall and Strack [87] assumed that it is sufﬁcient to regard direct particle contacts only, if the simulation timestep is selected small enough. In this case, particle movement results from the forces acting on an individual particle at the contact points to other particles or the pipe wall. Herein, the calculation of normal and tangential contact forces is carried out separately. The contact model developed for this purpose by Cundall and Strack is given in Fig. 12. The basic elements are a spring with the spring constant k, a dashpot with the damping constant Z and a frictional element with the friction coefﬁcient mfr. The arrangement of the elements resembles a Voigt-body, which is commonly known from rheology. At this point, no full account of the further equations used for the DEM simulations will be given. This can be either found in McGlinchey et al. [90] or Frye [73]. Here, the focus is laid on the results obtained from an analysis of DEM simulations with respect to the stress intensity in dense phase pneumatic conveying.

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Fig. 12. Model for calculation of (a) normal and (b) tangential contact forces in DEM simulations.

The raw data for extracting information about the process function was created by Don McGlinchey and Jiansheng Xiang of Glasgow Caledonian University. They developed the code the DEM simulations were carried out with and applied the DEM to model the two-dimensional particle motion in plug ﬂow conveying in a straight pipe of D ¼ 80 mm in diameter and l ¼ 8 m in length. As particle properties, the values of a polyethylene (PE 2420 H) were used. This polymer exhibits the lowest Young’s modulus of all polymers investigated and thus enabled them to carry out the simulations in a manageable time (cf. Ref. [90]). The simulation parameters thus were xp ¼ 3.28 mm, rp ¼ 919.3 kg/m3, Ep ¼ 260 MPa, np ¼ 0.3 and mfr ¼ 0.3. The bulk solids load ratio was m ¼ 69.4, the coefﬁcient of restitution e ¼ 0.8 and the gas velocity parallel to the pipe axis u ¼ 4.26 m/s. For data analysis with respect to the process function, a single plug extending from x ¼ 5.5 to x ¼ 8 m was selected. The ﬁrst step was to verify whether the particles forming the plug are indeed ﬁxed relative to each other. For this purpose, the spatial distributions of vertical particle velocity vp and horizontal particle velocity up in the pipe were determined. As the results given in [73] show, the particles are indeed ﬁxed to each other, thus supporting the conclusions drawn in Section 4.2.1. The plug velocity was determined to up ¼ 2.0 y 2.3 m/s. This is in good agreement with velocities determined experimentally by Va´squez et al. [81], who under comparable boundary conditions found up ¼ 0.9 y 1.9 m/s. The stress intensity can be deduced from the total force in the vertical direction Fy,tot each particle experiences. The corresponding distributions are given in Fig. 13. In Fig. 13(b), the particles are represented by points which are shaded according to the vertical force they experience, x and y are the horizontal and vertical positions of the particles in the pipe. In the legend, only the lower boundary values of the intervals are given, while the upper boundary corresponds to the value given in the preceding row of the legend. Since the forces most particles experience lie between Fy,tot ¼ 28 and 34 mN, the force values of this interval, which has been marked with an arrow, were omitted in the bottom diagram to show additional details.

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Fig. 13. (a) Cumulative number distribution Q0 as well as (b) spatial distribution of total vertical force Fy,tot experienced by each individual particle in the plug (Fy,tot,max ¼ 0.22 N).

It can be clearly observed that the particles in the outer layer of the plug experience the highest forces. Therefore, the assumption that the particles forming the shell of the plug experience the highest stresses, appears to be justiﬁed. If the same analysis as in Section 4.2.2 is carried out for the maximum force Fy,max ¼ 0.22 N with the parameters used in the simulations, a contact pressure pc,p/W ¼ 7.3 MPa is obtained. This value is in the same order of magnitude as the values determined from the experimental investigations and thus conﬁrms the correctness of the simulation results. Concluding, it can be said with respect to the process function in plug ﬂow conveying, that the particles are subjected to a stress mode of sliding friction. The stress intensities in terms of contact pressure between a single particle and the pipe are in the order of pc,p/W ¼ 6 y 8 MPa, while the plug velocity was determined to be up ¼ 1 y 2.5 m/s. At present, no conclusions can be drawn concerning the number of stress events individual particles experience, e.g. how often they form the plug shell. For this, the trajectories of all particles would have to be resolved individually, which due to the large number of particles and the immense amount of data that would have to be processed, is not yet feasible.

4.3. Conclusions from the determination of the process function In the preceding sections, the process function has been determined for dilute phase and plug ﬂow conveying. In dilute phase conveying, the particles are mainly stressed by oblique particle-wall impacts in pipe bends, while the stress mode in plug ﬂow conveying is sliding friction. Here, the main stresses are experienced by the particles forming the shell of a plug, which are thus sliding along the inner pipe wall. An analysis of the stress intensities has shown that in dilute phase conveying, the particles impact the wall under average impact angles of aW,av ¼ 221, with an average normal velocity component vp,n,av ¼ 16 m/s and – depending on the product – average tangential velocity components vp,t,av ¼ 36 y 41 m/s. In plug

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ﬂow conveying, the contact pressure between an individual particle and the pipe wall is pc,p/W ¼ 6 y 8 MPa, while the particle velocity lies between up ¼ 1 and 2.5 m/s. For both conveying modes, the number of stress events was not evaluated in the course of the simulations. At present, it is not possible to determine the process function in full detail and explicitly express it in mathematical equations. Nevertheless, the results are regarded suitable to serve as input parameters for experiments carried out to determine the material function.

5. DETERMINATION OF THE MATERIAL FUNCTION In contrast to the determination of the process function, which was focused on the determination of the stress conditions, the experiments carried out to determine the material function aim at identifying the mechanisms and material properties relevant for attrition. In the following sections, the materials used, the sample preparation as well as the experimental methods applied are presented and the obtained results are discussed. Finally, a qualitative model for attrition in pneumatic conveying which is based on the results of the determination of both process and material function is presented.

5.1. Test material Pneumatic conveying is largely applied to transport commodities like granular polymers. Since according to Domininghaus [85] many of the technically relevant polymers are thermoplastics, four different polypropylenes (PP 1040 N, PP 1100 RC, PP 1148 RC and PP 2500 H) as well as two polyethylenes (PE 2420 H and PE 5031 L) were chosen for the experiments. Some of their material properties, i.e. particle shape, particle diameter xp, particle density rB, Vickers microhardness HV, Young’s modulus E yield stress sy and J-integral value JQd are given in Table 1. All of the six polymers mentioned, are semi-crystalline in structure. Besides that, polymers possessing an amorphous structure, i.e. polymethylmethacrylates (PMMA G7 and PMMA G55) and polystyrenes (PS 144 C and PS 158 K) were chosen for the experiments. Their material properties are given in Table 1 as well. Here, however, instead of yield stress sy the breakage stress sbr is contained. More details on the polymers can be found in Frye [73]. As a matter of course, the properties of these polymers were also used in the determination of the process function. Besides the experiments with polymers, further experiments with crystalline sodium and citric acid were carried out. The results were used to verify whether the attrition models developed by Ghadiri and Zhang [23,24] as well as Gahn [28],

Tradename

Shape

xp (mm)

rp (kg/m3)

Hv (MPa)

E (MPa)

sy(sbr) (MPa)

JQd (kJ/m2)

PP 1040 N PP 1100 RC PP 1148 RC PP 2500 H PE 2420 H PE 5031 L PMMA G7a PMMA G55a PS 144 Ca PS 158 Ka

Elliptical Elliptical Elliptical Elliptical Elliptical Elliptical Cylindrical Cylindrical Cylindrical Cylindrical

4.04 4.00 4.11 4.06 3.28 3.46 3.20 3.03 3.46 3.46

869.9 869.7 867.8 896.3 919.3 946.3 1191.8 1203.9 1053.1 1059.8

100.5 88.4 96.0 68.0 61.5 84.9 208.9 178.0 172.7 178.8

2000 1500 1650 1100 260 1000 3200 3100 3300 3300

40 34 35 23 11 26 72 62 42 55

2.04 2.09 1.92 na. na. 4.00 0.77 0.65 1.54 1.99

a

New Concept for Addressing Bulk Solids Attrition

Table 1. Overview of different material properties of the polymers used for the determination of the material (as well as process) function

Instead of yield stress sy, breakage stress sbr is given for these polymers.

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which were presented in Section 2.3.2, can be applied to attrition in pneumatic conveying. Details on this will be given in Section 5.4.1.

5.2. Sample preparation and determination of attrition rate Since the amount of attrition polymers exhibit is usually small, sample preparation is of major importance for the experiments. It has to be ensured that the samples used for the experiments are as free of attrition as possible, to rule out systematic measurement errors. For the single-particle experiments, which are described in the following section, the polymer particles were put on a sieve and carefully cleaned with pressurized air. Since the polymers did not contain any measurable amount of attrition upon delivery, no additional cleaning measures like washing and drying were taken prior to the experiments. After stressing the particles, the attrition rate A is calculated according to equation (18). Herein, Mp,0 is the initial sample mass, while Mp,st is the mass without attrition after stressing the particles. Finally, Mp,a is the mass of attrition. A¼

M p;0 M p;st M p;a ¼ M p;0 M p;0

ð18Þ

To determine Mp,st, the particles were cleaned in the same manner as before the experiments. Since attrition formation is a highly statistical process, in all cases, each experiment was repeated three times under identical conditions to obtain statistically signiﬁcant results. To carry out the experiments on the process scale (cf. Section 5.4.4), signiﬁcantly larger product masses were necessary. While a maximum of Mp,0 ¼ 0.025 y 0.030 kg was used for the experiments on the single-particle scale, Mp,0 ¼ 25 y 50 kg was needed on the process scale. Consequently, it was not possible to determine the attrition rate by measurement of Mp,0 and Mp,st. Therefore, the mass of attrition Mp,a was measured on the process scale. For this purpose, the product sample was cleaned according to FEM-Standard 2482 [91]. Prior to the experiments, the product was washed with distilled water and subsequently dried. Then, Mp,0 was measured. After the experiment, the conveying pipeline and the product were washed again. All of the distilled water used was collected and then given on a ﬁlter of known weight. This ﬁlter was dried afterwards and weighed again. From the difference in mass, the mass of attrition Mp,a was calculated. With this value, A was again calculated according to equation (18). The application of this procedure on the single-particle scale was not possible, since the installations used did not provide any possibility to collect both initial particles and attrition.

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Fig. 14. Design of test rigs for (a) normal impacts, (b) sliding friction and (c) processoriented particle stressing.

5.3. Single-particle experiments 5.3.1. Experimental setup and parameters The setups of the different installations used for the attrition experiments are shown in Fig. 14. Figure 14(a) shows the installation for normal impacts which was originally developed by Marktscheffel and Scho¨nert [92]. By a vibratory feeder (1), the particles are individually fed into a rotor (2) of diameter DR ¼ 122 mm. The particles enter the rotor in its centre and are then accelerated towards the rotor perimeter in one of four radial channels. At the perimeter, the particles leave the rotor under an angle of 451 [93] and hit the impact ring (3). The geometry of this ring was chosen in such a way that it ensures normal impacts. The impact chamber (4), where particles and attrition are collected is completely lined with rubber to avoid damage by secondary impacts and is evacuated during the experiments in order to eliminate any effects due to viscous drag. According to equation (19), the impact velocity vp,i can be calculated from the rotor circumferential velocity vR,j which is derived from the number of revolutions of the rotor n and its diameter DR [32]. pﬃﬃﬃ pﬃﬃﬃ ð19Þ v p;i ¼ 2v R;j ¼ 2pDR n With the test rig for normal impacts, impact velocities of vp,i ¼ 10 y 140 m/s can be realized. Due to its described technical setup, it allows the deﬁned stressing of particles through single impacts with exactly adjustable impact velocity. For the experiments, samples with Mp,0 ¼ 0.025 y 0.030 kg, i.e. approximately 1000 particles were used. In Fig. 14(b), the sliding friction test rig is shown. Here, particles possessing a rectangular shape, which were injection moulded form the original particles, are pressed onto a rotating disc simulating the wall material. The rectangular shape was necessary to maintain a constant contact pressure throughout the experiments. With the normal load Fn, the number of revolutions n, the distance

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r and the contact time tc, the stress conditions sliding velocity vs and distance s as well as contact pressure pc are controlled. Here, each experiment was carried out with single particles and repeated three times under identical conditions. In the pipe-bend installation shown in Fig. 14(c), the particles are stressed under conditions that are closely related to those in industrial conveying setups. It is designed as a negative pressure system and the dimensions are exactly the same as those used in the numerical simulations to determine the process function for dilute phase conveying. The particles are introduced into the pipe centrally and individually by a special feeder which allows particle acceleration to exactly deﬁned particle entrance velocities. As in the normal impact experiments, samples of Mp,0 ¼ 0.025 y 0.030 kg were used. It has already been stated that the stress conditions determined from the process function were to be used as input parameters for the single-particle experiments. This was done wherever possible. Due to the low attrition rates, sometimes it was however necessary to diverge from these values in order to obtain reproducibly measurable attrition rates. Further details on how the experimental parameters were chosen can be found in [73].

5.3.2. Discussion of experimental results In Fig. 15(a), the results for the normal impact experiments with the polypropylenes are given. They were conducted with impact velocities of vp,i ¼ 40 m/s.

Fig. 15. Attrition rates for polypropylenes as a function of (a) mass-speciﬁc kinetic energy Em,kin for normal impact experiments, (b) sliding velocity vs during sliding friction experiments and (c) number of impacts ni for pipe-bend experiments.

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The attrition rate is plotted as a function of mass-speciﬁc kinetic energy Em,kin. The energy values given are the sums of three, six and nine consecutive impacts. The error bars depict the standard deviation of the average value that was obtained from three experiments with identical parameters. For the results of the sliding friction experiments given in Fig. 15(b), these deviations were omitted for reasons of clarity. Here, the deviation ranged from 1 to 15% of the average value. In only one case did it rise to 20%. The standard deviation for these experiments was thus in the same order as for the normal impact and pipe-bend experiments. If the results of the normal impact experiments of Fig. 15(a) are compared to those of the sliding friction experiments in Fig. 15(b), which were obtained on a sandblasted vanadium steel surface at a contact pressure pc ¼ 61 kPa and for a sliding distance s ¼ 1500 m as a function of sliding velocity vs, it becomes obvious that the attrition behaviour of the polymers relative to each other differs between the two stress modes. While PP 1100 RC is most severely affected by attrition in the normal impact experiments, after PP 2500 it is the second least attrited in the sliding friction experiments. This leads to the conclusion that, depending on the stress mode, different material properties are responsible for attrition formation. Thus, a comparison of the results obtained in normal impact and sliding friction experiments with those from the pipe-bend experiments can be used to identify the relevant stress mode in the pipe bend and therefore to indirectly determine the effective attrition mechanisms. The attrition rates measured in the pipe bend for an average gas velocity of u ¼ 41 m/s and particle entrance velocities of up,0 ¼ 40.2 y 42.2 m/s are given as a function of number of impacts ni3 in Fig. 15(c). It is obvious that the attrition behaviour of the polymers relative to each other does not – as expected based on the stress mode oblique impact – resemble that of the normal impact experiments. In fact, a good agreement in relative attrition behaviour is found for the pipe bend and sliding friction experiments, which implies that the mechanisms in effect in pure sliding friction also dominate attrition in the pipe bend. These ﬁndings corroborate the conclusions drawn from the process function for dilute phase conveying that due to the low impact angles and high-tangential velocity components sliding friction cannot be neglected in dilute phase conveying. The results for polyethylene are summarized in Fig. 16. Here, instead of the sandblasted vanadium steel surface, a sample with an untreated surface, which in topography more closely resembles the surface of the pipe bend, was used in the sliding friction experiments. The contact pressure was increased to pc ¼ 0.62 MPa and the sliding distance was decreased to s ¼ 200 m. In the 3 It was assumed that only the ﬁrst particle wall impact in the bend is responsible for attrition in the pipe bend experiments. Thus, the number of impacts ni is equal to the number of passes of the particles through the bend.

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Fig. 16. Attrition rates for polyethylenes as a function of (a) mass-speciﬁc kinetic energy Em,kin for normal impact experiments, (b) sliding velocity vs during sliding friction experiments and (c) number of impacts ni for pipe-bend experiments.

pipe-bend experiments, the particle entrance velocities ranged from up,0 ¼ 43.0 y 45.3 m/s while the gas velocity was again set to u ¼ 41 m/s. If the results are compared to those obtained for the polypropylenes, a similar picture is obtained. Again, the attrition behaviour of the polyethylenes relative to each other in the pipe-bend experiments resembles that of the sliding friction experiments, while differences to the results from the normal impact experiments are found. This indicates that sliding friction is the dominant stress mode in attrition of the polyethylenes in the pipe bend as well. Finally, Fig. 17 contains the results for polymethylmethacrylates and polystyrenes. The normal impacts were once again carried out with vp,i ¼ 40 m/s, while the stress conditions in the sliding friction experiments were set according to those for the polyethylenes. For the polystyrenes it was not possible to determine attrition rates in sliding friction experiments for vs46 m/s, since the particles were fully attrited before the sliding distance s ¼ 200 m was reached. In the pipe-bend experiments, the particle entrance velocities lay between up,0 ¼ 40.8 and 43.8 m/s, the gas velocity was once more u ¼ 41 m/s. In contrast to the results presented so far, the attrition rates measured for PMMA and PS show a different picture. Here, despite minor deviations for PMMA G55, the attrition behaviour of the polymers relative to each other is similar in the pipe bend and normal impact experiments, while differences are found for pipe bend and sliding friction experiments. In case of these polymers, the mechanisms responsible for attrition under pure normal impact conditions are also dominant and thus responsible for attrition in the pipe bend. These at ﬁrst sight contradictory ﬁndings will be enlightened in Sections 5.4 and 5.5.

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Fig. 17. Attrition rates for polymethylmethacrylates and polystyrenes as a function of (a) mass-speciﬁc kinetic energy Em,kin for normal impact experiments, (b) sliding velocity vs during sliding friction experiments and (c) number of impacts ni for pipe-bend experiments.

In conclusion, the experiments conducted to determine the material function have shown that attrition of PP and PE granules when stressed in a pipe bend is mainly caused by the same mechanisms that are in effect under the stress mode of sliding friction. PMMA and PS granules on the other hand are attrited in the pipe bend by mechanisms in effect under normal impact conditions. Through the systematic approach followed in the experiments, it was thus for the ﬁrst time possible to elucidate the dominant mechanisms responsible for attrition of different polymers which were stressed in a pipe bend under almost identical stress conditions.

5.3.3. Experimental validation of process function for dilute phase conveying Besides identiﬁcation of the relevant attrition mechanisms, the single-particle experiments in the pipe bend were also used to experimentally validate the ﬁndings obtained for the process function in dilute phase conveying. For this purpose, the test rig shown in Fig. 18 was developed. It allows for the impact of particles with a deﬁned velocity onto an impact plate installed in a vacuum chamber. The impact angle can be adjusted by pivoting the plate. The particles are accelerated by impulse transfer to avoid any stressing prior to the impact. Right before the particle outlet, their velocity is measured with photo-electric guards. To be able to stress several particles without evacuating each time, a magazine for up to 20 particles is installed inside the vacuum chamber. The

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Fig. 18. Test rig with pivoting impact plate for experimental validation of results obtained for the process function for dilute phase conveying.

impact angle as well as particle metering is done by computer from outside the chamber. For the validation of the process function, two polymers PP 1040 N and PS 144 C, i.e. one polymer mainly attrited by mechanisms in effect under sliding friction conditions and one mainly attrited by mechanisms in effect under normal impact conditions, were chosen. The impact velocity vp,i was determined from the 50% values of the cumulative number distributions of the normal and tangential impact velocity components vp,n and vp,t. As wall impact angles aW, the 50% values of the corresponding distributions were taken as well. Consequently, PP 1040 N was stressed at vp,i ¼ 41 m/s and aW ¼ 221, while PS 144 C was stressed at vp,i ¼ 44.6 m/s and aW ¼ 211. As in the pipe-bend experiments, the attrition rates were determined after three, six and nine consecutive impacts. The attrition rates obtained in the setup shown in Fig. 18 were then compared to those found in the pipe-bend test rig of Fig. 14(c) and are given in Fig. 19. The comparison shows that with slight deviations for PP 1040 N, the attrition rates measured in the impact plate test rig correspond well with those obtained in the pipe bend. The comparatively higher standard deviations in the impact plate experiments are caused by the lower particle number that was included in the determination of the attrition rates. Consequently, ﬂuctuations in the strength of individual particles have a more pronounced effect. In general, the results proof that the ﬁndings presented in Section 4.1 for the process function in dilute phase conveying are quantitatively correct.

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Fig. 19. Comparison of the attrition rates obtained in the pipe bend and impact-plate test rigs as a function of impact number ni for (a) PP 1040 N and (b) PS 144 C.

5.4. Continuum mechanical material properties To date, little is known about the material properties of inﬂuence on attrition behaviour. Consequently, in a ﬁrst step, the existing attrition models by Ghadiri and Zhang [23,24] as well as Gahn [28] were analysed with respect to their applicability to the results discussed in the previous sections. This will give a ﬁrst indication about material properties with relevance to attrition. Afterwards, different properties of the polymers are analysed to identify those, which are suitable to explain the already discussed experimental ﬁndings. Here, properties known from fracture and continuum mechanics as well as thermo-mechanical properties are considered.

5.4.1. Application of existing attrition models on results obtained in singleparticle experiments Both the models of Ghadiri and Zhang [23,24] as well as that developed by Gahn [28] possess the advantage that it is differentiated between stress conditions and material parameters. This allows for judgement of the relevance of the material properties used. The equation for the dimensionless amount of attrition per stress event x (cf. equation (2)) can be re-written as x ¼ aZ

with

x¼

Va V

and

ZGhadiri ¼

rp x p H K 2c

v 2p;i

ð20Þ

Herein, Va is the attrited volume, while V is the initial volume. In ZGhadiri, the material properties particle density rp, particle diameter xp, hardness H and fracture toughness Kc as well as the impact velocity vp,i are summarized. The only

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ﬁtting parameter of the model, a has to be constant for a given material. The model of Gahn can be written in the same manner (cf. equation (21)). It can be seen that the structure of both models is similar and that the main difference lies in the exponents of material properties and stress parameters. x ¼ an Z with

x¼

Va V

4=3

and

ZGahn ¼

rp x p H 2=3 K 2c

8=3

v p;i

ð21Þ

Both models were originally developed for normal impacts. Therefore, the results from the test rig to simulate normal impacts as well as from the pipe-bend test rig (oblique impacts) were used. To determine x, the attrition rate A was divided by the number of impacts ni (cf. equation (22)). In this context it was exploited that for a constant particle density, the ratio of attrited volume to initial volume is identical to the attrition rate A as deﬁned in equation (18). x¼

1 A ni

ð22Þ

For the calculation of Z, the material properties given in Table 1 were used. For the normal impact experiments, the impact velocity was set to vp,i ¼ 40 m/s. In case of the pipe-bend experiments, the 50% value from the cumulative number distribution of the normal impact component vp,n was used. Since the fracture toughness values Kc were not known, the J-integral values JQd were used instead. If it is assumed that the polymers show elastic behaviour under the stress conditions of the experiments, according to [29,84], Kc can be calculated from JQd, Young’s modulus E and Poisson ratio n as given in equation (23). sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E J Qd ﬃ EJ Qd Kc ¼ ð23Þ 2 ð1 n Þ From the determined x and Z values, the ﬁtting parameters a and a were calculated. Since the attrition rates were determined for three, six and nine impacts, three values of a, a were determined for each material. It was thus possible to check, whether a, a in fact remained constant. For the evaluation of the models, the average values of a, a were used. It was not possible to measure the JQd values for PP 2500 H and PE 2420 H, which is why they could not be included in the analysis. Instead, additional experiments were carried out with sodium chloride (NaCl) and citric acid. The reason for this was that both the model by Ghadiri and Zhang as well as that by Gahn were developed for semi-brittle and crystalline materials, respectively. The material properties needed for the calculation were taken from publications of Zhang and Ghadiri [24] and Gahn [28]. The attrition rates obtained for the crystalline materials for normal impact as well as pipe-bend experiments are given in Fig. 20, along with the curves obtained from the models of Ghadiri and Zhang and Gahn, respectively. Due to the determination of a and a from the experimentally

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Fig. 20. Description of measured attrition rates for NaCl and citric acid in (a) normal impact experiments and (b) pipe-bend experiments by the models of Ghadiri and Zhang [23,24], equation (20) and Gahn [28], equation (21).

determined dimensionless amount of attrition x and the dimensionless parameters ZGhadiri and ZGahn, the product of a and ZGhadiri equals the one of a and ZGahn, leading to identical curves for both models. Consequently, the general applicability can be determined, but it is not possible to decide which model is more accurate. Figure 20 shows that the attrition rates measured in normal impact experiments are well described by the models. In the pipe bend, deviations due to the distributions of the stress conditions are observed, the general tendency is however reﬂected correctly. The results thus clearly show the suitability of both models to describe attrition of crystalline materials due to both normal and oblique impacts. For the brittle polystyrenes, a qualitatively similar picture is obtained. As shown exemplary for PS 158 K in Fig. 21, the description of the attrition rates by the models is acceptable. But it has to be borne in mind that the calculations to determine the model constants were based on the averaged attrition rates. If the deviations would have been considered by means of error propagation, larger relative deviations in a and a would have been observed. A different picture is obtained if the analysis is carried out for the polymethylmethacrylates, polypropylenes and polyethylenes. Here, standard deviations of a/a of up to 100% were observed. This clearly indicates that for these materials effects are of inﬂuence on attrition that are not covered in the models. This is graphically undermined by Fig. 22, which exemplary contains the results for PP 1100 RC. One possible explanation for the observed deviations is that fatigue due to repeated stresses plays a different role than for the brittle materials. Zhang and Ghadiri explicitly stated in [24] that such effects are not accounted for in the model. However, Kalman et al. [94] have shown that fatigue can

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Fig. 21. Description of measured attrition rates for PS 158 K in (a) normal impact experiments and (b) pipe-bend experiments by the models of Ghadiri and Zhang [23,24], equation (20) and Gahn [28], equation (21).

Fig. 22. Description of measured attrition rates for PP 1100 RC in (a) normal impact experiments and (b) pipe-bend experiments by the models of Ghadiri and Zhang [23,24], equation (20) and Gahn [28], equation (21).

play an important role in attrition or breakage of bulk solids. Besides that, the discussion in Sections 4.1 and 5.3 have shown that mechanisms contribute to attrition, e.g. sliding friction in oblique impacts which are not reﬂected by the material properties used by Ghadiri and Zhang or Gahn. Here, an increased understanding of the inﬂuence of material properties on attrition behaviour is necessary to develop a quantitative attrition model for polymers exhibiting viscoelastic material behaviour. To evaluate which of the two models describes attrition more accurately, additional experiments were carried out with citric acid. Since both models differ in the exponent of the impact velocity vp,i and this is the only parameter that can be

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Fig. 23. Inﬂuence of impact velocity vp,i on attrition rate A and evaluation of the models by Ghadiri and Zhang as well as by Gahn with respect to their suitability to describe this attrition rate.

varied independently, normal impact experiments were carried out with vp,i ¼ 12 y 24 m/s. For each velocity, only a single impact was carried out. For the evaluation of both models, the measured attrition rates given in Fig. 23 were approximated by equation (24). For the model of Ghadiri and Zhang, b was set to b ¼ 2, while for the model of Gahn b ¼ 2.67 was used. The parameter a contains the material properties and constants a respectively a according to equations (20) and (21). A ¼ av bp;i

ð24Þ

As the stability index R2 given in Fig. 23 shows, the model of Ghadiri and Zhang describes the attrition rates measured in the normal impact experiments more accurately than the model of Gahn. Whether this result is of general nature still has to be veriﬁed by additional experiments. As the previous discussion has shown, the attrition models by Ghadiri and Zhang are only of limited applicability to describe the attrition behaviour of polymers. While both models work well for crystalline materials and brittle polymers like polystyrenes, large deviations are found for visco-elastic polymers. The main reason for this is that no material properties reﬂecting this visco-elastic behaviour are contained in the models. Therefore, additional material properties were analysed to identify the relevant properties.

5.4.2. Hardness and fracture mechanical properties The starting point for the analysis of additional material properties were hardness and Young’s modulus, which are often used for material characterization in fracture mechanics. Besides that, hardness is among those properties that are used

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Fig. 24. Correlation between Vickers microhardness HV and Young’s modulus E for PP, PE, PMMA and PS.

in attrition models developed for brittle materials. The hardness of the polymers was measured with a Vickers micro hardness tester. The obtained values are given in Table 1. A comparison of Vickers hardness HV and Young’s modulus E of the polymers with the measured attrition rates did not show any correlation, but it was found that the hardness values and the values of Young’s modulus are approximately linearly related (cf. Fig. 24). Hardness and Young’s modulus are thus not independent properties. Flores et al. [95] found a similar correlation for polyethylenes and an analysis carried out by Frye [73] for a large number of polymers also corroborates these results. It thus does not seem feasible to include both hardness and Young’s modulus in an attrition model. In general, based on the ﬁndings discussed in Section 5.3.2 it has to be assumed that only material properties that in their respective measurement procedures reﬂect the dynamic nature of attrition processes are suitable for describing attrition formation. One option to obtain results about the material behaviour under dynamic stresses is an analysis of the stress-strain-behaviour as obtained from tensile tests. The results of such tests are given in Fig. 25. Herein, s is the measured tensile stress, while e is the relative strain. The stress velocity vst in the tensile tests was set as high as possible. PP and PE were tested at vst,PP/PE ¼ 350 mm/s, while testing of PMMA and PS was only possible at vst,PMMA/PS ¼ 60 mm/s. For higher velocities, these materials failed too fast to record the stress-strain diagrams. Although the stress velocities are lower than in attrition processes, interesting parallels are found between the stress-strain diagrams and the attrition experiments. All PP and PE granules, where sliding friction dominates attrition formation, exhibit a signiﬁcant zone of plastic ﬂow before failure. The measured tensile stresses are much lower than for the PMMA and PS granules. These show a quick increase in tensile stress up to failure without any plastic ﬂow. Only PS

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Fig. 25. Comparison of stress-strain diagrams from tensile tests for (a) PP and PE (vst,PP/PE ¼ 350 mm/min) and (b) PMMA and PS (vst, PMMA/PS ¼ 60 mm/min).

158 K exhibits a comparatively small zone of plastic ﬂow. The analysis of the stress-strain diagrams thus shows differences in material behaviour, which can be connected to the different attrition mechanisms identiﬁed in the determination of the material function. It is a clear advantage that instead of focusing on one property, the material behaviour is measured over a wider range including both elastic and visco-elastic respectively plastic regimes. The fracture toughness Kc used by Ghadiri and Zhang [23,24] in their model is also of interest in attrition processes. However, it is not suitable to describe the attrition behaviour of visco-elastic polymers since for these the plastic deformation at the crack tip cannot be neglected and the linear fracture mechanical approach is not applicable [96].

5.4.3. Thermo-mechanical properties Intensity, i.e. velocity and interfacial temperature – which can signiﬁcantly increase, e.g. during the stress mode sliding friction – are two important parameters in attrition. Both are of inﬂuence on the material-speciﬁc reaction of a bulk solid on the stress conditions. Dynamic mechanical analysis (DMA) offers a possibility to measure the mechanical behaviour of polymers as a function of these two parameters. In DMA-measurements, a sample of deﬁned geometry is stressed by a periodical sinusoidal stress. As shown in Fig. 26, rectangular bars of 32 mm 6 mm 2 mm were used for the experiments. From the stress modes available (cf. [97]), the single cantilever ﬂexure mode was chosen. For the measurement, both ends of the sample are ﬁxed in clamps. One of these clamps is ﬁxed, while the other one can be moved vertically up and down by an inductor with a certain amplitude eAmp and frequency f. The frequency can be varied during the experiments to study its inﬂuence on the material properties.

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Fig. 26. Setup of a DMA-measurement in the single cantilever ﬂexure mode.

The optical grid shown in Fig. 26 serves to determine the clamp position during the experiments. The movement of the clamp results in the deformation of the polymer sample and the force necessary to induce this deformation is recorded. From it, together with the sample dimensions, the stress sAmp is calculated. For ideally elastic materials, deformation eAmp and stress sAmp are exactly in phase, viscoelastic materials on the other hand exhibit a phase lag d between these two quantities. From eAmp and sAmp the complex modulus of elasticity E can be calculated according to equation (25). With the phase lag d this modulus can be divided into the storage modulus E0 and the loss modulus E00 (cf. equations (26) and (27)). The storage modulus is proportional to the energy that is stored elastically in the material. The loss modulus corresponds to the energy that is dissipated in the material during one load cycle. ﬃ sAmp qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ E ¼ ð25Þ ¼ ½E 0 2 þ ½E 00 2 Amp E 0 ðoÞ ¼ E cosd

ð26Þ

E 00 ðoÞ ¼ E sin d

ð27Þ

To be able to investigate the inﬂuence of temperature, the measuring cell can be either heated electrically or cooled with liquid nitrogen. Through this, a temperature range from W ¼ 150 y 6001C can be covered. To minimize measurement time and effort, a deﬁned temperature proﬁle was programmed to simultaneously determine the inﬂuence of frequency and temperature in one single measurement. The resulting curves for storage E0 and loss modulus E00 as a function of temperature are given in Fig. 27 for one polymer of each polymer group and a frequency f ¼ 10 Hz. From the measurements, the typical viscoelastic behaviour of the polymers becomes obvious. The polymer chains are ﬁxed relative to each other at low temperatures. Thus the amount of energy that is stored elastically is

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Fig. 27. Comparison of storage (E0 ) and loss (E00 ) modulus for one PP, PE, PMMA and PS each as a function of temperature W.

high which is reﬂected in a high value of the storage modulus. The corresponding values of the loss modulus are low. The mobility of the polymer chains increases with increasing temperature and consequently the storage modulus decreases while the loss modulus increases. In the glass transition regime, the maximum of energy dissipation is reached (maximum of loss modulus) and the storage modulus falls rapidly. From the results obtained in the DMA measurements, the glass transition temperatures Wg were determined. These are given in Table 2. For the determination from the storage modulus, the tangent method was applied, while the absolute peak maximum was evaluated for the loss modulus. According to Ehrenstein et al. [97], the observed deviations in glass transition temperatures lie within the normal tolerance. If the glass transition temperatures are compared to the results of the attrition experiments carried out in the pipe-bend test rig, it becomes obvious that all polymers (PP, PE), for which the attrition mechanisms in effect under pure sliding friction experiments were identiﬁed as dominant, were stressed above their respective glass transition temperatures. For those possessing a glass transition temperature above the stress temperature (PMMA, PS), the attrition mechanisms in effect under normal impact conditions were identiﬁed as relevant. Since the pipe-bend experiments were carried out under ambient conditions, the stress temperature was approximately Wst ¼ 251C. If the attrition rates obtained in the pipe bend after nine consecutive impacts are plotted as a function of the storage and loss modulus values determined at W ¼ 251C and f ¼ 10 Hz, the graphs given in Fig. 28 are obtained. For the polymers stressed above their respective glass transition temperatures, decreasing values of E0 and E00 lead to decreasing attrition rates. Apparently, there is a correlation between the thermo-mechanical behaviour and the

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Table 2. Glass transition temperatures Wg as obtained from storage (E0 ) and loss (E00 ) modulus values

Wg (1C) Tradename

E0

E00

PP 1040 N PP 1100 RC PP 1148 RC PP 2500 H PE 2420 H PE 5031 L PMMA G7 PMMA G55 PS 144 C PS 158 K

17.54 15.04 15.16 12.43 122.32 107.31 107.69 90.12 92.13 107.84

15.08 15.10 15.08 9.99 114.82 109.81 110.20 95.12 95.17 110.20

Fig. 28. Attrition rates measured in pipe bend after nine impacts as a function of (a) storage (E0 ) and (b) loss (E00 ) modulus values determined at W ¼ 251C and f ¼ 10 Hz.

mechanisms in effect under sliding friction stresses. To verify this hypothesis, the attrition rates obtained for PP and PE in the sliding friction experiments with an untreated steel surface at a contact pressure pc ¼ 0.61 MPa, s ¼ 200 m and vs ¼ 6 m/s were plotted against the corresponding storage and loss modulus values (Fig. 29). Here, the modulus values were determined at W ¼ 501C since as Frye [73] has shown, this is a reasonable approximation for the temperature at the contact interface. The regression lines shown in Fig. 29 indicate that the correlation between attrition rates and storage as well as loss modulus is approximately linear for the

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Fig. 29. Attrition rates measured in sliding friction experiments at pc ¼ 0.61 MPa, s ¼ 200 m and vs ¼ 6 m/s as a function of (a) storage (E0 ) and (b) loss (E00 ) modulus values determined at W ¼ 501C and f ¼ 10 Hz.

polypropylenes. The attrition rate increases with an increasing amount of elastically stored energy as well as with an increasing amount of dissipated energy. A detailed explanation of these ﬁndings, which is based on the attrition mechanisms known from tribology, which were discussed in Section 2.3.2, will be given in Section 5.5. While PE 5031 L also follows the described trend, although deviations are observed for the loss modulus, PE 2420 H does not ﬁt at all. For this polymer, which possesses the lowest storage and loss modulus values of all polymers, additional mechanisms apparently contribute to attrition, which are not reﬂected in the thermo-mechanical behaviour.

5.4.4. Experiments on the process scale Since the scope of the present research is to further the understanding of attrition processes and to develop simpliﬁed test methods to predict the amount of attrition that will be encountered as a result of the conveying process, establishing a link between the single-particle experiments and pneumatic conveying installations is of great importance. For this purpose, attrition experiments were carried out in a pilot–plant scale conveying installation at the Chair of Process Engineering of Disperse Systems at the TU Mu¨nchen Weihenstephan headed by Prof. Karl Sommer. In these experiments, which were conducted for dilute phase as well as plug ﬂow conveying, stress conditions closely related to those of industrial conveying setups were realized. The total length of the conveying installation, which is shown in Fig. 30, is Ltot ¼ 103 m. The pipe is made of stainless steel and has a diameter of D ¼ 50 mm. In total nine 901 pipe bends with a radius of curvature rB ¼ 0.525 m

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Fig. 30. Setup of pilot plant scale pneumatic conveying system.

Table 3. Process parameters average gas velocity u and bulk solids load ratio m as well as resulting pressure drop Dptot in pilot plant conveying installation

Dilute phase conveying

Plug ﬂow conveying

Tradename

u (m/s)

m

Dptot (bar)

u (m/s)

m

Dptot (bar)

PP 1040 N PMMA G7 PS 144 C

38.67 38.73 38.70

7 6 6

0.739 0.653 0.699

0.42 0.55 0.53

195 149 179

1.296 1.430 1.363

are present in the installation, three of them vertical, the rest horizontal. Due to the large amount of material that was required for the experiments, these were only performed for three polymers, i.e. PP 1040 N, PMMA G7 and PS 144 C. The process parameters of the experiments are given in Table 3. In Fig. 31(a) and (b), the measured attrition rates A are given for plug ﬂow and dilute phase conveying in the pilot plant scale installation. These are compared to

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Fig. 31. Comparison of the attrition rates A obtained in the pilot–plant scale installation for (a) dense phase conveying, (b) dilute phase conveying to (c) those measured in the pipebend installation.

the results obtained in the pipe-bend installation for three, six and nine consecutive impacts with gas and particle entrance velocities of approximately 41 m/s (Fig. 31(c)). It has to be emphasized that the attrition rates cannot be compared quantitatively due to the varying stress conditions between the respective experiments. For plug ﬂow conveying, no signiﬁcantly differing attrition rates were obtained. If at all, it can be reasoned that PP 1040 N shows a slightly lower attrition in this mode of conveying. For dilute phase conveying however, the relative attrition behaviour is very similar to that observed in the pipe-bend installation. It is thus possible to conclude that the same attrition mechanisms are in effect. Consequently, the pipe-bend installation can be regarded as a suitable experimental setup to simulate the stress conditions that are encountered in pneumatic conveying installations which are operated in dilute phase. An analysis of the attrition fragments that were formed as a consequence of dilute phase pneumatic conveying corroborates the conclusions drawn concerning the governing material-speciﬁc attrition mechanisms. As can be seen from Fig. 32(a) and (b) both PS 144 C and PMMA G7 show small attrition fragments. These are typical for attrition by normal impact stresses. This observation is in agreement with the material-speciﬁc attrition mechanisms that were identiﬁed from the experiments on the single-particle scale for both PS 144 C and PMMA G7. The attrition of PP 1040 N has a completely different form. Here, angel hair is found. This is an indication for high temperatures at the particle-wall interface during impacts. A possible explanation for these is the dominance of sliding friction in attrition. This again veriﬁes the conclusions drawn from the singleparticle experiments.

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Fig. 32. Analysis of the attrition fragments observed after dilute phase pneumatic conveying for (a) PS 144 C, (b) PMMA G7 and (c) PP 1040 N.

Fig. 33. Three-level model of attrition processes based on the ﬁndings from the process and material functions.

5.5. Qualitative model of attrition in pneumatic conveying Based on the results of the determination of the process and material functions, a three-level model which in particular pays attention to the multiscale nature of attrition processes was developed. The top level of this model constitutes of the stress mode which in Fig. 33 is exemplary given for oblique impacts as observed in dilute phase conveying.

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Through the stress mode as well as the stress intensity and the number of stress events, the process function is deﬁned. Only if these quantities are known for an attrition process, can experiments on the single particle and continuum mechanics scales deliver a deeper understanding of attrition formation. Results from these two scales are reﬂected in the material-speciﬁc reaction of different bulk solids on the stresses of the process function. The experiments to determine the material function clariﬁed that attrition formation and thus the material-speciﬁc reaction to the stresses of the process function, varies with the materials used, i.e. the material properties. It was shown through the experiments that in dilute phase conveying PP and PE granules are mainly attrited by the same mechanisms in effect under pure sliding friction stress conditions. Consequently, the corresponding material-speciﬁc attrition mechanism was termed Friction. The PMMA as well as the PS granules on the other hand are attrited by mechanisms observed under normal impact stress conditions, thus the corresponding attrition mechanism was termed Impact. However, it has to be stated at this point that the identiﬁed material-speciﬁc attrition mechanisms shown in Fig. 33 can only be considered as boundaries of a spectrum. Depending on the material properties, both mechanisms contribute to attrition on varying scales. By applying DMA, it was possible to show that the glass transition temperature is a key parameter in the identiﬁcation of the dominating material-speciﬁc attrition mechanisms. Materials stressed in their glass transition regimes are attrited by Friction, while Impact is dominating for materials whose glass transition temperature lies above the stress temperature. These ﬁndings can be explained with the temperature-dependent material behaviour of the bulk solids. The materials are brittle below their glass transition, and thus most of the impact energy is stored elastically in the particles. This leads to the emergence of a three-dimensional stress ﬁeld, which reaches far into the particles. These stresses lead to the formation of attrition fragments due to growth of microcracks, accumulation of internal ﬂaws or just by locally exceeding the material strength. Due to only minor plastic deformation, the surface asperities of the pipe wall do not penetrate far into the material. Consequently, the tangential impact velocity component is of minor importance because its main effect results in an asymmetry of the stress ﬁeld. On a microscopic scale, attrition fragments due to this material behaviour are formed by chipping, which has been described by Ghadiri et al. [22–24]. If the stress intensities are high, fragmentation can even be the consequence. In Fig. 34(a), chipping observed on a PMMA G55 particle after nine impacts in the pipe bend is shown. This behaviour was also found for PMMA G7, PS 144 C and PS 158 K. For PP and PE on the other hand, no chipping could be detected (cf. Fig. 34(b)). Fragmentation of the polymer particles was never observed in any of the experiments.

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Fig. 34. Comparison of attrition induced changes on (a) a PMMA G55 particle and (b) a PP 1040 N particle.

In case Tst lies above Tg, the materials are stressed in their glass transition regime and react viscoelastic. Here, the three-dimensional stress distribution develops in the immediate vicinity of the contact region. Due to plastic deformation by relative motion of the polymer chains, no stresses reaching into the material develop, but the wall surface asperities penetrate into the polymer and remove attrition fragments due to the acting vp,t. On a microscopic scale, attrition is caused by abrasion, surface disruption or adhesion. By taking REM pictures of the polymer surfaces before and after the sliding friction experiments, abrasion was identiﬁed as one of the mechanisms contributing to attrition of PP 1040 N (cf. Fig. 35). As discussed in Section 2.3.2, according to Czichos and Habig [42], different mechanisms leading to attrition by abrasion can be distinguished. A comparison of the unstressed surface with the stressed surface shows signiﬁcant furrows. Therefore, both micro plowing and micro fatigue contribute to attrition. Besides that, a chip which is a result of micro cutting can be observed in the right picture of Fig. 35. Which of the three mechanisms dominates is mainly a function of the angle of the surface asperities relative to the sliding surface [42]. For the other PP and PE granules, similar ﬁndings were obtained. With the attrition model, the correlation, which was found between storage as well as loss modulus and attrition rates found in the sliding friction experiments for PP (cf. Fig. 29) can be explained. For this purpose, the microscopic attrition

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Fig. 35. Comparison of the contact surface of PP 1040 N before and after a sliding friction experiment on vanadium steel with a sandblasted surface.

Fig. 36. Interrelation of tribological attrition mechanisms abrasion, surface disruption and adhesion with the material properties of the polymers.

mechanisms abrasion, surface disruption and adhesion are shown in Fig. 36 for the contact of a single-surface asperity and the polymer surface. The formation of attrition fragments due to abrasion can be correlated with the plastic deformation of the polymer. A measure for this is the loss modulus, which is proportional to the energy dissipated through viscous deformation. The higher the loss modulus, the higher the viscous deformation. This leads to a deeper penetration of the surface asperities of the steel into the polymer surface which in turn results in higher attrition rates. For surface disruption on the other hand, the number and distribution of cracks, dislocations or voids close to the surface is relevant. Surface disruption is caused by elastic stressing of the material through the surface asperities of the steel. The

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induced compressive (sc) and tensile (st) stresses can exceed the material strength at the mentioned material imperfections which ultimately leads to the formation of attrition fragments. On top of that, repeated elastic stressing of the surface leads to fatigue thus worsening the effect of surface disruption. To which extend surface disruption leads to attrition can be qualitatively determined by the storage modulus. A low value of the storage modulus corresponds to low attrition rates due to surface disruption since according to the theory of elasticity, for a constant outer stress, the elastic deformation is higher which leads to lower internal stresses. The approach to adhesion is more complex, since this attrition mechanism is inﬂuenced by molecular interactions. Consequently, surface properties of both the polymers and the wall material determine to which extend adhesion contributes to attrition. These complex interactions at contact surfaces are not well understood, which is the reason why, at present, no material properties can be directly related to adhesion.

6. BRIDGING THE GAP BETWEEN ACADEMIC RESEARCH AND INDUSTRIAL NEEDS So far, the results presented predominantly originate from academic research. As was stated at the beginning of this chapter, a lot of empiricism is still involved, when new pneumatic conveying lines are planned. This is particularly the case, when the bulk solid that is to be conveyed is a new product whose conveying and attrition characteristics are unknown. Commonly, such a product is tested in pilot–plant scale conveying installations to obtain suitable data for scale-up. This procedure is time consuming and cost intensive. Besides that, large amounts of product are required, which can be difﬁcult to obtain or, in the worst case, are not available at all. From this, especially in industry, speciﬁc needs arise, which are discussed in this section. The scope is to point out, where academia and industry can join forces in order to achieve signiﬁcant progress in attrition research. To reduce time consumption and expenditures in connection with the design of pneumatic conveying systems and to develop a sound estimate of the attrition that will be encountered, the development of lab-scale tests is favoured by industry. Such lab-scale tests have to meet several requirements. As discussed above, the stress conditions leading to attrition can widely vary (from impacts to sliding friction). Consequently, these conditions, in particular stress mode and intensity, have to be reproduced in the lab-scale tests. It is thus likely that it will not be possible to develop a single test that covers all stress conditions. Besides that, the tests have to generate signiﬁcant attrition in order to lead to reproducible results. This can be either realized by increasing the stress intensity, which is sometimes

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Fig. 37. Outline of possible cooperation between academia and industry in attrition research.

problematic or by increasing the number of stress events, which is usually favoured. To ensure well-deﬁned stress conditions, it is necessary to conduct singleparticle experiments. On the other hand, a large number of particles has to be stressed in order to obtain statistically signiﬁcant results. It will be one of the biggest challenges in the development of suitable lab-scale tests, to unite these two opposing requirements. Besides that, care has to be taken that the tests are feasible, i.e. easy to operate, operable with small amounts of product, reliable and, of course, economic. A possible way to combine academic research with industrial needs is outlined in Fig. 37. The state of knowledge in attrition research, which has been covered in this chapter, serves as a foundation for further research. Furthermore, the concept of process and material function has proven suitable to further the understanding of attrition formation. The methods for determination of these two functions however have to be continuously reﬁned in ongoing research. Here, academia and industry should join efforts to maximize the success of research on attrition. From an industrial standpoint, it is not possible to provide the manpower needed to successfully advance in attrition research. This ‘‘full-time’’ research has to be carried out by academic researchers. However, industry can assist with practical experience, test material or pilot-plant possibly even industrial scale test facilities. Furthermore, experienced personnel to carry out experiments in these facilities can be provided. It is believed that through the cooperation between academia and industry, which bundles the respective strengths, signiﬁcant progress towards a better understanding of attrition processes, towards the development of the desired labscale tests and hopefully even towards the long-term goal – the development of a quantitative attrition model – can be achieved.

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7. SUMMARY Although pneumatic conveying has long been used to transport a large variety of bulk solids, the problem of product degradation due to attrition is still largely unresolved. It has been shown that one reason for this is the multiscale nature of attrition processes. Only if experimental as well as theoretical results obtained on the process, single particle, continuum and molecular scales are combined, can the understanding of the mechanisms leading to attrition be furthered. Based on this insight, a concept to investigate attrition formation was presented in which it is distinguished between a process and a material function. The process function summarizes the inﬂuence of all process parameters and as a result provides the stress conditions in terms of stress mode, intensity and number of stress events. The material function on the other hand contains the material properties which govern the attrition behaviour and as a result supplies an attrition function. Both functions were determined by applying theoretical and/or experimental methods. For dilute phase conveying, it was found that the impact conditions in a pipe bend are considerably different from what is commonly expected. The impact angles were determined to lie between aW ¼ 5 and 351, with an average value of aW ¼ 221. Consequently, the tangential impact velocity components are considerably higher than the normal ones. For dense phase conveying, sliding friction between the particles forming the shell of a plug and the pipe wall was identiﬁed as the relevant stress mode. The experiments carried out to determine the material function revealed that the dominating attrition mechanisms differ between PP and PE particles on one hand and PMMA and PS particles on the other. By applying DMA, the glass transition temperature of the polymers was identiﬁed to be a key factor in the determination of the prevailing attrition mechanism. Based on these ﬁndings, a qualitative three-level model of the attrition process involving stress mode, material-speciﬁc attrition mechanisms and basic (microscopic) attrition mechanisms was developed. Finally, the results obtained in academic research were compared to industrial needs with respect to attrition in pneumatic conveying. It was shown that a close cooperation between industry and academia offers the possibility to achieve signiﬁcant progress in the understanding of attrition research. Nomenclature Latin symbols A A a a

attrition rate (–) empirical parameter according to Neil and Bridgwater [16] (s/m) ﬁtting parameter (sb/mb) characteristic size of zone of plastic deformation according to Gahn [28] (m)

New Concept for Addressing Bulk Solids Attrition

a b c c cm,p D d E E E E0 E00 Ei,crack

e F f fMat G H JQd Kc Kn Kt k k k l M _ M m m Np/W n ni np PB p _ Q Q0

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radius of contact circle (m) ﬁtting parameter (–) speed of sound (m/s) ﬁtting parameter of Weibull function according to Salman et al. [31] (m/s) ratio of mass of disperse phase and mass of continuous phase (–) diameter distance energy (J) Young’s modulus (Pa) complex modulus from DMA-measurements (Pa) storage modulus from DMA-measurements (Pa) loss modulus from DMA-measurements (Pa) minimum energy according to Potapov and Campbell [50] that is required to initiate a crack which completely propagates through a particle (J) coefﬁcient of restitution absolute value of force vector (N) frequency (Hz) mass-speciﬁc material parameter introduced by Vogel [32] (kg/(J m)) shear modulus (Pa) hardness (Pa) J–integral value (J/m2) fracture toughness (N/m3/2) normal material stiffness according to Potapov and Campbell [50] (N/m) tangential material stiffness according to Potapov and Campbell [50] (N/m) number of stress events according to Vogel [32] (–) spring constant (N/m) turbulent kinetic energy (m2/s2) length (m) mass (kg) mass ﬂow (kg/s) ﬁtting parameter according to Salman et al. [31] (–) empirical parameter according to Gwyn [18] (–) number of total particle-wall impacts in pipe-bend test rig (–) number of revolutions (s1) number of impacts (–) number of particles (–) breakage probability (–) pressure (Pa) heat ﬂux (W) number distribution (–)

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R2 r S s T t u V v vp,i vp,n vp,t vR,j vs Wm,kin Wm,min w x x, x0 xp y y, y0 yA z z, z0

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stability index (–) radius (m) (interparticle) spacing (m) sliding distance (m) absolute temperature (K) time (s) absolute value of velocity (in x-coordinate direction) (m/s) volume (m3) absolute value of velocity (in y-coordinate direction) (m/s) absolute value of particle impact velocity (m/s) absolute value of impact velocity normal to tangential plane to impact point (m/s) absolute value of impact velocity parallel to tangential plane to impact point (m/s) circumferential (rotor) velocity (m/s) sliding velocity (m/s) mass-speciﬁc kinetic energy at the moment of impact according to Vogel [32] (J/kg) minimum mass-speciﬁc kinetic energy at the moment of impact according to Vogel [32] (J/kg) absolute value of velocity (in z-coordinate direction) (m/s) Cartesian coordinate direction (m) transformed Cartesian coordinate directions (m) particle diameter (m) Cartesian coordinate direction (m) transformed Cartesian coordinate directions (m) ratio of mass of attrition and initial mass (–) Cartesian coordinate direction (m) transformed Cartesian coordinate directions (m)

Greek symbols a a a, a aW b bmax D d e e Z Z

angle in global x-y-z-coordinate system (1) impact angle (1) ﬁtting parameters (–) wall impact angle (1) empirical parameter according to Neil and Bridgwater [16] (–) speciﬁc fracture plane energy (J/m2) difference (–) phase lag (–) strain (–) dissipation rate of turbulent kinetic energy k (W/kg) damping coefﬁcient (N/(ms)) parameter for summarizing material properties and stress conditions (–)

New Concept for Addressing Bulk Solids Attrition

Zf W m mfr n r s sc sx sy, sbr t f j1, j2 o x

dynamic ﬂuid viscosity (Pa s) Celsius temperature (1C) bulk solids load ratio (–) friction coefﬁcient (–) Poisson ratio (–) density (kg/m3) (normal) stress (Pa) compressive failure stress according to Neil and Bridgwater [16] (Pa) axial stress (Pa) yield respectively breakage stress (Pa) shear stress (Pa) empirical parameter according to Neil and Bridgwater [16] (–) angles in transformed coordinate systems (1) angular frequency (s1) dimensionless amount of attrition per stress event (–)

Vectors F Fp,f g N S u v x

force pressure force exerted by particle collective on ﬂuid ﬂow acceleration due to gravity normal vector surface vector ﬂuid velocity velocity position in Cartesian coordinates

Indices 0 Amp a av B br c c circ cr e F f fr

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starting value amplitude attrition, attrited average bend breakage contact compressive circumferential critical entrance ﬁlter ﬂuid, gas friction

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g g i i int j kin l m max min n n p p/p p/W R sp st t tot V W y

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glass transition gravity initial particle i intact particle j kinetic layer mass speciﬁc maximum minimum normal integration timestep particle particle-particle particle-wall rotor sidepressure stress tensile total Vickers wall y-coordinate direction

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SUBJECT INDEX Abrasion, 916–917, 919, 927, 1022–1023, 1031–1032, 1036, 1038–1039, 1043, 1162–1163, 1208–1209 Academic research, 1211–1212 Adhesion, 1162–1163, 1165, 1208–1210 Agglomerate Cohesion, 741–745, 747–749, 751–755, 757–761, 763, 765–767, 769–771, 773, 775, 777–779, 781, 783, 785–787 Failure, 855 Formation, 741–744, 746, 750–753, 759–760, 762, 765–766, 777–778, 780–782 Mass, 744–747, 749, 754–763, 765–772, 774–787 Porosity, 743, 748–749, 751, 754, 756–757, 761, 765, 771, 787 Strength, 743–744, 747–751, 760–761, 772, 776–778 Aggregate Average mass, 759, 762–763, 772–773, 777, 779, 783 Average porosity, 757 Breakage, 741, 752, 760–762 Growth, 742, 745, 752–753, 778–782, 784–786 Mass distribution, 759, 777, 779, 781 Strength, 743–744, 747–751, 760–761, 772, 776–778 Aggregation Diffusion-limited, 741–742, 745, 773, 777, 779 Reaction-limited, 741, 747, 775, 779 Agitating ball mill, 492 Agitation mills, 510 Agitator geometry, 256 Air jet, 422–423, 425, 427, 429, 431, 433 Air jet mill, 494, 498, 501, 883, 905 Alumina, 567–570, 581–582, 584–585, 589–591, 593–594, 597, 972–973, 1085, 1090, 1092

Aluminium oxide, 1121, 1123, 1126, 1130–1132, 1136, 1146 Analytical solutions, 74, 75, 77, 647, 793, 795, 801–802, 804, 824, 832, 973, 1107 Annular cell, 95, 97, 102–104, 114–115 Annular gap geometry, 256–257, 275, 312–313, 326 Ash, 1020, 1024–1025 Attrition, 743–744, 752, 758, 761–762, 769, 771, 916–919, 926–937, 1019–1027, 1029–1033, 1035–1041, 1043–1049, 1149–1171, 1173, 1175, 1177, 1179, 1181, 1183–1199, 1201–1215 Autoadhesion, 1055, 1060, 1064, 1086, 1111, 1113 Derjaguin, Muller and Toporov (DMT) theory, 1115, 1090, 1055, 1090 Johnson, Kendall and Roberts (JKR) theory, 1066 Autogenous grinding, 252, 358–362 Autogenous milling, 480 Ball Mass, 540–542, 547–549 Media mill, 493, 498, 501 Mill grindability, 535, 548 Milling, 437, 441, 459, 465, 471, 475, 477, 479, 481–482, 510, 529, 535–536, 538–540, 542, 546, 548 Barite, 437, 452–455, 463–464, 472, 474, 476–479, 482–484 Batch grinding equation, 879 Batch operation, 265–267, 333–334, 493, 501 Binder, 1056–1059, 1073, 1081–1082, 1086–1087, 1091–1092, 1095–1096, 1101–1109, 1113 Bond’s law, 529, 531 Bond’s Work index, 13–14, 529, 531, 535, 549

1220 Born interaction, 551, 560–562, 600 Bread, 384, 386, 388–391, 412–415 Breakage, 87–89, 91, 93–95, 97, 99, 101–103, 105, 107–115, 1019–1020, 1023–1024, 1031–1037, 1039–1040, 1043, 1049, 1055, 1057–1059, 1061, 1063, 1065, 1067, 1069, 1071, 1073, 1075, 1077, 1079, 1081, 1083, 1085, 1087, 1089–1093, 1095–1105, 1107, 1109, 1111, 1113–1116 Energy, 63, 611, 630 Equation, 397–399, 401, 403–405, 410, 412–413, 415 Function, 511, 884, 886, 888, 891, 896, 899 Parameters, 439, 444, 448, 452, 454, 455, 457–463 Pattern, 838–839, 854–860 Probability, 33–36, 511, 896, 899, 906, 909 Break-up, 593, 760, 787, 980 Brittle fracture, 4, 638 Bubble cap, 1026, 1030 Bubbling, 1029 Calcination, 1024–1025, 1037–1039 Calcite, 437, 447–453, 463–464, 472, 474, 476–480, 482–484 Catalyst, 1020, 1022–1023, 1043, 1048 Cereal bioreﬁnery, 396, 414 Cermet, 525 CFBC, 1023, 1040 Chipping, 917, 928, 1158–1159, 1164, 1166, 1207 Chromite and ceramic raw materials, 455, 456 Circuit operation, 263, 266, 329–332 Circulating, 1021–1023, 1025, 1032, 1041, 1043, 1048 Classiﬁcation, 434, 873–874, 876, 878, 885–891, 898–900, 905, 907, 909 Clinker, 437, 455, 458, 463, 467, 482–483 Closed-circuit grinding, 501 Coagulation, 552, 562, 564, 567, 574, 576 Coal, 457 Coefﬁcient of restitution, 17, 28–31, 1103, 1105–1106, 1108, 1113

Subject Index Cohesion, 741–745, 747–749, 751–755, 757–761, 763, 765–767, 769–771, 773, 775, 777–779, 781, 783, 785–787 Collision Frequency, 741, 744, 752–758, 760–763, 765, 768–769, 777–779, 782, 784–785 Time, 745, 749, 758–759, 762, 769, 771–772, 774–775, 777–784 Colloid mills, 510 Combustion, 1020, 1022–1025, 1032, 1036, 1043, 1049 Combustor, 1021, 1023, 1040–1041 Comminution, 60–63, 117–123, 125, 127, 145–146, 148–152, 154, 157, 159, 162–164, 179, 184–191, 198, 202, 207–208, 210, 214, 216–218, 873, 880–881, 884–886, 892, 894, 896–898, 901–902, 905, 907, 910–911 Compaction, 941–945, 948–956, 962–966, 968 Modelling, 964 Simulator, 941–942, 952–956, 965, 968 Compression, 1121–1141, 1143–1147 Strength, 925, 928 Test, 18–21, 1121, 1123–1124, 1127–1130, 1132–1133, 1136, 1140 Computer simulation, 838–839, 843, 845, 847, 862 Computational Fluid Dynamics, 892, 929, 1166 Connection Constraint, 771, 778, 782, 784, 786 Frequency, 741, 744, 752–758, 760–763, 765, 768–769, 777–779, 782, 784–785 Number of, 741, 753, 755, 757, 761, 763, 767, 769, 777–779, 781, 783, 787 Connectivity, 1037 Contact angle, 1062–1063, 1068, 1074, 1076–1078, 1110–1111, 1113, 1116 Contact radius, 1067–1068, 1071, 1084, 1088, 1114 Continuous Distribution Kinetics Modeling, 971, 985

Subject Index Continuous operation, 493, 501 Crack, 1122, 1125–1128, 1130, 1132–1141, 1146–1147 Hertzian, 1122, 1125, 1127–1128, 1130, 1137, 1146 Median, 1132–1133, 1135–1139 Propagation, 5, 1125–1127, 1137, 1139–1141, 1146–1147 Crazing, 1122, 1134–1135 Critical speed, 540–542, 550 Critical surface tension, 1062–1063 Crushing, 117–127, 129, 131, 133, 135, 137, 139, 141–155, 157, 159, 161, 163–177, 179–187, 189, 191, 193, 195, 197–199, 201, 203, 205, 207, 209, 211, 213–218, 606, 610–611, 613, 631 Parameters, 117, 148–150, 153 Cryogenic grinding, 502 Crystal Growth, 971–975, 977, 979, 981, 983, 985 Cut point, 505 Cyclone, 1019, 1021–1023, 1032, 1041, 1043 Damage, 605 DAPRAL, 568–570, 584–585 Davison, 1043–1044 Debranning, 430 Density distribution, 941–942, 956–957, 961–963, 965–968 Diametric compression, 1055–1056, 1082, 1090–1092, 1097, 1099–1100 Die ﬁll, 943–944, 949, 953, 962, 965, 968 Disc agitator, 256–257 Discrete element method (DEM), 61, 510, 839, 846, 847, 989, 1004, 1056, 1099, 1149, 1165, 1181 Disintegration, 119, 251, 254, 271–273, 277, 283–284, 296–303, 305, 496, 794, 838, 845, 851, 853–855, 857–858, 860, 1023, 1039, 1095–1096, 1100–1101, 1133, 1147 Dispersive interaction, 1063 Distribution, 252, 258, 260, 262–263, 265, 269–271, 273–274, 284, 286–287, 292, 297, 303–304, 311, 317, 321–324, 326–335, 342–345, 358–361, 368, 370–372, 374, 378–379

1221 Distributor, 1021, 1023, 1028, 1037, 1049 DLVO theory, 562–563, 572, 589 Donor acceptor model, 556 Double impact tests, 69–72 Double layer, 551, 557–561, 564, 572 Drop weight testing, 9–12 Drucker-Prager cap model, 964, 967, 968 Dry grinding, 493–494, 500–501 Dry ball mills, 879, 888 Dustiness, 1027, 1035, 1045, 1048 Ductile-brittle transition, 54–55, 921 Dwell time, 941–942, 944, 949–950, 954–955 Dynamic Mechanical Analysis, 1199 Dynamic yield stress, 1102–1103, 1106–1108, 1113 Ejection, 941, 943–945, 949, 951, 954 Elastic strain energy, 532 Electrostatic interactions, 551, 554 Electrosteric, 316, 562, 564 Elutriation, 1019, 1025, 1027, 1029, 1039, 1044–1045, 1048 Energy transfer, 17–18, 30–33, 251–252, 256–257, 264, 269–271, 274, 279–283, 295–296, 306, 309, 313, 364–365, 369–370, 372–375, 378–380 Energy transfer factor, 251–252, 274, 281–283, 306, 309, 313, 364–365, 369–370, 372–375, 380 Energy utilisation, 44–45, 49, 271–273, 292–293, 320, 378 Energy-related materials, 525–526 Erosion, 919, 1029, 1035 ESR spectra, 522–523 Failure, 661, 663, 665–669, 671, 673–675, 677–680, 683–688, 690–696, 698–700, 702, 704, 706, 708, 710–712, 718, 720–721, 724–727, 729–734, 736–737 Mode, 1121, 1123, 1125–1127, 1129–1131, 1133–1135, 1137, 1139–1141, 1143, 1145 Zone, 89, 91–92, 94, 97, 110, 113 Fatigue, 917, 919, 921, 924–926

1222 FCC, 1020, 1022–1024, 1030–1031 Feed mass, 540–542, 546–549 Feed properties, 498, 499 Feed size, 529, 535, 540–549 Filling ratio, 251, 262, 276, 284, 286, 304–306, 313, 333, 338–340, 342, 344–348, 359, 361, 364, 376, 380 Fine grindability, 529, 546–547 Fine grinding, 529, 531, 533–537, 539–541, 543, 545–547, 549 Fine grinding mills, 509–510 Flow sheeting for solid processes, 893 Fluid catalytic cracking (FCC), 1020 Fluid energy, 421, 494 Fluidisation, 1020–1022, 1025–1028, 1030, 1033, 1041 Fluidised bed, 1019–1027, 1029–1033, 1035–1037, 1039–1045, 1047–1049 Attrition, 1019–1053 Breakage, 1019–1020, 1023–1024, 1031–1037, 1039–1040, 1043, 1049, 1051 Granulation, 1022–1023, 1032–1033, 1035 Jet, 1019, 1031, 1041–1046, 1048, 1050 Jetmill, 1029 Milling, 1029 Shear, 1020, 1022, 1031, 1039, 1044, 1049 Size reduction, 1020, 1034–1035 Fluidised bed type mill, 488, 494, 495, 499 Forces Breakup, 761 Hydrodynamic, 763–764, 778, 782, 786 Formulation, 915–916 Fractal dimension, 743, 745, 747, 750, 753, 761, 771, 785 Fractionation, 385, 396, 410, 412, 414 Fracture, 605, 607, 613, 617–619, 621–625, 627, 743–745, 760–761, 763 Fracture energy, 33–35, 529, 532–534, 548 Fragmentation, 41–45, 922, 924, 927, 929–930, 971, 974, 977, 986, 1022–1024, 1035–1041, 1043, 1047, 1049, 1158, 1163, 1166, 1207 Irreversible, 762–763

Subject Index Models of, 741, 763 Rate of, 741, 749–750, 753–754, 757–760, 762, 765, 772, 777, 779, 781 Frequency Mass, 744–747, 749, 754–763, 765–772, 774–787 Pore volume, 755, 757 Friability, 941–942, 956–957, 960–962, 966–968 Friction, 513–516, 527, 609, 942, 944–945, 949, 953, 955–956, 962–969, 1055–1056, 1059–1060, 1069–1072, 1083, 1093, 1107–1109, 1114, 1116 Adhesion mechanism, 1070 Amonton’ Law, 1070, 1072 Coulomb’s law, 1070, 1072, 1116 Sliding criteria, 1077 Static friction (adhesive peeling), 1055, 1071 Frictional coefﬁcient, 515 Generalized method of moments, 828 Gill plate, 1022–1023, 1029 Glass, 1121, 1123–1130, 1134, 1141, 1143–1144, 1146 Glass transition temperature, 1201–1202, 1207, 1212 Grain trade, 385, 391 Granulation, 862, 869, 1022–1023, 1032–1033, 1035, 1056–1060, 1073, 1080, 1084, 1090–1091, 1102, 1104–1105, 1113 Granule, 1055, 1057–1061, 1063, 1065, 1067, 1069, 1071, 1073, 1075, 1077, 1079, 1081–1083, 1085–1116, 1125–1126, 1133, 1138–1141, 1145–1147 Binderless, 1060, 1091–1092, 1094, 1110, 1121, 1144–1145, 1147 Calcium Carbonate, 1121, 1126, 1136–1140, 1142, 1144, 1147 Dry, 1141, 1143 Fertilizer, 1121, 1126, 1133, 1139, 1147 Solid, 1055–1057, 1059–1064, 1066–1067, 1069, 1074–1075, 1077, 1080–1083, 1085–1086, 1088–1089, 1091–1092, 1094, 1096, 1099, 1101, 1106, 1110–1116

Subject Index Wet, 1056–1057, 1060, 1073–1074, 1087, 1092, 1094, 1100, 1102–1104, 1106–1109, 1113, 1121, 1141, 1145, 1147 Granule strength, 1055, 1058–1060, 1082–1083, 1086–1088, 1090–1091, 1096–1097, 1102, 1112–1113 Ensemble elastic modulus, 1055, 1083–1086, 1089 Kendall’s theory, 1055, 1087 Rumpf’s theory, 1055, 1081, 1086–1088, 1090 Grinding, 605, 639, 641–647, 649, 652–653, 655, 657–658, 660 Kinetics, 879–880, 891 Rate constant, 515–516, 541–545 Time, 251, 263–265, 267, 269, 274, 276, 278–279, 283–285, 287, 289, 294, 333, 358–360, 377 Grinding media, 251–253, 255–259, 261–267, 274–290, 294–296, 298–301, 304–307, 309–311, 313, 315–322, 327, 329, 333, 336–362, 366–369, 372, 374, 376–380 Density, 286–288, 290, 295, 338, 341, 343, 348, 366 Size, 262, 284–290, 294–296, 299–300, 304, 320–321, 327, 333, 338–339, 345–346, 348–349, 352–353, 366–368, 374 Growth regime map, 1109 Hamaker constant, 553–554, 566, 598 Hardgrove grindability index, 529, 535–536 Hardness, 605, 613, 616–617, 621–624, 631, see also Kernel hardness Hertz theory, 38–40, 1064 Heterogeneity, 661, 666, 668–669, 671–673, 677, 686, 688–694, 699–700, 706, 734, 736 Heubach, 1044–1045, 1048 High-pressure roll mill, 496 High-speed hammer mills, 883, 886 Holmes’s law, 529, 531 Hydration interactions, 561 Impact, 6–18, 21–32, 421–426, 1022–1023, 1025, 1031–1032, 1037,

1223 1039–1041, 1045–1049, 1055–1056, 1058, 1081–1082, 1087, 1092–1096, 1098–1106, 1108–1109, 1115 Energy of balls, 515 Grinding, 424, 500, 892, 894 Load cell, 21–32, 51, 53, 55 Mill, 488, 490, 498, 505, 510 Tests, 69–72, 79 Indentation, 605, 613–618, 621–622, 627, 629 Industrial granules, 922 Industrial needs, 1211–1212 Industrialisation, 941, 952, 956 Inter-aggregate link, 763 Interface energy, 1061–1063, 1100, 1111, 1116 Solid-liquid, 1062, 1110 Jet, 1019–1020, 1025, 1028–1032, 1041–1045, 1047–1049 Jet mills, 510 Kernel hardness, 388, 396, 404, 405, 407, 412, 416 Kick’s law, 383–384, 529–530 Lime, 1020, 1039 Limited breakage, 793, 795, 813, 832–833 Liquid bridges, 1055, 1057, 1059–1060, 1073–1074, 1077–1078, 1080–1081, 1086, 1107, 1113 Static capillary force, 1055, 1074 Viscous junction, 1055, 1078 Loss modulus, 1200–1203, 1209, 1213 Machine function, 894 Markov Chain, 645–647, 660 Material Function, 896, 1149, 1167–1168, 1175, 1184, 1191, 1199, 1207, 1211–1212 Processing, 509, 523, 526 Properties, 11, 118, 215, 487, 495, 498–499, 507, 514, 605–606, 613, 623–624, 629–630, 663–664, 670, 673, 691–692, 699, 710, 734, 749, 885, 896–897, 902, 907, 928, 948, 967, 997, 1007, 1029, 1034, 1045, 1049, 1095, 1150, 1152, 1155,

1224 1158–1161, 1164, 1166–1167, 1180, 1184–1185, 1189, 1193–1194, 1196–1199, 1207, 1209–1210, 1212, 1214 Testing, 952 Mechanical activation, 517–518, 525–526 Mechanical properties, 606, 612–613, 622–624, 627–629 Mechanochemical phenomena, 509–510, 516–517 Mechanochemical doping, 524, 525 Mechano-chemistry, 590 Media packing, 252, 258, 284, 342–343, 345–346 Media motion, 509–510, 512 Method of moments, 827–828 Micronisation, 433, 434 Mill related stress model, 251, 269, 366, 371 Milling, 605–613, 615, 617–619, 621–625, 627–631, 1029 Milling circuits, 874, 877, 879, 883, 891, 893, 899, 903, 905, 909, 911 Mill models, 900 Mill operation mode, 905 Mixer, 1021 Modell, 239, 261, 279, 323, 325 Modelling levels, 877 Monte Carlo method, 639, 641, 643, 645, 647–649, 651–653, 655, 657, 659–660 Morphological parameters, 437, 439, 441–443, 445, 447, 449, 451, 453, 455, 457, 459, 461, 463, 465, 467, 469, 471, 473, 475, 477, 479, 481, 483–484 Morphology, 421, 431 Multiple granule test, 926 Nanomilling, 551–553, 555, 557, 559, 561, 563, 565, 567, 569, 571, 573–575, 577, 579, 581, 583, 585–587, 589–591, 593, 595, 597–599 Nano-particles, 252, 314, 316, 509, 516, 526 Narrow particle size distribution, 905 Newton-Reynolds diagram, 375 N-mixers in series approach, 883–884 Non-dispersive interaction, 1063 Normal force, 512

Subject Index Nozzle, 1021–1022 Number of conﬁgurations, 741 Number of stress events, 251, 269–271, 274, 276, 283–285, 287, 292–293, 295, 303, 307, 311, 315, 321, 332, 368, 379 Numerical modelling, 664, 734 Numerical simulations, 69, 76, 82–83 Open-circuit grinding, 504 Operating parameters, 251–252, 257, 261–262, 264, 269–270, 283–284, 288–290, 299, 303–304, 316, 322–323, 326–327, 333–335, 339, 341, 345–346, 348–349, 351–354, 362, 364–367 Operation mode, 252, 263, 321–322, 327–328, 330–331, 333, 335, 346 Optimum grinding condition, 529, 540, 542 Optimum stress intensity, 271–273, 301–302, 379 Oriﬁce, 1029–1032, 1042–1044, 1048 Ostwald ripening, 972–975, 984–985 Particle Compounds, 989, 995, 1004, 1016 Modiﬁcation, 506 Porosity, 743, 748–749, 751, 754, 756–757, 761, 765, 771, 787 Size, 742–744, 747–753, 757–758, 761–762, 764–765, 771, 781 Size distribution, 393, 397–399, 401–403, 405–407, 409, 411–414, 422, 424, 430, 473, 749, 751, 753, 757, 761, 781, 1024–1025, 1029, 1037, 1039, 1046–1047 Passage mode, 263, 265–267, 303, 327–328, 330 Pearling, 383, 412–415 Pendulum testing, 12–18 Perfect mixing grinding approach, 883 Perforated plate, 1022, 1025–1026, 1029, 1042 Pharmaceutical, 421, 426, 430–431, 433 Phase change, 509, 517 Pin-counter pin agitator, 256 Pipe bend, 1152, 1160, 1170, 1173, 1175, 1177, 1183, 1189, 1191, 1193, 1195, 1201–1203, 1207, 1212 Planetary ball mill, 493

Subject Index Plastic Deformation, 1123, 1125, 1133, 1136–1137, 1139, 1141–1142, 1146–1147 Pneumatic conveying, 1149, 1151–1153, 1156, 1159–1161, 1167–1168, 1175, 1177, 1181, 1184, 1186, 1203–1205, 1212 Pneumatic transport, 916, 926, 929, 932, 937 Polyelectrolyte, 572 Polymethylmethacrylate, 1133 Polymorphic transformation, 517 Population balance, 429–430, 641, 643–644, 648–649, 879, 884, 891, 893 Powder-bed attrition-type mill, 498 Powder paints, 905 Power draw, 252, 264, 266, 335–336, 338–342, 362–366, 375–378 Power-number diagram, 252, 336–338 Pre-compression, 941–943, 949, 951, 953, 968 Press, 6, 50, 61, 148, 497, 942–943, 949–950, 952–955, 957–958, 962, 1000 Primary ash particle, 1024–1025 Process scale, 1149, 1155–1156, 1186, 1203 Product sizes, 123–124, 316, 438, 483 Product related stress model, 251, 270 Production capacity, 251, 263–264, 274, 303–304, 375 Pull-off force, 1063–1064, 1068–1069, 1114 Punch velocity, 941–942, 949–950, 954–955, 965 Quartz, 437, 444–447, 457–463, 470–471, 473–476, 482–484 Radical, 521–522 Rate process, 509, 511 Reference state method, 899 Regenerator, 1022 Repeated impact test 926, 929–934, 936–937 Repeated loading, 924 Repeated test/tester, 926, 929, 937 Residence time distribution (RTD), 883 Reynolds-number, 259, 336–340, 376

1225 Rheology, 551–552, 577–579, 581–583, 598, 1181 Riser, 1021–1023, 1032, 1041 Rittinger’s law, 529–530 Rod milling, 465, 475, 477, 479, 481–483 Roller milling, 383, 385, 387–389, 391, 393, 395–399, 401–403, 405, 407, 409–411, 413–415, 488, 494, 498, 501, 510, 880, 891 Rotor cutters, 1001 Rotor mills, 230, 245 Rotor shears, 998 Rotor shredders, 1001–1002 Rubber-elastic and ductile material behaviour, 993–995, 997–998, 1016 Scale-up, 252, 338, 362, 364–366, 369–370, 375 Sectional methods, 829 Selection function, 511 Shear, 87–89, 99–107, 109, 111, 113–115, 1031 Shear deformation, 837–838, 862–863 Shear-type mills, 510 Simulation, 261, 437, 459, 462–463, 465–468, 482–483 Simulation of media (balls) motion, 509 Simulator, 60, 461–462, 827, 952–954, 965 Size ratio, 862–869 Size distribution, 411, 509, 511 Size reduction, 411, 509, 518 Solid Bridges, 1055, 1057, 1080–1082, 1096 Concentration, 276, 348 Sim, 873, 893–894, 896, 898–899 State electrolyte, 525 State reaction, 510, 516, 518 Waste, 989, 995 Solvation forces, 1109–1112 Solvation interactions, 561 Sorbent, 1019–1020, 1025, 1036–1037, 1039–1040 Speciﬁc energy, 8, 10, 64, 251, 258, 260, 264–268, 270–272, 274, 278, 281, 283, 287–303, 305–314, 317–321, 335, 342, 346, 355–356, 361–375, 377–378, 411 Spheres, 71–72, 74–76, 79–81

1226 Spouting bed, 1023 Spreading pressure, 1062, 1116 Stability factor, 576, 593 Stabilization, 551–552, 561–562, 564, 568, 570–574, 597–598 Steric interaction, 565, 600 Steric stabilization, 551, 562, 564, 568, 570–573, 597 Sticking probability, 783 Stirred media mill, 251–259, 261–263, 265, 267, 269, 271, 273–275, 277–279, 281–285, 287, 289, 291, 293, 295, 297, 299, 301, 303, 305, 307, 309, 311–317, 319, 321, 323–325, 327, 329–331, 333, 335–337, 339, 341–347, 349, 351, 353, 355, 357–359, 361–367, 369–371, 373–375, 377, 379, 551–552, 571, 573–574, 577, 586, 593 Stokes deformation number, 1103 Storage modulus, 1200–1201, 1213 Strength, 36–38, 606, 610, 612–613, 625, 631–632 Stress Conditions, 1149, 1155–1157, 1159–1161, 1167–1168, 1170–1171, 1173, 1175, 1177, 1184, 1191, 1193, 1195, 1199, 1203, 1205, 1207, 1211–1212, 1214 Energy, 251–252, 269–270, 274–275, 278–281, 283–293, 295–304, 306, 309, 311–314, 361, 364–372, 374–375, 379 Event, 1155–1156, 1159, 1161, 1167–1168, 1177, 1183–1184, 1193, 1207, 1211–1213, 1215 Frequency, 251, 269–270, 274–276, 315, 321, 372, 379 Intensity, 251, 270–276, 278, 281, 283–284, 301–303, 315, 320–321, 379, 1149, 1155, 1167, 1179–1181, 1207 Mode, 1149, 1155, 1158, 1165, 1167–1168, 1177, 1179, 1183, 1189, 1191, 1199, 1207, 1212 Models, 251, 265, 269, 274, 276, 283 Structural interactions, 561 Sulphation, 1024, 1037–1040 Surface Abrasion, 1022–1023, 1036, 1038–1039

Subject Index Charge, 555–560 Disruption, 1163, 1209 Energy, 841–845, 849–851, 854, 857–858, 861, 1055, 1061–1063, 1068, 1072, 1084–1085, 1111, 1116 Suspension, 551–553, 564, 567, 571, 574, 577–582, 586–594, 597–598 Tablet, 941–943, 945–946, 948–951, 954–969 Compaction, 941, 943, 949, 962, 964, 968 Formulation, 941, 951, 955, 959 Friability, 960 Microstructure, 941–942, 956–957, 962, 967 Press, 949–950, 954–955, 958 Strength, 941–942, 945–946, 948–949, 951, 955, 968 Tabor parameter, 1069 Talc, 421, 431–433, 437, 463, 479, 481–484 Tangential force, 512 Tensile strength, 532 Terminator, 1019, 1022, 1032, 1037, 1041 Tip speed, 251, 260, 262, 264, 279, 284, 286–288, 290, 294, 296, 299, 301, 304, 315, 326–327, 333–334, 339–340, 345–346, 348–349, 351, 353, 364, 366, 374, 377, 379 Toner, 421, 428, 433–434 Toughness, 613, 623 Translatory shears, 998 Transport behaviour, 252, 311, 321–324, 327, 330, 378 Tribology, 1158, 1162–1163, 1203 Tumbling ball mill, 492, 495, 507, 514, 516 Universal mills, 229, 245 Van der Waals interaction, 551–553, 563 Vertical spindle mills, 886 Vibrating ball mill, 492, 507 Void Distribution, 744, 749, 751–754, 757–762, 765, 774, 776–777, 779–781, 786 Volume, 741, 743, 750, 752–758, 761

Subject Index Wear, 252, 263, 265, 267, 294, 304, 316–317, 319–320, 342, 347–358, 360, 362, 378–379 Wear of grinding media, 252, 265, 267, 347–348, 358 Wear of mills, 252, 347 Weber number, 1101, 1115 Weibull coefﬁcient, 922 Wet ball mills, 880, 888 Wet grinding, 493, 498, 501 Wetting, 1062–1063, 1078, 1112 Hysteresis, 1078 Wheat ﬂour, 383–416

1227 Work of adhesion, 1061–1064, 1066, 1068, 1072–1073, 1085–1086, 1089, 1102, 1111, 1115 Yield stress, 573, 579, 581–582, 584, 592, 600 Shear, 719, 725, 732, 734, 743, 750–751 Young’s equation, 1062, 1068 Young’s modulus, 38, 605, 613–618, 622, 630 Zeolite, 437, 454–456, 463, 465, 482–483 z-Potential, 556, 559, 581–582, 588–590

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