FLOTATION KINETICS AND RESIDENCE TIME STUDIES ON COAL FLOTATION

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FLOTATION KINETICS AND RESIDENCE TIME STUDIES ON
COAL FLOTATION
A Thesis Submitted for the Degree of
Doctor of Philosophy
by
Djafar Sarghini
The Centre for Minerals Engineering
School of Mines
Faculty of Applied Science
The University of New South Wales
CONTENTS
Page
ACKNOWLEDGMENT...................................................................................................vi
ABSTRACT......................................................................................................................vii
LIST OF SYMBOLS.........................................................................................................viii
LIST OF TABLES...............................................................................................................x
LIST OF FIGURES............................................................................................................xii
CHAPTER 1 INTRODUCTION 1
1.1 FLOTATION AS AN INTERACTIVE SYSTEM....................................................2
1.1.1 Chemical Factors............................................................................................. 2
1.1.2 Equipment Factors..........................................................................................3
1.1.2.1 Flotation Machines........................................................................... 3
1.1.2.2 Flotation Circuits.............................................................................. 6
1.1.3 Operational Factors.........................................................................................7
1.2 COAL FLOTATION..................................................................................................7
1.2.1 Chemical and Petrographic Composition of Coal...........................................8
1.2.2 Impurities Associated With Coal.................................................................... 8
1.2.3 Natural Floatability of Coal............................................................................ 9
1.3 FLOTATION FLOWSHEET DEVELOPMENT.....................................................11
1.4 SCALING-UP FROM LABORATORY BATCH TESTS TO
PLANT SCALE........................................................................................................ 11
1.4.1 Flotation Rate............................................................................................... 12
1.4.2 Residence Time Distribution in Flotation Process........................................ 13
1.5 OBJECTIVE AND SCOPE OF THE INVESTIGATION....................................... 14
1.6 ARRANGEMENT OF THE THESIS...................................................................... 16
CHAPTER 2 KINETICS OF FLOTATION 18
2.1 INTRODUCTION AND BACKGROUND............................................................ 18
2.1.1 Modelling of the Flotation Process................................................................18
2.1.1.1 Batch Approach...............................................................................20
2.1.1.2 Continuous Approach........................................................................... 20
2.1.1.3 Batch Tests and Their Limitations.......................................................21
2.1.2 A Review of Existing Models for Flotation Process........................................23
2.1.3 Shortcomings of the Models................................................................................27
2.2 DETERMINATION OF THE FLOTATION RATE CONSTANT
FROM BATCH TESTS................................................................................................... 27
2.2.1 Ultimate Recovery................................................................................................. 28
2.2.2 Induction Time or Lag Time................................................................................29
2.2.3 Data Fitting............................................................................................................ 30
2.3 EVALUATION OF RATE MODELS........................................................................... 31
2.3.1 Model Parameters................................................................................................. 33
2.3.2 Requirements for Model Assessment................................................................. 33
2.4 MODELS SELECTED FOR THIS STUDY................................................................. 35
2.5 THE EFFECT OF OPERATIONAL VARIABLES ON
FLOTATION RATE......................................................................................................... 35
2.5.1 Particle Size Distribution......................................................................................36
2.5.2 Pulp Density........................................................................................................... 37
2.5.3 Impeller Speed.......................................................................................................39
2.5.4 Aeration Rate......................................................................................................... 39
2.5.5 Reagents................................................................................................................. 40
2.6 SUMMARY AND CONCLUSIONS..............................................................................43
CHAPTER 3 RESIDENCE TIME DISTRIBUTIONS IN
FLOTATION......................................................................44
3.1 INTRODUCTION AND BACKGROUND.................................................................44
3.1.1 Mixing Pattern of Fluid Flowing Through A Reactor......................................44
3.1.2 Flow Models for Flotation Cells........................................................................ 50
3.1.3 Residence Time Distribution................................................................................ 53
3.1.4 Applications of Residence Time Distribution to Flotation Cell Design........53
3.1.5 The Effects of Different Variables on RTD of Particles in Flotation
Process................................................................................................................... 54
3.2 RTD MEASUREMENT AND CHARACTERISATION........................................... 54
3.2.1 Measurement of RTD....................................................................................54
3.2.2 Tracer Considerations...................................................................................57
3.2.2.1 Liquid Tracers.................................................................................57
3.2.2.2 Solid Tracers...................................................................................58
3.2.3 Characterisation of Measured RTD..............................................................58
3.3 REVIEW OF RTD STUDIES..................................................................................62
3.3.1 RTD Tests Under Non-Flotation Conditions................................................ 65
3.3.1.1 Works Concerned With Liquid Phase............................................65
3.3.1.2 Works Concerned With Solid Phase..............................................66
3.3.1.3 Works Concerned With Liquid and Solid Phases..........................67
3.3.2 RTD Tests Under Flotation Conditions....................................................... 68
3.3.2.1 Works Concerned With Liquid Phase............................................. 68
3.3.2.2 Works Concerned the Solid Phase................................................ 69
3.4 SUMMARY AND CONCLUSIONS....................................................................... 70
3.5 OBJECTIVES AND METHOD OF RTD MEASUREMENT
IN THIS STUDY......................................................................................................71
CHAPTER 4 MODELLING OF CONTINUOUS
FLOTATION PROCESS 72
4.1 INTRODUCTION.....................................................................................................72
4.2 A REVIEW OF EXISTING MODELS FOR CONTINUOUS FLOTATION....... 73
4.2.1 Recovery Models Assuming Perfect Mixing................................................. 73
4.2.2 Recovery Models Assuming General Mixing............................................... 76
4.3 APPLICATION OF MODELS.................................................................................77
4.3.1 Direct Approach............................................................................................ 77
4.3.2 Indirect Approach.........................................................................................79
4.4 COMPARISON OF RECOVERIES OBTAINED FROM DIFFERENT
METHODS...............................................................................................................80
4.5 SUMMARY AND CONCLUSIONS....................................................................... 82
iii
CHAPTER 5 EXPERIMENTAL PROCEDURES,
RESULTS AND DISCUSSION 83
5.1 BATCH TESTS FOR KINETICS DATA COLLECTION........................................83
5.1.1 Introduction........................................................................................................... 83
5.1.2 Experimental Apparatus.......................................................................................83
5.1.3 Experimental Procedure.......................................................................................84
5.1.3.1 Coal Sample........................................................................................... 84
5.1.3.2 Sample Preparation................................................................................ 85
5.1.3.3 Test Procedure.......................................................................................86
5.1.3.4 Tree Flotation......................................................................................... 90
5.1.3.5 Variables Considered, Results and Discussion.................................. 96
5.1.4 Calculating Flotation Rate Constant................................................................ 106
5.1.5 Evaluation of the Kinetics Models and Selection of the Best Model.........109
5.1.6 Acquiring Data Using Selected Models.............................................................. 11
5.1.7 Summary and Conclusions..................................................................................112
5.2 RESIDENCE TIME DISTRIBUTION TESTS IN A PILOT SCALE CELL......113
5.2.1 Introduction..........................................................................................................113
5.2.2 Experimental Apparatus..................................................................................... 113
5.2.3 Experimental Procedure..................................................................................... 114
5.2.3.1 Tracers...................................................................................................114
5.2.3.2 Test Procedure..................................................................................... 114
5.2.3.3 Fluid RTD Determination...................................................................115
5.2.3.4 Solid RTD Determination...................................................................115
5.2.4 Assessment of the RTD Results Using Existing Criteria............................... 116
5.2.5 Variables Considered, Results and Discussion................................................ 121
5.2.5.1 Particle Size...........................................................................................123
5.2.5.2 Feed Rate.............................................................................................. 125
5.2.5.3 Aeration Rate....................................................................................... 127
5.2.6 Data Utilisation....................................................................................................129
5.2.7 Summary and Conclusions................................................................................. 131
5.3 PILOT SCALE CONTINUOUS FLOTATION TESTS............................................ 133
5.3.1 Introduction.........................................................................................................133
iv
5.3.2 Experimental Apparatus..............................................................................133
5.3.3 Experimental Procedure.............................................................................. 133
5.3.4 Method of Recovery Estimation................................................................. 134
5.3.5 Calculations.................................................................................................135
5.3.6 Variables Considered, Results and Discussion............................................ 136
5.3.6.1 Size Distribution of Feed and Collector Dosage......................... 138
5.3.6.2 Pulp Density.................................................................................. 139
5.3.6.3 Frother Dosage and Aeration Rate.............................................. 141
5.3.6.4 Feed Rate...................................................................................... 142
5.3.7 Summary and Conclusions...........................................................................144
CHAPTER 6 RECOVERY MODEL EVALUATION 145
6.1 CONTINUOUS FLOTATION PROCESS SIMULATORS................................. 145
6.2 OVERVIEW........................................................................................................... 146
6.3 COMPARISON OF PREDICTED AND OBSERVED RECOVERIES............... 148
6.3.1 Sensitivity of the Simulators........................................................................ 149
6.4 IMPOSING MODIFIERS TO THE LABORATORY RATE CONSTANT....... 151
6.5 SUMMARY AND CONCLUSION........................................................................159
CHAPTER 7 CONCLUSIONS AND
RECOMMENDATIONS 160
7.1 FLOTATION KINETICS...................................................................................... 161
7.2 RESIDENCE TIME DISTRIBUTION IN THE FLOTATION CELL................. 162
7.3 SCALE-UP FROM LABORATORY TO CONTINUOUS OPERATION.......... 163
7.4 CONTINUOUS FLOTATION TESTS.................................................................. 164
7.5 PREDICTIVITY OF THE MODELS................................................................... 164
7.6 RECOMMENDATIONS FOR FURTHER STUDIES.......................................... 165
REFERENCES................................................................................................................166
APPENDIX A.................................................................................................................. 176
APPENDIX B.................................................................................................................. 184
APPENDIX C.................................................................................................................. 196
APPENDIX D.................................................................................................................. 198
ACKNOWLEDGMENTS
After praising God who gave me the ability to think and work, I wish to express my 
sincere thanks to the following persons and organisations for their support and assistance 
during the course of this study:
Dr A. C. Partridge for his academic supervision, encouragement, guidance, advice, 
and discussion throughout the research work.
Dr T. Tran for his help and encouragement as my co-supervisor.
All staff of Mining Engineering Department and the Centre for Minerals 
Engineering, especially Ms Ling Lau and Mr Jin Song for their help.
All postgraduate students in the centre for their friendship and their help in any form.
Ministry of Culture and Higher Education, Islamic Republic of Iran for providing 
scholarship for this study.
Also very special thanks to my wife, my children, my brother, and my sister for their 
patience, tolerancesand love.
Last but not the least, I wish to pay respects to my late parents for all sacrifices they 
made for me, may their souls rest in peace.
vi
ABSTRACT
Many researchers have investigated the residence time distribution (RTD) of particles in 
continuous flotation processes but few of them have linked the RTD data with kinetics 
data to formulate a predictive recovery model. While flotation has been studied as a rate 
process and different models have been proposed, only a few workers have considered 
continuous processing, and the RTD effects have been mostly ignored.
In this study, rate constants were determined for various batch operating conditions for a 
NSW coal and RTDs were measured over a range of pilot-scale operating conditions 
with the objective of incorporating the data into models of continuous flotation. Data 
were used to test five continuous flotation recovery models (Simulators 1-5) by 
comparing predicted recoveries with experimentally measured coal recoveries from a 
pilot-scale continuous flotation cell.
A comparison of the predicted and observed recoveries showed the necessity for 
modifications to the rate constants from batch tests, before using them in the simulators. 
The modification factors (Xm) for adjusting the rate constants were found to be different 
for each simulator. As, for each simulator, the mean of individual Xm values was used to 
modify the simulator, the range and dispersion of Xm were key considerations for 
selecting preferred models. This was accomplished using the standard deviation and 
relative deviation of each set of modifiers.
Comparison of these statistical measures showed that Simulator 2, which is based on a 
first-order rate model with a rectangular distribution of floatabilities, was the best, as the 
individual modifiers for that simulator were not widely dispersed so their mean could be 
used with good reliability. The predicted recoveries using Simulator 2, after application 
of the mean modifier, are shown to correlate closely with recoveries obtained in 
pilot-scale practice.
vii
LIST OF SYMBOLS AND ABBREVIATIONS
RTD Residence Time Distribution
CFSTR Constant Flow Stirred Tank Reactor
E(t) RTD function
E\n Dimensionless RTD function
t' Dimensionless time
X Mean residence time
tE Effective residence time
t The mean
G2 Variance
V Nominal volume of a flotation cell
Veff Effective volume of a flotation cell in operation
0 Volumetric flowrate of pulp or water into the cell
c Concentration of tracer in RTD studies in each sample
Qo Initial concentration of tracer in a vessel
Nd Vessel dispersion number
Pe Peclet number (the inverse of dispersion number)
a RTD shape parameters for a flotation cell
b RTD shape parameters for a flotation cell
n RTD shape parameters for a flotation cell
K Rate constant of flotation
Kf Rate constant of fast floating components
Ks Rate constant of slow floating components
viii
(j) Fraction of slow floating material in the pulp
Ki Rate constant for lower size of a mineral
Ku Rate constant for upper size of a mineral
Roc Ultimate recovery in a flotation operation
R Recovery of a flotation process
Xm Rate constant modifier for continuous flotation
ix
LIST OF TABLES
Page
CHAPTER 1................................................................................................... 1
1.1 Major types of flotation machines......................................................................................4
CHAPTER 2............................................................................................................................ 18
2.1 Percentage recovery of coal in batch flotation tests on successive refloats............... 29
2.2 Models investigated by Dowling et al. (1985)............................................................... 32
2.3 The selected models to be applied to batch data............................................................35
CHAPTER 3............................................................................................................................44
3.1 Proposed models for possible conditions by Cholette and Cloutier (1959)................. 48
3.2 The RTD studies on liquid phase.....................................................................................64
3.3 The RTD studies on solid phase.......................................................................................64
CHAPTER 4............................................................................................................................72
4.1 Calculation of recovery for a reaction with certain RTD (K=0.307)..........................78
4.2 The rate constants obtained for a sample of coal using different rate models........... 82
4.3 Calculated recoveries from different models for one coal sample on the
basis of the different rate equations and for two RTDs................................................ 82
CHAPTER 5........................................................................................................................... 83
5.1 Analysis of the coal fed to the Westcliff plant................................................................. 85
5.2 Size distribution of the coal sample along with ash content of each
size fraction..........................................................................................................................85
5.3 Characteristics of eight samples produced by the sample preparation
procedure.............................................................................................................................86
5.4 Result of a tree flotation test............................................................................................ 93
5.5 Cumulative yield and cumulative ash data obtained from tree flotation.................... 94
5.6 Yield as a function of particle size and collector dosage (Frother 180 g/t)...............97
5.7 The selected models to be fitted to batch data............................................................. 107
5.8 The result of model evaluation for different tests.......................................................110
5.9 Comparison of predicted R*, values with actual yields............................................... 110
5.10 Predicted parameters by the selected models for selected tests (K is min'1).......... 111
5.11 Size analysis of different type of coal used as solid phase tracer.............................116
5.12 The predetermined conditions for different RTD tests...............................................122
5.13 Acquired data from preliminary tracer information for Test No 19
(t =1.7 min and x = 1.75).............................................................................................. 130
5.14 Acquired data from different tests to be used in recovery prediction..................... 131
5.15 Weight of coal recovered in concentrates and tailings during
3rd, 4th, and 5th minutes of five different tests............................................................. 136
5.16 Results of a continuous flotation test (Test No 1)...................................................... 136
5.17 The predetermined conditions for different recovery tests....................................... 137
5.18 Size analysis of different type of coal............................................................................138
5.19 Continuous flotation tests with different size coal and different
collector dosage................................................................................................................ 139
5.20 Recovery of continuous flotation tests at different pulp densities............................. 140
5.21 Continuous flotation tests at different frother dosages and different
aeration rates..................................................................................................................... 142
5.22 Continuous flotation tests at different feed rate...........................................................143CHAPTER 6..........................................................................................................................145
6.1 Predicted values of recovery with Simulator 1 using both solids
and liquid RTDs................................................................................................................ 148
6.2 Observed and predicted recoveries for different tests................................................150
6.3 Comparison of rate constant modifiers (Xm) for different simulators.................... 153
xi
LIST OF FIGURES
Page
CHAPTER 1.............................................................................................................................. l
1.1 Schematic illustration of a mechanical flotation cell.........................................................5
1.2 Concept of contact angle between bubble and particle................................................. 10
1.3 Variation of contact angle with carbon content of coal................................................ 10
1.4 Rate plot of flotation data.................................................................................................. 13
CHAPTER 2............................................................................................................................18
2.1 Confidence limits for K fitting different models to the results of
two different tests...............................................................................................................34
2.2 The effect of particle size on coal flotation yield........................................................... 37
2.3 Typical effect of pulp density on the yield of coal in normal condition......................38
2.4 Impeller speed zones of best flotation performance for different
particle sizes........................................................................................................................40
2.5 The effect of collector dosage on the rate of coal flotation.........................................42
2.6 The effect of frother dosage on the rate of coal flotation............................................ 42
CHAPTER 3........................................................................................................................... 44
3.1 Two types of ideal reactors............................................................................................ 45
3.2 Plug-flow, mixed-flow and typical flotation cell flow, all with x = 1.5.......................48
3.3 Schematic view of perfect-mixers-in-series model........................................................50
3.4 RTD curves for the tanks-in-series model......................................................................50
3.5 Mixing model proposed for an incompletely-baffled bank of cell...............................52
3.6 Treating a bank of cells as a large cell with several mixing zones instead
of individual perfect-mixers-in-series.............................................................................. 52
3.7 Stimulus-response techniques used to study flow in vessels........................................55
3.8 Typical E and F cuives for tracer flowing through a vessel.........................................56
3.9 A typical C curve for flotation cell's RTD.......................................................................59
3.10 A typical E curve for flotation cell's RTD.......................................................................59
xii
3.11 Means and Variances of distributions for different conditions.................................... 60
3.12 Finding the tE/r ratio from plotting F(t) versus t curve................................................61
3.13 Typical RTD data fitted by equation 3.5.........................................................................62
CHAPTER 4..........................................................................................................................72
4.1 Residence time distribution in a reactor......................................................................... 78
4.2 RTD curve for test A......................................................................................................... 81
4.3 RTD curve for test B......................................................................................................... 81
CHAPTER 5.......................................................................................................................... 83
5.1 Schematic illustration of experimental set-up for batch tests......................................84
5.2 Schematic display of the deflector block in the cell.......................................................90
5.3 Schematic display of a tree flotation test........................................................................92
5.4 The relationship of yield/ash............................................................................................. 95
5.5 Effect of pulp density on yield, ash content of concentrate, and
rate constant........................................................................................................................99
5.6 Effect of impeller speed on yield, ash content of concentrate, and
rate constant...................................................................................................................... 100
5.7 Effect of air flowrate on yield, ash content of concentrate, and
rate constant...................................................................................................................... 102
5.8 Effect of collector dosage on yield, ash content of concentrate and
rate constant...................................................................................................................... 104
5.9 Effect of ff other dosage on yield, ash content of concentrate and
rate constant...................................................................................................................... 105
5.10 Rate plot of flotation data............................................................................................... 106
5.11 Fitting data from a certain test to the selected models.............................................. 108
5.12 Schematic diagram of the experimental set-up for RTD tests.................................. 114
5.13 Values of t and a2 for four different tests.................................................................. 118
5.14 Values of tE/r ratio for four different tests...................................................................119
5.15 Values of a, b, and n for four different tests............................................................... 120
5.16 Values of tE/zratio for different size coal.....................................................................124
5.17 Effect of feed flowrate on RTD of different size coal and water............................. 126
xiii
1285.18 Effect of aeration rate on RTD of different size coal and water
5.19 Effect of collector dosage on differently sized coals.................................................139
5.20 Effect of changes in pulp density on recovery............................................................ 141
5.21 Effect of ff other dosage on recovery in different aeration rate................................142
5.22 Recoveries obtained with different feed rates.............................................................143
CHAPTER 6..........................................................................................................................145
6.1 Predicted values of recovery with Simulator 1 using both solids
and liquid RTDs............................................................................................................... 147
6.2 Recovery prediction by Simulator 1 (before and after applying the modifier).........154
6.3 Recovery prediction by Simulator 2 (before and after applying the modifier)........ 155
6.4Recovery prediction by Simulator 3 (before and after applying the modifier)........ 156
6.5 Recovery prediction by Simulator 4 (before and after applying the modifier).........157
6.6 Recovery prediction by Simulator 5 (before and after applying the modifier).........158
xiv
Chapter 1: Introduction Flotation as An Interactive System
CHAPTER 1 
INTRODUCTION
Flotation is arguably the most important mineral processing technique. Mining of low 
grade and complex ore bodies became increasingly economical with the development of 
flotation early this century. The introduction of the flotation process revolutionised 
mineral industries and must be considered to be a major step in man's technological 
advance. In the last five decades great advances have been made both in the chemical 
aspects of flotation and equipment development and today flotation can be used to 
achieve specific separations from complex ores and was initially developed to treat the 
sulfides of copper, lead and zinc. However the field of flotation has now extended to 
include non-sulfide metallic minerals, industrial minerals and fine coal (Wills 1992).
The history of flotation development has seen several different flotation methods but 
froth flotation has been the most successful and dominant. Exploitation of differences in 
surface reactivities of different minerals is the basis of concentration by the flotation 
process. The finely ground ore is suspended in water in a flotation cell by agitation, while 
reagents are added to enhance differences in reactivities of minerals. Air as fine bubbles is 
introduced through the suspension, the bubbles attaching onto selected particles and 
raising them to the surface of the pulp where they form a froth layer. The unselected 
particles stay in suspension and finally pass out of the cell as tailings.
The theory of flotation is complex and has been reviewed comprehensively by a number 
of authors. Sutherland and Wark (1955), Gaudin (1957), Klassen and Mokrousov 
(1963), Apian (1980), and King (1982) have contributed invaluably towards the 
understanding of the process. The basic aspects, such as flotation kinetics, flotation
1
Chapter 1: Introduction Flotation as An Interactive System
chemistry and flotation modelling, have been the subjects of many works in recent years 
(For example, Arbiter and Harris 1962a, Lynch et al 1981, Leja 1982, Barbery 1984, 
Crozier 1984, Inoue 1984, Dowling et al., 1985, Harris 1986, and Fuerstenau 1988).
1. 1 Flotation as An Interactive System
Many complicated physical and chemical interactions are involved in the flotation 
process. Therefore, to obtain useful correlations, it is necessary to consider the flotation 
process as an interactive engineering system. A combination of effects related to three 
factors (chemistry, equipment, and operation) determines the flotation system's response. 
A change in any one of these factors will result in a change in the overall flotation 
response (Klimpel et al., 1986).
1. 1 .1 Chemical Factors
The interaction of chemical reagents with the mineral particles' and air bubbles' surfaces 
to achieve selectivity and stability is the main intent in flotation. Kelly and Spottiswood 
(1982) have classified the most frequently used reagents in flotation into four groups: 
collectors, frothers, activators and depressants. The desired metallurgical performance 
can be achieved by manipulating the dosage of each class of reagents.
Collectors are the most important of the flotation reagents. They are usually organic 
molecules or ions which selectively adsorb on mineral surfaces to render them 
hydrophobic. The adsorbed collector layer enables the selected mineral surface to form a 
thermodynamically stable air-water-mineral interface and ensures the occurrence of 
attachment between the conditioned mineral surface and air bubble on bubble-particle 
collision.
2
Chapter 1: Introduction Flotation as An Interactive System
Frothers control the characteristics of the froth. They are water-soluble organic reagents 
which are adsorbed at the air-water interface to render bubbles the ability to form a 
stable froth. The froth should be stable enough to prevent partial froth breakage during 
the froth removal process. Frothers also give rise to fine bubbles in the pulp, enhancing 
collection.
Activators are generally inorganic substances which increase selectivity by enhancing 
collector adsorption on a certain mineral surface, thus rendering it more hydrophobic. 
Depressants check or prevent collector adsorption by a certain mineral and thus also 
provide better selectivity. They include both organic and inorganic substances.
1. 1 .2 Equipment Factors
All factors in this class can be related to flotation cell design and configuration of the 
cells in a bank. Unlike the other factors, the equipment factors cannot be optimised by 
operators as easily as other factors and any change in these factors will be costly and 
time consuming.
1. 1.2.1 Flotation Machines
Many machines have been developed for flotation (Table 1.1). All seek to establish a 
large air-water interface at which hydrophobic particles become concentrated while 
hydrophilic particles are retained in the pulp phase. Conventional (mechanical) cells 
(Figure 1.1) concentrate floatable particles by vigorous mixing of a slurry into which air 
is dispersed as fine bubbles, but recent advances have seen the introduction of flotation 
columns, in which bubbles rise through a tall column of quiescent pulp, and even units 
(such as the Jameson cell (Jameson 1988)) in which a froth is first formed by intense
Chapter 1: Introduction Flotation as An Interactive System
mixing of air and feed pulp, followed by the separation of non-floatable particles away 
from the froth in a separate stage.
In spite of different designs, the mechanically agitated cell has remained the most 
common unit for coal and minerals flotation.
Table 1 1 Major types of flotation machines (Kelly and Spottiswood 1982).
Machine
Mechanical Pneumatic
Size (m3) Impeller 
diameter (mm)
Capacity*
(m3/h)
Agitair 0.01-28.3 114-1016 1.3-3800
Denver 0.01-36.1 114-838
Krupp 0.5-5 381-719 3.2-21.6
Outokumpu 2.7-16 378-759 36-1540
Sala 2.7-10,7 771 12.7-38.1
Wemco-Fagregren 0.03-12 89-660 0.2-106
Maxwell 4.2-56.6 457-1070 32-422
Nagahm 0.05-8.1 102-832 0.01-3.4
Cyclo-cell 6.8-22.6 None
Davcra 5.4-34 None
Flotaire 12.0-42 None
Column 2.5-40.1 None 14-230
* Based on ore having a density of 2700 kg/m3 in a 25% solids slurry under normal conditions.
4
Chapter 1: Introduction Flotation as An Interactive System
Concentrate
Stator
Impeller
Tailings
Figure 1.1 Schematic illustration of a mechanical flotation cell.
The result of flotation is partly determined by the flotation machine characteristics but 
the selection of a particular type of flotation machine for an application on the basis of 
the machine characteristics is not a straightforward assignment. The main factors to be 
considered in this regard are (Kelly and Spottiswood 1982): metallurgical performance, 
as represented by grade and recovery, capacity, operating costs per tonne of feed and 
ease of operation.
5
Chapter 1: Introduction Flotation as An Interactive System
1.1.2.2 Flotation Circuits
Numerous circuit designs incorporating multi-stage flotation with different degrees of 
complexity have traditionally been in use to achieve maximum grade and recovery in ore 
flotation. Certain basic types of circuit configurations can be recognised based on the 
flotation behaviour of the ores. These basic types can be combined to develop the 
specific flowsheet configurations to meet the needs of the operation.
In coal flotation, the more usual approach has been that of a single stage or 'rougher- 
only' circuit (Arnold and Apian 1986). However, some complicated circuits have also 
been proposed to produce a low ash concentrate or to reject morepyritic sulfiir. Firth et 
al. (1979) have shown that the use of stage-wise reagent addition can increase the yield 
of a poorly floating coal.
Regardless of circuit design, a flotation circuit consists of several banks of flotation cells 
arranged in series and/or parallel, whereas a single bank has flotation cells essentially 
arranged in a series configuration. The cells composing a bank can be totally separated 
from each other (multi-compartment type), or can be a long trough with some agitators 
in it with no baffling or partial baffling between them. The first type is called 'cell-to-cell 
design' in which the stream of pulp is from cell to cell. The second one is called an 'open- 
flow-trough' in which the stream of pulp transfers through the bank with some short- 
circuiting and back-mixing.
6
Chapter 1: Introduction Coal Flotation
1.1.3 Operational Factors
These factors can be changed and optimised by the plant operator to get the desired 
output. Feed rate, particle size, pulp density, air flow and froth removal rate are the most 
important operational factors which can affect flotation results.
1.2 Coal Flotation
Coal flotation was first used in the United States in 1918 to beneficiate anthracite coals 
and had been in use for bituminous coals only rarely until about 1950 (Matoney et al., 
1988). Currently flotation is used to clean the fine coal fraction (usually -0.5mm). The 
principle of coal flotation is similar to that of minerals. However, an essential difference 
between ore and coal flotation is that for ores the entire tonnage has to be ground to 
flotation feed size but for coal comminution is not usually required. Some amount of fine 
coal is originally introduced to the coal washery and some fine coal is produced during 
different stages of coal beneficiation.
Due to the high cost of the flotation process and subsequent dewatering cost on one 
hand and low price of products on the another hand, flotation has not been widely 
practiced in coal beneficiation in the past. Nowadays the price for clean coal is high and 
modem mining methods produce more fines. In some cases more than 25% of the run- 
of-mine coal falls into the fine-sized fraction and has to be considered as a valuable 
material to be treated. Hence, economic considerations, development of modem mining 
methods and environmental issues have greatly enhanced the desirability to clean fine 
coal.
7
Chapter 1: Introduction Coal Flotation
1. 2 .1 Chemical and Petrographic Composition of Coal
Coal is a combustible sedimentary rock resulting from the degradation and alteration of 
vegetable matter under elevated temperatures and pressures over a geological age. Coals 
vary in chemical composition from one to another. Due to their extraordinarily complex 
carbon chemistry, coals cannot be presented by a uniquely defined chemical structure. 
However it is accepted that coals are chiefly composed of C, H, O, S and N plus 
associated mineral matter.
The combustible matter in coal has been shown to be composed of different constituents 
called macerals. Different macerals are known to behave differently in the flotation 
process (Given 1975). The old classification of vitrain, clarain, durain and fusain, which 
is based on petrographic composition, has given way to a more complicated grouping 
based on the reflectance of these petrographic constituents (Apian 1976).
Coal is characterised by its rank and type. The nature of the original plant material 
determines the type of coal, while the degree of diagenesis determines the rank. The rank 
of coal is the stage reached by a coal during its coalification.
1. 2 .2 Impurities Associated With Coal
Run-of-mine coal is accompanied by ash forming components which can be described in 
two classes. Inherent mineral inclusions in the coal seam, which include the mineral 
matter associated with the coal from the coalification period, are in the first class. In the 
second class are included shale, clay, sandstone, or other forms of rock, adulterating the 
raw coal from adjoining strata.
8
Chapter 1: Introduction Coal Flotation
Among all impurities associated with coal the most crucial ones are those containing 
sulfur. Sulfur is present as an impurity in coal in four forms: elemental, sulphate, organic 
and pyritic sulfur. Since the sulphate and elemental sulfur contents are usually low and 
the organic sulfur is intimately associated with coal, basically only pyritic sulfur may be 
removed by flotation or other physical separation methods (Apian 1976).
1. 2 .3 Natural Floatability of Coal
Some minerals, such as graphite, molybdenite, sulfur and talc, are naturally hydrophobic, 
because their surfaces are non-polar. The same is true for coal. But the majority of 
minerals tend to be hydrophilic due to the main bonds holding the solid together being 
ionic or covalent, so that their surfaces are highly polar (Fuerstenau and Raghavan 
1977). Hydrophilic minerals can be made hydrophobic by a reaction (i.e. adsorption) 
with a collector. The reaction occurs only selectively with certain minerals.
The natural floatability of coal differs from coal to coal. Usually high rank coal shows a 
better floatability but as the rank of coal decreases it becomes more and more difficult to 
be floated. Brady and Gauger (1940) evaluated the natural floatability of coal by using 
the value of the contact angle. The extent to which water is displaced from the surface of 
a solid is measured in terms of contact angle, and the latter is used to quantify 
floatability; a greater contact angle means better floatability (Figure 1.2). Different 
parameters determine the value of contact angle and the most important one is carbon 
content. Figure 1.3 shows the variation of contact angle with carbon content ot coal 
Brown (1962) observed the maximum contact angle at 89% carbon, up to this value the 
coal becomes more hydrophobic and floatable and above 89% carbon the hydrophobicity 
decreases slightly.
Chapter 1: Introduction Coal Flotation
Water
Water
System 1 System 2
02 >01: The solid *n system 2 is more hydrophobic
Figure 1.2 Concept of contact angle between bubble and particle.
Carbon content of coal (%)
Figure 1.3 Variation of contact angle with carbon content of coal (Brown 1962).
10
Chapter 1: Introduction Scaling-Up from Batch Tests to Plant Scale
1. 3 Flotation Flowsheet Development
The aim of flotation flowsheet development is to determine the number, arrangement and 
hydrodynamic characteristics of industrial scale flotation machines which will yield, 
economically, the same metallurgical output (recovery and grade of concentrate) as that 
obtained in a laboratory experiment. Besides the kinetic data, the optimised flotation 
time, which gives every desired particle the chance to float, plays a main role in flotation 
flowsheet design. The optimum flotation time for maximum economic recovery is 
determined from laboratory tests. Agar et al. (1980) have identified three criteria for 
specifying the optimum flotation separation time and various simulations have been 
presented by researchers in this regard (For example, King 1976 and van Orden 1986), 
but most of them do not include the specific problem of scale-up from laboratory to plant 
data. It may be surprising that no accepted method of determining scale-up factors exists 
and in some cases a time multiplier method is used and a reference time is the time taken 
to obtain a given recovery at laboratory scale; the reference time is multiplied by a factor 
between two and three to obtain the industrial flotation time (Barbery et al., 1986).
1. 4 Scaling-Up from Laboratory Batch Tests to Plant Scale
Much attention has recently been focused on the scale-up from laboratory batch tests to 
continuous commercial-scale operation. For scale-up of laboratory tests to commercial 
scale, flotation rate and residence time distributions arethe key parameters to predict the 
performance of the flotation process (Degner 1986, van Orden 1986). These two 
parameters will be discussed in more detail in Chapters 2 and 3, respectively, and a brief 
introduction is presented in the following sections as a background to the forthcoming 
discussion on these subjects.
11
Chapter 1: Introduction Scaling-Up from Batch Tests to Plant Scale
1. 4 .1 Flotation Rate
Flotation is often assumed to be analogous to a chemical reaction where reactant A is 
converted into product B at some rate K. Under steady state operating conditions the 
action which takes place in a flotation cell can be considered similar to a chemical 
reaction, a general rate equation describing the operation may be formulated as:
dC_
dt
- -KCn 1.1
where C is the concentration of floatable component in the pulp. Time is represented by 
t, K is the flotation rate constant and n is the order of reaction. Equation 1.1 adequately 
describes the flotation process only if the mineralised froth is transferred from the pulp to 
the concentrate launder and the amount of mineral which returns to the pulp body in the 
cell from the froth layer is negligible.
A simple approach to determine the flotation rate constant is to use cumulative recovery 
data and plot the percent of material remaining in the cell versus time on a semi-log 
graph (Bushel 1962, Rastogi and Apian 1985). For the first few froth increments, this is 
generally a straight line, indicating a first-order process (Figure 1.4). The first-order rate 
constant, K, is determined from the slope of the line. Typically, however, the curve levels 
off after much of the valuable constituent has been floated.
12
Chapter 1: Introduction Scaling-Up from Batch Tests to Plant Scale
Time (nun)
Figure 1.4 Rate plot of flotation data.
The order of reaction in the flotation process has been argued since the beginning of its 
application. Second and higher-order equations have been proposed. Modifications to 
the first-order equations have been made to enable them to differentiate between slow- 
floating and fast-floating particles or fine and coarse particles. A comprehensive 
discussion of this matter is presented in Chapter 2.
1.4.2 Residence Time Distribution in Flotation Process
After obtaining the flotation rate, the determination of the time available for different 
particles (by size or by relative density) to stay in the flotation cell is necessary for scale- 
up. This may be approximated simply by using the cell volume l and flow rate Q (mean
13
Chapter 1: Introduction Objective and Scope of the Investigation
residence time, x - V/Q). However a better approach is to determine the residence time 
distribution (RTD) of a particular material by a systematic investigation. The RTD 
describes the time for all fractions of a material as they pass through a reactor. Much 
information exists (For example, Barbery et al., 1986, van Orden 1986, and Bourassa et 
al., 1988) but still more is needed to use the measured RTDs in scale-up and design of 
flotation circuits.
Both the solid and fluid residence times in a cell can be measured using various tracer 
techniques. Generally, the solid residence time would be best measured by irradiating or 
otherwise tagging a sample of the feed material or by using other materials as tracers 
with characteristics similar to the feed solids. Water residence time is readily measured 
using a soluble tracer.
A flotation cell is more complex than chemical reactors because both tailings and 
concentrate streams exist, i.e. one input and two outputs. Woodburn et al. (1971) and 
Gardner et al. (1985) have shown that the effect of the concentrate's RTD on the pulp 
body's RTD is minimal, so that measuring the tailings RTD reasonably approximates that 
of the cell. This assumption has been used in most RTD studies (For example, van Orden 
1986 and Klimpel 1987).
1. 5 Objective and Scope of the Investigation
The observed flotation rate in batch tests cannot predict the continuous flotation result 
reasonably well and scale-up from batch to plant scale flotation involves some problems. 
Limited information is available on the analysis of the mechanisms of scale-up but it is 
believed that by using RTD data in conjunction with rate data the actual results can be 
better approximated. Therefore systematic research towards the understanding of the
14
Chapter 1: Introduction Objective and Scope of the Investigation
problem and modelling the continuous flotation process using both batch flotation 
kinetics and RTD data are the primary objectives of this study. An experimental study of 
the chemical and operational factors affecting the flotation rate, RTD and recovery was 
considered necessary to get an optimised condition to verify different models. The scope 
of the investigation is summarised as follows:
1. Carrying out laboratory batch flotation testwork to study and optimise the major 
factors affecting the flotation rate and obtaining rate data under the optimised conditions.
2. Carrying out a series of RTD measurements on a pilot scale continuous flotation cell 
to study and optimise the major factors affecting the residence time of particles within a 
cell and generate RTD data under the optimised conditions.
3. Modelling of a continuous flotation process using both the batch flotation rate data 
and the pilot scale continuous flotation RTD data for coal particles.
4. Carrying out a series of recovery measurements for pilot scale continuous flotation 
operation to study and optimise the major factors affecting the recovery of coal in a 
continuous flotation process.
5. Comparing observed recovery data with data predicted by the recovery models, from 
step 3, to select more reliable models and make necessary modifications on them.
15
Chapter l: Introduction Arrangement of the Thesis
1. 6 Arrangement of the Thesis
After the introductory first chapter, the remaining six chapters are arranged as follows:
In Chapter 2, a review of different approaches to flotation modelling are presented. 
Existing rate models are described and their short-comings are discussed. Methods of 
rate constant determination and relevant considerations are mentioned and a detailed 
study on the method of model evaluation is presented. A brief discussion on the effect of 
chemical and operational variables on flotation rate is also presented.
In Chapter 3, some discussions about residence time distributions and their measurement 
in the flotation process are made and some considerations in the selection of tracers are 
pointed out. A comprehensive review of research works in the RTD field is detailed 
while outlining the different approaches towards RTD measurement. After reviewing the 
various approaches, a systematic approach is outlined for RTD data collection and 
comparing the resulting RTDs for each test.
In Chapter 4, a literature review about recovery prediction in flotation is presented. 
Some existing recovery models are discussed and some methods with better reliability 
are proposed.
Chapter 5 deals with the laboratory set-up, test arrangements, experimental procedures 
and analysing the results. Testing programs, experimental equipment and properties of 
coal tested are discussed in this chapter. Also in this chapter, an investigation is described 
on the effect of chemical and operational factors on the rate constant, RTDs of particles 
and recovery of the flotation process and the most influential factors are outlined. Data 
collected for testing the models are presented in this chapter.
16
Chapter 1: Introduction Arrangement of the Thesis
Iii Chapter 6, the experimental data from Chapter 5 are utilised. Using Matlab 
programming (a commercial software package), the data obtained for K and RTD are 
used in different models to compute (predict) recoveries for different conditions. 
Comparisons of the predictedrecoveries with real recoveries are made, the most 
promising models are selected and necessary modifications to them are proposed.
Finally, in Chapter 7, conclusions are drawn and recommendations are made for further 
investigation.
17
CHAPTER 2
KINETICS OF FLOTATION
2.1 Introduction and Background
Flotation kinetics is the study of variation in the quality and quantity of froth overflow 
product with flotation time and the quantitative identification of rate controlling 
variables. It has been the subject of many investigations for several decades. Flotation of 
hydrophobic particles is accepted as a rate phenomenon and it is believed that the rate of 
the process depends mainly on the physico-chemical properties of particles such as their 
size, surface composition, specific gravity, and shape.
2.1.1 Modelling of the Flotation Process
Modelling of flotation is a basic requirement for quantitative understanding of the 
process. The study of flotation kinetics can lead to the development of rate models which 
are useful in evaluating the performance of a flotation system Most of the kinetic models 
for flotation are based on the analogy of a chemical reaction. Collision of hydrophobic 
particles and air bubbles in a flotation cell is assumed to be analogous to collision of 
molecules in a chemical reactor (Lynch et al., 1981).
Rate models can be developed either by the analysis of the mechanism of flotation 
process or by direct fitting of equations to recovery-time data. Arbiter and Harris 
(1962a) classified the methods for obtaining flotation rate equations into four categories:
1) Empirical: Curve fitting to recovery-time data, which is the most common method.
Chapter 2: Kinetics of Flotation Introduction and Background
2) Semi-empirical: Solving a probable differential equation with appropriate boundary 
conditions.
3) Analogue: Application of equations developed for chemical reactions, with or without 
modifications.
4) Analytic: Application of hydrodynamics principles to a particle-bubble encounter 
hypothesis to obtain equations for the probability and frequency of encounters, particle 
capture and solids mass removal rate from the cell.
In laboratory scale studies of the flotation process, test data describing the kinetics of the 
process can be obtained by conducting the tests either in a conventional batch mode or 
by continuous flotation operation. But, for the development of a reliable rate model, the 
investigator has to decide the testing mode on the basis of the individual approach 
adopted towards modelling and hence assess the relative merits and demerits of each 
testing procedure. Some researchers have found batch flotation more efficient than 
continuous flotation (For example, Tewari and Biswas 1969, Harris and Chakravarti 
1970, and Ball and Fuerstenau 1970). In contrast, others considered batch flotation 
testing as a process incapable of providing satisfactory kinetics data concerning the 
change in mineral concentration inside the cell with time (For example, Brown and Smith 
1954, Jowett and Ghosh 1966, and Bourassa et al., 1988)
Small-scale techniques have also been proposed for flotation experiments. However, the 
majority of researchers have preferred conventional batch and continuous, rather than 
small scale, methods in their work. Small scale techniques include beaker tests, the 
Hallimond tube, small 50-gram flotation cell and microfloat agitator (Fichera 1990).
19
Chapter 2: Kinetics of Flotation Introduction and Background
2. 1.1-1 Batch Approach
Kinetic models for the flotation process have been under investigation for many years 
and much of the experimental data have been obtained from the investigation of the 
process in a single batch cell. Laboratory batch testing for flotation circuit design and 
model development has been carried out on a routine basis in a large number of 
laboratories throughout the world. Batch tests have been accepted as the most efficient 
and practical device for flotation process development. Usually a sample, assumed to be 
representative of the material to be processed in a plant, is ground and diluted to obtain 
the required solid concentration. The resulting slurry is transferred into a laboratory 
batch flotation machine, having a volume which can vary from 0.5-8 litres (Barbery et al., 
1986).
The froth products, obtained at various times, are collected to assess the time response 
of the material in the laboratory machine. The tests are performed with different 
combinations of variables. The froth products, and the solids remaining in the cell are 
filtered, dried, and analysed for the various components of interest.
2. 1.1.2 Continuous Approach
Those who used continuous flotation tests to obtain kinetics data are in a minority. The 
method was first introduced by Schuhmann (1942). Brown and Smith (1954) criticised 
batch testing and showed that batch testing can lead to faulty conclusions. They believed 
that a continuous flotation process quickly reaches a steady state in which the variables 
to be measured do not change with time and, therefore, can be easily measured and 
correctly correlated. Bull (1965) also claimed that continuous flotation procedures can 
lead to steady state operation very rapidly and hence give more reliable results. He
20
Chapter 2: Kinetics of Flotation Introduction and Background
carried out all his tests using a continuous flotation cell. Small-scale continuous flotation 
tests were performed by Jowett and Ghosh (1966) in a single cell to determine the 
optimum operating conditions and kinetic correlations of the flotation process for n cells 
in series. Bourassa et al. (1988) earned out a pilot plant study to compare the kinetic 
behaviour observed in the pilot plant to that of the laboratory batch tests, to calibrate 
pilot plant kinetic data with laboratory results.
In a continuous flotation test the required amounts of material and water are mixed 
together in the feed tank for some time, then reagents are added and a further 
conditioning time is allowed. The conditioned pulp is then fed into the flotation cell. 
When pulp volume in the cell reaches a required level, the impeller, with ah inlet open, 
and the froth remover are both switched on. Over predetermined intervals, successive 
tailings and concentrate samples are collected. After reaching steady state condition the 
recovery of the process is calculated. Having recovery of the process (R) and mean 
residence tune (t), and assuming perfect mixing condition in the cell, the flotation rate 
(K) is obtained from the following equation (Bull 1965):
t-Rt
2.1.1.3 Batch Tests and Their Limitations
Inherent system transients make batch testing unreliable. As a batch test pioceeds, the 
volume of the pulp in the cell decreases because of froth lemoval. Addition of puie watei 
or frother-in-water solution to the cell, either at irregular intervals oi at the same iate as 
it is being removed, may keep the volume of the cell steady to some extent but density 
and frothing properties of the pulp will still vary with time. As the batch test advances,
21
Chapter 2: Kinetics of Flotation Introduction and Background
the size distribution of particles in the pulp body also changes. Fine and high-grade 
particles of a mineral generally float before coarse or low grade particles of the same 
mineral, leaving behind an increased percentage of the latter fractions for later stages of 
flotation (Brown and Smith 1954).
One of the other difficulties associated with batch testing techniques is the poor 
reproducibility of results. The application of instrumented batch flotation cells decreases 
the variability in the results but the results of automated batch tests are still subject to 
significant variability. Bazin et al. (1995) concluded from a statistical study of the 
differences between replicate tests that the principal source oferror was the unavoidably 
stochastic nature of the flotation process. These differences were due to changes in pulp 
volume, reagent concentration, pulp density and the size distribution of the particles and 
air bubbles in the cell during operation. According to their results, the preparation and 
analysis of flotation samples and operator error are only responsible for a small portion 
of the observed variability of the flotation performances.
It is not within the scope of this work to evaluate the relative merits of batch testing and 
continuous testing and there is much argument on this point elsewhere (For example, 
Jowett and Ghosh 1966). However, as a matter of fact, it should be mentioned that the 
batch tests do not adequately simulate continuous flotation circuits. Hydrodynamic 
characteristics of the industrial flotation machines are not scaled down properly tor 
existing laboratory batch machines and hence batch tests are unsuitable as the principal 
means of deciding the flowsheet design. In spite of this deficiency, batch testing is 
unavoidable and the mineral processing industry depends heavily on it to obtain model 
parameters, design flowsheets and evaluate reagents.
22
Chapter 2: Kinetics of Flotation Introduction and Background
2.1.2 A Review of Existing Models for Flotation Process
Some researchers have suggested that the flotation process follows a first-order rate 
equation in most cases (For example, Klassen and Mokrousov 1963, Tomlinson and 
Fleming 1963, Harris and Chakravarti 1970, and Dowling et al., 1985), while others 
claim second or higher-order rate (For example, Arbiter 1951, Horst 1958, and Bennett 
et al., 1958). The main discussion has been between first and second-order, but some 
works reported orders which varied from one to six in their experiments (For example, 
Bogdanov et al., 1964). Although the argument regarding the order of the process has 
never been resolved and a particular model has not been accepted by all flotation 
researchers, most authors agree with the simple first-order model for the flotation 
process. A survey by Inoue (1984) showed the use of the first-order equation in the 
majority of cases.
First-order kinetics models are widely accepted in literature. They are convenient and 
simple and have been used satisfactorily for decades. Among the first-order rate 
equations, those in which the rate constant is not necessarily single-valued are generally 
preferred. In this form the rate constant varies for different floatable components 
(Imaizumi and Inoue 1963).
A comprehensive review of works on flotation kinetics published up to 1961 was 
provided by Arbiter and Harris (1962a) and a more recent extensive review is that of 
Ahmed and Jameson (1989). In the present work the order of process is not the matter 
of discussion but some selected rate equations from first and second-order classes are 
reviewed and considered to evaluate tests results.
23
Chapter 2: Kinetics of Flotation Introduction and Background
Zuniga (1935) was the first who found that his experimental data could be fitted by an 
equation of the form:
where C is the concentration of the mineral in the cell at time t, C0 is the initial 
concentration of mineral, and AT is a flotation rate constant. The equation was derived by 
the integration of:
which is analogous to the simplest form of a first-order chemical reaction and shows that 
flotation rate is proportional to the amount of floatable mineral in the cell.
Following Zuniga’s model, numerous other investigations were carried out to produce 
new data which were found to be fitted by equations based on simple first-order models. 
Schuhmann (1942) accepted a first-order relationship for the flotation process and 
defined K as: K = Pc.Pa.F, where Pc and Pa are the probabilities of collision and adhesion 
respectively between particle and bubble and F is the froth stability factor.
In the early works, only one rate constant was assumed for the flotation process, but 
workers subsequently introduced the concept of different K values for different 
components of a feed. Sutherland (1948) also accepted the first-order rate equation for 
flotation and proposed that the total flotation rate is the sum of the rates of each size 
class of particles. He presented his model as:
C = C0e~Kt (2.2)
(2.3)
n
(2.4)
24
Chapter 2: Kinetics of Flotation Introduction and Background
where /' (=1, 2,.......n) represents the size classes.
Also some researchers introduced the concept of “floatable fraction”, which attempts to 
quantify the fraction of the mineral present in the feed which can be recovered by 
flotation. Gaudin (1957) pointed out that the rate constant varies over the flotation 
period and that particles with better floatability would float early. The least floatable 
particles would continue to be recovered for a longer period and consequently be a large 
part of the product recovered towards the end. Hence the concept of a distributed 
floatability model developed:
m
C = ZC<,he-K>' (2.5)
1
where h (=1, 2,.......m) represents the floatability classes.
Imaizumi and Inoue (1963) developed a double distributed parameter model, which 
describes both particle size and floatability distribution dependence of the rate constant:
n m
C = ZIC„/% (2-6)
1 1
where / and h have the meanings stated earlier.
Kelsall (1961) described particles as having either slow or fast floatability and considered 
two groups of floatable components with high and low flotation rate constants in a cell. 
He proposed the following equation for a batch laboratory cell, where (j) is the fraction of 
the components with low rate constant, Ks, and Kf is the rate constant of fast floating 
components.
25
Chapter 2: Kinetics of Flotation Introduction and Background
C = C0(l - ?t)exp(-A'//) + CJexp(-Kj) (2.7)
Arbiter and Harris (1962a) developed a kinetics model for flotation process which dealt 
with the cell contents in two different phases, the pulp and froth phases. They assumed 
first-order kinetics and perfect mixing conditions for both phases in the cell and hence 
they defined two different rate constants to describe the interaction between the two 
phases: 1) the rate constant for mass transfer from pulp to froth and 2) the rate constant 
for mass transfer from froth to pulp. If the rate constant for the return of solids from 
froth to the pulp is zero or negligible then their equation becomes a well known first- 
order equation. Harris et al. (1963) justified this model experimentally using the steady 
state continuous flotation of coal. Bull (1965) also showed evidence of return of 
floatable material from the froth to the pulp. Several authors (Imaizumi and Inoue, 1963 
and Woodbum and Loveday, 1965) rejected the two-phase model on the basis that the 
return rate is negligible because of the high froth removal rate. They believed that the 
complete experimental verification of the two-phase model is practically impossible. Ball 
and Fuerstenau (1970) acknowledged that a calculable solids return rate from the froth 
phase was present but indicated that it was insignificant compared to the rates of transfer 
of material from pulp to froth.
There are more models in the literature, some of which are discussed in the forthcoming 
sections. A review of rate models selected by different workers to deal with continuous 
flotation tests is presented in Chapter 4.
Chapter 2: Kinetics of Flotation Determination of the Flotation Rate Constant
2. 1 .3 Shortcomings of the Models
Almost all of the models discussed in previous section are useful in describing the 
performance of a single batch scale flotation process but cannot adequately predict the 
behaviour of a continuously operating plant. Absence of a model versatile enough to 
describe quantitatively the behaviour of an operating flotation plant, having a complex 
flow configuration, was emphasisedby King (1972) and Lynch et al. (1981). The 
majority of the models utilise data acquired from batch tests while the behaviour of 
particles and conditions of operation in industrial units differ from those existing in 
laboratory batch cells. The difference is mainly due to the fact that the recovery of a 
mineral in a continuous flotation process is controlled not only by the distribution of rate 
constants of particles, but also by the distribution of their residence times in the cell 
(Lynch et al., 1981). To predict the recovery of a continuous flotation process, a 
knowledge of the residence time distribution (RTD) of the materials flowing through the 
cell is as essential as the kinetics data.
2. 2 Determination of the Flotation Rate Constant from Batch Tests
The flotation rate constant is independent of time and characterises the whole course of 
the mineral separation by flotation. The value of the rate constant is dependent upon all 
variables which control the process, and is changed by significant changes in any of them. 
The simplest approach of determining the flotation rate constant has been explained in 
Section 1.4.1, but it is a common practice to fit the available data to a model using 
computer programs. The programs provide a fit to the data and calculate the model 
parameters for the best fit.
27
Chapter 2: Kinetics of Flotation Determination of the Flotation Rate Constant
Before calculating the rate constant, two important parameters should be defined. The 
amount of'ultimate recovery' and the duration of'lag time' have an important effect on 
calculated rate constant.
2.2.1 Ultimate Recovery
All particles in a cell are not equally floatable and some cannot be floated at all. Hence, 
even after prolonged flotation times under intense flotation conditions some material may 
still remain in the cell. Thus, it may be preferable for the value chosen for Co to refer only 
to that portion of the feed which will eventually float. This idea led to the concept of a 
'terminal concentration', Coo, which can be determined by sequentially attempting to 'float' 
the tailings of the preceding batch until negligible material floats in the last step (Ball and 
Fuerstenau 1970).
Morris (1952) suggested that the maximum percentage recovery obtained in a batch test, 
conducted under conditions simulating the continuous test, may be taken as the total 
percentage of floatable material in the feed. Jowett and Safvi (1960) showed that it is not 
an easy matter to decide how much of the solids in a pulp are floatable, because the 
floatable particles can be recovered from the pulp only as long as they are present in 
quantities which give a good chance of collision and adhesion. To explain their point, 
they floated a coal sample in a batch test, then carefully dried the concentrate collected at 
a low temperature. They repulped the material recovered from froth, then reconditioned 
and refloated it. It was anticipated that all the material would be recovered with froth, 
but in fact about the same percentage recovery as before could be obtained. Even two 
further refloats did not improve the results (Table 2.1). The significant feature of their 
results was that 100 percent recovery was never obtained, even after a very long
28
Chapter 2: Kinetics of Flotation Determination of the Flotation Rate Constant
residence time and several refloats which might be expected to reject all the non-floatable 
material. In the present study the 'ultimate recovery' is a model parameter and can be 
obtained by fitting the best line to the kinetics data .
Table 2.1 Percentage recovery of coal in batch flotation tests on successive refloats.
(Jowett and SafVi 1960)
Test % Recovery (Sample 1) % Recovery (Sample 2) % Recovery (Sample 3)
No. in each test of original in each test of original in each test of original
1 86.8 86.8 85.3 85.3 72.5 72.5
2 85.2 71.5 84.2 71.8 76.8 55.8
3 79.6 56.8 83.5 59.2 74.0 41.3
4 88.4 50.3 76.8 45.0 69.0 28.5
5 - - 82.7 37.0 60.0 17.0
6 - - 67.0 24.9 57.4 9.8
7 - - 65.5 16.3 - -
2. 2 .2 Induction Time or Lag Time
Sutherland (1948) was the first to express the difficulty of establishment of a reference 
zero time in batch testing. He pointed out that the froth begins to form some while after 
switching on the air, so this moment cannot be marked as time zero, indicating the need 
for a correction factor to the time. This time correction is known as the induction time or 
lag time (the term 'lag time' will be used in this presentation to avoid confusion with the 
'induction time' term specifying the time required for a particle to get attached with a gas 
bubble after collision).
After initiating the air flow, several seconds also pass before a full depth of loaded froth 
is present on top of the pulp. The time to establish this equilibrium froth depth is usually 
several seconds. Agar et al. (1980) pointed out the problem and the necessity of adding a
29
Chapter 2: Kinetics of Flotation Determination of the Flotation Rate Constant
further (negative) correction to the measured time and proposed a modification to the 
first-order flotation rate model to take the total lag time (cp) into account as follows:
R = ^{l-exp[-A:(f+ <p)]} (2.8)
Rastogi and Apian (1985) also considered the induction tune and showed that it varies 
with test conditions, and tests with higher flotation rates give insignificant lag time. Agar 
(1985) emphasised the relatively insignificant influence of the time correction factor cp 
and suggested that the inclusion of a time correction factor has little impact on the 
calculated results.
In this study the lag time correction is not applied, because the flotation rate is high and 
flotation is actually started before the opening of air flow as some very hydrophobic 
solids float during the conditioning period and cause a positive correction which 
effectively neutralises the effect of lag time.
2.2.3 Data Fitting
To determine the ultimate recovery, rate constant and other model parameters, the 
Datafit1M program (version 1.1c) was used in this study. The fundamental approach of 
parameter estimation in this program is the minimisation of the sum of squares of 
differences between the observed and calculated recoveries at a given time.
Estimated parameters by different models vary widely in value and in this regard the 
results should be assessed to ascertain how well the models give meaningful predictions 
of the parameters.
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Chapter 2: Kinetics of Flotation Evaluation of Rate Models
2. 3 Evaluation of Rate Models
Different mathematical flotation kinetics models have been introduced, as described 
above, but most of them explain the process under a certain set of conditions and no 
single model can represent the flotation process under all applicable conditions. Hence, it 
is necessary to evaluate the models for their reliability in describing a certain flotation 
system. Dowling et al. (1986) reported flotation tests with a copper ore using various 
collectors and brothers to produce different time-recovery data. They used the data to 
evaluate thirteen rate models from literature. Table 2.2 presents the title and 
mathematical form of each model. Goodness-of-fit and confidence limits for the models' 
parameters were chosen as criteria to evaluate the models. Although all the models 
tested gave reasonably good fits to the experimental data, large differences existed in the 
confidence limits of the parameters from model to model and hence some models clearly 
appeared to be better than others.
Dowling’s results showed the flotation of the copper ore under the test conditions was a 
first-order process and the best overall agreement between the observations and 
estimations was obtained for the first-order model with a rectangular distribution of 
particle floatabilities (Model 2). Model 1 which is a classical first-order model stood the 
second best.