Contents
About the Contributors
Preface to the Second Edition
Preface to the First Edition
Introduction
1 Particle Size Analysis
1.1 INTRODUCTION
1.2 DESCRIBING THE SIZE OF A SINGLE PARTICLE
1.3 DESCRIPTION OF POPULATIONS OF PARTICLES
1.4 CONVERSION BETWEEN DISTRIBUTIONS
1.5 DESCRIBING THE POPULATION BY A SINGLE NUMBER
1.6 EQUIVALENCE OF MEANS
1.7 COMMON METHODS OF DISPLAYING SIZE DISTRIBUTIONS
1.8 METHODS OF PARTICLE SIZE MEASUREMENT
1.9 SAMPLING
1.10 WORKED EXAMPLES
2 Single Particles in a Fluid
2.1 MOTION OF SOLID PARTICLES IN A FLUID
2.2 PARTICLES FALLING UNDER GRAVITY THROUGH A FLUID
2.3 NON-SPHERICAL PARTICLES
2.4 EFFECT OF BOUNDARIES ON TERMINAL VELOCITY
2.5 FURTHER READING
2.6 WORKED EXAMPLES
3 Multiple Particle Systems
3.1 SETTLING OF A SUSPENSION OF PARTICLES
3.2 BATCH SETTLING
3.3 CONTINUOUS SETTLING
3.4 WORKED EXAMPLES
4 Slurry Transport
4.1 INTRODUCTION
4.2 FLOW CONDITION
4.3 RHEOLOGICAL MODELS FOR HOMOGENEOUS SLURRIES
4.4 HETEROGENEOUS SLURRIES
4.5 COMPONENTS OF A SLURRY FLOW SYSTEM
4.6 FURTHER READING
4.7 WORKED EXAMPLES
5 Colloids and Fine Particles
5.1 INTRODUCTION
5.2 BROWNIAN MOTION
5.3 SURFACE FORCES
5.4 RESULT OF SURFACE FORCES ON BEHAVIOUR IN AIR AND WATER
5.5 INFLUENCES OF PARTICLE SIZE AND SURFACE FORCES ON SOLID/LIQUID SEPARATION BY SEDIMENTATION
5.6 SUSPENSION RHEOLOGY
5.7 INFLUENCE OF SURFACE FORCES ON SUSPENSION FLOW
5.8 NANOPARTICLES
5.9 WORKED EXAMPLES
6 Fluid Flow Through a Packed Bed of Particles
6.1 PRESSURE DROP–FLOW RELATIONSHIP
6.2 FILTRATION
6.3 FURTHER READING
6.4 WORKED EXAMPLES
7 Fluidization
7.1 FUNDAMENTALS
7.2 RELEVANT POWDER AND PARTICLE PROPERTIES
7.3 BUBBLING AND NON-BUBBLING FLUIDIZATION
7.4 CLASSIFICATION OF POWDERS
7.5 EXPANSION OF A FLUIDIZED BED
7.6 ENTRAINMENT
7.7 HEAT TRANSFER IN FLUIDIZED BEDS
7.8 APPLICATIONS OF FLUIDIZED BEDS
7.9 A SIMPLE MODEL FOR THE BUBBLING FLUIDIZED BED REACTOR
7.10 SOME PRACTICAL CONSIDERATIONS
7.11 WORKED EXAMPLES
8 Pneumatic Transport and Standpipes
8.1 PNEUMATIC TRANSPORT
8.2 STANDPIPES
8.3 FURTHER READING
8.4 WORKED EXAMPLES
9 Separation of Particles from a Gas: Gas Cyclones
9.1 GAS CYCLONES – DESCRIPTION
9.2 FLOW CHARACTERISTICS
9.3 EFFICIENCY OF SEPARATION
9.4 SCALE-UP OF CYCLONES
9.5 RANGE OF OPERATION
9.6 SOME PRACTICAL DESIGN AND OPERATION DETAILS
9.7 WORKED EXAMPLES
10 Storage and Flow of Powders-Hopper Design
10.1 INTRODUCTION
10.2 MASS FLOW AND CORE FLOW
10.3 THE DESIGN PHILOSOPHY
10.4 SHEAR CELL TEST
10.5 ANALYSIS OF SHEAR CELL TEST RESULTS
10.6 SUMMARY OF DESIGN PROCEDURE
10.7 DISCHARGE AIDS
10.8 PRESSURE ON THE BASE OF A TALL CYLINDRICAL BIN
10.9 MASS FLOW RATES
10.10 CONCLUSIONS
10.11 WORKED EXAMPLES
11 Mixing and Segregation
11.1 INTRODUCTION
11.2 TYPES OF MIXTURE
11.3 SEGREGATION
11.4 REDUCTION OF SEGREGATION
11.5 EQUIPMENT FOR PARTICULATE MIXING
11.6 ASSESSING THE MIXTURE
11.7 WORKED EXAMPLES
12 Particle Size Reduction
12.1 INTRODUCTION
12.2 PARTICLE FRACTURE MECHANISMS
12.3 MODEL PREDICTING ENERGY REQUIREMENT AND PRODUCT SIZE DISTRIBUTION
12.4 TYPES OF COMMINUTION EQUIPMENT
12.5 WORKED EXAMPLES
13 Size Enlargement
13.1 INTRODUCTION
13.2 INTERPARTICLE FORCES
13.3 GRANULATION
13.4 WORKED EXAMPLES
14 Health Effects of Fine Powders
14.1 INTRODUCTION
14.2 THE HUMAN RESPIRATORY SYSTEM
14.3 INTERACTION OF FINE POWDERS WITH THE RESPIRATORY SYSTEM
14.4 PULMONARY DELIVERY OF DRUGS
14.5 HARMFUL EFFECTS OF FINE POWDERS
15 Fire and Explosion Hazards of Fine Powders
15.1 INTRODUCTION
15.2 COMBUSTION FUNDAMENTALS
15.3 COMBUSTION IN DUST CLOUDS
15.4 CONTROL OF THE HAZARD
15.5 WORKED EXAMPLES
16 Case Studies
16.1 CASE STUDY 1
16.2 CASE STUDY 2
16.3 CASE STUDY 3
16.4 CASE STUDY 4
16.5 CASE STUDY 5
16.6 CASE STUDY 6
16.7 CASE STUDY 7
16.8 CASE STUDY 8
Notation
References
Index
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Library of Congress Cataloging-in-Publication Data
Rhodes, Martin.
Introduction to particle technology / Martin Rhodes. – 2nd ed. p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-01427-1 (cloth) – ISBN 978-0-470-01428-8 (pbk.)
1. Particles. I. Title.
TP156.P3R48 2008
620′.43–dc22
2007041699
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
ISBN 978-0-470-01427-1 (Cloth) ISBN 978-0-470-01428-8 (Paper)
About the Contributors
Dr Karen P. Hapgood holds a BE and PhD in Chemical Engineering from the University of Queensland, Australia. Her PhD was on granulation processes and she continues to research in this area and related areas of powder technology. Karen worked for Merck & Co., USA for 5 years, where she worked on designing, troubleshooting and scaling up tablet and capsule manufacturing processes. Karen is currently a Senior Lecturer at the Department of Chemical Engineering at Monash University, Australia.
George Vincent Franks holds a Bachelor’s degree in Materials Science and Engineering from MIT (1985) and a PhD in Materials Engineering from the University of California at Santa Barbara (1997). George worked for 7 years in the ceramic processing industry as a process development engineer for Norton Company and Ceramic Process Systems Incorporated. His industrial work focused mainly on near net shape forming of ceramic green bodies and nonoxide ceramic firing. He has research and teaching experience at the Universities of Melbourne and Newcastle in Australia. George is currently Associate Professor in the Department of Chemical and Biomolecular Engineering at the University of Melbourne and the recently formed Australian Mineral Science Research Institute. His research interests include, mineral processing, particularly flocculation, advanced ceramics powder processing, colloid and surface chemistry, ion specific effects, alumina surfaces and suspension rheology.
Jennifer Sinclair Curtis received her BS degree in chemical engineering from Purdue University and PhD in chemical engineering from Princeton University. Jennifer has an internationally recognized research program in the development and validation of numerical models for the prediction of particle flow phenomena. Jennifer was a recipient of the NSF Presidential Young Investigator Award, the Eminent Overseas Lectureship Award by the Institution of Engineers in Australia and ASEE’s Sharon Keillor Award for Women in Engineering. She currently serves on the Editorial Advisory Board of the AIChE Journal, Powder Technology, and the Journal of Pharmaceutical Development and Technology. Jennifer has also served as a Trustee of the non-profit Computer Aids for Chemical Engineering Corporation and on the National Academy of Engineering’s Committee on Engineering Education. Jennifer is currently Professor and Chair of the Chemical Engineering Department at the University of Florida.
Martin Rhodes holds a Bachelor’s degree in chemical engineering and a PhD in particle technology from Bradford University in the UK, industrial experience in chemical and combustion engineering and many years experience as an academic at Bradford and Monash Universities. He has research interests in various aspects of gas fluidization and particle technology, areas in which he has many refereed publications in journals and international conference proceedings. Martin is on the editorial boards of Powder Technology and KONA and on the advisory board of Advanced Powder Technology. Martin has a keen interest in particle technology education and has published books and CDROM on Laboratory Demonstrations and directed continuing education courses for industry in the UK and Australia. He was co-founder of the Australasian Particle Technology Society. Martin has a Personal Chair in the Department of Chemical Engineering at Monash University, Australia, where he is presently Head of Department.
Preface to the Second Edition
It is 10 years since the publication of the first edition of Introduction to Particle Technology. During that time many colleagues from around the world have provided me with comments for improving the text. I have taken these comments into consideration in preparing the second edition. In addition, I have broadened the coverage of particle technology topics – in this endeavour I am grateful to my co-authors Jennifer Sinclair Curtis and George Franks, who have enabled the inclusion of chapters on Slurry Transport and Colloids and Fine Particles, and Karen Hapgood, who permitted an improved chapter on size enlargement and granulation. I have also included a chapter on the Health Effects of Fine Powders – covering both beneficial and harmful effects. I am also indebted to colleagues Peter Wypych, Lyn Bates, Derek Geldart, Peter Arnold, John Sanderson and Seng Lim for contributing case studies for Chapter 16.
Martin Rhodes
Balnarring, December 2007
Preface to the First Edition
Particle technology is a term used to refer to the science and technology related to the handling and processing of particles and powders. Particle technology is also often described as powder technology, particle science and powder science. Powders and particles are commonly referred to as bulk solids, particulate solids and granular solids. Today particle technology includes the study of liquid drops, emulsions and bubbles as well as solid particles. In this book only solid particles are covered and the terms particles, powder and particulate solids will be used interchangeably.
The discipline of particle technology now includes topics as diverse as the formation of aerosols and the design of bucket elevators, crystallization and pneumatics transport, slurry filtration and silo design. A knowledge of particle technology may be used in the oil industry to design the catalytic cracking reactor which produces gasoline from oil or it may be used in forensic science to link the accused with the scene of crime. Ignorance of particle technology may result in lost production, poor product quality, risk to health, dust explosion or storage silo collapse.
The objective of this textbook is to introduce the subject of particle technology to students studying degree courses in disciplines requiring knowledge of the processing and handling of particles and powders. Although the primary target readership is amongst students of chemical engineering, the material included should form the basis of courses on particle technology for students studying other disciplines including mechanical engineering, civil engineering, applied chemistry, pharmaceutics, metallurgy and minerals engineering.
A number of key topics in particle technology are studied giving the fundamental science involved and linking this, wherever possible, to industrial practice. The coverage of each topic is intended to be exemplary rather than exhaustive. This is not intended to be a text on unit operations in powder technology for chemical engineers. Readers wishing to know more about the industrial practice and equipment for handling and processing are referred to the various handbooks of powder technology which are available.
The topics included have been selected to give coverage of broad areas within easy particle technology: characterization (size analysis), processing (fluidized beds granulation), particle formation (granulation, size reduction), fluid-particle-separation (filtration, settling, gas cyclones), safety (dust explosions), transport (pneumatic transport and standpipes). The health hazards of fine particles or dusts are not covered. This is not to suggest in any way that this topic is less important than others. It is omitted because of a lack of space and because the health hazards associated with dusts are dealt with competently in the many texts on Industrial or Occupational Hygiene which are now available. Students need to be aware however, that even chemically inert dusts or ‘nuisance dust’ can be a major health hazard. Particularly where products contain a significant proportion of particles under 10 μm and where there is a possibility of the material becoming airborne during handling and processing. The engineering approach to the health hazard of fine powders should be strategic wherever possible; aiming to reduce dustiness by agglomeration, to design equipment for containment of material and to minimize exposure of workers.
The topics included demonstrate how the behaviour of powders is often quite different from the behaviour of liquids and gases. Behaviour of particulate solids may be surprising and often counter-intuitive when intuition is based on our experience with fluids. The following are examples of this kind of behaviour: When a steel ball is placed at the bottom of a container of sand and the container is vibrated in a vertical plane, the steel ball will rise to the surface.
A steel ball resting on the surface of a bed of sand will sink swiftly if air is passed upward through the sand causing it to become fluidized.
Stirring a mixture of two free-flowing powders of different sizes may result in segregation rather than improved mixture quality.
Engineers and scientists are use to dealing with liquids and gases whose properties can be readily measured, tabulated and even calculated. The boiling point of pure benzene at one atmosphere pressure can be safely relied upon to remain at 80.1°C. The viscosity of water at 20°C can be confidently predicted to be 0.001 Pa s. The thermal conductivity of copper at 100°C is 377W/m · K. With particulate solids, the picture is quite different. The flow properties of sodium bicarbonate powder, for example, depends not only on the particle size distribution, the particle shape and surface properties, but also on the humidity of atmosphere and the state of the compaction of the powder. These variables are not easy to characterize and so their influence on the flow properties is difficult to predict with any confidence.
In the case of particulate solids it is almost always necessary to rely on performing appropriate measurements on the actual powder in question rather than relying on tabulated data. The measurements made are generally measurements of bulk properties, such as shear stress, bulk density, rather than measurements of fundamental properties such as particle size, shape and density. Although this is the present situation, in the not too distant future, we will be able to rely on sophisticated computer models for simulation of particulate systems. Mathematical modelling of particulate solids behaviour is a rapidly developing area of research around the world, and with increased computing power and better visualization software, we will soon be able to link fundamental particle properties directly to bulk powder behaviour. It will even be possible to predict, from first principles, the influence of the presence of gases and liquids within the powder or to incorporate chemical reaction.
Particle technology is a fertile area for research. Many phenomena are still unexplained and design procedures rely heavily on past experience rather than on fundamental understanding. This situation presents exciting challenges to researchers from a wide range of scientific and engineering disciplines around the world. Many research groups have websites which are interesting and informative at levels ranging from primary schools to serious researchers. Students are encouraged to visit these sites to find out more about particle technology. Our own website at Monash University can be accessed via the Chemical Engineering Department web page at http://www.eng.monash.edu.au/chemeng/
Martin Rhodes
Mount Eliza, May 1998
Introduction
Particulate materials, powders or bulk solids are used widely in all areas of the process industries, for example in the food processing, pharmaceutical, biotechnology, oil, chemical, mineral processing, metallurgical, detergent, power generation, paint, plastics and cosmetics industries. These industries involve many different types of professional scientists and engineers, such as chemical engineers, chemists, biologists, physicists, pharmacists, mineral engineers, food technologists, metallurgists, material scientists/engineers, environmental scientists/ engineers, mechanical engineers, combustion engineers and civil engineers. Some figures give an indication of the significance of particle technology in the world economy: for the DuPont company, whose business covers chemicals, agricultural, pharmaceuticals, paints, dyes, ceramics, around two-thirds of its products involve particulate solids (powders, crystalline solids, granules, flakes, dispersions or pastes); around 1% of all electricity generated worldwide is used in reducing particle size; the impact of particulate products to the US economy was estimated to be US$ 1 trillion.
Some examples of the processing steps involving particles and powder include particle formation processes (such as crystallization, precipitation, granulation, spray drying, tabletting, extrusion and grinding), transportation processes (such as pneumatic and hydraulic transport, mechanical conveying and screw feeding) and mixing, drying and coating processes. In addition, processes involving particulates require reliable storage facilities and give rise to health and safety issues, which must be satisfactorily handled. Design and operation of these many processes across this wide range of industries require a knowledge of the behaviour of powders and particles. This behaviour is often counterintuitive, when intuition is based on our knowledge of liquids and gases. For example, actions such as stirring, shaking or vibrating, which would result in mixing of two liquids, are more likely to produce size segregation in a mixture of free-flowing powders of different sizes. A storage hopper holding 500 t of powder may not deliver even 1 kg when the outlet valve is opened unless the hopper has been correctly designed. When a steel ball is placed at the bottom of a container of sand and the container is vibrated in the vertical plane, the steel ball will rise to the surface. This steel ball will then sink swiftly to the bottom again if air is passed upwards through the sand causing it to be fluidized.
Engineers and scientists are used to dealing with gases and liquids, whose properties can be readily measured, tabulated or even calculated. The boiling point of pure benzene at atmospheric pressure can be safely assumed to remain at 80.1°C. The thermal conductivity of copper can always be relied upon to be 377 W/m · K at 100°C. The viscosity of water at 20°C can be confidently expected to be 0.001 Pa s. With particulate solids, however, the situation is quite different. The flow properties of sodium bicarbonate powder, for example, depend not only of the particle size distribution, but also on particle shape and surface properties, the humidity of the surrounding atmosphere and the state of compaction of the powder. These variables are not easy to characterize and so their influence on the flow properties of the powder is difficult to predict or control with any confidence. Interestingly, powders appear to have some of the behavioural characteristics of the three phases, solids, liquids and gases. For example, like gases, powders can be compressed; like liquids, they can be made to flow, and like solids, they can withstand some deformation.
The importance of knowledge of the science of particulate materials (often called particle or powder technology) to the process industries cannot be overemphasized. Very often, difficulties in the handling or processing powders are ignored or overlooked at the design stage, with the result that powder-related problems are the cause of an inordinate number of production stoppages. However, it has been demonstrated that the application of even a basic understanding of the ways in which powders behave can minimize these processing problems, resulting in less downtime, improvements in quality control and environmental emissions.
This text is intended as an introduction to particle technology. The topics included have been selected to give coverage of the broad areas of particle technology: characterization (size analysis), processing (granulation, fluidization), particle formation (granulation, size reduction), storage and transport (hopper design, pneumatic conveying, standpipes, slurry flow), separation (filtration, settling, cyclones), safety (fire and explosion hazards, health hazards), engineering the properties of particulate systems (colloids, respirable drugs, slurry rheology). For each of the topics studied, the fundamental science involved is introduced and this is linked, where possible, to industrial practice. In each chapter there are worked examples and exercises to enable the reader to practice the relevant calculations and and a ‘Test Yourself’ section, intended to highlight the main concepts covered. The final chapter includes some case studies–real examples from the process industries of problems that arose and how they were solved.
A website with laboratory demonstrations in particle technology, designed to accompany this text, is available. This easily navigated resource incorporates many video clips of particle and powder phenomena with accompanying explanatory text. The videos bring to life many of the phenomena that I have tried to describe here in words and diagrams. For example, you will see: fluidized beds (bubbling, non-bubbling, spouted) in action; core flow and mass flow in hoppers, size segregation during pouring, vibration and rolling; pan granulation of fine powders, a coal dust explosion; a cyclone separator in action; dilute and dense phase pneumatic conveying. The website will aid the reader in understanding particle technology and is recommended as a useful adjunct to this text.
In many powder handling and processing operations particle size and size distribution play a key role in determining the bulk properties of the powder. Describing the size distribution of the particles making up a powder is therefore central in characterizing the powder. In many industrial applications a single number will be required to characterize the particle size of the powder. This can only be done accurately and easily with a mono-sized distribution of spheres or cubes. Real particles with shapes that require more than one dimension to fully describe them and real powders with particles in a range of sizes, mean that in practice the identification of single number to adequately describe the size of the particles is far from straightforward. This chapter deals with how this is done.
Regular-shaped particles can be accurately described by giving the shape and a number of dimensions. Examples are given in Table 1.1.
The description of the shapes of irregular-shaped particles is a branch of science in itself and will not be covered in detail here. Readers wishing to know more on this topic are referred to Hawkins (1993). However, it will be clear to the reader that no single physical dimension can adequately describe the size of an irregularly shaped particle, just as a single dimension cannot describe the shape of a cylinder, a cuboid or a cone. Which dimension we do use will in practice depend on (a) what property or dimension of the particle we are able to measure and (b) the use to which the dimension is to be put.
If we are using a microscope, perhaps coupled with an image analyser, to view the particles and measure their size, we are looking at a projection of the shape of the particles. Some common diameters used in microscope analysis are statistical diameters such as Martin’s diameter (length of the line which bisects the particle image), Feret’s diameter (distance between two tangents on opposite sides of the particle) and shear diameter (particle width obtained using an image shearing device) and equivalent circle diameters such as the projected area diameter (area of circle with same area as the projected area of the particle resting in a stable position). Some of these diameters are described in Figure 1.1. We must remember that the orientation of the particle on the microscope slide will affect the projected image and consequently the measured equivalent sphere diameter.
Table 1.1 Regular-shaped particles
Figure 1.1 Some diameters used in microscopy
Figure 1.2 Comparison of equivalent sphere diameters
If we use a sieve to measure the particle size we come up with an equivalent sphere diameter, which is the diameter of a sphere passing through the same sieve aperture. If we use a sedimentation technique to measure particle size then it is expressed as the diameter of a sphere having the same sedimentation velocity under the same conditions. Other examples of the properties of particles measured and the resulting equivalent sphere diameters are given in Figure 1.2.
Table 1.2 compares values of these different equivalent sphere diameters used to describe a cuboid of side lengths 1, 3, 5 and a cylinder of diameter 3 and length 1.
The volume equivalent sphere diameter or equivalent volume sphere diameter is a commonly used equivalent sphere diameter. We will see later in the chapter that it is used in the Coulter counter size measurements technique. By definition, the equivalent volume sphere diameter is the diameter of a sphere having the same volume as the particle. The surface-volume diameter is the one measured when we use permeametry (see Section 1.8.4) to measure size. The surface-volume (equivalent sphere) diameter is the diameter of a sphere having the same surface to volume ratio as the particle. In practice it is important to use the method of size measurement which directly gives the particle size which is relevant to the situation or process of interest. (See Worked Example 1.1.)
Table 1.2 Comparison of equivalent sphere diameters
A population of particles is described by a particle size distribution. Particle size distributions may be expressed as frequency distribution curves or cumulative curves. These are illustrated in Figure 1.3. The two are related mathematically in that the cumulative distribution is the integral of the frequency distribution; i.e. if the cumulative distribution is denoted as F, then the frequency distribution dF/dx. For simplicity, dF/dx is often written as f(x). The distributions can be by number, surface, mass or volume (where particle density does not vary with size, the mass distribution is the same as the volume distribution). Incorporating this information into the notation, fN(x) is the frequency distribution by number, fS(x) is the frequency distribution by surface, FS is the cumulative distribution by surface and FM is the cumulative distribution by mass. In reality these distributions are smooth continuous curves. However, size measurement methods often divide the size spectrum into size ranges or classes and the size distribution becomes a histogram.
Figure 1.3 Typical differential and cumulative frequency distributions
Figure 1.4 Comparison between distributions
For a given population of particles, the distributions by mass, number and surface can differ dramatically, as can be seen in Figure 1.4.
A further example of difference between distributions for the same population is given in Table 1.3 showing size distributions of man-made objects orbiting the earth (New Scientist, 13 October 1991).
The number distribution tells us that only 0.2% of the objects are greater than 10 cm. However, these larger objects make up 99.96% of the mass of the population, and the 99.3% of the objects which are less than 1.0 cm in size make up only 0.01% of the mass distribution. Which distribution we would use is dependent on the end use of the information.
Many modern size analysis instruments actually measure a number distribution, which is rarely needed in practice. These instruments include software to convert the measured distribution into more practical distributions by mass, surface, etc.
Table 1.3 Mass and number distributions for man-made objects orbiting the earth
Relating the size distributions by number, fN(x), and by surface, fS(x) for a population of particles having the same geometric shape but different size:
Fraction of particles in the size range
Fraction of the total surface of particles in the size range
If N is the total number of particles in the population, the number of particles in the size range x to x + dx = NfN(x)dx and the surface area of these particles = (x2αS)NfN(x)dx, where αS is the factor relating the linear dimension of the particle to its surface area.
Therefore, the fraction of the total surface area contained on these particles [fS(x)dx] is:
where S is the total surface area of the population of particles.
For a given population of particles, the total number of particles, N, and the total surface area, S are constant. Also, assuming particle shape is independent of size, αS is constant, and so
(1.1) 
where
Similarly, for the distribution by volume
(1.2) 
where
where V is the total volume of the population of particles and αV is the factor relating the linear dimension of the particle to its volume.
And for the distribution by mass
(1.3) 
where
assuming particle density ρp is independent of size.
The constants kS, kV and km may be found by using the fact that:
(1.4) 
Thus, when we convert between distributions it is necessary to make assumptions about the constancy of shape and density with size. Since these assumptions may not be valid, the conversions are likely to be in error. Also, calculation errors are introduced into the conversions. For example, imagine that we used an electron microscope to produce a number distribution of size with a measurement error of ±2%. Converting the number distribution to a mass distribution we triple the error involved (i.e. the error becomes ±6%). For these reasons, conversions between distributions are to be avoided wherever possible. This can be done by choosing the measurement method which gives the required distribution directly.
In most practical applications, we require to describe the particle size of a population of particles (millions of them) by a single number. There are many options available; the mode, the median, and several different means including arithmetic, geometric, quadratic, harmonic, etc. Whichever expression of central tendency of the particle size of the population we use must reflect the property or properties of the population of importance to us. We are, in fact, modelling the real population with an artificial population of mono-sized particles. This section deals with calculation of the different expressions of central tendency and selection of the appropriate expression for a particular application.
The mode is the most frequently occurring size in the sample. We note, however, that for the same sample, different modes would be obtained for distributions by number, surface and volume. The mode has no practical significance as a measure of central tendency and so is rarely used in practice.
The median is easily read from the cumulative distribution as the 50% size; the size which splits the distribution into two equal parts. In a mass distribution, for example, half of the particles by mass are smaller than the median size. Since the median is easily determined, it is often used. However, it has no special significance as a measure of central tendency of particle size.
Table 1.4 Definitions of means
| g(x) | Mean and notation |
| x | arithmetic mean,  |
| x2 | quadratic mean,  |
| x3 | cubic mean,  |
| log x | geometric mean,  |
| 1/x | harmonic mean,  |
Many different means can be defined for a given size distribution; as pointed out by Svarovsky (1990). However, they can all be described by:
(1.5) 
where 
is the mean and g is the weighting function, which is different for each mean definition. Examples are given in Table 1.4.
Equation (1.5) tells us that the mean is the area between the curve and the F(x) axis in a plot of F(x) versus the weighting function g(x) (Figure 1.5). In fact, graphical determination of the mean is always recommended because the distribution is more accurately represented as a continuous curve.
Each mean can be shown to conserve two properties of the original population of particles. For example, the arithmetic mean of the surface distribution conserves the surface and volume of the original population. This is demonstrated in Worked Example 1.3. This mean is commonly referred to as the surface-volume mean or the Sauter mean. The arithmetic mean of the number distribution 
conserves the number and length of the original population and is known as the number-length mean 
:
(1.6) 
As another example, the quadratic mean of the number distribution 
conserves the number and surface of the original population and is known as the number-surface mean 
:
(1.7) 
A comparison of the values of the different means and the mode and median for a given particle size distribution is given in Figure 1.6. This figure highlights two points: (a) that the values of the different expressions of central tendency can vary significantly; and (b) that two quite different distributions could have the same arithmetic mean or median, etc. If we select the wrong one for our design correlation or quality control we may be in serious error.
Figure 1.5 Plot of cumulative frequency against weighting function g(x). Shaded area is 
Figure 1.6 Comparison between measures of central tendency. Adapted from Rhodes (1990). Reproduced by permission
So how do we decide which mean particle size is the most appropriate one for a given application? Worked Examples 1.3 and 1.4 indicate how this is done.
For Equation (1.8), which defines the surface-volume mean, please see Worked Example 1.3.
Means of different distributions can be equivalent. For example, as is shown below, the arithmetic mean of a surface distribution is equivalent (numerically equal to) the harmonic mean of a volume (or mass) distribution:
(1.9) 
The harmonic mean 
of a volume distribution is defined as:
(1.10) 
From Equations (1.1) and (1.2), the relationship between surface and volume distributions is:
(1.11) 
hence
(1.12) 
(assuming ks and kv do not vary with size)
and so
which, by inspection, can be seen to be equivalent to the arithmetic mean of the surface distribution 
[Equation (1.9)].
Recalling that d Fs = x2kSd FN, we see from Equation (1.9) that
which is the surface-volume mean, 
[Equation (1.8) – see Worked Example 1.3].
Summarizing, then, the surface-volume mean may be calculated as the arithmetic mean of the surface distribution or the harmonic mean of the volume distribution. The practical significance of the equivalence of means is that it permits useful means to be calculated easily from a single size analysis.
The reader is invited to investigate the equivalence of other means.
In this distribution, shown in Figure 1.7, particle sizes with equal differences from the arithmetic mean occur with equal frequency. Mode, median and arithmetic mean coincide. The distribution can be expressed mathematically by:
(1.13) 
where σ is the standard deviation.
To check for a arithmetic-normal distribution, size analysis data is plotted on normal probability graph paper. On such graph paper a straight line will result if the data fits an arithmetic-normal distribution.
This distribution is more common for naturally occurring particle populations. An example is shown in Figure 1.8. If plotted as dF/d(log x) versus x, rather than dF/dx versus x, an arithmetic-normal distribution in log x results (Figure 1.9). The mathematical expression describing this distribution is:
(1.14) 
Figure 1.7 Arithmetic-normal distribution with an arithmetic mean of 45 and standard deviation of 12
Figure 1.8 Log-normal distribution plotted on linear coordinates
where z = log x, 
is the arithmetic mean of log x and σz is the standard deviation of log x.
To check for a log-normal distribution, size analysis data are plotted on log-normal probability graph paper. Using such graph paper, a straight line will result if the data fit a log-normal distribution.
Dry sieving using woven wire sieves is a simple, cheap method of size analysis suitable for particle sizes greater than 45 μm. Sieving gives a mass distribution and a size known as the sieve diameter. Since the length of the particle does not hinder its passage through the sieve apertures (unless the particle is extremely elongated), the sieve diameter is dependent on the maximum width and maximum thickness of the particle. The most common modern sieves are in sizes such that the ratio of adjacent sieve sizes is the fourth root of two (eg. 45, 53, 63, 75, 90, 107 μm). If standard procedures are followed and care is taken, sieving gives reliable and reproducible size analysis. Air jet sieving, in which the powder on the sieve is fluidized by a jet or air, can achieve analysis down to 20 μm. Analysis down to 5 μm can be achieved by wet sieving, in which the powder sample is suspended in a liquid.
Figure 1.9 Log-normal distribution plotted on logarithmic coordinates
The optical microscope may be used to measure particle sizes down to 5 μm. For particles smaller than this diffraction causes the edges of the particle to be blurred and this gives rise to an apparent size. The electron microscope may be used for size analysis below 5 μm. Coupled with an image analysis system the optical microscope or electron microscope can readily give number distributions of size and shape. Such systems calculate various diameters from the projected image of the particles (e.g. Martin’s, Feret’s, shear, projected area diameters, etc.). Note that for irregular-shaped particles, the projected area offered to the viewer can vary significantly depending on the orientation of the particle. Techniques such as applying adhesive to the microscope slide may be used to ensure that the particles are randomly orientated.
In this method, the rate of sedimentation of a sample of particles in a liquid is followed. The suspension is dilute and so the particles are assumed to fall at their single particle terminal velocity in the liquid (usually water). Stokes’ law is assumed to apply (Rep < 0.3) and so the method using water is suitable only for particles typically less than 50 μm in diameter. The rate of sedimentation of the particles is followed by plotting the suspension density at a certain vertical position against time. The suspension density is directly related to the cumulative undersize and the time is related to the particle diameter via the terminal velocity. This is demonstrated in the following:
Referring to Figure 1.10, the suspension density is sampled at a vertical distance, h below the surface of the suspension. The following assumptions are made:
Figure 1.10 Size analysis by sedimentation
Let the original uniform suspension density be C0. Let the suspension density at the sampling point be C at time t after the start of settling. At time t all those particles travelling faster than h/t will have fallen below the sampling point. The sample at time t will therefore consist only of particles travelling a velocity ≤ h/t. Thus, if C0 is representative of the suspension density for the whole population, then C represents the suspension density for all particles which travel at a velocity ≤ h/t, and so C/C0 is the mass fraction of the original particles which travel at a velocity ≤ h/t. That is,
All particles travel at their terminal velocity given by Stokes’ law [Chapter 2, Equation (2.13)]:
Thus, equating UT with h/t, we determine the diameter of the particle travelling at our cut-off velocity h/t. That is,
(1.15) 
Particles smaller than x will travel slower than h/t and will still be in suspension at the sampling point. Corresponding values of C/C0 and x therefore give us the cumulative mass distribution. The particle size measured is the Stokes’ diameter, i.e. the diameter of a sphere having the same terminal settling velocity in the Stokes region as the actual particle.
A common form of this method is the Andreason pipette which is capable of measuring in the range 2–100 μm. At size below 2 μm, Brownian motion causes significant errors. Increasing the body force acting on the particles by centrifuging the suspension permits the effects of Brownian motion to be reduced so that particle sizes down to 0.01 μm can be measured. Such a device is known as a pipette centrifuge.
The labour involved in this method may be reduced by using either light absorption or X-ray absorption to measure the suspension density. The light absorption method gives rise to a distribution by surface, whereas the X-ray absorption method gives a mass distribution.
This is a method of size analysis based on fluid flow through a packed bed (see Chapter 6). The Carman–Kozeny equation for laminar flow through a randomly packed bed of uniformly sized spheres of diameter x is [Equation 6.9]:
where (−Δp) is the pressure drop across the bed, ε is the packed bed void fraction, H is the depth of the bed, μ is the fluid viscosity and U is the superficial fluid velocity. In Worked Example 1.3, we will see that, when we are dealing with non-spherical particles with a distribution of sizes, the appropriate mean diameter for this equation is the surface-volume diameter , which may be calculated as the arithmetic mean of the surface distribution, .
In this method, the pressure gradient across a packed bed of known voidage is measured as a function of flow rate. The diameter we calculate from the Carman–Kozeny equation is the arithmetic mean of the surface distribution (see Worked Example 6.1 in Chapter 6).
Particles are held in supension in a dilute electrolyte which is drawn through a tiny orifice with a voltage applied across it (Figure 1.11). As particles flow through the orifice a voltage pulse is recorded.
The amplitude of the pulse can be related to the volume of the particle passing the orifice. Thus, by electronically counting and classifying the pulses according to amplitude this technique can give a number distribution of the equivalent volume sphere diameter. The lower size limit is dictated by the smallest practical orifice and the upper limit is governed by the need to maintain particles in suspension. Although liquids more viscous than water may be used to reduce sedimentation, the practical range of size for this method is 0.3–1000 μm. Errors are introduced if more that one particle passes through the orifice at a time and so dilute suspensions are used to reduce the likelihood of this error.
Figure 1.11 Schematic of electrozone sensing apparatus
This method relies on the fact that for light passing through a suspension, the diffraction angle is inversely proportional to the particle size. An instrument would consist of a laser as a source of coherent light of known fixed wavelength (typically 0.63 μm), a suitable detector (usually a slice of photosensitive silicon with a number of discrete detectors, and some means of passing the sample of particles through the laser light beam (techniques are available for suspending particles in both liquids and gases are drawing them through the beam).
To relate diffraction angle with particle size, early instruments used the Fraunhofer theory, which can give rise to large errors under some circumstances (e.g. when the refractive indices of the particle material and suspending medium approach each other). Modern instruments use the Mie theory for interaction of light with matter. This allows particle sizing in the range 0.1–2000 μm, provided that the refractive indices of the particle material and suspending medium are known.
This method gives a volume distribution and measures a diameter known as the laser diameter. Particle size analysis by laser diffraction is very common in industry today. The associated software permits display of a variety of size distributions and means derived from the original measured distribution.
In practice, the size distribution of many tonnes of powder are often assumed from an analysis performed on just a few grams or milligrams of sample. The importance of that sample being representative of the bulk powder cannot be overstated. However, as pointed out in Chapter 11 on mixing and segregation, most powder handling and processing operations (pouring, belt conveying, handling in bags or drums, motion of the sample bottle, etc.) cause particles to segregate according to size and to a lesser extent density and shape. This natural tendency to segregation means that extreme care must be taken in sampling.
There are two golden rules of sampling:
Since the eventual sample size used in the analysis may be very small, it is often necessary to split the original sample in order to achieve the desired amount for analysis. These sampling rules must be applied at every step of sampling and sample splitting.
Detailed description of the many devices and techniques used for sampling in different process situations and sample dividing are outside the scope of this chapter. However, Allen (1990) gives an excellent account, to which the reader is referred.
WORKED EXAMPLE 1.1