Mean particle diameters: From statistical definition to physical understanding

Mean Particle Diameters

From Statistical Definition to Physical

Mean Particle Diameters

From Statistical Definition to Physical

Understanding

Proefschrift

ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,

op gezag van de Rector Magnificus prof.dr.ir. J.T. Fokkema, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 18 november 2008 om 12.30 uur

door

Maarten ALDERLIESTEN

Ingenieur fysische techniek, HTS Dordrecht geboren te Leiden

Dit proefschrift is goedgekeurd door de promotor: Prof.dr. A. Schmidt-Ott

Samenstelling promotiecommissie: Rector Magnificus, voorzitter

Prof.dr. A. Schmidt-Ott, Technische Universiteit Delft, promotor Prof.dr. K. Heiskanen, Helsinki University of Technology, Finland Prof.dr. G. J. Witkamp, Technische Universiteit Delft

Prof.dr. K. Sommer, Technische Universität München, Duitsland Prof.dr. P.F. Dunn, University of Notre Dame, USA

Prof.dr. R. Finsy, Vrije Universiteit Brussel, België Prof.dr. S. Luding, Universiteit Twente

Prof.dr. S.J. Picken, Technische Universiteit Delft, reservelid ,

© Maarten Alderliesten, 2008

ISBN 978-90-71382-59-8

All rights reserved. No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission from the author.

The author encourages the communication of scientific contents and explicitly exempts the use for scientific, non commercial purposes, provided the proper citation of the source. Parts of the thesis are published in scientific journals and copyright is subject to different terms and conditions.

On the cover: The background is the cover of Particle and Particle Systems

Characterization (Courtesy of Wiley-VCH Verlag GmbH & Co., Weinheim).

Design and layout: Maarten Alderliesten and Annelies ter Laare.

The Road goes ever on and on

Down from the door where it began.

Now far ahead the Road has gone,

And I must follow, if I can,

Pursuing it with eager feet,

Until it joins some larger way

Where many paths and errands meet.

And wither then? I cannot say.

Bilbo Baggins

In: John Ronald Reuel Tolkien.

The Lord of the Rings, Part I,

The Fellowship of the Ring

vi

Summary

Mean Particle Diameters

From Statistical Definition to Physical Understanding Maarten Alderliesten

Mean particle diameters are important for the science of particulate systems. This thesis presents a rigorous mean particle diameter definition system, called the Moment-Ratio (M-R) definition system, and provides a general statistical and physical basis. The standardization area deals with the basic aspects of particle size analysis and demonstrates the lack of a general basis. The standardization area, therefore, is a source of basic issues to investigate. With hindsight, particle size analysis is perceived as empirical, with insufficient scientific coherence, with pragmatic approaches and personal preferences. The incoherence is observed in, e.g., the definition and calculation of mean diameters, the usage of statistical quantities like medians and modes of size distributions, and the designations of mean diameters types, etc.

Chapter 2 first introduces the Moment-Ratio (M-R) definition system of mean particle diameters, which is based on moments of the number density distribution of particle sizes. Next, the DIN/ISO definition system is described. This definition system was one of the main subjects of the course “Grundlagen und moderne Verfahren der Partikelmesstechnik”, given by the late Prof. Kurt Leschonski at the University of Clausthal. The M-R definition system is shown to be a simple and complete definition system. Many types of mean diameters defined according to the M-R system are not merely statistical parameters, but quantities of physical relevance because of causal relationships with, e.g., physical product properties. They are pre-eminently suitable to represent the dispersed phase properties in property and process functions, i.e., in functional relationships between dispersed phase properties and product or process properties.

After an overview of designations of mean particle diameters in the particle size literature, chapter 3 presents the development of an unambiguous, coherent system of designations for all mean diameter types for which terms are available in the natural languages. This nomenclature has a sound philosophical basis, which clearly conveys the physical meaning of mean particle diameters. A first step is taken to remove ambiguities in formalism and differences in the nomenclature for particle size distributions.

In the fourth chapter the ISO prescribed integration of histograms is compared with the summation of histogram data. The mathematical equations of the integration method appear to be difficult to apply in daily practice and their complexity easily hides the physical background and meaning of a mean particle diameter. Summation of histogram data tends to be more accurate and clear.

vii On the basis of the previous findings, chapter 5 introduces a method to derive theoretically the proper type of mean diameter describing a product or process property. Examples from the areas of evaporation, heat transfer, and turbulent two-phase flow, illustrate the theoretical approach. One of the examples given demonstrates that the particle diameter should not be ‘concealed’ in dimensionless numbers such as the Reynolds number in an early stage of the modeling process when a distribution of sizes of particles has to be taken into account.

In chapter 6 the M-R definition system is used as an instrument for an empirical selection of the proper type of mean diameter. This is illustrated by a number of examples, mainly from the areas of high shear granulation in detergent processing and pharmaceutics. The first of these examples is concerned with a visual ranking of photographs of bubble size distributions of chocolate mousse samples. The selected type of mean bubble diameter suggests that our visual system, i.e., eyes + brain, has a logarithmic response. Another method for searching systematically for the proper type of mean particle diameter is not known.

Chapter 7 summarizes the conclusions of the previous chapters. Additionally, the physical background of the M-R and ISO/DIN definition systems is compared to investigate the applicability of the ISO/DIN system for empirical selection and theoretical derivation of mean diameters. Further, this chapter gives some recommendations for further research.

viii

Table of Contents

Definitions, symbols and abbreviated terms

xiii

1. Introduction

1

2. Evaluation of Definition Systems

7

2.1 Introduction 7

2.2 Moments of a Distribution 8

2.3 Definition of Mean Particle Diameters 10

2.3.1 Moment-Ratio Notation 10

2.3.1.1 Definition of Dp,q 10

2.3.1.2 The Standard Deviation 11

2.3.1.3 Relationships between Mean Diameters Dp,q 12

2.3.2 German (DIN) Notation System 14

2.3.2.1 Definition of xk,r 14

2.3.2.2 The Standard Deviation 15

2.3.2.3 Relationships between Moments and between Mean

Diameters xk,r 16

2.3.3 Evaluation of Definition Systems 17

2.3.3.1 Relationship between Dp,q and xk,r 17

2.3.3.2 Geometric Mean Diameters 18

2.3.3.3 The Order of Mean Diameters 19

2.3.3.4 The Standard Deviation 22

2.3.4 Interchanging Mean Diameters and Moments 22

2.4 Discussion 24

2.5 Conclusion 25

2.6 Acknowledgements 26

Appendix A2.1 Derivation of Geometric Mean Diameters Dp,q 27

Appendix A2.2 Derivation of Geometric Mean Diameters x0,r

for the German (DIN) Notation System 28

Appendix A2.3 Proof of the statement Dp,0≤Dm,0 if p ≤m 29

3. Standardization of Nomenclature

31

3.1 Introduction 31

3.2 Naming the Set of Mean Particle Diameters Dp,q 33

3.3 Systematic Nomenclature for Mean Particle Diameters 33

3.4 Physical Relevance of the Nomenclature 38

3.5 Alternative Standardized Nomenclature Systems 39

3.6 Discussion 40

3.7 Acknowledgements 40

Appendix A3.1 Particle-size Distributions: their Notation and

ix

Methods for Estimating Mean Particle Diameters from

Histogram Data

45

4.1 Introduction 45

4.2 Density Distributions used to compare Summation and Integration

Methods to estimate Mean Diameters 46

4.3 Analytical Values of Mean Diameters and the Summation and the

Integration Methods to Estimate Mean Diameters from Histograms 49

4.3.1 Computation of the Analytical Values of Mean Diameters 49

4.3.2 Estimation of Mean Diameters from Histograms according

to the Summation (M-R) Approach 50

4.3.3 Estimation of Mean Diameters from Histograms according

to the Integration Approach 51

4.4 Comparison of Summation and Integration Methods 53

4.4.1 Introduction 53

4.4.2 Comparison of Mean Diameters of Uniform Density

Distributions 54

4.4.3 Comparison of Mean Diameters of Triangular Increasing

and Decreasing Density Distributions 56

4.4.4 Comparison of Mean Diameters of Double-triangular

Density Distributions 59

4.4.5 Comparison of Mean Diameters of Lognormal Density

Distributions 61

4.4.6 Integration and the Shape of Histograms when Converting

to Other Particle Quantities 63

4.5 Discussion on the Summation and Integration Estimation Methods 65

4.5.1 Accuracy of the Estimation Methods 65

4.5.2 Integration and Histogram Shapes 66

4.5.3 Complexity of the Integration Equations 66

4.5.4 Comparison of Integration and Summation 67

4.5.5 Order of a Mean Diameter 67

4.6 Conclusions 68

4.7 Acknowledgements 68

5. Theoretical Derivation of the Proper Type of Mean

Particle Diameter describing a Product or Process

Property

69

5.1 Introduction 69

5.2 Estimation of Mean Diameters According to the M-R Method 71

5.3 Procedure to Derive the Proper Type of Mean Diameter 72

5.4 Three Examples of Deriving Theoretically the Proper Type of Mean

Diameter 72

5.4.1 Example 1: Evaporation of Particles or Droplets 73

5.4.1.1 Langmuir Evaporation of Particles 73

5.4.1.2 Generalization of Langmuir Evaporation Equation 74

5.4.2 Example 2: Heat Transfer to or from Particles 75

x

5.4.3 Example 3: Resistance Force of a Turbulent Two-Phase

Flow and a Distribution of Particle Sizes 84

5.5 Discussion 88

5.6 Conclusions 89

5.7 Acknowledgements 89

6 Empirical Selection of the Proper Type of Mean Particle

Diameter describing a Product or Material Property

91

6.1 Introduction 92

6.2 Procedure for Selection of the Proper Type of Mean Particle Diameter 93

6.2.1 Selection 93

6.2.2 Scales of Measurement 94

6.3 Five Examples to Describe the Proper Mean Diameter Selection

Procedure 95

6.3.1 Example 1: Visual Ranking of Photographs of Air Bubble

Size Distributions 96

6.3.1.1 Experimental Data on Mean Bubble Sizes and Visual

Ranking 96

6.3.1.2 Comparison between Visual Ranking and Mean Diameter

Rankings 100

6.3.1.3 Physical or Physiological Meaning of D2,2 102

6.3.2 Example 2: Binder Concentration and the Granule Size

Distributions 103

6.3.2.1 Experimental Data on Binder Concentration and Granule

Size Distributions 103

6.3.2.2 Relationship between Binder Concentration and Granule

Mean Diameters Dp,q 104

6.3.2.3 Physical Meaning of D3,2 106

6.3.3 Example 3: Distribution of Total Ingredient Mass over the

Granule Sizes 110

6.3.3.1 Experimental Data on Ingredient Masses and Granule Size

Distributions 110

6.3.3.2 Relationship between Ingredient Mass and Granule Mean

Diameters Dp,q 110

6.3.3.3 Physical Meaning of D6,3 112

6.3.4 Example 4: Granulation Process Yield and the Granule Size

Distribution 115

6.3.4.1 Experimental Data on Process Yield and Granule Size

Distributions 115

6.3.4.2 Relationship between Process Yield and Granule Mean

Diameters Dp,q 115

6.3.4.3 Physical Meaning of D4,3 117

6.3.4.4 Subdivision of Process Yield Data Points 118

6.3.5 Example 5: Liquid-Solid Ratios and the Granule Size

xi

distributions. 120

6.3.5.2 Relationship between liquid-solid ratio and granule mean

diameters Dp,q 120

6.3.5.3 Physical meaning of D5,3 122

6.3.5.4 Non-aqueous granulation and D3,2 123

6.4 Discussion 124

6.4.1 Character of the Method 124

6.4.2 Validity of the Method 126

6.4.3 Accuracy of the Method 126

6.4.4 Limitations of the Method 127

6.4.5 Miscellaneous Remarks 128

6.5 Conclusions 128

6.6 Acknowledgements 129

Appendix A6.1 Similarity between Individual Rankings 130

Appendix A6.2 Agreement between Two Rankings 131

Appendix A6.3 Results of two permutations of ranks 132

7. Discussion and conclusions

135

7.1 Definition systems 136

7.2 Theoretical derivation and empirical selection of proper mean

diameter types 138

References

141

List of publications

147

Samenvatting

148

Appendix Particle Size Distribution Measurement

Methods

151

A.1 Introduction 151

A.2 General Requirements for Methods Measuring a Size Distribution 152

A.2.1 Sensitivity of the Measuring Method 152

A.2.2 Truncation of Measured Size Distribution 152

A.2.3 Presence of ‘foreign’ Particles 153

A.2.4 Systematic Deviations of Measuring Instruments 153

A.2.5 Influence of Particle Shape 153

A.2.6 Physical Properties of Particle Size Measurement

Instruments 155

A.2.6.1 What do Particle-Sizing Instruments Actually Measure? 155

xii

Principles 158

A.2.6.4 Effects of Other Deviations from Assumption in Table A4 160

A.2.6.5 Choosing the Right Particle Sizing Instrument for the

Application 161

A.3 An Overview of Particle Size Distribution Measurement Methods 161

A.3.1 Light Diffraction/Scattering 162

A.3.2 Microscopy (optical & electron) 165

A.3.3 Sedimentation (gravity & centrifugal) 166

A.3.4 Sieving (dry & wet) 169

A.3.5 Zone sensing (electrical & optical) 171

A.4 References 172

Curriculum Vitae

179

xiii

Definitions, Symbols and Abbreviated Terms

Definitions

central moment of a distribution n-th moment around the arithmetic mean size; the 2nd central moment is the variance of a distribution (see also: moment, raw moment).

cumulative (size) distribution fraction of quantity of particles equal to of less than the corresponding particle size.

equivalent sphere sphere that has the same property as the observed particle in relation to a given measurement principle.

frequency distribution/ set of particle sizes and their frequency of

frequency density function occurrence collected from measurements over

a population of particle sizes (sample

distribution); hypothetical set of frequencies of the population of particle sizes (population

distribution).

geometric mean particle exponential of the arithmetic mean of log-

size/diameter transformed particle sizes for a given population or sample of particles, weighted

according to number, volume, etc.

geometric standard deviation exponential of the standard deviation of the distribution of log-transformed particle sizes.

histogram diagram of contiguous rectangular bars proportional in area to the frequency of particles within the particle size interval.

kurtosis kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution.

lognormal distribution particle sizes are lognormally distributed if the logarithm of the particle sizes is normally distributed.

mean particle size/diameter arithmetic mean particle size for a given population or sample of particles, weighted according to number, volume, etc.

median (size/diameter) middle value in a distribution, above and below which lie an equal number of values.

mode of distribution value of the particle size at the maximum of the frequency distribution or of the histogram.

xiv

is the expected value of the n-th power of the deviations from a fixed particle size (see also: raw moment, central moment).

normal distribution distribution which is also known as Gaussian distribution and as bell-shaped curve because the graph of its frequency distribution resembles a bell.

number distribution distribution by number of particles as a function of their size.

order of mean diameter sum of the subscripts p and q of the mean diameter Dp,q.

particle size/diameter diameter of some defined equivalent sphere.

population set of particles concerning which statistical inferences are to be drawn, based on a representative sample taken from the population.

process function mathematical equation (model) that relates the quality (e.g. size distribution) of a (particulate) product to the conditions of its production process.

property function mathematical equation (model) that relates a property of a particulate product to its particle size (distribution).

raw moment of a distribution n-th moment around the origin of the particle size axis; the 1st raw moment is the arithmetic mean size (see also: moment, central moment).

sample part of a population of particles.

size class class of particle sizes having an interval between two stated size limits.

skewness characterizes the degree of asymmetry of a distribution around its mean; a distribution is skew to the right, when the long tail is on the right side of the distribution.

specific surface area surface area of an ensemble of particles per unit mass of material, as measured by a specified analysis method.

standard deviation measure of the spread or dispersion of a set of particle size data; square root of the variance.

surface area surface area of particles, incl. open pores, as measured by a defined method/technique.

xv function of their size.

variance measure of spread or dispersion of a set of particle size data; square of the standard deviation.

volume distribution distribution by volume of particles as a function of their size.

Symbols and Abbreviated Terms

A frontal area of a particle exposed to a flow ABS absolute value

cp particle specific heat at constant pressure

CD dimensionless drag coefficient or friction factor

d granule nucleus diameter

D (equivalent) particle diameter (M-R)

Di midpoint diameter value of the i-th size class

Dmax upper boundary of a distribution (M-R)

Dmin lower boundary of a distribution (M-R)

Dmod mode of a distribution

q p

D , mean particle diameter of a sample of particles (M-R)

e type of quantity of particles

fr(D) frequency density distribution (M-R)

h binder layer thickness

h heat transfer coefficient

k proportionality factor

k number of classes of a histogram

k power of x in moments or subscript (DIN/ISO)

I physical intensity of a stimulus

m number of classes of a histogram

mi mass of a particle in the i-th size class

mij mass of the i-th ingredient in the j-th formulation

Mj mass of the j-th powder formulation

Mij mass of the j-th powder formulation in the i-th size class

xvi

M-R Moment-Ratio definition system

n total number of ingredients of a product formulation

ni number of particles in i-th size class ( = f0(Di) )

nij number of granules of the j-th powder formulation in the i-th size class

O order of a mean particle diameter Dp,q (O = p + q )

p, q powers of D in moments or mean diameter subscripts indicating the same (M-R)

PY Process Yield (percentage of mass in granules < 1500 µm)

Qr(x) cumulative distribution (DIN/ISO)

qr(x) frequency density distribution (DIN/ISO)

r type of quantity of particles

r radial distance from particle origin

rp,q correlation coefficient of relationship between η and Dp,q

R strength of sensation of stimulus I experienced by man

s species of quantity of particles

S particle surface area

Si rank totals of i-th photograph

S(d2) sum of squared rank differences

Stv viscous Stokes' number (dimensionless)

sln D standard deviation of a sample of log-transformed particle diameters σln D standard deviation of a population of log-transformed particle diameters

t time

T temperature

vr particle velocity

Vp particle volume

wi weighting factor of i-th object

x (equivalent) particle diameter (DIN/ISO) r

k

x , mean particle diameter (DIN/ISO)

xmax upper boundary of a distribution (DIN/ISO)

xmin lower boundary of a distribution (DIN/ISO) δ heat penetration depth

δp,q relative difference between Dp,q and ∆p,q

∆p,q mean particle diameter of a population of particles (M-R) ∆Qr,i particle quantity in the i-th class (DIN/ISO)

ε granule (nucleus) porosity η physical product property λ thermal conductivity

µ fluid viscosity ρ particle mass density

ρi specific density of the i-th ingredient

ρpij mass density of granules of the j-th formulation in the i-th size class ρ binder specific density of the binder

ρSD solids density of all ingredients of a formulation ΦM,binder mass fraction of binder

1

1

Introduction

From the Earth on which we live to the interstellar world we want to explore, we are confronted with particles of many sizes and size distributions. The word ‘particles’ is a general term for grains, granules, drop(let)s, gas-bubbles in liquids or solids, pellets, etc. Particles can be defined as entities separated from their surrounding medium by an interface. The particle material may be called the dispersed phase which is surrounded by a continuous phase, which in many consumer products is mostly air or water. Particles are everywhere in our everyday life: cements, creams, flour, hailstones, raindrops, sand, bubbles in soda-water, sprays (aerosols), sugar, toothpaste, washing powders, etc. Even the Universe can be considered as a seemingly infinite collection of (macro) particles: interstellar dust, planetary systems, galaxies, etc. A number of chemical and physical processes specifically deal with particulate materials: heterogeneous catalysis, emulsification, sedimenta-tion, atomizasedimenta-tion, granulasedimenta-tion, grinding and spray-drying. Other processes deal with particulates, e.g., absorption and drying. Dispersed phases in many consumer and industrial products determine to a large extent manufacturing and properties of those products. To support product and process design, modeling and control, one therefore requires a coherent and scientifically sound apparatus of statistical methods and techniques for characterizing particles.

Examples of the influence of particle or droplet sizes on physical/chemical/ physiological behavior of products or processes might give some feeling for the behavior of particulates. With decreasing size, particles become chemically more reactive (dust explosions!), optically more active, etc., since the specific surface of the particles increases. To appreciate the influence of particle or droplet sizes on physical/chemical/physiological behavior of products or processes we give a brief description of three examples from quite different areas:

(i) Hiding power of paints. The hiding power of a paint depends on the opacity of the paint film. The opacity is largely influenced by the relative refractive indices of the pigment (dispersed phase) and the medium (continuous phase) as well as by the (distribution of) particle sizes and of the pigment. For white pigments there is little or no absorption of light, so that the hiding power of white paints depends entirely on the scattering of incident light. The higher the refractive index of the pigment relative to that of the medium and the nearer the particle size to the optimum, the greater the scatter and the greater the opacity of the paint. The maximum scattering of e.g. rutile titanium dioxide pigment particles occurs at a size of about 0.24 µm (spherical particle shape). For a good hiding power of this pigment, the usual size

2

range aimed at is 0.2 to 2 µm diameter (Herdan, 1960, pp. 206-210; Cadle, 1965, pp. 322-329).

(ii) Microbiological safety of margarines. A margarine is a water-in-oil emulsion, which means that an aqueous phase is dispersed in an oil phase. Most of the water volume is in droplets of typically 5-20 µm in diameter. Microorganisms in a margarine can live only in the aqueous phase and, therefore, are present only in the water droplets. Given a usual level of contamination with microorganisms, only a very limited fraction of droplets contains a microorganism. The growth of the number of microorganisms in a droplet is limited by the droplet volume: the larger a droplet, the larger the number of microorganisms in the droplet after full outgrowth. Hence, given an initial contamination of the aqueous phase, the microbial safety of a margarine can be increased by reducing the sizes of droplets of the aqueous phase. Other product properties such as taste, however, may require opposite requirements of the water droplet size distribution (Verrips and Zaalberg, 1980; Verrips et al., 1980).

(iii) Aerosols. Medical aerosols from inhalers and nebulizers are widely used for treating diseases of the respiratory system, particularly asthma. Particle size plays an important role in controlling where the aerosol particles will deposit in the lungs. Particles of sizes larger than 10 µm are most likely to deposit in the mouth and throat. Particles smaller than 5 µm in diameter deposit more frequently in the lower airways and are appropriate for pharmaceutical aerosols. However, for very fine particles below 0.5 µm in diameter there is a chance of escaping deposition altogether and being exhaled. With many nebulizers only 10% of the prescribed dose may reach the lung. Several factors, including patient factors, appear to affect nebulizer output and drug delivery. Particle size in this special context refers to

aerodynamic particle size, meaning the size of a spherical, unit density particle that settles with the same velocity as the particle in question (O’Callagham and Barry, 1997; Mitchell and Nagel 2004; Hoogendoorn and Tukker, 2001).

A basic property common to all dispersed phases is their particle size distribution, i.e. the frequency of occurrence of particles of every size present. Mean characteristics of such size distributions of particles are of great interest and can be studied physically and statistically. Knowledge of mean size characteristics, however, is of little value unless adequate correlation has been established with product properties of specific interest or with processing variables that can be controlled (Irani and Callis, 1963). A correlation between a physical product property and a mean size characteristic may be derived as follows:

Suppose that a physical product property P depends upon the sizes Di (i=1,...,N) of all N particles present in that product. Furthermore, suppose that the contribution of the i-th particle to P is proportional to Di

p

, where p is real. If particle-particle interactions are negligible with respect to the property of interest, the total contribution of all particles to P, then, is proportional to the sum of all these individual contributions. Thus P ∝ ΣDi

p

. Alternatively, this may be written as P ∝ NM'p where

' p

M ≡ ΣDi

p

/N stands for the p-th moment of the size distribution of the particles. Notice that P now depends on N, the total number of particles. To correct for this we can divide by N, but we may also opt for other

3 quantities Q than the total number, e.g. total volume. The latter quantity is proportional to the third moment of the size distribution since Q ∝ ΣDi

3

= NM3' . More generally, the total quantity Q∝Σ Di

q

= NM'q, is proportional to the q-th moment of the size distribution, where q = 0, 1, 2, 3 for the total number, diameter, surface, volume of the particles, respectively. Hence, the total contribution per unit of amount of particles is a ratio of two moments of the size distribution of the particles, P ∝ (NM'p)/(NMq' )= M'p/M'q. A ratio of two moments of a particle size distribution, transformed to the unit of length, can be considered to be a mean particle diameter, Dp,q ≡(M'p/M'q)1/(p-q). This moment-ratio definition system for mean particle diameters produces quantities which can represent the physical contribution of the population of particles to the physical product property P.

The above derivation demonstrates that these mean diameters Dp,q are not merely statistical parameters, but can be quantities of physical relevance, pre-eminently suitable to represent physical properties of the dispersed phase in so-called property functions and process functions. A property function is a functional relationship between a product property and the corresponding dispersity characteristics; similarly, a process function relates the process parameters and the dispersity characteristics (Rumpf, 1967; Polke, 1987; Borho et al., 1991; Polke, 1993; Müller et al., 2001; Schubert and Engel, 2004; Peukert et al., 2005; Hounslow and Reynolds, 2006). Property and process functions apply not only to the end products but to every stage of a process. The particle properties of the end product affect the quality, for example the color strength, while those of the intermediates influence processing properties, for example the filtration resistance and subsequent process stages such as washing and granulation. The correlations between product properties and dispersity parameters have to be determined experimentally (Polke, 1987). It is an assertion of this thesis that mean particle diameters defined according to the rules discussed above play an important role in property and process functions, a role that is rarely understood and appreciated.

In physical models of a product property which depends on the nature of the dis-persed phase, mean diameters can represent the physical contribution of this dispersed phase. Such models will be optimal if the mean particle diameter of choice corresponds physically to the product property in a causal manner. Any other type of characteristic size parameter of the dispersed phase will then give sub-optimal models. An example of a ‘sub-optimal’ type of parameter is statistical parameters such as median diameters and size distribution percentiles in general. These parameters are statistical measures for locating a distribution, based only on a ranking of particles according to their size. From their definition it follows that, e.g., the median value of a number density distribution will not change when some or more of the largest particles increase in size ten-fold. Therefore, the median does not correspond to a physical product property in a causal way. A statistical and physical study of mean diameters requires an adequate knowledge of both Statistics and Physics. This was recognized earlier by Herdan who, in the preface to his book, stated that “there is scarcely another body of scientific data more in need of statistical criticism and refinement than that of particle size determinations and correlations” (Herdan, 1960, p. vii). In daily practice, the selection of size

4

distributions and distribution parameters of dispersed phases to model physical product or process properties, is subjective and, therefore, a weak point in existing knowledge in the particulate area. We believe that this is caused mainly by a poor knowledge of particle size distribution parameters physically related to product or process properties and a poor understanding of the physical meaning of statistical quantities. This shortcoming may also underlie the existence of various definition and notation systems, and various nomenclatures of statistical size distribution parameters, which, therefore, provide a rather incoherent body of physical/statistical knowledge. The still inspiring paper of Mugele and Evans (1951) is one of the first papers demonstrating relationships of mean particle diameters to a physical product or process property, although these authors introduce such relationships very briefly. Their approach stimulates the use of size distribution parameters of physical relevance, which are suitable to represent physical dispersed phase properties in property and process functions.

These considerations provide the framework for this thesis. The first part of three chapters, viz., chapter 2, 3 and 4, evaluates current definition systems and practices, to come to a firm and physically suitable instrument for further development of the particle size analysis area. Two developments are described in the second part of the thesis, viz., chapters 5 and 6. A short overview of the chapters is given below. − Evaluation of mean particle diameter definition systems (Chapter 2)

Two definition systems for mean particle diameters are reviewed. These are: (i) The system by which mean diameters are expressed as a ratio between two moments of the number density distribution of particle sizes, as in, e.g., British Standard 2955; (ii) The system described in, e.g., DIN 66141 and ISO 9276-2, in which mean diameters are defined on the basis of moments of different types of particle-size distributions The latter definition system was also one of the main subjects of the course “Grundlagen und moderne Verfahren der Partikelmess-technik”, given by the late Prof. Kurt Leschonski at the University of Clausthal. − Development of an unambiguous nomenclature system for mean particle

diameters (Chapter 3)

Different names are sometimes used for one and the same mean diameter or, conversely, the same or very similar names are assigned to different mean diameters. A nomenclature system has been developed to overcome this observation.

− Evaluation of two methods to estimate mean particle diameters from histogram

data (Chapter 4)

Statisticians have devised histograms to visualize a distribution of observations in a sample, and use summation over the intervals of a histogram to calculate sample characteristics. Currently, ISO recommends integration of histograms. Both methods have been evaluated.

5 − Theoretical derivation of the type of mean diameter related to a physical product

or process property (Chapter 5)

In theoretical studies various approaches are followed to introduce a parameter of the size distribution of a dispersed phase into theoretical models of physical/chemical product or process properties. A theoretical method has been developed to derive the proper type of mean diameter. Examples from the areas of evaporation, heat transfer, and turbulent two-phase flow, illustrate this method. − Empirical selection of the type of mean diameter related to a physical product or

process property (Chapter 6)

A method was developed to select empirically the proper type of mean diameter describing a product or process property. Examples, mainly from the granulation area, are used to illustrate the method.

The meaning of specific terms used in this thesis is given in the list of ‘Definitions, Symbols and Abbreviated Terms’ on page xii.

The thesis is completed with an introductory survey of current methods in particle size analysis (see Appendix on Particle Size Distribution Measurement Methods).

7

2

Evaluation of Definition Systems

Abstract

Mean particle diameters are important for the science of particulate systems because they provide information on particle-size distributions in such a way that they can be related to physical or physiological processes (e.g., seeing) or product properties. There are different notation systems for mean diameters, which may cause much confusion. This equally applies to their nomenclature. The present chapter is concerned with the 'Moment-Ratio' and German (DIN) notations. At the end, the Moment-Ratio notation is recommended for standardization, requiring a more extensive contribution of statistics to the science of particulate systems.

2.1

Introduction

The purpose of mean particle diameters is to provide information on particle-size distributions in a way suitable for relating them to physical or physiological processes or product properties. Reduction of data from a complete particle-size distribution into one or more mean diameters requires a profound knowledge of their definition and their physical meanings. Therefore, understanding the physical character of mean particle diameters would benefit from a generally-accepted mathematical definition of mean diameters and also from descriptions of mean diameters that are unambiguous and convey their physical meaning. In practice, however, users of particle size analysis data are greatly hampered by different definition systems for mean particle diameters, since these systems have quite different properties. The different names used for mean diameters in the literature add to this confusion. This unsatisfactory situation can be highlighted by the following examples.

(i) Mean particle diameters are denoted as Dp,q in one definition system, whereas another denotes xk,r. Although similar notations are used, the definition of the subscripts differs and so does the meaning of the mean diameters. Only Dp,0 is identical to xp,0.

(ii) Although geometric mean particle diameters may have distinct and important physical or physiological meanings, they are not defined in one of the definition systems.

(iii) Some scientists consider mean particle diameters as basic quantities, whereas others consider moments of particle-size distributions as the relevant ones.

8

Apparently, both views are thought to be equivalent but they are not (see under 3.4).

(iv) Mean diameters and median diameters of particle-size distributions are frequently used alternately, although they have quite different (physical and/or statistical) meanings.

(v) The contribution of statistics to the science of particulate systems seems insufficient. Probably, statistics is a science scientists are not always quite familiar with. Drawing conclusions about the properties of a population of particle sizes on the basis of properties of samples, requires statistical knowledge of relationships between sample and population properties. In fact, the science of particulate systems is very much a science for which statistics is a suitable tool.

(vi) ANSI has not published any material on mean particle diameters. Recently, ASTM published Standard Practice E 2578 (2007).

Such cases show the necessity to come to a clear definition and nomenclature. Below we will review the two systems for the definition of mean particle diameters. These are:

− The system by which mean diameters are expressed as a ratio between two moments of the number density distribution of particle sizes (see for example British Standard 2955); this notation system will be referred to as: Moment-Ratio notation;

− The system described in e.g. DIN 66141, in which mean diameters are defined on the basis of moments of different types of particle-size distributions (e.g. number, surface or volume distribution); this system is called German (DIN) notation. First, statistical moments will be dealt with in brief, thereby distinguishing between populations and samples of particle sizes, because moments form the basis of any system describing particle size quantities.

2.2

Moments of a Distribution

Statistically, the moments of distributions (Mood et al., 1974) are the basis for defining mean diameters and standard deviations. Moments of a population of particle sizes have to be distinguished from moments of a sample taken from that population.

If a random variable X has a continuous density distribution f(x), the r-th moment of

X about the origin, usually denoted by µr' is defined as

dx x f xr r' : ∫ ( ) +∞ ∞ − = µ (1) where 1 ) ( = ∫ +∞ ∞ − dx x f (2)

9 '

r

µ will be referred to as the r-th raw moment of X or simply as the r-th moment of

X. The (arithmetic) mean of X, µx, is the first moment µ1'. The r-th moment of X about µx, defined as dx x f x x r r : ∫ ( ) ( ) +∞ ∞ − − = µ µ (3)

is known as the r-th central moment. Central moments are denoted without a prime. The second central moment of X

dx x f x x) ( ) ( 2 2 ∫ +∞ ∞ − − = µ µ (4)

is the variance of X, denoted by σ2x. Its square root σx, is called the standard deviation. The arithmetic mean, µx, and the variance, σ2x, characterize the normal distribution function.

A random sample, containing N elements from a population of particle sizes, enables estimation of the moments of the population distribution as defined above. The r-th (raw) sample moment, denoted by Mr', is defined to be

∑ − = i r i i r N n X M': 1 (5) where =

∑

i i n

N , Xi is the midpoint of the i-th interval and ni is the class frequency. '

r

M is an unbiased estimator of µr'. The (arithmetic) sample mean M1' of the particle size X is mostly represented by X . The r-th sample moment about the mean

X , denoted by Mr , is defined by ∑ ₋ = − i r i i r N n X X M : 1 ( ) r

M is an estimator of the population parameter µr. Unlike the raw sample moments, the sample moments about the mean are not always unbiased estimators of the population moments about the mean. The best-known example is the 2nd sample moment about the mean, the sample variance M2, which underestimates the population variance σ2x. An unbiased estimator is obtained when M₂ is multiplied by N /(N - 1). Thus, the sample variance s2x has to be calculated from the formula

1 ) ( 2 2 − − = ∑ N X X n s i i i x (6)

Its square root sx is an estimator of the standard deviation σx, although not unbiased: 1 2 2 − ∑ − = N X N X n s i i i x (7)

10

Lognormal distribution (Aitchison and Brown, 1957, p.9; Herdan, 1960, p.81): If X

is lognormally distributed (X > 0), then Y = ln X has a normal distribution. The parameters µy (the arithmetic mean) and σy2 (the variance) characterize the lognormal distribution of X, similar to the normal distribution function. The parameters µy and σy of Y can be estimated by Y and sy, which are defined similarly to X and sx. Note that the lognormal distribution is skew to the right. It can be shown that exp(Y) is identical to the geometric mean Xg of the sample of

X: N ni i i i i i g N nY X X =exp_ −1∑ _= Π

The standard deviation sy can be expressed in X as:

{

}

1 ) / ( ln 2 − ∑ = N X X n s i g i i y (8)

2.3

Definition of Mean Particle Diameters

For the discussion of the Moment-Ratio and the German (DIN) notations, shape coefficients are not taken into account. They can be neglected as they influence the notation systems in the same way. In particle-size analysis, however, their importance is, of course, beyond any doubt.

2.3.1

Moment-Ratio Notation

2.3.1.1 Definition of Dp,q

For a continuous number density distribution f0(D)*) of particle sizes D, the mean diameter ∆p,q of D is defined as 1/(p - q)-th power of the ratio of the p-th and the q-th raw moment of q-the number density distribution of q-the particle sizes:

) /( 1 0 0 0 0 , ) ( ) ( q p q p q p dD D f D dD D f D − ∞ ∞           = ∆ ∫ ∫ (9)

An obvious choice for estimating the mean diameter ∆p,q from a sample, is Dp,q:

*) _{The ( )} 0 D

f notation is used in order to avoid any confusion with respect to the differential operator, although f0(d) would be in agreement with the convention to use capital letters to denote random variables (Mood et al., 1974).

11 ) /( 1 ' ' , q p q p q p M M D −       = (10) where − ∑ = i p i i p N n D

M' 1 , see Eq. (5). Thus ) /( 1 , q p i q i i i p i i q p D n D n D −           = ∑ ∑ , if p ≠ q (11)

Though p and q may have any real value, we restrict p and q in this thesis to integer values. For equal values of p and q it is possible to derive from Eq. (11) (see Appendix A2.1) that

          = ∑ ∑ i p i i i i p i i p p D n D D n D ln exp , , if p = q (12) If p = 0, then N ni i i i i i i i D n D n D _= Π       ∑ ∑ = ln exp 0 , 0 . 0 , 0

D is the well-known geometric mean diameter. Note that the physical dimension of any Dp,q is equal to that of D itself.

2.3.1.2 The Standard Deviation

According to Eq. (7), the standard deviation of the number density distribution of a sample of particle sizes can be estimated from

1 2 0 , 1 2 − ∑ − = N D N D n s i i i x (13)

which can be rewritten as 2 0 , 1 2 0 , 2 D D c s= − with ) 1 /( − = N N c .

In practice, N >> 100, so that c ≈ 1. Hence 2 0 , 1 2 0 , 2 D D c s≈ − (14)

12

The population standard deviation σlnD of a lognormal number density distribution of particle sizes D can be estimated by slnD (see Eq. (8)):

{

}

1 ) / ( ln 0,0 2 ln − ∑ = N D D n s i i i D (15)

In particle-size analysis, the quantity sg, ]

[ exp ln D

g s

s = (16)

is frequently referred to as the geometric standard deviation (Herdan, 1960, p.81) although it is not a standard deviation in its true sense.

2.3.1.3 Relationships between Mean Diameters Dp,q

Using Cauchy's inequality (Kendall and Stuart, 1958), it can be shown (see Appendix A2.3) that

0 , 0 , m p D D ≤ , if p ≤ m (17) and that q p q p D D _{−1, −1}≤ , . (18)

Differences between mean diameters decrease according as the spread of the particle sizes D decreases. The equal sign applies when all particles are of the same size. Thus, the differences between the values of the mean diameters provide already an indication of the dispersion of the particle sizes.

Another relationship very useful for relating several mean particle diameters has the form

(

)

(

)

(

)

q c c q c p c p q p q p D D D , − = , − / , − . (19) In particular, if c =0

(

)

q q p p q p q p D D D , = _,0/ _,0 − . (20)

For example for p = 3 and q = 2: D_3,2=D33,0/ D22,0.

Eq. (19) seems particularly useful when a specific mean diameter cannot be measured directly. Its value may be calculated from two other, but measurable mean diameters.

Eq. (11) also shows that p

q q

p D

D , ≡ , , (21)

because exchange of the subscripts p and q in Eq. (11) results in identical right sides of that equation. In spite of its simplicity, this symmetry relationship plays an

13 important role in the use of Dp,q. The sum O of the subscripts p and q is called the order of the mean diameter Dp,q (Mugele and Evans, 1951)

O = p + q . (22)

Its importance will become clearer further on.

We can now construct a diagram of Dp,q for integer values of p and q (Figure 1).

Fig. 1: Diagram of Mean Particle Diameters Dp,q. The arrows pointing downwards connect mean diameters having the same p - q value. The arrows pointing upwards connect mean diameters of the same order O = p + q.

For lognormal particle-size distributions, there exists a very important relationship between mean diameters:

] 2 / ) exp[( ln2 0 , 0 ,q D p =∆ p+q σ ∆ (23)

where σlnD is the standard deviation of the log-transformed particle sizes D. This equation can be derived from the moments of the lognormal distribution (Aitchison and Brown, 1957, p.8). Analogous to Eq. (23), the relationship

] 2 / ) exp[( ln2 0 , 0 ,q D p D p q s D = + (24)

14

is a good approximation for a sample if the number of particles in the sample is large

(N > 500), the standard deviation σlnD < 0.7 and the order of Dp,q not larger than 10. These conditions can be derived from an estimation method described by Finney (1941).

For lognormal particle-size distributions, the values of the mean diameters of the same order are, therefore, equal. Conversely, an equality between the values of these mean diameters points to lognormality of a particle-size distribution. For this type of distribution a mean diameter Dp,q can be rewritten as Dj,j, where j = (p + q)/2 =

O/2, if O is even. Dj,j can be considered as a location parameter (viz. the geometric mean) of the j-th moment distribution of the number density distribution (see Aitchison and Brown, 1957, p. 12).

2.3.2

German (DIN) Notation System (Rumpf and Ebert, 1964;

DIN 66 141; Leschonski, 1984)

The acronym DIN is the abbreviated name of the German Institute for Standardiza-tion (Deutsches Institut für Normung) and is used in the names of its standards, like DIN 66 141. This section does not present full details of the DIN system, but only gives an overview of aspects necessary for a comparison (in Section 2.3.3) of this system with the M-R system. It is not intended to present an introduction to the DIN system.

2.3.2.1 Definition of xk,r

The frequency density of the variable x is denoted by qr(x) and the cumulative distribution is denoted by Qr(x), in which x characterizes a physical property uniquely related to a particle size, such as a linear dimension, a projected area or a volume. For the definition of mean particle diameters it is necessary that the meaning of x be restricted to (equivalent) particle diameters. The subscript of qr(x) represents the measure used for expressing the type of quantity of the particle size x (Table 1).

Table 1: Types of quantities of a particle size x.

Subscript r Type of quantity

0 number distribution 1 length distribution 2 area, surface distribution 3 volume, weight distribution

15 The relationship between qr(x) and Qr(x) is given by

x d x Q d x qr( )= r( )/ or = ∫ x x r r x q y d y Q min ) ( ) ( .

The k-th raw moment Mk,r of x is defined by ∫ = max min , ( ) x x r k r k x q x dx M (25) Note that 1 ) ( max min , 0 = ∫ = x x r r q x dx M (26)

is a normalizing condition. The moment Mk,r is called a complete moment. Incom-plete moments are not considered, as they do not give rise to extra aspects of comparison.

The mean diameter xk,r is defined by k r k r k M x , = , (27)

2.3.2.2 The Standard Deviation

The k-th central moment of the number distribution qo(x) is defined by

∫ − = max min 0 0 , 1 0 , ( ) ( ) x x k k x x q x dx m , (28)

similar to Eq. (3) (M-R system). The 2nd central moment, the variance, is therefore 2 0 , 1 0 , 2 max min 0 2 0 , 1 0 , 2 (x x ) q (x)dx M (M ) m x x − = − = ∫ (29)

Eq. (29) can be generalized to hold for any density function qr(x) by replacing the subscript 0 by the subscript r. Hence, the variance sr2 of the distribution qr(x) is

2 , 1 , 2 2 , 2 r sr M r (M r) m = = − (30)

For normal size distributions qr(x), the standard deviation sr can also be estimated from: r r r r r x x x x s = 84, − 50, = 50, − 16, (31)

where x50,r is the 50%-point (or median) of the cumulative distribution Qr(x). The quantities x16,r and x84,r are the 16%- and 84%-points (or percentiles), respectively, of this cumulative distribution.

For lognormal size distributions qr(x), the standard deviation s can be calculated from:

16 ) / ( ln ) / ( ln x84,r x50,r x50,r x16,r s= = (32)

The geometrical standard deviation sg is obtained from ) ( exp s sg = (33) Hence, r r g x x s = 84, / 50,

2.3.2.3 Relationships between Moments and between Mean Diameters xk,r

A relationship between mean diameters xk,r, can be derived from the relationship between different particle-size distributions. If qr(x) and qt(x) are two size distributions, then t t r t t r x x t t r t t r r M x q x dx x q x x q x x q , max min ) ( ) ( ) ( ) ( − − − − = =

∫

(34)

The denominator normalizes the distribution function qr(x). Substitution of Eq. (34) in Eq. (25) gives t t r t t r k r k M M M , , , − − + = (35)

From Eq. (35) and Eq. (27) it follows that

t r t t r t r k t t r k k r k x x x − − − + − + = , , , Hence, if t = 0, then r r k r k r k r k x x x ++,0 = , ,0 (36)

Putting t = 0 in Eq. (35) gives a simpler relationship between moments:

0 , 0 , , r r k r k M M M = + (37)

For a lognormal particle-size distribution qr(x), the mean diameter xk,r is expressed in terms of the median x50,r of the distribution and of the geometrical standard deviation sg (Eq. (33)): ) 2 / ln exp( 2 , 50 ,r r g k x k s x = (38)

17 A diagram of xk,r for integer values of k and r is shown in Figure 2 in which mean diameters having the same averaged quantity but originating from different size distributions, are found on a horizontal line.

Fig. 2: Diagram of mean particle diameters xk,r.

2.3.3

Evaluation of Definition Systems

2.3.3.1 Relationship between Dp,q and xk,r

The relationship between these mean diameters can be found by expressing xk,r in terms of moments of the number density distribution according to Eq. (37) and using Eq. (10):

(

)

(

k rr

)

k r r k r k k r k D M M M x , 0 , 0 , , , = = + = +

This leads to the important relationship r

p r

k D

x , = , if p = k + r and k ≠ 0. (39) The case k = 0 refers to geometric mean diameters and is dealt with in the next section. A consequence of the definition in the DIN notation of mean diameters on the basis of different types of particle-size distributions (number, length, etc.), is rather complicated interrelationships between mean diameters. For example, it has been shown (Leschonski, 1984) that x−1,3=x1,2, both being inversely proportional to the specific surface of a sample. In fact, the mean diameters x−1,3 and x1,2 are both

18

equal to and have the same physical meaning as D3,2 and are only two different designations in the DIN notation for one and the same mean particle diameter. This redundancy may hamper the physical interpretation of mean particle diameters.

2.3.3.2 Geometric Mean Diameters

The German (DIN) system has no class of mean diameters with the same properties as the class of geometric mean diameters Dq,q in the M-R system. This statement is based on the following observations:

− The approach to define mean particle diameters on the basis of moments of various density functions (e.g. number, volume) seems to hide the existence of geometric mean diameters. Substitution of Eq. (26) in Eq. (27) may lead to the conclusion that geometric mean diameters cannot even exist within the frame-work of the DIN notation, because M0,r =1 by definition and

apparently does not exist;

− The Median, z50 (DIN 66 141) or x50,r (Rumpf and Ebert, 1964; Leschonski, 1984), is used for estimation of both parameters of a lognormal distribution function, i.e., for the location parameter exp(µy) and for the dispersion parameter exp(σy).

The absence of a definition of geometric mean diameters in the German (DIN) notation has serious consequences for (i) the derivation of a general relationship between mean diameters of lognormal size distributions, and (ii) description of relationships between geometric mean diameters and physical/physiological product and process properties.

Case (i). If a geometric mean diameter is the relevant one for description of a physical or physiological process or product property under investigation, the choice of the median diameter of that distribution should be considered as erroneous. It may hide or obscure a relationship between a process or product property and its relevant particle-size distribution. In the German (DIN) system for example, it is impossible to find an equivalent for D_2,2 to describe the result of a visual ranking of micro-graphs of air-bubble distributions (Alderliesten, 1991) (see Chapter 6 for more details).

Case (ii). In the Moment-Ratio notation, Eq. (24) shows that the mean diameter q

p

D , can be expressed in terms of the standard deviation slnD and of any other mean diameter. In the German (DIN) notation, however, the mean diameter xk,r is expressed only in terms of the median and the geometrical standard deviation of its density function qr(x), see Eq. (38). Using an equation similar to Eq. (24) to relate estimates of the median diameters of various distribution functions in order to overcome the limitations of Eq. (38), is doubtful.

These two cases show that in any notation system, geometric mean particle diameters must be defined because these mean diameters have several important aspects for particle size analysis, e. g.,

− their distinct physical meanings necessary for relating the geometric means to physical, physiological process or product properties;

0 0 , 0 , 0r = M r = 1 x

19 − their property to be the central values of lognormal particle-size distributions,

which values are influenced by the size of each individual particle. These properties are not shared by the median diameter:

− The median is only a purely statistical measure for locating a distribution, based only on a ranking of particles according to their sizes. From its definition it follows that e.g. the median value of a number density distribution will not change when some or more of the largest particles are given a ten-fold size increase. Therefore, the median has no physical meaning;

− The median of a sample is not the best estimator for the central value of a lognormal size distribution, let alone for the geometric mean of a population of particle sizes not lognormally distributed.

Hence, a median should not be used in this context. Obviously, any other percentile of a size distribution has statistical properties similar to those of the median. To illustrate further the absence of geometric mean diameters in the DIN system, a derivation of this class of mean diameters, written in terms of the DIN notation, is presented in Appendix A2.2.

2.3.3.3 The Order of Mean Diameters

Using Eqs. (22) and (39), the order O of xk,r is defined by

O = k + 2r (40)

For a quick decision about the order of a mean particle diameter, this definition is less easy to use in daily practice than the expression O = p + q. It is, for example, easier to see that D3,2 and D1,4 are of the same order as x1,2 and x3,1.

But this is not the only drawback. This can be illustrated by an example from the lit-erature (Orr and DallaValle, 1959) citing the results of a microscopic measurement of a sample of fine quartz (Table 2). The notation of the class boundaries in Table 2 was chosen to remove any doubts as to the classification of a particular particle size. A histogram of these data is shown in Figure 3.

We now put a number of values calculated for the mean particle diameters Dp,q in the diagram belonging to Dp,q (cf. Figure 4).

We do the same for the values of xk,r (Figure 5). For the sake of completeness, also a couple of geometric mean diameters calculated from the formula derived in Appendix A2.2, are inserted.

20

Table 2: Microscopically-measured frequency distribution of a sample of fine quartz (Orr and DallaValle, 1959).

Class Number Mid-point [µm] Range of sizes [µm] Freq. ni Cum. freq.

∑

i i n Cum. fraction N n i i/

∑

1 2 1.5 - < 2.5 10 10 0.05 2 3 2.5 - < 3.5 35 45 0.225 3 4 3.5 - < 4.5 43 88 0.44 4 5 4.5 - < 5.5 44 132 0.66 5 6 5.5 - < 6.5 22 154 0.77 6 7 6.5 - < 7.5 18 172 0.86 7 8 7.5 - < 8.5 12 184 0.92 8 9 8.5 - < 9.5 7 191 0.955 9 10 9.5 - < 10.5 4 195 0.975 10 11 10.5 - < 11.5 5 200 1.00

Fig. 3: Particle size vs. frequency of fine quartz sample (data from Table 2).

21 Fig. 4: Diagram of Dp,q for fine quartz sample. The

order of the mean diameters is indicated by arrows.

Fig. 5: Diagram of xk,r for fine quartz sample. Order of an xk,r: (●) O = 3; (■) O = 4; (▲) O = 5; (▼) O = 6. Note that mean diameters of the same order are connected by a 'knight's move'.