Identification and control of wet grinding processes - Application to the Kolwezi concentrator

Identification and control of wet grinding

processes - Application to the Kolwezi

concentrator

December 2013

Ecole Polytechnique de Bruxelles

Thèse présentée par

Moïse MUKEPE KAHILU

en vue de l’obtention du grade de Docteur en Sciences de l’Ingénieur

Promoteur :

Prof. Michel KINNAERT Co-promoteurs :

(metal) set by the consuming market, control is often applied on the mineral processing whose product, the ore concentrate, constitutes the input material of the extractive metallurgy. Therefore much attention is paid on mineral processing units and especially on concentration plants. As the ore size reduction procedure is the critical step of a concentrator, it turns out that controlling a grinding circuit is crucial since this stage accounts for almost 50 % of the total expenditure of the concentrator plant. Moreover, the product particle size from grinding stage influences the recovery rate of the valuable minerals as well as the volume of tailing discharge in the subsequent process. The present thesis focuses on an industrial application, namely the Kolwezi concentrator (KZC) double closed-loop wet grinding circuit. As any industrial wet grinding process, this process offers complex and challenging control problems due to its configuration and to the requirements on the product characteristics. In particular, we are interested in the modelling of the process and in proposing a control strategy to maximize the product flow rate while meeting requirements on the product fineness and density.

A mathematical model of each component of the circuit is derived. Globally, the KZC grinding process is described by a dynamic nonlinear distributed parameter model. Within this model, we propose a mathematical description to exhibit the increase of the breakage efficiency in wet operating condition. In addition, a relationship is proposed to link the convection velocity to the feed ore rate for material transport within the mills.

All the individual models are identified from measurements taken under normal process operation or from data obtained through new specific experiments, notably using the G41 foaming as a tracer to determine material transport dynamics within the mills. This technique provides satisfactory results compared to previous studies.

Based on the modelling and the circuit configuration, both steady-state and dynamic simulators are developed. The simulation results are found to be in agreement with the experimental data. These simulation tools should allow operator training and they are used to analyse the system and to design the suitable control strategy.

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This thesis is the result of a four-year work performed partly at the Université Libre de Bruxelles (ULB) in Belgium and partly at the Université de Lubumbashi (UNILU) in the Democratic Republic of Congo. The project budget and the Belgian stay were financially supported by the Belgian Technical Cooperation (BTC) which is therefore gratefully acknowledged.

As this thesis cannot be the result of a single person, I would like to thank all the people who helped me to achieve this work.

First of all, my greatest thanks are due to my promoter, Professor Michel Kinnaert, head of the Control Engineering and System Analysis Department of ULB, who initiated me in research. He devoted a considerable part of his limited time on this thesis and guided me with his relevant ideas and advices. Thanks for encouraging me to systematically think in a critic and in an analytic way.

Many thanks are due to my co-promoters, Professor Pierre Kalenga Ngoy Mwana and Professor Jean-Marie Moanda Ndeko for their interest in my work and for their precious advices.

I also owe my best thanks to Professor Alain Delchambre, president of ULB, who met me in my country during his supporting teaching at the Electromechanical Department of UNILU. He was always open to me and put me in touch with Professor Michel Kinnaert. Many thanks for supporting and encouraging me. A friendly personal contact resulted from these interactions.

My great thanks are also for Professor Raymond Hanus for having received me in the Control Engineering and System Analysis Department and for his precious ideas and encouragements.

I would like to express my respectful acknowledgement to Professor Philippe Bogaerts, president of this thesis committee, for his pertinent contributions during the board for the thesis follow-up.

I gratefully acknowledge Professor Emmanuele Garone, for his relevant ideas on optimization and Model Predictive Control (MPC) issues.

Moreover, I would like to thank the personnel of Kolwezi Concentrator (KZC) for their help and collaboration during the experimental work and the data acquisition.

I would like to express my gratitude to my colleagues from the Control Engineering and Systems Analysis Department for their encouragements and their contribution to the good atmosphere of the lab, especially during the famous coffee breaks.

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permanent and immeasurable support. Many thanks to my sisters and brothers, especially to Léon Zeka, and to my friend Daniel Kasongo. Thanks are also due to all my family in-law and to all the MUSKA Engineering’s people.

Finally, I wish to express all my admiration and all my gratefulness to my beloved wife Nicole Ihemba who has accompanied me with her strong love during these four years. She has suffered uncounted days of separation, accepting this difficult situation with an incredible patience. Without her encouraging support, it would have been much harder for me to finalise this work.

To Monica and Jovic, my beloved children, thank you for their patience and unconditional love. As my offspring, they are my heart.

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Chapter I: INTRODUCTION ... 1

I.1 Motivations ... 1

I.2 State of the art ... 2

I.3 Contributions of this study ... 3

I.4 Outline ... 5

Part I: PROCESS MODELLING AND SIMULATION

Chapter II: WET GRINDING CIRCUIT OF THE KOLWEZI CONCENTRATOR ... 9

II.1 General concepts of mineral processing ... 9

II.1.1 Objectives and processes of mineral processing ... 9

II.1.2 Grinding process ... 10

II.2 Grinding process of the Kolwezi concentrator ... 11

II.2.1 Preliminary description ... 11

II.2.2 Objective of KZC ... 12

II.2.3 Dry crushing ... 12

II.2.4 Wet grinding ... 12

II.2.5 Flotation ... 15

II.2.6 Thickening and filtration ... 15

II.2.7 Current control mode ... 15

Chapter III: STATE OF THE ART FOR GRINDING PROCESSES MODELLING ... 17

III.1 Fragmentation or breakage modelling ... 17

III.1.1 Introduction ... 17

III.1.2 Energy-based models ... 18

III.1.3 Phenomenological models ... 21

III.2 Material transport modelling ... 26

III.2.1 Notion of material transport within a mill ... 26

III.2.2 Residence time distribution ... 26

III.2.3 Distributed parameter model ... 27

III.3 Complete model of a grinding mill process ... 28

III.4 Modelling of hydrocyclone classification ... 29

III.4.1 Selectivity function (Tromp curve) of a hydrocyclone classifier ... 29

III.4.2 Recycled and circulating loads ... 30

III.4.3 Experimental determination of a tromp curve ... 31

III.4.4 Mathematical models of a hydrocyclone classifier... 31

Chapter IV: MODELLING OF THE KZC GRINDING PROCESS ... 33

IV.1 Modelling purpose ... 33

IV.2 Rod mill model ... 34

IV.2.1 Mathematical model ... 34

IV.2.2 Model parameters ... 35

IV.3 Hydrocyclone classifiers model ... 36

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IV.4 Ball mill model ... 36

IV.4.1 Mathematical model ... 36

IV.4.2 Model parameters ... 38

IV.5 Model of the other components... 38

IV.6 Overall representation ... 38

Chapter V: PARAMETER IDENTIFICATION AND SYSTEM SIMULATION .... 41

V.1 Identification procedure ... 41

V.1.1 Modelling objectives ... 41

V.1.2 Experimental field (Measurements) ... 41

V.1.3 Model structure ... 42

V.1.4 Optimization procedure ... 42

V.2 Mill transport parameters ... 43

V.3 Mill grinding parameters ... 46

V.4 Hydrocyclone classifier parameters ... 52

V.5 Steady-state simulator ... 53 V.5.1 Simulator implementation ... 53 V.5.2 Simulation results ... 56 V.6 Dynamic simulator ... 58 V.6.1 Simulator implementation ... 58 V.6.2 Simulation results ... 59

Part II: CONTROLLER DESIGN AND VALIDATION

Chapter VI: STATE OF THE ART FOR WET GRINDING PROCESSES CONTROL ... 67

VI.1 Typical wet grinding process ... 67

VI.1.1 Description ... 67

VI.1.2 Dynamic and steady-state features ... 69

VI.1.3 Typical variable pairing ... 70

VI.2 Instrumentation ... 71

VI.2.1 General concepts ... 71

VI.2.2 Sensors ... 71

VI.2.3 Actuators ... 72

VI.2.4 Instrumentation for the typical wet grinding circuit ... 72

VI.3 Decentralized control strategies ... 73

VI.3.1 Classical control ... 74

VI.3.2 Advanced control ... 76

VI.4 Centralized control strategies ... 78

VI.4.1 Decoupling control... 78

VI.4.2 Model-based Predictive Control... 80

VI.4.3 Adaptive and nonlinear control ... 82

VI.5 Discussion ... 82

Chapter VII: ANALYSIS OF THE SYSTEM MODEL ... 85

VII.1 Model linearization ... 85

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VII.1.3 Discrete-time linearized model ... 90

VII.2 Operating point optimization ... 94

VII.3 Pairing of controlled and manipulated variables ... 95

VII.3.1 Problem statement ... 95

VII.3.2 Steady-state criteria ... 96

VII.3.3 Dynamic criteria ... 98

VII.4 Interaction effects ... 101

VII.5 Uncertainty characterization ... 101

VII.5.1 General concept of uncertainty ... 101

VII.5.2 System model uncertainties ... 104

VII.6 Discussion ... 108

Appendix VII.1 ... 110

Appendix VII.2 ... 111

Chapter VIII: DECENTRALIZED CONTROL FOR THE KZC GRINDING PROCESS ... 113

VIII.1 DIMC Structure ... 113

VIII.1.1 Control issues ... 113

VIII.1.2 Control objectives ... 115

VIII.1.3 Presentation of the DIMC ... 116

VIII.2 Robust DIMC design ... 119

VIII.2.1 Robustness problem ... 119

VIII.2.2 Duality between PI-SP and DOB design ... 122

VIII.2.3 Robust design of DOB and PI-SP ... 127

VIII.3 Application to KZC grinding process ... 132

VIII.3.1 Instrumentation of the KZC grinding circuit ... 132

VIII.3.2 Simulation on the linearized model ... 133

VIII.3.3 Progressive implementation ... 137

Chapter IX: CONCLUSIONS AND PERSPECTIVES ... 143

IX.1 Conclusions ... 143

IX.1.1 Modelling and simulation ... 143

IX.1.2 Control ... 144

IX.2 Future research directions... 145

IX.2.1 Modelling and simulation ... 145

IX.2.2 Control ... 146

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Acronyms

AG: Autogeneous

ANN: Artificial Neural Networks

1 : Ball mill 1 2 : Ball mill 2 D: Decoupler

DC: Decoupling Control DI: Density Indicator

DIMC: Double Internal Model Control DNA: Direct Nyquist Array

DOB: Disturbance Observer DRGA: Dynamic Relative Gain Array Ds: Distributor

FI: Flow Indicator

FOTD: First Order with Time Delay FV: Flow control Valve

1 : Hydrocyclone 1 2 : Hydrocyclone 2

IAE: Integral of Absolute Error IMC: Internal Model Control

ISA: Instrumentation Systems and Automation Society KZC: Kolwezi Concentrator

LI: Level Indicator MC: Multivariable Control

MPC: Model-based Predictive Control MIMO: Multiple-Input Multiple-Output MPT: Multi-Parametric Toolbox MOL: Method Of Lines

NMPC: Nonlinear Model Predictive Control ODE: Ordinary Differential Equation OP: Operating Point

PC: Pre-Compensator

PI: Proportional-Integral, Pressure Indicator PID: Proportional-Integral-Derivative

PI-SP: Proportional-Integral Smith Predictor PDE: Partial Differential Equations

PSA: Particle Size Analyser PSD: Particles Size Distribution RGA: Relative Gain Array

: Rod mill

XIV SISO: Single-Input Single-Output

SP: Smith Predictor

SVD: Singular Value Decomposition VSD: Variable Speed Driver

WI: Weight Indicator

Notations

Arabic alphabet

: Dynamic matrix

: Multivariable MPC dynamic matrix : Ore grindability

: Input matrix

, : Breakage, repartition or distribution function ∗ _{: Triangular fragmentation matrix}

: Transfer function of controller, output matrix : Circulating load of a hydrocyclone classifier : Mass of constituent

: Tracer concentration in the material at the outlet of the mill : Recycled load of a hydrocyclone classifier

: Uniform diffusion coefficient,Laplace transform of disturbance signal on output

: Ore grindability (inverse of ore hardness) : Feed particle size variations

: Size for which the hydrocyclone selectivity is 25 % : Size for which the hydrocyclone selectivity is 75 % : Size expressing the product fineness

, : Hydrocyclone classifier cut-point : Diffusion coefficient of constituent

!: Laplace transform of estimated lumped disturbance brought back to the manipulated variable

" : Total lumped disturbance

" : Estimated total lumped disturbance

#_" : Error between _" and _"

$,%&,'!" : Rod mill output slurry density

(: Mean value of the solid ore density in Kolwezi copper belt ) : Water density

⋀ : Matrix defining the relative gain array ∆, : Additive uncertainty

∆-, : Radius of disk .

XV / : Laplace transform of error signal

/0 : Specific breakage energy (according to Bond)

E2 : Specific breakage energy (according to Charles)

E3 : Specific breakage energy (according to Hukki)

/₄: Specific breakage energy (according to Kick) /5 : Specific breakage energy (according to Rittinger)

/₆₇ : Specific breakage energy (according to Svensson and Murkes)

8 : Matrix transfer function between the controlled variables and the manipulated variables

9 : Probability of fragmentation of grains according to their sizes

8 : (;, <)">entry of8

? : Matrix transfer function between the controlled variables and the external disturbances

@

_A : Corrected coefficient matrix

? : (;, <)"> entry of?

: Material hold-up in the mill per unit of length : Initial spatial profile of the hold-up

B : Identity matrix, Cutoff imperfection coefficient

C

: Cost function

CD6 : Least squares cost function

K₂ : Charles’s constant F : Sampling instant

FG : Additional parameter in the selection function

K3 : Hukki’s constant

H₄: Kick constant

HIJ,K : Proportional gain of nominal PI controller

H5 : Rittinger constant

H₆₇ : Constant of Svensson and Murkes

FL,MN : Supplementary parameter in the convective velocity model

O : Length of the mill

P : Column vector expressing the masses per particles size : Mass of particles in the ;"> size interval

Q : Laplace transform of measurement noise signal RS : number of spatial grid points

RT : number of spatial intervals U : Transfer function of the plant

V: Laplace complex variable

U

: Mass fraction of material crossing through the sieve of mesh ; UW: Cumulated particles size distribution of material after fragmentation

UG : Peclet number

XVI

X: Error weighting

Y: Transfer function of DOB low-pass filter YG : Water flow rate

YZ: Feed mass flow rate

Y_Z,%GW : Feed mass flow rate at a reference state

Y7 : Feed mass flow rate of a hydrocyclone classifier

Y&∗ : Mass flow rate coming out of the rod mill

YI : Product mass flow rate of a hydrocyclone classifier

Y₅ : Recirculating mass flow rate of a hydrocyclone classifier Y%&,'!" : Rod mill output slurry flow rate

[ : Mass flow rate of constituent

\ : Input weighting matrix

: Mass fraction of material retained on the sieve of mesh ; ] : Diagonal selection matrix

] : Specific rate of breakage or selection function of interval size ;

^

: Time

^

, : Mean of the residence time distribution ^G : Median of the residence time distribution _( : Sampling period

` : Fresh ore feed rate

` : Rod mill feed water rate, dilution water flow rate for the typical wet grinding circuit

`a : Dilution water flow rate `a,& : Dilution water mass flow rate

b : Uniform convective velocity b : Convection velocity of constituent

b%GW : Uniform convective velocity at a reference state

c : Total mass of particles dZ, : Feed mass fraction of size e

c : Bond’s index, characteristic of the material d : Mass fraction of particles of size ;

d : Initial spatial the mass fraction of size e f: Vector of state variables

g: Spatial coordinate along the mill axis h : Laplace transform of output signal

i : Product particle size or fineness

i : Product flow rate, circulating load for the typical wet grinding circuit ia : Product density

h& : Vector of measured variables

i$ : Predicted output

h%: Reference or setpoint

XVII e : Size of particles in the <"> size interval ‖… ‖_l : l− Rno

Greek alphabet

p : Parameter depending on material in the selection function q_, : Cumulative breakage function

r : Vector of deviations betweenh& and h

r

: Probability of particles of size e to cross through the discharge grid s : Sensitivity function

t : Parameter depending on material in the cumulative breakage function u : Dimensionless selection function of interval size ;

: Constituent of a tubular chemical reactor

v : Hydrocyclone selectivity function or tromp curve w : Delay of the tracer impulse

w : Second hydrocyclone model parameter w : (;, <)">Relative gain

wM: Time constant of DOB low-pass filter

wM,K : Time constant of nominal DOB low-pass filter

x : Complementary sensitivity function

.

: Family of plants

y : Standard deviation of the residence time distribution z : Time constant

z{ : Time delay

| : Vector of model parameters

|} : Vector of model parameter estimates

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II.1: Simplified chain of processes in mineral processing and metallurgical plants

for a metallic ore ... 10

II.2: Typical wet grinding circuit... 11

II.3: Flowsheet of the KZC grinding circuit ... 13

II.4: Rod Mill ... 14

II.5: Hydrocyclone Classifiers ... 14

II.6: Ball Mills ... 14

III.1: Classification into size intervals ... 21

III.2: Mass flow rates at the hydrocyclone ends ... 30

IV.1: Representation of the KZC grinding process as a MIMO system ... 39

V.1: Laboratory scale slurry flotation ... 44

V.2.a): RTD fitting of the rod mill ... 45

b): RTD fitting of the ball mill... 45

V.3.a): Rod mill PSD fitting-simple validation ... 49

b): Evolution of the PSDs for the three intervals of size classes within the rod mill 49 V.4.a): Rod mill PSD fitting-cross validation... 50

b): Evolution of the PSDs for the three intervals of size classes within the rod mill 50 V.5.a): Ball mill PSD fitting-cross validation ... 51

b): Evolution of the PSDs for the three intervals of size classes within the ball mill 51 V.6.a): Hydrocyclone 1 fitting curve – simple and cross validations ... 52

b): Hydrocyclone 2 fitting curve – simple and cross validations ... 53

V.7: Evolution of the PSDs for the three intervals of particle sizes within the rod mill .. 57

V.8: Steady-state characteristics of the system ... 58

V.9: Time-mill axis evolution of the rod mill content ... 60

V.10: Time evolution of the product PSD ... 61

V.11: Time evolution of the product flow rate ... 61

V.12: Time evolution of the product density ... 61

V.13: Time evolution of the product fineness variations around nine operating points 62 V.14: Time evolution of the product flow rate variations around nine operating points . 62 V.15: Time evolution of the product density variations around nine operating points .. 63

VI.1: Typical wet grinding circuit: single-stage closed-loop ball mill grinding ... 68

VI.2: Representation of the typical wet grinding process as a MIMO system ... 69

VI.3: Typical wet grinding circuit with its instrumentation-ISA norm ... 72

VI.4: PID control ... 76

VI.5: IMC diagram ... 77

VI.6: DOB based control ... 78

VI.7: Decoupling control ... 79

VI.8: Philosophy of the MPC scheme ... 80

VII.1: Principle of the initial steady-state operating point selection ... 86

VII.2: Step variations of manipulated variables ... 88

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VII.5: Time evolution of the product flow rate after step variations of input variables .. 90

VII.6: Time evolution of the product density after step variations of input variables ... 90

VII.7: Comparisons between continuous-time and discrete-time models ... 92

VII.8: Frequency evolution of relative gains ... 99

VII.9: DNA with Gershgorin’s bands for the first choice of variable pairing ... 100

VII.10: DNA with Gershgorin’s bands for the second choice of variable pairing ... 100

VII.11: Interaction effects between manipulated and controlled variables ... 102

VII.12.a): Additive uncertainty representation ... 104

b): Multiplicative uncertainty representation ... 104

VII.13: Neighbourhood space around the nominal operating point ... 104

VII.14: Step variations of manipulated variables ... 105

VII.15: Variations of output variables after step variations of manipulated variables around the nominal and the eight boundary operating points ... 106

VII.16: Family of uncertain plants ... 108

A-VII.1: Output variations after step change of feed particle size distribution ... 110

VIII.1: Standard form of PI-SP ... 116

VIII.2: IMC form of PI-SP ... 117

VIII.3: Standard form of DOB ... 118

VIII.4: IMC form of DOB ... 118

VIII.5: DIMC scheme ... 119

VIII.6: Block diagram for definition of sensitivity ... 120

VIII.7: Block diagram for DOB design ... 123

VIII.8: Block diagram for PI-SP design... 124

VIII.9: Design of DOB and PI-SP ... 128

VIII.10: Reconstruction of lumped disturbance on the product fineness for both nominal and uncertain models ... 129

VIII.11: Reconstruction of lumped disturbance on the product flow rate for both nominal and uncertain models ... 129

VIII.12: Reconstruction of lumped disturbance on the product density for both nominal and uncertain models ... 129

VIII.13: Bode’s diagram of transfer function between the total lumped disturbance and its estimate for the channel 1 ... 130

VIII.14: Bode’s diagram of transfer function between the total lumped disturbance and its estimate for the channel 2 ... 130

VIII.15: Bode’s diagram of transfer function between the total lumped disturbance and its estimate for the channel 3 ... 130

VIII.16: Step response of product fineness ... 131

VIII.17: Step response of product flow rate ... 131

VIII.18: Step response of product density ... 131

VIII.19: KZC wet grinding circuit with its instrumentation-ISA norm ... 132

VIII.20: Time evolutions of fresh ore feed rate and product fineness for both nominal and mismatch cases – linear simulation ... 134

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XXIII

IV.1.a): Rod mill transport parameters ... 35

b): Rod mill grinding parameters ... 36

IV.2: Hydrocyclone parameters ( = 1,2) ... 36

IV.3.a): Ball mill transport parameters ... 38

b): Ball mill grinding parameters ... 38

V.1: Internal transport parameters estimation for rod/ball mill ... 44

V.2: Transport parameters estimation for rod/ball mill ... 44

V.3: Grinding parameters estimation for rod/ball mill ... 48

V.4: Parameter estimation for hydrocyclone classifiers ... 52

V.5: Steady-state simulation results ... 56

V.6: Signs of gains between manipulated and output variables ... 59

VI.1: Dynamic and steady-state features of the typical grinding circuit... 70

VI.2: Typical variable pairing ... 70

VII.1: Initial steady-state operating point ... 86

VII.2: Optimal operating point ... 95

A-VII.2.1: Neighbourhood space around the nominal operating condition after +/- 10 % variations of manipulated variables ... 111

A-VII.2.2: Identification of boundary parameters of the linearized system model ... 111

VIII.1: Step changes of setpoints ... 115

VIII.2: Step-sinusoidal variations of external disturbances ... 115

VIII.3: Duality between PI-SP and DOB design ... 126

VIII.4: Tuning parameters of the controllers ... 127

VIII.5: Performance indices in step setpoint changes ... 127

VIII.6: Performance indices in step setpoint changes for all the control system – linear simulation ... 134

VIII.7: Gains of frequency response between external disturbances and controlled variables at frequency equal to 10 [rad/h] in the nominal case ... 137

VIII.8: Gains of frequency response between external disturbances and controlled variables at frequency equal to 10 [rad/h] in the mismatch case ... 137

VIII.9: Reduction ratio of gains of frequency response at frequency of 10 [rad/h] obtained thanks to the DOB and the PI-SP in the nominal case ... 137

VIII.10: Reduction ratio of gains of frequency response at frequency of 10 [rad/h] obtained thanks to the DOB and the PI-SP in the mismatch case ... 137

VIII.11: Reduction ratio of gains of frequency response at frequency of 10 [rad/h] obtained thanks to the DOB and the PI-SP in the nominal case for one closed- loop configuration ... 140

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loop configuration ... 142 VIII.14: Reduction ratio of gains of frequency response at frequency of 10 [rad/h] obtained thanks to the DOB and the PI-SP in the mismatch case for two

1

Chapter I: INTRODUCTION

I.1 Motivations

Mineral processing refers to the set of techniques and methods for extraction of the valuable minerals contained in the blocks of ore coming from the mine. The raw material is thus valorised. This can be performed only after reducing the coarse blocks by means of mineral preparation methods into a size such that one can separate efficiently each valuable mineral from useless minerals. This separation resorts to physical, chemical or physicochemical methods and allows obtaining a concentrate of maximal value and the lowest possible tailing.

Using mineral processing methods constitutes a permanent challenge in the mineral and metal industry. In a first stage, the process has to be designed. The second step is concerned with the starting up and the commissioning of the installation combined with adjustments of some parameters. Once the running-in is completed, the plant operator should keep the process under control and, in order to maintain or increase the profit margin, he has to optimize it. This requires a deep knowledge of the process behaviour.

To satisfy the requirements on the final product (metal) set by the consuming market, one can either make changes and/or modifications on the mineral processing stage or on the extractive metallurgy step. Changing some values of parameters and variables of the extractive metallurgy may induce a high cost (Bouchard, 2001). Thus, control is often applied on the mineral processing whose product, the ore concentrate, constitutes the input material of the extractive metallurgy. This is why much pressure and attention are paid on mineral processing units and especially on concentration plants.

As the ore size reduction procedure is the critical step of a concentrator, it turns out that controlling a grinding circuit will be of much benefit and crucial for all the concentrator. Indeed, the grinding process accounts for almost 50 % of the total expenditure of the concentrator plant (Chen et al., 2008). Moreover, the product particle size from grinding stage influences the recovery rate of the valuable minerals as well as the volume of tailing discharge in the subsequent process. This is the first justification for studying an efficient way to optimize and control a grinding process such as that of the Kolwezi concentrator.

An industrial wet grinding process offers complex and challenging control problems, among which:

2 and characteristics;

- the difficulty to highlight the influence of the water on the process efficiency; - the presence of a physical closed-loop, i.e. the material rejected by a classifier

is recycled back to a ball mill inlet;

- the existence of nonlinear relationships between state variables such as the material hold-up and the particle size distribution within a mill.

The above five stated issues constitute the second motivation of this study.

I.2 State of the art

Several studies and investigations have been carried out on the mineral grinding processes, from their modelling to their optimization and control. Thus, much scientific literature has been devoted to this framework in the last thirty years. In the reference books by Lynch (Lynch, 1977) and Austin (Austin et al., 1984) followed by King (King, 2001), many studies and implementations have been performed and regularly surveyed. Other very interesting and precious works have been achieved by academic and/or industrial specialists. We can quote, with references therein, (Hulbert, 1989; Hodouin & Del Villar, 1994; Duarte et al., 1999; Pomerleau et al., 2000; Boulvin, 2001; Hodouin et al., 2001; Liu & Spencer, 2004; Lepore, 2006; Chen et al., 2008; Chen et al., 2009; Ozkan et al., 2009; Weig & Craig, 2009; Yang et al., 2010; Hodouin, 2011). Most of these studies are based on a typical grinding process. This typical configuration is a single closed-loop grinding circuit consisting mainly of one ball-mill, one pump sump and one hydrocyclone classifier.

Modelling

Deriving a model for a process is drastically linked to the expected use of the model system. In the case of grinding circuits, one can distinguish two main classes of models; on the one hand, steady-state or static models used for dimensioning or optimizing the circuit, and on the other hand, dynamic models allowing to proceed with dynamic simulation and the design of a control strategy.

Initially, most of the models found in the literature were static and/or empirical. For breakage operations, in these empirical models (Rittinger, 1857; Kick, 1883; Bond, 1952), the energy consumed is the main variable to be minimized. Hence, such models are useful for the design of circuits.

3

Many simulation techniques and packages already exist for flowsheet simulation in the mineral processing industry (Liu et al., 2004; Reyes-Bahena, 2001; Napier-Munn & Lynch, 1992) but most of them are either based on a steady-state analysis, for instance METSIM, USIM PAC, MODSIM, Limn and JKSimMet, JKMRC Simulator, SIMBAL, GSIM, CAM and Bruno, or on simple lumped parameter dynamic models, e.g. Aspen Dynamics, SysCAD and MinOOcad (Herbst & Blust, 2000).

Control

In the literature, one can find several studies devoted to the challenge of grinding circuits control. The investigations in this framework cover many aspects from theoretical up to practical considerations. The related results include classical control as well as advanced control techniques from decentralized Proportional-Integral-Derivative (PID) control to centralized Model-based Predictive Control (MPC) schemes (Hodouin, 1994; Duarte et al., 1999; Pomerleau et al., 2000; Chen et al., 2008; Chen et al., 2009; Weig & Craig, 2009; Yang et al., 2010; Hodouin, 2011). Focusing on the typical wet grinding process, most of these control strategies are based on a two input-two output scheme. The controlled variables are typically the product particle size and the circulating load while the manipulated variables are usually the fresh ore feed rate and the dilution water flow rate.

I.3 Contributions of this study

The current dissertation is focusing on an industrial grinding plant, namely the grinding process of the “KolweZi Concentrator”, KZC in short, from the Democratic Republic of Congo. This grinding circuit consists mainly of one rod-mill in an open-loop and two ball mills, one pump sump, one distributor and two hydrocyclone classifiers in a double closed-loop. Hence, the KZC grinding process has a more complex

topology than the typical one and offers more challenging issues in modelling and

control.

Therefore in this thesis, we have developed and extended to wet operating/working condition and to mineral processing the complex model used in dry cement grinding. Thus, a phenomenological approach combining population mass balance and pulp dynamics transport is used for rod mill and ball mills modelling. This leads to nonlinear partial differential equations (PDE) containing six parameters for each mill. While the repartition function is almost the same for both dry grinding and wet grinding, a mathematical description is proposed for the selection function

which should be higher in wet grinding than in dry grinding (Ozkan et al., 2009).

Another mathematical relationship which describes the link between the

convective velocity and the feed ore rate has been suggested. A steady-state

model with two parameters is sufficient to describe the hydrocyclones classifiers (King, 2001) while the pump sump is modelled by a perfect mixer. The other components of the KZC flowsheet are simply considered as time delay elements.

4

KZC circuit can be considered as a Multi-Input Multi-Output (MIMO) dynamical system with couplings, time delays, strong external disturbances such as ore grindability (inverse of ore hardness) and feed ore particle size distribution, and internal disturbances caused by model mismatches. This control study includes explicitly

the control of the product density which has usually been neglected in the control of

the typical grinding process. Yet, the density of the pulp from the grinding stage influences significantly the performance of the subsequent stage. The process under study is thus considered as a 3x3 MIMO system.

All model parameters are determined from experimental data by using the nonlinear least squares algorithm. It should be noted that, due to the unavailability of sensors on the process, we could not perform on-line measurements of the variables in real-time. Hence we could not use a colorant neither a radioactive tracer to evaluate the transport dynamics of material within the mills. An alternative approach has been

developed on the basis of the G41 foaming as tracer. This new experimental

procedure provided satisfactory results in comparison with previous studies (Rogovin et al., 1988). Similarly due to a lack of instrumentation, Tyler series sieves have been employed for measurements of particle size distributions.

For global simulation, all the individual models are connected according to the circuit configuration provided by the flowsheet of the installation. Both a steady-state simulator and dynamical simulator based on three classes of particle sizes have been developed within the MATLAB/Simulink software. The resulting steady-state simulator is in good agreement with the recorded data, and the dynamic simulator exhibits the expected qualitative behaviour. Both tools could allow operator training. Finally, the dynamical simulator has been analysed in order to highlight the steady-state and dynamic features of the process. These features characterize the main control issues and led us to choose a suitable control structure and design the corresponding controller.

The process model has been linearized around a steady-state operating point. A two-step approach has been used to determine the operating point around which linearization has been performed. First, an initial estimate of the optimal operating point is determined on the basis of a simulation study. Thus, a linearized model of the process is determined around this initial operating point. Next, a correction to the initial operating point has been computed by solving a convex optimization problem. The optimization problem consisted in maximizing the product flow rate while meeting specific constraints on product quality and accounting for actuator limitations. To do so, it has been assumed that the linearized model remains valid for the considered correction range.

5

internal and external disturbances. The proposed DIMC structure is composed of a

Proportional-Integral (PI) Smith Predictor (SP) and a Disturbance Observer (DOB)

on each channel of the paired variables. A duality between the PI-SP and the DOB has been established from a design point of view. A robust design has been performed to allow the controller to work properly on the actual plant whose model is uncertain. The validation of the control scheme has been made on the basis of simulation results from the linearized model. A progressive implementation of the

control structure is also explained in the framework of the KZC installation. This is

helpful to show the benefit of the control if the plant does not possess all the required instrumentation.

In summary, our contributions in the framework of wet grinding processes can be listed as follows:

- the generalisation of the grinding model to a complex topology containing a double closed-loop;

- the mathematical description of the selection function which takes into account the positive effect of water on the efficiency of the grinding;

- the mathematical description of the link between the convective velocity and the feed ore rate;

- the use of the G41 foaming as tracer yielding a new experimental method to determine the dynamics of material within a mill;

- the derivation of an analytic solution of the steady-state model in both open-loop and double closed-open-loop configurations, and its use to estimate the grinding model parameters;

- the control of the product density as the third controlled variable; - the proposition of a DIMC as a suitable decentralized control scheme;

- the use of the PI-SP and the exploitation of the duality between the PI-SP and the DOB for controller design;

- the use of a robust performance criterion to design the controller and the DOB; - the possibility of a progressive implementation of the control scheme.

I.4 Outline

This dissertation is divided into two parts: process modelling and simulation (chapters II, III, IV and V), and controller design and validation (chapters VI, VII and VIII).

First part

Chapter II provides the complete description of the Kolwezi concentrator and especially its grinding stage which is a double closed-loop wet grinding circuit. The flowsheet is presented and the main components are depicted.

In Chapter III, we present an overview of the state of the art in the modelling of grinding processes. This state of the art is focused on the grinding phenomenon and the classification by hydrocyclone.

6

different models. Experiments and measurements performed on site are explained. The obtained data and the least squares algorithm are employed to estimate the optimal parameters. Finally, we present both the steady-state simulator as well as the dynamic simulator and we discuss the simulation results.

Second part

In Chapter VI, a survey of the various control schemes used in the framework of grinding circuits is presented. Based on the typical wet grinding circuit, this survey covers the state of the art of the main control methods from decentralized controllers to multivariable controllers.

Chapter VII presents a systematic analysis of the process under study. Aspects as system model linearization, operating point optimization, variables pairing, interaction effects and uncertainty characterization are studied. This chapter ends by highlighting the main features of the process.

Part I :

9

Chapter II: WET GRINDING CIRCUIT OF THE

KOLWEZI CONCENTRATOR

Introduction

The present chapter aims at describing the industrial plant which is the object of this study, namely the grinding process of the Kolwezi concentrator. First, the framework of the current dissertation is defined. To this end, the general concepts of mineral processing are presented. The goal and the chain of processes in mineral processing are briefly given. After, the emphasis is made on the critical aspect of grinding stage. The typical grinding flowsheet, most encountered in the literature, is particularly highlighted. Finally, the wet grinding process of the Kolwezi concentrator is presented. While the other stages of the plant are briefly explained, the grinding one is described and specified. This description will later be our reference in the current study, on modelling and control, especially in the comparison with the typical circuit.

.

II.1 General concepts of mineral processing

II.1.1 Objectives and processes of mineral processing

A raw ore cannot be used as such as a final product for industrial or commercial uses. It needs to be treated for preparing usable materials that can be either specific minerals released from the ore, or more usually metals, alloys, or compounds such as oxides. The aim of mineral processing for most of ores is to concentrate the valuable minerals contained in raw ores for the subsequent stages such as metal extraction, fabrication of consumption products. Usually, minerals are first liberated from the ore matrix by comminution and particle size separation processes, and then separated from the tailings using processes capable of selecting particles based on their physical or chemical properties, such as surface hydrophobicity, specific gravity, magnetic susceptibility, chemical reactivity, and colour. The processes in mineral processing can thus be classified into the following distinct categories (Hodouin, 2011):

- minerals liberation processes (crushing, grinding and size classification);

- minerals separation processes (flotation, magnetic or gravimetric separation, sorting, leaching, etc.);

- concentrate pre-treatment for subsequent metal extraction (drying, agglomeration, sintering, etc.);

10

But in general, the transformation chain in mineral processing is a technically coherent sequence of processes. This chain is shown in Figure II.1 for a metallic ore.

Figure II.1: Simplified chain of processes in mineral processing and metallurgical plants for a metallic ore

The current dissertation is just focusing on one operation among the above processes chain. Therefore, the following subsection is concerned with the grinding step since it is the critical stage of the mineral processing chain.

II.1.2 Grinding process

The grinding process is a size reduction operation often used in the mineral industry to liberate the valuable minerals from the tailings (Pomerleau et al., 2000). It is a fundamental operation process, and in many respects the most important unit operation in a mineral processing plant (Chen et al, 2008). Indeed, grinding process represents almost half of the total operating costs associated with the mining operation, and the product particle size greatly influences the recovery rate of the valuable minerals and the volume of tailing discharge in the subsequent processes. Low qualified rate of product particle size can cause unacceptable economic loss and could be harmful for pollution control. This is why the grinding process is the critical stage of a mineral concentrator plant.

The true objective of grinding is to obtain a proper liberation of the minerals coupled with a proper particle size distribution in order to maximize recovery and concentrate value without overgrinding, while keeping specific comminution energy and reagent consumption as low as possible, taking into account the prevailing economic conditions. These objectives being highly complex, simpler ones must be formulated. For effective concentration, the grinding process has simply to maintain the following output variables stable, mainly including the product particle size distribution, circulating load and mill solid concentration (Chen et al., 2008).

11

Figure II.2: Typical wet grinding circuit

The next section presents the wet grinding circuit which is the object of the current study. This circuit is compared to the typical one in order to underline the differences leading to the additional challenges characterized in subsequent chapters.

II.2 Grinding process of the Kolwezi concentrator

II.2.1 Preliminary description

The industrial plant, object of this study, is the “KolweZi Concentrator”, KZC in short. This plant belongs to Gécamines which is a Democratic Republic of Congo state company. Gécamines is the greatest mining and mineral processing company in the DRC and is acting in the Katanga province. Its head quarter is located in the Lubumbashi town. The Kolwezi town, where the KZC is situated, constitutes the so-called “west group” of Gécamines.

KZC is a mineral processing installation dealing especially with mineral concentration. This concentrator is fed by raw material coming from mines and produces a copper-cobalt concentrate after rejecting the tailings.

Four steps summarize the KZC process: dry crushing and screening, wet grinding and classification, flotation, thickening and filtration.

12

II.2.2 Objective of KZC

KZC receives dolomitic or siliceous blocks of ore from three different deposits: - Deposit K1: the ore of this deposit is rich in copper and cobalt with a high grade

in manganese.

- Deposit K2: this ore is poor in copper and cobalt but rich in manganese.

- Deposit K3: the ore contained in this deposit is rich in copper but poor in cobalt and manganese.

The production capacity of KZC is actually of 50 /ℎ/ (fifty tons-dry per hour and per rod mill).

The feed mean grades are of 2.5 % in copper and 0.2 % in cobalt.

The required grades of the produced concentrate are of 15 − 18 % in copper and 0.98 % in cobalt.

The target of KZC is to produce the best copper-cobalt concentrate in terms of quantity and quality.

II.2.3 Dry crushing

The dry crushing section is the first step of the KZC process and it is also the first stage of the ore mechanical preparation or size reduction.

This section is fed by coarse blocks of 1500 to 1800 as size and should feed into the wet grinding section the ore with 19 as maximum size.

The KZC dry section is a three sub-steps size reducing procedure: primary crushing by the 1/10 Arbed jaw crusher, secondary crushing by the 7 standard Symons cone crusher and tertiary crushing by the 7 short head Symons cone crusher.

II.2.4 Wet grinding

The wet grinding section is the critical step of KZC. Its flowsheet is shown in Figure II.3. This circuit is composed mainly of one rod mill, two ball mills, two hydrocyclones classifiers, one sump pump, one slurry distributor and several pipes and channels for material transport. With reference to the mineral processing language, this kind of flowsheet can be called “double-closed loop modified traditional circuit” (Bouchard, 2001) or “double-closed loop preclassification circuit” (Hodouin et al., 1994).

After the dry crushing and screening stage, the copper-cobalt ore is sent to a stock pile. Then, a set of conveyor belts transport the raw material to the wet grinding circuit. The ore is fed into the rod mill together with the feed water.

13

Ds

hydrocyclone classifier whose underflow stream containing the larger particles is recycled back to a ball mill for regrinding while the overflow stream containing the finer particles is considered as one part of the product.

The set made by the sump pump, the pulp distributor, the two hydrocyclones classifiers and the two ball mills constitutes the double closed-loop circuit. The process product is thus the sum of the two overflow streams produced by the classifiers.

Figure II.3: Flowsheet of the KZC grinding circuit

Caption

- HC 1: Hydrocyclone 1 - HC 2: Hydrocyclone 2 - Ds: Distributor

This second step of the KZC mechanical preparation process is simply called “milling” and its aim is just that of providing the subsequent flotation section with a maximum flow rate of pulp characterized by:

- 80 % of particles with size smaller than the reference size of 74 ; - a density ranging between 1.3 and 1.4.

The Particles Size Distribution (PSD) is measured by means of 48 and 200 sieves of Tyler sieves series having respectively 0.3 and 0.074 as mesh dimension. One litre of pulp is poured on the two sieves and the method consists in checking if nothing is retained on the 48 sieve and around 20 % should be retained on the 200 sieve.

The pulp density is deduced from the weighing of one litre of pulp by means of a Marcy balance.

This milling circuit is composed of the following unitary operations: fine milling by rod mill, size classification by hydrocyclones and ultra-fine milling by ball mills. Figures II.4, II.5 and II.6 show respectively the photographs of the rod mill, the hydrocyclones classifiers and the ball mills in use at KZC, with characteristics therein.

14

Figure II.4: Rod Mill

Dimensions: 7’x12’

Brand: Marcy (Metso Minerals)

a) b) Figure II.5: Hydrocyclone Classifiers

Kind: Static classifier

a) Tricône b) Simple Figure II.6: Ball Mills

Kind: Ball Mill 8’x84’’ or Tricône 9’x3’x6’x8’

15

II.2.5 Flotation

Flotation is the mineral concentration method employed at KZC. The role here is to concentrate the mineral or grow up the grade of useful mineral with an acceptable efficiency. The input of this section is the hydrocyclones overflow and the outputs are the concentrate sent to the last step of thickening and filtration, and the tailing which is taken out.

The characteristics of the flotation cells are the following:

• Brand: Auto Kumpo

• Composition: - 20 primary cells. - 20 secondary cells. - 16 tertiary cells.

- Capacity per cell: 8 .

- Motor: Asynchronous, 25 , < 1000 / , 550 .

II.2.6 Thickening and filtration

This last step of the KZC process is concerned with removing out the water contained in the concentrated pulp coming from the flotation. For this purpose, a solid-liquid separation method combining thickening and filtration is used.

One usually completes the process by drying the mineral mud in order to obtain a dry concentrate.

II.2.7 Current control mode

Currently, there is no control system installed on the KZC plant and furthermore, there is no instrumentation on the plant except two valves for the rod mill feed water and the dilution water. The process is both manually controlled and not optimised. Hence, the KZC target is never reached. Therefore in this work, we are going to propose an automatic control structure in order to maintain the KZC wet grinding process on the optimal operating point maximizing the production while meeting constraints on the product fineness and density.

Conclusion

17

Chapter III: STATE OF THE ART FOR GRINDING

PROCESSES MODELLING

Introduction

The goal of this chapter is to give an overview of the different modelling approaches used in the scope of grinding circuits. By observing the typical and the KZC grinding circuits, we realise that two main operations are involved, namely the mill grinding and the hydrocyclone size classification. The grinding operation within a mill may be split into two elementary operations, i.e. the fragmentation and the material transport. Therefore, the fragmentation or breakage models are presented first. Next, the chapter addresses the issue of modelling the transport dynamics of material within a mill. After, the two models are combined to form the final model of a grinding mill process. The last section is concerned with the modelling of the size classification of particles by the hydrocyclone. The models described in this chapter will be employed as reference from which the appropriate model of the KZC wet grinding plant will be derived.

III.1 Fragmentation or breakage modelling

III.1.1 Introduction

Nowadays, several kinds of breakage machines are used in industry. The energy consumed by those machines constitutes a very high cost (Boulvin, 2001). Moreover, only 2 to 20 % (according to the grindability of the material and the kind of installation) of the energy supplied is efficiently dedicated to the breakage (Labahn & Kohlhaas, 1983) while the rest is dissipated trough heat.

Deriving a dynamical model of the process should allow more understanding of its behaviour and eventually improving its performance.

The attempts to quantify the breakage phenomena have been the object of several theoretical studies including energy-based approaches, morphological aspects

and the study of transformations occurring in the breakage devices (Jdid et al., 2006). Most of the mathematical models found in the literature are steady-state and empirical

(Rittinger, 1857; Kick, 1883 and Bond, 1952). These models have been derived after many years of experiments and observation works. In these models, the energy consumed for the fragmentation constitutes the main variable to be minimised. The steady-state models proposed in the literature should allow the circuit design and they are also called “energy-based models”.

18

explained by the “disappearance or death” and “appearance or birth” of particles within each particle size interval or class. This is the phenomenological modelling.

In the following subsections, the two approaches introduced above are detailed as a state of the art for the modelling of the particles fragmentation or breakage.

III.1.2 Energy-based models

Three main theories have been reported in order to describe the relationship between the energy consumed by the material and the reduction of its size during the fragmentation process. Nowadays, the assumptions proposed to determine this relationship are not rigorously proved because one does not yet know how to measure the quantity of energy really absorbed by the particles during their fragmentation. One can only measure the total energy consumed by the fragmentation device or equipment.

In 1857, Von Rittinger postulated that the specific energy for the breakage of a given material is directly proportional to the amount of specific surface newly created. The specific energy is defined as the quantity of energy to be supplied in order to grind a unit mass of the material.

This results in the following relationship:

= ( − )

(3.1) where : Specific breakage energy (according to Rittinger).

: Rittinger constant.

: Specific surface of the material after fragmentation. : Specific surface of the material before fragmentation.

The Rittinger constant represents in fact the specific energy of breakage per unit of specific surface produced. The Rittinger relationship can also be expressed with regard to the size ‘z’ of particles:

=

( ) −

( )

(3.2) where ( ) : Cumulated particles size distribution of material before fragmentation. ( ) : Cumulated particles size distribution of material after fragmentation. As in practice it is difficult to perform the integration from = 0; one commonly uses the following pseudo-Rittinger form:

=

_{, !}

−

_{, !}

"

(3.3) where _,#$ : Size corresponding to 80 % of passing in the initial particle size

19

_,#$ : Size corresponding to 80 % of passing in the final particle size distribution.

The drawback of this law lies in the fact that Rittinger did not take into consideration the material deformation before its fragmentation. The amount of surface produced might be proportional to the required work only if it is proportional to the applied stress multiplied by the deformation length.

Kick proposed in 1883 a very simple model for breakage of solid particles. This model expresses the specific breakage energy of a homogeneous material with regard to initial and final particles sizes:

%

=

%

log

, !_{, !}

"

(3.4) where _%

: Specific breakage energy (according to Kick).

_% : Kick constant.

The required work to reduce a given mass of material is therefore the same for a given reduction ratio regardless of the initial size of the material. This is obviously not compatible with practice. In addition, the material is not really homogeneous and the fragmentation depends on its imperfections (cracks, dislocations, etc.).

Since none of the two previous laws was in agreement with the overall results observed during industrial fragmentation operations, Bond proposed in 1952 a law whose form is similar to the pseudo-Rittinger’s, by analysing several experimental results. The general form of this law is given by:

)

= *

₊ $_{, !}

−

₊ $_{, !}

"

(3.5) where ₎

: Specific breakage energy (according to Bond).

* : Bond’s index, characteristic of the material.

Originally based on the Griffith’s fractures theory, Bond’s study has been proved physically unfounded after analysis. Nevertheless, this law is still regularly applied in practice for the sizing of breakage/grinding facilities and it is a compromise between those of Rittinger and Kick. With no theoretical value, it has become an empirical reference in the world of grinding: Bond’s index is used universally. This index is determined by a grindability test (Bond testing) performed according to a standardized procedure in a laboratory scale mill (Austin and al., 1984).

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./

=

./

0

$$₁

− 0

$$₂

"

(3.6) where _./

: Specific breakage energy (according to Svensson and Murkes).

_./ : Constant of Svensson and Murkes.

₃ : Final mean size (in µm) of particles which can be calculated approximately as the geometric mean of final sizes

_{, $}, _,4$, _,5$, _,6$ and _,7$.

: Initial mean size (in µm) of particles which can be calculated as the

geometric mean of initial sizes

_{, $}, _,4$, _,5$, _,6$ et _,7$.

8 : Parameter to be determined for each material and for a given fragmentation

mode.

Charles proposes a general law as follows:

9

= −

9: (3.7) where E_< : Specific breakage energy (according to Charles).

K_< : Charles’s constant.

After integration of this relationship, one derives: - for 8 = 1, the Kick’s law;

- for 8 = 1.5, the Bond’s law; - and for 8 = 2, the Rittinger’s law.

For Hukki, the relationship between the energy required for particles size reduction is a composite relationship of those of Rittinger, Kick and Bond. This can be expressed by:

A

= −

A :(B) (3.8) where E_C : Specific breakage energy (according to Hukki).

K_C : Hukki’s constant.

D( ) : Probability of fragmentation of grains according to their sizes.

In addition to the assumptions made by the authors of the first three energy-based laws, it takes into account the probability of fragmentation D( ) of grains according to their original dimensions. This probability is equal to unity for large particles and tends to zero for ultrafine particles.

21

from crushing to the ultrafine grinding. However, many fragmentation tests performed on different materials show that Rittinger's law is consistent with fine to ultrafine fragmentation, that the Kick’s law applies well in the case of a coarse fragmentation and the Bond’s law covers areas of coarse to fine grinding. The latter law is also used for mill sizing (Jdid et al., 2006). The other laws are generally difficult to handle and face particularly complex calculations and/or tests to determine some parameters (proportionality constant, value of the exponent 8, etc.). For fine to ultrafine grinding, whatever the law in question, the energy consumption increases monotonically but not always linearly with the fineness of the particles.

III.1.3 Phenomenological models

In contrast to energy-based models, phenomenological models are not aiming at determining the energy consumed by the operation of fragmentation. They allow a more physical description of this process (Boulvin, 2001).

To characterize the evolution of particle size distribution resulting from a fragmentation process, mathematical models have been developed based on the concept of RL Brown (1941) that does not consider the material to be broken as a whole but as a population of fragments varying in size.

This population is divided into disjoint size classes. Each size class or interval includes particles of similar size. Grinding a given size class is characterized by the disappearance or death of some of the fragments belonging to the broken class and by the creation or birth of smaller fragments going to be added to other classes.

Figure III.1 shows an example of a classification into size intervals. The idea of Brown sparked of course other works.

Figure III.1: Classification into size intervals

First came the stochastic formulation of grinding by Epstein in 1947 (Jdid et al., 2006). This author considers that a fragmentation operation breaks up into a series of unit steps, each consisting of two principal operations which are the selection of a fraction of the material and the fragmentation of that fraction. He introduced the concept of fragmentation probability ( ), defined as the probability of a particle of size to be fragmented at the EFG step of the fragmentation process and the notion of

particle size distribution H_,I( , _I) which is the cumulative mass distribution of particles of size < _I coming from the fragmentation of a unit mass of size _I .

. . .

22

Then, other researchers among whom Broadbent and Callcott, Sedlatschek and Bass, Austin Klimpel, and Luckie and their co-workers Reid, Herbst and Fuerstenau (Jdid et al., 2006), used the concept of Epstein to obtain results covering modelling, simulation and prediction of particle size distribution of the breakage or fragmentation process.

Studying the fragmentation process is therefore separated in two fundamental functions, i.e. the selection function and the breakage function. Both functions are regarded as time-continuous and particles size-discrete. They are presented in the next two paragraphs.

The fragmentation model described below was presented for the first time by Sedlatschek and Bass (Boulvin, 2001). To describe the kinetics of fragmentation, each size class is characterized by a coefficient ( ) called specific rate of breakage or selection function. In developing this model, a key assumption is introduced: it is assumed that the selection function of a given size class is not influenced by other classes. This assumption leads to linear equations.

The selection function provides the proportion of particles, from a given size interval, which is broken per unit time. It expresses thus the speed of mass evolution within this size interval during the fragmentation process.

If we denote by K the mass of particles in the EFG size interval at time L, the mass variation of this size interval per unit time yields :

: (F)

:F

=

− K (L)

(3.9)

with 0 ≤ ≤ 1

The proportionality coefficient ( ) gives the specific rate of breakage of interval size E and its dimension is the inverse of time.

If * is the total mass of particles and N (L) the mass fraction of particles of size E, (3.9) can be written:

:[P (F) Q]

:F

=

− N (L)*

(3.10)

If * is constant, we simply obtain: :P (F)

:F

=

− N (L)

(3.11)