Particle Measurements Using Fluctuations in the - DiVA Portal

Particle Measurements Using Fluctuations in theRegular Transmittance of Light Through a Particle

Dispersion

Concentration and Particle size — Theory, Measurement Principles and Applicationsfor Pulp and Paper Production

STAFFAN RYDEFALK

Doctoral ThesisStockholm, Sweden 2009

Particle Measurements Using Fluctuations in the Regular Transmittance of Light Through a ParticleDispersionConcentration and Particle size — Theory, Measurement Principles and Applications for Pulp andPaper Production

© Staffan Rydefalk, 2009

TRITA-IIP-09-05ISSN 1650-1888ISBN 978-91-7415-254-8

KTH School of Industrial Engineering andManagement

SE-100 44 StockholmSWEDEN

Academic thesis for the degree of Doctor of Technology in Production Engineering to be presented withdue permission for public examination and criticism in Brinellsalen, M311, KTH Campus Valhallavägen,Brinellvägen 68, Stockholm at the Kungliga Tekniska Högskolan (KTH) on June 15, 2009, at 10.00.

Chairman: Prof. Lars Berglund, KTHSupervisor: Prof. Lars Mattsson, KTHAssistent supervisor: Ph.D. Marie-Claude Béland, InnventiaOpponent: Prof. John Gregory, University College LondonExamination committee: Assoc. prof. Göran Manneberg, KTH

Ph.D. Petter Kolseth, Stora EnsoProf. Carl-Gustaf Ribbing, Uppsala University

Printed in 2009 by Universitetsservice US AB

To my loved ones

Abstract

The regular transmittance of light or similar radiation through a flowing suspensionof particles fluctuates because of the random occurrence of particles in the beam.In the work presented here, a theory for this fluctuating behaviour with the empha-sis on dispersions of mm-length slender cylindrical particles having circular crosssections is given. The particles in question are wood pulp fibres, which as a firstapproximation are considered to have a cylinder shape. Four possible measurementprinciples are described theoretically and experimentally. The four principles arefor the measurement of concentration, length distribution characterized as lengthclasses, mean length, and mean width. The usefulness in industrial process moni-toring of two of these principles is exemplified with pulp measurements. In order toestimate model errors, numerical simulations were used. Although other techniquessuch as image analysis may compete, the technique presented here is attractive be-cause of the simplicity of the measurement device used.

Keywords: Particle concentration; Particle size; Cylindrical particles; Large particles; Cellulose fibres;Wood pulp fibres; Suspensions; Particle length; Particle length distribution; Particle width; Randommedia; Turbid media; Optical transmittance; Transmittance fluctuations; Online measurement; Opticalsensors; Pulp; Paper; Fillers; Retention; Density measurement; Fluctuance; Theory; Experiment

v

Sammanfattning

Direkttransmissionen av ljus eller liknande strålning genom en rinnande partikelsus-pension fluktuerar på grund av den slumpvisa förekomsten av partiklar i ljusstrålen.Det arbete som presenteras här redovisar en teori för det fluktuerande beteendet.Betoningen är på dispersioner av millimeterstora slanka cylindrar med cirkulärttvärsnitt, speciellt med tanke på att pappersmassafibrer kan, som första approxi-mation, anses vara cylinderformade. Fyra mätprinciper beskrivs teoretiskt och ex-perimentellt. Dessa är för mätning av koncentration, längdfördelning i form avlängdfraktioner, medellängd och medelbredd. Användbarheten av två av dessa mät-principer för att följa industriella processer exemplifieras med mätningar på pap-persmassafibrer. Även om andra tekniker såsom bildanalys konkurrerar, är denteknik som presenteras här attraktiv på grund av mätgivarens enkelhet.

vii

Foreword

The completion of this thesis was not made according to the cookbook saying thatthe thesis is written at the end of a four year period of research and theoreticalstudies. I will not tire the reader with all details, but a short explanation of thedelay of this thesis is necessary.

When I accepted the offered position at the industrial research institute STFI inthe mid 1970s it was agreed that it would be possible for me to make the necessarytheoretical studies for doctoral degree and publish scientific articles as a part of myjob. However, in industrial research it sometimes happens that the publication ofresearch results is stopped due to commercial interests. This is what happened toessential parts of my research work and it was not until the beginning of the 1990sthat it was possible to start to prepare a thesis.

The confidentiality was, however, relaxed reluctantly and the thesis had to be amonograph in Swedish, ready in 1992. Even if a monograph in Swedish was possiblein theory it was not an acceptable format according to the examination committeeappointed to evaluate the thesis. Therefore the thesis in Swedish was withdrawn.However, I eventually got permission to write a new monograph in English. At thesame time the theory was reworked. The resulting thesis (TRITA-ILA 95-01) wasused for my degree of licentiate 1995.

After the licentiate degree there was no funding for a continuation to doctoraldegree. Therefore, the four articles and the report constituting Papers 3, 4, 5, 6, 7and the Appendices were written in my spare time. STFI (now Innventia AB) haspartially sponsored this work by providing linguistic revision and page fees. Sincespare time is a very limited resource it has taken more than a decade from thelicentiate thesis to the completion of the present thesis.

Stockholm 2009Staffan Rydefalk

ix

Acknowledgements

When you drink from a stream,think of its sources.

Chinese saying

A lot of people were throughout the years involved in the work now summarizedas this thesis. Many have already been acknowledged in the included papers andin my licentiate thesis and I think of all these with a special gratitude. Withoutyou I would not have reached this far. This group of people includes a number ofcolleagues and former colleagues, people in the academia, photographers, financialsupporters and relatives.

In addition, I want to acknowledge a number of people for their contributionsto this thesis, viz.: Marie-Claude Béland for valuable discussions and linguistic re-view, Göran Stemme for valuable advice, and Lars Mattsson for supervising thefinalisation of this thesis. A number of people helped me with pictures and illus-trations: Joanna Hornatowska, Rolf Thapper, Barbro Brynte, Mikael Rigdahl, andHans Christiansson. Joanna Hornatowska, Einar Jung, Thorulf Pettersson, Ulla-Britt Mohlin, and Inge Lundqvist gave valuable comments on different parts of themanuscript.

I also want to thank my mother for her support and ceaseless encouragementto finish my thesis and for showing me by example that learning is more than aproject. It is a lifestyle. Last but not least, I wish to thank my dear wife Birgittaand my children for their involvement and for forgiving the father in the familyfor taking up so much of his spare time studying courses, developing software andwriting the articles and the thesis.

xi

List of papers

Paper 1Rydefalk, S., T. Pettersson, E. Jung, and I. Lundqvist, “The STFI Opticalfibre Classifier,” in EUCEPA International Mechanical Pulping Conference,(EUCEPA, Oslo, 1981), Session III: No 5 16p.

In the work leading to Paper 1, Rydefalk investigated the validity of the ideaof an optical fibre length classifier, evaluated different design alternatives,participated in the construction and building of prototypes and participatedin the transfer of technique at the commercialisation stage. Rydefalk alsomade the presentation at the EUCEPA conference.

Paper 2Lindström, T., S. Rydefalk, and L. Wågberg, “The Development of an Inte-grated Retention Control System,” in SPCI 84. The World Pulp and PaperWeek. New available techniques, (SPCI and Adforum, Stockholm, 1984), p.492-496.

In Paper 2, Rydefalk’s contribution was the part concerning the retentionmeasurement system.

Paper 3Rydefalk, S., “Fluctuations in the regular transmittance of dispersions ofstraight circular cylinders with a diameter much larger than the wavelengthof the radiation,” J. Opt. Soc. Am. A 15, 1689-1697 (1998).

Paper 4Rydefalk, S., “Theory of fluctuations in the regular transmittance through adispersion of large cylindrical particles: extension to higher concentrations,”J. Opt. Soc. Am. A 16, 2737-2745 (1999).

Paper 5Rydefalk, S., “Assessment of the mean and variance of the random regulartransmittance through a dispersion of large cylinders using numerical simu-lations,” Appl. Opt. 47, 993-1001 (2008).

xiii

xiv LIST OF PAPERS

Paper 6Rydefalk, S., “Validation of a simulation algorithm for the mean and varianceof the random regular transmittance through a dispersion of large cylinders,”KTH-TRITA-IIP-09-04, KTH 2009.

Paper 7Rydefalk, S., “Applications of the theory of fluctuation in the regular trans-mittance through a dispersion of large cylindrical particles to concentrationand size measurements,” Appl. Opt. 47, 4509-4521 (2008).

Contents

Contents xv

List of symbols and abbreviations xix

I Thesis 1

1 Introduction 31.1 Research on industrial process-related measurement techniques at

STFI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Problem statement . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Innovation contributions . . . . . . . . . . . . . . . . . . . . . . 4

2 Background 52.1 Concentration and size metrology . . . . . . . . . . . . . . . . . . 52.2 Concentration and size metrology in this thesis . . . . . . . . . . 132.3 The particle model and real particles . . . . . . . . . . . . . . . . 14

2.3.1 Wood pulp fibres . . . . . . . . . . . . . . . . . . . . . . . 142.3.2 Modelling considerations . . . . . . . . . . . . . . . . . . . 14

2.4 Light scattering by particles and especially wood pulp fibres . . . 152.4.1 Single scattering . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 Optical properties and models of wood pulp fibres . . . . 172.4.3 Scattering from many particles . . . . . . . . . . . . . . . 182.4.4 Optical cross sections and efficiencies . . . . . . . . . . . . 18

2.5 Fundamental concepts . . . . . . . . . . . . . . . . . . . . . . . . 212.5.1 Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.5.2 Concentration . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.3 Calculation of the mean of particle properties . . . . . . . 222.5.4 Particle density . . . . . . . . . . . . . . . . . . . . . . . . 232.5.5 Orientation and projection . . . . . . . . . . . . . . . . . 23

2.6 Statistical notation and sums with random number of terms . . . 252.7 Outline of the material presented . . . . . . . . . . . . . . . . . . 26

xv

xvi CONTENTS

2.7.1 Chronological remarks . . . . . . . . . . . . . . . . . . . 262.7.2 Conceptual outline . . . . . . . . . . . . . . . . . . . . . . 28

3 The random transmittance at low concentrations 313.1 General particle geometry . . . . . . . . . . . . . . . . . . . . . . 313.2 Large slender cylindrical particles . . . . . . . . . . . . . . . . . . 333.3 Examples of other particle geometries . . . . . . . . . . . . . . . 35

4 The random transmittance at higher concentrations 394.1 The lognormal model . . . . . . . . . . . . . . . . . . . . . . . . . 404.2 Comparison with other theoretical models . . . . . . . . . . . . . 43

4.2.1 Lambert-Beer-Bouguer approach . . . . . . . . . . . . . . 444.2.2 The TP theory . . . . . . . . . . . . . . . . . . . . . . . . 444.2.3 Gregory’s theory . . . . . . . . . . . . . . . . . . . . . . . 45

4.3 Two experimental studies . . . . . . . . . . . . . . . . . . . . . . 464.3.1 Experimental details . . . . . . . . . . . . . . . . . . . . . 464.3.2 Study 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.3.3 Study 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

5 The random transmittance for intermediate particle lengths 495.1 Connecting special cases . . . . . . . . . . . . . . . . . . . . . . . 495.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505.3 Simulations and results . . . . . . . . . . . . . . . . . . . . . . . 52

5.3.1 Comparisons with the theoretical predictions . . . . . . . 525.3.2 The response function of V 2 and ϕ . . . . . . . . . . . . . 53

6 Measurement principles based on random transmittance 556.1 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . 56

6.1.1 Sample set . . . . . . . . . . . . . . . . . . . . . . . . . . 566.2 Triple-beam unit . . . . . . . . . . . . . . . . . . . . . . . . . . . 566.3 Single-beam unit . . . . . . . . . . . . . . . . . . . . . . . . . . . 576.4 Triple beam unit used as instrument model . . . . . . . . . . . . 586.5 Concentration measurement . . . . . . . . . . . . . . . . . . . . . 58

6.5.1 Total concentration . . . . . . . . . . . . . . . . . . . . . 586.5.2 Concentration measurement with small particle suppression 59

6.6 Length classification . . . . . . . . . . . . . . . . . . . . . . . . . 616.7 Length measurement . . . . . . . . . . . . . . . . . . . . . . . . . 636.8 Width measurement . . . . . . . . . . . . . . . . . . . . . . . . . 656.9 General considerations . . . . . . . . . . . . . . . . . . . . . . . . 68

7 Examples of industrial applications 697.1 The work in a 2009 perspective . . . . . . . . . . . . . . . . . . . 697.2 Applications of the particle length classifier . . . . . . . . . . . . 69

7.2.1 Application in mechanical pulping . . . . . . . . . . . . . 70

CONTENTS xvii

7.2.2 Application in pulp beating . . . . . . . . . . . . . . . . . 727.3 Application of the concentration meter . . . . . . . . . . . . . . . 72

7.3.1 The fine paper process . . . . . . . . . . . . . . . . . . . . 727.3.2 The measurement task . . . . . . . . . . . . . . . . . . . . 757.3.3 The measurement solution . . . . . . . . . . . . . . . . . . 767.3.4 The first prototype and pilot plant tests . . . . . . . . . . 777.3.5 After the prototype . . . . . . . . . . . . . . . . . . . . . 79

8 Comments on measurement uncertainty 83

9 Conclusions 87

10 Summary of Paper 1 89







References 103

II Appendices 113

Appendix A – Three-dimensional isotropic orientation of straightcircular cylinders 115

Appendix B – A system for the selective measurement of fibre andfiller concentrations in the manufacture of fine paper grades 121

Appendix C – Description of pulpsamples 155

Appendix D – The mean and variance of the number of particlesat random particle orientation 163

Appendix E – The TP theory 167

xviii CONTENTS

III Included Papers 173

Paper 1 175

Paper 2 193

Paper 3 201

Paper 4 213

Paper 5 225

Paper 6 237

Paper 7 253

List of symbols and abbreviations

Page numbers in normal font refer to the first appearance of the symbol. Additionalpage numbers in italic refer to pages where important complementary informationabout the meaning of the symbol may be found. Random variables and theirsample values are usually denoted as the upper and lower case versions of a letteror symbol, e.g. if X is a random variable, x denotes the corresponding samples.The expected value of X is denoted μX or EX(X) and the variance is denoted vX ,σ2X or varX(X). E is the expectation operator and var is the variance operator.

Experimental estimates are marked with a hat, i.e. μX is an estimate of μX . Arandom function X(t) has the time average 〈x(t)〉t. E(X |Y ) is the expected valueof X on the condition that Y = y.A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42

Linear light attenuation – the negative natural logarithm of themean regular transmittance, A = − lnμT.

a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32Path length through measuring cell at transmittance measurement,usually the distance between the entrance and exit windows.

A0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50Area of the model measuring cell perpendicular to the opticalbeam.

b . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Length unit in a comparison of different particle shapes.

C C1, C2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Extinction cross section – random variable having the expectedvalue μC , the variance vC or σ2

C , and the square mean EC(C2).c c1, c2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Extinction cross section – deterministic or sample value of C.cabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .19

Absorption cross section of a particle – sample or constant.Cb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .31

Light beam cross section. If it is circular, Cb = Db2 π/4.

Cb,p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51The beam cross section expressed in pixel units.

cext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Extinction cross section of a particle – sample or constant.

xix

xx List of symbols and abbreviations

csca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19Scattering cross section of a particle – sample or constant.

D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35, 65Diameter of a cylindrical particle – random variable having theprobability density function fD(d), the expected value μD, thevariance vD or σ2

D, and the expected value of the reciprocal valueμ1/D.

d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15The diameter of a cylindrical model particle or the width of a realparticle (fibre) – sample or constant.

Db . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31Diameter of the optical beam.

dc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Diameter of a spherical particle being optically equivalent to acertain particle of another shape.

dc,G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Diameter of a spherical particle having an equivalent mean geo-metrical cross section as a particle of another shape.

dc,G2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36Diameter of a spherical particle for which dc,G2 is equivalent tothe square mean geometrical cross section of a particle of anothershape.

di . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Diameter of cylinder particle or fibre number i.

fA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51The surface fraction of the optical beam covered by particle crosssections.

G . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20, 23, 34Geometrical cross section (the silhouette) of a particle on a planeperpendicular to propagation direction of the incident light – ran-dom variable having the expected value μG and the variance vG orσ2G.

g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20, 23Geometrical cross section (the silhouette) of a particle on a planeperpendicular to propagation direction of the incident light – sam-ple or constant.

Gp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51The number of pixels having a non-zero value, thus representingthe union of one or several particles projections.

k1, k2, k3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Coefficients 1/(kBL a) for each beam of a triple beam unit describ-ing the sensitivity of ϕ to pulp concentration Spulp. The index 1represents the thinnest beam.

kA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18, 32Coefficient used in the expression for the attenuation of the meanof the transmitted light.

List of symbols and abbreviations xxi

kB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Coefficient used in the expression for the variance of the transmit-ted light.

kBL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34The value of kB for long particles.

kfiller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Coefficient describing the sensitivity of Δρ to the filler concentra-tion Sfiller.

kp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Coefficient describing the sensitivity of the density difference Δρto the concentration of particles.

kpulp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Coefficient describing the sensitivity of Δρ to the pulp concentra-tion Spulp.

kV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .34Coefficient in the expression for the variance of the transmittanceof light.

kμ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Coefficient used in the expression for the attenuation of the meanof the transmitted light.

L . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35, 63Length of a cylindrical particle – random variable having the prob-ability density function fL(l), the expected value μL and the vari-ance vL or σ2

L.l . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .15

The length of a cylindrical model particle or the width of a realparticle (fibre) – sample or constant.

l1, l2, l3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56Triplet of lb values for the three channels of a triple beam sensor,where index 1 represents the thinnest beam.

l1, l2, l3, l4, l5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54The five lb values used in the numerical simulations.

lb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34Particle length corresponding to the knee point of the responsecurve for vT, V 2 and ϕ when these are regarded as functions of theparticle length.

li . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Length of cylinder particle or fibre number i.

m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40Number of nonoverlapping layers.

mL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Mean particle length as determined by the optical length meter.

mLF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64The mL value derived from the fitted model.

xxii List of symbols and abbreviations

mLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64The mL value derived from the simple model.

mp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23The weight of a particle.

N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31The number of particles interacting with the optical beam – ran-dom variable having the expected value μN and the variance vNor σ2

N . In this thesis N is usually considered to have a binomial orPoisson distribution.

n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44The number of particles interacting with the optical beam – sampleor constant.

OMFL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63The Optical Mean Fibre Length, the output signal from the parti-cle length meter.

OMFW . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .66The Optical Mean Fibre Width, the output signal from the particlewidth meter.

p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Product of concentration and path length S a or S z.

p0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40The value of p at the exit from the interaction zone.

q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Number of solid components in the suspension.

Qabs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Absorption efficiency of a particle.

Qext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Extinction efficiency of a particle.

Qobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59Observed extinction efficiency in an experiment.

Qsca . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Scattering efficiency of a particle.

r(l) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64The length response function of the optical fibre length meter.

rF(l) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64The r(l) function derived from the fitted model.

rS(l) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64The r(l) function derived from the simple model.

S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18, 22Particle mass-volume concentration (solids concentration) definedas dry weight of solid material per unit volume of suspension.

Sfiller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Filler concentration of the sample suspension.

List of symbols and abbreviations xxiii

Slong . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Concentration of particles in the long fraction as measured by theoptical length classifier.

Smedium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Concentration of particles in the medium fraction as measured bythe optical length classifier.

SN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22Number concentration defined as number of particles per unit vol-ume of suspension.

Spulp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Pulp concentration of the sample suspension.

Sshort . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62Concentration of particles in the short fraction as measured by theoptical length classifier.

t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Time.

U(c1, c2 . . .) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .50The union of the areas c1, c2 . . .

V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33Coeffiecient of variantion of the transmittance, V = σT/μT.

Vp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15The volume of a particle.

W (p) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .41The natural logarithm of the transmittance as a function of p.

x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Cartesian coordinate axis, perpendicular to the optical axes and y.

y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Cartesian coordinate axis, perpendicular to the optical axis and x.

z . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Cartesian coordinate axis, parallel to and having the same directionas the optical axis.

Δa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43The thickness of nonoverlapping layers of the path length a.

Δp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40The thickness of one nonoverlapping layer of the variable p.

Δρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76The density difference between a suspension and its liquid compo-nent (the dispersion medium).

ζ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Spherical coordinate. The azimuth angle between the projection ofthe fibre axis on the xy-plane and the x-axis – sample or constant.

θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23Spherical coordinate. The polar angle between the fibre axis andthe optical axis (z-axis) – sample or constant.

xxiv List of symbols and abbreviations

Λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24The ratio between the projection and the length of a cylinder orfibre – random variable having the probability density functionfΛ(λ), the expected value μΛ, the variance vΛ or σ2

Λ and the squaremean E(λ2).

λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24The ratio between the projection and the length of a cylinder orfibre – sample or constant.

ν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42Extra parameter used to calculate ϕν .

ρL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76The density of the liquid of a suspension (the dispersion medium).

ρp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15Particle density defined as dry weight per unit particle volume ofwet particles. Cavities are included in the volume.

T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18, 31–33, 40–42Internal transmittance – random variable (Capital τ . Not to beconfused with the Latin letter T !). T has the expected value μTand the variance vT or σ2

T.Ti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Internal transmittance T through layer i – random variable.T(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

Internal transmittance T as a random function of time.τ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33, 44

Internal transmittance – sample or constant.φ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

The radiation flux reaching the detector of the tranmittace sensorwhen the measurement cell is filled with the dispersion.

φ(t) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33The radiation flux φ as a function of time.

φ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33The radiation flux reaching the detector of the transmittace sensorwhen the measurement cell is filled with the pure medium of thedispersion.

ϕ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42The fluctuance – a function of the coefficient of variation V forthe transmittance T proportional to the concentration S. ϕ =ln(V 2 + 1).

ϕ1, ϕ2, ϕ3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56The ϕ values from the three channels of the tripple beam sensor,where index 1 represents the thinnest beam.

ϕ1F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58The ϕ1 value calculated using the fitted model for the a transmit-tance sensor having a fine beam compared with the length of theparticles.

List of symbols and abbreviations xxv

ϕ1S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58The ϕ1 value calculated using the simple model for the a transmit-tance sensor having a fine beam compared with the length of theparticles.

ϕν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42The fluctuance calculated using the extra linearisation parameterν.

suffix S . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58The suffix S refers to calculations based on the simple responsemodel.

suffix F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58The suffix F refers to calculations based on the fitted responsemodel.

Part I

Thesis

1

Chapter 1

Introduction

1.1 Research on industrial process-related measurementtechniques at STFI

The work presented in this thesis started at the Physics Department at what wasthen called STFI – Svenska Träforskningsinstitutet1 – a name that has beenchanged several times since then. The company is today called Innventia AB.One of the main research activities at the Physics Department was the develop-ment of measurement techniques for the pulp and paper industry. Several newmeasurement principles were invented and tested. Prototype measurement deviceswere built and used both in the laboratory and online in pulp or paper mills. Basedon these unique measurement possibilities research activities in the area of processcontrol could be carried out.

A number of the measurement principles studied had to do with particle concen-tration, particle size and other particle properties in the pulp and paper processes.Most of these were based on optical techniques, with particle measurements mostlymade on particles suspended in water. In Section 2.7.1, more information is givenregarding research activities at the Physics Department that were closely relatedto the work presented in this thesis.

1.2 Problem statement

One of the important industrial goals was to be able to monitor critical proper-ties of particles and particle suspensions online in order to better understand andpossibly control the pulp and paper manufacturing processes. The classification ofcellulose fibres into length categories, separate concentration measurements of cel-lulose fibres and filler particles mixed into the same suspension, and a verification of

1In English referred to as The Swedish Forest Products Research Institute.

3

4 CHAPTER 1. INTRODUCTION

the preliminary theory behind these and two other related measurement principleswere the research problems to be solved.

1.3 Research objectives

The common theme of the papers included in this thesis is the theory and applica-tion of how the regular transmittance2 of light (or other similar radiation) is affectedby particles passing the measurement zone at random. Along this development line,the research objectives were:

1. To investigate the validity of the idea of an optical fibre length classifier andto evaluate different design alternatives.

2. To solve the problem of separately measuring the concentrations of paper-making pulp fibres and filler particles in the short circulation of fine papermachines.

3. To independently verify the theory behind the measurement principles.

4. To find an approximate expression for the connection of two theoretical specialcases regarding the particle length response.

5. To describe the area of applicability and sources of uncertainty of four mea-surement principles that are based on the theory.

1.4 Innovation contributions

It was a clear ambition that the measurement principles should be made availableto the industry as commercial products. Rydefalk was involved in this innovationprocess in different ways. Regarding item 1 above, Rydefalk was involved in theconstruction and building of prototypes and in the technology transfer at the com-mercialisation stage. Regarding item 2 above, Rydefalk led the development of aprototype and tested it online in the pilot plant paper machine FEX demonstratingits usefulness to monitor the fine paper process. Rydefalk was also involved in thecommercialisation process.

2The difference between transmittance and regular transmittance is explained in Section 2.2on page 13.

Chapter 2

Background

(In the beginning) There wasEru, the One, who in Arda iscalled Ilúvatar. . .

The opening words ofSilmarillion by JRR Tolkien.

2.1 Concentration and size metrology

To measure and control the concentration, size (or size distribution) and shape (orthe distribution of a shape parameter) of particles is important in many industries,but the exact meaning of these three concepts varies with the type of particlesstudied and the reasons for studying them. For spherical particles the situationis comparatively simple, but even for these highly symmetrical particles one hasto decide the exact meaning of the fundamental concepts, such as concentration.Several possible definitions exist: volume fraction, weight fraction, weight-volumefraction etc. For a more general particle one also has to decide what is meant bysize and shape. In this thesis, the interest is directed towards the measurementof particles in the pulp and paper processes. These particles are further describedbelow. The measurement principles described may, within the limits of the theory,be applied to other types of particles.

In the pulp- and paper-making processes, the solid material constituting theend product, the paper, is usually carried part of the way through the processas particles suspended in a liquid. The characteristics of the particle suspension(often referred to as the slurry) are important for process control and for end prod-uct properties. For process control, characteristics such as concentration, volumeflow, mass flow, density, viscosity and temperature are important as these affect

5

6 CHAPTER 2. BACKGROUND

runnability1 and contribute to end product properties. Examples of propertiesaffecting runnability are drainage resistance, which limits the speed of the papermachine, and aggregation or propensity for aggregation. End product propertiesare paper strength, printability etc.

The solid phase of a suspension often consists of different types of particleshaving different influence on runnability and end product. These constituents maybe cellulose fibres from different wood types such as pine, spruce and birch, and fromdifferent pulping processes such as kraft pulping and refiner mechanical pulping,Fig. 2.1. The fibres from early wood and late wood2 are different. There are otherstructures than cellulose fibres in the wood, such as vessel cells, that are found in thepulp. Pulping produces fines, small irregular particles, to varying degree dependingon type of process and of process variables. Also the fibres may be divided in classesaccording to size, Figs. 2.23 to 2.3, and the lumen of the fibres may be collapsed, Fig.2.4. When wood fibres are not properly separated small needle like bundles of fibrescalled shives are found, Fig. 2.5. In the processing, some of the crystalline fibre wallmaterial is separated forming fibrils, which are extremely thin and slender threadsof cellulose. Fillers (Fig. 2.6) and coating pigments (Fig. 2.7) are also suspended in aliquid when entering the papermaking process. These are usually mineral particlesin the micrometer or sub-micrometer range. Sometimes pulp and mineral particlesare transported together in the same suspension.

Therefore, depending on the constituents and their provenance, several typesof geometric forms are found. Undamaged wood fibres may be considered to behollow cylinders or collapsed hollow cylinders. Damaged fibres may form band-shaped fragments or short irregular fragments. Vessel cells are thin-walled andmuch wider than the wood fibres. Fillers and pigments may have a variety offorms, e.g. small flakes, irregular compact particles and nearly spherical particles,but are in general much smaller than fibres.

For the characterization of the fibres one may use measurement of particle lengthor length distribution, width or width distribution, wall thickness, curvature, oc-currence of bends and kinks, mechanical bending stiffness and surface fibrillation.Also chemical properties such as the kappa number, which is a measure of thelignin content in the fibres, are used for fibre characterisation. The same goes forthe liquid phase, which is usually aqueous and may contain added chemicals andcarry-over chemicals from the production or extracted matter from the solid frac-tion. The liquid phase constitutes the chemical environment of the particles andmay greatly influence the behaviour of the suspension.

It is not the ambition here to give an exhaustive account of all particle mea-surement methods, but a few comments are needed to put this thesis in a wider

1Runnability is the combination of properties of the material entering a process making itpossible to run that process fast, accurate and without interrupts caused by the material.

2Early wood is the portion of a tree’s annual ring formed during the spring whereas late woodis formed during the summer. The fibres of early wood have thinner walls and larger lumens(inner cavity) than the fibres of late wood. Annual rings are visible in Fig. 2.1(a).

3Figs. 2.2 and 2.3 are found at the end of this chapter on pages 29 and 30.

2.1. CONCENTRATION AND SIZE METROLOGY 7

(a) (b)

� � � � � � � � � � � � � �

(c) (d)Figure 2.1. Wood material and pulp fibres. In (a) and (b) wood samples fromspruce and birch are shown. The wood samples were cut such that one surface isparallel with the annual rings, one is radial and one is horizontal. In (a) the verticalstructure due to the tracheids is clearly visible. The small holes in the horizontalsurface are the lumens of the tracheids. One annual ring is visible in the horizontalsurface (late wood is the dense portion of the annual ring). The most obviousdifference between the spruce (a) and the birch (b) samples is the large vessel cellsin birch here seen as large cavities between the tracheids. In the pulping process thetracheids are separated such that the pulp fibres are disintegrated intact or damagestracheids. (c) and (d) show optical microscopy images with examples of pulp fibresand other pulp particles from mechanical pulping (c) and chemical pulping (d). (c)shows a thermomechanical pulp made from softwood (spruce). The particles werestained with Herzberg solution, which makes the lignin appear yellow. (d) showsa mixture of particles from two fully bleached sulphate pulps made from Europeanpine and birch. The particles were stained with Graff “C” solution enhancing thedifference between the two fibre types, the dark particles are birch. Photo: JoannaHornatowska.


Figure 2.4. Scanning electron microscope images of fully bleached kraft pulp fibresfrom softwood. The three pictures show the same sample region at different magni-fications. The collapse of the fibre lumen causes an indentation along the fibre. Itis visible especially in the highest magnification. The images provided by Ulla-BrittMohlin, STFI-Packforsk.

perspective. In an industrial process it is often difficult or impossible to measureparticle properties, such as concentration and dimensions, according to their def-initions. Various indirect methods have therefore been invented from which it ispossible to determine particle concentration and size under certain conditions.

Fibre concentration is an important process variable for the pulp and paperindustry since it is directly linked to process efficiency and production. Manyinstruments exist on the market. Measurement principles are based on e.g. me-chanics, acoustics, γ-radiation, optics and possibly chemistry, Fig. 2.8. Among themechanical principles filtration and sieving are found especially for laboratory useand principles based on fluid mechanical forces and pressure differences are usedonline. Principles based on mechanical vibration are also suited for online use [1–3]. Acoustical principles use e.g. changes in ultrasonic attenuation or phase. Manyoptical principles exist, Fig. 2.9. They use optical phenomena, such as light atten-uation, low angle scattering, dynamic low angle scattering, back scattering, otherdynamic effects, chromatic effects (simultaneous use of different wavelengths), and


Figure 2.5. Shives from mechanical pulp. Shives are bundles of fibres not separatedproperly in the pulping process. The longest shive is about 1 cm.

changes in the polarization state [4]. The performance of different available con-centration meters can be found in the Literature [5–7]. Fladda, Zetterberg andEinarsson [5] made an evaluation of five concentration meters in 1975. The meterswere a Tellusond TD 102 built on 90◦ scattering, a Partech two-gap light attenu-ation sensor, an Anacon 303 back scatter sensor, a Lowcon MEKL 6 built on thepolarisation principle in [4], and a Fiberlog counting the rate of fibres by low anglescattering [8]. In a report on concentration control in 1982 [6], Gavelin presentsa number of concentration measurement principles and their pros and cons. Theprinciples mentioned are based on viscosity, shear force, flow resistance, pressurerecovery after a venturi tube, ultra sonic attenuation, light attenuation, light reflec-tion, optical fibre counting, the Total Power principle (the TP principle) [9], andpolarisation. In 1988, Jack, Bentley and Barron made an experimental evaluationof four optical concentration meters [7]. These were Kajaani LC-100 based on po-larisation, Cerlic ACM based on attenuation of 950 nm optical radiation, Monitekmodel 240/34 based on light attenuation, and Biobic model CS-1 built on backscatter. All these compilations are concerned about the considerable measurementerrors that are seen when concentration meters are used in situations for which theyare not suited. Parameters that may cause errors are flow rate of the sample, thedegree of beating, the fibre type, presence of filler particles, temperature, pH-value,air bubbles, changes in the relative content of different constituents (length classes,fines, fillers etc), degree of bleaching, and absorption in the liquid phase (e.g. fromblack liquor, the waste liquid from a completed sulphate pulp cook or a soda pulp


(a)

(b)Figure 2.6. (a) Filler particles of clay (kaolin). (b) The filler particles are mixedinto the paper sheet. They are here seen as bright areas in a cross section of a papersheet. The images were provided by Mikael Rigdahl and Joanna Hornatowska.


Figure 2.7. Scanning electron microscope image of a calcium carbonate papercoating. The paper coating improves the optical and printing properties of thepaper. The image was provided by Joanna Hornatowska.

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(b)Figure 2.9. Examples of different set-ups for optical measurement of the concen-tration of suspended particles. Boxes containing the letter D symbolise detectorpositions. (a) shows regular transmittance (0◦), low angle scattering, high anglescattering (90◦), and back scattering. (b) shows a method invented at PFI, Norway,using the change in the polarisation state due to the presence of cellulose in themeasurement cell.

2.2. CONCENTRATION AND SIZE METROLOGY IN THIS THESIS 13

cook).Particle size meters belong to one of two main categories of measurement prin-

ciples: single particle measurements and ensemble measurements. In the first,measurements are made on single particles. Different types of particle counters andimage analysis methods belong to this category. Examples of particle counters arethe Coulter counter and the Kajaani FS-100. The L&W FiberMaster [10] is anexample of an image analysis instrument.

In the second category, ensemble measurement, particles are not measured oneat the time. Instead a volume containing many particles is used to determine meansize or size distribution. Particle size distribution may be determined as the prob-ability density function (PDF) or be characterized using the relative concentrationof particles in different size classes. For off-line purposes, sedimentation and sievingtechniques, such as the Bauer-McNett classifier, are used for cellulose fibres. En-semble measurements are used in different light scattering based techniques as wellas laser Doppler techniques and acoustical techniques. Optical particle size distri-bution measurement sometimes use a priori information about type of distributionto be expected. Given a distribution described by an analytical expression takinga few parameters affecting its exact shape, optimization of these parameters withrespect to measured data often serves a considerable simplification of the problem.However, it is important that the chosen distribution model is possible to adjustto fit the actual one. Examples of distributions used are normal, lognormal andRosin-Rammer distributions [11].

2.2 Concentration and size metrology in this thesis

The concentration and size measurement principles presented in this thesis arebased on the fluctuating regular transmittance through a dispersion of slenderstraight circular cylinders. Regular transmittance is the ratio between the partof the transmitted light flux that is still collimated after passing the sample andthe incident light flux, whereas transmittance usually means the ratio of light fluxtransmitted in all directions and the incident light flux. The particles are woodpulp fibres for which the slender straight circular cylindrical shape serves as a firstapproximation. The measurement principle for concentration makes it possible tomeasure concentration independent of both particle width and length within pre-defined limits. Three principles for different size meters are presented:

• a fibre classifier giving the concentration or relative concentration of particlesin a number (usually three) of particle length classes

• a particle length meter giving a measure of the weighted mean length up toa predefined maximum length

• a particle width meter giving a measure of the particle diameter for idealparticles and of the particle width for non-ideal particles provided that theparticles are not too short.


The three principles for particle size are all ensemble measurements. All four prin-ciples are based on a common theory presented in this thesis.

2.3 The particle model and real particles

For convenience, three categories of real particles are referred to in this thesis, viz.fibres, fines and fillers, meaning fibres and fines found in wood pulp and mineralfillers used in paper manufacture. These particles have already been introduced.However, additional information is found in this section. The fibres are as a firstapproximation approximated as slender straight circular homogeneous cylinders.These idealized particles are referred to as cylinders.

2.3.1 Wood pulp fibres

The emphasis in this thesis is measurement of the more or less intact cellulose fibresfound in the pulp, but some attention is also given to the possibility to measurethe concentration of mineral fillers and cellulose fibres separately. Fibres have adiameter much larger than the wavelength of the light used. The fibres also have acomplex internal structure, which is not included in the cylinder model. The fibresoriginate from the tracheids of the wooden raw material of the pulping process, Fig.2.1. The different pulping techniques aim to separate the tracheids such that theseform the fibres of the pulp. The pulping process may be mechanical or chemicalor a combination. Chemical pulping processes preserve much of the natural sizedistribution of the tracheids, whereas mechanical pulping processes makes consid-erable damage to the tracheids resulting in shortened, splitted and bent fibres aswell as a large fraction of small fragments, Fig. 2.3. These fragments constitutethe majority of the particles in the middle and fines fractions. Before reaching thepaper machine, the pulp may be bleached in order to improve the optical propertiesof the paper and beaten in order to improve the strength properties. The bleachingremoves the lignin covering the unbleached fibres. Beating makes the fibres swell,which improves the contact between the fibres in the paper sheet at the expense ofa weakening of the fibres.

The natural size distribution of the tracheids in the wooden raw material differsaccording to type of wood and the part of the plant used. It is also affected by theconditions under which the tree has grown. Although the natural size distributionof the tracheids is the most important factor for the size distribution of the fibresin the pulp, many other factors also have a great influence, examples of which arefound in Appendix C.

2.3.2 Modelling considerations

The fact that fibres are much more complex than the ideal cylinder particles affectsthe definitions of particle volume, the particle density, the probability of interaction

2.4. LIGHT SCATTERING BY PARTICLES . . . 15

with the light beam, and the definition of the particle orientation state. It alsoaffects the complexity of the optical properties of the particles.

The model particle has a diameter d, a length l, and a volume Vp = π d2 l/4,whereas fibres have a width that varies along the particle and a length that some-times is hard to define exactly due to curvature or unclean rupture. Moreover,fibres rarely have circular cross sections. Therefore, it is better to use the wordwidth instead of diameter for a fibre. Thus, for a fibre, the variables d and l onlyresult in the correct volume and do not describe the exact shape of the particle.

The cylindrical model particle is homogeneous and non-swelling, which makesit possible to define the density of the particle ρp as the ratio between its mass andits volume. The fibre has a lumen. The fibre wall thickness varies between fibresand within fibres and the wall material may swell more or less with the dispersionliquid.

The probability of interaction between the particle and the light beam is some-what different for the cylindrical model particle and the fibre, e.g. when the fibre iscurved. For the slender cylinder having a circular cross section, only the azimuthand the polar angles are needed in order to describe the orientation of the particlesince it is axially symmetrical. To describe the orientation state of a more generalless symmetrical particle, a third angle is needed. For a non-circular cylinder, thethird angle is the rotation around its own axis.

2.4 Light scattering by particles and especially wood pulp fibres

There is a vast corpus of literature on light scattering by particles. Four textbooksare recommended here for more in-depth studies of this subject. These are:

• Van de Hulst 1957 Light scattering by small particles [12],

• Kerker 1969 The scattering of light and other electromagnetic radiation [13],

• Bohren and Huffman 1983 Absorption and Scattering of Light by Small Par-ticles [14] and

• Mishchenko, Hovenier and Travis 2000 Light scattering by nonspeherical par-ticles [15].

Via these textbooks, the original works by e.g. Lord Rayleigh, Mie etc. may befound. If not otherwise stated, the following summary regarding light scatteringrefers to these books.

This section includes subsections on single scattering, scattering from wood pulpfibres, scattering from many particles, and a discussion on optical cross sections andefficiencies.


2.4.1 Single scattering

The typical single scattering experiment comprises a plane electromagnetic wavetravelling in the z direction in right-handed Cartesian coordinate system. Usuallyat the origin, a particle is placed. The part of the wave hitting the particle isaffected and so is the wave in the neighbouring region, whereas the wave far offfrom the z axis is unaffected. The affected part of the wave may be scattered orabsorbed. The electromagnetic field is described by Maxwell’s equations and thetheoretical description of the scattering and absorption by the particle is a solutionto the vector wave equation for the time-harmonic electric field. Analytical exactsolutions are found only for a small set of particle shapes, such as homogenousspheres, homogenous infinite circular cylinders and spheroids. These solutions usethe Separation of Variables Method, SVM. Light scattering from somewhat morecomplex particles, such as particles that are stratified in the radial direction mayalso be solved by SVM. A number of different approximate numerical solving tech-niques make it possible to calculate the scattering from other types of particles.The most important techniques mentioned in [15] are

ADA Anomalous Diffraction ApproximationDDA Discrete Dipole ApproxiamtionFDM Finite Difference MethodFDTDM Finite Difference Time Domain MethodFEM Finite Element MethodFIEM Fredholm Integral Equation MethodGOA Geometric Optics ApproximationGPMM Generalised Point Matching MethodME-GPMM Multiple-Expansion Generalised Point Matching MethodMOM Method of MomentsPMM Point Matching MethodPT Perturbation TheoryRA Reyleigh ApproximationRGA Reyleigh-Gans ApproximationRT/MC Ray Tracing and Monte Carlo method, a special case of GOASIEM Surface Integral Equation MethodSM Superposition Method (for aggregates of spheres and speroids)SVM Separation of Variables MethodTMM T -Matrix MethodVIEM Volume Integral Equation Method

These methods are designed for a variety of validity regions and the accuracy andthe computational speed has to be considered. Many of the methods run intodifficulties when the particle is large compared with the wavelength. An exceptionis GOA, which only works with large particles. The methods VIEM, MOM, DDA


and FIEM are called Integral Equation Methods4.The evolution of the methods has enabled the solution of more complex systems,

such as particles having inclusions, randomly oriented particles, rough particles andparticles having a stochastic shape. However, very few are able to handle woodpulp fibres since these are usually too large compared with the wavelength. In thisthesis, a very simple approach is made. The light scattered in different directions isnot measured. Instead the effect of the presence of the particles on the transmittedplane wave is used. This effect is a decrease in the light flux caused by a combinationof light absorbed in the fibre and the total amount of light scattered in all directionby the particles.

2.4.2 Optical properties and models of wood pulp fibresThere are essentially two reasons to study the optical properties of wood pulp fibres.The first is to understand how the optical properties of the final product, the paperor the print, are affected by the optical properties of the ingoing particles. Thisfield of science is outside the scope of this thesis, which will focus on the secondreason, viz. to devise different measurement methods for non-optical properties ofthe fibres, such us concentration, length, width, cell-wall thickness etc. Propertiesthat affect e.g. the mechanical properties of the paper.

The obvious first approximation of the fibres is the infinite cylinder, for whichthe scattering can be calculated using the SVM. This was done by Lord Rayleighalready in 1881 [16]. Another geometric shape that might be used as an approxima-tion for wood fibres and for which SVM can be used is the prolate spheroid. In 1994Mishchenko and Travis used the TMM to study the light scattering from randomlyoriented and polydisperse spheroids [17]. However, this study did not include verylarge particles having high aspect ratios, which is required for modelling wood pulpfibres. A third geometric shape that can be used as a model for wood pulp fibres isthe finite cylinder. Mishchenko, Travis and Macke used the TMM, also on randomlyoriented finite cylinders [18], but the size parameter and the aspect ratio did notextend to values appropriate for wood pulp fibres. For large cylinders, GOA can beused instead of TMM. GOA was used by Takano and Tanaka in 1980 [19]. Saarinenand Muinonen [20] used GOA together with a particle model for wood pulp fibresthat took into account that the length, width and cell wall thickness should beconsidered as random variables in order to make the model realistic. However, thislevel of complexity is not needed for the theory presented in this thesis due to theexperimental configuration used, see Section 2.4.4.

4 The total electric field E(r) at the space coordinate r is described by the Fredholm integral,

E(r) = Einc(r) + k2∫V

d3r′(

1 +1k2∇∇

) exp(i k|r− r′|)4π|r− r′|

[m2(r′)− 1

]E(r′) ,

where Einc(r) is the incident field, k is the wave number in free space, m(r) is the refractive index,r′ is the coordinates of an interior point, and 1 is the identity dyad.

[m2(r′)− 1

]E(r′) is the

interior polarisation current, which can be regarded as the source of the scattered field.


2.4.3 Scattering from many particles

Light scattering experiments usually involve more than one particle. For a collectionof well separated particles in a small volume, the scattering from each particle isindependent of the scattering from the other particles and the theory for singlescatterers still applies. If the interaction volume is increased while maintaining thenumber of particles per unit volume, the incident light beam becomes attenuatedalong its path through the cloud of particles. Therefore, the flux of the scatteredlight also decreases along the path. The incident beam decreases as exp(−z S kA),where z is the path length, S is the concentration of particles, and kA is a constantwhich depends on how efficiently each particle attenuates the beam. The productS kA is called the turbidity or the light attenuation coefficient. The dimensionof kA depends on how the concentration is defined, see 2.5.2. The measurementprinciples treated in this thesis all operate within a concentration range for whichthis attenuation relation is valid.

For higher concentrations, a multiple scattering situation occurs where the in-cident light on a particle in the cloud from other scatterers becomes significantlyimportant and the original light beam becomes less important. If the number ofparticles per unit volume is increased further, the distance between the particlesbecomes so small that the scattering no longer is independent and when particlesare packed into sheets or layers, as they are in paper, they come in partial opticalcontact with each other and some of their scattering ability is lost.

For clouds of particles, the Radiative Transfer Equation (for a brief introductionsee Chapter 1 Section XII in [15]) is used for the relation between the opticalproperties of the individual particles and the optical properties of the cloud. Forpaper, the simpler Kubelka-Munk approach [21–23] was adapted by pulp and paperresearchers [24–27] and is still often used despite its limitations and the fact thatmore advanced methods exist and others are being developed.

2.4.4 Optical cross sections and efficiencies

General light scattering experiments involve many scattering angles, different po-larisation states, a coherent light source illuminating the particle from differentangles and tests at different wavelengths. The experiment of interest in this thesisis the random regular transmittance, T, which is much simpler since the regulartransmittance of a light beam through an interaction zone, where scattering par-ticles are contained, is the ratio between the radiation flux leaving the zone as aplane wave to the incident radiation flux, which is also a plane wave. Furthermore,this thesis only deals with the regular transmittance of non-coherent and unpolar-ized light, which is nearly but not ideally monochromatic, and the medium of thedispersion is assumed to be non-absorbing, i.e. T = 1 when no particles are present.With particles in the interaction zone, T decreases below 1 due to scattering by andabsorption in the particles. By definition, scattering here means even the slightestdiversion of the direction of light caused by the particles. The experimental set-up


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(b)Figure 2.10. Experimental set-up for measurement of the regular transmittancethrough dispersion of particles. (a) Shows the ideal set-up where no scattered lightreaches the detector, which makes it possible to measure regular transmittance ac-cording to its definition. (b) Shows a non-ideal set-up that can be used to improvethe signal to noise ratio but at the same time allows some of the light scattered atsmall angles to reach the detector, which makes the measured transmittance higherthan the regular transmittance if the sample scatters light. Details of the implemen-tation of the sensor are given on page 46.

is shown in Fig. 2.10(a). A small light source and a small detector are placed in thefocal planes of two collimating lenses. The light beam, passing through the glasscell containing a flowing dispersion of particles, is approximately parallel and theangle within which the detector accepts scattered light is very small.

An alternative experimental set-up used in this work for very thin light beamsis shown in Fig. 2.10(b). It comprises a small light source focused in the middle ofthe glass cell and a detector imaged into the same point in the cell. This set-upincreases the amount of available light and thereby improves the signal to noiseratio, but light scattered in small angles reach the detector giving a somewhathigher transmittance value than the ideal regular transmittance.

The particles are assumed to scatter and absorb light independent of each other,i.e. the distance between the particles has to be sufficient for the wave-particle in-teraction to take place as if no other particles were present. The scattering propertyof interest is how the transmitted plane wave is affected by the presence of eachparticle. This property is summarised by the optical extinction, scattering and ab-sorption cross sections cext, csca, and cabs, all having area as their dimension. Theydepend on the size, shape and orientation of the particle and on the wavelengthand polarisation of the incident light.


For a given particle in a fixed position and orientation in relation to the in-cident light, cext describes the power removed from the incident beam in relationto the incident power. The exact wording in [15] is: “the product of the extinc-tion cross section and the incident monochromatic energy flux is equal to the totalmonochromatic power removed by the particle from the incident beam”.

The definition of csca is: “the product of the scattering cross section and theincident monochromatic energy flux gives the total monochromatic power removedfrom the incident beam resulting solely from light scattering by the particle”.

The difference between the two is cabs:

cabs = cext − csca . (2.1)

The relation between the size of the particle and its ability to affect the beamis described by the extinction, scattering and absorption efficiencies Qext, Qscaand Qabs, where the particle size is described as the geometrical cross section, g,which is the geometrical projection of the particle on a plane perpendicular to thepropagation direction of the incident light. With the exception of a few specialcases, g is a sample value of a random geometrical cross section G.

The efficiencies are defined as

Qext = cext/g (2.2)Qsca = csca/g (2.3)Qabs = cabs/g . (2.4)

WhenQext is studied as a function of particle size and shape, three kinds of solutionsare often proposed depending on the ratio of particle size to the wavelength of thelight used. If the ratio is very small, the scattering is of Rayleigh character. ForRayleigh scatterers, Qext increases with increasing particle size. If the ratio is closeto 1, the scattering is of Mie character. For Mie scatterers, Qext is often a non-monotonous function of the size-to-wavelength ratio. If the particle size is muchgreater than the wavelength, geometrical optics together with Fraunhofer diffractionmay be used.

Characteristic of Qext for the geometrical optical case is that Qext = 2 andthat Qext is independent of the geometrical cross section and of the shape andorientation of the particle. The value 2 may seem counter-intuitive, but in rayoptics it corresponds to a situation where all rays within the geometrical crosssection are either absorbed or more or less change direction due to the presence ofa particle. In addition the particle causes the light close to the rim of the particleto be diffracted due to Fraunhofer diffraction.

The value of Qext describes the light attenuation efficiency in a highly idealisedtransmission optical system with a ideally collimated beam and zero acceptanceangle. In practice, the light attenuation is less than in the ideal case since lightscattered at small angles is also detected. This affects the observed Qext valueespecially when the particle is large compared with the wavelength of light sincemost of the light scattering is mainly at small angles for these particles.

2.5. FUNDAMENTAL CONCEPTS 21

Wood pulp fibres are much larger than the wavelength of light. Therefore,Qext = 2 will be assumed in most cases. The fines fraction in the pulp can usuallybe considered to be non-spherical Mie scatterers. However, very thin fibrils maybe regarded as Rayleigh scatterers. In Paper 7 and Appendix B measurements arealso made on mineral filler particles. These are non-spherical Mie scatterers.

The extinction cross section cext is proportional to the forward scattering am-plitude of the radiation. This relation is based on the principle of conservation ofenergy and is referred to as the optical theorem. Two recent publications [28, 29]discuss the nature of cext, csca, cabs and the optical theorem. They describe asituation in the far field, but at the same time the distance from the particle tothe detector is only less or equal to 100 times the diameter of the particle. In thepresent work, the distance between particle and detector is two orders of magnitudelarger. The particles used as examples are also very different from the particles inthe pulp and paper processes. The implication for the present work is, therefore,unclear.

2.5 Fundamental concepts

2.5.1 DispersionA dispersion is a heterogeneous two-phase system comprising

• a continuous phase referred to as the dispersion medium, which may be a gas,a liquid or a solid

• a dispersed phase, which may be gas bubbles, liquid droplets or solid particles.

All combinations of dispersion media and dispersed phases exist except gas-gassystems. Several of these combinations have their own names such as emulsions(liquid-liquid systems) and aerosols (gaseous dispersion medium). Dispersions occurin many familiar forms such as cream, foam, haze and smoke. In this thesis, onlydispersions of solid particles in a liquid are considered.

The size of the dispersed particles may range from the limit where a furtherfragmentation of the dispersed phase would make it form a true solution to thelimit where gravity dominates. In the latter case the particles settle directly onthe bottom or at the surface of the liquid when left alone and the density of theparticles is different from the density of the liquid. If the particles are sufficientlysmall, they will not settle, but are kept moving around by Brownian motion. Largerparticles settle after a while when the dispersion is left alone by sedimentation orfloatation. These dispersions are usually called suspensions.

If all particles in the dispersion have the same shape, size and density, the dis-persion is monodisperse or monosized. However, size distributions narrow enoughto classify the dispersion as monodisperse are seldom seen. Most natural and ar-tificial particle populations contain particles having a variety of sizes and shapes.These dispersions are polydisperse, polysized or heterodisperse.


2.5.2 ConcentrationThe concentration of a substance dissolved or dispersed in a liquid may be definedin different ways. Common alternatives are:

• the ratio between the volume of the substance and the total volume

• the ratio between the mass of the substance and the total mass

• the ratio between the mass of the substance and the total volume

• the ratio between the number of particles and the total volume.

In the first two cases, concentration is a dimensionless quantity. In the third case ithas the dimension mass/volume, i.e. the same dimension as density. In the fourthcase, the dimension is volume−1.

In the case of solid particles dispersed in water, the third definition, the mass-volume concentration, is often preferred. Probably because of the laboratory pro-cedure used to determine the solids content of a suspension of particles. A knownvolume of the dispersion is filtered and the retained particles on the filter are driedand weighed. If nothing else is stated, this is the definition used in this thesis. Thesymbol S is from here reserved for the mass-volume concentration.

The fourth way to express concentration, the number concentration, is also afundamental concept in this thesis. Under the condition that the particles arewell dispersed in the water, the number concentration is calculated from the mass-volume concentration S as

SN = S

ρp Vp, (2.5)

where ρp is the density of the particles and Vp is the particle volume.If the particles are straight circular cylinders with the diameter d and the length

l, the number concentration is calculated as

SN = 4Sπ ρp d2 l

. (2.6)

2.5.3 Calculation of the mean of particle propertiesNote that a consequence of the choice of mass-volume concentration above, smallparticles of a given concentration and density represent a higher number concen-tration than large particles having the same concentration and density. Therefore,it is common to calculate, e.g., the mean length of cylindrical particles as weightedaverages5 with the intention to obtain a mean value with a weighting function cor-responding to the gravimetrical weight of the particles. If both the length li and

5A weighted average mW of a random variable X with the weighting function W (x) is calcu-lated as

mW =

∑n

i=1W (xi)xi∑n

i=1W (xi)

2.5. FUNDAMENTAL CONCEPTS 23

the width di of each particle i are measured and the density is the same for allparticles, a weighting function li di2 is the ideal choice. When di is not available,li, li2 or li3 are often used. 6

2.5.4 Particle densityThe particle density ρp is used in Eqs. (2.5) and (2.6) without any further expla-nation, and in the case of homogenious particles with volumes that are unaffectedby the dispersing medium the notion of the density of macroscopic particles is notvery complicated. It is obvious what is meant by the particle weight mp and theparticle volume Vp such that ρp = mp/Vp may be determined. However, the woodpulp fibres are not homogenious. The lumen, the pores and openings in the fibrewall and the fact that the material in the particle may be more or less swollen bythe medium complicates the definition. In order to make a definition ρp, whichworks with Eq. (2.5) and with the way in which the mass-volume concentration isdetermined, mp is defined as the weight of the particle with the dispersion mediumremoved (the dry weight) and Vp is defined as the volume of the particle as itappears in the dispersion (the wet volume).

2.5.5 Orientation and projectionThe (deterministic) geometrical cross section, g, of a particle is the projection, i.e.the silhouette, of the particle on a plane perpendicular to the optical axis. g dependson the particle orientation if the particle is not spherical. Usually the orientationhas a random nature which means that even in a monodispersion of non-sphericalparticles the cross sections of the particles will have a random nature. The randomgeometrical cross section is denoted G. This effect is pronounced for slender andflat particles whereas the cross section of compact particles is less affected by theorientation.

Fig. 2.11 shows the coordinate system used in this thesis to describe the positionand orientation of slender circular cylinders. The lateral position is given by (x, y),the azimuth angle by ζ and the polar or tilt angle by θ. The z axis is parallelwith the optical axis, which makes the (x, y) plane the projection plane, i.e. g isthe projection of the particle onto the (x, y) plane, making g independent of z. Acylindrical particle has a rotational symmetry around its axis, this rotation angleis not included in the description.

For a slender cylinder small enough to be contained within the beam, a simplifieddescription of the orientation state can be used. When the cylinder is slender, the

or if the sample contains fi observations of the value xi the equation may be written

mW =

∑n

i=1W (xi) fi xi∑n

i=1W (xi) fi.

6W (l) = l means that all particles are assumed to have equal width and W (l) = l3 means thatthe particles are assumed to have a width proportional to their length.


�

�

��

�

� � � � � � � � �

Figure 2.11. Definition of the coordinates used to describe the orientation of acylinder particle. The projection plane is parallel with the xy plane and perpendic-ular to the optical axis (the z axis). The azimuth and the polar angles are definedin relation to the x and z axes.

end surfaces of the cylinder contribute very little to g, which gives g = λ l d, whereλ is sin θ, l is the length and d is the diameter of the cylinder. In the case of randomorientation, the orientation state is described by the random variable Λ for whichλ is a corresponding event or sample value. In this thesis two special orientationstates are used: flat and isotropic. The flat orientation state approximates thesituation when the cylindrical particles are contained between two narrowly spacedwindows parallel with the xy plane. Although the azimuth is random, this gives aconstant value of geometrical cross section since the particles are parallel with theprojection plane. The isotropic orientation state approximates a situation when theparticles are free to or forced to orient randomly in three dimensional space. Forthis special case the expected value μΛ of Λ is π/4, the variance vΛ = (32−3 π2)/48and the square mean EΛ(Λ2) = 2/3, the derivation of which is found in AppendixA. In a real sensor the orientation state is not expected to be exactly one of thesetwo states. The way in which the fibre orientation of a flowing suspension is affectedby being close to the wall containing the flowing suspension was recently studiedby Carlsson [32].

2.6. STATISTICAL NOTATION AND SUMS WITH RANDOM NUMBER OFTERMS 25

2.6 Statistical notation and sums with random number of terms

The convention in this thesis is to denote random variables by upper case letters,while sample values, events or constant values of the same quantities are denotedby the corresponding lower case letters, e.g. are cext and Cext the deterministicand random versions of the extinction cross section7. The expected value of arandom variable X is denoted μX or, if X is an expression too complicated to putin a subscript, the expression EX (X) is used. Note that in statistics the expectedvalue and the mean value are different concepts. The expected value refers to thetheoretical probability distribution, whereas the mean value refers to the outcomeof an experiment. When the mean value is used as an estimate of the expectedvalue the notation μX is used. The mean of the experimental values x1 · · ·xn iscalculated as (x1 + · · ·+ xn)/n.

The variance of a statistical distribution is denoted vX , σX2 or varX (X) . Thevariance of experimental data is calculated as

[(x1 − μX)2 + · · ·+ (xn − μX)2

]/(n−

1) and is denoted vX or σX 2.If X is a partially ergodic random function of time X(t), its properties do not

change with time and the estimates of μX and vX can be calculated as time averages〈x(t)〉t and

⟨x2(t)

⟩t− 〈x(t)〉2t . 8

In this thesis sums

Y = X1 +X2 + · · ·+XN (2.7)

of independent and equally distributed random variablesXi are used, where N alsois a random variable. The expected value μY = EY (Y ) and variance vY = varY (Y )can be determined in two steps. In the first step the number of terms in the sumis considered to be constant, N = n, giving the conditional expected value andvariance E (X |N = n) and var (X |N = n). In the second step the condition isremoved using a theorem from mathematical statistics which says that

EY (Y ) = EN[EY (Y |N)

](2.8)

varY (Y ) = EN[varY (Y |N)

]+ varN

[EY (Y |N)

], (2.9)

see e.g. Lindgren [30] pages 115 and 130 or Råde and Westergren [31] page 4319.Applications of this theorem are found in Paper 3 and Paper 7.

7The capital letter Q is, however, used in this thesis for the efficiencies although Q is regardedas deterministic and q would be more proper. However, Q is used in the major references and wastherefore used also in Paper 3 – Paper 7.

8A time average is calculated as 〈x(t)〉t = 1t0

∫ t00x(t)dt, where t0 is the time over which the

average is taken.9To be precise, it is not required that the terms in Eq. (2.7) are independent for Eqs. (2.8)

and (2.9) to be valid. However, independence is assumed in the theory in Chapter 3.


2.7 Outline of the material presented

2.7.1 Chronological remarks

When the Physics Department at STFI was formed in 1969, one of the first ac-tions taken was a survey of the needs for new measurement techniques in the pulpand paper industry.10 This was the start of a long period of metrology researchand development with the emphasis on specialised measurement techniques for thisindustry branch. The emphasis was also on measurement techniques having thepotential to be adapted to quantitative online assessment. With the new mea-surement techniques it was possible to study and control the industrial processesin new ways. Consequently, process modelling and process control soon becamemajor activities. When a measurement principle proved useful in the process, itwas commercialised. There were usually restrictions imposed on the commerciali-sation process by the Swedish industry, who wanted a technological advantage inexchange for providing funds for the research. Commercial interests also led to avery restrictive publication policy.

The concept in the metrology research was to take advantage of recent devel-opment in light sources (such as lasers and powerful LEDs), detectors, electronics,data processing, computers etc. The knowledge base in these areas covered by theresearch team was considerable.

Most of the measurement principles studied were based on ideas from within theteam, but evaluation and use were also made of other techniques such as sensors forash content [33], drainage properties, and commercial concentration (consistency)sensors [5]11.

In the area of measuring pulp fibre properties using optics, a laser based equip-ment [34] for fibre characterisation was built in the early 1970s. However, it waslaboratory equipment. Considerable simplifications had to be made in order touse the knowledge obtained in the industrial situation resulting in the Fiberlogpresented in 1974 [8, 35, 36]. It signalled an alarm when the industrial effluentscontained fibres. A shive analyzer soon followed in 1975 [37–40]. Shives are smallbundles of cellulose fibre that were not separated in the pulping process. A shiveanalyser is useful both to monitor the pulping process and the screening process,which separates shives from the pulp.

Another track of development was to use the statistical properties of regularlytransmitted and low-angle scattered light. The basic principles were establishedin a master’s thesis work in 1975 [41]. The main purpose was to find a way tomeasure the fibre concentration independent of the fibre size distribution. Thisconcentration meter was called the TP-meter (1977) [9, 42, 43]. Its principle wasbuilt on a linear combination of the mean and the variance of the detected light.

10It is not the ambition here to give a full account of the activities at the Physics Department.Only those activities which impact on the work presented in this thesis are mentioned.

11In this section several references are made to papers written in Swedish. Papers, which arenot strictly scientific in character, have also been included.

2.7. OUTLINE OF THE MATERIAL PRESENTED 27

Using the same equipment, but combining the signals differently, it was possible tomeasure a weighted mean fibre length12 (1977) [44, 45].

The 1975 theory part in [41] is referred to as the TP-theory. It suggested othermeasurement principles that could be considered for patent applications. Thereforethe theory was subjected to a strict classification and the content was known only toa few persons in the research team and Lundqvist’s master’s thesis was not handedover to the KTH library13. Eventually new measurement principles inspired by theTP-theory were developed. The optical fibre classifier was patented in 1979 [46, 47]and measurement of length and width was patented 1985 [48, 49]. The opticalclassifier was presented to an industrial audience in 1981 [50], a presentation thatconstitutes Paper 1 of this thesis. It was followed by a report in Swedish in 1982[51]. The fibre classifier was commercialised and was eventually included in thePulp Quality Monitor together with the shive analyzer [52].

With the new measurement possibilities, processes were studied, modelled andautomatically controlled in a number of research projects, e.g. screen room researchreported 1977–1982 [53–59]. Mechanical pulping can produce pulps having particlesin considerable size range. Therefore the shive analyzer and the fibre classifier weremuch used in research activities on mechanical pulping, reported 1978–1980 [60–63].Research continued also after commercialisation [64–68].

A consequence of the understanding of the principle of the optical fibre classifieris that it is possible to construct a meter that measures concentration independentof the particle length and width as long as the particles may be approximated asslender cylinder being longer than the diameter of the optical beam and having awidth smaller than the beam diameter. This is possible for pulp fibres and a beamhaving a diameter of about 0.1 mm. The measurement is not sensitive to fines inthe pulp nor to filler particles because of their much smaller size. This insensitivityproperty was used in combination with a high resolution density meter to developa retention measurement system, RMS. The wire retention on a paper machine isdetermined from the particle concentrations in the headbox and in the white water.Thus, the RMS was essentially a concentration meter. The unique property wasthat pulp and filler concentrations were measured separately. The RMS was firstpresented in 1984 [69], which is Paper 2 of this thesis, and a patent was granted in1986 [70, 71]. The RMS was also reported in two STFI reports [72, 73] and in alicentiate thesis [74] preceding this thesis. The theory was later published [75, 76](Paper 3 and Paper 4 of this thesis) in 1998 and 1999. The licentiate thesis isincluded in condensed form in this thesis as Appendix B, with overlaps with Paper3 and Paper 4 removed. The RMS was used in the control of fine paper manufactureand was also commercialized [77–79].

The Kappa number is a measure of the lignin content of a pulp. Lignin absorbsUV radiation. It is possible to detect this absorption by using the directly trans-

12The concentration meter, the TP-meter, and the fibre-length meter are not the same as theconcentration meter and the length meter presented in this thesis.

13December 16, 2008, the master’s thesis was finally made available to the KTH library. There-fore, this is the formal date of publication of the TP-theory.


mitted radiation and radiation scattered at a low angle using an equipment, similarto the TP-meter, but with a UV radiation source. This was used in the Kappameter which turns up in publications from 1985 [80–84].

Image analysis was used for the measurement of fibre length and width alreadyin the 1980s. At the end of the 1980s, the performance of image analysis hardwarehad reached such a level that length, width and form measurements could be madewithin the 40 ms time span between the frames of a video camera. In about 1990,the image analysis based system Fibermaster had been developed and commercial-ized [10, 85–90]. Because of its ability to give detailed information of the fibrepopulation the focus shifted from the optical fibre classifier to this new equipment.

In 2008, the theoretical publications [75, 76] were complemented with one pub-lication covering the transition between the response characteristics of two specialcases in the theory [91] and another with detailed information on the four mea-surement principles for concentration, length classes, mean length, and mean widthdeduced from the theory [92]. These two publications constitute Paper 5 and Pa-per 7 of this thesis. In addition, a validation of the methods used in Paper 5 wasreported separately [93] in 2009, included in this thesis as Paper 6.

2.7.2 Conceptual outlineThe presentation in Chapters 3 to 8 does not follow chronological order. Thestarting point is the regular transmittance, T, through a suspension of particlesin a set-up as in Fig. 2.10a. Chapter 3 gives expressions for the mean, varianceand coefficient of variation of T when the particle concentration is low enough forthe probability of overlapping particles to be a very low. Special attention is givensamples consisting of mm-sized slender cylinders which are either very short or verylong compared with the beam diameter of the transmittance sensor. In Chapter 4,the concentration of particles is allowed to increase such that overlapping of particlesis allowed, but it must still be low enough to avoid multiple scattering. Chapter4 also includes a comparison with other theories and two experimental examples.Chapter 5 focuses on cylindrical particles and the transition between two specialcases of the theory given in Chapter 3. The transition is modelled using numericalsimulations. In Chapter 6, four measurement principles based on the theory arestudied in more detail. Implementation considerations and experimental tests arepresented. Chapter 6 is closely connected to Paper 7. In Chapter 7, the industrialmeasurement potential of two of the measurement principles is illustrated. Theprinciples used are the length classifier and a measurement system involving asmall-particle-suppressed concentration measurement. The examples are found inPaper 1, Paper 2, and Appendix B. In Chapter 8, measurement uncertainties arediscussed.

2.7. OUTLINE OF THE MATERIAL PRESENTED 29

Figure 2.2. Fibres and fines from a fully bleached softwood kraft pulp. In (a) to(d), size classes were separated using a Bauer-McNett apparatus. The separationsieves are characterized by their mesh/inch number. (a) shows fibres retained on 16mesh/inch sieve. (b) is the class 16–30, (c) is 30–50 and (d) is 50–100. (e) showsthe fines fraction, which was separated as the accept through a sieve having 200 μmopenings.


Figure 2.3. Fibres and fines from a groundwood pulp. In (a) to (d), size classes wereseparated using a Bauer-McNett apparatus. The separation sieves are characterizedby their mesh/inch number. (a) shows fibres retained on 30 mesh/inch sieve. (b) isthe class 30–50, (c) is 50–100 and (d) is 100–200. (e) shows the fines fraction, whichwas separated as the accept through a sieve having 200μm openings. Comparingthe middle fractions in (b)–(c) with the corresponding fractions (c)–(d) in Fig. 2.2,it is evident that the particles here were subjected to much more damage during thepulping process than were the kraft pulp particles.

Chapter 3

The random transmittance at lowconcentrations

. . . and there was light.

Genesis 1:3.

3.1 General particle geometry

Starting from an ideal set-up for the measurement of the regular transmittance Tthrough a dispersion as in Fig. 2.10(a) and assuming additivity of the extinctioncross sections, the random regular transmittance depends on the sum of the at-tenuations caused by the individual particles in a subvolume of the sample that iscontained in the light beam such that

T = 1− 1Cb

(C1 + C2 + · · ·+ CN ) , (3.1)

where Cb is the cross section of the parallel light beam, C1 . . . CN are the randomextinction cross sections1 of the particles, and N is the random number of particlesinteracting with the beam. In this thesis the beam cross section Cb is assumed tobe circular with diameter Db, but other shapes such as elliptical and rectangularmight be considered. Cb is assumed to be the same at different positions along theoptical axis in the cell, i.e. the beam is assumed to be well collimated. A highlyscattering dispersion must also be avoided, since it will successively broaden thebeam throughout the path due to scattering. The particle cross section Ci = QiGi,

1 As pointed out in Section 2.4.4, it is important that the acceptance angle of the detector isvery close to zero. The increase in the observed value of T when the acceptance angle is largerthan zero can also be seen as a decrease in the observed extinction cross sections Ci. Therefore,if the transmittance is measured with a non-ideal set-up as in Fig. 2.10(b), the term attenuationcross section may be better than extinction cross section in order to indicate that Ci < Cext,i.

31

32 CHAPTER 3. . . . LOW CONCENTRATIONS

c.f. Eq. (2.2). Here nothing is assumed about the dimensions and the geometricalshape of the particles.

Requirements for the additivity assumption and thereby for Eq. (3.1) to be validare (1) the concentration is sufficiently low to avoid overlaps of cross sections, (2)the particles do not emit radiation at the wavelength used, (3) the particles do notflocculate, (4) the distance between the particles is much larger than the wavelength,and (5) the radiant flux of the incident optical beam is uniformly distributed acrossthe beam cross section and, when there are no particles present, throughout theinteraction volume.

If the random optical cross sections of the particles have independent and iden-tical distributions2 with the expected value μC and the variance vC and usingEqs. (2.8) and (2.9), the expected value μT and variance vT of the random regulartransmittance T are

μT = 1− μN μCCb

, (3.2)

vT = vC μN

Cb2 + μ

2C vN

Cb2 , (3.3)

where Cb is the beam cross section, and μN and vN are the expected value and thevariance of the number of particles N , which has an arbitrary distribution with thenatural numbers as its event space.

If the sample is well-dispersed, N can be approximated with a Poisson Dis-tribution3 having μN ≡ vN ≡ aCb S/(ρp Vp), where a is the path length of thebeam though the dispersion and the sample is a monodispersion of particles havingan (almost) arbitrary shape and S/(ρp Vp) is the number concentration, Eq. (2.5).This leads to

μT = 1− aS μCρp Vp

, (3.4)

vT = aS

ρp Vp CbEC

(C2) , (3.5)

where EC(C2) = μ2

C + vC . In Paper 44 this relation is expressed as

μT = 1− kA aS , (3.6)2 There are situations where this assumption may be debated. The particles may e.g. inter-

act more or less with the walls of the measurement cell. Such effects are neglected here. Theassumption also means that Eqs. (3.2) and (3.3) are conditional on the monodispersity of thesample.

3If a fixed number of particles are spread out in a large volume, such that the probabilityof finding a certain particle in any specific position is uniformly distributed over the volume, thenumber of particles in a sub-volume is binomially distributed. If the sub-volume is small comparedwith the large volume, the binomial distribution may be approximated by a Poisson distribution.The assumption implies that uneven distributions of the position and volume exclusion effects arenot covered by this analysis. Note that N is the link between the transmittance and the particleconcentration, size parameters and density, which requires that the particle density ρp and theparticle volume Vp be the same for all particles. This is again a monodipersity condition.

4In Paper 4, x is used for the path length instead of a which is used here.

3.2. LARGE SLENDER CYLINDRICAL PARTICLES 33

vT = kB aS , (3.7)

Where S is the mass-volume concentration. Note that these equations may be usedfor other types of sensors, where the effect on a signal can be described by Eq. (3.1).A simple example is the percentage of unaffected pixels in an image from a digitalcamera viewing particles on an even background. Moreover, the radiation used inthe transmittance sensor need not be light or optical radiation as in this thesis.μT and vT can be estimated from measurements of T on a flowing dispersion

provided that the random function T(t) is partially ergodic, where t is the time.Note that the estimates μT and vT are essentially insensitive to changes in the flowrate. However, there may be an indirect dependence caused by influence of flowrate on the particle orientation state. The magnitude of the resulting uncertaintycan be assessed by computing μT and vT for orientation states which are believedto be limiting cases for the flow variations in question. However, if the flow rateinterval within which the flow changes from laminar to turbulent is avoided andif the flow rate is kept fairly constant, its influence on kA and kB is reduced andthe associated measurement uncertainty will usually be acceptably small for manytechnical applications. If T is influenced by flow-caused concentration gradients viathe number concentration, it is preferable to determine it experimentally.

In practice, μT and vT are estimated from sample values τ = φ/φ0, where φis the radiation flux reaching the detector when the beam has passed through thedispersion and φ0 is the radiation flux passing the same path length of the pureand particle-free medium of the dispersion. If the particles are moving and if thesampling of T(t) is discrete, the measurement of the flux must be done in a shortenough time that the particles can be considered to be in fixed positions. If, on theother hand, T(t) is ergodic and μT and vT are estimated from time averages, thenthe bandwidth of φ(t) must be considered in order to obtain a good estimate of vT.

Note that both φ and φ0 need to be determined. This means that the measure-ment uncertainties of both will affect the uncertainty of the observed values of μTand vT. However, in a case where the goal of the measurement is to utilise the re-lation between the variance and the properties of the dispersion, this can be partlyavoided by using the square of the coefficient of variation V 2 = vT/μT

2 instead ofvT. This can be done since V 2 ≈ vT at low concentrations. Furthermore, V 2 isindependent of φ0. Thus, in this case, φ0 does not need to be determined at all,which reduces the measurement uncertainty and simplifies the measurement set-upto a great extent.

3.2 Large slender cylindrical particles

In the special case of the ideal cylindrical particles having a length l and a diameterd much larger than the wavelength of light and a circular beam cross section Cbwith diameter Db, two special cases can be recognized: one where l � Db, and theother where l � Db. In both cases it is assumed that d � Db. The resulting kAand kB values are


kA = 4QμΛ

π ρp d, (3.8)

kB =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

16Q2 l

π2 ρpDb2EΛ

(Λ2) l� Db

128Q2 μΛ

3 π3 ρpDbl� Db

, (3.9)

where5 the extinction efficiency Q is constant and Λ is a random variable repre-senting the random orientation expressed as the sine between the cylinder axis andthe optical axis, i.e. the polar angle. Two types of orientation states are discussedin this work: one where each particle is oriented in a plane perpendicular to theoptical axis (flat orientation state), and the other where the orientation is isotropicin such a way that μΛ = π/4 and vΛ = (32 − 3 π2)/48, a derivation of whichis found in Appendix A. If the relations for the two special particle length casesin Eq. (3.9) are extrapolated towards each other as in Fig. 3.1(a), they intersectat lb = 8μΛDb/

[3 π (vΛ + μΛ

2)]. Note that lb and Db are closely related since

8Db/(3 π) ≤ lb ≤ Db.6 The expression in Eq. (3.9) for kB when l � Db is denoted

kBL. The response curve in Fig. 3.1(a) is referred to as the simple response modeldue to the fact that the sharp knee on the curve is obviously a simplification. Amore realistic response curve having a smooth transition between the special caseswas derived in this work and presented in Paper 5. Note that the curve does notinclude the limiting case of l→ 0, since this would cause conflicting model require-ments, since l→ 0 implies that d→ 0 due to the assumption of the particles beingslender, but d → 0 is not allowed since d is required to be much larger than thewavelength of the light used.

The derivation of the short particle alternative in Eq. (3.9) is straightforwardaccording to the method outlined above. However, the derivation of the long particlealternative needs special attention. Firstly, the random geometrical cross section Gof the particle is limited such that G/d ≤ Db. Thus, the distribution of G dependson the circular shape of the beam and the projected axis of the particle beinga randomly positioned corda across the beam cross section. Secondly, the way todetermine μN and vN is different. The distribution is still assumed to be a Binomial(or a Poisson) distribution, but the probability of an interaction between a particleand the beam is determined as a combination of the particle cross section and thebeam cross section. When the particle is thin but long compared with the beamdiameter, μN = aS Db l μΛ/(ρp Vp) instead of μN = π aS Db

2/(4 ρp Vp), which is

5In Paper 3, kμ = 4 aQμΛ/(π ρp) = a d kA and kV = 128 aQ2 μΛ/(3 π3 ρp Db) = a kB areused instead of kA and kB in the simplification of the expression for μT and vT.

68/(3 π) ≈ 0.85. If the flow rate through the instrument is changed such that the orientationstate changes from flat to isotropic, the values of kA and kB are reduced by 21 % and lb is increasedby 18 %. Thus, changes in the orientation state have a moderate influence on the measurements.

3.3. EXAMPLES OF OTHER PARTICLE GEOMETRIES 35

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1.0

1.2

Sensor response for vT

Particle length in lb units

(a)

Var

[T] i

n k B

L a S

uni

ts kBL

a S

lb

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1.0

1.2

Sensor response for ϕ


(b)

ϕ in

kB

L a S

uni

ts

kBL

a S

lb

Figure 3.1. Response curves for the variance vT at low particle concentrationsand ϕ and the fluctuance (presented in Chapter 4) for higher concentrations. (a) isbased on Eqs. (3.7) and (3.9) and shows the relation between vT and the cylinderlength valid for low concentration. The vT scale is expressed in kBL aS units, wherekBL is the value for kB for long particles. The different types of linear behaviour forthe short and long particle categories were extrapolated towards each other. Thelines intersect at the particle length lb, which is closely related to the beam diameterDb. (b) is based on Paper 4 and shows the corresponding response curve for thefluctuation property ϕ defined in Eq. 4.20. The ϕ scale is also expressed in kBL aSunits. (a) and (b) are shown together in order to emphasize that the response curvefor ϕ is the same as the response curve for vT at low concentrations.

the corresponding relation for short particles. The resulting kB has the interestingproperty of being independent of both the particle length and the particle diameter.In the derivation in Paper 3, a procedure to determine μN when then interactionarea is random is briefly described. A more complete description for the case oflong slender cylinders is given in Appendix D.

If the sample is a polydispersion such that the length is a random variable L andthe width is a random variable D, Eq. (3.8) is modified such that d is replaced byμ1/D and second order terms are introduced in both alternatives for vT in Eq. (3.9).When using vT, the magnitude of these terms should be considered.7

3.3 Examples of other particle geometries

In Paper 3 and Paper 7 the effect of a few particle shapes on kA and kB wereinvestigated. In Paper 3, the same S, ρp and Vp were chosen for the dispersionof particles of the different shapes. A comparison of the sensitivity in 1 − μT toconcentration can then be made merely by comparing μC for the different shapes.In the same way, the sensitivities in vT to increasing concentration can be com-pared using the E

(C2) values of the particles. In the case of particles which are

7For short cylinders the second order term relative to the first order term islb kμ

2 S v1/D/(a kBμL

). For long cylinders it is kμ2 S v1/D/ (kV). kμ is defined in footnote

5 on page 34.


large compared with the wavelength, one can instead use expected value μG andthe square mean E

(G2)of the random geometrical cross section G. Five types of

particles were considered: (i) spheres with the diameter b, (ii) cylinders with diam-eter b and length 2 b/3, (iii) cylinders with diameter b (2/3)1/2 and length b, (iv)cylinders with diameter 0.237 b and length 11.856 b and (v) cylinders with diameter3.218 b and length 0.0644 b. Thus, all particles have the same volume and all μGvalues contain a factor b2 and all E

(G2) contain the factor b4. Provided that the

orientation state is isotropic, calculation of μG and E(G2) for these particles gives:

μG/b2 E

(G2) /b4

i sphere 0.785 0.617ii compact cylinder 0.692 0.480iii compact cylinder 0.754 0.572iv slender cylinder 2.218 5.299v disc-shaped cylinder 1.908 6.773.

In these examples, the values for the compact particles are of the same order ofmagnitude, whilst the slender and disc-shaped cylinders have a more pronouncedeffect on the transmittance. The effect of the particles being nonspherical can beseen in the deviation of vG/μ2

G from 1. The effect can also be seen as a differencein the corresponding spherical diameter dc when the calculation is based on μG,denoted dc,G, and when it is based on E

(G2), denoted dc,G2 . Calculations for the

cylindrical particles give:vGμ2G

− 1 dc,G dc,G2

ii compact cylinder 0.001 0.939 0.939iii compact cylinder 0.008 0.980 0.981iv slender cylinder 0.078 1.680 1.712v disc-shaped cylinder 0.860 1.559 1.820.

This shows that the orientation effect is limited for compact particles, but thatvariations in T are to a great extent orientation-induced for a particle shapes suchas disc-shaped cylinders.

For spheres much smaller than the light beam diameter, the values for μC andvC are μC = Qd2 π/4 and vC = 0, which leads to

μT = 1− 3 aQS2 ρp d

(3.10)

vT = 3 π aS Q2 d

8 ρpCb, (3.11)

which is consistent with the results in Ref. [94]. Note that vT is proportional to d.Another class of particles treated in Paper 3 is particles much larger than the

light beam diameter. This is a limiting case giving μC = Cb and vC = 0 formany particle shapes. As an example a monodispersion of very large spheres with

3.3. EXAMPLES OF OTHER PARTICLE GEOMETRIES 37

diameter d� Db was used. Giving:

μT = 1− 3 aS2 ρp d

(3.12)

vT = 3 aS2 ρp d

, (3.13)

which shows that vT decreases with increasing particle diameter in this case. There-fore, the response function is different for spheres and slender cylinders.

In Paper 7 the response in vT to very small irregular particles (mineral fillersmuch smaller than the light beam) relative to the response to large cylinders havingl � lb and d � Db was studied. The model used for the small particles wasmonosized spheres having a random extinction efficiency.

Chapter 4

The random transmittance at higherconcentrations

Nec scire fas est omnia (We arenot granted to know everything).

Horace.

The results in Chapter 3 are valid only at low concentrations. In this chapter,that requirement is relaxed by presenting a lognormal model providing a consider-able extension of the validity range regarding concentration of the results obtainedin the previous chapter. The lognormal model can handle a concentration rangesimilar to that of the well known notion of turbidity. The name lognormal modelrefers to the result of the model, where the random regular transmittance T isdescribed as a lognormal random variable. It is obvious that Eq. (3.1) is only validas long as the cross sections Ci do not overlap. A negligible probability for overlapis expected as long as the product aS is small, e.g. by having a short path length,a low concentration or both. If aS increases, so does the overlap probability. Theconsequence of an increase in aS on the expected value μT, the variance vT andthe square of the coefficient of variation V 2 of the transmittance is the theme ofPaper 4, where the symbol p = aS is introduced1. The general shape of μT andvT as a function of p is shown in Fig. 4.1, where experimental data for a sampledispersion are shown as dots. Experimental estimates of μT and vT are in thisthesis written μT and σT

2. Initially both μT and σT2 change in proportion to the

concentration, but with increasing p the deviation from linearity increases. μTclosely follows an exponential reduction while σT

2 reaches a maximum and thendecreases. The non-linearity of these curves is primarily due to the overlapping ofparticle cross sections. If the concentration is increased even further, the effectsof multiple scattering, volume exclusion, particle interactions and different second

1See footnote 3.4 on page 32.

39

40 CHAPTER 4. . . . HIGHER CONCENTRATIONS

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � / ' � 0 � 1 2+ � + + + + � + + 3 + � + + 4 + � + + 5 + � + + � + � + � + + � + � 3

��

+ � + + +

+ � + + 6

+ � + � +

+ � + � 6

+ � + 3 +

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � / ' � 0 � 1 2

+ � + + + + � + + 3 + � + + 4 + � + + 5 + � + + � + � + � + + � + � 3

��

+

�

+ � �

+ � 4

+ � 3

+ � 5

7 � 8 7 � 8

Figure 4.1. Experimental statistical parameters of the transmittance. (a) showsthe sample mean and (b) shows the sample variance plotted against p, the productof concentration and light interaction path length. The sample is a dispersion ofthermomechanical pulp fibres in water. The dots indicate experimental values. Thefitted curves are derived from Eqs. (4.15) and (4.16). This means that two parametershave been determined in order to make the curves fit. The path length was 1 mmand the beam diameter was ∼0.1 mm.

order effects affect the shape of the μT and σT2 curves, c.f. Section 2.4.3. These

effects are not treated in this thesis.

4.1 The lognormal model

To deal with the situation of overlapping particle cross sections with increasing p, pis divided into a number of small non-overlapping intervals which correspond to thedivision of a into a number of layers perpendicular to the optical axis. The randomregular transmittance over the full path length is considered to be the product ofthe random regular transmittance through the individual layers. The layers arealso chosen thin enough to allow the use of the Eqs. (3.6) and (3.7), which bysubstitution of p for aS can be written:

μT = 1− kA p (4.1)vT = kB p . (4.2)

The variable p is 0 at the entrance of the beam into the measuring cell and p0at the exit. The interval [0, p0] is divided into m small intervals Δp = p0/m. Thetransmittance at p0 may then be written

T(p0) =m∏i=1

Ti(Δp) , (4.3)

where Ti(Δp) is the random transmittance through layer i.

4.1. THE LOGNORMAL MODEL 41

The expected value E [Ti(Δp)] and the variance var [Ti(Δp)] of Ti(Δp) may bewritten

E[Ti(Δp)] = 1− kA Δp (4.4)var [Ti(Δp)] = kB Δp . (4.5)

The random variable W is defined as W (p) = ln [T(p)] . Taking the logarithms ofboth sides of Eq. (4.3) gives:

W (p0) =m∑i=1Wi(Δp) . (4.6)

It is further assumed that the transmittance of each of the m non-overlapping layersis independent and has identical probability distributions. This implies that W (p)has independent increments. The Wiener process is the only nontrivial, regularand continuous stochastic process with the property of independent increments[95]. Moreover, if W (p) is a Wiener process then T(p) has a lognormal distribution[96, 97]. To identify the expected value E[Wi(Δp)] and the variance var[Wi(Δp)]for Wi(Δp), results from the theory of lognormal distributions are used, viz.:

E[Ti(Δp)] = exp{E[Wi(Δp)] + 12 var[Wi(Δp)]} (4.7)

var[Ti(Δp)] = exp{2E[Wi(Δp)] + var[Wi(Δp)]} ×(exp{var[Wi(Δp)]} − 1) . (4.8)

For sufficiently small Δp, the following approximation can be made:

E[Ti(Δp)] = 1 +E[Wi(Δp)] + 12

var[Wi(Δp)] (4.9)

var[Ti(Δp)] = var[W (Δp)] . (4.10)

Together with Eqs. (4.1) and (4.2)

E[Wi(Δp)] = −(kA + 1

2kB

)Δp (4.11)

andvar[Wi(Δp)] = kB Δp (4.12)

are obtained. The expected value and the variance of W (p0) at the exit point p0,E [W (p0)] and var [W (p0)], are the sums of the corresponding parameters for theindividual layers which result in

E[W (p0)] = −(kA + 1

2kB

)p0 , (4.13)

var[W (p0)] = kB p0 . (4.14)


Consequently, the random regular transmittance T = T(p0) at p = p0 has theexpected value

μT = exp(−kA p) (4.15)

and the varianceσT

2 = exp(−2 kA p) [exp(kB p)− 1] . (4.16)

The square of the coefficient of variation, V 2 = σT2/μT

2, becomes:

V 2 = exp(kB p)− 1 , (4.17)

which is a continuously growing function of kB p but independent of kA.When the relations in Eqs. (4.15) and (4.17) are used in instruments for particle

characterisation, measurement results proportional to kA p and kB p are usuallypreferred. For the average behaviour of T, the linear light attenuation is defined as

A = − ln(μT) (4.18)

which gives the relationA = kA p . (4.19)

Comparing with the deterministic Lambert-Beer-Bouguer law, A corresponds to2.303 times the absorbance (sometimes referred to as the Napierian absorbance) italso corresponds to the turbidity multiplied by the path length a.

For the fluctuating behaviour of T, a quantity ϕ can be calculated. ϕ is definedas

ϕ = ln(V 2 + 1) , (4.20)

which gives the relationϕ = kB p . (4.21)

In Paper 4, the name fluctuance2 was proposed for the quantity ϕ. In the same wayas V 2, ϕ can be determined without knowing the flux φ0 entering the measurementcell.

In addition, a phenomenological equation is used in situations where the re-quirements leading to Eq. (4.21) only are partially fulfilled. This equation can alsobe used in a situation where the exact relation between the amplifications in themean and variance estimating branches of the sensor electronics is unknown. Inthese situations Eq. (4.21) may give a nonlinear relation between ϕ and p. Anapproximately linear relation may then be obtained using:

ϕν = ν ln(V 2

ν+ 1

), (4.22)

where ν > 0 is a parameter which is adjusted in order to make ϕν approximatelyproportional to p. The initial slope of ϕν is the same as for ϕ and is independent

2A drawback with the introduction of the word fluctuance here is that the word is used inother disciplines such as dermatology.

4.2. COMPARISON WITH OTHER THEORETICAL MODELS 43

of the chosen value of ν. An example of the use of this equation is found in Section4.3.

A comprehensive discussion of the theoretical results for higher concentrationsis found in Paper 4. Some of the discussion is related in this section. An interestingfeature of dividing the variable p in a number of layers Δp instead of dividingthe path length a into a number of Δa is that it takes care of any variations inthe particle concentration along the optical path through the sample. In Paper 4,it is explained how the upper limit of p in the low-concentration theory may bedetermined provided that kA and kB are known.

A requirement of the lognormal theory is that the parameters kA and kB must beindependent of p. This means that variables such as particle dimensions, extinctionefficiency, orientation, density and the beam cross section need not be affected bychanges in p. This is also true for the way in which the positions of the particlesare randomised. A higher concentration may mean a higher probability for theparticles to form flocs. Note also that polydispersity may introduce a term in kBthat depends on p.

The main objection to the continuous lognormal model presented here is thatthere is a small probability that T exceeds unity. This non-physical behaviour isnot a severe problem for two reasons. (1) It is only significant for very low valuesof p, and for these values of p the low-concentration relationships can be used; and(2) the mean value μT never exceeds unity.

4.2 Comparison with other theoretical models

Already in the 1950s, Hubley, Robertson and Mason [98, 99] used the AC and DCcomponents of the detector signal from a transmission optics set-up in order tostudy flocculation of large particles (pulp), but it was not possible to trace whattheory was used, thus it is uncertain how their work was supported theoretically.No comparison with the theory presented in this thesis can therefore be made.

However, in this section, three other theories are briefly described for comparisonwith the theory presented in Paper 3 and Paper 4. These are:

1. the Lambert-Beer-Bouguer approach based on a deterministic model

2. the TP theory developed in 1975 by Pettersson, Lundqvist and Fladda [41].TP is the acronym for Total Power referring here to the use of the meanand variance, which carry the total power of all frequency components of arandom function

3. the theory for monosized spheres by Gregory published 1985 [94].


4.2.1 Lambert-Beer-Bouguer approachThe traditional way to extend the concentration range is the Lambert-Beer-Bouguer[100] model, which can be derived from the deterministic form of Eq. (3.1)

τ = 1− 1Cb

(c1 + c2 + · · ·+ cn) , (4.23)

where ci is the deterministic attenuation cross section and n is the deterministicnumber of particles interacting with the light beam. If all cross sections are equalin area and if the number concentration is defined as SN = n/(Cb a), Eq. (4.23) fora thin layer dz can be written

τ = 1− SN dz c , (4.24)

i.e. the difference in transmittance over dz is

dτ = −SN c dz . (4.25)

The solution to this simple differential equation for a path length a using theboundary condition that τ = 1 for z = 0 is

τ(a) = exp(−SN c a) , (4.26)

where SN c is the turbidity. A great advantage of this approach is that the interpre-tation of the cause of the radiation attenuation is maintained. The negative termin Eq. (4.24) for low concentrations is simply converted to a negative exponent forhigher concentrations.

Note that both the Lambert-Beer-Bouguer’s law and the lognormal model arecontinuum models although the scattering particles are discrete. Usually there is nodisadvantage in using a continuum model here. The important difference betweenthe models is that the Lambert-Beer-Bouguer model does not model the variance,whereas the lognormal model gives the variance and the coefficient of variationexplicitly.

At still higher concentrations, Eq. (4.25) does not apply because the scatteringalso introduces a considerable radiation flux in the −a-direction. In each infinites-imal layer da, the flux is both subtracted from and added to the forward andbackward fluxes. In this case the theories of Schuster [101] and Kubelka and Munk[21–23] are often used, although more contemporary theories exist. Note that thementioned theories are also continuum models and that the variance of the trans-mittance is not modelled.

4.2.2 The TP theoryThe theory here referred to as the TP theory got its name from Total Power, whichwas a reference to the fact that both the DC and the RMS components of the

4.2. COMPARISON WITH OTHER THEORETICAL MODELS 45

detector signal were used. DC and RMS is another way to express the mean andthe standard deviation of the signal. The only available reference of this theoreticalmodel is a part of a master’s thesis written in Swedish by Lundqvist [41]. Sincethat text may not be easily accessible, a presentation of the relevant parts of theTP theory is included in Appendix E.

The TP theory assumes a flow-through cell where the suspended particles pass aparallel light beam having a square cross section. The derivation results in equationssimilar to Eqs. (4.15) to (4.17) and coefficients similar to kA and kB. A comparisonbetween the TP theory and the lognormal model shows that the differences betweenthe two models are that T is regarded as a snapshot in the lognormal model andas a random function of time in the TP theory, that the shape of the beam crosssection differs, and that the effect of the random orientation of the particles only isaccounted for in the lognormal model, see Appendix E for details. The conclusion isthat the lognormal model verifies the essential conclusions of the TP theory, whichwas one of the objectives of this work.

4.2.3 Gregory’s theoryAn approximate alternative to the lognormal model was presented by Gregory [94].In short, he assumes that Eq. (4.26) is true also for the case of a random numberof particles. He also assumes that the particles are identical spheres so that allextinction cross sections are the same. Exchanging in Eq. (4.26) the variable n forthe random variable N and the variable τ for the random variable T = τ(N), theexpected value μT and the standard deviation σT of T were calculated by Gregoryas

μT ≈ τ(μN ) , (4.27)

σT ≈ 12

[τ(μN + σN )− τ(μN − σN )] , (4.28)

where μN and σN are the expected value and the standard deviation of N . Further,N is considered to have a Poisson distribution with the expected value and varianceequal to SN Cb a. This method leads to approximate expressions for the expectedvalue and the variance:

μT ≈ exp(−SN c a) , (4.29)

σT ≈ exp(−SN c a) sinh[(SN Cb a)1/2 c/Cb

]. (4.30)

The square of the coefficient of variation is then:

V 2 = σT2/μT

2 ≈ sinh2[(SN Cb a)1/2 c/Cb

]. (4.31)

However, this method requires that the cross section c is the same for all particles.Therefore, it cannot be used if the particles are non-symmetrical or polysized insuch a way that the variation in T is due to a significant extent to variations in theextinction cross sections.


4.3 Two experimental studies

Paper 4 presents two experimental studies. These are given to illustrate the de-gree of agreement between the theoretical model and the observed random regulartransmittance of flowing particle dispersions. The first study illustrates a goodagreement, whereas the second illustrates a less ideal situation.

In both studies, dispersions of fibres from paper-making pulps were used. Inthe first study, the sample was taken from a thermomechanical pulp (TMP) whereScandinavian spruce was used as raw material, c.f. Fig. 2.1(c). In the second studya mixture of two bleached kraft pulps was used. One of them originated from pineand the other from birch and they were mixed in equal proportions, c.f. Fig. 2.1(d).

4.3.1 Experimental details

From a container, a diluted dispersion of fibres was fed to the flow-through cell of thetransmission sensor via a dispersion unit in order to ensure an even fibre suspensionfree of flocs. The uncertainty of the concentration value based on filtering, dryingand weighing was approximately ±5 %.

The transmission sensor with a layout as in Fig. 2.10(b) had plane parallelwindows. The distance between the windows, i.e. the path length, was 1 mm.The radiation beam was created by focusing a 0.1 mm diameter radiation sourcehaving the wavelength 860 nm in the middle of the glass cell. The half cone anglewas 3.4◦. The detector was a Si diode with an active diameter of 0.1 mm. Thebandwidth of the detector-amplifier combination was about 70 kHz. The flow-rateof the dispersion was approximately 1 m/s.

The variance of the detector signal was estimated using an analogue RMS esti-mating circuit preceded by a high-pass filter (cut-off frequency 100 Hz). The outputfrom this circuit was a low frequency signal that was sampled by a computer andsummed. The expected value of the detector signal was estimated by letting thesignal pass a low-pass filter followed by computer sampling and summing. Thesampling frequency was 1 Hz and one estimate was based on 30 samples. Clearwater was used as the τ = 1 reference and τ = 0 was adjusted with the light sourceturned off.

4.3.2 Study 1

In study 1, the thermomechanical pulp (TMP) had a concentration between 0.7and 11.5 kg/m3. The particles in a dispersion of TMP are usually divided intothree fractions or particle types: (i) a fibre fraction consisting of intact or almostintact cellulose fibres, (ii) a middle fraction consisting of fractured and damagedfibres of intermediate size and, (iii) a fines fraction consisting of irregular smallparticles. Since the lignin is not removed, the fibres are (partly) coated with lignin[see Fig. 2.1(c)] and are more light absorbing than bleached fibres.

4.3. TWO EXPERIMENTAL STUDIES 47

Plots of the experimental statistical parameters μT and σT2 against p are shown

in Fig. 4.1 on page 40 together with curves based on Eqs. (4.15) and (4.16). Thecurves were obtained by using the averages of the experimental values of kA andkB which may be calculated at each value of p using Eqs. (4.19) and (4.21). Theseaverages were 250 and 25.6 m2/kg for kA and kB respectively and the individualvalues showed no obvious dependence on p. The order of magnitude of the averagesis reasonable considering the parameters included in kA and kB. Because of thegood agreement between experiments and theory in Fig. 4.1, there was also anagreement between A, V 2 and ϕ and their theoretical dependence on p.

4.3.3 Study 2

In study 2, the particles used were a mix of fibres from two bleached kraft pulps.The concentration range was 0.6 to 9.0 kg/m3. The fibres had a size distributionwhich resembles the population of fibres in the wood chips much more than the sizedistribution of the TMP particles did, see Fig. 2.1(d). Any lignin remaining on theparticles was removed chemically by bleaching. This makes these fibres flexible,which may cause the lumen to collapse and make them take on a band-like shape.The bleaching also makes the fibres transparent and imparts to them a greatertendency to form flocs. The population was complicated by the fact that the fibresoriginated from two types of wood.

Fig. 4.2 shows the experimental values μT and σT2 plotted against p together

with curves fitted to the data in the same way as in study 1. Because of a de-pendence of kB on p, it was not possible to obtain a better fit with this averaging

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � / ' � 0 � 3 2

+ � + + + + � + + 3 + � + + 4 + � + + 5 + � + + � + � + � + + � + � 3

��

+ � +

+ � 3

+ � 4

+ � 5

+ � �

� � +

� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � / ' � 0 � 3 2

+ � + + + + � + + 3 + � + + 4 + � + + 5 + � + + � + � + � + + � + � 3

��

+ � + +

+ � + �

+ � + 3

+ � + 9

+ � + 4

7 � 8 7 � 8

Figure 4.2. Results from Study 2. Experimental values μT (a) and σT2 (b) of the

expected value μT and the variance σT2 of the transmittance T plotted against p,

the product of path length and concentration. The sample is a dispersion of bleachedkraft pulp fibres in water. The large dots are the experimental values and the solidcurves are fitted to the data on the basis of the averages of the experimental valuesof kA and kB, which were determined using Eqs. (4.19) and (4.21). For comparison,the corresponding curves from study 1 are plotted as dotted curves.


� � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � / ' � 0 � 3 2

+ � + + + + � + + 3 + � + + 4 + � + + 5 + � + + � + � + � + + � + � 3

��

�

+ � + +

+ � + 6

+ � � +

+ � � 6

+ � 3 +

+ � 3 6

+ � 9 +

+ � 9 6

��

� � � � � � � � ��

� � �

Figure 4.3. Comparison between ϕ defined in Eq. (4.20) and ϕν defined inEq. (4.22). The sample is a dispersion of bleached kraft pulp fibres in water. Thesolid dots are the experimental ϕ values and the circles are the experimental ϕνvalues. The line fitted to the ϕ data is based on the solid curve in Fig. 4.2(b) andthe curve fitted to the ϕν data is a regression line forced to pass the origin.

method. Consequently, V 2 and ϕ fitted poorly to their theoretical relationshipswith p.

There may be at least three reasons for the deviation. The first is that some ofthe parameters that kB relies on, such as Q, μΛ or Db, depend on p. The secondreason is that polydispersity may cause kB to increase with p, as is discussed inPaper 3. A third reason may be some dependence between the transmittances ofdifferent virtual layers.

However, study 2 was included to illustrate that it is possible to use Eq. (4.22)in this case. The parameter ν was adjusted in order to make ϕν approximatelyproportional to p. In Fig. 4.3, Eq. (4.22) was applied to the data of study 2 withν = 0.171. Note that the concentration range, for which ϕν is approximatelyproportional to p, is larger than the corresponding range for ϕ.

Chapter 5

The random transmittance for intermediateparticle lengths

neko no me no youLike a cat’s eye – rapid in changeand unpredictable.

Japanese saying.

5.1 Connecting special cases

The lognormal model has this far been restricted to two special cases for the be-haviour of the variance vT and the square of the coefficient of variation V 2 of thetransmittance T through suspensions of slender cylindrical particles having a diam-eter much greater than the wavelength. The two special cases concern the relationbetween the length of the cylindrical particle and the diameter of the optical beamand are seen in the expression for the coefficient kB in Eq. (3.9), which is also be-hind the shape of the length response curves in Fig. 3.1. Note that the coefficientkA is not affected by the relation between the particle dimensions and the opticalbeam diameter Eq. (3.8). It was from the beginning assumed that the transitionbetween the special cases is smooth, but for convenience the simple model, justconnecting the two cases by straight lines, was used in applications. In order toestimate the error made by this simplification, an investigation was carried out forintermediate lengths of the model particles. The investigation was performed bynumerical simulation and is presented in Paper 5. In order to validate simulationresults these were, wherever possible, compared with the theoretical predictions ofμT, vT and V 2. In addition, an alternative way to calculate the geometrical cross

49

50 CHAPTER 5. . . . INTERMEDIATE PARTICLE LENGTHS

sections was also developed in Paper 6. It was implemented and compared withthe algorithm used in Paper 5. This comparison of algorithms is found in Paper 6.

5.2 Method

Fundamental for the intermediate length simulations was Eq. (3.1), but in orderto handle overlapping particles, it had to be changed to a more general form sinceoverlapping particles violate the additivity assumption upon which it is based. Thesample value τ of T is calculated as

τ = 1− 1CbU (c1, c2, . . . , cn) , (5.1)

where U (c1, c2, . . . , cn) is the union of the different extinction cross sections ci areason the xy plane, and Cb is the cross section of the optical beam. In the special caseof non-overlapping cross sections, the equation converts into Eq. (4.23).

In the simulation, sample values τ of T were generated by generation of samplevalues n of N and ci of Ci. The τ values were generated a number of times suchthat μT, σT

2 and V 2 could be calculated. The number of interacting particles,n, were drawn from a Poisson distribution having a concentration-dependent meanvalue1. The generation of the optical cross sections ci, here considered to be anarea on the xy plane, was made by randomly generating the particle length, width,orientation and position of the particles.

The simulation model is thoroughly described in Paper 5 and Paper 6. In themodel, a number of instantaneous states, characterized by the number of particlestheir positions and orientations, were generated inside the transmission sensor,shown in Fig. 5.1. The figure is another way to view the idealised sensor in Fig.2.10(a). The different steps of the simulation model were:

a) The number of particles n in each test was generated as independent samplesof a Poisson(μN ) distribution, where μN was determined from the numericalconcentration given in Eq. (2.6) and the volume A0 a, where A0 is an areaperpendicular to the optical axis somewhat larger than the beam cross section,Fig. 5.1. The number of tests in each simulation was chosen such that thetotal number of particles was about 3000 in most simulations.

b) The optical cross section ci for each cylindrical particle was generated con-sidering the limit defined by the rim of the beam. The extinction efficiencyQ was applied under item e) below since Q was assumed to be the same forall particles. Thus, only the geometrical cross section was generated at thisstage of the simulation. The simulated particles were distributed according toa uniform statistical distribution within a virtual box centred on the optical

1An important feature of the Poisson distribution is that its variance is the same as its meanvalue.

5.2. METHOD 51

( � � � � � � � ��

& � � � ��

* � � � � ��

: � � � � � � � � � � � � � � � � � � � � � � � � � �� %� � � � � � � � � � ; � �� # � � � �� # � � � � � � # � � � # � �

�

� �

� � � � � � � � � � � �

�

� : � � � � � � � � � ��

�

Figure 5.1. Model of the transmittance sensor. The sample particles were placedat fixed but randomly chosen positions and orientations in a cell with a larger crosssection than the beam. The extension of the cell area A0 must provide the possibilityto generate particles just reaching the rim of the beam from outside. The area A0was centred on the axis of the light beam. Cartesian coordinates x, y and z weredefined such that z coincided with the optical axis.

beam and having the dimensions A0 a. The position of the particle was con-sidered to be the position of its centre point. In order to handle overlappingparticles in an appropriate way, the geometrical cross section of the particleswere inserted into a digital image coinciding with A0. The azimuth and po-lar angles were randomly generated according to the orientation states flator isotropic, c.f. Section 2.5.5 and Appendix A. The extension of the crosssection was adjusted such that only the part of the cross section within thebeam contributed to the transmittance calculation.

c) The summation of the cross sections was made such that overlapping crosssections were taken into consideration. In the digital image used for thesummation, the pixel value 0 means that no particle was shielding the lightin the corresponding position. For each particle “covering” the pixel its valuewas increased by 1.

d) The fraction fA of the beam cross section covered by particle cross sectionswas calculated using the number Gp of pixels having a value exceeding 0 andthe beam cross section area expressed in pixels Cb,p such that

fA = Gp

Cb,p. (5.2)

e) The instantaneous transmittance τ , the simulated sample of T, was calculatedas

τ = 1−QfA , (5.3)


where theoretically, Q = 2 for particles which are large compared with thewavelength of the optical radiation used, provided that the incident beamis parallel and the acceptance angle of the detection system is 0, i.e. onlylight parallel with the z axis is recorded. In Paper 5, Q was chosen to unity,corresponding to a small finite acceptance angle.

The steps a)–e) were repeated a number of times letting the number of parti-cles vary according to the statistical distribution of N . Values τ from each testwere stored and the experimental mean and variance were calculated. A separateverification of the software was also made and can be found in Paper 6, where aprocedure involving the implementation of an alternative algorithm is described.

5.3 Simulations and results

The simulations can be divided into two categories, viz. simulations with the pur-pose to validate the simulation technique, and simulations showing the responsefunction, V 2/S as a function of particle length, for a length range which includesintermediate particle lengths. The response function is usually normalized suchthat it is unity for very long particles, i.e. it is expressed in kBL aS units.

5.3.1 Comparisons with the theoretical predictionsThe simulated expected value of μT was consistent with the following importanttheoretically predicted properties of μT:

(i) μT decreases with concentration,

(ii) μT is proportional to 1/d,

(iii) μT depends on the orientation state of the particles,

(iv) μT is independent of the particle length,

(v) μT is independent of the beam diameter Db.

The square of the coefficient of variation, V 2 = σT2/μT

2, also has easily distin-guishable theoretically predicted properties. The simulation technique was testedon the following properties and found consistent with the theory.

(i) V 2 is initially almost linearly dependent on concentration,

(ii) V 2 is independent of the particle width d,

(iii) V 2 depends on the orientation state,

(iv) V 2 depends on the particle length as described in Fig. 3.1,

(v) V 2 depends on the beam diameter Db.

5.3. SIMULATIONS AND RESULTS 53

The properties of ϕ are the same except that the initial linearity (property i) isextended to larger values of kB S a.

5.3.2 The response function of V 2 and ϕThe main result from Paper 5 is an approximate response function fitted to thesimulation data for V 2 and thereby also for ϕ. It covers the full range of particlelengths which can be used when evaluating deviations between the simple responsefunctions and the real response in different applications. A cubical Bézier function[102] was chosen as the general form of the response function since the parametersof the Bézier function are easily interpreted. Therefore, it is easy to formulatesuitable conditions for these parameters.

The determination of the approximate response function was made in threesteps.

1. Simulated response functions for five different lb values, examples of whichare shown in Fig. 5.2, were fitted to individual Bézier functions.

2. The fitted functions were modified such that values were limited not to exceedunity and regions having positive curvature were replaced by linear interpo-lations.

3. The length scale of each function was modified making the position of lb forthe different curves to coincide. Excluding outliers, the set of functions wasused as input to a new curve fit of a common Bézier function approximatelyvalid for all simulations.

The fitted response curve resulting from step 3 above expressed in the parameterform of the Bézier curve is described as(

l/lbϕ/kBL

)= (1− t)3

(00

)+ 3 t (1− t)2

(0.690.98

)

+3 t2 (1− t)(

2.300.92

)+ t3

(7.21

), (5.4)

where the parameter t is chosen such that 0 ≤ t ≤ 1. For l > 7.2 lb, ϕ/kBL ≡ 1.kBL is the kB value for very long particles. The Bézier curve fit reasonably to thesimulations and can therefore be used when estimating measurement errors causedby the use of the response function of the simple model. The fitted and the simplemodels are compared in Fig. 5.3.


0 1 2 3 4 50

0.5

1

1.5Orientation state = Flat

Particle length [mm](a)

Res

ponc

e ϕ r

l1

↓

l2

↓

l3

↓

l4

↓

l5

↓ D

b = 0.1 mm

0.51.0

2.0

3.0

SimulatedSimple theoretical

0 1 2 3 4 50

0.5

1

1.5l1

↓

l2

↓

l3

↓

l4

↓

l5

↓

Particle length [mm](b)

Res

pons

e ϕ r

Orientation state = Isotropic

Db = 0.1 mm

0.5

1.02.0

3.0

SimulatedSimple theoretical

Figure 5.2. Response functions for ϕ as a function of particle length of the trans-mittance of beams having different diameters through a dispersion of cylindricalparticles having a diameter of 10 μm and different orientation states. In (a) theorientation state is flat and in (b) it is isotropic. The simulated values (solid lines)are compared with the simple model (dotted lines). In this figure the general lengthdependence and the dependence on the lb (which is closely related to the beam diam-eter Db) of ϕ are exemplified. The response function is defined as ϕr = ϕ/(kBL aS)making the curves tend to unity for long particles. Five different beam diameters(0.1, 0.5, 1.0, 2.0 and 3.0 mm) were used. The different beam diameters result infive lb values for each orientation state and which are denoted l1, l2, l3, l4 andl5. The concentration and the particle diameter were constants in this simulation.Despite the irregularities in the simulation curves for long particles, the simulationsexhibit the same general behaviour as the simple model with the exception that thetransition from short particles to long particles is smooth in the simulation.

0 0.5 1 1.5 2 2.50

0.2

0.4

0.6

0.8

1.0

1.2

Fitted and simple responses for ϕ


ϕ in

kB

L a S

uni

ts

kBL

a S

lb

Simple modelFitted model

Figure 5.3. Comparison of the response function for ϕ for the simple and the fittedmodels.

Chapter 6

Measurement principles based on randomtransmittance

To measure is to know.

Lord Kelvin

In this chapter, measurement principles aiming to assess the four different prop-erties of a particle suspension are presented. They are all based on the behaviourof the random regular transmittance. The assessed properties are:

• Concentration: Two apspects of the concentration meter are considered,viz. the possibility to measure concentration of slender cylindrical particlesindependent of their length and width, and the possibility measure concentra-tion without interference from under-sized particles such as fines and fillers.The concentration meter is built on a transmission sensor having a very thinoptical beam. The idea was implemented and used e.g. in the measurement ofconcentration in the short circulation of paper machines making fine paper.

• Length classification: The possibility to use a set of transmission sensorshaving different beam cross sections for fibre length classification was firstmentioned in [41] and the patents are from 1979 and 1982 [46, 47]. The clas-sifier used here had three transmission sensors with different beam diameters.The usefulness of this measurement principle was shown e.g. in Paper 1 andthe theoretical base is derived in Paper 3 and Paper 7.

• Mean length: For a transmission sensor having a beam with a large crosssection, ϕ/S is a measure of the particle length [41] as long as the particlesare shorter than the beam diameter. Early attempts to use this involved aratio between ϕ and A [44, 45]. However, a superior way to obtain a particlelength signal is to take the ratio between the ϕ values of a wide beam sensorand a fine beam sensor, where the latter serves as a concentration measure.Patents describing this solution are from 1985 and 1988 [48, 49].

55

56 CHAPTER 6. MEASUREMENT PRINCIPLES . . .

• Mean width: Already in [41] it was stated that S/[− ln(μT)] is a measureof the fibre diameter. In Paper 7, it is shown that S is determined as the ϕvalue from a transmission sensor with a small beam cross section. Patentsdescribing this measurement principle are from 1985 and 1988 [48, 49]. Themeasured quantity is here referred to as the width instead of the diameter fortwo reasons: the particles of interest were not circular cylinders and when theparticle population is polydisperse with respect to the diameter, the measuredquantity is not the mean diameter even if the particles were circular cylinders.It is the reciprocal value of the mean of the reciprocal value of the diameter.

6.1 Materials and methods

6.1.1 Sample set

Throughout this chapter, the measured data refers to a set of samples that wasused when testing the measurement principles. The samples are different types ofwood pulps. The set was put together with the ambition to include samples of agreater variety than can be expected in e.g. an online position in a pulp productionunit. The set of samples is further described in Appendix C.

6.2 Triple-beam unit

In the experiments described in this thesis, two types of optical units were used.One had three optical beams with different diameters. The other had a single beam.

In the triple-beam unit, the dispersion of fibres passes three transmission sensorswith different beam diameters corresponding to three lb values, l1 < l2 < l3, whichwere chosen to suit the fibre population of interest. For cellulose fibres a reasonablechoice is l1 ≈ 0.1 mm, l2 in the range 0.5–1.5 mm and l3 in the range 3–6 mm.Here (l1, l2, l3) = (0.1, 0.8, 3.6) mm.

The outputs are indexed in accordance with the beam numbers, viz. ϕ1, ϕ2 andϕ3, and so are the calibration constants k1, k2 and k3, where ki = 1/(kBL, i a),where the symbol kBL denotes the kB value for l � lb. The orientation state ofthe particles is assumed to be the same in the three transmission sensors. Thetriple-beam unit is shown in Fig. 6.1(a). For convenience, the three transmissionsensors are referred to as the fine, middle and coarse channels.

Before reaching the triple beam unit, the sample was stirred and diluted asindicated in Fig. 6.1(b). The triple beam unit was placed under the tank containingthe diluted dispersion and the hydrostatic pressure caused the dispersion to flowthrough the cell. The cell of the triple beam unit was 10 mm wide in the opticaldirection. The fine channel had a layout as in Fig. 2.10(b) and is already describedin Section 4.3.1 on page 46. The middle and coarse channels had a layout as inFig. 2.10(a) and were apart from that similar to the fine channel.

6.3. SINGLE-BEAM UNIT 57

The limit angle for scattered light to reach the detector in the fine channel was6.8◦ (the cone angle of the optical beam). This means that the observed extinctionefficiency Qobs was considerably smaller than the theoretical Q value. It also meansthat Qobs will decrease somewhat with increasing particle diameter. In the middleand coarse channels, the limit angle for scattered light to reach the detector wasmuch smaller. If the optics were otherwise ideal, the limit would be 0.3◦ whendetermined from the finite size of light source and detector. Consequently, Qobswas close to the theoretical Q value and its dependence on d was small.

A detailed investigation of Qobs is not made here since the measurement appli-cations were calibrated using other laboratory methods. For similar reasons it is notalways necessary to determine any difference in the mean and in the RMS-branchesof the signal-processing electronics since only the calibration constants and the νvalues are affected.

� � � � � � � � �

� � � � � � � � � � '

� � � � � �

� � � � � � � � � � � �

� � � � � �

� � � � � � � �� # � � � �

(a) (b)Figure 6.1. The triple beam system used in several of the experiments in Paper7. (a) shows the triple beam optical unit, where three optical transmittance sensorshaving different beam diameters are arranged along a common flow-through samplecell. The cylindrical body of the unit has a diameter of 10 cm and a height of 21 cm(b) shows the principle of the triple beam measurement system.

6.3 Single-beam unit

In order to include wet-end samples from a paper machine without dilution, aninstrument having a shorter path length (1 mm) was constructed. This allowed usto increase the range of concentration measurements. The instrument was equipped


with the fine channel optics. To make the sample homogeneous and to ensure thatthe suspension flowed smoothly through the cell, it passed through a dispersionunit immediately before the cell. This also had the advantage of minimising therisk of re-flocculation.

6.4 Triple beam unit used as instrument model

Numerical calculations were carried out to show the effect of using the more realisticfitted response model instead of the simple model, which were introduced in Chapter5 and are illustrated in Fig. 5.3. In these calculations, a triple beam instrumentmodel was used. It had a somewhat different set of beam diameters than the realtriple beam unit. The lb values were chosen such that (l1, l2, l3) = (0.1, 1.0, 3.0)mm. When a variable refers to either of the simple or the fitted models, suffixes Sand F were used, e.g. ϕ1S and ϕ1F. Together with the instrument model, a specificparticle length and width distribution was used. The length distribution is shownin Fig. 6.2.

6.5 Concentration measurement

6.5.1 Total concentrationThe most obvious demand on a concentration meter is that it shall give a con-centration measure which is equally sensitive to the different particle types in thesuspension of interest, i.e. the total concentration. Although this is usually not ful-filled completely, the concentration meter presented here is insensitive to particlelength and width over a considerable range.

To understand the principle, consider ϕ from the fine channel or the single beamunit. Fig. 6.2 shows that the response of ϕ to concentration is constant for a widerange of particle lengths. An example of the probability density function (PDF) forthe particle lengths of a sample of cellulose fibres is shown in the same figure. Theresponse is constant for almost all fibres of the sample population. At the sametime ϕ is insensitive to the width of the fibres. This means that a concentrationsensor built on this ϕ signal is independent of particle length for almost all particles,provided that the following conditions are met:

• The beam must be thinThe majority of the particles must have a length greater that the beam di-ameter, i.e. l > Db.

• The particles must be thinFor Eq. (3.9) to be valid it is required that d � Db. Particles that are toothick will cause a problem.

• Common kBL.

6.5. CONCENTRATION MEASUREMENT 59

0 2 4 6 80

0.2

0.4

0.6

0.8

1.0

Particle length [mm]

Res

pons

e fu

nctio

ns

Pro

babi

lity

dens

ity fu

nctio

n [m

m−

1 ]

↑The response

curves when lb = 0.1 mm

← PDF

0

0.1

0.2

0.3

0.4

0.5

Figure 6.2. The concentration meter characteristics. Response functions for k1 ϕS1(solid line) and k1 ϕF1 (dashed line) plotted together with a weighted probabilitydensity function, PDF, of a sample of wood pulp fibres (dotted line, y-axis to theright). The index 1 refers to a transmission sensor having a thin beam. The responseis independent of particle length for almost all fibre lengths represented in this type ofsample. The PDF, which only serves as an example, was determined using a KajaaniFS-200 fibre length meter, which gives the number of particles at each length. Thiswas converted into a PDF for the concentration (weight/volume) using a simpleassumption about the dependence of the width on the length.

It is not enough that kBL, the kB value for long particles, is the same for allparticle lengths, variables upon which it depends must also be constant. Suchvariables are the observed extinction efficiency Qobs, the orientation state andthe density of the particles

Fig. 6.2 shows the response curve of both the simple model (k1 ϕS1) and thefitted model (k1 ϕF1). The measurement error due to the response curve dependson the fraction of particles having a length close to the beam diameter. For thegiven example the error is about 1 %.

The concentration measurement principle was tested on the set of wood pulpsamples. The set was divided into to subsets, one with only fully bleached pulpsand the other with unbleached and mechanical pulps. The results are shown inFigs. 6.3(a) and 6.4(a). In Figs. 6.3(b) and 6.4(b) the linear light attenuation isalso shown as a function of concentration. It is obvious that the use of ϕ forconcentration measurements is a great improvement compared with the use of thelinear light attenuation, which is used in turbidity sensors.

6.5.2 Concentration measurement with small particle suppressionThe low sensitivity to small particles may be utilised as a possibility to suppresssmall particles in the concentration measurement. However, in e.g. the case ofthe fines fraction of a papermaking pulp or in the case of filler particles added tosuch a pulp, the smallest particles in the dispersion are irregular and cannot bedescribed as slender cylinders with a diameter larger than the wavelength. On thecontrary, these particles are sometimes used to improve the light scattering of the


0 0.2 0.4 0.6 0.80

1

2

3

4

5

Concentration [kg/m3](a)

ϕ 1 [arb

. uni

ts]

pinebirchline fit

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Concentration [kg/m3](b)

Line

ar li

ght a

ttenu

atio

n

pinebirch

Figure 6.3. Test of the concentration measurement principle on fully bleachedwood pulp samples. Measurements were made using the fine channel of the triple-beam unit. The samples were from group 1 (pine) and group 2 (birch) described inAppendix C. (a) the fluctuance ϕ1 and (b) the linear light attenuation A plottedagainst manually determined concentration.

0 0.2 0.4 0.6 0.80

1

2

3

4

5

6

Concentration [kg/m3](a)

ϕ 1 [arb

. uni

ts]

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

0.8

1

Concentration [kg/m3](b)

Line

ar li

ght a

ttenu

atio

n

� � � � � � 6

� � � � � � 4

� � � � � � 9 � � � � � � 5

� � � � � � <

� � � � � � ��

Figure 6.4. Test of the concentration measurement principle on unbleached andmechanical wood pulp samples. Measurements were made using the fine channelof the triple-beam unit. The samples were from group 3 to group 8, described inAppendix C. (a) the fluctuance ϕ1 and (b) the linear light attenuation A plottedagainst manually determined concentration.

6.6. LENGTH CLASSIFICATION 61

0 5 10 150

1

2

3

4

5

6

7

Total particle concentration [kg/m3]

k 1 ϕ1 [k

g/m

3 ]pulppulp+claypulp+clay

Figure 6.5. The discrimination of small particles in concentration measurementusing the k1 ϕ1 concentration meter signal is illustrated with papermaking pulp aslarge particles and filler clay as small particles. Clay particles of this type are usedas filler in fine papers. At two pulp concentrations, clay was added without changingthe pulp concentration. k1 ϕ1 is plotted against manually determined concentration.The small clay particles add to the total concentration but have almost no effect onthe k1 ϕ1 signal. Measurements were made using the single-beam unit.

paper and the size is therefore chosen to be comparable with the wavelength oflight. Therefore, the response curve in Fig. 6.2 does not apply. Consequently, asimple theoretical analyses is carried out in Paper 7 enabling a comparison of thesensitivity of ϕ and A to two hypothetical samples, one corresponding to fibres andthe other corresponding to fillers. It was concluded that the sensitivity of ϕ to thefiller is only 0.006 to that of fibres, which gives a very good ability to suppress thefillers in the measurement of fibre concentration, Fig. 6.5. This is a fundamentalobservation for the application in Paper 2. In contrast, the light attenuation, A, isabout 25 times more sensitive to filler than it is to fibres.

6.6 Length classification

When the work with the optical particle length classifier started, the Bauer-McNettclassifier [103], Fig. 6.6, was the equipment usually used for the purpose of particlelength classification in the pulp and paper industry. The Bauer-McNett classifierincludes a number of sieving stages through finer and finer sieves. The sieving rejectof each step is filtered, dried and weighed. The finest fraction, the sieving acceptof the finest sieve, is usually estimated as the difference between the dry weight ofthe sample added to the apparatus and the sum of the dry weights of the rejectfractions.

The purpose of the development of an optical length classifier was to be ableto make a similar characterisation of the sample but online in the manufacturingprocess, something the Bauer-McNett apparatus is not adapted to.

The optical length classifier gives the length distribution of the particles as theconcentration of particles in three length classes. Its principle is based on three


Figure 6.6. The Bauer-McNett apparatus, which is a cascaded sieving system forthe separation of pulp fibres into different length classes. The insert shows a smallpart of one of the sieves as compared with a match.

linear combinations of ϕ1, ϕ2 and ϕ3 from the triple-beam unit

Sshort = l2l2 − l1 (k1 ϕ1 − k2 ϕ2) , (6.1)

Smedium = l1l1 − l2 k1 ϕ1

+ l2 (l1 − l3)l22 − l2 l3 − l1 l2 + l1 l3

k2 ϕ2

− l3l3 − l2 k3 ϕ3 , (6.2)

Slong = − l2l3 − l2 k2 ϕ2 + l3

l3 − l2 k3 ϕ3 , (6.3)

where Sshort, Smedium and Slong correspond to the concentration of short, mediumand long particles respectively.

For this measurement principle, the difference between the simple and the fittedmodels is large. The response functions for the three length classes calculated fromboth the simple and the fitted models are shown in Fig. 6.7 together with thesample length distribution. The two models for the optical length classifier clearlydepend on the choice of model for the individual transmittance sensor. If therelative concentrations in the length classes, e.g. Sshort/(Sshort + Smedium + Slong),are calculated using the same sample length distribution and the two classifier

6.7. LENGTH MEASUREMENT 63

0 2 4 6 80

0.1

0.2

0.3

0.4


Pro

babi

lity

dens

ity fu

nctio

n [m

m−

1 ]

Leng

th r

espo

nse

↑Simple model

↑Fitted model

← PDF

short medium long

0

0.25

0.5

0.75

1

Figure 6.7. Fibre length classifier. Length classifier responses for Sshort, Smediumand Slong in Eqs. (6.1) to (6.3) according to both the simple (dashed line) and thefitted (solid line) models plotted together with the same PDF as in Fig. 6.2 (dottedline, y-axis to the right) in order to show how much of the total concentration maybe found in each of the length classes.

models, the different outcomes of the simple and fitted models are:Simple Fitted

Short: 8 % 23 %Medium: 38 % 39 %Long: 54 % 38 % .

This shows that this set of linear combinations of ϕ1, ϕ2 and ϕ3 works as aclassifier, but the interpretation of the result should be made according the morerealistic fitted model.

In Paper 1 three experimental examples are given. The first of these involvesmeasurements on fibre fractions produced in a Bauer-McNett apparatus. The othertwo were production-process related and will be further discussed in Chapter 7. Thefibre fractions extracted from a Bauer-McNett apparatus have overlapping lengthdistributions although the mean length differs between the fractions1. The lengthclass responses of the optical length classifier also overlap. Therefore, the fibrefractions are expected to turn up in more than one class. Fig. 11 of Paper 1confirms this behaviour.

6.7 Length measurement

The principle of the particle length meter is based on the ratio of two ϕ values, onefrom a coarse channel and one from a fine channel. Measurements on pulp fibresaccording to this principle are sometimes referred to as OMFL, Optical Mean FibreLength. Considering the response curves, the ratio mL = l3 k3 ϕ3/(k1 ϕ1) is ameasure of the length of the particles in the dispersion.

1The fractions are characterized by the mesh number of the separation. R30 means that thesample is retained on a sieve having 30 meshes per inch. 30/50 means that the particles havepassed the 30 meshes per inch sieve, but they were retained on a 50 meshes per inch sieve, etc.


0 2 4 6 80

0.1

0.2

0.3

0.4


Pro

babi

lity

dens

ity fu

nctio

n [m

m−

1 ]

Leng

th r

espo

nse

↑Fitted model

l1

↓

l3

↓Simple model

↓

← PDF

0

1

2

3

4

Figure 6.8. Length measurement. Length response function, r(l), for mL =l3 k3 ϕ3/(k1 ϕ1) according to both the simple (dashed line) and the fitted (solidline) models plotted together with the same PDF as in Fig. 6.2 (dotted line, y-axisto the right) showing that the mean length in this example is underestimated by thesimple model since the longest particles of the population exceed l3.

In Fig. 6.8 the length responses r(l) are plotted using both the simple and thefitted models. The coarse and the fine channels are those used in the numericalexperiments. The responses are shown together with the same particle length dis-tribution as in Fig. 6.2. Provided that only a small variance component of T comesfrom the variance of the length, the resulting mL may be calculated as

mL =∫ ∞

0r(l) fL(l) dl , (6.4)

where fL(l) is the volume weighted PDF of the length distribution, r(l) may berS(l) for the simple model and rF(l) for the fitted model. The resulting mean lengthvalues for the PDF in Fig. 6.2 are mLS = 1.96 mm and mLF = 1.54 mm. Note thatthe usual mean length, which is obtained when r(l) is set to l, is 2.12 mm. Theoptical method gives lower mean values due to the choice of lb being smaller thanthe longest particles.

For monodispersions the non-linearity of rF(l) may be compensated for by cal-ibration, but this may be difficult for polydispersions.

One experiment, using the samples described in Appendix C, uses data collectedat the same time as that for the concentration experiments of Figs. 6.3 and 6.4.Two laboratory methods were used for comparison: manual measurement in amicroscope and the Kajaani-FS200 instrument2. Weighted mean length valueswere used (as explained in footnote 2.5 on page 22). The weighting function l d2was used with the microscopic data and l was used with Kajaani data3.

The three methods are plotted against each other in Fig. 6.9. The scale ofmL was adjusted to the microscopic data instead of trying to determine l3 k3/k1.

2The Kajaani-FS200 instrument determines the fibre length optically while the fibre passes acapillary in the instrument.

3This instrument did not provide fibre diameter data.

6.8. WIDTH MEASUREMENT 65

The microscopic data are based on the measurement of ∼300 particles and ∼10000particles were measured in the Kajaani instrument. Fig. 6.9(c) shows that theagreement between the microscopic and the Kajaani methods is of the same orderof magnitude as the agreement between these and the ϕ3/ϕ1 method (the OMFLmethod), which means that the usefulness of ϕ3/ϕ1 for the characterisation of theparticle length is about the same. One of the reasons for the scatter of the data inFig. 6.9 is that the three methods have different sensitivities for the short particles.

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

Length Microscope [mm](a)

Leng

th s

igna

l Trip

le s

enso

r [m

m]

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

Length Kajaani−FS200 [mm](b)

Leng

th s

igna

l Trip

le s

enso

r [m

m]

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

Length Microscope [mm](c)

Leng

th K

ajaa

ni−

FS

200

[mm

]

� � � � � � �

� � � � � � 3

� � � � � � 9

� � � � � � 4

� � � � � � 6

� � � � � � 5

� � � � � � <

� � � � � � �

� � � � � �

Figure 6.9. Comparison of three length measurement methods: microscopemethod, Kajaani-FS200 and the length signal ϕ3/ϕ1 from the triple beam unit.The sample groups referred to in the legend are described in Appendix C.

6.8 Width measurement

The width sensor for slender cylinders is built on the fact that the linear lightattenuation A is proportional to 1/d and S, and that ϕ1 from a sensor with athin beam can be used to determine the concentration S provided that a negligiblefraction of the sample has a length smaller than l1. Moreover, A is independent ofthe particle length and ϕ1 is independent of d. Thus, the ratio ϕ1/A can be usedas a measure of d. However, if the particles do not have identical diameters, i.e.


0 10 20 30 40 0

0.01

0.02

0.03

0.04

0.05

0.06

Particle width [μm]

Pro

babi

lity

dens

ity fu

nctio

n [μ

m−

1 ]

Wid

th r

espo

nse

Simplemodel

↓← Fitted model

PDF →

0

10

20

30

40

50

60

Figure 6.10. Particle width measurement. Response for different fibre widths(fibre diameters) from a particle width meter based on ϕ1/A according to both thesimple (dashed line) and the fitted (solid line) models plotted together with thePDF (dotted line, y-axis to the right) for a width distribution calculated from thelength distribution in Fig. 6.2 using a simplified assumption regarding the width-length dependence. A proper width signal was obtained for almost all particles ofthe sample.

the diameter is a random variable D, this changes the interpretation somewhat:

1μ1/D

= 3 π2Db

32Qϕ1

A, (6.5)

were μ1/D is the expected value of the reciprocal random particle diameter D. Since1/μ1/D is usually not the same as μD this measurement principle is here referredto as a width measurement and not a diameter measurement. Measurements onpulp fibres based on this principle are sometimes referred to as OMFW, OpticalMean Fibre Width, measurements. In Fig. 6.10, the response curves for ϕ1/Aare shown for a non-random particle width d together with a sample PDF for thewidth distribution. The PDF was calculated from the length distribution usedpreviously using the same simple assumptions of the dependence of the width onthe length, except that the diameter was allowed to exceed 40 μm. For this sampledistribution the mean width was 28.4 μm whereas the reciprocal value of the meanof the reciprocal width was 20.6 μm. The simple model here results in a widthsignal of 20.7 μm and the fitted model results in 21.4 μm. This calculation confirmsthat if a negligible fraction of the particle population has a length shorter than lb,ϕ1/A is a rather good measure of the particle width provided that the differencebetween 1/μ1/D and μD is considered.

Two experiments to illustrate the width measurement are included using samplesfrom the set described in Appendix C. The triple beam unit was used. In the firstexperiment, microscopic width measurements were used for comparison. However,since these manual measurements are unreliable if the sample has a significantfines fraction, only those samples having no fines fraction were used. Fig. 6.11(a)shows a direct comparison between the two methods. The second experiment givesindirect evidence. A systematic change of the fines fraction content in the sample

6.8. WIDTH MEASUREMENT 67

0 0.02 0.04 0.06 0.080

2

4

6

8

Width Microscope [mm](a)

Wid

th s

igna

l Trip

le s

enso

r [a

rb. u

nits

]

group 1group 2group 4group 5group 8line fit

0 5 10 15 20 25 302

2.5

3

3.5

4

4.5

Fines fraction [%](b)

Wid

th s

igna

l Trip

le s

enso

r [a

rb. u

nits

]

0 5 10 15 20 25 300

20

40

60

80

100

120

Fines fraction [%](c)

Rel

ativ

e re

spon

se [%

]

S/Aϕ/S

Figure 6.11. Width signal from the triple beam unit. (a) Comparison with mi-croscopically determined particle widths for samples where the fines fraction hasbeen removed. The groups referred to in the legend are described in Appendix C.(b) Comparison of width signals from samples having different fines fraction con-tents but otherwise being identical samples. (c) S/A and ϕ/S as a function of finescontent for the samples in (b).

affects the width signal correspondingly. Therefore, the width signals of the foursamples constituting group 8 in Appendix C were plotted against the fines fractioncontent in Fig. 6.11(b). Note however that the decrease in the width signal withfines fraction content is due both to an increase in A and a drop in concentrationresponse of ϕ1 because a greater fraction of particles is smaller than the beam. Inthis particular experiment, the concentrations are known. In Fig. 6.11(c) S/A andϕ1/S are plotted as functions of the fines content. In order to make the curvescomparable they were normalized to their values for 0 % fines content. Fig. 6.11(c)shows that 1/A has a higher sensitivity to the presence of fines than ϕ1. Since thefines fraction is rather heterogeneous with regard to size and shape, the differencein fines content is not only a change in fibre width, but the observation supports theidea of using ϕ1/A as a width measurement. The result also indicates that ϕ1/Aalso might be used to monitor fines content variations. Both experiments supportthe width measurement principle.


6.9 General considerations

As with all measurement techniques, it is important to be aware of the regime ofapplication and parameters that can interfere with the measurement results of themeasurement principles described in this chapter. A detailed account is given inPaper 7. The following items are given attention:

• Sampling

• Drift

• Flow rate

• Relation between mass and optical cross section

• Polydispersions

• Flocculation

• Concentration range

• Particle shape

• Particle orientation

Chapter 7

Examples of industrial applications

The hieroglyph for papyrus.

The measurement principles were developed in a context where new measure-ment techniques specific for the pulp and paper industry were sought, especiallymeasurement techniques having online potential. This chapter gives examples ofthis measurement-in-industrial-production aspect discussed mainly in Paper 1, Pa-per 2 and Appendix B. The examples are applications of the particle length classifierand the concentration measurement with suppression of small particles.

7.1 The work in a 2009 perspective

From a technical perspective, the time elapsed seems very long since the work inPaper 1, Paper 2 and Appendix B was done. However, that work serves as agood example of how the measurement principles described in this thesis may beapplied industrially. In addition Appendix B shows that although the optical unitof an online meter is comparatively simple, the measurement system built aroundit may be rather complex. Additional functions are usually needed such as takingout representative samples from the process, preparing them for measurement and,when the measurement is done, sending the results to a computer for use in processmonitoring or control.

7.2 Applications of the particle length classifier

Paper 1 was written for a conference on mechanical pulping and the presentationwas adapted to this audience. The paper includes one example of measurementson pulp fractions produced with the Bauer-McNett apparatus and two examples

69

70 CHAPTER 7. EXAMPLES OF INDUSTRIAL APPLICATIONS

of using the length classifier for the characterization of pulps in relation to processparameters. The first example, which was already mentioned in Section 6.6, wasused as verification that the measurement principle worked as expected. The secondexample concerned production of mechanical pulp and the third concerned beatingof pulp.

7.2.1 Application in mechanical pulpingMechanical pulping

Mechanical pulp [104] was traditionally manufactured as stone groundwood pulp(SGW), i.e. the wooden logs were cut into shorter pieces, which were pressed to-wards a rotating grindstone. This pulping process damaged much of the woodfibres and the pulp produced contained a lot of fragmented fibres and a consid-erable amount of fines. A typical use of SGW was newsprint paper. Later, thegrindstones were replaced by disc refiners, where the wood (now cut into chips)was pressed between rotating discs having a bar pattern on the grinding surfaces.The refiner mechanical pulp produced, RMP, was of higher quality than SGW andthe process could be better controlled. The refiner technique was further developedby pre-heating the wood chips by steam (thermomechanical pulp – TMP) that canbe preceded by a chemical treatment of the wood chips (Chemithermomechanicalpulp – CTMP).

At the time of Paper 1, pulp production was moving from SGW towards RMPand TMP. One of the new possibilities to control the production was the discclearance, the distance between the two refiner discs. This could be adjusted duringproduction to control the pulp quality especially if proper online measurements ofthe pulp could be made. The composition of the pulp in length classes, similar tothose obtained by the Bauer-McNett classifier, was one of the potentially usefulmeasurements.

Experiment and result

The experiment in Example 2 of Paper 1 aimed to investigate if disc clearancechanges gave observable responses in the optical length classifier measurement. Thedisc refiner was a conical disc refiner having two variable clearances: one of its planepart and one of its conical part. The clearance of the conical part was changed insteps. Results from the optical classifier are shown in Fig. 7.1. A narrowing of thediscs resulted in a higher production of the short fraction1 at the expense of a lowerproduction of the long fraction, whereas the medium fraction is barely affected.

For comparison, a table of laboratory measurements on the same pulp sampleswas included in Paper 1. For convenience it is reproduced here as Table. 7.1.A CSF (Canadian Standard Freeness) tester determines the rate of drainage of a

1In Paper 1, the classes presented by the instrument are denoted fine, medium and coarse.Here the classes are denoted short, medium and long.

7.2. APPLICATIONS OF THE PARTICLE LENGTH CLASSIFIER 71

0.5 0.55 0.6 0.65 0.720

25

30

35

40

45

50

Disc clearance [mm]

Leng

th c

lass

ifica

tion

[%] Long

MediumShort

Figure 7.1. The relative fibre content in the three optical length classes as afunction of the disc clearance in the manufacture of refiner mechanical pulp.

dilute pulp suspension, which is an important pulp property with regards to thebehaviour of the pulp on the paper machine. The rate of drainage is affected bythe work done on the fibres during refining or beating. A high CSF value meansthat the pulp is easily de-watered. Tear index is a strength property related tothe growth of a cut in the paper when tension is applied. The light scatteringcoefficient is determined according to the Kubelka-Munk equations through tworeflectance factor measurements on laboratory sheets made from the pulp: onemeasurement on a single sheet over a black light trap and one on an opaque pad ofthe laboratory sheets. The light scattering coefficient is expressed in m2/kg units.It is an important property of the pulp for the reflectance and opacity of the finalpaper.

Both CSF and the light scattering coefficient are affected by changes in the discclearance, and in both cases there is a strong relation to the changes in the particlelength distribution.

disc CSF tear light scatteringclearance index coefficient

[mm] [ml] [Nm2/kg] [m2/kg]0.53 112 7.7 50.60.60 160 7.9 45.50.63 161 8.0 43.80.67 174 7.6 44.9

Table 7.1. Some complementary data on the samples of Fig. 7.1


7.2.2 Application in pulp beatingPulp beating

Beating by mechanically treating the pulp slurry is a widely used way to improvethe properties, such as strength and formation, of paper [104]. Beating makes thepulp fibres more flexible and more inclined to bind to each other. However, toomuch beating reduces the tear index, the light scattering coefficient and makes thepulp hard to dewater on the paper machine. There are several types of beatingequipment used, e.g. conical and disc mills. Beating can be made using low con-centration2 (LC) pulp (2–6 % by weight) or high concentration (HC) pulp (25–30 %by weight), giving different pulp properties. The amount of work done on the pulpis characterized by the specific energy and the specific load. The specific energy isthe power imposed on the pulp per ton (kWh/ton). The specific load is the ratiobetween the energy and the length of the edges of the bar pattern of the beatinggear (Ws/m).

One of the measurements often used to characterize the beating effect on thepulp is the Schopper-Riegler test. The Schopper-Riegler value (◦SR) is a measure-ment of the rate at which a diluted pulp suspension may be de-watered. A highSR value means that the pulp is difficult to dewater.

Experiment and results

In this application, a high yield chemical pulp was beaten in a pilot plant. Thebeating was done in four stages: first a high concentration stage then three lowconcentration steps. Samples for testing were taken from the original pulp andafter each beating step. The SR number, the specific energy and the specific loadare found in Table 7.2.

Fig. 7.2 shows that the effect of the beating steps can be followed as changesin the optical length classes. The general trend is that increased beating will sig-nificantly enlarge the medium fraction. The short fraction also increases. Conse-quently, the long fraction decreases.

7.3 Application of the concentration meter

7.3.1 The fine paper processThe development of the paper making process has been continuously ongoing sincethe time of Ts’ai Lun3 until today. It is not the intention here to give an exhaustiveoverview over this, but a brief description of the situation in the mid 1980s maybe beneficial in order to understand why it was of special interest at that time to

2The word consistency is often used (e.g. in Paper 1) instead of concentration with referenceto pulp slurries.

3Ts’ai Lun, , is said to be the inventor of paper about AD 105 in China.

7.3. APPLICATION OF THE CONCENTRATION METER 73

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+

3 +

4 +

5 +

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. � � � � � � � � � � � �

" � � �

� � � � �

. � � �

Figure 7.2. The relative content in the three optical length classes as a function ofpulp beating. Increasing sample numbers represent increased beating.

make concentration measurements in the short circulation of a fine-paper machine.Making fine paper, fully bleached chemical pulp, filler, retention chemicals and otheradditives are mixed with recirculating white water, as shown in Fig. 7.3(a). Themixture is fed to the headbox, which distributes the suspension onto the wire cloth.An example of a plane wire machine4 is shown in Fig. 7.3(b). Most of the pulp fibresand some filler particles are retained on the wire, forming the paper web, while mostof the water and some particles pass through the wire, forming the white water.The white water is recirculated to dilute the incoming pulp and additives, thusclosing the loop of the short circulation. Under normal conditions, the concentrationof the pulp suspension fed into the short circulation is approximately 35 kg/m3.Apart from the pulp fibres and fillers, the water which carries the pulp into theshort circulation also carries dissolved organic and inorganic substances. Dissolvedsubstances affect, among other things, the effectiveness of the retention chemicals

4Note that modern paper machines often use another construction than the plane wire type.

sample SR-number operation specific specificnumber energy load

[◦SR] [kWh/t] [Ws/m]10 12 unbeaten 0 –11 12 HC 123 –12 17 LC 157 0.8413 24 LC 192 0.8414 35 LC 225 0.84Table 7.2. Some complementary data on the samples of Fig. 7.2


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(b)Figure 7.3. (a) The short circulation of a fine paper machine. The stock consistingof pulp, filler, retention chemicals and other additives is mixed into the white waterof the short circulation and is distributed by the headbox onto the wire. Most ofthe water and some particles pass through the wire, forming the white water, whichis collected in the wire pit. The reuse of white water for dilution closes the shortcirculation. (b) A section of a plane wire on a running paper machine.

and the density of the suspension.At the time the investigations of this thesis were initiated, there was an in-

creased interest in a more efficient closing of the paper making process. The waterseparated from the main particle flow in the paper machine was to a higher degreerecovered and reused as dilution water at the wet end of the machine. The reuse ofwater made the concentration of substances such as dissolved organic substances,metal ions and small particles increase in the short circulation of the machine. Thisprovided a new chemical environment which threatened to decrease the retention of


filler particles, thus decreasing the fine-paper quality and the cost efficiency of theproduction. The quality aspect was mainly the improvement of the optical proper-ties of the paper caused by the filler. The cost efficiency aspect was that filler wasless expensive than pulp and that it needs less drying energy. These driving forceswere the background to the development of both more efficient retention chemi-cals and measurement techniques in order to make the process more controllableand observable and thereby improving runnability.5 The single component reten-tion chemicals such as cationic starch and polyacrylamide (PAM) were replaced bytwo-component systems such as combinations of starch and silica and starch andbentonite (a special form of clay).

The development of the fine paper process also required a better monitoringand control of the process. It was especially important to measure the particleconcentrations in the headbox and in the white water. Moreover, the measure-ments needed to be selective with regards to pulp and filler particles. The need foronline measurements was the starting point of the work on particle concentrationspresented in Paper 2 and Appendix B.

7.3.2 The measurement task

At the beginning of the work, the only commercially available products for mea-suring the concentration of fillers in paper were ash content control systems. Thesewere usually based on an ash content measurement on the paper web combinedwith the measurement of linear light attenuation in the short circulation. Ashcontent measurement on a paper web usually utilised the detection of γ-radiationattenuation through the paper. These systems did not measure the concentrationsof different particle types.

Other attempts to make concentration measurements in the short circulationhad been unsuccessful. A few years earlier, Kallmes had presented a retentionmeter [105], but it was flawed from a measurement point of view. At the end of the1970’s, STFI had also made some unsuccessful attempts to use a modified so-calledTP-meter for measurements in the short circulation.

An overview of the concentration meters available for use in the pulp and paperindustry [6] in the early 1980s covers measurements based on mechanical and op-tical properties of the suspension. Mechanical properties mentioned are viscosity,shear force, flow resistance and ultrasonic attenuation. Optical techniques coveredwere mean light attenuation, mean back scattering, fluctuating back scattering,fluctuating low angle scattering and polarisation state changes. A technique notmentioned in [6], but known at the time, was to use the density of the suspensionas a measure of concentration [1–3]. None of these measurement principles solvedthe problem of selective concentration measurement of pulp and filler.

5In practice, there is an upper limit to how high the filler content can be because a high fillercontent decreases bending stiffness of the paper. In addition, the retention chemicals needed foran extreme filler content can make the paper non-uniform, i.e. give it a bad formation.


One goal of the work presented here was to solve the problem of separatelymeasuring the concentrations of papermaking pulp fibres and filler particles in theshort circulation of fine paper machines. For the work presented in Paper 2 andAppendix B this objective was realized as the following research activities

• to develop a technique having the potential of online use in the short circula-tion of a fine paper machine to make direct, reliable and separate concentra-tion measurements of bleached chemical pulp and one type of filler mixed inan aqueous suspension

• to implement this technique as an online measurement system

• to test it experimentally on a pilot paper machine.

7.3.3 The measurement solutionThe introduction of a new measurement principle comprising a combination of anoptical fibre concentration meter, described in Section 6.5.2 and a high resolutiondensity meter [3, 106] made it possible to measure the concentrations of pulp andfiller separately.

The density change Δρ when adding particles to a liquid is a measure of theparticle concentration having the unusual property of being independent of the sizeand shape of the particles provided that the particles have the same density ρp,which also has to be different from the density ρL of the liquid. The relation is

Δρ = kp S , (7.1)

where kp is a sensitivity coefficient and S is the concentration. For a suspension ofparticles all having the same ρp and the surrounding liquid has the density ρL,

kp = 1− ρLρp. (7.2)

If the suspension contains particles of q different types each having its own ρp,i,Eq. (7.1) becomes

Δρ =q∑i=1

(1− ρLρp, i

)Si , (7.3)

i.e. Δρ is a linear combination of the concentrations of the different particle types.The derivation of the above equations rely on the same principle Archimedes usedin the anecdote about the golden crown, Fig. 7.4.

In the short circulation of the fine paper process considered in Appendix B,there were two particle types, viz. pulp and filler. Therefore,

Δρ = kpulp Spulp + kfiller Sfiller . (7.4)


Figure 7.4. The anecdote sais that taking a bath, Archimedes realised that thevolume of an object having an irregular shape can be measured as the volume ofwater spilling over from an already full container when the object is submerged intothe water. This insight made it possible for him to determine the volume of a goldencrown and thereby determine its density, which revealed that the metal was not puregold.

As shown in Fig. 6.2, the optical concentration meter, based on the fluctuationsof the regular transmittance of optical radiation of a thin beam through the sus-pension, effectively suppresses the influence of the filler particles. Fines are alsosuppressed, but that was not a problem in this application since the pulp used onlyhad a very small fines fraction. Therefore,

Spulp = k1 ϕ1, (7.5)

which solved half the problem and made it possible to subtract the influence on Δρof Spulp such that

Sfiller = 1kfiller

(Δρ− kpulp k1 ϕ1) , (7.6)

which solved the second half if the problem.The first presentation of the principle was made in Paper 2. The principle was

later presented in much more detail in [74], which is reproduced in condensed formin this thesis as Appendix B.

7.3.4 The first prototype and pilot plant testsA prototype measurement system based on this principle was constructed and isshown in Fig. 7.5(a) together with the later commercial system, Fig. 7.5(b). Theprototype was tested on the pilot paper machine FEX [107], Fig. 7.6. Test resultsshowed that the measurement principle functioned as expected. Appendix B coversthe principle, the design and the first test of this measurement system. The mea-surement system built around the optical and density units controlled the process


(a) (b)Figure 7.5. (a) The prototype measurement system used in the experiments inAppendix B. (b) The commercial measurement system.

sampling, prepared the samples for measurement, controlled the measurements,presented and stored measured data properly, and implemented necessary sequencecontrol.

For the optical concentration meter, the sample had to be diluted approximately10 times in order to fit the concentration range of the meter. The dilution wascontinuous using two pumps working at different speeds. The sample also had tobe well dispersed. For this a dispersion unit preceded the optical unit.

The density of the sample was affected by the presence of air bubbles in theliquid. Therefore, the sample was pressurised using a diaphragm pump in a closedloop through the density meter. Moreover, the density of the sample and the densitymeter itself were sensitive to the sample temperature. A temperature compensa-tion method was provided with the instrument, but its temperature range was toolimited. Therefore, a special temperature compensation was developed involving atemperature meter having a resolution of approximately 0.01 ◦C.

The principle of the mechanical layout is found in Fig. 7.7(a). All valves, pumps,measurement devices etc. were computer controlled. Sampling from the process wasmade alternately from the headbox and the white water. It was also possible to runcleaning liquid through the system. The raw signals from the optical and densityunits were registered by the computer and both concentrations and retention were


Figure 7.6. The pilot plant paper machine FEX. Photo: Rolf Tapper.

calculated, Fig. 7.7(b). The temperature compensation of the density meter wasalso computer controlled.

One example of the result from online measurements using the prototype mea-surement system is found in Fig. 7.8, the concentration of pulp fibres and fillerparticles are shown as a function of time. The measurements were made during afine paper experiment on FEX. For comparison, laboratory data are also plotted inthe graphs.

7.3.5 After the prototype

As already indicated, the measurement technique was developed into a commer-cial product, Fig. 7.5(b), which was successfully used on full-scale paper machines[78, 79]. The work was led by E. Stenberg and involved among other things thedevelopment of a new optical unit [108] for the fibre concentration sensor. In thenew unit the light path length through the sample was reduced to 1 mm, whichmade it possible to measure on the samples in the short circulation without dilutingthe sample first. This optical unit is referred to as the single-beam unit in Section


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(b)Figure 7.7. (a) The mechanical layout of the measurement system. (b) The princi-ple of the measurement signal flow from the optical and density units to the outputfrom the measurement system.

6.3. A sign that reveals that concentration and retention measurement got muchattention at this time is the competing systems [109–115] appearing on the marketonly a few years after the presentation of the STFI system.


9:00 10:00 11:00 12:00 13:00 14:000

1

2

3

4

5Fibre concentration

Time [h]

Con

cent

ratio

n [g

/l]

headboxlab hbwhite waterlab ww

(a)

9:00 10:00 11:00 12:00 13:00 14:00 0

2

4

6

8

10Filler concentration

Time [h]

Con

cent

ratio

n [g

/l]

headboxlab hbwhite waterlab ww

(b)Figure 7.8. The concentrations of pulp fibres (a) and filler particles (b) measuredindependently during a test of the prototype measurement system on the pilot plantpaper machine FEX. Samples were taken for laboratory testing 8 times. In thelegend, laboratory measurement are denoted “lab hb” for the headbox samples and“lab ww” for the white water samples. The concentration values for these samplesare also shown in the graphs.

Chapter 8

Comments on measurement uncertainty

In these matters the onlycertainty is that nothing iscertain.

Pliny the Elder

The origin of measurement uncertainties in the measurement principles de-scribed in Chapter 6 may be divided into two categories:

• Sensor-induced

• Model-induced.

Regarding the sensor-induced uncertainties there is a difference between thesensitivities of ϕ and A to errors in the DC and AC components of the detectorsignal when measuring on a suspension of particles. Measurements based onlyon ϕ has a smaller demand on the DC than if A is used. A is sensitive to slowchanges in the light source radiance, the detector sensitivity and absorption in thedispersing medium. Strictly, slow variations are a part of the AC component of thesignal, but since estimation of the AC and DC components has to be made duringa limited time, there are variations in the measured data which show as smallchanges in the reference DC level. A depends essentially on the ratio between theDC component and the reference DC level determined when there are no particles inthe measurement cell. This means that the reference DC level has to be frequentlyupdated in order to avoid errors in A. The fluctuations in the detector signal causedby the fibres passing the light beam have much higher frequency components and aretherefore found in the AC component. ϕ depends essentially on the ratio AC/DC,but not on the reference DC level, which relaxes the demand on updates of thereference DC level. However, it is important that the reference DC level and theDC component when measuring particles are unbiased. It is also important not toadd any AC component in the frequency range of the AC estimation. Unwanted AC

83

84 CHAPTER 8. COMMENTS ON MEASUREMENT UNCERTAINTY

components may arise from noise in both light source and detector and air bubblesin the dispersing medium. This is especially true at low and high concentrationas was pointed out by Gregory [94]. For measurements involving A, such as thewidth measurement, the demand on the DC component must be considered. Thefundamental requirements to avoid sensor-induced uncertainties are

• Sample treatment. It is important that the sample fed to the transmis-sion sensor is representative. This has to be considered all the way from thesampling in the process, through the dilution system and to the flowing char-acteristics through the sensor. If dilution is used, the dilution ratio needsto be known. The flowing conditions may affect the orientation state of theparticles and can possibly also distort the size distribution of the particles.

• Light source. The light source needs to be very stable in order not tointroduce an extra AC component and, if A is used, it is essential that slowdrift of the radiance is eliminated or compensated for. Changes in the spectralcharacteristics of the light source must also be avoided.

• Transmittance path. The transmittance path involves the apertures, lensesand the measurement cell. The design of the optics affects important sensorproperties. Factors such as the parallelism of the light beam, the beam crosssection shape and size, evenness of the light across the beam cross section,the stray light level, and the acceptance angle of the detector need to be wellknown and controlled. Especially in the fine channel, the finite acceptanceangle is probably the dominating sensor-induced error.

• Detection system. The detection system must give voltage output whichis linear to the radiation input. It must be unbiased, have a low noise level,and have a constant sensitivity, which is unaffected by e.g. the temperature.It must not add any noise (AC component) to the signal.

The model-induced uncertainties are not so easy to describe in general termsbecause of the variety of measurement principles and the variety of possible samplesto measure. A good understanding of the measurement principles is necessary inorder to estimate the magnitude of these uncertainties. Paper 7 was written forthis purpose. In Paper 7 considerations regarding each measurement principle arepresented as well as general considerations for this group of measurement principles.

The model-induced uncertainties have two origins

• the particle model and

• the sensor model.

The particle modelled as a straight slender cylinder is obviously a simplificationfor wood fibres, Fig. 2.1(c)–(d), and the clay particles in Fig. 2.6 are far fromspheres. Still, it is possible to use the described measurement principles, but the

85

interpretation of the results become uncertain to a higher degree than would be thecase with ideal particles.

The sensor model is highly idealized. The use of the simple response modelinstead of the fitted model is an obvious deviation between model and reality thatcauses a measurement error. Situations where the cross section is not constant alongthe optical axis and the illumination is uneven throughout the cross section are otherexamples of deviations between model and reality that increase the uncertainty.

The triple and single beam sensors were never included in a test such as [6].However, in e.g. Paper 7, results are compared to those obtained with laboratoryconcentration measurement, measurement of length and width using a microscope,and length measurements using the Kajaani-FS200. Unfortunately, these referencemethods have rather large uncertainties, which make it difficult to estimate the un-certainty by comparison. Since the concentration measurement may be consideredto be the most fundamental of the four measurement principles, the following willonly concern concentration of pulp and especially the pulps described in AppendixC. In the measurement of these samples, the different concentrations were obtainedboth by adding samples of laboratory-determined concentration and by automaticdilution in two steps of each of these samples. A more detailed analysis of thedata behind Figs. 6.3 and 6.4 shows that the sensitivity of ϕ1 and A to concen-tration differs somewhat between the different sample groups. It is to be expectedsince the combined effect of several error components can be regarded as system-atic if the samples are similar. In Table 8.1, the sensitivity coefficients for differentsample groups and combinations of groups are shown. The sensitivity coefficientfor ϕ1 is a kB and for A it is a kA. They were determined using linear regressionforced through the origin, i.e. it is assumed that ϕ1 = 0 and A = 0 when S = 0.The uncertainty of the experimentally determined sensitivity coefficients a kB anda kA is characterized by the standard errors sϕ and sA. In order to facilitate thecomparison, the relative standard errors are also shown.ϕ1 has its lowest sensitivity for bleached kraft pulp without fines (the fines-free

samples in groups 1 and 2) and 62 % higher sensitivity for the groundwood samplesin group 8 for which ϕ1 has its highest sensitivity. The standard error data can beused to see whether or not the differences in a kB are significant. Groups 3–5 aremerged in the analysis since all samples originate from the same unbleached kraftpulp. Note that probably due to the low sensitivity to fines, a kB is somewhat higherfor the samples group 3–5 without fines than for those with fines. Surprisingly theopposite situation is found for the bleached pulp in groups 1–2. However, the effectis rather small and may be caused by the special fibre shortening method used.

Since the linear light attenuation A is sometimes used for concentration moni-toring and commercial instruments exist, a comparison with the characteristics ofa kA is of interest. A has its lowest sensitivity for the thermomechanical samples ingroup 7 and a 167 % higher sensitivity for the groundwood samples in group 8 forwhich A has its highest sensitivity. There is a pronounced difference between a kAfor groups 1 and 2, which means that the idea of using a concentration measure Afor mixtures of group 1 and 2 is not a very good one, whereas ϕ1 may be used in

86 CHAPTER 8. COMMENTS ON MEASUREMENT UNCERTAINTY

that way. Mixtures of group 7 and 8, both mechanical pulps, are problematic forboth ϕ1 and A, but using A is meaningless here, whereas ϕ1 might be used if arather large uncertainty can be accepted.

In measurement techniques where indirect measurements are used, it is commonthat an unsuitable application of the technique gives unacceptably large uncertain-ties. It can be concluded that the use of ϕ1 for concentration measurement is notan exception from that. In an ideal situation when it is only the concentrationthat varies and not any other pulp property, both ϕ1 and A have reproducibilitythat is better than the laboratory method, i.e. a few percent. When deviating fromthe ideal situation ϕ1 in general is less sensitive to the deviations than A is andthere are situations such as the measurement of pulp concentration when fillers arepresent, Fig. 6.5, where ϕ1 performs well but where it would be meaningless to useA.

Sample group a kB = ϕ1S sϕ

sϕa kB

a kA = AS sA

sAa kA

1 5.88 0.09 1.6 % 1.45 0.05 3.2 %2 5.76 0.10 1.7 % 1.81 0.02 1.0 %

1–2 5.85 0.07 1.2 % 1.53 0.04 2.7 %3–5 7.65 0.13 1.7 % 1.65 0.03 1.6 %6 7.05 0.07 1.0 % 1.45 0.02 1.2 %7 6.82 0.04 0.6 % 1.09 0.01 0.9 %8 8.96 0.16 1.8 % 2.90 0.11 3.7 %

7–8 7.40 0.18 2.4 % 1.58 0.14 9.1 %1–2 no fines 5.55 0.13 2.4 % 1.37 0.11 7.9 %3–5 no fines 8.57 0.21 2.4 % 1.69 0.06 3.5 %

Table 8.1. The concentration sensitivity of the optical concentration signal ϕ1and the linear light attenuation A tested with the samples described in AppendixC. The experimentally determined sensitivity coefficients are denoted a kB and a kAand their respective standard error are denoted sϕ and sA. In order to facilitatecomparisons the relative standard errors are also shown.

Chapter 9

Conclusions

The theory presented in this thesis successfully describes fundamental properties ofthe fluctuating behaviour of the random regular transmittance through a suspensionof particles. Four measurement principles based on the theory are presented for thecase that the suspended particles are straight slender cylinders having a diametermuch larger than the wavelength. The measurement principles are: concentrationmeasurement independent of particle length and width, presentation of the lengthdistribution of the particles as length classes, measurement of the mean particlelength, and measurement of mean particle width.

For simplicity, the measurement principles were presented here for the use withinone application area viz. the pulp and paper industry. However, the theory pre-sented is general in character. Eq. (4.17) especially has a generality close to thatof the Lambert-Beer-Bouguer law. The importance of the dimensional relation be-tween the particles and the cross section of the light beam were shown for slendercylinders and it is possible to make the same analysis for other types of particlesgiven the theoretical tools presented. Example of such an analysis is given forspheres.

Experimental evidence is given for all four measurement principles, which workas expected considering the sometimes rather large deviation between the idealizedparticles of the theory and the actual particles used in the experiments.

The measurement techniques presented were developed due to their high indus-trial relevance. Most of the work on the industrial application of the measurementprinciples was made by others and is therefore not included in this thesis. However,examples showing the industrial relevance are given in Chapter 7.

87

Chapter 10

Summary of Paper 1

Title: The STFI Optical fibre Classifier.Authors: Staffan Rydefalk, Thorulf Pettersson, Einar Jung, and Inge Lundqvist.Published: 1981 in EUCEPA International Mechanical Pulping Conference, (EU-CEPA, Oslo, 1981, 16 p.).

Paper 1 is a contribution to the International Mechanical Pulping ConferenceEUCEPA held in Oslo 1981. The attendees to this conference were mainly experts inmechanical pulping and not experts in measurement science. The presentation wasadapted to the audience. Therefore, the description of the measurement principleis made intuitively and in the application examples, knowledge of the processes,equipment and standard measurements is taken for granted.

The purpose of the presentation was to introduce the measurement principle,to show that it worked as expected in experimental examples, and to show thatexpected changes in the particle population showed up in the fibre classificationresults. It was also important to show that the measurement principle was imple-mented as a working prototype.

Paper 1 describes how the fibre classification signals are composed of three ϕsignals from transmittance sensors having different diameters. Three experimentalexamples are included: measurements on Bauer-McNett fibre fraction, measure-ment on mechanical pulps produced with different clearances between the refinerdiscs, and a high yield chemical pulp when processed through a number of beatingstages.

89

Chapter 11

Summary of Paper 2

Title: The Development of an Integrated Retention Control System.Authors: Tom Lindström, Staffan Rydefalk and Lars Wågberg.Published: 1984 in SPCI 84 – The World Pulp and Paper Week. New availabletechniques (The Swedish Association of Pulp and Paper Engineers, Stockholm,Sweden, 1984, 492-496).

New available techniques was a conference arranged by SPCI, Svenska Pappers-och Cellulosaingenjörsföreningen (The Swedish Association of Pulp and Paper En-gineers), in conjunction with a large exhibition at Mässan Älvsjö, Stockholm 1984.The attendees to this conference were mainly experts on pulp and paper processesand not experts in measurement science. The presentation was adapted to theaudience. Therefore, the description of the measurement principle was made in-tuitively while knowledge of the processes, equipment and standard measurementswas taken for granted.

Paper 2 contains descriptions of two measurement systems, which, when com-bined and complemented with proper control equipment, constitute an integratedcontrol system for the wet end of a fine paper machine with emphasis on the re-tention of particles. The first of the measurement systems described is automatictitration equipment for colloid titration. With this technique, the adsorption ofan oppositely charged (usually cationic) polyelectrolyte onto the particles in thepulp suspension is determined. This is important information to understand theresponse of the particles to retention agents. This part of Paper 2 has no connectionwith the rest of this thesis.

In contrast, the second measurement system is a part of this thesis. The systemmeasures the concentration of fibres and filler particles separately although fibresand filler are mixed in the same suspension. It is built on two sensors: one opticalsensor measuring the fibre concentration and one measuring the density of the sus-pension. If the density of the suspension medium can be determined, the separateconcentration can be calculated. Knowing these concentrations in the headbox andthe white water, the retention of fibres and filler can be calculated.

91

Chapter 12

Summary of Paper 3

Title: Fluctuations in the regular transmittance of dispersions of straight circularcylinders with a diameter much larger than the wavelength of the radiation.Author: Staffan RydefalkPublished: June 1998 in J. Opt. Soc. Am. A (vol. 15, no. 6, 1998, 1689-1697).

The expected value and the variance of the regular transmittance of a dilutedispersion of slender straight circular cylinders with a diameter much larger than thewavelength of the radiation were studied theoretically in relation to the dimensionsof the cylinders and the diameter of the radiation beam used. Both monodisperseand heterodisperse dispersions were considered. The study was limited to twoparticle categories: (i) particles that were short and (ii) particles that were longbut thin compared with the diameter of the radiation beam. The expected value wasa function of concentration and particle diameter. The variance was proportionalto the concentration. The variance was also proportional to particle length in thecase of short particles but not in the case of long particles. The results explainedinteresting possibilities for the characterization of dispersions of large cylindricalparticles shown earlier in industrial online applications.

93

Chapter 13

Summary of Paper 4

Title: Theory of fluctuations in the regular transmittance through a dispersion oflarge cylindrical particles: extension to higher concentrationsAuthor: Staffan RydefalkPublished: November 1999 in J. Opt. Soc. Am. A (vol. 16, no. 11, 1999, 2737-2745).

The theory presented in Paper 3 was valid only for very low concentration×path-length products p. In practical applications, such as online applications, it is oftendesirable to work at higher p values in order to avoid complicated sensor con-structions. This paper extended of the theory toward higher values of p. Theextended theory was based on the assumption of the independence of the regulartransmittance of parallel layers of the dispersion. The extended theory showed thatinformation about the dispersion found in the expected value and variance at verylow values of p could also be obtained for higher values of p with a few simpleexpressions. Therefore applications possible in the case of low p values were alsopossible for higher p values. Two experimental examples were included to facilitatethe discussion of the theory presented.

95

Chapter 14

Summary of Paper 5

Title: Assessment of the mean and variance of the random regular transmittancethrough a dispersion of large cylinders using numerical simulationsAuthor: Staffan RydefalkPublished: March 2008 in Applied Optics (vol. 47, no. 7, 2008, 993-1001).

The theory in Papers 3 and 4 covered, among other things, expressions for theexpected value μT and the variance σT

2 of T in the two extreme cases when thecylinders were much shorter or much longer than the diameter of the optical beamused. Intermediate lengths were not treated. In this Paper numerical simulationwas used to demonstrate the random behaviour of T for intermediate cylinderlengths. The simulation results were consistent with the theory and provided areliable estimate of the measurements produced by this analysis process. The resultof the simulation was summarized as a fitted Bézier function model. The advantageof the simulation was primarily in estimating measurement errors caused by thepresence of intermediate length particles in measurement applications.

97

Chapter 15

Summary of Paper 6

Title: Validation of a simulation algorithm for the mean and variance of the randomregular transmittance through a dispersion of large cylinders.Author: Staffan Rydefalk.Published: Technical report KTH 2009 (KTH-TRITA-IIP-09-04, 13 p).

This Paper was a supplement to Paper 5. The simulations covered how theexpected value μT and the variance σT

2 of T were related to the width and lengthof the cylinders. In this Paper the validation of the simulation software is described.The validation was made in two steps. Three different algorithms were used for thevalidation. Parallel use of the algorithm implementations showed only negligibledeviations between their results when a large number of random data were used.This was taken as strong evidence that the algorithms and their implementationswere correct.

99

Chapter 16

Summary of Paper 7

Title: Applications of the theory of fluctuation in the regular transmittance througha dispersion of large cylindrical particles to concentration and size measurementsAuthor: Staffan RydefalkPublished: September 2008 in Applied Optics (vol. 47, no. 25, 2008, 4509-4521).

In this Paper, four measurement principles based on the theory of fluctuationin the regular transmittance through a dispersion of large cylindrical particles werepresented. The principles concerned the measurement of particle concentration,particle length classes, particle length, and particle width. A detailed description,experimental demonstrations, and a number of considerations needed to understandthe region of validity of the different principles as well as their uncertainty, werepresented. The basic ideas of the measurement principles within their respectiveregions of validity were supported.

101

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[35] J. Hill, C Lundgren, Mätningar på fiberhaltiga avlopp med fiberlog vid Fröv-ifors Bruk. SSVL delprojekt 4, STFI-meddelande B 266, STFI, Stockholm1974, (in Swedish).

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[37] J. Hill, H. Höglund, E. Johnsson, “Evaluations of screens by optical measure-ments,” TAPPI 58(1975):10, pp 120–124 .

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