The Terra Incognita of sampling: Grouping and segregation

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THE

TERRA INCOGNITA

OF SAMPLING:

GROUPING AND

SEGREGATION

Dosti S. Dihalu

Delft University of Technology Faculty of Applied Sciences

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iii

The Terra Incognita of Sampling:

Grouping and Segregation

Proefschrift

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

op gezag van de Rector Magnificus Prof. ir. K.C.A.M. Luyben, voorzitter van het College voor Promoties,

in het openbaar te verdedigen op dinsdag 12 juni 2012 om 12:30 uur

door

Dosti Shaam DIHALU

Scheikundig ingenieur geboren te Rotterdam

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iv Dit proefschrift is goedgekeurd door de promotor: Prof. dr. H.T.Wolterbeek

Copromotor: Dr. ir. P.Bode

Samenstelling promotiecommissie:

Rector Magnificus voorzitter

Prof. dr. H.T. Wolterbeek Technische Universiteit Delft, promotor

Dr. ir. P. Bode Technische Universiteit Delft, copromotor

Prof. dr. K.H. Esbensen GEUS and Aalborg University

Prof. R.C.A. Minnit University of Witwatersrand

Prof. dr. C. Pappas Technische Universiteit Delft

Prof. em. dr. I.T. Young Technische Universiteit Delft

Dr. B. Geelhoed adviseur

Prof. dr. P. Dorenbos Technische Universiteit Delft, reservelid

All rights reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without prior permission from the publisher. ISBN 978-1-61499-074-1 (print)

Keywords: Grouping and Segregation, Sampling Theory, LTS, Multi-Axial Shape Factor

Published and distributed by IOS Press under the imprint Delft University Press

Publisher IOS Press Nieuwe Hemweg 6b 1013 BG Amsterdam The Netherlands tel: +31-20-688 3355 fax: +31-20-687 0019 email: info@iospress.nl www.iospress.nl LEGAL NOTICE

The publisher is not responsible for the use which might be made of the following information.

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v

Voor Susy en Reshano

The research described in this thesis was supported by Dutch Technology Foundation STW, which is the applied science division of NWO, and the Technology Program of the Ministry of Economic Affairs, under STW grant 7457. The Nederlands Forensisch Instituut,, Deltares, Nutreco, Hosokawa Micron B.V. and Schering Plough are members of the users’ committee of this project.

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6.4 Simulation Study ... 69 6.5 Results ... 71 6.6 Discussion ... 74 6.7 Conclusion ... 75 6.8 References ... 75 7 A New Multi‐Axial Particle Shape Factor‐ application to particle sampling ... 77 7.1 Introduction ... 77 7.2 Experiments ... 81 7.3 Simulations ... 84 7.4 Conclusions ... 86 7.5 Outlook ... 86 7.6 References ... 86 8 Incorporation of a Fractal Breakage Mode into the Broken Rock Model ... 88 8.1 Introduction ... 89 8.2 Methodology ... 91

8.3 Conclusions and Outlook ... 95

References ... 96 9 The Input, Core, and Auxiliary Parameters of the New Sampling Theory ... 97 9.1 Introduction ... 97 9.2 Input Parameters ... 98 9.3 Core Parameters ... 103 9.4 Auxiliary Parameters ... 108

9.5 The Decision Support System ... 110

9.5.1 Purpose of the Decision Support System (DSS) ... 110

9.5.2 Structure of the DSS ... 111

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ix 10 Conclusion ... 119 11 Discussion ... 122 12 Final Remarks ... 123 Appendix A: History of Sampling Theories ... 124 A.1 Brunton (1895) ... 124

A.2 The binomial models (1928, 1935, 1945)... 124

A.3 Kassel and Guy (1935) ... 125

A.4 Visman (1947) ... 125 A.5 Ingamells (1973, 1976) ... 126 A.6 Gy (1953, 1979, 2004) ... 127 Appendix B: Sampling Experiment ... 128 Appendix C: Circle Assigning Algorithm ... 134 Appendix D: Finite Population Correction ... 135 Appendix E: Var(FSE) vs X of the FBRM ... 136 List of Symbols ... 137 References ... 142 Summary ... 146 Samenvatting ... 147 Acknowledgements ... 148 Curriculum Vitae ... 150 List of Publications ... 151

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2

REPRESENTATIVENESS

Sampling is a necessary physical handling step in many fields of industry and the science behind it has its basis in chemistry, physics, and mathematics. The fact that sampling has both a practical and a theoretical fundament makes this topic rather complex, as in practice both aspects need to be balanced to reach an optimal point from an economical point of view. Throughout the years, much effort is spent by designers and mechanical engineers to optimize the practical side of sampling; this has resulted in a wide range of (automated) sample takers, computer simulations of the sample takings, and the development of a various number of sampling strategies for specific circumstances. Equivalently, many chemists, physicists, and mathematicians have focused on the theoretical basis of sampling, which has resulted in different sampling theories, varying from empirical to fundamental theories.

The main idea of taking a correct sample includes the withdrawal of a sufficiently enough amount of material for it to be representative for the total amount of material. From a practical point of view, it is essential to know the minimum amount of material that can be regarded as being representative. Simultaneously, it is of importance to know how accurate an obtained sample is, i.e., preferably, the deviation between the true properties of the total batch and the amount that has been withdrawn as a sample needs to be known. The representativeness of a sample indicates to what extent the sample resembles the properties of the original batch from which the sample was taken. Ideally, the representativeness needs to be high (yet in accordance with the fitness for intended use), so that an analysis of the sample can give clear information about the –mostly a multiple times larger amount than the sample- total batch. Taking correct, representative samples will have positive influence on the time-consuming, and therefore also the economical part of the sampling process.

(Pehlken 2011) regards the sampling procedure as the consecutive steps of :

0. Sampling Strategy

This step deals with the selection of the batches to be sampled and the time and place of sample taking.

1. Sample Taking

Typical questions that are raised during this step are: how many samples are to be taken, which tools are used and from which locations in the batch are they withdrawn?

2. Sample Preparation

During this step it is decided what will be done with the sample before it is made available to the laboratory for analysis.

3. Sample Analysis

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sub-sampled, which might be necessary due to the sample size and the maximum handling weight of the analysis devices used, these operations can cause an increased total sampling error.

Besides this, the sample can be wrongly analyzed due to technical malfunction of the device that is used. Moreover, each analysis technique inevitably includes a source of error itself.

The final result of the sampling procedure indicates the amount or ratio of the property of interest in the total batch. This can be e.g. the iron content in cereals, the number of maligneous copper-stained cells in a dog liver, the concentration of zirconium particles of a specific size in a binary mixture of large and small zirconium particles, et cetera. In general, this measured substance value Xmeasured can be expressed as (see also (Pehlken 2011)) :

Xmeasured = Xtrue + e0 + e1 + e2 + e3 (1.1)

With

Xmeasured = the measured value, rendered from the complete sampling analysis procedure

Xtrue = the true value of the property of interest

e0 = the error introduced during the sampling strategy step

e1 = the error introduced during the sample taking step (this includes the so-called “fundamental sampling error”)

e2 = the error introduced during the sample preparation step

e3 = the error introduced during the sample analysis step.

If the random errors are statistically independent of each other, the variance of the measured value is the accumulated value of all random errors, i.e.:

(1.2)

With

= the variance of the measured substance Xmeasured

= the variance of e0 = the variance of e1 = the variance of e2 = the variance of e3.

If the error that is introduced in sampling step 0 is left out, the most influential error is the error of sample taking (Pehlken 2002; Pehlken 2011):

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≫ ≫ (1.3)

This can clearly be understood if one reminds that typically a batch consists of several kg-tons of material that are reduced to analytical samples of grams, milligrams or even smaller.

The practical problem is that not every particle from the batch has the same probability of being included in the analytical sample, especially if the particle is bigger than the analytical sample. However, this problem can in theory be overcome by the selection of appropriate sampling tools. By doing so, it can be said that each particle type has a constant first-inclusion probability πi, whereas a non-constant first-inclusion probability indicates that wrong sampling tools are used. The latter case however, is a pure practical problem and is not the focus of this thesis.

Besides the probability of an individual particle type to be included in a sample, another interesting parameter is the probability that a pair of particles types is included in the sample, which can contain particles from many different types in total. In the case of independencies between particles of, say, type i and j, the second-order inclusion probability πij would simply be the multiplication of πi and πj. However, in practical cases (Geelhoed, Koster-Ammerlaan et al. 2009), it has been shown that the second-order inclusion probability is unequal to the product of πiand πj, i.e.:

(1.4)

Intuitively, this makes sense if one takes into account the attraction or repulsion that can exist between particle types due to specific particle properties, e.g. particle size difference, magnetism, electrostatics, et cetera. This was also discussed by Geelhoed (Geelhoed 2007). It has to be noted here that the sampling taking step plays a big role in the value of the second-order inclusion probability. If particles are withdrawn individually in a random manner, the inter-particle relations are vanished and the second-order inclusion probability of particle types i and j will simply be the multiplication of their first-order inclusion probabilities. If one is specifically interested in the second-order probability of components in a mixture, current sampling devices, e.g. sampling lances or sample splitters, could well be used to conserve the inter-particle attractions or repulsions, since a group of particles is withdrawn at once. However, due to the sample taking step, the original clustering status of the particles can change unintentionally. One way to overcome this issue is by not physically withdrawing particles, but to digitally capture the spatial status of the particles. State-of-the-art image analysis software can thus be used to automatically discriminate particle types from each other and mostly also the particle locations can be obtained directly, without interfering the true locations of the particles. It can be concluded that an increased or decreased probability to be included in the sample is due to the grouping and segregation between particles in the population.

It is specifically the impact of this behavior of particles in the practice, characterized by grouping and segregation to which the research described in this thesis has been directed;

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In a variogram, different variance terms are plotted against the variances of the sets and the lower limits of asymmetric confidence ranges (e.g. 95% or 99%), enabling to estimate directly where orderliness in a sample space dissipates into randomness (Merks 2011) The formula for the j-th variance term of a set of bulk samples is :

2 (1.5)

With:

xi = i-th result of ordered set of test results

xi+j = (i+j)-th result of ordered set of test results

n =total number of test results (x) = j-th variance term of ordered set 2[n-j] = number of degrees of freedom.

(iii) Modeling Approach

The modeling approach, which is commonly known as the “bottom-up” approach, uses information on the particle level, whereas in a “top-down” approach the estimation is rather done without information at the level of the individual particles. The most common modeling based sampling theory is the theory constructed by Pierre Gy from 1979 on (Gy 1979).

The interactions, volumetric and statistical properties of the particles present in the batch influence the probability of being included in the sample. Pierre Gy’s theory can be classified as one from this category, and the same holds for the generalization of Gy’s theory that can take into account dependencies between particle selections(Geelhoed 2005; Geelhoed 2006; Geelhoed 2007), which plays an significant role in this thesis. Gy’s variance estimator is denoted as (Gy 1979):

1 _(1.6)

Due to its complexness in terms of practical obtainability, Gy has simplified and approximated terms of this equation by introducing a set of new parameters. This simplification can be written down as :

(1.7)

In the literature, a further simplification of this relation of Gy can be found. The variance is namely not dependent on D3 in every case, but rather on Db:

(1.8)

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V = the relative variance

q = the inclusion probability of each particle

cbatch = the concentration of property of interest in the population

Mbatch = the (total) mass (or weight) of the population

mi = the mass of the i-th particle in the population

ci = the concentration of property of interest in the i-th particle of the population

f = shape factor (unitless quantity)

g = size range factor (unitless quantity)

l = the liberation factor (unitless quantity)

c = the mineralogical factor (specified in the same units as density, e.g. g/cm3 or kg/m3₎

D = nominal size dimension of the particle (specified in units of length)

M = mass of the sample (specified in units of mass)

K = fg ℓ cD3-b

b = an exponent that can be selected freely, but in practice will be set to a value that describes how the variance prediction changes with D.

Since the first publications, various suggestions have emerged for expanding and improving Gy’s original model, and a selection of contributions in recent years comprise, e.g. the generalization of the theory of Gy (Geelhoed 2005; Geelhoed 2007), the liberation factor (Francois-Bongarcon and Gy 2002) and a new shape factor (Dihalu and Geelhoed 2011).

(iv) Resampling Approach

Resampling approach techniques are not based on distributional assumptions of conventional statistical analysis, but they rather rely upon random resampling of available data, viz. by replacing the population partition function by the sample partition function (Bellocchi, Libro et al. 2011). The resampling techniques have been developed since the 1930, but have gained an increased interest along with the development and availability of computers with high processor speeds. Major contributions have been made by Fischer(Fisher 1935), Pearson (Pearson 1937), and Snedecor (Snedecor and Cochran 1967).

The main advantage of this approach is its relative simplicity and potential application to circumstances in which conventional statistical theory does not yield a satisfactory answer due to complexness (Bellocchi, Libro et al. 2011). The most common resampling techniques are

 Monte Carlo methods;

These procedures are mostly used to investigate the spatial distribution of objects by implying a model-based distribution of parameters based on random extractions ((Bellocchi, Libro et al. 2011),(Barker 1965)).

 Randomization methods

These techniques include the random permutations of all elements of a dataset and they are especially to be used for sample comparisons (Bellocchi, Libro et al. 2011), (Edgington 1964)).

 Bootstrap methods

The bootstrap methods are based on the resampling of the original data with replacement ((Bellocchi, Libro et al. 2011), (Efron 1979)).

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(v) Empirical Approach1

Essentially, each empirical approach entails the use of an a priori proposed equation, that relates easy-to-determine quantities (e.g. the sample mass and the top size of the particles) to the sampling variance using one or more empirical parameters..

The advantage of the empirical approach is its general applicability to various kinds of materials without the requirement of profound knowledge about the material to be sampled. The down side is that the empirical approach is not an independent approach (Geelhoed 2011), because the empirical factors must be calibrated and during this calibration the sampling variance must be estimated or calculated using one of the other approaches. Once, however, for a specific type of material and mode of sampling the empirical parameters are known, the empirical approach is easy to apply and involves little extra effort.

OBJECTIVE

The research described in this thesis is directed to material sampling following the modeling approach. The overall objective is to arrive at an improvement of the theory of Gy by introducing fundamental rather than empirical relationships for describing the influence of grouping and segregation on the sampling variance and for estimating the minimum sample mass.

Even though grouping and segregation of particles has frequently been acknowledged as a potentially major contributor to both sampling properties, it has only been taking into account in an empirical manner until now.

Pioneering work by Geelhoed (Geelhoed 2005; Geelhoed 2006; Geelhoed 2007) forms the starting point of the here described new modeling approach. The sampling variance is calculated after an independent measurement of the particle properties involved. The relation between particle properties and the sampling variance has been studied, and attention has been paid to the determination of the property values. The different aspects of this new theory are discussed and both theoretically and experimentally verified. Proposed adaptations and practical scope of application will be examined for a selection of cases.

The research has been conducted as part of the STW project “A decision support system (DSS) for the sampling of industrial mixtures of particles and improved sampling standards (STW 7457)” , in which it was aimed to make use of practical obtainable types of parameters in order to estimate information like the minimum sample mass and the sample variance. The theoretical and practical findings will to be implemented in a Decision Support System software package.

CHAPTER OVERVIEW

In Chapter 2, an outline is given of various sampling theories, introducing the basic principles of the theory of Gy which currently is the sampling theory with the biggest impact in the field. From a scientific point of view, the theory will be reviewed on the basis of correctness, empirical aspects and potential points of improvement will be shared.

1

The term “empirical” is often also used to indicate what is here called the “ANOVA”approach. Here, however, the term “empirical” refers to a fundamentally different approach.

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In Chapter 3 the focus is on the details of the new sampling theory. The new parameter for the dependent selection of particles, will be discussed comprehensively together with the parameters that are necessary for applying the new theory.

In Chapter 4, the relevance and the definition of grouping and segregation is explained within the framework of sampling of particulate materials. A first method will be discussed that can be applied in order to quantify the grouping and segregation (Cij) with the purpose of calculating the sampling variance of a mixture: Line Transect Sampling (LTS).

In Chapter 5, the added value of the new theory that includes the parameter for the dependent selection of parameters is compared with the theory of Gy. A profound experiment was done to verify the predicted variance.

In Chapter 6, an alternative method is described for quantifying the grouping or segregation. The major difference with the previously mentioned LTS method, is that this method has a more statistically basis, but can currently not yet be applied for the calculation of the sampling variance. The Ripley’s K-method can be applied in order to find to what extent a selection of particles can be qualified as being grouped or segregated.

Every theory that uses individual particle masses ( ) requires determination of . Gy proposed to use the Brunton shape factor, but in Chapter 7 a new, multi-axial shape factor will be proposed. This new shape factor can also be applied in the generalized theory of Gy. The possible investment of more effort in the size determination that is necessary for this method is discussed and justified.

In Chapter 8 the focus is on the relationship between particle sizes and sampling variance during particle size reduction. In the past, a so-called Broken Rock Model was introduced that gave information about the broken pieces of a particle after it is broken. The broken pieces were of the same size, which is a good starting point. In this chapter, a new breakage model is introduced; the breakage method of this model includes a mode based on fractals. Each particle breaks into pieces, based on fractals. This results in particle pieces of unequal size. Both methods are experimentally examined and compared.

Finally, in Chapter 9 the new theory and the practical insights that have been obtained during this research will be combined by the construction of a Decision Support System software package. Depending on the particles of interest, the particle properties, the required accuracy, and the operator’s background knowledge, this program can assist the user to find the minimum sample mass and the sampling variance. The chapter will be concluded with a worked-out case in which a typical use of the DSS is guidingly discussed.

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2 The Theory of Gy: Limitations and

Improvements

2

One of the most used variance estimators for the fundamental error from the modeling approach is the theory developed by Pierre Gy. After a number of French scientific publications about this topic, he first made his new theory available to the worldwide sampling community in his 1979 publication. The impact of this work has been significant; even nowadays this book is regarded as the number one source of information for sampling engineers and process operators. The scientific value of this work has been groundbreaking at its introduction and the impact to the practice of sampling in many areas has been unprecedented. Still, the development of new technologies, recent experimental results and novel insights have made clear that there is room for improvement and expansion of the 1979 theory as will be outlined here.

In this chapter, the focus will be completely on only one part of Gy’s theory, i.e. the discrete model with special attention to the approach therein for evaluation of the sampling variance and the grouping and segregation error (GSE).

2.1 Introduction

Since 1979 Gy’s theory is increasingly being used in all areas where sampling plays a role.

Gy’s theory consists roughly of three main parts:

(i) The continuous selection model

(ii) The discrete selection model

(iii) Practical rules for the dimensions and operating speeds of sampling tools.

In the continuous selection model, Gy’s theory attempts to capture or describe the temporal variation of an industrial or natural process. Gy only uses the variogram for this purpose, while in the theory of signal processing other tools are more common, notably the Fourier transformation. In order to unify the two completely different models (that is the continuous and discrete model), Gy assumes that the variogram has a so-called “nugget”, which contains the variance components necessary to make the continuous and the discrete model compatible. There is no experimental proof that this is a valid methodology and that both models are indeed compatible. And it has also been argued that the nugget, in one school of thought at least, should always be zero (Clark, 2009). In the discrete selection model, Gy derives a formula for the sampling variance that is based on a simplified model of the sample drawing process in which independent particle selections are assumed. Gy derives, without giving references to previous work, a basic equation for the sampling variance that was however already derived in 1935 by Kassel and Guy. Gy subsequently simplifies the basic equation as a product of several factors. But because Gy’s equation does not have to represent the reality other factors which are considered irrelevant by Gy, may still have a strong influence on the sampling variance. Moreover, Gy assigns default values to his parameters, causing the abolishment of performing measurements in many practical scenarios.

2

A manuscript based on this text is published in VIXRA (accessible from http://vixra.org/abs/1203.0081) : D.S.Dihalu and B.Geelhoed , A Critique of Gy’s Sampling Theory (2012)

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As mentioned before, in order to unify the two completely different models, viz. the continuous and discrete model, Gy assumes that the variogram has a nugget, which contains the variance components necessary to make the continuous and the discrete model compatible. There is no experimental proof that this is a valid methodology. With respect to third part of Gy’s theory (namely the practical rules for the dimensions and operating speeds of sampling tools) modern results obtained using Discrete Element Modeling (DEM) simulations show that Gy’s rules, which are based on approximating the material “flow” by the movement of a single spherical particle, are not necessarily applicable (Cleary et al, 2005; Cleary et al, 2007; Robinson and Cleary, 2009; Robinson and Sinnot, 2011). Since during sampling normally more than one particle is involved and due to the inter-particle forces that are thus not taken into account by him, the practical rules of Gy can be considered to be a simple starting point, but may not be taken over too strictly.

2.2 Limitations and Improvements

As Pierre Gy indicated himself: “ The time when “honest sampling” was made of “good judgment” and “practical experience” is over” (see (Gy 1979), p.8).

A sampling theory would be needed without correction factors, empirical constants, and non-measurable parameters. For decades, many have made valuable contributions within the framework of sampling theories. A condensed chronological overview of the most significant contributions is given in Appendix A to introduce the underlying ideas, the various attempts and the degree of overlap that sometimes occurs.

The reader will understand that there is not an universal sampling theory developed hitherto, and therefore the term “TOS” or “Theory of Sampling” is currently irrelevant if it is related to one of the, yet incomplete, theories.

The theory of Gy will be critically examined. The main focus of this criticism will be on the work that was published by Gy in 1979 and 1982 (Gy 1979; Gy 1982), being the most comprehensive bundle of work shared by Gy in a non-French language. The various topics of this work will be discussed in terms of usefulness, potentials for improvement, and flaws.

2.3 Discrete

Model

of

Gy (1953, 1979, 2004)

Gy takes for his theory Poisson sampling as the basis. He elaborates on this topic in a scientific manner and he uses a Poisson model with a sample mass varying from 0 to

Mbatch, which is not a realistic assumption from a practical point of view. In Geelhoed 2004 a new model was proposed that uses a constant sample mass instead, taking the Finite Population Correction into account (see Appendix D).

The contributions of Pierre Gy to the theory of sampling cover a time period approximately from 1953 to 2004 (see e.g. Gy, 1953, 1979, 2004). Starting with an

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equation similar to Eq. 1.6 (of which Gy (1979) presents a mathematical derivation), Gy (1979) showed that Eq. 1.6 can be written as:

V(csample) =cbatch2 f g l c D3 / Msample (2.1)

where:

cbatch = the concentration in the batch the sample was taken from

f = (Brunton’s) shape factor

g = a size range factor

l = a liberation factor

c = a mineralogical composition factor

D = a typical maximum diameter

Msample = the mass (or weight) of a sample.

The precise definitions of the factors can be found in Gy (1979). The parameters are more practically obtainable than the fundamental variance estimator of Gy, but it needs to be noted that these factors are somewhat empirical. However, since it was not Gy’s intention that these definitions are used in practice, their associated error can vary extremely. The variance in the estimated f can be as high as 100%, g 50%, etc. The propagation of error can be written down as:

2 2 2 2 2 2 2 2 cbatch f g 3 Msample V c D batch sample V c f g c D M      _   _  _  _  _   _{ } _       _ _ _ _ _ _        

This indicates that if the error contribution of one parameter is high, the total (relative) error becomes at least equal to the error contribution of this one parameter, even if the remaining errors are smaller. Hence, it is not possible to compensate one high error contribution by minimizing the error contribution of that parameter. If, for instance, a big error is involved in the estimation of c, it might be a waste of time and energy to estimate any other parameter very accurate, since all error contributions are summed.

Due to the lack of the range of number of the parameters, the final result, i.e. the sampling variance, can be estimated, albeit with a potential high accompanied error; the final result is manipulated with the empirical factors.

The sampling variance is inversely proportional to the sample mass. The following equation for the minimum required sample mass is obtained:

Mmin = f g l c D 3 / α2 (2.2)

where:

α = the maximum allowable relative standard deviation.

It is noted here that the above equation is similar to, the equation of Brunton, especially if one considers that the factor ρ ( cH – cav)/(cav) in Brunton’s equation is comparable to Gy’s mineralogical composition factor c. However there are also differences between both equations.

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2.4 Consequences of Empirical Factors

One of the main points of concern is the fact that the theory of Gy is generally considered to be a pragmatic theory, whereas it is fair to conclude that this theory has a strong empirical character.

By the introduction of several factors to Gy’s formula he makes it possible to adjust the end result towards a result closer to reality. The need for this is obvious: these factors are always case-specific and they can thus be applied only empirically. Moreover, the correctness of the first estimates of the factors have to be verified in retrospective, which is a weakness of the theory.

Recently, it was shown by a few case studies that the deviation between Gy’s prediction for the sampling variance and minimum sample mass and experimental results can be significant (Geelhoed 2010),(Geelhoed, Koster-Ammerlaan et al. 2009).

People who adhere the theory of Gy explain this difference due to the absence of or incorrect set of factors used in the theory, even though these factors are not well-defined, always case-specific and they can thus be applied only empirically. Moreover, these factors can only be estimated afterwards, and a therefore lacking predictive power.

As Gy pointed out himself, sampling is a multidisciplinary, delicate, but complex topic. It is therefore unfortunate that the whole setup of the theory with its empirical factors and the absence of the derivation of these numbers does not contribute to its clarity.

2.5 Empiricity in Gy’s Discrete Model

Gy has attempted to link up the discrete and the continuous model with each other. The deviation between both models is the nugget that is obtained graphically when both models are compared.

The major point of concern regarding the discrete model lies in the fact that Gy uses empirical correction factors in order to adjust the end result towards experimentally obtained values. As an example, it can be mentioned that Gy does not concern dependent particle selections, which may result in grouped or segregated particles. In order to still account for grouping and segregation, Gy introduced two factors that are independent on the particle properties.

From a mathematical point of view, it is remarkable to observe the similarity of Gy’s basic formula with the formula originally developed by Kassel and Guy in 1935 (Kassel&Guy, 1935).

In his 1979 book, Gy shows how to derive the following result from the equation Kassel and Guy constructed.

V(csample) =cbatch2 f g l c D3 / Msample

Even though its practical use is much more accessible than Eq. 1.6, the criticism on Gy’s derivation is related to the numerical values of each parameter that is introduced here. Gy only gives directions on how to find the values, but he does not give exact scientific

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grounds for the derivations of his parameters. He does give, however, typical values – i.e. default values- that the parameters may have, supposedly based on experimental experiences. This empirical character indicates that there is room for improvement in Gy’s theory. The absence of a scientific base of Gy’s parameters enables one to use irrelevant default values in specific cases, causing potentially large errors in the variance estimation procedure. Gy describes the misinterpretation of his work in several publications (Francois-Bongarcon and Gy 2002), but the author of this thesis holds Gy at least partially responsible due to empirical nature of the parameters, combined with the absence of the scientific background in its derivation.

One such an improvement relates to the variance estimators: the variance estimators of Gy’s model use population information, whereas the (Glass and Geelhoed 2003) model gives predictions on the sampling variance based on the sample information. The empirical factors of Gy’s theory cannot be calculated in a different way than by comparison of Gy’s prediction for the fundamental sampling variance with an observed or measured variance.

Another improvement lies with the aspect of grouping an segregation. While Gy puts a lot of attention on the selected parameters, two major parameters are not included in the basic formula. From experimental verifications (Geelhoed, Koster-Ammerlaan et al. 2009) it is concluded that the grouping and segregation of particles can have a large influence on the sampling variance. Gy does indeed define a set of parameters that should account for the grouping or segregating behavior that a collection of particles may have, but again the scientific basis is not included. Both parameters can be regarded as the most severe empiric factors of his theory: grouping factor Y and segregation factor Z (in the literature also denoted by γ and ε respectively). Again, it seems that the value of both parameters are more intuitively to be determined, even though it is claimed sometimes that these are phenomenological parameters, the value thereof to be determined experimentally, and thus difficult to apply a priori before the sample taking starts.

If we consider the VGy to be the variance estimation of Gy (by application of Eq. 1.6) and

Vobs the true observed variance, Gy proposed the following correction:

1 (2.3)

Even though the grouping factor, Y, is well-defined as the approximate number of particles in a group within a sample, the Z factor is very vaguely defined. Since factors Y and Z can vary between 0 and infinity ( 0, ∞ ), their product can become a very high number in theory. Moreover, due to the fact that Z is not well specified, there can be a big error involved in the combination of YZ. Furthermore, it is a fact that the YZ can only be determined afterward. In that respect, the use of the grouping and segregation factors by Gy can be considered to be an ‘empirical quotient’ that can only be used after VGy and

Vobs are known and it seems to have as its only goal to equate both to each other. In several publications (e.g. Francois-Bongarcon and Pitard (2011)), it is suggested that it was never the intention of Gy that these factors could or should be estimated, due to the fact that they were possibly only introduced in order to derive a series of practical

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procedures which are meant to counteract GSE effects in sampling practice. However, the lack of the scientific basis of the segregation factor makes this suggested goal also unreachable in practice.

This problem has been overcome by development of grouping and segregation factors with a more scientific basis like Cij for the dependent selection of particles and fn (Dihalu

and Geelhoed 2011), allowing for predictive use in the estimation of the sampling variance. In Chapter 4, 5, and 7 it is described how the quality of these parameters has been experimentally verified.

2.6 Conclusion

and

Final

Remarks

It has been pointed out that the theory of Gy is not the only sampling theory of interest. Furthermore, it has been shown that Gy was evidently inspired by the equation presented by Kassel and Guy of 1935.

The use of empirical factors to adjust the predicted values with the experimental values are a major point of concern in Gy’s theory for the variance prediction. The reason why Gy does not appropriate the use of Eq. 1.6 for practical use is the fact that this equation requires information at the level of the particles, which is difficult to obtain in practice. Therefore, Gy has propagated to use Eq. 2.1 instead. However, the parameters f,g,l, and c are also fundamentally based on information at the level of the particles. The corresponding values of these parameters can only be assigned by adjustments afterwards. Furthermore, the absence of a grouping and segregation in the fundamental variance estimator equation implicitly indicates that Gy did not find the influence thereof significantly enough to be included. Even though it is claimed in several publications (e.g. Francois-Bongarcon and Pitard (2011)) that Gy did introduce additional grouping and segregation parameters only to enable the derivation of a series of practical procedures to counteract GSE effects in sampling practice, the lack of a of scientific basis of the segregation factor makes also this suggested goal unreachable.

Summarizing, the main conclusions can be listed as follows:

 The theory of Gy fails to provide convincing theoretical and/or experimental proof that two of its major theoretical parts, namely the discrete selection model and the continuous selection model, are compatible and unconflicting.

 Gy’s rules for the dimensions and operating speeds of sampling tools are based on outdated considerations, whereas nowadays more realistic DEM simulation methods are available. When these are applied to the question of sampling tool design, very different conclusions than those of Gy are certainly to be expected.

 The parameters f,g,l, and c in Equation 2.1 are fundamentally based on information at the level of the particles. Their values can only be estimated by adjustment (‘calibration’) using experimental values and this process has to be applied for sampling of each new type of material. As such, the predictive value is limited and wrong use may end up in wrong sampling.

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The absence of a scientifically based and predictive factor for the grouping and segregation in the fundamental variance estimator equation is urgently needed to make correct assessments of the sampling variance for granular material under realistic conditions. The empirical factors of Gy’s theory cannot be calculated in a different way than by comparison of Gy’s prediction for the fundamental sampling variance with an observed or measured variance. This all contributes to the unnecessary complexness of the theory of Gy, which should and could be avoided.

2.7 References

1. Clarke, I. (2009) Statistics or geostatistics? Sampling error or nugget effect? Fourth World Conference on Sampling & Blending, Conference Proceedings The Southern African Institute of Mining and Metallurgy, 2009, page 13-18.

2. Cleary, PW, Robinson, GK, Sinnot, MD (2005) Use of granular flow modeling to investigate possible bias of sample cutters. Second World Conference on Sampling & Blending, conference proceedings, p. 69-81.

3. Cleary, P, Robinson, GK, Owen, P, Golding, M. (2007) A study of falling-stream cutters using discrete element modeling with non-spherical particles. Third World Conference on Sampling and Blending, conference proceedings, p.17-32. (ISBN 978-85-61155-00-1).

4. Dihalu DS, Geelhoed B. (2011) A new multi-axial particle shape factor--application to particle sampling. Analyst, 136(18):3783-8.

5. Esbensen, KH, Paoletti, C, Minkkinen, P. , Pitard, F. (2009) Developing meaningful international standards – Where do we stand today? The world’s first horizontal (matrixindependent) standard – First foray. Fourth World Conference on Sampling & Blending, Conference Proceedings The Southern African Institute of Mining and Metallurgy, 2009, page 63-64.

6. Esbensen, KH, Minkkinen, P (2011) Illustrating sampling standards –How to guarantee complete understanding and TOS-compliance? Fifth World Conference on Sampling & Blending, Conference Proceedings, Gecamin Ltda, p.57-63.

7. Francois-Bongarcon and Pierre Gy (2002) The most common error in applying ‘Gy’s Formula’ in the theory of mineral sampling, and the history of the liberation factor, The Journal of The South African Institute of Mining and Metallurgy, p.475-479. 8. Bongarcon & Pitard (2011): “Demystifying the Fundamental Sampling Error and the

Grouping and Segregation Error for Practitioners”. p. 39-56 in. Proceedings WCSB5 (2011), Santiago de Chile Oct. 25-28, 2012.

9. Geelhoed, B, Glass, HJ (2004) 'Estimators for particulate sampling derived from a multinomial distribution', Statistica Neerlandica, vol. 58, no. 1, 57-74.

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10. Geelhoed, B, 2007. Variable second-order inclusion probabilities as a tool to predict the sampling variance, Third World Conference on Sampling and Blending, conference proceedings, p.82-89.(ISBN 978-85-61155-00-1), http://arxiv.org/ftp/arxiv/papers/0911/0911.1472.pdf.

11. Geelhoed, B, Koster-Ammerlaan, M.J.J., Kraaijveld, G.J.C., Bode, P., Dihalu, D.S., Cheng, H. (2009), An experimental comparison of Gy’s sampling model with a more general model for particulate material sampling. Southern African Institute of Mining and Metallurgy, 2009. Symposium Series S59. page 27-38 ISBN: 978-1-920211-29-5. 12. Geelhoed, B (ed) (2010) Approaches in Material Sampling, Delft University Press,

152 pp.

13. Geelhoed, B. (2011) Is Gy’s formula for the Fundamental Sampling Error accurate? Experimental evidence. Minerals Engineering 2011; 24(2): 169-173.

14. Gy, P (1953) Erreur commise dans le prelevement d’un echantillonsur un lot de minerai.Congres des Laveries des Mines Metalliques Francaises. R. Ind. Min. 36: 311-345.

15. Gy, P (1964) Le principe d’equiprobabilite. Ann. Min. (Dec. 1964): 779-794.

16. Gy, P (1975) Theorie et pratique de l’echantillonage des matieres morcelees. Editions PG, Cannes, France. 597 pp.

17. Gy P. (1979 & 1982) Sampling of Particulate Material: Theory and Practice. Elsevier: Amsterdam, 431 pp.

18. Holmes, RJ (2011) Challenges of developing ISO sampling standards. Fifth World Conference on Sampling & Blending, Conference Proceedings, Gecamin Ltda, p.383-392.

19. Kassel LS, Guy TW (1935) Determining the correct weight of sample in coal sampling. Industrial and Engineering Chemistry Analytical Edition 1935; 7(2):112--115.

20. Robinson, GK and Cleary, PW (2009) Some investigations of Vezin sampler performance. Fourth World Conference on Sampling & Blending, Conference Proceedings The Southern African Institute of Mining and Metallurgy, 2009, page 219-229.

21. Robinson, GK, Sinnot, M. (2011) Discrete element modeling of square cross-belt samplers with baffles. Fifth World Conference on Sampling & Blending, presentation, http://www.sampling2011.com/evento2011/index.php?option=com_content&task=vi ew&id= 57

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The segr

3.1

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Basic

ing the cou ghts in the s as Gy’s m will focus on important fe uping and s behave an thermore, ea uitively, it erences betw ween each usion proba icle type is h particle w ng step is t er inclusion bability. The ombination mple taken.

3 N

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any notable new samplin

nto account ore, all part of the same ass mi and ass properti or repulsion first-order a ndicates the cles in the b n the sampl are indepen the second as the prob particle of

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ated on and sentially the differences ng theory. the influen ticles in the e particle cl concentratio ies or even n of particle and second probability atch are ide le, if the sam ndent on the d-order inc bability of f another typ ng and d new e same s. First nces of e batch lass T. on ci . n size e types d-order y that a entical, mpling e first-clusion finding pe in a

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In the case all particles in the batch can be selected independently from each other, i.e. a certain paired combination of two particle types does not have an increased or decreased probability, the relation between the first-order inclusion probability and the second-order inclusion probability can be noted as :

ij i j

   (3.1)

However, in the case that a certain pair of particle types has an increased or decreased probability of being found together in a sample, this effect is accounted for by the insertion of the parameter for the dependent selection of particles, Cij. The relation between the first-order and second-order inclusion probabilities and Cij is given by (Geelhoed 2005):

(1 )

ij i j Cij

    (3.2)

Cij is a function of first- and second-order inclusion probabilities. It is mentioned in literature (see e.g. (Deville 1992)) that the inclusion probabilities are strictly positive. It is also assumed that the first and second-order inclusion probabilities can vary between classes but not within classes. If Cij has a positive value (with Cij = 1 as a maximum number), from Eq 3.2 it follows that πij<πiπj, implying that particles of type i and j are segregated. If Cij has a negative value, πij>πiπj, and particles of type i and j are grouped. An estimator based on a Taylor linearization and an estimator based on Horvitz-Thompson estimator are firstly proposed using Cij (Geelhoed 2006), respectively written down as: 2 2 2 1 1 1 1 ˆ ( )T s T i i( i s) T T i j ij i j( i s)( j s) i i j s V c N m c c N N C m m c c c c M       _     _ 





 (3.3) and

 

2 2 2 1 1 1 1 ˆ 1- 1-T T T ij i j i j i j i i i HT s i ii i i ij C N N c c m m N m c V c M  C   C    _  _ 





 (3.4) in which

= the variance estimator based on a Taylor linearization

= the variance estimator based on Horvitz-Thompson estimator for the variance

T = the number of particle classes

Ni = the number of particles belonging to the i-th particle class in the sample

mi = the mass of a particle belonging to the i-th class

ci = the mass concentration of the property of interest in a particle of type i

cs = the mass concentration of the property of interest in the sample

Ms = the mass of the sample

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In the new theory, at the moment only the standard particle types are taken into account and the effects of interstitial fluids or gases are not modeled.. The main focus is on the solid particles of any size. It has not been investigated yet if the theory also holds for liquids and gases.

Population is regarded as a constant value, i.e. the number of particles is not subject to change during the sample taking step. This set of assumptions is constructed in order to let the new theory be realistic and practically sound.

3.2 The Parameter for the Dependent Selection of Particles

3

A generalization of Gy’s 1979 model for the variance of the fundamental sampling error has been proposed by Geelhoed (Geelhoed 2005). This generalized model can potentially lead to a better correspondence with actual sampling processes. However, a new parameter is required by the new model: ‘the parameter for the dependent selection of particles’(Cij). It is important that this new parameter, which is at the basis of the new theory, is well understood in the sampling and blending community. In order to clarify the meaning of Cij and to prevent potential misinterpretation of Cij, we discuss the difference between Cij and an intuitively associated, but different, parameter: ‘the parameter for the dependent selection of particle types’(Mij). In this contribution, the theoretical difference between both parameters is clarified. Simulations demonstrate that, numerically, the difference between Cij and Mij can be large. Some additional advantages of the approach which uses Cij are discussed: (i) the approach based on Cij is more general; (ii) in all cases where Mij is known, Cij can be determined as well, while the reverse is not true; (iii) an equation already exists that relates the sample variance to the value of Cij.

3.2.1 Introduction

During the random sampling of particulate materials, each particle has a probability of becoming part of the sample. From a statistical point of view, however, it cannot be axiomatically assumed that the particle selections are independent processes (although this is often implicitly assumed when simple models for sampling are constructed). The parameter for the dependent selection of particles is a quantitative measure of the dependency between particle selections. This parameter was introduced in the 2005 generalization of Gy’s theory (Geelhoed 2005).

In the mathematical definition of this new parameter, the first and second-order inclusion probabilities of the particles (see definitions below) are used. If the particles in the population are visualized as being numbered, the inclusion probability of the i-th particle in the population is denoted by the symbol πi and represents the probability that the i-th particle will end up in the sample. Similarly, the second-order inclusion probability of the pair consisting of the i-th and j-th particle of the population is denoted by the symbol πij and represents the probability that this pair will end up in the sample. More information

3

This paragraph is published as an article in: B.Geelhoed and D.S. Dihalu, Clarification of the concept of dependent selection of particles, Fourth World Conference on Sampling & Blending, The Southern African Institute of Mining and Metallurgy, pp.39‐42 (2009)

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about the concept of inclusion probability can be found in literature (see e.g.(Deville and Sarndal 1992)).

The purpose of this short section is to clarify the meaning of the ‘parameter for the dependent selection of particles’ (Cij) and to clarify the distinction between this parameter and a different, but intuitively associated, parameter, ‘the parameter for the dependent selection of particle types’ (Mij).

3.2.2 Theory

To simplify the description here, particles are classified into T particle classes. Within a class, all particles have the same particle properties. It is also assumed that the first and second-order inclusion probabilities can vary between classes but not within classes. Denoting the class number of the x-th particle of the population by n(x), this assumption is formulated as:

πi = πj whenever n(i)=n(j)

πij = πkl whenever n(i)=n(k) and n(j)=n(l). The notation can now be simplified by defining:

κn(i) = πi

κn(i)n(j) = πij

Hence, κt denotes the first-order inclusion probability of a particle of class t and κst denotes the second-order inclusion probability of a particle pair consisting of a particle belonging to class s and t. The parameter for the dependent selection of particles, denoted here by the symbol Cij, can then be defined as:

κij = κi κj (1 – Cij) (3.5)

It is immediately visible from the above equation that Cij describes the deviation from the

case when particles are selected independently, because when Cij =0 it follows that κij =

κiκj, which would be the result for independent selections (probabilities of independent events can be multiplied in order to get the probability of both events happening). Eq. 3.5 is equivalent to:

Cij = 1 – κij /( κi κj) (3.6)

Dependent selection of particles can to some extent be mimicked by modeling the drawing of a sample by Markov chain. The probability that the first particle in such a chain belongs to class j is denoted as pj. The most natural choice for pj is to select the ‘steady state’ probability, defined below.

However, a crucial feature of the Markov chain is that, given that a particle of type j is selected during the n-th selection, the probability that the particle that is selected during

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the (n+1)-th selection is of type i may depend on both i and j and is denoted by pij. It is now possible to define the steady state, pi, by:

The above equation is the ‘steady state condition’, and if it is met, the probability that the

n-th particle in the chain (for arbitrary n smaller than or equal to the number of particles

in the chain/sample) belongs to class i is also given by pi, hence the term ‘steady state’. The parameter pij may be larger or smaller than pi, and, here, a new parameter Mij is proposed (‘the parameter for the dependent selection of particle types’) that governs this increase or decrease of selection probability with respect to pi. Using this new parameter

Mij, the probability that the next particle that is selected is of type j is denoted by pj(1–

Mij). The probability Pij that for a sequence of two particles the first is of type i and the

second of type j is given by: Pij = pipij. Using the definition of Mij, Pij is then seen to be equal to:

Pij = pi pj (1 – Mij) (3.7) This is equivalent to:

Mij = 1–Pij/(pi pj) (3.8) It is proposed here to reserve the symbol Cij for ‘the parameter for the dependent

selection’ defined by Eq. 3.6 and Mij for the parameter defined by Eq. 3.8. Although Eq.

3.6 has the same structure as Eq. 3.8, the latter relates to a sequence of particle types under the assumption of a restrictive model (a Markov chain) while the former relates to the probability of including in the sample a specific pair of particles independent on the specific model of sampling. Numerically, the differences between Cij and Mij are large, possibly several orders of magnitude in certain cases. In order to give an example of this, Eq.3.9 below is used. Under certain general conditions (see conditions 1 to 4 in (Geelhoed 2005)), it has been proven that the parameter for the dependent selection of particles can be related to the covariance’s and expected values of the numbers of particles belonging to the different particle classes in a sample:

Cij = δij/E(Ni) – Cov(Ni, Nj)/( E(Ni) E(Nj) ) (3.9) Where δij is the Kronecker delta which is one when i=j and zero otherwise, Ni and Nj represent the numbers of particles in a sample belonging to the i-th and j-th particle class respectively, Cov(.,.) represents the covariance and E(.) the expected value operator. Eq. 3.9 and its derivation were given in (Geelhoed 2005).