The relationship between hydrodynamic variables

and particle size distribution in flotation

Thèse

Ali Vazirizadeh

Doctorat en génie des matériaux et de la métallurgie

Philosophiæ Doctor (Ph.D.)

Québec, Canada

© Ali Vazirizadeh, 2015

iii

Résumé

La flottation industrielle est un procédé continu qui se déroule souvent en plusieurs étapes

et dans lequel des particules d'une espèce de minéral donnée (généralement celles d'intérêt),

présentes en différentes tailles, rencontrent une grande quantité de bulles de gaz

(normalement de l'air) pour produire des agrégats bulles-particules minérales, qui sont

extraits du dispositif de flottation (colonne ou cellule) en tant que produit de valeur

(concentré). Le contenu en bulles est décrit par les conditions hydrodynamiques régnant

dans le réacteur de flottation. Celles-ci sont reconnues pour leur influence sur la

performance de la flottation.

Ce projet de recherche porte sur deux sujets majeurs. Le premier est l'analyse de l’impact

des particules solides sur les variables hydrodynamiques et l’effet de ces variables

hydrodynamiques sur la récupération d’eau au concentré. Pour ce faire, l'effet du solide sur

la distribution de la taille des bulles et le taux de rétention de l’air, ainsi que la corrélation

entre la distribution de taille des bulles et le taux de rétention de l’air dans une colonne de

flottation ont été étudiés. L'effet du taux de rétention de l’air, de la dimension des bulles et

du taux surfacique de bulles (Sb) sur la quantité d’eau extraite au concentré a ensuite été

analysé.

Le second sujet traite de l'utilisation des variables hydrodynamiques pour la modélisation

de la cinétique du procédé de flottation selon distribution granulométrique des particules

introduites. La surface inter-faciale de bulle (Ib) est introduite à cet égard comme une

variable hydrodynamique fournissant plus d'informations sur la distribution de taille de

bulle que le taux surfacique de bulles qui est plus couramment utilisé. De plus, la

corrélation entre la constante cinétique, la taille des particules et certaines variables

hydrodynamiques a été analysée en utilisant une projection de structures latentes (PSL).

Les résultats indiquent que l'importance relative des variables hydrodynamiques pour la

modélisation de la cinétique de flottation dépend de la distribution granulométrique des

particules. Finalement, les variables hydrodynamiques suggérées pour chaque classe

granulométrique considérée ont été utilisées pour produire des modèles de régression

mono-variable de la constante cinétique.

v

Abstract

Industrial flotation is a continuous and often multistage process, where particles of a given

mineral species (usually the targeted one), present in different sizes, encounter a large

amount of gas bubbles (normally air) to produce mineral–bubble aggregates, which are

removed from the flotation device (cell or column) as a valuable product (concentrate). The

bubble content inside the cell is characterized by the prevailing hydrodynamic conditions

(known as gas dispersion variables), which in turn are known to influence the flotation

performance.

This research project deals with two major topics. The first one is identifying the effect of

mineral particles on hydrodynamic variables, and the effects of hydrodynamic variables on

the final water recovery. For this purpose, the effect of solid particles on the bubble size

distribution and gas hold-up, as well as the correlation between bubble size distribution and

gas hold-up in column flotation were studied. It is followed by an assessment of the effect

of the gas hold-up, bubble size and bubble surface area flux (Sb) on the amount of water

reporting to the concentrate.

The second topic deals with applying appropriate hydrodynamic variables for flotation

modeling based on a given introduced particle size distribution. The interfacial area of

bubbles (Ib) is introduced to address this issue as a hydrodynamic variable providing more

information about the size distribution of bubbles than the commonly used bubble surface

area flux. The correlation between the flotation rate constant and particle size as well as

given hydrodynamic variables using a Projection to Latent Structures (PLS) has been

analyzed. Results suggest that the relative importance of hydrodynamic variables for

flotation rate modeling depends on the particle size distribution. Finally the suggested

hydrodynamic variables for each of the various particle size-classes considered were used

to produce single variable models for the flotation rate constant.

vii

Table of contents

Résumé .................................................................................................................................. iii

Abstract .................................................................................................................................... v

List of Tables .............................................................................................................................. xi

List of Figures ........................................................................................................................... xiii

Nomenclature ........................................................................................................................... xvii

Acknowledgments .................................................................................................................... xxi

Foreword .............................................................................................................................. xxiii

Chapter 1 Introduction ................................................................................................................ 1

1.1 Principles of flotation ....................................................................................................... 1

1.2 Hydrophobic and hydrophilic particles ............................................................................ 1

1.3 Use of reagents in flotation practice ................................................................................ 2

1.3.1 Collectors .......................................................................................................................... 2

1.3.2 Modifiers ........................................................................................................................... 3

1.3.3 Frothers ............................................................................................................................. 3

1.4 Bubble-particle collection ................................................................................................ 4

1.4.1 Bubble-Particle Collision .................................................................................................. 5

1.4.2 Particle attachment ............................................................................................................ 7

1.4.3 Particle detachment ........................................................................................................... 8

1.4.4 Particle entrainment .......................................................................................................... 9

1.5 Flotation devices ............................................................................................................ 10

1.5.1 Mechanical cells ............................................................................................................. 10

1.5.2 Flotation columns ........................................................................................................... 11

1.6 Flotation modeling ......................................................................................................... 15

1.6.1 Performance justification ................................................................................................ 15

1.6.2 Kinetic constant modeling .............................................................................................. 18

1.6.3 Residence time measurement in a flotation column ....................................................... 21

1.7 Gas dispersion properties ............................................................................................... 22

1.7.1 Gas hold-up (εg) .............................................................................................................. 22

1.7.2 Superficial gas velocity (Jg) ............................................................................................ 22

viii

1.7.3 Bubble surface area flux (Sb) .......................................................................................... 23

1.7.4 Bubble size distribution .................................................................................................. 26

1.7.5 Gas dispersion properties range and effects ................................................................... 27

1.8 Relation between the kinetic constant and gas dispersion properties ............................. 29

1.9 Problems associated to the use of a single Sb value........................................................ 30

1.10 Assumptions made in this experimental work................................................................ 33

1.10.1 Hydrophobicity ............................................................................................................... 33

1.10.2 Solid particle residence time .......................................................................................... 33

1.10.3 Ultimate recovery (R∞) ................................................................................................... 34

1.10.4 Froth depth and its effect on the final recovery ............................................................. 35

1.11 Experimental set-up and results...................................................................................... 35

1.12 Objectives ....................................................................................................................... 36

1.13 Outline of the thesis ........................................................................................................ 37

Chapter 2 Effect of particles on the bubble size distribution and gas hold-up in column

flotation ................................................................................................................................. 41

2.1 Introduction .................................................................................................................... 42

2.2 Experimental set-up ........................................................................................................ 43

2.3 Results and discussion .................................................................................................... 48

2.3.1 Solid particles on the bubble size distribution ............................................................... 48

2.3.2 Effect of solid particles on the gas hold-up .................................................................... 55

2.3.3 Discussion ...................................................................................................................... 59

2.4 Conclusion ...................................................................................................................... 61

Chapter 3 The effect of gas dispersion properties on water recovery in a laboratory

flotation column ...................................................................................................................... 63

3.1 Introduction .................................................................................................................... 64

3.2 Results and discussion .................................................................................................... 66

3.2.1 Effect of gas dispersion properties on the water recovery ............................................. 66

3.2.2 Effect of hydrophobic particle size on the water recovery ............................................. 69

3.2.3 Effect of gas dispersion properties on the carrying capacity ......................................... 70

3.3 Conclusion ...................................................................................................................... 72

ix

Chapter 4 On the relationship between hydrodynamic characteristics and the kinetics of

column flotation. Part I: modeling the gas dispersion ............................................................. 73

4.1 Introduction .................................................................................................................... 74

4.1.1 Hydrodynamic variables and particle size ...................................................................... 75

4.1.2 Bubble surface area flux models ..................................................................................... 76

4.1.3 Bubble size measurement ............................................................................................... 77

4.1.4 Bubble size distribution .................................................................................................. 78

4.2 Equations of gas dispersion............................................................................................ 80

4.2.1 Normal distribution ......................................................................................................... 80

4.2.2 Log-normal distribution .................................................................................................. 82

4.3 Test procedure ................................................................................................................ 84

4.4 Results and discussion ................................................................................................... 85

4.4.1 Modeling the experimental bubble size distributions ..................................................... 85

4.4.2 Correlations between hydrodynamic variables ............................................................... 88

4.5 Conclusion ..................................................................................................................... 92

Chapter 5 On the relationship between hydrodynamic characteristics and the kinetics of

flotation. Part II: model validation ........................................................................................... 95

5.1 Introduction .................................................................................................................... 96

5.2 Test procedure ................................................................................................................ 99

5.3 Results and discussion ................................................................................................. 100

5.3.1 Particle size and kinetic constants ................................................................................ 100

5.3.2 Hydrodynamic variables and rate constant ................................................................... 101

5.3.3 Experimental validation ................................................................................................ 103

5.4 Interfacial area of bubbles and rate constant ................................................................ 107

5.5 Conclusion ................................................................................................................... 111

Chapter 6 Single variable rate constant models ...................................................................... 113

6.1 Introduction .................................................................................................................. 113

6.2 Flotation kinetics of fine particle size-class ................................................................. 113

6.3 Flotation kinetics of large particle size-class ............................................................... 118

6.4 Flotation kinetics of particle size-class spanning a wide range ................................... 120

6.5 Particle size effect on the kinetics of flotation ............................................................. 121

x

6.6 Conclusion .................................................................................................................... 122

Chapter 7 Thesis conclusion .................................................................................................. 123

7.1 Future work .................................................................................................................. 124

References ............................................................................................................................... 127

Appendix A .............................................................................................................................. 137

A.1 Sampling ............................................................................................................................ 137

A.2 RTD measurement ............................................................................................................. 138

A.3 Kinetic constant calculation .............................................................................................. 140

Appendix B .............................................................................................................................. 141

B.1 Selection of the shear water rate ........................................................................................ 141

B.2 Effect of the temperature and duration of the test on the bubble size ............................... 141

B.3 Relationships between hydrodynamic variables in two and three-phases ......................... 142

Appendix C .............................................................................................................................. 147

C.1 Solid Characteristics .......................................................................................................... 147

Appendix D .............................................................................................................................. 151

D.1 Radial gas dispersion analysis (Banisi et al., 1995) .......................................................... 151

Appendix E ............................................................................................................................... 153

Appendix F ............................................................................................................................... 155

F.1 Correlation between hydrodynamic variables .................................................................... 155

F.2 Regression methods ........................................................................................................... 157

F.3 Datasets .............................................................................................................................. 160

Table F.1 Experimental results ................................................................................................. 160

xi

List of Tables

Table 1.1 Yoon’s model parameters for different flow conditions ......................................... 6

Table 1.2 Example of a same Sb originating two different rate constants. ........................... 31

Table 1.3 Talc particle sizes measured by twice wet screening and Malvern 2000 ............ 34

Table 2.1 Summary of the experimental plan ....................................................................... 53

Table 2.2 Generated gas hold-up and d32 for gas-water and gas-slurry systems .................. 59

Table3.1 Experimental plan ................................................................................................. 67

Table 4.1 Experimental conditions for three types of BSD .................................................. 88

Table A.1 Measured RTDs of the tests ............................................................................... 140

Table B.1 Effect of temperature and duration of test on bubble size distribution .............. 142

Table B.2 Values of hydrodynamic variables generated by manipulating Jg and Jsw ....... 143

Table C.1 XRF results of quartz (%) ................................................................................. 147

Table C.2 Mineralogy analysis results ................................................................................ 148

Table C.3 XRF results for quartz particles (%) .................................................................. 149

Table F.1 Experimental results ........................................................................................... 160

Table F.2 Gorain's results ................................................................................................... 161

Table F.3 Massinaei et al. data base ................................................................................... 162

Table F.4 Kracht et al. data base ....................................................................................... 163

Table F.5 Jincai et al. data base .......................................................................................... 164

xiii

List of Figures

Figure1.1 Distribution of radial components of the particles and liquid velocity on the

bubble surface ......................................................................................................................... 7

Figure 1.2 Mechanical cell schematic (Fuerstenau et al., 2007) ........................................... 11

Figure 1.3 Flotation column schematic ................................................................................. 12

Figure 1.4 Three layers of the froth zone .............................................................................. 14

Figure 1.5 Typical grade–recovery curves for froth flotation ............................................. 17

Figure 1.6 Different bubble size distributions all having the same d32 ................................. 26

Figure 1.7 Different pairs of bubble size and gas velocity leading to a unique BSAF ........ 31

Figure 1.8 Two BSDs having same d32 originate different kinetic constant ........................ 32

Figure 1.9 Light beam passing through loaded bubble ......................................................... 35

erugiF 2.1 Schematic of the experimental set-up ................................................................. 44

Figure 2.2 Schematic of the εg sensor (Gomez et al., 2003) ................................................. 45

Figure 2.3 Schematic of a frit-sleeve sparger (Kracht et al., 2008) ...................................... 46

Figure 2.4 Example of an original captured image and detected bubbles .......................... 47

Figure 2.5 Effect of the solid percent on the bubble size distribution .................................. 49

Figure 2.6 Example of bubble images using 25 ppm F150, and for Jg = 1 cm/s .................. 50

Figure 2.7 Effect of the solid percent on the bubbles size histogram ................................... 52

Figure 2.8 Bi-modal distribution due to coalescence ........................................................... 53

Figure 2.9 The effect of the particle size on the bubble d32 at constant gas rate (1 cm/s) .... 54

Figure 2.10 The effect of solids on the parameters of the bubble size distribution .............. 57

Figure 2.11 Bubble size and gas hold-up correlation in two and three-phase system .......... 60

Figure 2.12 Bubble size and gas hold-up correlation in three-phase system ........................ 61

xiv

Figure 3.1 Effect of the mean bubble size (d32) on water recovery for constant as rate ...... 68

Figure 3.2 Effect of the gas hold-up on the water recovery for constant as rate .................. 68

Figure 3.3 Effect of the mean talc particle size on the water recovery ................................ 69

Figure 3.4 Effect of (a) the bubble size, (b) gas holdup, and (c) bubble surface area flux on

the carrying capacity ............................................................................................................ 71

Figure 4.1 Bubble size distribution histogram ..................................................................... 85

Figure 4.2 Example of a bubble size histogram for T1 distribution .................................... 86

Figure 4.3 Example of a cumulative bubble size T2 distribution fitted with multi-shape

density function .................................................................................................................... 87

Figure 4.4. Example of a cumulative bubble size T3 distribution fitted with a lognormal

density function .................................................................................................................... 88

Figure 4.5 Shows the correlation between Ibc and Ibm (a) for T2 distribution (b) for T3

distribution ........................................................................................................................... 89

Figure 4.6 Correlation between : (a) Ibc and Sb for T2 distributions, (b) Ibc and Sb for T3

distributions, (c) Ibm and Sb for T2 distributions, (d) Ibm and Sb for T3 distributions ........... 90

Figure 4.7 Correlation between: (a) Ibc and εg for T2 distributions, (b) Ibc and εg for T3

distributions, (c) Ibm and εg for T2 distributions (d) Ibm and εg for T3 distributions ............. 91

Figure 5.1. Relative importance of particle size (P.S.), εg, Sb and Ib .................................. 101

Figure 5.2. Importance of variables in VIP projection for three sizes of particles ............ 104

Figure 5.3 PLS regression for hydrodynamic variables .................................................... 105

Figure 5.4. PLS regression for hydrodynamic variables .................................................... 106

Figure 5.5 PLS regression for hydrodynamic variables ..................................................... 107

Figure 5.6 PLS regression for hydrodynamic variables ..................................................... 107

Figure 5.7 PLS regression for hydrodynamic variables using mixed-size class particles . 108

Figure 5.8 PLS regression for hydrodynamic variables ..................................................... 109

Figure 5.9 PLS regression for hydrodynamic variables using the datasets combined ...... 110

xv

Figure 6.1 Talc collection rate constant as a function of the bubble size ........................... 114

Figure 6.2 Talc collection rate constant as a function of Sb for two particle size-classes .. 114

Figure 6.3 Kinetic constant as a function of d32 for the collection zone and the overall

process ................................................................................................................................ 115

Figure 6.4 Nonlinear regression for d32 and kinetic constant, Predicted and actual values 117

Figure 6.5 Nonlinear regression for Sb and kinetic constant, Predicted and actual values . 118

Figure 6.6 Flotation rate constant as a function of εg for large particle size-class .............. 119

Figure 6.7 Predicted and actual kinetic constants for the 106/150 µm particle size-class . 120

Figure 6.8 Flotation rate constant as a function of Ib for large particle size-class .............. 120

Figure 6.9 Linear regression for Ib and k for large range particle size-class ...................... 121

Figure A. 1 shows the sampling points from flotation column .... ......................................137

Figure A. 2 schematic of sampler and its position in the column ....................................... 138

Figure A. 3 Example of measured RTD by conductivity cells; Left) tracer impulse in the

feeding point, Right) the detected response of tracer in the tailing point ........................... 139

Figure B. 1 Bubble size variation by shear water to a frit and sleeve sparger ... .................141

Figure B. 2 d32 variations over time .................................................................................. 142

Figure B. 3 a) Variations of d32 with the gas and shearwater rates, b) variations of εg with

the gas and shearwater rates and c) variations of Sb with the gas and shearwater rates .... 145

Figure C.1 Mineral liberation – talc....................................................................................148

xvii

Nomenclature

BSD bubble size distribution f(db) size distribution function

c concentrate assay Ib interfacial area of bubbles

C total concentrate weight Ibc calculated Ib

db bubble size Ibm predicted Ib from d32 and εg

db max maximum bubble size Jg superficial gas rate

db min minimum bubble size k flotation rate constant

d0 bubble size at Jg = 0 µ mean of the BSD

d10 mean diameter of the BSD PC collision probability

d32 Sauter mean diameter of the BSD PD detachment probability

EA attachment efficiency R recovery

Ec collision efficiency Re Reynolds number

Ek collection efficiency Rp particle radius

erf (x) error function Sb bubble surface area flux

εg gas hold-up t assays of the tailing

f assays of the feed tind induction time

F total feed weight tslide sliding time

Fa attachment force σ2 variance of the BSD

xix

“Whatever you look for-will become your core” 1

To my immortal beloved ones: my parents

1 Rumi, (1207-1273) “Rubaiyat”, verse 1815, line 4.

xxi

Acknowledgments

I would like to sincerely thank my supervisor, Professor Rene del Villar, for his guidance,

support, and insightful comments throughout the years. Under his tutelage, I have learned

to question myself and to think more sophisticatedly in order to perform quality research.

As a mentor, his contributions to my professional development are invaluable.

Special thanks are likewise expressed to Professor Jocelyn Bouchard, my Co-supervisor,

for providing perspective that contributed significant to this work. I am grateful for his

diligent efforts to mentor me in many directions throughout my years as a PhD student.

I would like to express my gratitude towards Professor Claude Bazin for their thought-

provoking comments.

I would like to extend my gratitude to Jonathan Roy and Alberto Riquelme Diaz for their

dedicated cooperation at the beginning of this research. Special thanks to Vicky Dodier for

assistance in setting up the apparatus and technical support. I would like to acknowledge

Dr. Massoud Ghasemzadeh Barbara and Professor Carl Duchesne for cooperating on the

statistical analysis of the data.

I acknowledge the financial support received from NSERC (Natural Sciences and

Engineering Research Council of Canada) and Corem, throughout my studies at Laval,

Finally, I am forever thankful for the encouragement and support from my family,

especially my parents, Dr. Damavandi and Dr. Amir Vasebi without whom this endeavor

would not have been possible.

xxiii

Foreword

Chapter 2 is based on the article "Effect of particles on the bubble size distribution and gas

hold-up in column flotation" submitted in October 2014 to the International Journal of

Mineral Processing authored by Ali Vazirizadeh, Jocelyn Bouchard and Yun Chen. This

article presents some explanations for the solid particle effects on bubble size and gas hold-

up through the investigation of micro-phenomena. My role in the preparation of this article

is that of main writer supervised by the co-authors for corrections.

Chapter 3 uses the material of the article "The effect of gas dispersion properties on water

recovery in a laboratory flotation column" accepted and presented by myself at the 2014

IMPC Conference, Santiago, Chile, from 20-24 October 2014. I participated in the writing

as the first author, followed by Jocelyn Bouchard and Rene del Villar who contributed to

the editing of this document. The main objective is to study the effect of the gas rate, gas

hold-up, bubble size and bubble surface area flux on the water recovery to the concentrate.

I performed the laboratory testwork and analysis under the supervision of professors.

Bouchard and del Villar.

Chapter 4 is based on the article "On the relationship between hydrodynamic characteristics

and the kinetics of column flotation. Part I: modeling the gas dispersion". The interfacial

bubble area concept is introduced in this article as a new hydrodynamic variable; its

correlations with the other hydrodynamic variables are presented. This article was accepted

for publication in Minerals Engineering in September (2014). I did all the laboratory, data

analysis and mathematical developments. I participated in the writing as the first author.

My co-authors, Professors Jocelyn Bouchard and del Villar corrected and edited the

manuscript.

Chapter 5 uses the material presented in the article "On the relationship between

hydrodynamic characteristics and the kinetics of flotation. Part II: model validation". The

correlation between the flotation rate constant and particle size as well as some

hydrodynamic variables is investigated in this article. This article was accepted for

publication in Minerals Engineering in November (2014). My contribution as the first

author was performing the testwork, analyzing the data and writing the manuscript. Mr.

Ghasemzadeh Barbara and Prof. Duchesne of Chem. Eng. Dept. contributed to the

statistical analysis using PLS. Professors Bouchard and del Villar supervised the data

analysis, as well as the preparation and edition of the article.

1

Chapter 1 Introduction

1.1 Principles of flotation

Froth flotation is a mineral separation method based on differences in water-repellency

characteristics of the various mineral species contained in aqueous slurry. The hydrophobic

particles attach to gas (generally air) bubbles injected at the bottom of the separation vessel.

As a result of their lower specific gravity, the bubble-particle aggregates ascend to the

vessel slurry surface, whereas the hydrophilic particles, that remain completely wetted, stay

in the liquid phase and therefore are removed from the bottom of the device. Froth flotation

can be practiced for a broad range of mineral separations, as it is possible to use chemical

treatments to selectively modify mineral surface characteristics so that they have the

required hydrophobicity for the separation to occur. Some examples are separation of

sulfide minerals from siliceous gangue, separation of coal from ash-forming minerals and

removing silicates from iron ores. Flotation is also used in chemical engineering fields such

as processing recycled printed papers, where ink (carbon) is separated from paper-pulp; it is

also used in the oil production from tar sands, where oil (carbon derived) is removed from

sand (siliceous gangue). Flotation is particularly useful for processing fine-grained ores that

are not amenable to other physical separation methods such as gravity- or magnetic-based

methods.

1.2 Hydrophobic and hydrophilic particles

In terms of their behaviour in an aerated liquid (water), particles can be classified in two

groups: those that are readily wettable (hydrophilic) and those that are water-repellent

(hydrophobic). If a mixture of hydrophobic and hydrophilic particles is suspended in water

contained in a device where air bubbles are injected, the hydrophobic particles will tend to

attach to the bubbles and float with them to the top of the device (called a flotation cell)

where they accumulate as a persistent froth, heavily loaded with the hydrophobic mineral,

wherefrom it can be removed as a product, usually a valuable mineral concentrate. Since

hydrophilic particles do not attach to the air bubbles, they will remain in the suspension,

flowing down to the bottom exit of the cell, as a refuse (tail).

2

1.3 Use of reagents in flotation practice

With very few exceptions, minerals are rarely suitable for froth flotation, because they are

naturally hydrophilic. Among the exceptions (natural hydrophobic minerals), it can be

mentioned coal and carbon derivatives (like oil, diamonds, graphite), native sulphur,

molybdenite and talc (used in this thesis as the hydrophobic component). Chemical

reagents are thus required to render hydrophobic the target (valuable) hydrophilic mineral

and also to preserve suitable froth characteristics.

1.3.1 Collectors

Collectors control the selective attachment of particles onto the bubble surface. They form a

monolayer on the particle surface, which essentially leads to generate a thin film of

nonpolar hydrophobic hydrocarbons, rendering the mineral surface hydrophobic. Selection

of the correct collector is important for an effective separation by flotation. Collectors are

generally classified depending on their ionic charge. They can be anionic, cationic, or non-

ionic. The anionic and cationic collectors contain a polar part, which selectively attaches to

the mineral surfaces and a non-polar part (organic), which projects out into the solution,

making hydrophobic the particle surface. Non-ionic collectors are hydrophobic products

(hydrocarbon oils, grease), which adhere to the surface forming a physical coat repelling

water. Collectors can either chemically cover the mineral surface (react with) or they can be

bonded on the surface by physical forces.

In the chemical coverage case, ions or molecules from the solution undergo a chemical

reaction with the surface, becoming strongly bonded. This process drastically changes the

nature of the surface. These kinds of collectors are highly selective, as the chemical bonds

are specific to particular atoms. Among them Xanthates are the most commonly used

collectors for sulfide mineral flotation.

In the case of physical coverage, ions or molecules from solution are reversibly linked with

the surface, the attachment is due to electrostatic attraction or Van der Waals bonding.

These collectors can be desorbed from the surface if some conditions change, such as

variations in pH or in the solution composition. The collectors, bonded through physical

coverage, are much less selective than the collectors chemically attached, as they might

3

adsorb on any surface that has the correct electrical charge or degree of natural

hydrophobicity. The collector available for the flotation of oxide minerals (oxyhydryl

collectors) are one of those. Typical examples are the fatty acid salts, e.g. sodium oleate,

the sodium salt of oleic acid.

1.3.2 Modifiers

Modifiers are chemical reagents changing the form the collector attaches to the mineral

surface. They are categorized in: (a) activators, which increase the adsorption of collector

onto a given mineral, (b) depressants, that prevent collector from adsorbing onto a mineral

and (c) pH-regulators.

The latter are probably the simplest type of modifiers, since the surface chemistry of most

minerals is determined by the pH. For example, minerals in general exhibit a negative

charge under alkaline conditions and a positive surface charge under acidic conditions.

Since each mineral change from negative charge to positive charge at some particular pH, it

is possible to control the degree of attraction of collectors to their surfaces simply by pH

adjustments. Acids are used generally to adjust pH in the low range, while alkalis such as

lime (CaO or Ca(OH)2) are used to rise pH. Pyrite can be separated from chalcopyrite by

regulating the suspension pH to an appropriate value.

The possibility of collector attachment increases by means of activators, i.e. activators

facilitate the task of a collector. A classical example of an activator is the use of copper

sulfate as an activator for sphalerite (ZnS) flotation with xanthate collectors. The copper

ions replace the zinc ions on the surface and the xanthate gets fixed more effectively to the

newly formed ”copper” surface, rendering the Cu-coated ZnS particle hydrophobic.

Depressants act by preventing collectors from attaching onto particular mineral surfaces,

thus they show the opposite effect of activators. For example, cyanide ions act as

particularly useful depressant for pyrite (FeS2) during sulfide minerals flotation.

1.3.3 Frothers

Frothers are compounds that help changing the size and stabilizing the generated bubbles,

so that they remain well dispersed in the slurry and form a stable froth layer, which can be

4

removed before bubble bursting. Many types of organic compounds serve this objective.

However, some of them have also collecting properties (e.g. amines, carboxylic acids, etc.).

In such cases, their dosage control might become cumbersome (same reagent having two

different actions). This is why the most commonly used frothers are alcohols, which do not

have collecting properties. The most commonly frother used in plant practice are the

methyl isobutyl carbinol (MIBC) and water-soluble polymers, such as polypropylene

glycols and polyglycols (F150). The polypropylene glycol frothers in particular are very

versatile and they can be manipulated to a wide range of froth properties. Some other

frothers available are natural product such as cresols and pine oils; however most of them

are considered obsolete, because of their cost, and are not as widely used as the synthetic

ones.

1.4 Bubble-particle collection

The basic flotation separation mechanism is hydrophobicity, which is the ability of a given

particle to get attached to an air bubble after a collision. However, such separation

mechanism is not perfect, as some undesired particles (low-grade non-liberated) might

attach to the bubbles or are entrained in the bubble wake entering into the froth zone.

Inversely, some hydrophobic particles might detach from the air bubbles during the

transport of the bubble-particle aggregates to the froth zone or due to bubble coalescence in

it, thus returning back to the pulp phase; also some hydrophobic particles do not attach to

the bubbles because of their size (too small or too large) and the number of available

bubbles (amount of bubble surface).

In a flotation device, a mineral particle is collected by an air bubble through one of two

main mechanisms: 1) particle-bubble attachment due to the particle surface hydrophobicity,

or 2) entrainment into the wake of the bubble and the boundary layer which does not relate

to the particle hydrophobicity.

The particle collection by attachment occurs due to bubble-particle collision followed by

adhesion of the hydrophobic particle on the surface of air bubble. The efficiencies of both

steps determine the final collection efficiency by attachment. However, there is another

micro process to be considered, called detachment. The detachment usually occurs when

5

the particle-bubble is disrupted by flow turbulence or bubble coalescence. Consequently,

the collection efficiency, Ek, is given by:

Ek = Ec EA (1- PD) (1-1)

where Ec is the collision efficiency, EA is the attachment efficiency, and PD is the

detachment probability.

1.4.1 Bubble-Particle Collision

The collision efficiency is the fraction of all hydrophobic particles swept out by the

projected area of bubbles that collide with the bubbles.

Yoon (1993) explained that the collision efficiency is strongly affected by both the particle

size and bubble size as well as by the system turbulence. He introduced a model between

collision efficiency and bubble size and particle size under different hydrodynamic

conditions:

( / )n

c p bP A d d (1-2)

where dp is the particle size diameter, db is the bubble mean size diameter, A and n are

model parameters depending on the flow conditions. Table 1.1 indicates the value of those

parameters for each flow condition, depending on the bubble Reynolds number (Re). He

quantified how increasing the bubble size decreases the probability of collision regardless

of the flow conditions.

6

Table 1.1 Yoon’s model parameters for different flow conditions

Flow condition A n

Stokes 2/3 2

Intermediate 0.723 4

2 15

Re 2

Intermediate 0.56

3 (3 /16)1

2 1 0.249

Re

Re 2

Potential 3 1

Based on experimental observation, Finch and Dobby (1990) indicated that when particle

size increases, collision efficiency also increases, but attachment efficiency decreases. This

is in accordance with Yoon's model (Eq. 1-2) and common knowledge. Dai et al. (2000)

came to the same conclusion as Finch and Dobby (1990), based on reviewing the previous

collision models.

Independently of the size of the particle being floated, large bubbles exhibit lower

collection efficiencies than fine bubbles. However, using very fine bubbles does not

improve the process selectivity because the possibility of particle-bubble collision for fine

and coarse particles are relatively similar (Dobby and Finch, 1987).

Weber and Paddock (1983) studied the particle-bubble collision based on the equation of

motion of spherical particle relative to a spherical bubble rising in an infinite pool of liquid.

According to their study, the collision efficiency is affected by two main steps: first, by

intercepting collisions, occurring for neutrally buoyant particles exactly following the fluid

streams, and second, by gravitational collisions which would occur for particles with

assumed zero dimension and finite settling velocity. The efficiency of both steps increase

by increasing the particle size and decreasing the bubble size.

7

The collection efficiency improves as a result of increasing the collision independently of

the attachment efficiency. Therefore, regarding Weber and Paddock's conclusion, the

collection efficiency would increase by generating finer bubbles.

1.4.2 Particle attachment

A key concept in the estimation of particle–bubble attachment probability is the induction

time, tind, first introduced by Sven-Nilsson (1934). Basically, the induction time is defined

as the sliding time of a particle on the bubble surface required for thinning and draining the

liquid film interposed between the particle and the bubble, until its rupture and actual

contact between particle and air bubble. In the flotation process, the induction time can be

interpreted as a threshold sliding duration (Nguyen and Schulze, 2003). When a particle

meets a bubble, it will first deviate from its initial trajectory, due to fluid forces, and then it

will slide on the bubble surface for a short period of time called tslide. If tslide equals or

exceeds tind, then attachment is expected to occur. The tslide depends on the distribution of

bubble-particle collision angles, the maximum bubble-particle contact angle and the

particle sliding velocity (Finch and Dobby, 1990). The particle collision angle is

determined from the angle made by the fluid streamlines and the bubble in the vicinity of

the bubble surface (Figure 1.1).

The maximum contact angle corresponds to the angle where the radial component of

particle settling velocity (directed toward the bubble surface) is equal to the radial

component of the liquid velocity (oriented away from the bubble surface).

Figure1.1 Distribution of radial components of the particles and liquid velocity on the

bubble surface

θ θmax

Radial component of particle settling velocity

Radial component of the liquid velocity

8

Moreover, Yoon (1993) stated that surface chemistry and system hydrodynamic conditions

have an enormous effect on the induction time, which is mainly determined by the particle

hydrophobicity. According to Yoon, For example, for two same-size mineral particles, the

strongly hydrophobic could have an induction time of around 15 ms, whereas the second

one, weakly hydrophobic, would require 40 ms for induction.

The reduction of particle size, for a given bubble size, may have an effect on flotation

selectivity. Longer sliding times give more chances to any particle to attach to bubbles,

irrespective of their degree of hydrophobicity. In other words, less hydrophobic particles

(containing some gangue at the surface) can have similar chances to attach to the bubble as

highly hydrophobic (richer) particles. Consequently, both sorts of small particles may

attach to the bubble and selectivity is then reduced.

On the other hand, selectivity increases if bubble size diminishes since the sliding time may

be too short compared to the induction time (Weber and Paddock, 1983).

While the collision process is more determined by physical parameters, adhesion is the

result of both physical and chemical factors. Since both the bubble size and particle size

affect collision and adhesion, the effect of these two variables is more pronounced on

collection efficiency.

1.4.3 Particle detachment

A particle will be detached from an air bubble if the detachment force exceeds the

attachment force. The maximum attachment force of a single particle can be calculated

from:

cosa pF =π σ R 1- θ (1-3)

where Fa is the attachment force, Rp is the particle radius, σ the liquid surface tension and θ

the three-phase contact angle (Nutt, 1960).

The detachment force is the sum of gravitational forces, shear forces and external vibratory

forces, the latter depending on the particle mass, the vibration amplitude and frequency

(Cheng and Holtham, 1995). The particle size and flow turbulence in flotation – the latter

mostly determined by the flotation device – affect on each individual detachment force.

9

However, in general, the probability of detachment is less important in a flotation column

than in a mechanical cell, because the flow turbulence in the flotation column is much

lower than in a mechanical cells (Finch and Dobby, 1990). In addition, the probability of

detachment of fine particles (less than 100 μm) is mostly negligible.

1.4.4 Particle entrainment

According to Cilek (2009), the entrainment mechanism not only refers to the recovery of

hydrophilic particles but also to hydrophobic particles recovered without being attached to

air bubbles. The particle entrainment is directly related to the fraction of water reporting to

the concentrate, called ‘water entrainment’. There is agreement in that the wake of the

bubble swarm is responsible for the water entrainment to the froth. The volume of the

bubble wake is a function of the bubble size and its rising velocity, as well as the liquid

viscosity, therefore the water entrainment should be determined by such variables.

The role of frother type and concentration (as the manipulated variable in flotation) on

water entrainment can be illustrated through its effect on the bubble size and velocity

(Ekmekçi et al., 2003). Since frother modifies the bubble size and its velocity, it indirectly

determines the bubble wake and consequently the volume of water reporting to the

concentrate.

In addition, it appears that gas dispersion properties in general (such as the bubble size, gas

hold-up, etc.) play a major role, similar to the frother effect, on the particle entrainment

through their effect on bubble wake (Nelson and Lelinski, 2000; Phan et al., 2003;

Rodrigues et al., 2001; Yoon, 2000).

Yianatos et al. (2009) observed a selective entrainment of fine particles (less than 45 μm) in

a large industrial cell as. On the other hand, the recovery of coarse particles (larger than

150 μm) by entrainment is considered negligible, less than 0.1% (Yianatos et al., 2009;

Zheng et al., 2006). Experimental evidence also confirms that if the liberation size is

smaller, grinding product (cyclone overflow) must be finer which might imply a greater

content of fine gangue particles, easier to be entrained than coarser ones (Guler and

Akdemir, 2012).

10

Regarding to the micro-processes presented in this section (1.4), it seems that the bubble

size and particle size, as well as the flow conditions in the flotation device, play the most

important role on particle recovery when particles have the same hydrophobicity.

1.5 Flotation devices

The flotation process is accomplished in a device having two main roles: keeping the pulp

in suspension and providing air bubbles. The device should also provide a mean to

evacuate the two products: concentrate (loaded froth) and tails (depleted mineral pulp).

There are two main flotation devices well accepted in the industrial practice, flotation

columns (and other similar devices) and mechanical cells. The latter are usually connected

in banks having enough cells to assure the required particle residence time for adequate

recovery. Subsequently, various banks, each having its particular targets of concentrate

grade and recovery, are interconnected to form a flotation circuit, capable of attaining the

desired overall metallurgy (product).

1.5.1 Mechanical cells

A mechanical flotation cell is basically a cylindrical or rectangular vessel or tank fitted with

an impeller. The impeller function is to mix thoroughly the slurry to keep particles in

suspension, and also to disperse the injected air into fine bubbles, providing conditions

promoting bubble-particle collisions. It is worth mentioning that mechanical cells operate

either on self aspiration mechanism (no air feed rate control) or controlled compressed air.

Formed bubble-particle aggregates rise up through the cell by buoyancy and are removed

from it into an inclined drainage box called "concentrate launder". Water sprays will help

breaking the aggregates to make possible the pumping of the concentrate downstream.

Particles that do not attach to the bubbles are discharged out from the bottom of the cell.

A mechanical cell requires the generation of three distinct hydrodynamic regions for

effective flotation. The region close to the impeller encompasses the turbulent zone needed

for solids suspension, dispersion of gas into bubbles, and bubble-particle interaction. Above

the turbulent zone lies a quiescent zone where the bubble-particle aggregates moves up in a

relatively less turbulent area. This zone also helps in sinking the amount of gangue minerals

that may have been entrained mechanically. The third region above the quiescent zone is

11

the froth zone serving as an additional cleaning step as a result of bubble coalescence and

other phenomena (Fuerstenau et al., 2007). Figure 1.2 shows a typical schematic of

mechanical cell.

Figure 1.2 Mechanical cell schematic (Fuerstenau et al., 2007)

1.5.2 Flotation columns

A flotation column is typically a tall vertical cylinder or rectangular with no mobile parts

(agitator), fed with a mineral pulp (top third of column), air bubbles is injected (always

generated by controlled compressed air) at its very bottom. These bubbles rise-up in

counter-current with the descending flow of pulp, so that the contained hydrophobic

particles are able to attach to the air bubbles. The so-formed bubble-particle aggregates are

carried upwards by a buoyancy effect. The zone where this process takes place is called the

collection zone, corresponding to 75% to 90% of the total column height. The hydrophilic

particles and some non-collected hydrophobic particles move downwards throughout the

collection zone, being finally discharged through the tailing outlet at the bottom of the

column. The ascending bubble-particle aggregates accumulate in the upper part of the

12

column before overflowing into a launder as a concentrate. This zone, which stands from

the pulp-froth interface up to the top of the column, is called the froth zone or cleaning. Its

thickness generally varies between 10% and 25% of total column height (Zheng, 2001 ),

although in specific applications it can be as shallow as 40 cm. Figure 1.3 shows a

schematic of a flotation column unit.

Figure 1.3 Flotation column schematic

1.5.2.1 Bubble generating systems

The bubble size is essentially determined by the type of frother and the mechanism used for

bubble generation. It can be modified through the frother concentration, and gas feed rate to

the device. The bubble generation system is another feature which distinguishes flotation

columns from mechanical cells. In the case of flotation columns the most commonly used

for bubble generation method is through internal spargers near to the bottom of the device.

There are two categories of internal spargers: porous spargers and single or multinozzle

spargers (Finch and Dobby, 1990). Whereas the former are used at laboratory scale, the

second method is the most common at industrial scale. A variant has been introduced in the

FeedConc

Tail

Water

Air

Wash

Water

13

so-called reactor-separator columns, in that gas and slurry are brought into intense contact

in an external section (called reactor), with a very short residence time. Then the aeratered

pulp passes to a quiescent zone (separator) where bubble aggregates separate from settling

gangue particles, each stream exiting the separator at a different outlet. The amount of

slurry introduced to the sparger is usually a small fraction of tails flow rate (around 10%)

(Massinaei et al., 2009).

In the mechanical cells, air enters to the device through a concentric pipe surrounding the

impeller shaft. The rotating impeller tips create a high-shear zone where air is broken up

into a dispersion of bubbles, when passing through a static set of bars around the impeller.

The bubbles are deviated outwards from the impeller tips and dispersed throughout the

solid-liquid mixture (slurry) into the next zone (quiescent) of the cell (Fuerstenau et al.,

2007).

1.5.2.2 Froth zone, wash water and bias rate

Another feature distinguishing the flotation column from the mechanical cell is the addition

of wash water above or slightly inside the froth zone. A fraction of this flow of fresh water

travelling down the froth zone, would wash-out the hydrophilic particles entrained into the

froth zone, thus avoiding their recovery to the concentrate. The remainder of the water

injected helps the concentrate overflowing into its launder. To assess the wash water

cleaning performance, the bias rate concept – i.e. the fraction of the wash water flow rate

going downwards through the pulp-froth interface – was introduced by Finch and Dobby

(1990).

14

Figure 1.4 Three layers of the froth zone

The froth zone normally consists of three regions: an expanded bubble bed (next to the

interface), a packed bubble bed above the previous one and a conventional draining froth at

the top (Figure 1.4). As for practical considerations (plugging of water nozzles) wash water

is sometimes sprayed above the froth and therefore the conventional draining froth does not

establish properly (Finch and Dobby, 1990).

Bubbles pass through the collection zone and enter the expanded bubble bed after colliding

with the first layer of bubbles, which defines a very distinct interface. These incoming

bubbles have a relatively homogeneous and small size and remain spherical all the way up.

Bubbles colliding with the interface, generate shock pressure waves promoting collisions

up through the expanded bubble bed. This phenomenon seems to be the main cause of

bubble coalescence, promoting film thinning and finally its rupture.

The packed bubble bed region expands to the wash water inlet level. The fractional liquid

content is lower than in the expanded bubble region and bubbles keep a relatively spherical

shape, but the range moves towards larger bubbles. Bubbles rise upward with close to

plug-flow conditions, thus promoting a good distribution of the wash water.

Expanded bubble bed

Packed bubble bed

Conventional draining froth

15

Above the wash water inlet level lays the conventional draining froth, therefore the net flow

of water is ascending here is negative. The main aim of this region is to convert froth

vertical motion into horizontal motion to collect the froth.

1.6 Flotation modeling

1.6.1 Performance justification

There is no universal way for expressing the effectiveness of a separation, but several

useful indices exist for evaluating the quality of the flotation process. The following are the

most commonly used.

Ratio of concentration: defined as the weight of feed divided by the weight of concentrate,

that is:

ratio of concentration = F/C (1-4)

where F is the total feed weight and C is the total concentrate weight .

The drawback of this performance index is the need of weight values. Although from

laboratory experiments these data can be obtained, in the industrial practice it is unlikely

that the ore is weighed, only assays are available. However, is possible to express the ratio

of concentration in terms of ore assays. From the definition of the ratio of concentration

(F/C) and using the following mass balance equations, it can be calculated:

F= C+T (1-5)

Ff=Cc +Tt (1-6)

where f, c, and t are assays of the feed, concentrate, and tailings, respectively. Rearranging

Eq. (1-5) and Eq. (1-6) gives:

c tF

C f t (1-7)

Percent recovery: The percent recovery is the percent of mineral in the original feed that is

recovered in the concentrate. By means of weights and assays it can be calculated by:

16

100C c

RF f

(1-8)

or by replacing weights by assays (from material balances):

100

f tcR

f c t

(1-9)

where R is the percent recovery.

Enrichment ratio: The enrichment ratio is directly calculated from the assays as c / f.

Mass pull: The mass pull is the inverse of the ratio of concentration, thus:

f tC

F c t (1-10)

Grade-recovery curves

Although the above mentioned indices are useful for comparing flotation performance for

different conditions, a “grade–recovery curve”, constructed from concentrate grade and

recovery values at different operating conditions, is a more useful tool for inferring the

optimal operating conditions. This is a graph of the recovery of the valuable mineral against

the concentrate grade for given operating conditions, and it is particularly practical for

comparing separations where both the grade and the recovery are varying. A set of grade–

recovery curves is shown in Figure 1.5.

17

Figure 1.5 Typical grade–recovery curves for froth flotation

Controlling of grade and recovery in flotation processes has received significant attention

from researchers in the past years. Since more than hundred variables affect the flotation

process (Arbiter, 1962), knowing which variables have more influence, can help controlling

flotation performance variables, i.e. the grade and recovery. On the other hand, finding the

relationship between manipulated variables and controlled variables is imperative to find

out set-points leading to improved metallurgy. Therefore, some models for recovery

parameters have been proposed. The form of such models and their definition are presented

in the next section.

Another important parameter often used for cell/column design is the so-called carrying

capacity. The carrying capacity of the cell is calculated based on the recovered mass of

particles, both hydrophobic and hydrophilic, per unit cell surface per unit of time. This

Improving Performance

Pure

Mineral Assay (%) Feed

100

0

Reco

very

(%)

18

variable is directly related to the bubble surface available for collection, and therefore to the

bubble size.

2

1( )min

dC kgCarrying Capacity

dt A m

(1-11)

where A is the cross-section area of the flotation cell and t is the time unit and C is total

recovered mass.

1.6.2 Kinetic constant modeling

To model the flotation process, two main approaches are possible: the empirical modeling

and the phenomenological modeling.

In empirical models, input and output variables are related through suitable mathematical

equations. Statistical methods are used to define dependent and independent variables and

to estimate the curve fitting parameters. Obtained parameters do not necessarily have any

physical meaning and they are valid just for the limited conditions in which the model was

calibrated.

Phenomenological modeling provides an explanation for causes and effects, which are

related to physical and chemical conditions of the process. Phenomenological models are

divided in three main categories: kinetic, probabilistic and population balance models.

Probabilistic models are based on some sub-processes such as collision, adhesion and

detachment and can be used as a bridge between micro- and macro-models. Kinetic models

use the chemical reactor analogy and consider flotation as a reaction between bubbles and

particles (Polat and Chander, 2000). A good understanding about mixing conditions in the

flotation process is also required to obtain a reliable kinetic model. For instance, mixing

conditions in a laboratory column (5 cm diameter) are close to plug flow pattern, whereas

in industrial columns they are something between plug flow and perfect mixing patterns

(Finch and Dobby, 1990).

There is an expression relating the collection efficiency to the flotation rate constant. For

the situation of gas bubbles rising through a column of water containing hydrophobic

particles at a concentration cp and having collection efficiency Ek, and given a cubic volume

19

of water with side dimension L, the particle collection process is represented by the rate of

particle removal from the slurry by air bubbles. In other words, it can be calculated from

the rate of particle removed per bubble times the number of bubbles.

At a gas rate Qg and a slip velocity Usg (the velocity of gas relative to slurry), this

expression is equivalent to:

23

34 6

p gbsg p k

sgb

dc Qd LL U c E

dt Ud

(1-12)

After cancelling out some terms:

1.5

p g k

p

b

dc J Ec

dt d (1-13)

where 2

g

g

QJ

L .

Eq. (1-13) is equivalent to the expression of first-order rate process where the first order

rate constant kc is given by:

1.5 g k

c

b

J Ek

d (1-14)

It is very well accepted that the flotation process follows a first order kinetics with respect

to the concentration of the floatable particles c as indicated in equation (1-15).

dc

k cdt

(1-15)

By integration and introduction of the recovery definition, it is possible to obtain the

following expressions, respectively for plug flow conditions (Eq.1-16) and perfect mixing

conditions (Eq.1-17):

(1 )

ckR e

(1-16)

1

c

c

kR

k

(1-17)

20

where τ is the particle mean residence time. Equation (1-16) for plug flow and Equation

(1-17) for perfect mixture indicate that increasing the flotation kinetic constant leads to an

increase in the recovery.

In the case of plug flow conditions (or a batch flotation processes), equation (1-16) is

always valid since the kinetic constant is invariant. However in actual flotation conditions,

particle size and bubble size are present as a population distribution, which leads to

consider a kinetic constant distribution for proper flotation modeling. It is also possible

that the industrial flotation device exhibits a mixing behaviour which is neither plug flow,

nor perfect mixing, in which case the residence time distribution, obtained through tracer

tests, must be used.

In this regard, Polat and Chander (2000) applied a generalized form of equation (1-16)

shown in

0 0

(1 ) ( ) ( )ktRe F k E t dk dt

R

(1-18)

where R is the mineral recovery at time t and R∞ represents the ultimate recovery at infinite

time, E(t) is residence time distribution and F(k) is the kinetic-constant distribution function

for a continues process. This formulation is more appropriate since both the kinetic

constant distribution and the residence time distribution are being considered.

It is worth mentioning that the presented recovery models are for the collection zone only;

the froth zone recovery has not been considered so far. In fact, as a result of various events

taking place in the froth zone (e.g. bubble coalescence) some hydrophobic particle may

detach from the bubbles and return to the pulp zone, making the overall recovery lower

than that of the collection zone, calculated with the previous equations. To eliminate the

froth zone effect on the overall flotation recovery, a very shallow froth height (less than 15

cm) has to be used in the experimental determination of the kinetic constant, so that the

overall flotation recovery can be assumed to be equal to that of the collection zone. Since

the evaluation of k always implies a series of restrictions or inaccuracies, such as the model

will require to be compared to a reference value that is inaccurate. Based on, given all these

21

inaccuracies, a different approach would be worth to be explored. This will be tackled later

on in this thesis.

1.6.3 Residence time measurement in a flotation column

One of the most used methods for determining the residence time distribution (RTD) in

liquid or pulp systems, is the injection of a know amount of tracer (liquid or solid) at the

column feed port and to track its concentration with tailing flow as function of time.

Mean residence time can be calculated based on the time variation of tracer concentration

through:

exp

( )

( )

i iierimental

ii

t C t dt t C

CC t dt (1-19)

where τexperimental is the mean residence time (min), C(t) is the tracer concentration at time t.

The expected mean residence time is simply obtained from:

exp

eff

ected

V

Q (1-20)

where Veff (m3) is effective cell volume (collection zone without gas) and Q (m3/h) is feed

flow rate.

Various sorts of tracers have been used for RTD measurement, such as ionic salt tracers

(NaCl and KCl) and radioactive tracers (Br-82) for liquid RTD and MnO2 as a solid particle

tracer. However, liquid tracers are not as accurate as solid tracers for modeling of flotation

because the aim is to track particle behavior in flotation conditions through studying the

flotation rate. However, ionic salt tracers are simpler to use as their concentration can be

detected by conductivity measurements. These tracers are generally used for liquid RTD

measurements as a mean to evaluate the mixing conditions in the flotation device.

Radioactive tracers provide more accurate results since less amount of tracer is required

and the tracer is tracked by a non-invasive sampling system (measuring the radiation in the

output flow).

22

The best tracer for measuring the residence time of solid particles are the solid tracers, as

long as they have the same physical features (density, size and hydrophobicity) as the target

solid particles. There are some reports on the application of solid tracers like MnO2 for

tracking particle behavior and particle RTD in the flotation cells (Finch and Dobby, 1990).

Yianatos and Bergh (1992) have systematically used the radioactive tracer technique for

coding the different solid particles and measuring the RTD of hydrophilic particles in

industrial flotation columns and cells. More recently, Cole et al. (2010) presented a

Positron Emission Particle Tracking (PEPT) method which can be applied to particles in

froth flotation systems to observe the behavior of individual hydrophilic particles in a

mixed particle–liquid–gas system.

1.7 Gas dispersion properties

Gas dispersion properties have proved to be a key feature of the flotation process. Among

them, the most relevant are: the gas hold-up, superficial gas velocity, bubble size and

bubble surface area flux. This latter has received considerable attention and has been

reported to linearly correlate with the flotation rate constant and bubble carrying capacity,

both related to flotation performance (Gorain et al., 1998).

1.7.1 Gas hold-up (εg)

The volumetric fraction of gas in a given volume of the device is called the gas hold-up.

For a given column section, it can be estimated through the following equation:

g

Volume of bubbles

Volume of aerated pulp (1-21)

where volume could refer to the total volume of a given zone, in which case we are talking

of an overall gas hold-up) or part of it thus a local gas hold-up. The εg value strongly

depends on the prevailing gas rate and bubble size values (Gorain et al., 1998).

1.7.2 Superficial gas velocity (Jg)

The volumetric gas flow rate (cm3/s) divided by the cross-sectional area of the device (cm2)

is called the superficial gas velocity (cm/s), i.e.

23

g

g

QJ

A (1-22)

where Qg is the gas volumetric flow rate and A is the cell cross-sectional area. This

definition is quite useful, since is independent of the device size, therefore it is a scalable

value. For flotation devices the recommended Jg value should vary from 1-2 cm/s.

1.7.3 Bubble surface area flux (Sb)

Bubble Surface Area Flux (BSAF or Sb) is defined as follows:

surface area of bubbles generated per unit of time

cross sectional area of the columnbS (1-23)

Assuming that bubble sizes can be presented in m different discrete bubble sizes, then Eq.

(1-23) can be expressed mathematically as:

1

m

i bi

ib

cell

n S

SA

(1-24)

where Acell is the cell cross sectional area, ni is the number of bubbles of size i generated per

unit of time and Sbi is the surface area of a bubble of size i. Similarly, the volumetric gas

flow rate can be defined as follows:

1

m

g i bi

i

Q nV

(1-25)

where Vbi is the volume of an air bubble of size i. Then, Sb can be written as:

1

1

m

i big i

b m

celli bi

i

n SQ

SA

nV

(1-26)

Assuming that all bubbles are spherical, then2

bi iS d and 3 6bi iV d . Since

g g cellJ Q A , Sb can be calculated in terms of the superficial gas velocity Jg and a mean

bubble size, as

24

32

6 g

b

JS

d (1-27)

where d32 is the Sauter mean diameter defined as:

3

132

2

1

m

i i

i

m

i i

i

n d

d

n d

(1-28)

Another way to calculate d32 from a set of bubble sizes is:

3

132

2

1

b

b

N

i

i

N

i

i

d

d

d

(1-29)

where Nb represents the total number of collected bubbles.

1.7.3.1 Experimental models for Sb

Bubble size measurement in flotation devices is difficult enough to induce researchers to

look for simpler alternatives, among which the empirical modeling of Sb has been

considered as the most appropriate approach.

An empirical model was developed by Gorain et al. (1999), using extensive pilot and

industrial tests obtained with different cell types and sizes. It allows predicting the Sb in

mechanical flotation cells by

0.44 0.75 0.10 0.42

80123 ( )b s g sS N J A P (1-30)

where Ns is the impeller peripheral speed, Jg is the superficial air velocity, As is the impeller

aspect ratio and P80 is feed 80% passing size.

This model was shown to be acceptable for 20-150 µm particles in rougher, scavenger and

cleaner mechanical cells, but was not validated for flotation columns (Gorain et al., 1999).

Heiskanen (2000) evaluated Gorain’s model and formulated some criticisms. In particular,

he mentioned that "the measurement and computation of superficial gas velocity, and

partially also the bubble size may be biased in some conditions. This makes the bubble

25

surface area flux behave such that the final outcome is in doubt." Some other criticisms

were:

- the model validation did not address the different particle sizes at all;

- the results from Gorain’s evaluation show that increasing bubble size gives a higher

flotation rate constant and that poor dispersion gives a good flotation response, both

results being in contradiction with earlier finding and industrial experience;

- lastly, he suggested that the bubble surface area flux would need more validation using

different ore types and that the linear relationship between flotation rate constant and Sb

also required more research.

Finch et al. (2000) suggested another empirical relationship, this time to predict the Sb from

gas hold-up measurements:

5.5b gS (1-31)

This model was tested for mechanical cells and flotation columns, both at laboratory and

industrial scale, and it was deemed to be valid for Sb lower than 130 s-1 and εg lower than

25%. The problem of this model is the method used for bubble size estimation (drift flux

analysis), which does not have a good accuracy. This Sb prediction model was proposed

based on the fact that gas hold-up measurement is easier than that of the bubble size.

A third expression can be developed using the empirical model proposed by Nesset et al.

(2006) for estimating the Sauter mean bubble diameter from superficial gas velocity

measurements. They found that the d32 increases when Jg is increased according on:

32 0

n

gd d C J (1-32)

where C and n are the model parameters and d0 is determined by extrapolation of the d32

graph for Jg = 0. By replacing the d32 in equation (1-27), the following expression for Sb can

be obtained:

0

6 g

b n

g

JS

d C J

(1-33)

26

As can be seen, none of the proposed models consider the actual bubble size distribution

even though it can now be measured and consequently better definition of Sb could be a

targeted.

1.7.4 Bubble size distribution

The bubble size distribution has lately received a lot of attention from researchers.

However, very few mention the effect of bubble size distribution to the flotation process

performance. In fact, bubbles generated in flotation cells or columns show a wide range of

sizes i.e. a bubble size distribution, but most if not all available models use a mean value as

‘the bubble size’, which results in overlooking some distribution features.

Figure 1.6 (Maldonado et al., 2008c) illustrates how different bubble size distributions may

lead to a same Sauter diameter.

Figure 1.6 Different bubble size distributions all having the same d32

(Maldonado et al., 2008c).

Wongsuchoto et al. (2003) showed the relationship between the shape of bubble size

distribution and the superficial gas velocity. They reported that for superficial gas velocities

in the range of 20 to 40 cm/s, bubble size distributions correspond to a normal distribution

in their case of study.

d32 (mm) d32 (mm)

d32 (mm) d32 (mm)

27

Grau and Heiskanen (2005) presented a study on bubble size distributions in laboratory

mechanical cells. Their main objective was to evaluate the influence of operating

conditions, such as the pulp density, frother concentration, air flow rate and impeller speed,

on the bubble size distribution. The results were modeled by applying different distribution

functions (Rosin-Rammler, Tuniyama-Tanasawa, log-normal and upper-limit). It was

shown that the upper-limit function better fit the experimental data as compared to the other

functions. However, there was not a unique distribution function which could represent all

the generated bubble size distributions for all the flotation devices tested.

These modeling results for some gas dispersion properties have been used for optimizing

the flotation performance regarding particular types of particles. In the next section, some

previous research results about influence of gas dispersion on flotation are presented.

1.7.5 Gas dispersion properties range and effects

It has been demonstrated that excessive gas rate might lead to a loss of bubbly flow regime,

pulp-froth interface and positive bias in the case of flotation columns (Hernandez-Aguilar

et al., 2005). In fact, when the gas rate is increased, it first affects the bias rate, which might

turn from positive to negative, but then this is followed by the perturbation of the bubbly

flow regime inside the column and finally the pulp-froth interface might be lost, i.e. it

becomes impossible to detect.

Based on experimental evidence, Deglon et al. (2000) reported that the bubble surface area

flux should stand in the range of 50 to70 s-1. From laboratory scale work using two types of

minerals, Hernandez et al. (2005) demonstrated that Sb values up to 50 s-1 could be used in

full-scale mechanical cells before deteriorating the flotation results.

Yianatos and Henriquez (2007) made a survey of typical ranges of bubble size and

superficial gas velocity, from micro-scale to large industrial flotation cells and columns.

Their work showed that for typical gas velocities (Jg = 1.2 cm/s), the optimal bubble size in

the collection zone should be in the range of 1 to 1.5 mm to optimize the bubble surface

area flux, that is Sb should then be in the range 50 to 100 s-1. They concluded that to have

an acceptable flotation performance, bubbles should be generated in this particular size

range.

28

If the bubble size stands in the range of 0.5 to 1 mm, three main effects in flotation were

observed, namely

- the loss of the pulp-froth interface;

- lower bubble surface area flux;

- lower mineral carrying capacity.

On the other hand, increasing bubble size up to 1.5 to 2 mm led to

- lower bubble surface area flux;

- lower mineral carrying capacity;

- greater disturbances at the interface level (“boiling” effect);

- more mineral entrainment into the froth.

Increasing the Sb to reach higher flotation rates might encounter some limitations with

respect to increasing the bubble size and the gas rate. As a result, attention should be placed

on the most adequate range of bubble size and gas rate.

Beside these considerations about bubble size range, it is worth considering some studies

on the effect of other variables on the bubble size. The following paragraphs present some

of these works.

Gorain et al. (1999) reported that the particle size has an important influence on the bubble

size, and thus on Sb. Therefore, they considered the particle size as an independent variable

in their Sb model. The effect of other physical and chemical variables on the bubble size

was also reported. For instance, an increase in the pulp viscosity, pulp density and air flow

rate or a decrease in temperature is reported to produce larger bubbles. Bubble size could

decrease if frother concentration increases; on the contrary collector dosage does not have

any major effect on the bubble size. However it’s not clear whether the reported effect of

the particle size is due to changes in the pulp density (solids concentration) or pulp

viscosity (related to the presence of fine particles and therefore more particles for a given

tonnage) or due to the size of the particles itself. From the analysis of their study, it can be

concluded that the proposed Sb model needs more research to clearly ascertain the effect of

these variables.

29

On the same subject, O’Connor et al. (1990) experimentally demonstrated that using

particles smaller than 38 μm is more efficient for generating smaller bubbles than using

particle larger than 166 μm. However, this result might need more investigation because the

degree of control over other influential variables during the study leading to this conclusion

seems doubtful. It was also observed that the bubble size increases with increasing pulp

density through the addition of quartz particles, was also reported by Grau and Heiskanen

(2005). It is not clear though, whether the effect of the particle size occurs during or

following the bubble generation process.

1.8 Relation between the kinetic constant and gas dispersion properties

Gorain et al. (1998) proposed a linear relationship between the flotation rate constant and

superficial area flux for different particle sizes in a pilot mechanical cell. They found that

the slope of the obtained straight line was different for different size fractions: for smaller

particle sizes (less than 38 µm), the slope increases and vice-versa. This observation was

validated in mechanical cells in the pilot scale and the obtained model was used for

mechanical cell characterization.

They also reported a strong relationship between k and Sb at different froth depths. The

relationship was found to be linear for shallow froths (7 cm). For intermediate (30 cm) and

deep froths (45 cm), the relationship was found to be non-linear, therefore the froth depth

plays an important role on the overall kinetics.

The existence of a linear relationship between the flotation rate constant and froth depth

was also reported by Vera et al. (1999) and a linear model for this relationship was

presented. They also claimed that the collection-zone rate constant of the two evaluated

minerals (chalcopyrite and pyrite) increases as air flow rate increases, at the expenses of a

reduction in froth-zone recovery. This was associated to a detachment of particles from the

bubbles in the froth-zone. The impeller speed also increased the collection-zone rate

constant and decreased the froth-zone recovery for both minerals. Finally, they showed that

the solid concentration and the collector dosage have an important effect on the collection-

zone rate constant, both being related to the mass transfer rate from the collection-zone to

the froth-zone. They appeared to have an indirect influence on the froth-zone behavior by

increasing froth stability.

30

Heiskanen et al. (2000) performed some experiments proving that the rate constant starts

decreasing for coarse particles at high superficial gas rates (more than 2 cm/s) and that fine

particles (20-25 µm) require much higher superficial gas rate for effective flotation.

Before them, Yoon (1993) reported that decreasing the bubble size is a more effective mean

than increasing the gas rate to reach faster kinetics.

On the subject of the relation between the particle and bubble size, Diaz-Penafiel and

Dobby (1994) demonstrated that the kinetic constant for a laboratory flotation process

using small-bubbles (mean bubble size smaller than 1.2 mm, estimated using the drift flux

analysis) varies for different particle sizes (silica). Their work indicates that, for the same

bubble size, the flotation kinetic constant increases when particle size increases up to a

maximum value and then decreases.

Moreover, the flotation kinetic constant obtained using larger bubbles (mean bubble size

larger than 2 mm) exhibits the same trend than that with small bubbles up to a maximum

value, wherefrom further increase of particle size does not affect the kinetic constant value.

Their study has shown that small bubbles are not effective for very fine particles (smaller

than 5 µm) and the influence of the bubble size on flotation is greater as particles become

larger than this size.

1.9 Problems associated to the use of a single Sb value

So far, the Sb has always been considered as a global variable developed using an average

value of the BSD (d32), a value that can be originated from quite different distribution

shapes (Figure 1.6). This practice disregards other distribution features contained in the

measured data (variance, multimodes, etc.).

The application of a unique value of Sb for the determination of the flotation rate constant

values certainly leads to errors. For instance, a given Sb value can be obtained from

different combinations of mean bubble size and superficial gas velocity as demonstrated by

Figure 1.7. It is doubtful that all different pairs of bubble size and superficial gas velocity

leading to a single Sb value will transpose into a same kinetic constant.

31

Figure 1.7 Different pairs of bubble size and gas velocity leading to a unique BSAF value

This hypothesis has been proven in some of the tests completed in this PhD work (full

results will be presented in Chapter 5 and 6). In the two tests (flotation of talc as a

hydrophobic mineral) run under the same operating conditions, by changing the distribution

shape rather than the d32, two different values of the kinetic constant were estimated as

shown in Table 1.2. The only difference between them is the shape of bubble size

distribution as depicted in Figure 1.8.

Table 1.2 Example of a same Sb originating two different rate constants.

Particle size Jg (mm/s) d32(mm) Sb (1/s) τ (s) R∞ k (1/min) Recovery %

+75/-106 10 1.34 44.82 300 1 0.26 73

+75/-106 10 1.35 44.57 300 1 0.24 70

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2

10

12

14

16

18

20

d32

(mm)

Jg (

mm

/s)

Sb=120 (1/s)

Sb=100 (1/s)

Sb=80 (1/s)

Sb=60 (1/s)

32

Figure 1.8 Two BSDs having same d32 originate different kinetic constant

Although the variation in flotation rate value is rather small, it affects the flotation recovery

by 3% (in this example).

From this review of previous works, it seems that the influence of the bubble size (and

particularly the BSD) and particle size on the flotation rate constant has not been

comprehensibly studied, nor the modeling of the BSD.

Bridging the gap between micro and macro flotation models should also be addressed for a

better understand of the flotation process. Consequently, determining the relation between

collision, attachment and detachment efficiencies and the prevailing cell hydrodynamic

conditions having an effect on the kinetics, can be useful.

The effect of solid particles on the bubble size distribution and gas hold-up, as well as the

correlation between the bubble size distribution and gas hold-up should be considered as

well. Developing a better characterization of prevailing flotation conditions will help

understanding how hydrodynamic variables influence the particle collection process.

Finally, optimizing the collection efficiency (improving collision and attachment) does not

seem to improve the selectivity of the flotation process. The effect of hydrodynamic

variables on the flotation selectivity should thus be studied as well. These effects are partly

reflected on the amount of water reported to the concentrate (water recovery).

10 20 30 40 50 600

0.01

0.02

0.03

0.04

0.05

0.06

Bubble size (mm)

De

ns

ity

k= 0.24 (1/min)

k= 0.26 (1/min)

0.5 1.51 2 2.5 3

33

Based on the presented problems the objectives of this study have been defined and

presented hereafter. However some assumptions made should first be presented and

explained.

1.10 Assumptions made in this experimental work

1.10.1 Hydrophobicity

Talc is a natural hydrophobic mineral. As such, it was selected in order to disregard the

effect of variable hydrophobicity and collector efficiency in the evaluation of the

experimental rate constant. This way, the pulp could be reused continuously without the

above-mentioned limitations.

1.10.2 Solid particle residence time

The measured residence time of water is not the same as that of solid particles. The particle

residence time in flotation columns depends on the particle settling time, which is a

function of particle size, particle density, hydrophobicity, pulp percent solids, pulp

viscosity and the mixing conditions of the column. In the present study only the particle

size was modified, all remaining factors were kept constant during the experiments, thereby

particle residence time is only depended on particle size.

The particle size can be determined by means of screening, or measuring the settling time.

These two methods however do not address to the same size. Table 1.3 indicates the

differences between the measured values via either method.

Based on the concept of residence time measurement the measured particle size by settling

should be used however owing to hydrophobicity of talc, this residence time cannot be

accurate (attachment and detachment occur in the cell). It is worth mentioning that in this

research the particle size are defined as fine, middle, coarse and mixed as indicated on

Table 1.3.

34

Table 1.3 Talc particle sizes measured by twice wet screening and Malvern 2000

Particle

size

Measured particle size

by screening Measured particle size by settling

below screen size/ top

screen size (μm)

d80=80% of particles

passing this size (μm) below size/ top size (μm)

Fine 53/75 86 0.5/208

Middle 75/106 113 0.6/275

Coarse 106/150 137 0.6/416

Mixed 53/150 112 0.5/416

As results of obvious restrictions at Université Laval facilities, applying the most accurate

methods for residence time measurement, such as radioactive tracers, is not possible. The

only possible method is through injection of ionic solution tracers (Appendix A) and

measuring their concentration at the tailing port through electrical conductivity. Obviously,

this implies assuming that the particle residence time is equal to the water residence time.

1.10.3 Ultimate recovery (R∞)

Since all hydrophobic particles used in these experiments are fully liberated (pure talc) and

there is enough bubble surface for particle attachment, we can assume that R∞ is always

equal to 1. Figure C.1 (Appendix C) shows that all used (sizes) talc particles are free.

Moreover, Figure 1.9 shows a white spot in the center of loaded bubbles, which proves that

the light beam passes through a transparent part of bubble surface, indicating that bubbles

have enough surface area available for particle attachment. It should be noticed that

Figure 1.9 corresponds to a condition where a minimum bubble surface was generated.

Therefore, in cases where more available bubble surface exists, this assumption should hold

as well.

35

Figure 1.9 Light beam passing through loaded bubble

1.10.4 Froth depth and its effect on the final recovery

A fraction of the collected particles usually drains back from the froth zone. Therefore, the

overall particle recovery is not the same as the collection zone recovery. The amount of

drop-back depends on the bias rate and froth depth. If a very shallow froth zone is set in the

process control system and wash water is not introduced to the flotation column, we can

assume that the overall recovery is approximately the same as the collection zone recovery.

Therefore, a pulp sample taken from the concentrate launder can be considered as

representative of the collection zone concentrate.

In this research, samples were taken from the concentrate launder and at 10 cm below the

froth interface (this position was chosen to eliminate the effect of interface turbulence).

However for the purpose of flotation rate calculation, only overall concentrate values were

retained for data reconciliation using mass balance conservation equations.

1.11 Experimental set-up and results

The experimental work was conducted on a flotation column made of 5.6 cm internal

diameter acrylic tubes for a total height of 650 cm, continuously fed with a mineral pulp

from a 360 liter stirred plastic reservoir, where both column products were recycled to

assure a continuous operation. Air bubbles were provided to the column through a frit-and-

sleeve sparger placed at its bottom. This device, along with a given frother dosage, allowed

the modification of bubble size. The produced bubbles were sized by an adapted version of

McGill bubble viewer, for on-line measuring purposes. The used image processing method

36

allowed the detection of close to 100% of the bubbles present in the pictures taken, a major

improvement with respect to the presently used method. It is detailed in section 2.2. Froth

and pulp conductivities were measured at various heights of the upper part of the column

for detecting and controlling the pulp-froth interface control, and at a lower position in the

collection zone, for gas hold-up evaluation. More details about the set-up and instruments

are presented in section 2.2. Before starting the three-phase test phase, the column

hydrodynamic conditions were assessed using a two-phase system, to determine the most

appropriate range of flotation conditions for the used mineral particle. The determined

hydrodynamic conditions are presented in Appendix B. Various mixtures of almost pure

talc (used as natural hydrophobic mineral) and quartz (as a hydrophilic mineral) was used

in all tests to emulate real flotation conditions. More details about the solid characteristics

are presented in Appendix C. During the flotation tests samples of the feed, concentrate (at

the column upper lauder) and tailing (at the bottom of the column) stream were taken,

processed and analyzed for the evaluation of the flotation kinetic constant. The sampling

points and experimental practice are detailed in Appendix A. The obtained raw data and the

reconciled values (BILMAT), as well as the calculated flotation constants, can be found in

Appendix E. The proper estimation of the kinetic constant requires the knowledge of the

prevailing particle residence time distribution (RTD) in the column. The RTD measurement

method and its results are also explained in the Appendix A. Finally, it must be mentioned

that the experimental set-up included a process control system ensuring the data acquisition

and the automatic control of some operating variables, e.g. froth-pulp interface.

1.12 Objectives

As shown in the above literature review, most previous research work dealing with bubble

size characterization have not focused on the relationship between the hydrodynamic

variables and particle size. Although most researchers and practitioners agree that such a

relationship exists, the lack of an adequate method for on-line measuring the average

bubble diameter has prevented such studies. The on-line version of the McGill bubble

viewer implemented at Université Laval has opened a door for a study of the relationship

between hydrodynamic variables and the particle size. Consequently, the main objectives of

this PhD thesis are the following:

37

to understand the interactions between solid particles and hydrodynamic variables,

and the effect of hydrodynamic variables on the water recovery and carrying

capacity;

to evaluate other hydrodynamic variables potentially providing more information

about the size distribution than the bubble surface area flux (Sb);

to investigate the correlations between flotation rate constant and particle size as

well as hydrodynamic variables.

1.13 Outline of the thesis

This thesis was prepared using the material from four papers that have been either

published, accepted for publication or under journal review. A brief summary of each one is

presented below:

Chapter 2: Effect of particles on the bubble size distribution and gas hold-up in column

flotation

This is a study on the effect of solid particles on the bubble size distribution and gas hold-

up, as well as the correlation between the bubble size distribution and gas hold-up in

column flotation. The experimental results are compared with previously published

correlations and models. Different mechanisms introduced from the literature are discussed

and used to explain the working mechanisms.

Chapter 3: The effect of gas dispersion properties on water recovery in a laboratory

flotation column

This chapter addresses three objectives. The first one is to study the effect of the gas rate,

gas hold-up, bubble size and bubble surface area flux on the water recovery. The second

one is to investigate the effect of hydrophobic particle size on the water recovery. The last

one consists in characterizing the effect of gas dispersion properties on the column carrying

capacity.

Chapter 4: On the relationship between hydrodynamic characteristics and the kinetics of

column flotation. Part I: Modeling the gas dispersion. The interfacial area of bubbles (Ib) is

introduced in this chapter as a hydrodynamic variable providing more information about

38

the size distribution than the bubble surface area flux (Sb). Experimental evidence shows

that the bubble size distribution can exhibit lognormal shape, or even multi-modal and bi-

modal shapes. Later on, the correlation between the hydrodynamic variables – i.e. the

bubble size, bubble surface area flux and hold-up – and the interfacial bubble area is

studied.

Chapter 5: On the relationship between hydrodynamic characteristics and the kinetics of

column flotation. Part II: Model validation

In this chapter, the correlation between the flotation rate constant and particle size and

some hydrodynamic variables is investigated. This correlation exercise was conducted

using a statistical modeling method called Partial Least Squares (PLS), a very powerful tool

to demonstrate the relative importance of the independent variables being considered.

Finally, based on the obtained results, the relationship between the flotation micro-

processes and the relevant hydrodynamic variables is presented.

Chapter 6: Single variable rate constant models

This chapter proposes single variable models for the flotation rate constant based on the

hydrodynamic variables for each particle size-classes specifically suggested in Chapter 5.

Observed trends for each model are then discussed.

Appendix A: Kinetic constant calculation

Firstly, the procedure for sampling input/output streams is presented as well as bubble

sampling technique. Then, the method used for RTD determination and an example are

provided. Finally the kinetic constant calculation is detailed through two examples.

Appendix B: Preliminary tests

Before undertaking three-phase tests (liquid, solid and gas) some tests for the identification

of the hydrodynamic conditions within the experimental column were conducted. Among

them, it can be mentioned the determination of manipulated variable responses and ranges,

temperature and activation time and their effect on the frother efficiency, selection of levels

of shear water rate to the sparger. Finally, the method for determining the residence time

distribution and the obtained results are presented.

39

Appendix C: Solid characterization

This appendix presents the various analysis done on the solid particles used in the tests,

such as XRF, liberation degree and mineralogy.

Appendix D: Radial gas dispersion

The analysis of radial gas hold-up in a section of a column proposed by Banisi et al. (1998)

is presented in this appendix.

Appendix E: Gas dispersion properties

This appendix contains a table showing the effect of the particle size, frother concentration

and shear water rate on the measured hydrodynamic variables such as gas hold-up, Sauter

mean bubble diameter and mean of the bubble size distribution (d10).

Appendix F: Statistical analysis

This appendix is divided in three main parts. In the first one, the existing correlations

among various gas dispersion properties are analyzed. The second one briefly introduces

the Projection to Latent Structures (PLS) method. The last part provides the datasets used

for the validation of the model developed in Chapter 5.

41

Chapter 2 Effect of particles on the bubble size distribution

and gas hold-up in column flotation2*

Abstract

This chapter studies the effect of solid particles on the bubble size distribution and gas

hold-up, as well as the correlation between the bubble size distribution and gas hold-up in

column flotation. Experiments were conducted in two and three-phase systems using a

laboratory flotation column (5.6 cm internal diameter for a total height of 650 cm), and

mixtures of quartz (hydrophilic gangue) and talc (naturally hydrophobic mineral), classified

in four different size fractions. Results are compared with literature correlations and models

to reveal that hydrophobic particles affect the gas hold-up through three different

mechanisms modifying the Sauter mean diameter and rise velocity, namely (1) surface

interactions, and the joint antagonistic effect of (2) bubble loading and (3) coalescence.

Résumé

Ce chapitre étudie l'effet des particules solides sur la distribution de taille de bulle et sur le

taux de rétention de l’air, ainsi que la corrélation entre la distribution de taille de bulle et le

taux de rétention de l’air pour la flottation en colonne. Des expériences ont été menées dans

des systèmes à deux et à trois phases avec une colonne de flottation de laboratoire

(diamètre interne de 5,6 cm pour une hauteur totale de 650 cm), et des mélanges de quartz

(gangue hydrophile) et du talc (minéral naturellement hydrophobe), séparés en quatre

classes granulométriques. Les résultats sont comparés avec des corrélations et des modèles

de la littérature pour révéler que les particules hydrophobes affectent le taux de rétention de

l’air par trois mécanismes différents modifiant le diamètre moyen de Sauter et la vitesse

ascensionnelle des bulles, à savoir (1) les interactions de surface, et l'effet antagoniste du

(2) chargement des bulles et (3) le phénomène de coalescence.

2*Ali Vazirizadeh, Jocelyn Bouchard and Yun Chen "Effect of particles on the bubble size distribution and

gas hold-up in column flotation". International Journal of Mineral Processing, (2014) SUBMITTED

42

2.1 Introduction

In flotation systems, gas dispersion plays a critical role for particle collection (recovery)

and froth mass transport (selectivity). To evaluate this effect at the industrial scale, the

bubble surface area flux (Sb) is typically estimated from the plant superficial gas rate (Jg)

and bubble size distribution (BSD) measurements, where the complete BSD is compressed

into a single value, i.e. the Sauter mean diameter (d32). However, it is now criticized that (1)

a given Sb value can be obtained from different combinations of Jg and d32 (Vinnett et al.,

2012), and (2) a given d32 value can be obtained from different BSDs (Maldonado et al.,

2008c). A better approach consists in adequately parameterizing the overall BSD as

presented by Vazirizadeh et al. (2014).

The gas hold-up is another important parameter used to characterize the hydrodynamic

conditions of bubble column reactors (Luo et al., 1999). It is useful because it combines the

influence of both the bubble size and gas rate. It provides a holistic indication of the

hydrodynamic conditions because it is dependent on various factors such as frother type

and concentration, cell dimensions, operating temperature and pressure, as well as solid

phase properties and concentration.

Banisi et al. (1995) indicated that the presence of solid particles reduced gas hold-up in a

column operated under conditions relevant to flotation. This effect increased with solids

concentration over the range 0 – 15% v/v. Hydrophilic (silica and calcite) and hydrophobic

(coal) particles produced similar reductions in the gas hold-up. Banisi et al. (1995) explored

four possible mechanisms responsible for the gas hold-up reduction in the presence of

solids: (1) coalescence, (2) a shift in the density and viscosity of the pulp, (3) a change in

the radial gas hold-up and flow profiles, and (4) bubble wake effects. They concluded that

changes in gas hold-up due to the addition of solids were due to a combination of the two

latter mechanisms, i.e. a change in radial gas hold-up and flow profiles, and bubble wake

effects.

Gandhi et al. (1999) investigated hydrodynamic behavior of slurry bubble column at high

solids concentrations, and found that the axial distribution of slurry concentration followed

the classical sedimentation–dispersion model. They also showed that the effect of the gas

43

velocity on axial solids distribution were minimal over the range of gas velocities

investigated.

To date, most of the laboratory-based research has been conducted on gas-water (two-

phase) systems. Since one cannot conclude that results in two-phase systems necessarily

apply for slurries, ‘surrogate’ solids such as talc, silica and coal have been used. Although

such synthetic feed material are idealized/simplified compared to natural ores, they provide

important insights about the nature of three-phase systems in flotation (Kuan and Finch,

2010). Despite extensive research on the effect of solids on gas-liquid systems, results are

not conclusive and sometimes contradictory. Little is known about the physical

mechanisms underlying the observed macroscopic effects. The majority of literature on the

topic points to a decrease in gas hold-up with increasing solid concentration (Banisi et al.,

1995; Kuan and Finch, 2010; Reese et al., 1996). In other words, the presence of solids

tends to increase the rise velocity of a bubble swarm.

Most of the literature focuses on the effect of solid particles on a single hydrodynamic

parameter, either the gas hold-up (Banisi et al., 1995; Kuan and Finch, 2010) or bubble size

distribution (Gandhi et al., 1999; Grau and Heiskanen, 2005; O’Connor et al., 1990). In this

chapter, the effect of solid particles is studied not only on the bubble size distribution and

gas hold-up, but also on the relationship between these two hydrodynamic characteristics

jointly. Different mechanisms introduced from the literatures are discussed in light of

experimental results.

2.2 Experimental set-up

Experimental data were generated using a fully automated laboratory column flotation set-

up. The column was made of 5.6 cm internal diameter acrylic tubes for a total height of 650

cm is shown in Figure 2.1.

44

2.1 Schematic of the experimental set-up

A frit-and-sleeve sparger, detailed below, was located at the bottom of the column to

provide controlled air flow rates through local PID controllers. Temperature and absolute

pressure sensors, were mounted at the bottom of the column, to compensate for standard

condition flow rate measurements (referred to a given temperature and pressure). A

conductivity-based sensor (Gomez et al., 2003), installed in the middle of the column (350

cm from the tailings port), allowed measuring the gas holdup using the conductivity of both

the aerated and de-aerated slurry. Maxwell's equation relates the conductivity to the

concentration of a dispersed non-conducting phase (i.e. bubbles) in a continuous liquid

phase (pulp in this case) as:

(2-1)

1

2

3

4

5

14

6

7

8

9

10

11

12

13

1 Flotation column2 Level measurement cells3 Feeding pump4 Conditioning tank5 Aerated conductivity cell6 De-aerated conductivity cell7 Tailing pump8 Shear water pump9 Frit-sleeve sparger10 Image analyzer11 Column control system12 Slurry filter13 Filter pump14 Bubble viewer

(1 )

(1 0.5 )

d p

g

d p

k k

k k

45

where kd is the conductivity of the dispersed phase with air, and kp, the conductivity of air-

free dispersion (de-aerated slurry). Figure 2.2 depicts the schematic of the gas holdup

sensor.

Figure 2.2 Schematic of the εg sensor (Gomez et al., 2003)

In order to measure the bubble size, the McGill bubble viewer was adapted for on-line

operation. The column was operated under automatic control of froth-depth, gas rate, shear-

water and feed flow rate. The froth zone level was measured and controlled using a set of

eleven conductivity cells sensor as described by Maldonado, et al., (2008a). No wash water

was used in the experiments. The air flow rate is measured and controlled using a mass

flow controller, whose readings are converted to volumetric flow rates at reference

conditions of gas flow rate, which then are corrected for actual (test) temperature and

pressure conditions as

(2-3)

where P is the measured absolute pressure (cm H2O) at the bottom of column, T is the

temperature (°C) and is the superficial gas rate (cm/s) at 21°C and 1033.23 cm H2O.

The frit-and-sleeve sparger consisted of a porous stainless steel ring concentrically installed

within a cylindrical sleeve (Kracht et al., 2005) , as shown in Figure 2.3. The porous ring

ref

gJ

46

was 4 cm long and 2.5 cm in diameter and had a nominal pore diameter of 10 µm. The

cylindrical gap (annulus) between the frit and the sleeve was 1 mm. This design allowed an

external flow of water to circulate through the gap providing the necessary shear to reduce

the size of the bubbles generated in the device. This sparger thus provides an additional

degree-of-freedom to control the bubble size with the water velocity, i.e. Jsl, defined such

as:

(2-4)

Figure 2.3 Schematic of a frit-sleeve sparger (Kracht et al., 2008)

where Qsl is the volumetric water flow rate in the sparger, and A is the cross sectional area

of the column. The water flow rate to the sparger is manipulated by the speed of a gear

pump, using a PID controller implemented on a Moore 353.

In order to continuously monitor the bubble size, the McGill bubble viewer (Hernandez-

Aguilar et al., 2004) was adapted to the experimental column at 100 cm above feed port.

The bubbles are collected via a sampling tube and directed into a viewing chamber where

they are exposed under lighting conditions to be photographed. Parameters of the bubble

size distribution – such as the Sauter mean diameter (d32), mean and variance at actual

conditions (i.e. not corrected to reference conditions) – are then determined by image

analysis. With this regard, the Circular Hough Transform (CHT)-based algorithm presented

47

by Riquelme et al. (Riquelme, 2013a, b) was implemented for accurate bubble detection

(including large ones and clusters). Figure 2.4 shows an example of an original picture

taken which contains 800×600 pixels (left hand side), and the bubbles detected by the

CHT-based algorithm, circled in blue and marked by a red cross in the middle (right hand

side). Three frames per seconds have been taken to ensure every picture presented an

entirely ‘new’ sample of bubbles, and avoid multiple counts of the same bubble.

Figure 2.4 Example of an original captured image (left hand side), and detected bubbles

using the CHT-based algorithm, circled in blue and marked by a red cross in the middle

(right hand side)

In the upper section of the column a series of eleven 10-cm spaced stainless-steel

conductivity electrodes, i.e. 5.1 cm external diameter by 0.5 cm height flush-mounted rings

inside the column, allow the measurement of the pulp-froth interface with a precision of 1

cm (Maldonado et al., 2008b).

Quartz was used as a hydrophilic gangue and talc as a naturally hydrophobic mineral,

ground and classified into four size classes. The synthetic pulp fed to the column contained

4% solids (i.e. 40 % talc and 60% quartz), which is a relatively low percent solids, but

exhibiting high proportion of hydrophobic (talc) mineral.

For its constant efficiency over a rather long period of time, a polyglycol frother (F150)

was selected. The gas rate and feed rate were set at 1 cm/s for all tests. Talc, as a natural

hydrophobic mineral consumes a fraction of F150, which is absorbed on the particle

surface (Kuan, 2009). To avoid the results being affected by this interaction, samples

48

started being taken after one hour of column operation time, thus ensuring constant frother

efficiency over time. The operating principle consists in directing a sample of bubbles into

a viewing chamber. Each test allowed sampling populations approximately composed of

10,000 bubbles, thus ensuring representative distribution.

2.3 Results and discussion

2.3.1 Solid particles on the bubble size distribution

Figure 2.2 provides the generated bubble size cumulative distribution functions (CDF), and

Figure 2.3 shows the result of bubble images, respectively, as a function of solid addition in

the presence of a polyglycol frother (F150). The data at 0 % solids (w/w) (i.e. the gas–water

system) shows decreasing bubble sizes (Figure 2.5), and more uniform distributions

(Figure 2.6) with increasing shearwater to the sparger.

The most significant impact of solids is to increase the bubble size. This becomes apparent

when introducing 4 % (w/w) (40% talc and 60% quartz) in the presence of 25 ppm F150

(i.e. above the critical coalescence concentration (CCC)): d32 increased from 0.72 mm to

1.35 mm while introducing 0.2 cm/s shearwater (Jsw), and from 0.64 mm to 1.17 mm with

0.8 cm/s shearwater.

49

Figure 2.5 Effect of the solid percent on the bubble size distribution (CDF)

at different shearwater rate (a) 0 wt%, (b) 4 wt%

0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Qu

an

tity

Bubble size (mm)

Jsw

=0.2 cm/s

Jsw

=0.8 cm/s

(a)

0.5 1 1.5 2 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bubble size (mm)

Qu

an

tity

Jsw

=0.2 cm/s

Jsw

=0.8 cm/s

(b)

50

Figure 2.6 Example of bubble images using 25 ppm F150, and for Jg = 1 cm/s

(a) Jsw =0.2 cm/s, 0% wt, (b) Jsw =0.8 cm/s, 0% wt, (c) Jsw =0.2 cm/s, 4% wt (d) Jsw =0.8

cm/s, 4% wt

Kuan and Finch (2010) suggested that bubble coalescence in the presence of talc was the

explanation for this phenomenon. This was supported by bubble size distribution

experimental measurements. For instance, Figure 2.7 shows how the bubble size

distribution evolves from (a) a narrow mono-modal range in a water-only system to (b) a

widespread one following the introduction of talc particles (4 % (w/w) slurry). This result is

typical of coalescence: the coexistence of fine and coarse bubbles would come from a

coalescence-induced break-up phenomenon (Tse et al., 2003). Quinn and Finch (2012) also

reported bi-modal bubble size distributions in the absence of coalescence inhibiting agents,

probably resulting from bubble-bubble interactions. Figure 2.8 depicts an example of

coalescence-induced break-up leading to a bi-modal distribution.

Dippenar (1982) reported that hydrophobic solids can promote coalescence through a

bridging effect of hydrophobic particles between bubbles. In addition, coalescence could

occur because the frother concentration decreases due to frother adsorption by talc (as a

(a) (b)

(c) (d)

1mm

51

hydrophobic solid). With this regard, Kuan and Finch (2010) proposed the following

hypothesis about the adsorption of F150 by talc particles based on the structure of the

frother molecule. Talc particles, via their uncharged surfaces providing natural

hydrophobicity, tend to attach to the bubbles. On the other hand, the F150 molecule has

two end OH groups (hydrophilic) lying flat on the surface of the bubble to accommodate

the two hydrophilic groups (OH and H(C3H6O)). In this orientation, F150 is possibly

exposing its hydrocarbon backbone with talc attached to bubbles, thus inducing uptake by

talc surfaces through hydrophobic interactions.

According to the above-mentioned hypothesis, talc surfaces exposed to bubble surfaces

would play an important role on the adsorption of frother. Frother up-taken by particle is

obviously not a desirable effect in flotation. However, it must be taken into account in cases

where finer grind size is considered to improve the degree of liberation, as it will lead to a

larger total surface of particles in the slurry. Similarly, increasing the solids percent in a

flotation circuit, for instance to increase the production rate, could also lead to higher

frother consumption.

52

Figure 2.7 Effect of the solid percent on the bubbles size histogram (25 ppm F150, Jg= 1

cm/s and Jsw= 0.8 cm/s) (a) 0 % wt, (b) 4 % wt

To check the variation of frother uptake by increasing particle total available surface, a set

of tests have been conducted. Four talc particle size-classes (53/75 µm, 75/106 µm,

106/150 µm and 53/150 µm), each one combined with the same quartz particle size class,

were used. The amount of both the hydrophobic and hydrophilic added solids remained

constant. For each particle class, three frother concentrations (5, 15 and 25 ppm) were

applied in conjunction with three levels of shear water rate (0.2, 0.4 and 0.8 cm/s). The

shearwater rate provided an extra degree-of-freedom for modifying the mean bubble size,

0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Bubble size (mm)

Qu

an

tity

(a)

0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Bubble size (mm)

Qu

an

tity

(b)

53

particularly for tests conducted at a constant frother concentration. Table 2.1 summarizes

the experimental plan for each particle size class.

Figure 2.8 Bi-modal distribution due to coalescence (5 ppm F150, Jg= 1 cm/s, 4 % wt and

Jsw= 0.2 cm/s)

Table 2.1 Summary of the experimental plan

Test # frother concentration (ppm) Shearwater rate (cm/s)

1 5 0.2

2 5 0.4

3 5 0.8

4 15 0.2

5 15 0.4

6 15 0.8

7 25 0.2

8 25 0.4

9 25 0.8

Figure 2.9 presents the effect of the talc particle size on the d32 at a constant gas rate of

1 cm/s. Coalescence leads to an increase of both the d32 and standard deviation of bubble

0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

Bubble size (mm)

Den

sity

54

size distribution. As can be seen, the particle size does not show any effect on d32, and

therefore, increasing the available surface of hydrophobic mineral surprisingly did promote

coalescence.

Figure 2.9 The effect of the particle size on the bubble d32 at constant gas rate (1 cm/s)

Tests 4 to 9 in Table 2.1 were conducted using 15-25 ppm frother, which is higher than the

CCC of F150. Results could not reveal any significant effect of the particle size on

coalescence, perhaps because of the concentration of frother in the solution which remained

over the CCC. Even for the first three tests, conducted using 5 ppm frother concentration,

which is below the CCC of F150 (Finch et al., 2008), the particle size effect could still not

be observed. This suggests that other mechanisms could play an important role.

The bubble-particle collision efficiency is a possible explanation. Yoon (1993) introduced a

model linking the probability of collision (Pc) with the bubble (db) and particle (dp) size in

different hydrodynamic conditions as

( / )n

c p bP A d d (2-4)

where A and n are model parameters depending on the flow conditions. According to

equation 2-4, fine particles are less likely to collide with a given set of bubbles than coarse

particles. The effect of increasing the specific particle surface with decreasing particle sizes

1 2 3 4 5 6 7 8 9

1.2

1.4

1.6

1.8

2

Test number

d3

2 (

mm

)

53/75 m

75/106 m

106/150 m

53/150 m

(mm

)

55

would thus be compensated by a lower rate of collection. This assumption however would

remain to be further investigated experimentally using the percentage of solids as a factor,

instead of the particle size distribution to modify the total available hydrophobic mineral

surface.

2.3.2 Effect of solid particles on the gas hold-up

Different hypotheses have been postulated in the literature to explain the influence of

hydrophobic particles on the gas hold-up. The effects would be to:

promote bubble coalescence leading to a reduction of the gas hold-up (Banisi et al.,

1995; Kuan and Finch, 2010);

lead to bubble break-up, which in turn increases the gas hold-up (Yang et al., 2007);

increase the apparent viscosity, and thus the gas hold-up (Banisi et al., 1995; Finch

and Dobby, 1990);

increase bubble wake entrainment effects which decrease the gas hold-up (Banisi et

al., 1995);

change the radial gas hold-up and flow profiles, thus leading to a reduction of the

global gas hold-up (Banisi et al., 1995);

increase the weight of bubbles once attached, and therefore increase the gas hold-up

(Tsutsumi et al., 1991; Uribe-Salas et al., 2003).

Bubble coalescence and break-up

Bubble coalescence occurs due to the bridging effect of particles (Dippenaar, 1982) and/or

adsorption of frother on the hydrophobic particle surface (Kuan and Finch, 2010). Faster

rising larger bubbles thus appear in the population, leading to a reduction of the gas hold-

up. The bubble coalescence induced by solid particles could occur during the generation

process or progressively along the column.

In the present study, a frit-sleeve sparger depicted in Figure 2.3 was used. The mechanism

of frit-sleeve sparger does not permit particles to induce bubble coalescence during the

generation stage since the water injected to the sparger does not contain solid particles.

Therefore, the observed coalescence occurs along the collection zone.

56

Coalescence-induced break-up occurs as one form of coalescence mechanism in which very

fine bubbles are generated along with very large ones (Quinn and Finch, 2012; Tse et al.,

2003). This mechanism decreases the gas hold-up, as larger / faster rising bubbles are

created.

On the other hand, it was suggested that in a three-phase fluidization system, ‘sharp’

particle edges can penetrate and break some bubble surfaces. This would cause bubble to

break-up into smaller ones, thus leading to an increase in the gas hold-up. Yang et al.

(2007) reported this phenomenon for large particles (larger than 1 cm). The exact effect of

solids on bubble break-up is still not clear as Gandhi et al. (1999) had previously explained

the reduction in gas hold-up caused by solids addition leading to a reduction in the bubble

break-up rate. It must be emphasized though that their experiments were conducted at high

slurry concentration (40 % vol.), and using very high gas rate (26 cm/s) that are outside the

typical range of operation.

The coexistence of both large and small bubbles was demonstrated above in Figure 2.8.

Figure 2.10 now illustrates the effect of solid particles on bubble coalescence-induced

break-up. Results show that adding solids (4 % (w/w)) to the gas-water system and keeping

the gas rate and frother concentration constant increased both the d32 and standard deviation

of distribution (σ). As mentioned above, the addition of hydrophobic particles thus

promoted coalescence (larger d32) and resulted in a more widespread distribution, thus

supporting the coalescence-induced break-up phenomenon.

57

Figure 2.10 The effect of solids on the parameters of the bubble size distribution

(Jg = 1 cm/s, 25 ppm F150). σ is standard deviation, d10 is mean bubble diameter and d32 is

Sauter mean diameter

Viscosity and bubble wake

The presence of solid particle increases apparent viscosity, leading to the opposite effect on

the bubble velocity, and thus on the gas hold-up (Finch and Dobby, 1990).

Based on the equation of Roscoe (1952), the slurry viscosity (µsl) is calculated as

2.5(1 )sl l (2-5)

where µl is the liquid viscosity, and α is the volume fraction of solids in the slurry.

Increased slurry viscosity through the addition of solid particles can stabilize the bubble

wake. This, in turn leads to an increase of the bubble velocity, and thus a reduction of the

gas hold-up (Banisi et al., 1995).

In the present work though, µsl / µl =1.038. As can be seen, the value is small enough so the

effect of viscosity on the gas hold-up can be neglected.

Radial gas hold-up and flow profiles

Using a laboratory flotation column (4.47 m height and 10.18 cm in diameter), Banisi et al.

(1995) observed that the flat radial gas hold-up in the water-gas system changes to non-

0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.50.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

d32

Two-phase

Three-phase

(mm)

58

uniform profiles in the slurry-gas system (10% v/v calcite). They reported a significant

difference between the distribution parameters of the gas hold-up in the gas-water and gas-

slurry systems due to non-uniform profile using drift flux analysis. Appendix D details the

derivation of the distribution parameters. This mechanism could explain the lower gas hold-

up in the slurry system.

The radial gas hold-up was not evaluated in the current study. However, more recently

Rabha et al. (2013) obtained contradictory experimental results showing, using a similar

set-up (7 cm diameter and 150 cm height), that the radial gas hold-up remained flat

following the addition of 5% solids, but decreased to a lower value. The material used was

spherical glass particle (100 µm and ρp =2500 kg/m3).

It must be emphasized that Banisi et al. (1995) applied the general drift-flux analysis

method to study the radial hold-up, which is not very accurate compared to the ultrafast

electron beam X-ray tomography used by Rabha et al. (2013) .

Bubble loading

The added weight following hydrophobic particle attachment is known to reduce the bubble

velocity and consequently increase the gas hold-up (Uribe-Salas et al., 2003). In the present

case, talc, as a hydrophobic fraction of added solid, attached on the bubble surface and was

collected at the top of the column. The increase in gas hold-up reported in the present work

supports this mechanism. Table 2.2 provides test results in gas-water and gas-slurry

systems. The addition of the fine particle class did not have a significant effect on the gas

hold-up. However, the d32 almost doubled. The same trend is also observed for the other

particle size fractions. This observation supports the role of bubble loading on the gas

hold-up.

59

Table 2.2 Generated gas hold-up and d32 for gas-water and gas-slurry systems

(Jg = 1 cm/s, Jsw = 0.8 cm/s, 25 ppm F150)

Two-phase Three-phase

εg (%) d32 (mm) Class (µm) εg (%) d32 (mm)

12.95 0.63

53/75 12.48 1.17

75/106 10.85 1.30

106/150 10.46 1.18

53/150 9.34 1.31

2.3.3 Discussion

Among the presented mechanisms, coalescence (and coalescence/induced break-up) and

bubble loading can be used to explain the variations of gas hold-up with the bubble size for

two and three-phase systems as depicted in Figure 2.11. The first mechanism accounts for

decreasing gas hold-up and the second one, for the opposite effect.

By manipulating the shearwater rate, different bubble size distributions can be generated

while the gas rate and frother concentration remain constant (25 ppm F150, Jg =1 cm/s). As

can be observed in Figure 2.11, different combinations of gas hold-up and bubble sizes

were thus produced. Without great surprise, increasing the bubble size reduced the gas

hold-up in the two-phase operation.

Adding solids produced coarser and more spread out bubble size distributions (see

Figure 2.8) due to coalescence. In most of the cases, this led to a reduction of the gas hold-

up that can be explained by the dominant effect of the bubble size on the rise velocity. It is

worth mentioning that shifting the d32 and increasing the standard deviation of distribution

do not change the general shape of bubble size histogram, which can still be represented by

lognormal probability density function in this case.

On the other hand, comparing the obtained results of the gas hold-up in three-phase and

two-phase systems also revealed that in a few cases, notwithstanding coarser bubble size

distributions, the addition of solids led to higher gas hold-up values. The bubble–particle

60

attachment mechanism explains this behavior, i.e. the bubble rise velocity decreases as

bubble loading increases.

Figure 2.11 Bubble size and gas hold-up correlation in two and three-phase system

Results from the test conducted at different frother concentrations in the presence of solids

lead to the same conclusion: coalescence and bubble–particle attachment are the main

mechanisms explaining how the presence of solid affect the ties between the bubble size

and gas hold-up. Reducing the frother concentration below the CCC value revealed the

effect of having larger bubbles in the swarm. At the same time, decreasing the number of

bubbles due to coalescence, results in bubble overloading and decreases the rise velocity.

Figure 2.12 depicts how hydrophobic particles affect the correlation between the gas

hold-up and bubble size for three different frother concentrations. It is worth mentioning

that very similar bubble size distributions – and gas hold-up – were generated at 15 and 25

ppm as in both cases the concentration is the above the CCC.

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.54

6

8

10

12

14

16

d32

(mm)

g (

%)

Two-phase

Three-phase

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

bubble size

Qu

an

tity

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

Bubble size

Nu

mb

er

of

bu

bb

les

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

Bubble size

Nu

mb

er

of

bu

bb

les

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

Bubble size

Nu

mb

er

of

bu

bb

les

d32 (mm)

ε g(%

)

Medium bubble loading & medium bubble coalescence in three-phase

Minimum bubble coalescence in two-phase

Maximum bubble loading & minimum bubble coalescence in three-phase

61

Figure 2.12 Bubble size and gas hold-up correlation in three-phase system

for three frother concentrations

2.4 Conclusion

The experimental results presented in this chapter allowed studying the effect of solid

particles on the bubble size distribution and gas hold-up, but also on the relationship

between these two hydrodynamic characteristics jointly. Different mechanisms introduced

from the literature were analyzed leading to the following conclusions:

1. surface interactions: the effect of solid particles on the bubble size distribution can

be explained by coalescence through (1) the bridging effect of solid particles and/or

(2) the absorption of frother by hydrophobic surfaces;

2. bubble weight and volume: the effect of hydrophobic particles on the gas hold-up is

determined by a combination of two antagonistic mechanisms: (1) bubble loading,

which reduces the rise velocity and (2) bubble coalescence, which increases it;

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

bubble size

Qu

an

tity

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

bubble szie

Qu

an

tity

1 1.2 1.4 1.6 1.8 2 2.2 2.42

4

6

8

10

12

14

d32

(mm)

g (

%)

25 ppm

15 ppm

5 ppm

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

Bubble size

Nu

mb

er

of

bu

bb

les

10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

Bubble size

Nu

mb

er

of

bu

bb

les

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

Bubble size

Nu

mb

er

of

bu

bb

les

0 10 20 30 40 50 60 70

0

0. 05

0. 1

0. 15

0. 2

0. 25

Bubble size

Nu

mb

er

of

bu

bb

les

d32 (mm)

ε g(%

)

Medium bubble loading & medium bubble coalescence

in three-phase

Minimum bubble loading & maximum bubble coalescence in three-phase

Maximum bubble loading & minimum bubble coalescence

in three-phase

Medium bubble loading & maximum bubble coalescence in three-phase

62

3. resulting effect: increasing the bubble size will not lead to the reduction of the gas

hold-up if the effect of the bubble loading outweighs that of the bubble size on the

rise velocity.

Characterizing the role of each one of these two mechanisms in the determination of the

gas hold-up remains to be investigated.

63

Chapter 3 The effect of gas dispersion properties on water

recovery in a laboratory flotation column3*

Abstract

This chapter analyzes the effect of gas dispersion properties on water recovery in a

laboratory flotation column. The experimental work was carried out under a constant gas

flow rate to eliminate the gas volume effect, while other gas dispersion properties have

been varied by modifying the frother concentration and the shearing water introduced to the

frit-and-sleeve sparger. A constant froth depth and no wash water were used, in order to be

able to evaluate the exclusive correlation between the water recovery and the bubble

surface area flux and/or the bubble size. It is worth mentioning, that the particle size of the

hydrophobic particles did not show a significant effect on the amount of water going to the

concentrate. Not surprisingly, experimental results demonstrate that water recovery

increases with the bubble surface area flux and with the gas hold-up.

Résumé

Ce chapitre traite de l’influence de la dispersion des gaz sur la récupération de l'eau dans

une colonne de flottation de laboratoire. Dans cette étude, le travail expérimental a été

effectué sous un débit de gaz constant pour éliminer l'influence de la quantité volumique de

gaz injecté dans la colonne, tandis que d'autres propriétés de dispersion de gaz ont été

modifiées par un changement de la concentration de moussant et de l'eau de cisaillement

introduite dans l'aérateur fritté à chemise. Afin de pouvoir évaluer la corrélation entre la

récupération en eau et la densité de flux d'air et/ou la taille des bulles, la profondeur

d’écume a été gardée constante et aucune eau de lavage n'a été ajoutée dans le système. Il

est à noter que la taille des particules hydrophobes n'a pas montré d'effet significatif sur la

quantité d'eau allant au concentré. Sans surprise, les résultats expérimentaux montrent que

3*Ali Vazirizadeh, J. Bouchard and R. del Villar. "The effect of gas dispersion properties on water recovery

in a laboratory flotation column". IMPC Conference, Santiago, Chile, October 20-24, 2014.

64

la quantité d’eau récupérée dans le haut de la colonne dépend de la densité de flux d'air et

du taux de rétention de l’air.

3.1 Introduction

The overall recovery of particles in a flotation system is the result of the so-called ‘true

flotation process’ and the hydraulic entrainment. The first mechanism would lead to the

collection of hydrophobic particles only, whereas the entrainment mechanism not only

implies the recovery of hydrophilic particles but also increases the amount of hydrophobic

particles transferred to the froth (Cilek, 2009). Furthermore, Tao et al.(2000) and Zheng et

al. (2006) reported a direct relationship between the overall recovery and the entrainment

recovery.

Size by size particle entrainment is a complex phenomenon, which ultimately depends on

process operating conditions, cell design, and froth characteristics. This notwithstanding, it

appears to correlate well with the water recovery to the concentrate (Martínez-Carrillo and

Uribe-Salas, 2008; Yianatos and Contreras, 2010). Entrainment is a two-step process:

(1) transfer of the suspended solids from the pulp region to the pulp–froth interface, and

(2) transfer of the entrained particles in the froth phase to the concentrate. In both cases,

particle entrainment increases with the amount of water reporting to the concentrate, which

in turn increases with the gas rate (Martínez-Carrillo and Uribe-Salas, 2008; Zheng et al.,

2006). In fact, the wake of the bubble swarm is responsible for the water entrainment in the

pulp. Its volume is a function of the bubble velocity and size, as well as the liquid viscosity.

The role of the frother type and concentration can be illustrated through its effect on the

bubble size and velocity, and ultimately on water recovery. For instance, it has been

observed that frother SF-6008 reported more water to the froth than DOW-200 and TEB

(triethoxybutane)(Ekmekçi et al., 2003). They also reported a direct correlation between

water recovery and the amount of frother introduced in the cell.

It appears that the system hydrodynamic conditions play a major role on the entrainment

(Nelson and Lelinski, 2000; Phan et al., 2003; Rodrigues et al., 2001; Yoon, 2000). These

conditions are determined by the prevailing values of gas dispersion properties, which

include gas hold-up, bubble size and bubble surface area flux (Sb). This latter is linked to

particle recovery through the flotation kinetic constant (Finch and Dobby, 1990; Gorain et

65

al., 1995). The correlation between gas dispersion properties and hydrophobic particle

recovery has been considered only via the true flotation mechanism, as a result of the

flotation kinetic. Quantifying the relationship between particle entrainment and gas

dispersion properties thus remains to be addressed. Therefore, the correlation between the

water recovery and gas dispersion properties is a key element to answer the above

statement. With this regard, analyzing the link between the carrying capacity of a column –

which is expressed as the mass of solids to the concentrate per unit of column cross

sectional area – and gas dispersion properties is an interesting avenue. It would lead to the

same information, as would the investigation of the relationship between the recovered

water and gas dispersion properties.

Among the various variables having an effect on the water recovery to the froth zone, the

wash water flow rate plays a key role. Experimental evidence shows that increasing the

wash water flow rate has a substantial effect to reduce the water recovery down to a

minimum, i.e. a saturation point beyond which additional wash water is not beneficial (Tao

et al., 2000). Interestingly, the froth depth only has a marginal effect on water recovery

(Tao et al., 2000; Zheng et al., 2006). It is worth noting that for a constant froth depth, the

water transferred to the concentrate was also used to analyze the froth zone stability in the

presence of different reagents such as collectors, activators and depressants (Wiese et al.,

2011).

Yianatos et al. (2009) observed a selective entrainment of fine particles (size less than 45

μm) in a large cell. On the other hand, the recovery of coarse particles (larger than 150 μm)

by entrainment was considered negligible (less than 0.1%) (Yianatos et al., 2009; Zheng et

al., 2006). Experimental evidence also confirms that if the liberation size diminishes, the

gangue entrainment increases (Guler and Akdemir, 2012). It is also worth noting in this

work, that the hydrophilic particle size also affects the water recovery and the entrainment.

Lastly, it must be emphasized that the water recovery in the industry is estimated based on

steady state mass balances, i.e. for most cases considering the feed, tailings and concentrate

mass flow rates, the mass solid fraction, and the mineral assays (Yianatos et al., 2009).

This chapter investigates (1) the effect of gas dispersion properties on water recovery, and

therefore on particle entrainment, and (2) the role of hydrophobic particle size on the water

66

recovery. The correlation between the carrying capacity and the water recovery is also

studied, to confirm previous results. It is worth mentioning that all conducted tests were

done at constant froth depth and without using wash water, in order to allow isolating the

correlation between water recovery and bubble surface area flux and bubble size.

3.2 Results and discussion

3.2.1 Effect of gas dispersion properties on the water recovery

Table 3.1 summarizes the experimental plan to do such study. The effect of the bubble size

on the water recovery is illustrated in Figure 3.1, showing that the water recovery linearly

decreases with increasing bubble diameter.

Since the total volume of air passing through the column section remained constant,

decreasing the bubble size led to an increased number of bubbles which in turn intensified

entrainment in the wake and boundary layer surrounding bubbles, as described by

Martínez-Carrillo and Uribe-Salas (2008). At constant gas flow rate, the bubble surface

area flux is determined only by the bubble sizes. Thus for the same bubble volume,

increasing the total bubble surface translates into a larger total boundary layer volume –

which contains water and entrained particles – and therefore, to additional water recovered

in the concentrate.

67

Table 3.1 Experimental plan

Hydrophobic particle class Frother concentration (10-6 V) Shear water rate (cm/s)

1 15 0.2

1 15 0.4

1 15 0.8

1 25 0.4

1 25 0.8

2 15 0.2

2 15 0.4

2 15 0.8

2 25 0.4

2 25 0.8

3 25 0.2

3 25 0.4

3 25 0.8

4 15 0.2

4 15 0.4

4 15 0.8

4 25 0.2

4 25 0.4

4 25 0.8

The number of bubbles, their size and velocity determine the gas hold-up. On the other

hand, the water recovery is also related to the number of bubbles and their size, that is to

the gas hold-up. Experimental results, shown in Figure 3.2, illustrate that the gas hold-up

also exhibits a similar relationship with the water recovery. It is worth noting that, since the

gas hold-up can be monitored on-line in industrial cells, it could eventually be used as an

indication of both the water recovery to the concentrate and thus the particle entrainment.

68

Figure 3.1 Effect of the mean bubble size (d32) on the water recovery

for constant gas rate (1 cm/s) and different particle size-classes

Figure 3.2 Effect of the gas hold-up on the water recovery

for constant gas rate (1 cm/s) and different particle size-classes

y = -11.75x + 18.68R² = 0.84

0

1

2

3

4

5

6

1 1,2 1,4 1,6

Wat

er

reco

very

(%

)

d32 (mm)

y = 0.41x - 0.04R² = 0.71

0

1

2

3

4

5

6

0 5 10 15

Wat

er

reco

very

(%

)

Gas holdup (%)

69

3.2.2 Effect of hydrophobic particle size on the water recovery

The experimental plan considered four different particle classes of talc to investigate the

effect of hydrophobic particle size on the water recovery. Figure 3.3 reports the water

recovery as a function of the bubble size for these four different particle size classes. All of

them exhibit the same behavior, hence suggesting that the size class of the hydrophobic

mineral does not influence the water recovery. This also suggests that the particle

entrainment would mainly correspond to the size fraction under 53 µm. Accordingly, using

particle size fractions coarser than 53 µm does not indicate any dependency on entrainment

or water recovery as they are recovered by means of true flotation.

Figure 3.3 Effect of the mean talc particle size on the water recovery

0

1

2

3

4

5

6

1 1,2 1,4 1,6

Wat

er

reco

vey

(%)

d32 (mm)

75/106 Microns

53/75 Microns

106/150 Microns

53/106 Microns

70

3.2.3 Effect of gas dispersion properties on the carrying capacity

Since the carrying capacity (the recovered mass of particles, both hydrophobic and

hydrophilic, per unit cell surface per unit of time) for a column encompasses the overall

recovery of flotation via true flotation and entrainment, its correlation with gas dispersion

properties indirectly determines the role of gas dispersion properties on entrainment. It

must be emphasized that the carrying capacity of the column is different from the bubble

carrying capacity (bubble load), the latter being related to the maximum particle load on the

surface of the bubbles. In this research, the carrying capacity of the column is calculated

based on the recovered mass of particles containing both hydrophobic and hydrophilic

particles. Figure 3.4 shows the effect of gas dispersion properties on the carrying capacity,

for three size classes of talc. Both the gas holdup and the bubble surface area flux have a

positive effect on the recovered mass of particles, as a result of both entrainment and true

flotation whereas the effect is negative for d32.

As mentioned previously, the coarse hydrophobic particles used in this work, were

recovered by true flotation. Thus, the amount of hydrophobic particles transferred to the

concentrate increases with both the bubble surface, and gas holdup. On the other hand, the

total boundary layer volume increases with the total available bubble surface. This results

in additional water, and fine entrained particles (quartz) in the concentrate. In other words,

both the entrainment and true flotation mechanisms follow the same trend with respect to

gas dispersion properties.

71

(a)

(b) (c)

Figure 3.4 Effect of (a) the bubble size, (b) gas holdup, and (c) bubble surface area flux on

the carrying capacity

1

2

3

0,9 1,1 1,3 1,5 1,7 1,9 2,1 2,3

Car

ryin

g C

apac

ity

(gr/

min

cm

2)

d32 (mm)

+53/-75 microns +75/-106 microns +106/-150 microns

1

2

3

0 5 10 15

Car

ry c

apac

ity

(gr/

min

cm

2)

εg (%)

1

2

3

25 45

Car

ryin

g C

apac

ity

(gr/

min

cm

2)

Bubble surface area (Sb)

72

3.3 Conclusion

In this chapter, the relevance of gas dispersion properties and water recovery was studied.

For a constant froth depth and no wash-water addition, increasing the gas hold-up and the

bubble surface area flux linearly increased the water recovery to the concentrate, whereas it

decreases for larger bubbles. These correlations can be explained from the variation of the

total boundary layer volume surrounding the bubbles and their number.

The hydrophobic particle size classes did not show any effect on the water recovery and

consequently on the particle entrainment, since relatively coarse particle size classes were

used. Lastly, the carrying capacity – as defined in this chapter – showed a good correlation

with all gas dispersion properties, the Sauter mean diameter, the gas hold-up and the bubble

surface area flux. This allowed concluding that both the entrainment and the true flotation

mechanisms follow the same trend with respect to gas dispersion properties.

73

Chapter 4 On the relationship between hydrodynamic

characteristics and the kinetics of column flotation. Part I:

modeling the gas dispersion4*

Abstract

Modeling of flotation has been the subject of many investigations aiming at better

understanding the process behavior per-se, and as well for process design, control and

optimization purposes. With this regard, the importance of hydrodynamic characteristics,

either as manipulated or measured variables, is paramount. The interfacial area of bubbles

(Ib) is introduced in this chapter as a hydrodynamic variable providing more information

about the size distribution than the commonly used bubble surface area flux (Sb).

Experimental evidence shows that the bubble size distribution can exhibit normal,

lognormal, and even multi-modal shape. Unlike the Sauter mean diameter (d32) and Sb, the

interfacial area of bubbles is derived from the complete bubble size distribution, and takes

into account these specific characteristics. Fundamental expressions are proposed

characterize Ib using the population mean and standard deviation. Experimental results

indicate that for lognormal bubble size distributions, Ib correlates well with the gas hold-up

and d32. Chapter 5 analyzes the correlation of gas dispersion characteristics with flotation

rate constant.

Résumé

La modélisation de flottation a fait l'objet de nombreuses recherches visant à mieux

comprendre le comportement du processus en soi, et à des fins de conception, de contrôle et

d'optimisation. A cet égard, l'importance des caractéristiques hydrodynamiques, soit en tant

que variables manipulables ou mesurées, est primordiale. La surface interfaciale de bulles

(Ib) est introduite dans ce chapitre comme une variable hydrodynamique fournissant plus

d'informations sur la distribution que le taux surfacique de bulles (Sb) couramment utilisé.

4* Ali Vazirizadeh, Jocelyn Bouchard and René del Villar. "On the relationship between hydrodynamic

characteristics and the kinetics of column flotation. Part I: modeling the gas dispersion". Minerals

Engineering, (2014) IN PRESS

74

Des expériences ont démontré que la forme de la courbe de distribution de la taille de bulle

peut présenter une forme normale, log-normale, et même multimodale. Contrairement au

diamètre moyen de Sauter (d32) et à l’actuelle définition de la Sb, la surface interfaciale de

bulles (Ib) provient de la distribution complète de la taille des bulles, et tient compte de ces

spécificités. Des modèles fondamentaux sont proposés pour permettre la caractérisation Ib

utilisant la population moyenne et l’écart type. Les résultats expérimentaux indiquent que

pour une distribution log-normale de la taille des bulles, Ib est corrélé avec le taux de

rétention de l’air et d32. Le chapitre 5 analyse la corrélation des caractéristiques de

dispersion du gaz avec la constante cinétique de flottation.

4.1 Introduction

Hydrophobic particles present in a mineral pulp are collected in a flotation column by

injecting fine air bubbles through a sparger located at the bottom of the reactor. The

resulting bubble-hydrophobic particle aggregates rise through the collection zone of the

column, allowing their separation from the hydrophilic particles, which settle and flow-out

through the tailings port. The ascending bubble - particle agglomerates finally reach the

froth zone, where typically a counter current wash water stream helps to remove entrained

fine -hydrophilic gangue particles. Bubble coalescence occurs in the froth zone leading to

the loss of some hydrophilic particles but providing improved mineral selectivity. The

overall flotation process can then be considered composed of a reaction process (collection)

followed by a separation process (froth selectivity).

As any other reaction process, the rate of flotation can be characterized by a kinetic

constant, relating the outcome, recovery of valuable species (primary variable) to the

process inputs, bubble and particle concentration and characteristics (secondary variables).

Assuming sufficient air-surface availability, a first order kinetics with respect to the

valuable-particle concentration, is usually accepted for flotation columns. If plug-flow

conditions prevail (as in columns), the relationship between valuable-particle recovery and

flotation time in the collection (reaction) zone is given by

(1 )kR R e

(4-1)

75

where R represents the recovery of floatable particle, R∞ is the ‘ultimate’ recovery

(resulting from the existence of impossible-to-float valuable-mineral particles),τ is the

mean particle residence time, and k is the kinetic constant of such particles. A refinement of

this expression was proposed by Gorain et al. (1997), where the kinetic constant is

decomposed into two elements: (1) the inherent floatability, associated to the

hydrophobicity of the particles, and (2) the bubble surface area flux, representing the

availability of air surface for particle collection.

4.1.1 Hydrodynamic variables and particle size

Hydrodynamic variables have a significant role on the performance of flotation cells since

they affect the reaction rate and the mass transport (water and particles) by increasing the

specific area of the dispersed phase. For instance, it has been experimentally proven that the

rate constant decreases for coarse particles at high superficial gas rates and that fine

particles require much higher superficial gas rates for effective flotation (Heiskanen, 2000).

However, Newell and Grano (2006) reported that increasing the superficial gas velocity

leads to a linearly increased overall flotation rate constant, whereas Yoon (1993) stated that

decreasing the bubble size is more effective than increasing the gas rate to reach higher

recoveries.

Many studies have been conducted to determine the effect of the bubble surface area flux

(Sb) on the collection recovery. For instance, Gorain et al. (1997) obtained a linear

relationship between the flotation rate constant and the bubble surface area flux, the slope

increasing with decreasing particle sizes. This investigation allowed characterizing the

performance of mechanical cells, by using this model. Later on, the same authors reported

that the k – Sb direct relationship was a function of the froth depth. Only for shallow froths,

a linear relationship was found (Gorain et al., 1998) . It can then be concluded that froth

depth plays an important role on the overall kinetics.

The existence of a linear relationship between the flotation rate constant and the froth depth

was also confirmed by Vera and coworkers (1999), The authors also claimed that the

collection zone rate constant of the two evaluated minerals (chalcopyrite and pyrite)

increases with the air flow rate, observing though a reduction in froth-zone recovery. This

76

was associated to a detachment of particles from the bubbles in the froth-zone. This

observation led to the use of a shallow froth in the present work.

4.1.2 Bubble surface area flux models

The bubble surface area flux is defined as the surface of a given number of rising bubbles

per unit time and unit of cell cross-sectional area. Assuming same-size spherical bubbles, a

relation for Sb can be derived as a function of the superficial gas velocity Jg and the bubble

diameter db.

6 g

b

b

JS

d (4-2)

Since bubbles do not have the same size, it is customary to replace db by the Sauter mean

diameter d32

32

6 g

b

JS

d (4-3)

The d32 is a scalar value obtained from a bubble size density function or directly calculated

from observed data. However, important information related to the shape of the bubble size

distribution (BSD), such as multi-modal and tails behavior, is lost in this compression

exercise (Maldonado et al., 2008c). The general Sb definition suggests that the bubble size

distribution play an important role in the metallurgical performance of any flotation

process.

For uniform bubbles, an empirical relationship was presented by Finch & Dobby (1990) to

predict the bubble size from the superficial gas velocity. Finch and Dobby's model was

further developed by Nesset and co-authors (2006), who proposed an empirical model for

estimating the d32 based on do (bubble size at Jg = 0) and two parameters depending on the

bubble generation system, the used chemical reagents, and eventually, the slurry properties.

Based on the collected data from commercial flotation cells, they also reported that the d32

values are sensitive to the relative percentage of large bubbles (larger than 1.5 mm),

whereas d10 values are more sensitive to the amount of small bubbles (smaller than 1 mm).

Instead of using direct values of d32 from histograms describing the size distribution, other

77

authors have used the bubble size density function to evaluate the d32 (Grau and Heiskanen,

2005; Vinnett et al., 2012).

Gorain and coauthors (1999) presented a model to predict the Sb in mechanical cells as a

function of the impeller peripheral speed, the air flow rate per unit cell cross-sectional, the

impeller aspect ratio and the 80% passing size of the feed. This model seems adequate for

cells with forced air feed mechanisms. However, it is known to predict inaccurate values

for self aspirating cells (Gorain et al., 1999).

Heiskanen (2000) criticized Gorain’s model, claiming that “the measurement and

computation of superficial gas velocity, as well as the bubble size in some cases, could be

biased under some conditions”. Moreover, the model validation did not address the real

behavior of various particle sizes, as larger bubble sizes show a higher flotation rate

constant, and poor dispersion conditions give a good flotation response, both results

contradicting earlier research findings and industrial experience. Heiskanen concluded that

the bubble surface area flux needed a broader validation using different types of ore, and

that the linear k – Sb relationship required being further investigated.

4.1.3 Bubble size measurement

Cape Town University (CTU) presented the first attempts to measure bubble size using an

optical system. The experimental set-up uses a belled tube to collect ascending bubbles,

wherefrom they were driven to an inclined viewing chamber. The system combined with a

high intensity lighting and a camera placed at either sides of the viewing chamber to allow

taking pictures (O’Connor et al., 1990; Tucker et al., 1994).

The HUT bubble size sampler developed at Helsinki University of Technology has also

been applied by Grau and Heiskanen (2005) to measure the bubble size distribution in

mechanical cells . They had previously compared the HUT and CTU systems to measure

the bubble size and they found some limitations with the CTU method, “when the gas was

not efficiently dispersed” (Grau and Heiskanen, 2002). The bubble sampling system in the

CTU method also caused bubbles larger than 1 mm to break-up, hence leading to finer

bubble size distribution than the actual ones (Grau and Heiskanen, 2002).

78

The McGill bubble size analyzer is another visual technique which was introduced by

McGill University researchers (Vinnett et al., 2012). Its operational principle is based on

conveying a sample of bubbles (with some pulp) into a viewing chamber where they are

exposed to an appropriate light source and are photographed with a digital camera. An

automated image analysis procedure allows measuring the size of the collected bubbles.

The bubble viewer system consists of a sampling tube attached to the bottom of a sealed

viewing chamber (Hernandez-Aguilar et al., 2004). It is worth mentioning that the viewing

chamber must be periodically purged of the accumulated air, and periodically

cleaned/refilled with fresh water because of progressive slurry buildup, thus making online

measurement very challenging.

Until now, the use of the bubble viewer has been reported basically in offline process audits

(gas dispersion conditions in cells or columns). The only known application for bubble size

control (i.e. continuous db measurement) is the two-phase research work at Université

Laval (Maldonado et al., 2008c).

For process control purposes, Maldonado et al. (2008c) used the commercial software

program Image-J to analyze the bubble images. Based on circular shape detection, the

method however fails at recognizing clustered, large or elliptical bubbles. Vinnett et al.

(2012) proposed a semi-automatic methodology using the USM-IMA in-house software

program to address this issue. The method however is inadequate for continuous

monitoring because of the offline pre-processing step. The introduction of a novel approach

using the Circular Hough Transform (CHT) (Riquelme, 2013a) finally enabled both online

applications (e.g. process control) and the detection of all bubbles, regardless of their shape,

dimension or interaction (clustered, super-imposed, etc.).

4.1.4 Bubble size distribution

Although many bubble size distribution measurements have been reported lately in the

literature (Liu et al., 2013; Xu et al., 2012; Zhu et al., 2014), only a few mention their effect

on flotation performance (Grau and Heiskanen, 2005; Heiskanen, 2000; Vinnett et al.,

2012; Wongsuchoto et al., 2003). Bubble size distributions are now frequently off-line

measured in various flotation cell (mechanical and columns) audits, where they are first

transformed into a scalar, the mean bubble size, thus hiding some important distribution

79

features. Despite their availability, these BSDs have not been used as such in mathematical

models or practical applications.

Experimental data for typical operating conditions in mechanical cells suggests that the

bubble sizes follow a log normal distribution (Vinnett et al., 2012). In some instances, a

modification of the log normal distribution leading to an upper limit distribution is more

suitable to represent bubble size distributions (Grau and Heiskanen, 2005; Heiskanen,

2000). Wongsuchoto et al., (2003) showed that for higher superficial gas velocities, i.e. 20

cm/s to 40 cm/s, which is out of the typical range for flotation conditions, the bubble size

distribution exhibits a normal distribution.

Nesset et al.(2006) studied the relationship between the percentage of bubbles smaller than

1 mm and the d10. They found a unique consistent relationship for different data sets. They

also concluded that increasing frother concentration reduces both the mean and standard

deviation of bubble size distribution. Tucker et al. (1994) reported the same behaviour

when using three different types of frother: one pure and two commercial blends. Vinnett et

al. (2012) also observed a strong relationship between the BSD mean and standard

deviation, and noted that BSDs with a large mean value showed greater dispersion, thus

confirming the existence of a pattern in the shape of distribution. Recent work by Quinn

and Finch (2012) revealed that in absence of frother, bimodal distributions are obtained, i.e.

bubbles finer than 1 mm and bubbles larger than 3 mm.

Considering the abovementioned results, observations and analysis, it seems clear that the

whole bubble size distribution should be considered for modeling the flotation rate

constant. Ideally, the distribution would be narrow, but this is not always the case in

industrial practice (Hernandez-Aguilar et al., 2004; Yianatos et al., 2001; Yianatos et al.,

2012). The interfacial area of bubbles is a solution to characterize normal or lognormal

populations, especially when the bubble size distribution is represented by a multi-model

density function. This chapter proposes fundamental equations to characterize normal and

lognormal bubble size distributions, and also analyses the correlations between the different

hydrodynamic variables.

80

4.2 Equations of gas dispersion

Bubble populations in flotation devices are characterized by the total number of individuals

per unit of volume n, and their size distribution function f(db), between a minimum and a

maximum bubble size. Both the gas hold-up (εg) and interfacial area of bubbles (Ib) can be

calculated from the bubble size distribution.

While Sb indicates the total bubble surface area passing through a horizontal section, the

gas hold-up represents the gas volumetric fraction. The interfacial area of bubbles refers to

the overall surface area of bubble inside the device. It has lately been the subject of several

studies in bubble columns (García-Salas et al., 2008; Gómez-Díaz et al., 2008; Maceiras et

al., 2010). However, it has not been considered so far in flotation modeling.

The bubble interfacial area (m2/m3) and the hold-up of a given set of bubbles are

respectively defined as

max

min

2 ( )db

b

d

b b b bd

I n d f d d (4-4)

and

max

min

3 ( )d6

b

b

d

g b b bd

n d f d d

(4-5)

Depending on the type of distribution used, these equations can be further developed as

presented hereafter.

4.2.1 Normal distribution

If the f(db) is a normal distribution, equations (4-4) and (4-5) become

2

max 2

min

( )

32

2

1d

6 2

bb

b

dd

g b bd

n e d d

(4-6)

and

2

max 2

min

( )

22

2

1d

2

bb

b

dd

b b bd

I n e d d

(4-7)

81

Combining Equations (4-6) and (4-7) allows representing the interfacial area as a function

of the gas hold-up:

2

max 2

min

2

max 2

min

( )

32

2

( )

22

2

1d

2

16 d

2

bb

b

bb

b

dd

b bd

g

ddb

b bd

e d d

Ie d d

(4-8)

or simply

max

min

max

min

2 3

2 2

, , d

6 , , d

b

b

b

b

d

b b bdg

d

bb b b

d

f d d d

I f d d d

(4-9)

Solving for db leads to

max2 2

2 2

2 2

2 2

min

2

2 2 2 2 2

2

2 2

2 ( ) (2 3 ) ( )1

6(2 ) ( ) 2 ( )

bb b

b b

b

dd d

bb b

g

d db b

bd

de d d e erf

I de erf e d

(4-10)

In a normal distribution, 95% of the bubbles are in the domain - 2 and 2. Thus

assuming

min 2bd (4-11)

and

max 2bd (4-12)

Equation (4-10) becomes

32 2

4

32 2

4

1 6(2) (2 3 )

1 2

1 26(2) (2 )

2

g

b

erfe

Ierf

e

(4-13)

or simply

82

2

2 2

1 92.28

6 92.28 46.14

g

bI

(4-14)

Finally, Ib can be expressed as a function of εg and the parameters of bubble size

distribution (μ and σ) as

2

2 2

692.28

92.28 46.14

g

bI

(4-15)

4.2.2 Log-normal distribution

If the BSD is represented by a log-normal distribution, then Equations (4-3) and (4-4)

become:

2

max 2

min

(ln )

22

2

1d

6 2

bb

b

dd

g b bd

n e d d

(4-16)

2

max 2

min

(ln )

2

2

1d

2

bb

b

dd

b b bd

I n e d d

(4-17)

Combining again the expressions for Ib and εg leads to

2

max 2

min

2

max 2

min

(ln )

22

2

(ln )

2

2

1d

2

16 d

2

bb

b

bb

b

dd

b bd

g

ddb

b bd

e d d

Ie d d

(4-18)

Solving for db leads to

max2

2

min

9 23

2

22( )

3 ln( )

21

6 2 ln( )

2

b

b

d

b

g

b b

d

de erf

I de erf

(4-19)

83

Since

2

2

3 ln( )

21

2 ln( )

2

b

b

derf

derf

(4-20)

Equation (4-19) can be approximated as

292

2

6

g

b

e

I

(4-21)

hence

29

22

6g

bI

e

(4-22)

As result of the complexity of manipulating bubble size distribution functions, Bordel et al.

(2006) proposed the following simplified model, based on McGinnis and Little’s approach

(2002), to predict the interfacial area of bubbles (Ibm) from the bubble Sauter mean diameter

and gas hold-up

32

6g

bmId

(4-23)

However, Vinnett et al. (2012) reported that the d32 can be calculated using the second and

third momentum of the lognormal distribution as

22.5

32d e (4-24)

which demonstrates that the model proposed by Bordel and coworkers is not an accurate

simplification of equation (4-22).

Finch et al. (2000) developed a simple approximation linking the bubble surface area flux

and the gas hold-up

5.5b gS (4-25)

84

This relation was validated using various data sets corresponding to different flotation

devices were analyzed and the correlation seems appropriate for Sb lower than 130 s-1 and

εg lower than 25% (typical industrial flotation conditions). To estimate the mean bubble

size, used for calculating Sb, the drift flux analysis method was used. It must be emphasized

that this method exhibits a limited precision, and the correlation is not adequate in some

cases. With this regard, Finch and coworkers (2000) reported that the model did not fit

Gorain’s data base well.

According to Equations (4-23) and (4-25), the bubble surface area flux can be equal to the

specific interfacial area of bubbles under certain condition (see section 4.4.2). Regardless of

their units, Ib and Sb are supposed to be interchangeable variables particularly in their

relation with the flotation rate constant.

4.3 Test procedure

As a result of its constant efficiency over a rather long period of time, a polyglycol frother

(F150) was selected (Appendix B) for the experimental work. The gas rate and feed rate

were set at 1 cm/s for all tests. Talc, as a natural hydrophobic mineral consumes a fraction

of F150, which is absorbed on the particle surface (Kuan, 2009). To avoid the results being

affected by this interaction, samples started being taken after one hour of column operation

time, thus ensuring constant frother efficiency over time. Bubble size distribution was

measured using an on-line version of the McGill bubble size analyzer. The operating

principle consists in directing a sample of bubbles into a viewing chamber, where they are

exposed to an appropriate lighting source, and photographed with a digital camera. Each

test allowed sampling populations approximately composed of 10,000 bubbles, thus

ensuring representative distribution. Figure 4.1 shows a typical bubble size distribution

(histogram).

Bubble detection was performed with the CHT-based algorithm (Riquelme, 2013a, b),

using three frames per second to ensure that every picture presented an entirely ‘new’

sample of bubbles, thus avoiding multiple counts of the same bubble.

85

Figure 4.1 Bubble size distribution histogram

4.4 Results and discussion

4.4.1 Modeling the experimental bubble size distributions

In the current study, bubble size distributions have been generated through manipulating

the shear water rate and frother concentration. Table E.1 in Appendix E provides the

detailed experimental results. The generated bubble size distributions can be categorized in

three main types.

The first type (T1) is obtained using low frother concentration (5 ppm F150) and low

shear-water rate (0.2 cm/s), and exhibits a peculiar shape as shown in Figure 4.2. Both

very fine and large bubbles are observed as a result of the lack of frother. Thus, the bubble

size distribution cannot fit a single or a combination of probability density functions

(PDF), and Ib can only be calculated through the Bordel et al's model (using the d32 and

gas hold-up).

0 10 20 30 40 50 60 700

200

400

600

800

1000

1200

Bubble size (mm)

Qu

an

tity

Pixels

0.820.41 1.25 1.66 2.08 2.5 2.9

86

Figure 4.2 Example of a bubble size histogram for T1 distribution

Increasing the shear water flow rate, and maintaining the frother concentration at the

minimum value changed the bubble size distribution shape and generated the second type

of bubble size distribution (T2), exhibiting a multi-modal bubble size density function.

Mixing two or more PDFs allows fitting multi-shape bubble size density function. In this

study, four normal PDFs have been applied to cover the second type of bubble size

histograms using the maximum likelihood concept. Equation (4-15) thus requires being

adapted as follows

1 1 2 2 3 3 4 4

1 13b g

P P Q QI

(4-26)

where P and Q are the distribution weights, and

2

2 2

92.28

92.28 46.14

i ii

i i

(4-27)

Figure 4.3 illustrates a cumulative multi- shape bubble distribution with fitted model.

0 10 20 30 40 50 60 700

200

400

600

800

1000

1200

1400

1600

Bubble size (mm)

Qu

an

tity

Pixels

2.5 2.90.82 1.25 1.66 2.080.41

87

Figure 4.3 Example of a cumulative bubble size T2 distribution fitted with multi-shape

density function

The third category (T3) of distribution was obtained using operating conditions ensuring

the generation of small bubbles and little coalescence, i.e. with both higher shear- water

flow and frother concentration. In this case, the bubble size histogram can be represented

by a lognormal density function, thus allowing Ib to be calculated using Equation (4-22).

Figure 4.4 shows an example where the cumulative bubble size distribution can be properly

fitted to a lognormal density function.

0 10 20 30 40 50 60 700

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bubble size (mm)

Cu

mu

lati

ve

Measured

Model

0.41 1.25 1.66 2.08 2.5 2.90.82

Cu

mu

lati

ve f

req

uen

cy

88

Figure 4.4. Example of a cumulative bubble size T3 distribution fitted with a lognormal

density function

Table 4.1 summarizes the experimental conditions used to generate three main types of

BSD. These hydrodynamic conditions have been tested on three particle size classes.

Table 4.1 Experimental conditions for three types of BSD

BSD

Type

Frother

concentration

(ppm)

Shear rate

(cm/s)

Gas rate

(cm/s)

T1 5 0.2 1

T2 5 0.4, 0.8 1

T3 15, 25 0.2, 0.4, 0.8 1

4.4.2 Correlations between hydrodynamic variables

The calculated interfacial area of bubbles using Equation (4-26), denoted by Ibc, for T2

multi-shape bubble size density function does not correlated with Ibm (Equation (4-23)) as

10 20 30 40 50 600

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Bubble size (mm)

Cu

mu

lati

ve

Measured

Model

0.82 1.66 2.08 2.50.410 1.25

Cu

mu

lati

ve f

req

uen

cy

89

shown in Figure 4.5(left). This could be a result of compacting the whole distribution into a

single value, d32. When d32 does not properly represent the distribution, Ibm does not relate

to Ibc.

The other possibility could be that the PDF model does not fit the bubble size histogram

well. However, in the current study, the PDF completely lies down on the histogram,

therefore the distribution compaction on a single value must be the cause. This is consistent

with the observation for T3 distributions (lognormal), and Ibc, calculated with Equation

(4-22), correlates very well with Ibm (Figure 4.5(right)).

Figure 4.5 Shows the correlation between Ibc and Ibm

(left) for T2 distribution (right) for T3 distribution

According to Figure 4.5 though, Ibm and Ibc are clearly not interchangeable and

dissimilarities (slope ≠ 1) are caused by the inaccurate approximation used for the

denominator of equation (4-23) (see section 4.2.2).

Figure 4.6(a) reveals some correlation between Ibc and Sb, but with an important scattering.

As observed for T2 distributions, Ibc correlates with Sb for T3 distribution, but results also

present some scattering (Figure 4.6(b)). Unlike the correlation between Ibc and Sb for T2

systems, no relationship between Ibm and Sb can be reported here (Figure 4.6(c)). This may

be explained once more by the compaction of the distribution into a single value (d32).

0 20 40 60 800

10

20

30

40

50

60

70

80

Ibm

I bc

y=0.57x

R2=0.98

0 5 10 150

5

10

15

Ibm

I bc

y=0.0006x+0.95

R2=0.00034

90

In addition, based on the good correlation of Ibm and Ibc for T3 distributions, a similar

relationship between Ibc and Sb is expected for Ibm and Sb; this can be seen in Figure 4.6(d).

(a) (b)

(c) (d)

Figure 4.6 Correlation between : (a) Ibc and Sb for T2 distributions, (b) Ibc and Sb for T3

distributions, (c) Ibm and Sb for T2 distributions, (d) Ibm and Sb for T3 distributions

Figure 4.7 shows the correlations between the gas hold-up and both Ibc and Ibm for T2

(multi shape modal) and T3 (lognormal) bubble size distributions.

31.5 32 32.5 33 33.5

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

Sb

I bc

y=0.11x-2.45

R2=0.69

42 44 46 48 50

10

15

20

25

30

35

Sb

I bc

y=2.94x-89

R2=0.77

31.5 32 32.5 33 33.5

4

6

8

10

12

14

Sb

I bm

y=0.95x-24.3

R2 =0.06

42 44 46 48 50

20

30

40

50

60

Sb

I bm

y=3.76x-136.1

R2= 0.76

91

(a) (b)

(c) (d)

Figure 4.7 Correlation between: (a) Ibc and εg for T2 distributions, (b) Ibc and εg for T3

distributions, (c) Ibm and εg for T2 distributions (d) Ibm and εg for T3 distributions

Obviously, the very good correlation between Ibm and the measured gas hold-up arises

from the definition of Ibm itself, i.e. as a depended function of εg. As seen in Figures 4.7(c)

and 4.7(d), the value of the slopes are different, this being the result of the fact that bubble

sizes in T2 distribution are larger than those in the T3 distribution.

3.5 4 4.5 5 5.5

0.75

0.8

0.85

0.9

0.95

1

1.05

1.1

g %

I bc

y=0.12x+0.4

R2 =0.72

6 8 10 12

10

15

20

25

30

35

g %

I bc

y=3.42x-6.4

R2= 0.97

3.5 4 4.5 5 5.5

10

12

14

16

18

g %

I bm

y=3.25x

R2 = 0.98

6 8 10 12

20

25

30

35

40

45

50

55

60

g %

I bm

y=4.68x

R2= 0.96

92

On the other hand, Ibm fits the measured gas hold-up very well, but the model coefficient is

not exactly identical to that reported by Finch et al. (2000), i.e. 5.5 compared to 3.25 and

4.68 (see Equation (4-25)). It must be emphasized that the bubble surface area flux

calculated using Equation (4-26) is similar to interfacial area of bubbles for d32 values

~1.09 mm. In the present investigation, the average values of d32 are 1.86 mm and

1.28 mm for T2 and T3 distributions, respectively.

Finding the correlation between Ib and the presently used hydrodynamic variables, i.e. Sb,

gas hold-up and d32, is the first step for applying Ib as a hydrodynamic variable in the

flotation process. The next step consists in determining the relationship between Ib and the

flotation kinetics. Since the hydrodynamic variables relate to the flotation rate and Ib, the

flotation rate is expected to have a correlation with Ib. Ib also encompasses both the effect

of gas hold-up and bubble size, and also integrates the whole underlying bubble size

density function. As mentioned earlier, all this information is lost when applying a single

d32 value.

4.5 Conclusion

The interfacial area of bubbles can be considered as a solution to address the issue of

compacting the entire bubble size distribution into a single value (d32) for the purpose of

calculating the flotation rate constant. For normal or lognormal distributions, fundamental

expressions allow characterising Ib using the population mean and standard deviation,

which may lead to a more adequate estimate of the kinetic constant.

In fact, the fundamental equations of gas dispersion have been first presented as a possible

solution for some cases, where multi-shape bubble size density function and lognormal

density function have been well fitted to histograms. Three main classes of bubble size

histograms have been distinguished. The first class (T1) could not be represented by a

single or multi PDF's and has been only determined through Bordel et al.’s model. The

second type (T2) of histogram has been fitted to multi-shape density function and the third

one (T3) follows a lognormal density function. For the two last classes, histogram of the

inter-correlations between gas dispersion properties has been studied. A strong correlation

between gas dispersion properties – namely the gas hold-up and bubble surface area flux –

has been detected for bubble size distributions represented by lognormal density functions.

93

Results were not as convincing for distributions modeled using multi-shape density

function.

Chapter 5 will analyze the correlation of gas dispersion characteristics and the rate constant

of flotation.

95

Chapter 5 On the relationship between hydrodynamic

characteristics and the kinetics of flotation. Part II: model

validation5*

Abstract

The estimation of the flotation rate constant is generally considered difficult because of the

number of variables – not always measurable –required to determine its value. Among

them, the particle size distribution and hydrodynamic characteristics of the device are

considered key elements.

Chapter 4 introduced the interfacial area of bubbles (Ib) as a hydrodynamic variable

providing more information about the size distribution than the bubble surface area flux

(Sb). Fundamental expressions were proposed to characterize Ib using the population mean

and standard deviation. Experimental results indicated that for lognormal bubble size

distributions, Ib correlates very well with the gas hold-up and d32.

Chapter 5 investigates the correlation between the flotation rate constant and the particle

size as well as some hydrodynamic variables using a Projection to Latent Structures (PLS)

analysis. The tests were conducted under 'ideal' conditions (i.e. shallow froth, low mineral

concentration and pure mineral particles). Results suggest that for the fine particle sizes, the

bubble surface area flux (Sb) should be considered for the kinetic constant modeling. For

coarser particle, the gas hold-up (εg) is the determining parameter. In practice though, the

particle size distribution often lies between these two extreme cases, and can either span a

very large range or contains intermediate size particles. In such cases, the interfacial area

of bubbles (Ib) better correlates with the flotation kinetic constant.

5 *Ali Vazirizadeh, Jocelyn Bouchard, René del Villar, Massoud Ghasemzadeh Barvarz and Carl Duchesne.

"On the relationship between hydrodynamic characteristics and the kinetics of flotation. Part II: model

validation". Minerals Engineering, (2014) IN PRESS.

96

Résumé

L'estimation de la constante cinétique de flottation est généralement considérée comme

difficile à cause du nombre de variables qui ne sont pas toujours mesurables. Parmi ceux-ci,

la distribution granulométrique des particules et les caractéristiques hydrodynamiques son

considérées comme des éléments clés.

Le chapitre 4 a introduit la surface interfaciale de bulles (Ib) comme une variable

hydrodynamique fournissant plus d'informations sur la distribution de la taille que la

densité de flux d'air (Sb). Des modèles fondamentaux ont été proposés pour caractériser Ib

en utilisant la moyenne de la distribution de dimension et sont écart-type. Les résultats

expérimentaux indiquent que pour une distribution logarithmique de la taille de bulle, Ib est

corrélé avec le taux de rétention de l’air et le d32.

Le chapitre 5 étudie la corrélation entre la constante cinétique de flottation et la taille des

particules ainsi que des variables hydrodynamiques données en utilisant une projection de

structures latentes (PSL). Les tests ont été effectués dans des conditions idéales (écume

superficielle, une fraction solide faible et des particules minérales pures). Les résultats

suggèrent que pour les particules fines, la densité de flux d'air (Sb) doit être envisagée dans

la modélisation des constantes cinétiques. Pour les grosses particules, le taux de rétention

de l’air (εg) est le paramètre déterminant. Dans la pratique cependant, la distribution

granulométrique des particules se situe souvent entre ces deux cas extrêmes, elle peut soit

couvrir une très large gamme ou contenir des particules de taille intermédiaire. Dans ce cas,

la surface interfaciale de bulles (Ib) a une meilleure corrélation avec la cinétique de

flottation.

5.1 Introduction

In chapter 4 the interfacial area of bubbles (Ib) was introduced as an alternative variable for

flotation kinetics modeling. The interfacial area of bubbles refers to the overall surface area

of bubble inside of the flotation cell. While Sb indicates the total bubble surface area

passing through a section, the gas hold-up (εg) represents the gas volumetric fraction inside

of the device. The bubble interfacial area and hold-up of a given set of bubbles are

respectively defined as

97

(5-1)

and

max

min

3 ( )d6

b

b

d

g b b bd

n d f d d

(5-2)

where n is the total number of bubbles per unit of volume, f(db), their size distribution

function, and db min / db max are the minimum and a maximum bubble size respectively.

The correlation between Ib and other hydrodynamic variables was studied aiming at finding

a relationship with the flotation rate constant. Results demonstrated that Ib is related to both

gas hold-up (εg) and bubble surface area flux (Sb).

Ib could be considered as a compromise between εg and Sb for flotation modeling, since,

according to its definition, it encompasses both the bubble size and gas hold-up effects.

Investigating the ties between the kinetics of flotation and the hydrodynamic variables

requires an adequate model and an appropriate set of tests for calibration purposes.

Moreover, the effect of the particle size distribution must also be factored in, which has

been the case in this investigation.

For the purpose of model-parameter calibration, collection-zone timed-recovery values

must be available. Samples of feed, concentrate and tail flows, must then be obtained over

a given time span, for subsequent processing and analyses for the considered species. In the

case of a flotation column, the device used in this study, normal launder samples were

taken, as sampling of the collection zone was not possible in the small device used for this

study. Therefore, using such samples for the estimation of the overall flotation recovery, to

infer the true collection-zone recovery, can strictly be valid only when the froth zone is

very shallow.

The collection-zone recovery has already been modeled by various researchers. Among

them, Yianatos et al. (2005) used a rectangular distribution function of the rate constant (k)

and a tank in-series model for residence time distribution. Others have demonstrated that

the shape of the distribution function does not have a substantial effect on the average rate

constant (Polat and Chander, 2000). In the present work, a single value of the kinetic

constant was used to represents the rate constant distribution. In fact, the effect of the

max

min

2 ( )db

b

d

b b b bd

I n d f d d

98

particle size distribution and hydrodynamic variables – i.e. the gas rate, bubble surface area

flux, gas hold-up and bubble size – on the ultimate rate constant is the dominant one, and

largely outweighs that of using a rate constant distribution model.

With this regard, it has been reported that the flotation kinetics of single-size particles

varied significantly with the ultimate recovery (R∞) and the mean of the kinetic-constant

distribution of the process (Polat and Chander, 2000). The effect of the particle and bubble

size on micro processes, such as collision and attachment, has also been reported in several

studies(Dai et al., 2000; Nguyen, 1998; Sarrot et al., 2005; Weber and Paddock, 1983). The

presented micro-process models were applied to develop a particle-size dependent

flotation-rate model. For instance, Pyke et al. (2003) reported a characteristic bell-shape

flotation rate constant curve with respect to particle sizes (Ahmed and Jameson, 1985),

exhibiting a maximum at intermediate particle size between 8 µm and 80 µm.

Since an empirical kinetic model would subsequently be derived, an appropriate approach

is required to relate hydrodynamic variables to the kinetic constant value. However, it was

shown in Chapter 4 that εg, Sb and Ib are inter-correlated. Classical linear regression

methods are then not applicable because they assume the predictor variables are

uncorrelated. The multicolinearity in the predictor variables is also demonstrated in

Appendix F.1. The analysis shows how the inter correlation adversely affects the regression

technique performance.

For this reason, a multivariable regression using a Projection to Latent Structures (PLS) is

proposed. The technique is briefly introduced in Appendix F.2 PLS regression takes

advantage of the multicolinearity in the predictor variables, and allows in the present case

comparison of the relevance of εg, Sb and Ib jointly to predict the flotation rate constant.

This comparison accredits a new hypothesis about the effect of particle size on

hydrodynamic variable selection for kinetic modeling, which was corroborated using four

different datasets extracted from the literature. It is worth mentioning that the PLS models

were developed using custom scripts written in MATLAB R2010a (MathWorks)

environment and using the PLS Toolbox (Eigenvector Research).

99

5.2 Test procedure

The experimental set-up used for the experimental work was already described in

Chapter 2. Here, only the experimental design and the test procedure are presented.

Four mixtures of talc (hydrophobic component or valuable mineral) and quartz (hydrophilic

component or gangue), classified with wet screening (two cycles for talc), were used. In all

mixtures, quartz was present with the same complete particle size distribution, which was

combined to a ‘single class’ of talc. Three different single classes were used: +53 / -75 μm,

+75 / -106 μm, and +106 / -150 μm. Nine tests were also conducted using a complete

‘mixed’ size distribution. The relative amount of quartz and talc varied between mixtures

(between 40% and 60%). The talc was very hydrophobic, producing a voluminous froth.

Hence a solid concentration of 4% was used to prevent the froth from overflowing the

launder.

A full factorial design was used with three independent variables: the talc particle size class

(4 levels: fine, medium, coarse, and mixed), frother concentration (3 levels: 5, 15, and 25

ppm) and shear-water superficial velocity to the sparger (3 levels: 0.2, 0.4, and 0.8 cm/s).

Other experimental conditions were set as follows:

gas superficial velocity: Jg = 1 cm/s

type of frother: F150

froth depth: H = 10 cm

pulp conductivity: 1000 – 20,000 µS.

To be in this range small amount of sodium chloride is added to the pulp and no other

reagent is used during the test.

The froth layer was kept very shallow (at 10 cm) in all tests, in order to estimate, with

minimum error, the collection zone recovery from global column recovery values. No

wash-water was used.

After one hour of column operation, the system was deemed to be working under steady-

state conditions, and data collection (bubble pictures and gas holdup measurements) was

launched where samples of tails, feed and concentrate were taken. The pulp residence time

100

was calculated by means of direct measurement of column tailings flow rate, since the

column feed rate and froth height were kept constant throughout the test. One-liter tailing

and concentrate samples were taken simultaneously. This sample volume did not have a

significant effect on the feed composition, compared to the 350 liters in feed tank. .An XRF

analyzer was used to determine the talc content of the samples. Chemical analysis were

then reconciled using Bilmat™. The reconciled data allowed evaluating the recoveries,

which in turn allowed estimating the flotation kinetic constant from

𝑅 = 𝑅∞(1 − 𝑒−𝑘𝜏) (5-3)

assuming a plug flow behavior – acceptable for a small diameter laboratory column – and

where R is the recovery of floatable particle, R∞ represents the ‘ultimate’ recovery, τ is the

mean particle residence time, and k is the talc kinetic constant.

5.3 Results and discussion

Table F.1 of Appendix F present the experimental results used to develop the PLS model.

5.3.1 Particle size and kinetic constants

As presented in Appendix F.2, the Variable Importance on the Projection (VIP) allows

quantifying the weight of the independent variables in the PLS model. The VIP score

thresholds are as follows:

VIP > 1: for the most influential independent variables in the PLS model,

0.8 < VIP < 1: for moderately influential independent variables in the PLS model,

VIP < 0.8: for the least significant independent variables in the PLS model.

As can be seen in Figure 5.1, the particle size has a significantly higher VIP score than any

of the other variables, thus showing that the particle size effect is predominant for the

flotation system investigated.

101

Figure 5.1. Relative importance of particle size (P.S.), εg, Sb and Ib

5.3.2 Hydrodynamic variables and rate constant

Notwithstanding the abovementioned results, the particle size obviously cannot be

manipulated for controlling a flotation cell performance. Therefore, other variables must be

considered. Among other possible manipulated variables, the hydrodynamic characteristics

seem to be good candidates, since they present a strong correlation with the flotation rate.

The rate constant is correlated to the collection efficiency, which in turn depends on two

main phenomena: collision and attachment efficiencies. It has also been reported that the

particle size weights the value of these two antagonistic terms on the collection process

(Diaz-Penafiel and Dobby, 1994; Heiskanen, 2000). Thus, small particle size shows low

collision efficiency, but in return exhibit a high probability of attachment (Dobby and

Finch, 1987). On the other hand, the coarse particles are likely to collide with bubbles, but

detach readily.

5.3.2.1 Fine particles

Several theoretical studies have proven the direct relationship between collision efficiency

and particle size – bubble size ratio (Nguyen, 1998; Sarrot et al., 2005; Weber and

Paddock, 1983; Yoon, 1993). For four different flow regimes, Yoon (1993) determined that

the collision efficiency can be calculated from the particle size – bubble size ratio as

1 2 3 40

0.5

1

1.5

2

2.5

3

Variables

VIP

Sco

res f

or

k

Sb

g

IbP.S.

102

(5-4)

where Pc is the collision probability, dp the particle size, db the bubble size. A and n are

model parameters related to the Reynolds number.

For a constant fine-size particle distribution, Equation (5-4) indicates that the main factor

on the collision efficiency is the bubble size. On the other hand, bubble surface area flux is

directly determined by bubble size when the gas rate is constant. Consequently, under the

assumed conditions, collision efficiency would be a function of the bubble surface area flux

which means that for collecting fine-size particles, the bubble surface area flux should be

considered as the major influencing factor. Therefore, the kinetic constant of fine particles

would exhibit a close correlation with the bubble surface area flux.

5.3.2.2 Coarse particles

As mentioned above, the collection efficiency of coarse particle is mostly determined by

the attachment process. The basic assumption in analyzing the likelihood of attachment is

that it occurs when the film between the bubble and particle breaks and the three-phase

contact is created. The total time required, following the particle-bubble collision, is called

the induction time. A given particle will then attach to the bubble if the particle sliding time

on the bubble surface is longer than the induction time. As presented by Finch and Dobby

(1990), the attachment efficiency depends on particle sliding velocity. Since in actual

flotation, instead of a single-bubble / single-particle phenomenon, a bubble swarm and a

population of particles are implicated, the slip velocity (Usg) can be assumed to be equal to

the sliding velocity on average. The slip velocity is a function of the gas and pulp

superficial velocities, and gas hold-up, i.e.

(5-5)

where Jl is the pulp superficial velocity, and +/- stands for counter-current flow and co-

current flow respectively. Consequently, the attachment efficiency should be related to gas

hold-up when gas and pulp flow rate are kept constant.

n

p

c

b

dP A

d

(1 )

g lsg

g g

J JU

103

5.3.3 Experimental validation

To validate the above phenomenological analysis, a set of tests was completed for each

given single size-class of talc. It was expected that for fine size particles, Sb with k would

exhibits a better correlation than εg and Ib. However, for a coarse size class, the correlation

between the kinetic constant and the εg should be better than with the other hydrodynamic

variables.

The relative importance of the various hydrodynamic variables for their use in the modeling

of the flotation rate, for various single-size particles was studied. For each single-size class,

the VIP scores for εg, Sb and Ib were calculated. Figure 5.2 shows how the VIP scores of εg,

Ib and Sb for (a) fine, (b) mid-, and (c) coarse size classes. Results indicate that Sb has a

stronger effect than εg and Ib, on the kinetic constant, for the flotation of fine particles. As

the particle size increases, the importance of Sb decreases, whereas the importance of εg and

Ib increases, thus corroborating the phenomenological analysis, i.e. the collision efficiency

mainly determine the rate constants for fine particles.

When the particle size increases, the weight of εg in the relationship becomes more

important, but the experimental results do not reveal a clearly dominant effect, because the

‘coarse’ size fraction had a top size of 150 μm, which is far from being extreme for a

flotation feed. The validation of the phenomenological analysis was completed using data

extracted from the literature. With this regard, three datasets were examined.

104

(a) (b)

(c)

Figure 5.2. Importance of variables in VIP projection for three sizes of particles

(a) 53/75 µm, (b) 75/106 µm, (c) 106/150 µm

5.3.3.1 Gorain et al. dataset (1997)

A 2.8 m3 flotation cell designed at the JKMRC was used in this study to evaluate the zinc

cleaner circuit of the Hellyer Concentrator in Tasmania. This mineral exhibited a d80

between 20 and 25 microns. Gas dispersion properties and the flotation kinetic constant

Eg Sb lb0

0.5

1

1.5

2

2.5

Variables

VIP

Sco

res f

or

k

Ib

Sb

g

Eg Sb lb0

0.5

1

1.5

2

2.5

Variables

VIP

Sco

res f

or

k

Sb

Ib

g

1 2 30

0.5

1

1.5

2

2.5

Variables

VIP

Sco

res f

or

k

Ib

g S

b

105

were evaluated. The test results for a Chile-X impeller are shown in Table F.2 of

Appendix F.

The authors concluded that Sb presents an appropriate correlation with the flotation rate

constant, but the correlation with εg was also observed. According to their test conditions,

the particle size is considered as a fine size-class particle, and therefore their results would

confirm that Sb should be the best hydrodynamic variable for flotation rate modeling. This

is in agreement with the PLS analysis presented in Figure 5.3 in which the bubble surface

area flux exhibits the highest VIP score.

Figure 5.3 PLS regression for hydrodynamic variables using the dataset from Gorain et

al.(1997)

5.3.3.2 Massinaei et al. dataset (2009)

For their study, Massinaei et al. used the rougher feed of the flotation column at Miduk

concentrator inIran. The ore had a d80 of 100 microns and contained 54% of chalcopyrite

particles smaller than 30 microns. In other words, most of the hydrophobic particles were in

the fine-size range. Their work presented a set of the tests conducted in an industrial

column (Metso minerals CISA Microcell) of 12 m in height and 4 m in diameter, 150

m3 volume) and a pilot column (Plexiglas, 10 cm diameter, 400 cm height). The data-set is

presented in Table F.3 of Appendix F. The correlation between hydrodynamic variables

1 2 30.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

Variables

VIP

Sco

res f

or

k

Ib

g S

b0.1 0.2 0.3 0.4 0.5 0.6

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0.55

0.6

kmeasured

kp

red

icte

d

R2 = 0.963

3 Latent Variables

RMSEC = 0.027036

Calibration Bias = -5.5511e-017

Data

1:1

Fit

Estimated Error

106

and the kinetic constant is depicted in Figure 5.4, where the PLS regression results and VIP

scores are presented. Again, Sb exhibits the highest VIP score, thus supporting the

phenomenological analysis for fine particles.

Figure 5.4. PLS regression for hydrodynamic variables using the dataset of Massinaei et al.

(2009).

5.3.3.3 Kracht et al. dataset (2005)

In this case, samples of copper sulphide ore, taken from the rougher flotation feed of the

Andina Division, property of Codelco Chile, were used. The samples assayed 1.29 % Cu,

mainly chalcopyrite (95%) and chalcocite (4%). The particles exhibited a d80 of 182 m,

thus laying in the coarser end of the size range for flotation. The tests were conducted in

two types of batch flotation cell: an Outotec batch cell (3.7 L), and Labtech-Essa cell

(4.9 L). The experimental data are presented in Table F4 of Appendix F. Figures 5.5 and

5.6 provide the results of the PLS regression. The VIP scores for εg are this time

predominant in the regression, thus supporting the phenomenological analysis for coarse

particles.

1 2 30.75

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

Variables

VIP

Sco

res f

or

k

Ib

g S

b

0.1 0.15 0.2 0.250.1

0.15

0.2

0.25

kmeasured

kp

red

icte

d

R2 = 0.652

3 Latent Variables

RMSEC = 0.024286

Calibration Bias = -8.3267e-017

Data

1:1

Fit

Estimated Error

107

Figure 5.5 PLS regression for hydrodynamic variables using the dataset from in the Outotec

cell

Figure 5.6 PLS regression for hydrodynamic variables

using the dataset from the Labtech-Essa cell

5.4 Interfacial area of bubbles and rate constant

The interfacial area of bubbles (Ib) contains the main effects of εg and Sb. According to the

previously presented data-analysis, Ib does not offer any significant interest for its use in

modeling the kinetic constant when the particle size distribution is either fine or coarse, in

both cases within a narrow range. To unveil the influence of Ib on the kinetic constant, a

sample with a broader size distribution must be used.

1 2 30.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Variables

VIP

Sco

res f

or

Variables

k

Ib

Sb

g

0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.30.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

kmeasured

kp

red

icte

d

R2 = 0.662

2 Latent Variables

RMSEC = 0.084234

Calibration Bias = -1.1102e-016

Data

1:1

Fit

Estimated Error

1 2 30.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Variables

VIP

Sco

res f

or

k

gI

b Sb

R2 = 0.718

2 Latent Variables

RMSEC = 0.084308

Calibration Bias = 0

1 1.1 1.2 1.3 1.4 1.5 1.6 1.71

1.1

1.2

1.3

1.4

1.5

1.6

k

pre

dic

ted

kmeasured

Data

1:1

Fit

Estimated Error

R2=0.718

2 Latent Variables

RMSEC =0.084308

Calibration Bias = 0

108

Therefore, a mixed-size particle sample, composed of one third of each prepared size talc-

particles was fed to the column. It was expected that the interfacial area of bubbles would

exhibit a larger VIP score than Sb and εg. In the mixed size-class flotation test, the

collection rate of fine particles should be sensitive to collision efficiency, while for coarse

particles, the attachment efficiency should determine their collection rate. Therefore,

neither Sb nor εg should be appropriate for flotation rate constant modeling. Figure 5.7

shows the VIP scores of Sb, Ib and εg for these mixed size-class particles.

The weight of Ib in the relationship becomes more important than previously observed for

single class feeds. The validation of the abovementioned hypothesis was completed using

another dataset extracted from the literature, exhibiting a wide size distribution.

Figure 5.7 PLS regression for hydrodynamic variables using mixed-size class particles

5.4.1.1 Jincai et al. dataset (2013)

Jincai et al. (2013) used a pilot scale flotation column (15 cm diameter and 350 cm height

Plexiglas cylindrical column) for the treatment of oily waste water. In this case oil droplets

show the same behavior as hydrophobic particles and flotation can be applied for separating

water from oil. On the studied sample, oil droplets are distributed in various size fractions

with a d85 below 100 µm.

Eg Sb lb0.8

0.85

0.9

0.95

1

1.05

1.1

VIP

Sco

res f

or

k

Variables

Sb I

b

g

109

Table F.5 of Appendix F presents the dataset. The statistical analysis was done by means of

standard regression and PLS methods. Figure 5.8 shows the PLS regression results showing

again that Ib is the most influential variable in the regression for mixed-size particle

distributions.

Figure 5.8 PLS regression for hydrodynamic variables using the dataset from Jincai et al.

(2013)

A final PLS regression was performed using all the combined datasets, regardless of the

particle size, hydrophobicity, and machine type. Figure 5.9 shows the results of the

regression clearly showing that Ib is the most influential variable in the rate constant model.

1 2 30.9

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

1.08

1.1

Variables

VIP

Sco

res f

or

k

Sb

g

Ib

0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.20.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

kmeasured

kp

red

icte

d

R2 = 0.958

2 Latent Variables

RMSEC = 0.094246

Calibration Bias = 2.2204e-016

Data

1:1

Fit

Estimated Error

110

Figure 5.9 PLS regression for hydrodynamic variables using the datasets combined

Results thus show that in a typical flotation operation, where floatable particles are found in

all particle size classes, optimizing the value of Ib should be the main focus for optimizing

the recovery.

This notwithstanding, Sb and εg are still variables of interest in very specific applications.

For fine particle size classes (e.g. following a regrind or ultrafine grinding), measuring and

optimizing the mean bubble size and gas rate could be an appropriate tool to obtain the best

recovery. On the other hand, coarse particle flotation (e.g. carbonates, phosphates or

graphite), εg is the main variable influencing the kinetic constant. Gas hold-up is a function

of the gas rate, frother type, frother concentration and pulp chemistry.

It is generally accepted that the bubble size presents an inverse correlation with gas hold-up

but this conclusion is not always true. It was observed that two bubbles having the same

size but generated from two sort of frother may cause dissimilar gas hold-up (Tan et al.,

2013). In other words, it is possible that the same bubble size and gas rate show different

gas hold-up. Optimizing the coarse particle recovery thus should be performed through

controlling the gas hold-up at the proper target, and not necessarily with a reduction of the

bubble size.

1 2 30.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Variables

VIP

Sco

res f

or

k

g

Ib

Sb

0 0.5 1 1.5 2 2.50

0.5

1

1.5

2

2.5

kmeasured

kp

red

icte

d

R2 = 0.641

2 Latent Variables

RMSEC = 0.27642

Calibration Bias = -1.1102e-016

Data

1:1

Fit

Estimated Error

111

5.5 Conclusion

The hydrodynamic variables present a strong inter-correlation. Because of this correlation,

it is difficult; to use multivariable regression techniques to determine which hydrodynamic

variable is the most influential for the determination of the kinetic constant. This problem

can be solved with PLS regression, a method using the covariance structure between

independent variables.

This chapter proposed a phenomenological analysis to determine which hydrodynamic

variable is the most influential for recovering fine and coarse particle, based on results from

tests conducted under ideal conditions and from four datasets extracted from the literature.

Conclusions were that the flotation of fine particles mostly relies on Sb, while coarse

particle flotation is mainly determined by the value of εg. PLS regressions on original

experimental data (talc flotation) and various other data-sets (on oil drops, zinc and copper

sulphide ores) extracted from the literature, confirmed these conclusions.

This chapter also introduces the concept of interfacial area bubbles as a gas dispersion

variable that encompasses the properties of both εg and Sb, and shows good correlation with

the flotation kinetic constant, for general cases where floatable particles f are of any size

class. In such cases, optimizing the value of Ib should be the main focus for optimizing the

recovery.

113

Chapter 6 Single variable rate constant models

6.1 Introduction

The interfacial area of bubbles (Ib) was introduced in chapter 4 as a hydrodynamic variable

suitable for flotation rate modeling; its correlation with the other hydrodynamic variables

such as the gas hold-up and bubble surface area flux was then analyzed.

In Chapter 5, it was demonstrated that the use of hydrodynamic variables for flotation rate

modeling depends on the particle size range. Bubble size and bubble surface area flux

showed better correlation with the flotation rate of fine particle size-classes. On the other

hand, gas hold-up was more adequate to characterize the flotation rate of coarse particles.

This notwithstanding, particle size distributions in flotation systems typically span large

size ranges, and thus do not fall into either of the above mentioned extreme cases.

Experimental evidence showed that the interfacial area of bubbles, which encompasses both

the effect of the gas hold-up and bubble size, is more flexible and could be advantageously

used for flotation rate modeling in general applications.

In this chapter, the most suitable hydrodynamic variable for each particle size-classes is

used to produce single-variable models for the flotation rate constant. Observed trends for

each model are then discussed.

6.2 Flotation kinetics of fine particle size-class

The bubble surface area flux was recommended for modeling the kinetics of fine particle

size-class. Since the gas rate is constant in all the conducted experiments, the bubble size

exhibits the exact inverse effect of Sb on the flotation rate. Figure 6.1 and 6.2 show the

effect of the bubble size, and Sb respectively on the kinetic constant value of two particle

size-classes (53/75 µm and 75/106 µm). Both particle size-classes behave similarly: the

flotation kinetic constant increases when bubble size increases to a peak value and then

decreases with increasing bubble size.

114

Figure 6.1 Talc collection rate constant as a function of the bubble size

for two particle size-classes (Jg= 1cm/s)

Figure 6.2 Talc collection rate constant as a function of Sb for two particle size-classes

(Jg= 1 cm/s)

Interestingly, results also show that the use of smaller bubbles will not always improve the

metallurgical performance. A possible explanation for this result could be that coalescence

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 2.1

0.1

0.15

0.2

0.25

0.3

d32

(1/m

in)

k

actual

53/75 m

actual

75/106 m

(mm)

30 35 40 45 50

0.05

0.1

0.15

0.2

0.25

Sb

(1/m

in)

actual

actual

53/75 m (model)

75/106 m (model)

(1/S)

k

k

115

occurs in the froth zone and/or might be related to froth stability and particle drop-back. It

has been reported that both hydrophobic particle and bubble size affect froth stability and

consequently final flotation recovery (Aktas et al., 2008). The froth stability may be

determinant although a shallow froth was used to minimize the interference in the results.

The collection zone kinetics has been directly calculated by sampling the ‘concentrate’ just

below the interface. Comparing results for the collection zone kinetics and the overall

flotation kinetics, proves that the effect of froth zone and pulp-froth interface do not change

the observed trend between bubble size or Sb and the overall flotation rate. Figure 6.3

shows the flotation rate constant trend for both collection zone and overall cases.

Figure 6.3 Kinetic constant as a function of d32 for the collection zone and the overall

process (53/75 µm, Jg= 1 cm/s)

The explanation must therefore lie in the antagonistic effect of the bubble size:

(1) for fine bubbles, a lower rise velocities contribute to lower flotation rates (Koh and

Schwarz, 2008), and

(2) the decreasing kinetics for large bubbles comes from a reduction of the collision

efficiency in fine particle with larger bubble sizes.

1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

0.1

0.15

0.2

0.25

(mm)

(1/m

in)

actual

collection zone kinetics

actual

overall flotation kinetics

d32

k

116

A nonlinear least square algorithm was applied to model the effect of the d32 on the

flotation kinetic, leading to

2

1 32 2 32 3

2

32 1 32 2

p d p d pk

d q d q

(6-1)

for the 75/106 µm particle class coefficients (with 95% confidence bounds) are

p1 = 0.1571 +/- 0.02

p2 = -0.4138 +/- 0.07

p3 = 0.2751 +/- 0.04 q1 = -2.67 +/- 0.09

q2 = 1.792 +/- 0.12

and for 53/75 µm particle class are

p1 = 0.09121 +/- 0.04

p2 = -0.2461 +/- 0.11

p3 = 0.167 +/- 0.08

q1 = -2.724 +/- 0.18

q2 = 1.859 +/- 0.25

Similarly, empirical models for the effect of the Sb on the kinetic constant are given by

3 2

1 2 3 4b b bk p S p S p S p (6-2)

for 53/75 µm particle size-class, correlation coefficients (with 95% confidence bound) are

p1 = 0

p2 = -0.0007658 +/- 0.0005

p3 = 0.06372 +/- 0.04

p4 = -1.137 +/- 0.8

and for 75/106 µm particle size-class the correlation coefficients are

p1 = -5.309e-005 +/- 4e-005

p2 = 0.006034 +/- 0.004

117

p3 = -0.2192 +/- 0.1

p4 = 2.744 +/- 2.2

It must be emphasized that the magnitude of the bubble size or Sb effect on the rate constant

was affected by the particle size. For a constant bubble size, the larger particle size-class

exhibits a higher kinetic constant.

Figure 6.4 and Figure 6.5 compare the prediction of the models with the actual kinetic

constant, as calculated from experimental results.

(a) (b)

Figure 6.4 Nonlinear regression for d32 and kinetic constant – Predicted and actual values;

(a) particle size-class 53/75 µm and (b) particle size-class 75/106 µm

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

kactual

km

od

el

(1/min)

SSE: 0.0012

R2: 0.926

RMSE: 0.020

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

kactual

km

od

el

(1/mim)

SSE: 0.0002

R2:0.95

RMSE:0.007

(1/m

in)

(1/m

in)

118

(a) (b)

Figure 6.5 Nonlinear regression for Sb and kinetic constant – Predicted and actual values;

(a) particle size-class 53/75 µm and (b) particle size-class 75/106 µm

6.3 Flotation kinetics of large particle size-class

The gas hold-up is the hydrodynamic variable presenting the best correlation with the

flotation rate of large particle size. Consequently, the gas hold-up was applied for flotation

rate modeling of particles in the 106/150 µm class, and results are presented in Figure 6.6.

As expected, the kinetic constant increases with the gas hold-up, but the fluctuation remains

on a modest scale in the tested range. This seems to indicate that for large particles,

hydrodynamic variables have only a moderate influence on the flotation rate constant.

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

kactual

km

od

el

SSE: 0.0033

R2: 0.79

RMSE: 0.0256

(1/min)0 0.05 0.1 0.15 0.2 0.25 0.3

0

0.05

0.1

0.15

0.2

0.25

0.3

kactual

km

od

el

(1/min)

SSE: 0.0005

R2:0.97

RMSE: 0.0097

(1/m

in)

(1/m

in)

119

Figure 6.6 Flotation rate constant as a function of εg for large particle size-class

(106/150 µm, Jg= 1cm/s)

A linear model relates εg to flotation kinetics and linear least square algorithm is applied for

curve fitting. The obtained model is

1 2gk p p (6-3)

whose coefficients (with 95% confidence bound) are

p1 =0.008583 +/- 0.003

p2 = 0.2022 +/- 0.02

Figure 6.7 compare the prediction of the models with the actual kinetic constant value, as

calculated from the experimental results.

3 4 5 6 7 8 9 10

0.23

0.24

0.25

0.26

0.27

0.28

0.29

g

k

(%)

(1/m

in)

120

Figure 6.7 Predicted and actual kinetic constants for the 106/150 µm particle size-class

6.4 Flotation kinetics of particle size-class spanning a wide range

The effect of Ib on the flotation rate was investigated for a wider particle size distribution

(53/150 µm). As can be seen in Figure 6.8, the flotation rate significantly increased with Ib.

Figure 6.8 Flotation rate constant as a function of Ib for large particle size-class

(53/150 µm, Jg= 1cm/s)

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.05

0.1

0.15

0.2

0.25

0.3

kactual

km

od

el

(1/min)

SSE: 0.0008

R2: 0.83

RMSE: 0.0106

10 15 20 25 30 35 40

0.2

0.22

0.24

0.26

0.28

0.3

0.32

Ib

k

(1/m

in)

(1/m

in)

121

Flotation kinetics depend on Ib through a linear model and the curve fitting is done with

linear least square algorithm. By means of

1 2bk p I p (6-4)

whose coefficients (with 95% confidence bound) are

p1 = 0.002931 +/- 0.0012 p2 = 0.1722 +/- 0.025

Flotation kinetics of larger range particle size are linked to interfacial area of bubbles.

Figure 6.9 compares the prediction of the models with the actual kinetic constant, as

calculated from the experimental results.

Figure 6.9 Linear regression for Ib and flotation kinetics for large range particle size-class

6.5 Particle size effect on the kinetics of flotation

Particle size remains the most influential parameter on the flotation rate. As can be seen in

Figures 6.1 and 6.2, the flotation rate significantly increases with the particle size. This is in

agreement with the VIP scores presented in Figure 5.1and previous studies (Diaz-Penafiel

and Dobby, 1994; Karimi et al., 2014; Ofori et al., 2014). However, in real flotation

conditions, the particle size is typically not a manipulated variable for the flotation circuit,

due to the required degree of liberation. The grinding circuit performance and ore

mineralogy determine the final particle size. Consequently the hydrodynamic variables are

0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.05

0.1

0.15

0.2

0.25

0.3

kactual

km

od

el

(1/min)

SSE: 0.0031

R2: 0.80

RMSE: 0.0212

(1/m

in)

122

the only variables applicable to improve flotation performance for a given hydrophobicity

degree.

6.6 Conclusion

It was demonstrated that the relationship between the flotation rate constant and d32 (or Sb)

for fine particles, exhibits a maximum. The correlation fits to actual values with R2=0.93

for the 53/75 µm class, whereas for the 75/106 µm class the fit has a R2=0.95.

The kinetic constant for the 106/150 µm class linearly increases with increasing εg. The

same trend can be reported using Ib as the independent variable for a wider range of particle

size-class.

In the current study, the particle size has been considered as an important feature of mineral

particles. However, other features such as the particle hydrophobicity and the degree of

liberation are determinant on the flotation recovery.

The single-variable modeling introduces the appropriate hydrodynamic variable for

modeling and optimization with respect to the particle size. On the other hand, finding the

proper hydrodynamic variable to model the kinetic of flotation can better illustrate the role

of micro-phenomena (collision and attachment efficiencies) on the collection efficiency.

For example, when Sb presents high correlation with the flotation kinetics means that the

attachment efficiency plays a major role, while the effect of collision can be seen through

the relationship with εg.

123

Chapter 7 Thesis conclusion

Specific conclusions have been presented on each chapter, with cross-reference between

them. This Thesis conclusion chapter, summarizes the original contributions of the present

work, and provides some suggestions for future work.

The thesis addressed two main original contributions. The first one, presented in chapter 2

and 3 is related to the relationship between hydrodynamic variables, the water recovery and

carrying capacity in a laboratory flotation column. The second contribution consists in

better defining the relationships between hydrodynamic characteristics and the flotation

kinetics. This is described in chapters 4, 5 and 6.

Chapter 2 contains the effect of solid particles on the bubble size distribution; also gas

hold-up correlation in flotation column has been studied. It was demonstrated that bubble

coalescence affects the bubble size distribution through particle bridging effect and/or by

the frother absorption on talc (hydrophobic) particles. The solid particle effect on gas hold-

up is explained through the bubble loading and bubble coalescence. Consequently, it was

observed that a reduction in bubble size does not always lead to an increase on gas hold-up,

depending on which one of the latter factor is dominant.

In Chapter 3, the relevance of gas dispersion properties – as measured hydrodynamic

variables – on water recovery was studied. Tests were conducted in a laboratory scale

flotation column using a pulp with 4% solids content. They demonstrated a good

correlation between the water recovery and the gas hold-up, bubble size and bubble surface

area flux. It was demonstrated that increasing the gas hold-up and bubble surface area flux,

linearly increased the water recovery to the concentrate. The opposite effect was observed

for increasing bubble diameters. However, the hydrophobic particle size distribution did not

show any effect on the water recovery and consequently on the particle entrainment.

Modeling of gas dispersion properties in flotation was proposed in Chapter 4. The

fundamental equations of gas dispersion have been first developed as a possible solution for

some cases, where multi-shape bubble size density function and lognormal density function

have been well fitted to histograms. In this regard, the interfacial area of bubbles was

124

introduced as a hydrodynamic variable to address the issue of compacting the entire bubble

size distribution into a single value (d32) for the purpose of calculating the flotation rate

constant.

The inter-correlations between gas dispersion properties was also studied. A strong

correlation between gas dispersion properties – namely the gas hold-up and bubble surface

area flux – and interfacial area of bubbles has been detected for bubble size distributions

represented by lognormal density functions.

The difficulty of using multivariable regression techniques – because of strong co-linearity

between hydrodynamic variables – for flotation rate modeling was demonstrated in

Chapter 5. Projection to Latent Structures (PLS) was suggested to solve this problem as a

method using the covariance structure between independent variables.

A phenomenological analysis was presented to determine which hydrodynamic variable is

the most influential for recovering fine and coarse particle. Conclusions were that the

flotation of fine particles mostly relies on the bubble surface area flux, while coarse particle

flotation is mainly determined by the value of gas hold-up. Also, the interfacial area of

bubbles shows good correlation with the flotation kinetic constant for general cases,

floatable particles can be found in any size-class.

It was demonstrated that the flotation rate of fine particle size-class increases with the

bubble surface area flux to a maximum value and then decreases. The same trend was

reported for the relation between flotation rate and bubble size. Furthermore, the flotation

rate of coarse particles linearly increased by increasing the gas hold-up. Similar results

were reported for wider particle size distribution: increasing the interfacial area of bubbles

could directly increase the flotation rate constant. However, regardless of which

hydrodynamic is applied, the particle size shows a large influence on the flotation rate.

7.1 Future work

Future work should target the following issues.

Implementation of Microcell sparger instead of frit-sleeve sparger in the laboratory

and pilot scale flotation column to manipulate the bubble size. This device is used in

125

the industrial scale column and the extracted model for manipulating the bubble size

could then be applied to the industrial cells.

Validation of the model on the relationship between hydrodynamic variables and

flotation rate using original industrial data.

Investigating and modeling the froth zone and wash water effects on the final

recovery.

Modeling the kinetic constant distribution using the PLS technique. Bubble and

particle size distributions would be the input variables and the size-by-size flotation

rate constants would be the output.

Online measuring and controlling of bubble size distribution in slurry-gas systems

via manipulating the gas rate, frother concentration and the sparger shear-water rate.

127

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137

Appendix A

A.1 Sampling

Kinetic constant is estimated by means of recovery value and RTD. The mineral recovery is

calculated through Eq. (1-9). Representative sample for mineral assaying (from feed, tailing

and concentrate) assists to estimate more accurately the kinetic constant. Also for

estimating the kinetic constant of the collection zone, one sample per test has been taken

from 10 cm below the interface to avoid the effect of disturbance on sampling. It is worth

mentioning that the samples should be taken in the same time while the system is working

in the steady state conditions however due to system restrictions the sampling from feed

flow and collection zone have been done before sampling from the concentrate flow and the

tailing flow (feed sampling causes the system interruption). As seen in Figure A.1 the

sampling points are marked.

Figure A.1 Sampling points of the flotation column

Bubble sampling is also considered for investigation on the relationship between

hydrodynamic variables and the flotation rate constant. There is a region above the feeding

point and below the interface where axial mixing condition can be ignored. This district is

appropriate for bubble sampling. Besides, the sampling method should be considered. The

Collection zone sample

Tailing sample

Feed sample

Concentrate sample

138

bubbles should be extracted from the flotation column and transferred to the bubble viewer.

The angle of sampler, position of the aperture in the cross section of the column, sampler

diameter and sampler length should be considered for taking a representative bubble

sample. Based on that, the port of the bubble sampler was installed 50 cm above the

feeding point in the center of column cross section. The sampler is a hydrophilic tube (to

eliminate bubble attachment on the tube) with 12.5 mm aperture diameter and 60 cm

length. The sampler diameter is selected based on the maximum size of generated bubble

(two times larger). It is worth mentioning that the minimum detected bubble size is 0.1 mm

regarding to image resolution (1mm is equal 24 pixel). Figure A.2 shows the schematic of

bubble sampler. It is worth mentioning that the bubble viewer and bubble sampler have

been filled by water contains the same frother concentration as slurry in the column.

Figure A.2 Schematic of sampler and its position in the column

A.2 RTD measurement

The experimental methodology consists of introducing an impulse of tracer inside the

column at the feed entrance, and to measure the transient response (tracer concentration) at

tailing point. Experiments were done using a solution of NaCl as the liquid tracer. During

45º

50.4 mm

12.5 mm

600 mm

139

0 200 400 600 800 1000 1200 1400 1600

6150

6200

6250

6300

6350

6400

6450

6500

6550

6600

Time (s)

Me

asu

red

co

nd

uctivity (

mS

)

the RTD measuring the tailing flow did not re-circulate to the conditioning tank. The tracer

detector is the pair of conductivity cells in the feeding entrance and tailing point working in

the particular range of conductivity. Figure A.3 shows the tracer impulse in the feed and its

response in the tailing point. Table A.1 presents the measured RTDs were used for kinetic

constant estimation.

Figure A.3 Example of Measured RTD by conductivity cells; Left) tracer impulse in the

feeding point, Right) the detected response of tracer in the tailing point

200 400 600 800 1000 1200 1400 1600

1

2

3

4

5

6

7

x 104

Time (s)

Me

asu

red

co

nd

uctivity (

mS

)

140

Table A.1 Measured RTDs of the tests.

Particle size class (µm) 53/75 53/75 53/75 53/75 53/75 53/75 53/75 53/75 53/75

Frother (ppm) 5 5 5 15 15 15 25 25 25

Shear water rate (cm/s) 0.2 0.4 0.8 0.2 0.4 0.8 0.2 0.4 0.8

RTD 338 304 257 340 302 254 357 302 247

Particle size class (µm) 75/106 75/106 75/106 75/106 75/106 75/106 75/106 75/106 75/106

Frother concentration (ppm) 5 5 5 15 15 15 25 25 25

Shear water rate (cm/s) 0.2 0.4 0.8 0.2 0.4 0.8 0.2 0.4 0.8

RTD 337 307 257 334 303 254 343 302 255

Particle size class (µm) 106/150 106/150 106/150 106/150 106/150 106/150 106/150 106/150 106/150

Frother concentration (ppm) 5 5 5 15 15 15 25 25 25

Shear water rate (cm/s) 0.2 0.4 0.8 0.2 0.4 0.8 0.2 0.4 0.8

RTD 385 307 256 340 302 254 347 296 250

A.3 Kinetic constant calculation

After mineral assaying and data reconciliation, the flotation recovery is calculated. By

assuming R∞=1 and Eq. (A-1), the kinetic constant is estimated. The following examples

demonstrate the kinetic rate estimation.

ln(1 )100

60

R

k

(A-1)

Particle size (µm) Frother (ppm) Jsw (cm/s) R (%) τ (s) Kinetic constant (1/min)

75/106 15 0.8 66.5 254 0.26

53/75 15 0.4 56.1 302 0.16

141

Appendix B

B.1 Selection of the shear water rate

Determining Jsw values have been done by identifying the most effective Jsw values on

varying the bubble size in two-phase tests. In these tests, 20 ppm of F150 used as frother

and bubble size distributions (BSDs) were built by sizing 12,000 bubbles.

Figure B.1 Bubble size variation by shear water to a frit and sleeve sparger

Regarding to Figure B.1, the maximum bubble size variations occurs in 0.2 cm/s, 0.4 cm/s

and 0.8 cm/s, therefore these levels have been chosen for bubble size-particle size testing.

B.2 Effect of the temperature and duration of the test on the bubble size

The time required each set of three tests is over 8 hours. Therefore, the effect of the frother

should remain constant for this period of time. On the other hand, the frother should not be

sensitive in variation of the pulp temperature resulting from the action of the pumps and

agitator (the slurry is recycled). With this regard, F150 was tested for 8 hours of system

0 0.5 1 1.5 2 2.5 3 3.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Jsw

(cm/s)

(mm

)

data1

data2

data3

mean

variance

d32

shear water

&

bubble size

142

operation. By measuring the bubble size over time, the effect of temperature and test

duration was assessed. Table B.1 depicts the variation of the bubble size with time and

temperature. As seen, the effect of F150 remains constant for the duration of the test

Figure B.2 shows that the d32 is relatively stable for the duration of the test.

Table B.1 Effect of the temperature and duration of test on the bubble size distribution

Time (hour) 1 2 3 4 5 6 7 8

Temperature (oC) 19.4 19.6 20.2 20.9 21.4 21.9 22.5 22.8

d32 (mm) 0.89 0.781 0.791 0.782 0.780 0.774 0.785 0.773

mean (mm) 0.579 0.54 0.479 0.491 0.474 0.512 0.51 0.489

variance 0.0845 0.065 0.0718 0.0708 0.0691 0.0686 0.0706 0.0731

The system needed one hour of operation before reaching steady-state conditions thus

sampling have always been executed following at least one hour of test. In addition, it must

be emphasized that the slurry required on hour of conditioning.

Figure B.2 d32 variations over time

B.3 Relationships between hydrodynamic variables in two and three-phases

Studying the response of measured variables to manipulated variable changes allowed

determining the range of operation for hydrodynamic conditions in the experimental set-up.

0 1 2 3 4 5 6 7 80.75

0.8

0.85

0.9

Time (hours)

d32 (

mm

)

143

The shearwater rate to the sparger (Jsw) and the gas flow rate (Jg) are the manipulated

variables. The frother concentration remained constant (20 ppm of F150) in all the tests.

This notwithstanding, manipulating Jsw and Jg allowed controlling the bubble size and gas

hold-up (as measured hydrodynamic variables). Table B.2 shows values of hydrodynamic

variables generated by manipulating Jg and Jsw.

Table B.2 Values of hydrodynamic variables generated by manipulating Jg and Jsw

εg (%)

Jg (cm/s)/Jsw (%) 0.25 0.375 0.5 0.75 1 1.25

0 1.80 2.48 3.11 4.32 5.19 6.23

25 2.43 3.27 3.88 5.19 6.09 6.86

35 2.30 3.38 4.03 5.91 7.76 9.35

45 2.35 3.32 4.23 6.29 8.57 10.44

65 2.627 3.68 5.00 7.53 10.30 10.80

d32 (mm)

0 1.06 1.00 1.13 1.22 1.40 1.38

25 1.12 1.05 1.06 1.22 1.26 1.38

35 0.85 0.93 1.01 1.02 1.07 1.08

45 0.86 0.87 0.91 0.95 0.98 1.05

65 0.78 0.80 0.83 0.85 0.93 0.91

Sb (s-1)

0 14.13 20.83 26.34 36.83 42.69 53.99

25 13.27 21.27 28.19 36.82 47.43 54.18

35 17.59 24.08 29.60 43.99 56.03 69.34

45 17.27 25.57 32.62 46.91 60.65 71.24

65 19.03 27.87 35.95 52.39 64.25 82.27

mean (mm)

0 0.86 0.81 0.86 0.91 0.95 0.91

25 0.96 0.89 0.86 0.95 0.92 0.95

35 0.80 0.83 0.89 0.90 0.92 0.93

45 0.76 0.76 0.78 0.82 0.82 0.86

65 0.70 0.72 0.74 0.76 0.82 0.77

variance

0 0.07 0.06 0.10 0.12 0.18 0.18

25 0.07 0.06 0.07 0.11 0.14 0.18

35 0.03 0.04 0.05 0.06 0.07 0.08

45 0.03 0.04 0.04 0.05 0.06 0.07

65 0.028 0.02 0.03 0.03 0.04 0.05

144

Figure B.3 shows how modifying the gas rate and shearwater rate can generate a large

range of bubble size and gas hold-up values.

(a)

(b)

0 10 20 30 40 50 60

0.8

0.9

1

1.1

1.2

1.3

1.4

Jsw

(%)

d32 (

mm

)

Jg 0.25 cm/s

Jg 0.375 cm/s

Jg 0.5 cm/s

Jg 0.75 cm/s

Jg 1 cm/s

Jg 1.25 cm/s

data8

data9

data10

data11

data12

0 10 20 30 40 50 60

2

3

4

5

6

7

8

9

10

11

Jsw

(%)

g (

%)

Jg 0.25 cm/s

Jg 0.375 cm/s

Jg 0.5 cm/s

Jg 0.75 cm/s

Jg 1 cm/s

Jg 1.25 cm/s

data8

data9

data10

data11

data12

145

(c)

Figure B.3 a) Variations of the d32 with the gas and shearwater rates, b) variations of εg

with the gas and shearwater rates and c) variations of Sb with the gas and shearwater rates

0 10 20 30 40 50 60

20

30

40

50

60

70

80

Jsw

(%)

Sb (

1/s

)

0 10 20 30 40 50 60

2

3

4

5

6

7

8

9

10

11

Jsw

(%)

g (

%)

Jg 0.25 cm/s

Jg 0.375 cm/s

Jg 0.5 cm/s

Jg 0.75 cm/s

Jg 1 cm/s

Jg 1.25 cm/s

data8

data9

data10

data11

data12

147

Appendix C

C.1 Solid Characteristics

The solid used contains quartz as a hydrophilic mineral and raw rock from a talc mine

around Quebec City. Based on the chemical analysis and microscopy results, quartz is 99%

liberate and does not show any significant contamination. Table C.1 shows the chemical

analysis results.

Table C.1 XRF results of quartz (%)

SiO2 Al2O3 Fe2O3 MgO CaO Na2O

98.7 0.2 1.1 0.04 0.01 <0.04

K2O TiO2 MnO P2O5 Cr2O3 LOI

0.01 <0.04 <0.01 0.01 0.02 <0.02

Crushing, grinding and two stages of wet screening have been done to prepare the desired

liberated particle size of talc for three different particle size classes. Table C.2 shows the

mineralogy analysis results of each particle size class. Table C.3 presents the XRF results

of the particle size classes. Figure C.1 depicts the liberated information about each class of

talc particles.

148

Table C.2 Mineralogy analysis results

Mineral 53/75 m 75/106 m 106/150 m

Chrome-Chlorite (Kämmererite) 16.9 17.7 20.4

Mg-Fe carbonate 10.2 14.9 18.3

Talc 66.0 61.4 54.1

Ankerite 4.4 3.8 5.0

Serpentine group 0.1 0.3 0.4

Quartz 0.1 0.3 0.2

Pyrrhotite 1.2 0.8 0.4

Pentlandite 0.8 0.8 0.8

Fe-oxi 0.3 0.1 0.2

Electrum 0.0 0.0 0.0

Zircon 0.0 0.0 0.0

Apatite 0.0 0.0 0.1

Total 100.0 100.0 100.0

Figure C.1 Mineral liberation – talc

149

Table C.3 XRF results for talc particles (%)

SiO2 Al2O3 Fe2O3 MgO CaO Na2O

51.2 1.8 5.8 31.1 0.43 < 0.04

48 1.8 6.2 32.1 0.47 < 0.04

43.3 2 6.8 32.3 0.52 < 0.04

K2O TiO2 MnO P2O5 Cr2O3 LOI

< 0.01 < 0.04 0.05 < 0.01 0.31 11

< 0.01 < 0.04 0.05 < 0.01 0.37 13.2

< 0.01 < 0.04 0.06 0.01 0.42 15.8

151

Appendix D

D.1 Radial gas dispersion analysis (Banisi et al., 1995)

The distribution parameter (C0,) is defined as

0

1( )

1 1( )

g g sl

g g sl

J J dAAC

dA J J dAA A

. (D-1)

where A is the column cross-section and Jsl is superficial slurry velocity. C0 takes into

account the effect of non uniform (i.e. non-flat) flow and profile in the cross section of the

column.

The terminal velocity coefficient (K0) is calculated from

0

1(1 )

1 1(1 )

m

g g

m

g g

dAAK

dA dAA A

(D-2)

where m = 3 for both the gas-slurry and gas-water systems. K0 is a parameter encompassing

both the uniformity of gas hold-up profiles and the hindering effect of the gas bubbles on

bubble rise velocity.

The gas hold-up and superficial velocity profiles are flat in section A if C0 = K0 = 1.

The difference between C0 in the gas-water (C0 = 1.12) and gas-slurry (C0 = 2.21) systems

suggested a change from flat gas hold-up and flow profiles in the water-gas system to non-

uniform profiles in the slurry-gas system.

153

Appendix E

Particle size(µm) F150 (ppm) Shear water (cm/s) εg (%) µ σ2 d32 (mm)

53/75 5 0.2 4.02 1.66 0.48 2.07

53/75 5 0.4 5.17 1.57 0.34 1.92

53/75 5 0.8 5.55 1.40 0.30 1.78

53/75 15 0.2 5.72 1.03 0.18 1.40

53/75 15 0.4 7.54 0.96 0.14 1.30

53/75 15 0.8 8.69 0.89 0.14 1.28

53/75 25 0.2 7.1 0.97 0.17 1.36

53/75 25 0.4 9.8 0.86 0.12 1.21

53/75 25 0.8 12.48 0.85 0.11 1.17

75/106 5 0.2 3.13 0.99 0.89 2.17

75/106 5 0.4 4.25 1.41 0.48 1.92

75/106 5 0.8 5.25 1.44 0.34 1.80

75/106 15 0.2 5.85 1.09 0.19 1.47

75/106 15 0.4 8.35 0.99 0.14 1.32

75/106 15 0.8 9.93 0.94 0.14 1.30

75/106 25 0.2 6.85 0.96 0.16 1.35

75/106 25 0.4 9.68 0.92 0.13 1.26

75/106 25 0.8 10.85 0.92 0.15 1.30

106/150 5 0.2 2.54 1.21 0.82 2.11

106/150 5 0.4 3.59 1.36 0.49 1.91

106/150 5 0.8 5.87 1.32 0.40 1.80

106/150 15 0.2 4.30 0.99 0.17 1.38

106/150 15 0.4 6.54 0.98 0.15 1.33

106/150 15 0.8 8.47 0.99 0.15 1.34

106/150 25 0.2 4.58 0.94 0.17 1.35

106/150 25 0.4 7.93 0.94 0.13 1.24

106/150 25 0.8 10.46 0.91 0.11 1.18

53/150 5 0.2 2.21 1.40 0.77 2.10

53/150 5 0.4 3.14 1.48 0.45 1.92

53/150 5 0.8 4.53 1.53 0.28 1.83

53/150 15 0.2 4.69 1.05 0.17 1.39

53/150 15 0.4 7.25 1.02 0.13 1.31

53/150 15 0.8 8.00 1.00 0.12 1.27

53/150 25 0.2 5.11 1.01 0.21 1.45

53/150 25 0.4 7.85 0.99 0.16 1.33

53/150 25 0.8 9.34 0.96 0.15 1.31

154

Particle size(µm) F150 (ppm) Shear water (cm/s) Feed G. Cons. G. Tailing G. Recovery (%)

53/75 5 0.2 33.61 82.04 24.81 37.55

53/75 5 0.4 36.11 82.03 24.07 47.21

53/75 5 0.8 38.11 81.39 29.41 35.74

53/75 15 0.2 34.85 80.39 17.01 64.92

53/75 15 0.4 37.47 81.01 22.21 56.10

53/75 15 0.8 38.10 82.32 27.84 40.69

53/75 25 0.2 37.16 81.21 15.62 71.78

53/75 25 0.4 38.92 80.39 25.35 50.91

53/75 25 0.8 37.61 81.51 29.10 35.20

75/106 5 0.2 38.41 83.77 20.42 61.93

75/106 5 0.4 39.35 79.95 22.93 58.49

75/106 5 0.8 41.54 81.21 27.10 52.16

75/106 15 0.2 37.16 79.05 14.39 74.91

75/106 15 0.4 39.97 79.88 16.02 74.95

75/106 15 0.8 41.22 81.51 20.80 66.51

75/106 25 0.2 38.41 80.49 13.56 77.80

75/106 25 0.4 39.77 80.69 14.71 77.05

75/106 25 0.8 39.66 79.88 19.28 67.75

106/150 5 0.2 40.99 77.73 17.62 73.73

106/150 5 0.4 40.96 82.43 19.47 68.69

106/150 5 0.8 41.75 79.48 22.60 64.11

106/150 15 0.2 39.13 82.16 15.72 73.99

106/150 15 0.4 41.74 82.70 17.73 73.21

106/150 15 0.8 39.56 76.38 19.71 67.63

106/150 25 0.2 46.70 82.16 21.93 72.34

106/150 25 0.4 44.17 76.07 18.87 76.17

106/150 25 0.8 47.58 76.38 24.93 70.66

53/150 5 0.2 39.26 80.70 18.86 67.81

53/150 5 0.4 39.78 80.20 22.37 60.71

53/150 5 0.8 40.72 79.88 24.35 57.84

53/150 15 0.2 40.07 80.18 18.20 70.62

53/150 15 0.4 41.47 80.31 20.19 68.55

53/150 15 0.8 38.01 79.05 17.43 69.45

53/150 25 0.2 39.46 80.50 18.56 68.84

53/150 25 0.4 39.47 79.37 19.67 66.70

53/150 25 0.8 38.21 79.97 17.04 70.40

155

Appendix F

F.1 Correlation between hydrodynamic variables

The purpose of this section is to assess the capability of classical regression method to

predict k by means of the hydrodynamic variables. It has shown here that, due to

correlation between these parameters, and consequently the sensitivity of the (XTX)-1 to

small variations, the traditional regression methods are not the appropriate tools. Therefore

in this work PLS regression has been implemented.

Fine particle class of test work (53/75 microns)

When X is mean-centered and scaled to unit length, the XTX shows the correlation between

variables:

The following matrices show the sensitivity of (XTX)-1 to an arbitrary ‘small’ change,

Δ=0.01, in the diagonal elements. The regression could be applied only if ‘small’ changes

lead to ‘small’ changes in . Otherwise, the correlation between the input

variables would be too important, and the results would be biased.

In the present case,

and thus,

1.000 0.889 0.996

0.889 1.000 0.9185

0.889 0.9185 1.000

g b b

T

S I

X X

1( )TX X

1

281.995 45.718 322.904

( ) 45.718 13.805 58.223

322.904 58.223 376.142

TX X

1

72.499 11.048 82.368

( ) 11.048 7.693 18.070

82.368 18.070 99.650

TX X

156

As can be seen regression is not suggested for this data base because the changes between

diagonal elements of and is too large.

Coarse particle class of test work (106/150 microns)

The differences between the diagonal elements of and is more

than normal therefore the same condition as fine particle size class occurs in coarse particle

size class.

Gorain et al data base

The same results as test works is achieved therefore regression cannot be a good method for

modeling of kinetic constant.

1( )TX X 1( )TX X

1.000 0.780 0.984

0.780 1.000 0.856

0.984 0.856 1.000

g b b

T

S I

X X

1

55.45 12.660 65.384

( ) 12.660 6.623 18.123

65.384 18.123 80.835

TX X

1

35.040 7.713 41.075

( ) 7.713 5.296 12.120

41.075 12.120 51.782

TX X

1( )TX X 1( )TX X

1.000 0.901 0.727

0.901 1.000 0.888

0.727 0.888 1.000

b g b

T

I S

X X

1

6.143 7.414 2.115

( ) 7.414 13.671 6.747

2.115 6.747 5.452

TX X

1

5.356 6.173 1.585

( ) 6.173 11.625 5.831

1.585 5.831 5.025

TX X

157

F.2 Regression methods

In order to model a response matrix (Y) by means of a predictor matrix (X), the Multiple

Linear Regression (MLR) approach is usually applied, as long as the X-variables are few

and fairly uncorrelated (Wold et al., 2001). In multiple linear regression (MLR), the X links

to Y matrices directly using a linear relation:

𝐘 = 𝐗𝐁 + 𝐄 (F-1)

where B is the matrix regression coefficients and E consists of the regression residuals,

often assumed to be identically and independently normally distributed. A number of

procedures have been developed for estimation of B in linear regression. The simplest and

most commonly method is applying ordinary least squares (OLS) where the regression

parameters is estimated by minimizing the sum of squared residuals (E):

𝐄 = 𝐘 − 𝐗𝐁 (F-2)

�̂� = (𝐗𝐓𝐗)−𝟏𝐗𝐓𝐘 (F-3)

where �̂� is the estimated regression parameters.

However, since the variables describing the hydrodynamic conditions of flotation

(X-variables) are strongly correlated the alternatives can be PCR (principal component

regression) or PLS (Projection to Latent Structures analysis, also known as the Partial Least

Squares) to model (Y) by (X).

In PCR, the X matrix is replaced by the T matrix computed by the PCA (the dimensions

corresponding to the largest eigenvalues are only kept). In the next step, the multi-linear

regression can be expressed as:

𝐘 = 𝐓𝐁 + 𝐄 (F-4)

�̂� = (𝐓𝐓𝐓)−𝟏𝐓𝐓𝐘 (F-5)

Similar to MLR, B and E correspond to the regression parameters and residuals,

respectively. The matrix TTT is diagonal since the columns of T are orthogonal, so the

elements of the inverse are merely the mutual of the diagonal elements. Therefore, PCR

avoids the problems corresponded to colinearity.

158

However, in the PCR approach, the decomposition of X (i.e. projection onto a lower latent

variables subspace) is done without using information from Y, and this dimensional

reduction is not essentially the most predictive of Y. It is worth mentioning that merely

selecting the components with the highest variance (large eigenvalues) may as well be

problematic.

The third method discussed here is the Projection to Latent Structures (PLS). This method

was used throughout this work to build the empirical latent variable models linking the

Hydrodynamic variables to the flotation rate constant. PLS was selected in this work since

the columns of X were highly collinear (proved in the previous section of this appendix).

Due to simplicity and small volume of calculations, the PLS method has been extensively

implemented in different research fields such as product design and process analysis and

optimization (Camacho et al., 2008; Flores-Cerrillo and MacGregor, 2003; Jaeckle and

MacGregor, 1998; Kourti, 2005; Yacoub and MacGregor, 2004). In fact the PLS regression

relates two blocks of variables, X and Y, by maximizing the covariance between them

(Höskuldsson, 1988).

The purpose of the PLS regression is to explain the relationship between X (N×K) and Y

(N×M) matrices, within each data-set, by the following equations (Eriksson et al., 2001):

𝐗 = 𝐓𝐏′ + 𝐄 (F-6)

𝐘 = 𝐓𝐐′ + 𝐅 (F-7)

𝐓 = 𝐗𝐖∗ (F-8)

𝐖∗ = 𝐖(𝐏′𝐖)−1 (F-9)

where the P (K×r) and Q (M×r) are loading matrices that respectively represent the X and

Y spaces, and T (N×r) contains a set of orthogonal components (i.e. latent variables),

defined by linear combination of the X-variables collected in the weight matrix of

W*(K×r). The E (N×K) and F (N×M) matrices respectively represent the model residuals

for X and Y. A cross-validation procedure is often used to select the number of PLS

components or latent variables r (r < K).

159

Another advantage of PLS approach along with solving the colinearity problems is

commonly used metric for judging the importance of the X- variables on the Y. In order to

quantify the importance of the X variables in the PLS model, according to their

contribution to explain the variance of Y by each PLS component, the Variable Importance

on the Projection (VIP) is defined as (Chong and Jun, 2005):

VIPk = √K ∑ ((

wak‖𝐰𝐚‖

)2

(𝐪𝐚𝟐𝐭𝐚

′ 𝐭𝐚))ra=1

∑ (𝐪𝐚𝟐𝐭𝐚

′ 𝐭𝐚)ra=1

(F-10)

where K is the total number of variables, 𝐰𝐚𝐤 is the weight of kth variable in principal

component a, r is the number of principal components, and wa, ta and qa are the ath column

vector of W and T and Q respectively. According to Eriksson et al. (20012001), the

variables with a VIP greater than 1, are most influential in the PLS model, because the

average of the squared VIPs is equal to 1. The variables having a VIP between 0.8 and 1.0

correspond to those having a moderate influence, and VIP less than 0.8 correspond to the

least influential ones.

160

F.3 Datasets

Table F.1 Experimental results

Particle size(µm) F150 (ppm) Shear water (cm/s) εg d32 (mm) Sb Ib k (1/min)

53/75 5 0.2 4.02 2.07 28.91 11.61 0.07

53/75 5 0.4 5.17 1.92 31.30 16.18 0.12

53/75 5 0.8 5.55 1.78 33.75 18.75 0.10

53/75 15 0.2 5.72 1.40 42.16 24.12 0.18

53/75 15 0.4 7.54 1.30 45.787 34.54 0.16

53/75 15 0.8 8.69 1.28 46.87 40.76 0.12

53/75 25 0.2 7.1 1.36 44.23 31.39 0.22

53/75 25 0.4 9.8 1.21 49.71 48.98 0.16

53/75 25 0.8 12.48 1.17 51.27 63.98 0.10

75/106 5 0.2 3.13 2.17 27.68 8.67 0.17

75/106 5 0.4 4.25 1.92 31.28 13.28 0.17

75/106 5 0.8 5.25 1.80 32.75 17.21 0.17

75/106 15 0.2 5.85 1.47 40.71 23.83 0.23

75/106 15 0.4 8.35 1.32 45.478 37.99 0.27

75/106 15 0.8 9.93 1.30 46.10 45.77 0.26

75/106 25 0.2 6.85 1.35 44.32 30.36 0.30

75/106 25 0.4 9.68 1.26 47.66 46.13 0.21

75/106 25 0.8 10.85 1.30 46.06 50.00 0.27

106/150 5 0.2 2.54 2.11 28.38 7.22 0.24

106/150 5 0.4 3.59 1.91 31.44 11.30 0.23

106/150 5 0.8 5.87 1.80 33.38 19.61 0.24

106/150 15 0.2 4.30 1.38 43.56 18.75 0.24

106/150 15 0.4 6.54 1.33 45.25 29.59 0.26

106/150 15 0.8 8.47 1.34 44.82 37.96 0.26

106/150 25 0.2 4.58 1.35 44.57 20.41 0.22

106/150 25 0.4 7.93 1.24 48.25 38.25 0.28

106/150 25 0.8 10.46 1.18 50.93 53.27 0.29

Mixed 5 0.2 2.21 2.10 28.60 6.31 0.20

Mixed 5 0.4 3.14 1.92 31.17 9.79 0.18

Mixed 5 0.8 4.53 1.83 32.78 14.86 0.20

Mixed 15 0.2 4.69 1.39 43.17 20.26 0.22

Mixed 15 0.4 7.25 1.31 45.92 33.31 0.24

161

Mixed 15 0.8 8.00 1.27 47.22 37.80 0.28

Mixed 25 0.2 5.11 1.45 41.23 21.075 0.21

Mixed 25 0.4 7.85 1.33 45.19 35.48 0.22

Mixed 25 0.8 9.34 1.31 45.73 42.69 0.29

Table F.2 Gorain's results for Chile-X impeller k= flotation rate, 1/min; Sb = Bubble surface

area flux (1/sec), Jg = superficial gas velocity (cm/sec), d32 =Sauter mean bubble diameter

(mm), εg = Gas holdup (%), Ib interfacial area of bubbles %

Ib Jg εg d32 Sb k

21.41 0.68 3.64 1.02 40 0.10

28.29 1.94 6.13 1.30 89.534 0.23

45.14 2.77 11.06 1.47 113.06 0.26

54.57 2.34 13.46 1.48 94.86 0.21

34.47 0.65 5.17 0.90 43.33 0.10

55.20 1.42 8.74 0.95 89.68 0.20

58.36 2.99 11.77 1.21 148.26 0.43

78.42 3.07 16.86 1.29 142.79 0.43

50.15 0.64 6.77 0.81 47.41 0.12

55.30 1.38 8.48 0.92 90.00 0.22

73.07 2.52 13.64 1.12 135.00 0.40

85.28 3.03 17.34 1.22 149.01 0.50

49.06 0.8 6.46 0.79 60.76 0.14

71.24 0.98 9.38 0.79 74.43 0.18

92.20 1.68 15.06 0.98 102.86 0.33

108.38 3.06 20.05 1.11 165.40 0.54

162

Table F.3. Massinaei et al. data base

industrial

Ib Jg εg d32 Sb k

35.53 1.1 8.35 1.41 46.81 0.15

32.39 1.1 7.99 1.48 44.59 0.11

27.45 1.1 7.32 1.6 41.25 0.10

36.21 1.6 10.08 1.67 57.48 0.26

37.20 1.6 10.23 1.65 58.18 0.19

35.14 2 11.42 1.95 61.54 0.20

Pilot-

scale

12.41 1.1 16.34 0.79 83.54 0.13

10.23 1.1 14.49 0.85 77.65 0.15

9.06 1.1 13.59 0.9 73.33 0.15

20.24 1.6 26.32 0.78 123.08 0.17

19.44 1.6 25.6 0.79 121.52 0.17

17.94 2 28.1 0.94 127.66 0.19

163

Table F.4 Kracht et al. data base

Ib Jg εg d32 Sb k

Outotec cell

74.22 1.2 12 0.97 74.23 0.60

64.82 1.5 12.1 1.12 80.36 0.58

75.00 1.8 13 1.04 103.85 0.7

57.10 2.1 11.8 1.24 101.61 0.83

87.79 1.2 13.9 0.95 75.79 0.72

81.65 1.5 13.2 0.97 92.78 0.86

105.33 1.8 15.8 0.9 120 0.88

60.00 2.1 13.4 1.34 94.03 0.86

102.76 1.2 14.9 0.87 82.76 0.90

108.71 1.8 18.3 1.01 106.93 0.90

86.28 2.1 17.4 1.21 104.13 0.96

103.14 1.2 15.3 0.89 80.90 0.95

122.89 1.5 17 0.83 108.43 0.99

98.53 1.8 15.6 0.95 113.68 1.04

110.59 2.1 18.8 1.02 123.53 1.13

102.47 1.7 15.2 0.89 114.61 0.85

Labtech-Essa

cell

46.45 0.4 9.6 1.24 19.35 1.07

105.00 0.8 14.7 0.84 57.14 1.21

129.88 1.3 18.4 0.85 91.76 1.33

89.59 0.4 10.9 0.73 32.88 1.03

113.68 0.8 14.4 0.76 63.16 1.24

116.88 1.3 15 0.77 101.30 1.44

82.22 0.4 11.1 0.81 29.63 1.17

77.08 1.3 16.7 1.3 60 1.44

83.70 0.4 11.3 0.81 29.63 1.19

98.00 0.8 14.7 0.9 53.33 1.46

101.87 1.3 16.3 0.96 81.25 1.52

164

.

Table F.5 Jincai et al. data base

Ib Jg εg d32 Sb k

10.09 2.35 5.72 3.40 41.47 0.62

39.63 3.14 8.72 1.32 142.72 1.08

101.50 3.14 12.35 0.73 258.08 1.47

50.55 2.35 8.93 1.06 133.02 1.20

136.38 3.14 11.82 0.52 362.31 2.07

15.63 3.92 6.41 2.46 95.61 0.72

101.91 2.35 11.72 0.69 204.35 1.74

16.19 3.14 6.53 2.42 77.85 0.74

42.57 3.92 9.01 1.27 185.19 1.18