cell geometries
Carolina Vivian de Souza
Natural Resources Engineering, master's level (120 credits) 2020
Luleå University of Technology
Department of Civil, Environmental and Natural Resources Engineering
Lab experiments using different flotation cell geometries
by:
Carolina Vivian de Souza
Division of Minerals and Metallurgical Engineering (MiMer) Department of Civil, Environmental and Natural Resources Engineering Luleå University of Technology
Supervisor:
Vitalis Chipakwe Examiner:
Saeed Chehreh Chelgani
Luleå, Sweden 2020
Abstract
Due to the increasing demand for processing low-grade ores, larger volumes of material are being processed. Therefore, the size of flotation equipment has significantly increased for the past decades. The studies related to scale-up are and will remain to be crucial in terms of designing larger flotation equipment. One of the most important factors for flotation scaling- up is the “flotation rate constant”. Hence, the main aim of this investigation was to understand the scale-up criteria when the size of different laboratory-scale cells increases, using the Outotec GTK LabCell®. This was done by assessing the influence of impeller speed, as a hydrodynamic variable, on the flotation performance. Recovery was found to increase with an increase in the cell area to rotor diameter ratio. Flotation rate and recovery increased with an increase in the impeller speed until a certain point that it eventually decreased for the 2 l and 7.5 l cells. For the 4 l cell, the flotation rate and recovery decreased with increasing the impeller speed. The impeller speed of 1200 rpm allowed a successful scale-up based on the flotation rate constants and recovery when increasing the size of the cells. Maintaining the impeller speeds constant at 1300 rpm increased the flotation rate constants and recovery when increasing the cell size from both the 2 and 4 l cells to the 7.5 l cell. A further increase in the impeller speed to 1400 rpm also produced the flotation rate constants and recovery to increase as the cell size increased from both the 2 and 4 l cells to the 7.5 l cell. However, when increasing the cell size from 2 l to 4 l, good results were also observed for all impeller speeds. The products concentrate seem to become finer when decreasing the cell size, with only a few exceptions. The recovery of particles larger than 38 μm was found to differ considerably less among the different scales.
Keywords: cell hydrodynamics; flotation; impeller speed; scale-up; mechanical laboratory- scale flotation cells; flotation kinetic rate
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Content
Introduction ... 1
1.2 Aim and Objectives ... 2
Literature Survey ... 4
2.1 Flotation ... 4
2.2 Flotation equipment ... 8
2.2.1 Pneumatic cells ... 9
2.2.2 Mechanical cells ... 11
2.2.3 Laboratory flotation equipment ... 13
2.3 Scale-up of flotation process ... 17
2.3.1 Computational Fluid Dynamics (CFD) ... 19
2.3.2 Kinetic scale-up ... 22
2.3.3 Machine design scale-up ... 28
2.4 Influence of the impeller speed and design in flotation ... 31
2.5 Flotation of silicates ... 35
2.5.1 Commonly used reagents ... 35
Materials and Methodology ... 38
3.1 Flotation equipment ... 39
3.2 Sampling and sample preparation ... 41
3.2.1 Grinding and Sieving ... 41
3.2.2 Flotation reagents ... 43
3.3 Flotation tests ... 44
Results... 45
4.1 Recovery assessments ... 45
4.2 Kinetic assessments ... 50
4.3 Effect of Cell Size ... 52
4.4 Effect of Particle Size Distribution ... 53
4.5 Effect of impeller speed ... 58
Discussions and Conclusions ... 60
5.1 Effect of Cell Size ... 60
5.2 Effect of impeller speed ... 61
5.3 Effect of Particle Size Distribution ... 63
5.4 Conclusions ... 67
EIT Chapter ... 70
6.1 Recommendations for future work ... 70
6.2 SWOT Analysis ... 71
References ... 72
Appendices ... 76 ii
List of figures
Figure 1 Process of adsorption of a collector in a mineral surface and attachment to the air bubble
(extracted from Gupta and Yan, 2006). ... 7
Figure 2 Classification of collectors (extracted from Napier-Munn and Wills, 2005). ... 7
Figure 3 Schematic of a Jameson cell (extracted from Silva, 2005). ... 9
Figure 4 Schematic pneumatic cell model Imhoflot (extracted from Chaves, 2006). ... 10
Figure 5 Schematic of a column flotation (extracted from Mesa and Brito-Parada, 2019a). ... 10
Figure 6 Schematic of a mechanical flotation cell (extracted from Mesa and Brito-Parada, 2019). ... 11
Figure 7 Typical flow patterns in a mechanical flotation cell (Outokumpu) (extracted from Gorain et al., 1995). ... 12
Figure 8 Flotation cell designs (extracted from Chaves, 2006). ... 12
Figure 9 Impeller-stator designs. (extracted from (Chaves, 2006). ... 13
Figure 10 Schematic of a Hallimond tube (extracted from Wills & Napier-Munn, 2006). ... 14
Figure 11 Schematic of a laboratory bench-scale mechanical flotation equipment (extracted from Mesa and Brito-Parada, 2019). ... 14
Figure 12 Laboratory mechanical flotation cell (extracted from Wills & Napier-Munn, 2006)... 15
Figure 13 Denver Lab Cell D-1 flotation machine (extracted from McGill University, 2020). ... 16
Figure 14 Trend in the flotation tank size over the last century (y-axis is on a logarithmic scale) (extracted from Mesa and Brito-Parada, 2019a). ... 17
Figure 15 Power number for the diverse impellers Reynolds number (extracted from Mesa and Brito- Parada, 2019a). ... 29
Figure 16 Schematic of the methodology. ... 39
Figure 17 Outotec GTK LabCell® flotation machine (extracted from Mattsson et al., 2019)). ... 40
Figure 18 Outotec GTK LabCell® cells, rotor, and impellers used in this investigation. ... 41
Figure 19. Outotec GTK LabCell® cells dimensions for the cells used in this investigation. ... 41
Figure 20. Particle Size Distribution for the initial sample and flotation feed. ... 43
Figure 21 Cumulative recovery reached by the different cells at each impeller speed after 7 minutes of flotation. ... 46
Figure 22. Cumulative recovery over time for the 2-, 4- and 7.5 l cells. ... 46
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Figure 23. Non-cumulative recovery over time for the 2-, 4- and 7.5 l cells. ... 48
Figure 24 Cumulative water recovery for the different impeller speeds after 7 minutes of flotation. .. 49
Figure 25. Non-cumulative water recovery over time for the 2-, 4- and 7.5 l cells. ... 50
Figure 26 Particle size distribution of the flotation product for the 2 litre cell according to the different impeller speeds. ... 55
Figure 27 Particle size distribution of the flotation product for the 4 litre cell according to the different impeller speeds. ... 55
Figure 28 Particle size distribution of the flotation product for the 7.5 litre cell according to the different impeller speeds. ... 56
Figure 29 Cumulative recovery for the impeller speeds of 1200-, 1300-, and 1400 rpm. ... 58
Figure 30 Recovery for the different impeller speeds after 1, 3, 5, and 7 minutes. ... 59
List of tables
Table 1 Silicates major groups (adapted from Agapito Mendes et al., 2018). ... 35Table 2 Manufacturer recommended machine parameters for the different scales (extracted from Outotec, 2018). ... 39
Table 3. Grinding parameters for the Ball mill and Rod mill. ... 42
Table 4. Recoveries for the different concentrations tested using Armeen C as a collector. ... 43
Table 5 Kinetics parameters obtained from two different kinetics models for the 2-, 4-, and 7.5 l cells at the impeller speeds of 1200 rpm, 1300 rpm, and 1400 rpm... 51
Table 6 Ratios among the different cells. ... 52
Table 7 Flotation rate (k) index for the scale-up between different cells and impeller speeds. ... 53
Table 8 Recovery (R) index for the scale-up between different cells and impeller speeds. ... 53
Table 9 d80 for the 2-, 4-, and 7.5 litres cells at the impeller speed of 1200 rpm, 1300 rpm, and 1400 rpm... 53
Table 10 d80 ratios when increasing the cell size. ... 54
Table 11 Kinetics parameters as a function of particle size for the different cells and impeller speeds.56 Table 12 SWOT analysis regarding the project. ... 71
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List of appendices
Appendix 1. Calculations for the predicted recovery according to the flotation rate constant for the first-order kinetic model equation ...76 Appendix 2. Calculations for the predicted recovery according to the flotation rate constant for the second-order kinetic model equation ...77 Appendix 3. Linear regression for calculation of the flotation rate for the different cells and impeller speeds using the first-order model ...78 Appendix 4. Linear regression for calculation of the flotation rate for the different cells and impeller speeds using the second-order model ...81 Appendix 5. Cumulative Recovery of solids as a function of particle size in froth after 1-, 3-, 5-, and 7 minutes ...84 Appendix 6. Calculations for the flotation rate constant estimation rate as a function of particle size for the different cells and impeller speeds using the first-order kinetic model ...85 Appendix 7. Linear regression for calculation of the flotation rate as a function of particle size for the different cells and impeller speeds using the first-order kinetic model ...86
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Preface
The present thesis report is the outcome of the final stage of the master’s program in Resources Engineering – EMerald, which is jointly developed by the following universities: Université de Liège (Belgium), Université de Lorraine (France), Technische Universität Bergakademie Freiberg (Germany) and Luleå Tekniska Universitet (Sweden).
This study was carried out at Luleå Tekniska Universitet. The methodology for developing this investigation is divided into two different parts: sample preparation and flotation tests. The sample preparation involves materials handling, grinding, splitting, and sieving for the olivine material. Flotation tests are performed to investigate the influence of the impeller speed, as a hydrodynamic variable, during the scaling-up in a laboratory-scale. This influence is examined in terms of particle size, recovery, and the flotation rate constant.
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Acknowledgments
I would like to give my utmost gratitude and respect for all those people who provided me assistance, insight, and encouragement, both directly and indirectly, whether they know of their contribution or not.
I would like to record my thank you and appreciation to Professor Saeed Chehreh Chelgani and Vitalis Chipakwe for providing direction and encouragement throughout this work, Tack så mycket!
To all my colleagues, professors, coordinators, friends, and all the people involved in the Emerald Program. Thanks a lot! Special thanks to Carlos, for all the moments we shared together, and for all the memories I will never forget. To Mari, Natália, Antônio, Ervin, Pasindu, Vimbainashe, Anna, Luis, Ali, Neil, and Chelsea for all the chats, understanding, forbearance, and encouragement. Without your support, this work would not have been able to come to completion. It was a pleasure to go on this adventure with you all!
Special gratitude to my family. To my dearest parents, Adriano and Viviane, and sister, Lívia, for their unconditional love and support. To my grandmother, Graça, for being the reason I am still here. Without you, I would not have had the courage to go on and face the challenges I was thrown every day. Muito Obrigada!
I could not finish without thanking all those who have shared all these experiences with me, contributing to all the memories and to make this an unforgettable time. To Fredrik for being my support and safety when I most needed it in Sweden. To Nicole, for being my guide when I could not see the light. To all my friends, in Brazil and all around the world. Thank you very much, guys!
Lastly, to God. I owe this all to You.
Infinite thanks for the memories, trust, and friendship!
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Chapter 1
Introduction
Back in time, the grades of ore deposits were considerably higher, and the ore required fewer and simpler beneficiation operations before being sent to the smelters than nowadays.
Most high-grade ore deposits have reached exhaustion, and technologies involving the beneficiation of lower grade deposits continue to merge. Nowadays, the requirement for treating ores with such low grade and complex mineralization increased the demand for grinding the ore into finer size fractions to meet the liberation degree. Flotation has surged in order to concentrate these fine particles. However, these fine particles are still problematic material in terms of concentration.
One of the main challenges is higher throughput. Because of the increasing demand for processing the low-grade ores, larger volumes of material are being processed. Therefore, the size of flotation equipment has also significantly increased for the past decades. With this, many challenges have surged in terms of equipment performance, design, and operation.
These challenges are commonly associated with problematic pulp hydrodynamics and froth transportation (Mesa and Brito-Parada, 2019a). Thus, the studies related to scale-up are and will remain to be crucial in terms of designing larger flotation machines. For addressing these issues, it is important to well-understand the scaling-up process in flotation. One of the most important factors for flotation scaling-up is the “flotation rate constant”.
The flotation rate constant (k) is the recovery that can be reached through a specific interval of time. It is known to increase with an increase in particle size until reaching its maximum value. After that, the flotation rate decreases associated with a further increase in particle size (Horst, 1952).
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Determining the flotation rate constant has been considered in many investigations using different methods of assessing it. The following first-order equation is usually applied. It was first expressed by H. Garcia Zunica in 1935 (Horst, 1952).
( )
In Equation 1, r is the amount recovered, R is the original amount, t is the flotation time, and k is the flotation constant (Horst, 1952).
The flotation rate is dependent on many flotation factors, such as mineral properties and flotation hydrodynamic variables. Thus, all these factors can directly affect the flotation scaling up. For that, it is important to well-understand the hydrodynamics phenomena behind the scale-up process.
1.2 Aim and Objectives
This investigation, as a comparative study, is going to examine the impeller speed, as a hydrodynamic variable, and its effect on the flotation rate constant during the scaling-up in a laboratory-scale. In the first stage of this investigation, the impeller speed would be varied for different Outotec GTK LabCell® flotation cell sizes. This machine is a mechanical laboratory-scale batch flotation equipment that contains 2, 4, 7.5, and 12 litres plastic cells with its respective OK type rotors, impellers, and froth scrapers. It has only recently been introduced to the market; therefore, not many studies have been conducted regarding its operating conditions. The examined flotation experiments can show the influence of this variable on the flotation rate constant when the size of these cells increases. Assessing this influence would be an important step in defining the required impeller speed for each flotation cell.
This will be achieved through to the fulfillment of the following objective:
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Objective -
To provide a comparative study in order to understand the influence of the impeller speed, as a hydrodynamic variable, during the scaling-up in a laboratory-scale. For that, the impellerspeed is varied for different Outotec GTK flotation cell sizes, in order to assess the influence of the impeller speed when the size of these cells increases. This influence is going to be examined in terms of particle size, recovery, and the flotation rate constant.
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Chapter 2
Literature Survey
2.1 Flotation
Flotation is, in practice, a separation technique capable of separating minerals based on their surface property, which is the hydrophobicity degree. It is the most efficient separation technique for the particle size range between 25 to 100 μm, although it differs according to the mineral being floated (Alves dos Santos and Galery, 2018).
Froth flotation is currently the most common mineral treatment method in mineral beneficiation, due to its technical versatility and cost-viability. It was licensed in 1906 for the concentration of ores, but it can be additionally applied in different industrial sectors, for example, oil sands concentration, ionic flotation, algae separation, paper deinking, plastic reusing and water treatment (Mesa and Brito-Parada, 2019a)
Its principle is based on the surface chemistry of a material, in which hydrophobic mineral particles are separated through attachment to gas bubbles, ascending to create a froth layer, overflowing as the mineral-rich concentrate. The separation process consists of scattering small bubbles of gas, mostly air, in the interior of a flotation tank, which is also called a flotation cell. The flotation cell is filled with a mineral suspension in an aqueous media, to give a pulp. For improving the separation process, chemical reagents that modify surface properties of minerals can also be added in the process, acting as collectors, frothers, or regulators (Agapito Mendes et al., 2018).
The recovery of particles within the concentrate is mainly done though: true flotation, which is the selective attachment of a particle to an air bubble; entrainment of particles;
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and through aggregation, which is a physical entrapment among particles to an air bubble within the froth (Napier-Munn and Wills, 2005).
Selective attachment of particles to bubbles is the main goal through the flotation separation. In the true flotation, the reagents should selectively react with the surface of target minerals. However, the effectiveness of the separation among gangue and valuable minerals relies on the entrainment and entrapment degree, as for this case, valuable minerals and gangue are equally likely to be recovered (Napier-Munn and Wills, 2005). Entrainment is mostly affected by the particle size, as finer particles can easily flow upwards due to its lower gravitational forces (Boeree, 2014). But it can also be affected by pulp density, particle shape, and froth properties like stability, drainage, removal rate, and residence time (Flint, 2001).
According to Wang et al. (2015), coarser particles are more likely to settle at the bottom of the flotation cell, while the fine particles are more likely to be uniformly dispersed in the pulp phase. In a perfect mixing system, a higher number of entrained particles can be observed, as more material is entering the froth.
The particle size also plays an important role in the stability of the froth. The froth zone avoids the direct transport of the pulp to the concentrate defining the quality of the concentrate (grade) and the efficiency of the process. A stable froth increases entrainment as the drainage degree of particles back into the pulp zone is reduced, while an increase in residence time allows a higher degree of particle detachment before the froth flows into the launder (Boeree, 2014).
Typically, lower recoveries are associated with finer and coarser particles. In the case of coarser particles, it has been mostly associated with detachment. This can happen in the froth phase or the interface between pulp and froth. For finer particles, the collision probability is decreased due to the behaviour of particles with smaller masses, which is to follow water streamlines, decreasing recovery. In general, the impact of particle size
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in flotation relies on the hydrophobicity degree, which is highly conditioned by liberation, mineral texture, and reagent adsorption (Alves dos Santos and Galery, 2018).
All minerals are categorized in conformity to their surface properties into polar or non-polar groups. Minerals in the non-polar group have their surface categorized by the moderately weak molecular bonds. These have their covalent molecules adhered together through van der Waals forces, making them hydrophobic because the non- polar surfaces are not easily attached to water dipoles. On the other hand, minerals belonging to the polar group have a stronger covalent or ionic surface bonding. These are naturally hydrophilic because their surfaces easily react with water molecules, creating a stronger bond. These surface properties can be altered through the addition of chemical reagents (Napier-Munn and Wills, 2005).
Mineral separation is connected to the surface selective affinity, mostly modified by the reagents. Collectors are added to the pulp to selectively improve the hydrophobicity of the targeted minerals, which are in some cases, the material of value to be floated (Mesa and Brito-Parada, 2019a). Its molecular structure is characterized by a covalent molecular portion and an ionic portion, turning the collector into a surfactant, a compound with an amphipathic structure (Agapito Mendes et al., 2018).
The molecules of a collector can be either ionising or non-ionising compounds.
Ionising collectors can be cationic or anionic. These can be complex in terms of molecules' structure and are heteropolar. This means these are composed of a charged polar group and an uncharged non-polar group. The non-polar group commonly consists of a hydrocarbon chain that can be found in the form of oil in the commencement of a flotation process, this allows the mineral surface to repel water by covering it with a thin film. For the polar group, it can be an ionizing and hydrophilic compound, meaning that it dissociates into ions when in water. This can be altered in order to react with the specific surface of a mineral. This process can be seen on Figure 1,
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in which A shows when the collector dissolves in the aqueous phase, in B the adsorption of a collector in a mineral surface is presented, and in C, the insertion of an air bubble allows its attachment onto the hydrophobic surface (Gupta and Yan, 2006).
Figure 1 Process of adsorption of a collector in a mineral surface and attachment to the air bubble (extracted from Gupta and Yan, 2006).
The classification of ionising collectors is made according to the kind of ion, anion, or cation that creates the effect of repelling water. Figure 2 shows the different classification for collectors (Napier-Munn and Wills, 2005).
Figure 2 Classification of collectors (extracted from Napier-Munn and Wills, 2005).
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The recovery of these particles happens in the froth phase. Therefore, the froth should be stable and under control for reaching an effective separation. For an effective flotation process, the valuable mineral surface should be, or become hydrophobic and the gangue mineral surface should be or become hydrophilic. When the valuable minerals are hydrophobic and extracted in the floated fraction, it is called direct flotation. The opposite is also possible, and it is called reverse flotation, in which the gangue is floated and extracted in the floated fraction (Napier-Munn and Wills, 2005).
Flotation processes often happen in different stages (circuits). Rougher is the first stage, in which the concentrate is obtained together with a waste that is still rich in the mineral ore. The concentrate from this stage goes to a cleaning process, called cleaner stage, that allows the upgrading of the final concentrate. The waste from the rougher and cleaner stage generally follows to the scavenger stage for the possible recovery of valuable minerals and for obtaining a waste that is adequate for disposal. There is also the possibility of altering the stages in a circuit according to the requirements for the process (Agapito Mendes et al., 2018).
In order to create the appropriate conditions for an efficient flotation process, flotation machines are required. These are carefully selected to enhance the mixing performance for promoting particle-bubble collision and bubbles dispersion and production (Newell, 2006).
2.2 Flotation equipment
Before entering the flotation equipment, some material dressing processes are required to ensure the efficiency of the process, such as reducing the particle size through grinding, desliming, and conditioning.
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Independent of the size, flotation equipment can be mainly grouped into mechanic cells and pneumatic cells, however, these can also be classified as tank cells and flotation columns (Agapito Mendes et al., 2018)
2.2.1 Pneumatic cells
In pneumatic cells, the bubbles are created by infusing the air inside the cell at high pressure or speed. This can be done either by feeding the pulp and the air separately, as in the flotation column (Figure 7), or it can be done as in the Jameson cell (Figure 3) in which the pulp is injected together with the air at high pressure, intensifying the collision among bubbles and particles (Mesa and Brito-Parada, 2019a).
Figure 3 Schematic of a Jameson cell (extracted from Silva, 2005).
Figure 4 shows a schematic pneumatic cell model Imhoflot. The conic device at the top is to regulate the froth height. The cone is inserted or removed from the cylindric cell, increasing or reducing the available section for the froth. Agitation is provided by injected air, reducing pulp turbulence, which is an advantage for coarse and fine particles flotation (Chaves, 2006).
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Figure 4 Schematic pneumatic cell model Imhoflot (extracted from Chaves, 2006).
Column flotation is more frequently utilized in coal, phosphates, iron ore and base metal plants, as the cleaner stage. In this equipment, the air is usually inserted in the base of a tall cell employing a sparging system, while the pulp is fed close to the highest part of the column (Figure 5). This allows particles to settle due to gravity and the bubbles to rise due to its lightness and properties. In this equipment, the column height and the ratio between height and diameter are crucial, as the bubble-particle interaction is dependent on the space between the region where the air is inserted and the top of the column, where the pulp is fed (Mesa and Brito-Parada, 2019a).
Figure 5 Schematic of a column flotation (extracted from Mesa and Brito-Parada, 2019a).
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The main variables related to column flotation that influences the concentration process are primarily the airflow rate, water flow rate, the height of the froth, residence time, air hold up, bubble, and particle size, among others. These variables have a significant influence on the grade and recovery of the mineral of interest (Silva, 2005).
2.2.2 Mechanical cells
The first cells applied in flotation were Pneumatic cells, that nowadays is not very common and more broadly utilized in the industry for specific cases. Currently, mechanical cells comprise the major part of flotation equipment employed worldwide.
The purpose of a flotation tank is to inject air bubbles into the pulp, to enhance the likelihood of collision between the bubbles and the particles within the slurry, to ensure a stable pulp-froth interface, and to provide an adequate froth removal capacity (Mesa and Brito-Parada, 2019a).
Controlling the airflow rate, impeller speed and pulp level is vital for the optimization of the flotation process. As presented in Figure 6, mechanical cells contain an impeller to create a region with high turbulence aiming to maintain the particles in suspension, to provide bubble-particle collision, and to produce and disperse the bubbles. These can also be sub-divided into self-aerated and forced-air cells, according to the air introduction scheme applied. Both are broadly applied in treatment plants, although forced-air cells provide greater control of the supplied air (Mesa and Brito- Parada, 2019a).
Figure 6 Schematic of a mechanical flotation cell (extracted from Mesa and Brito-Parada, 2019).
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In a mechanical cell, for an efficient flotation to occur, the number of bubbles should be high and with a small diameter, in a way that it captures the higher number of particles as possible. For that, an impeller is used for generating bubbles, while a stator breaks the bubbles in an appropriate size. An example can be seen in Figure 7 (Chaves, 2006).
Figure 7 Typical flow patterns in a mechanical flotation cell (Outokumpu) (extracted from Gorain et al., 1995).
Many different cell geometries can be used, as presented in Figure 8. These can apply different impeller-stator designs, as presented in Figure 9.
Figure 8 Flotation cell designs (extracted from Chaves, 2006).
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Figure 9 Impeller-stator designs. (extracted from Chaves, 2006).
The simplest cell design is the Aker model. The impeller is a turbine, and the stator is fixed at the bottom of the cell. Outokumpu (OK type) presents a more sophisticated impeller-stator design as presented in the previous Figure 7.
2.2.3 Laboratory flotation equipment
The equipment previously introduced is presented on the industrial scale, but due to physical and financial aspects, trial runs and tests are commonly performed on the laboratory scale. These are smaller and adapted flotation equipment used to reproduce and accomplish a comparable performance to the industrial flotation procedures.
For the micro-flotation tests, either a Modified Partridge-Smith cell or a Hallimond tube can be applied. The first contains a frit at the bottom, a straight glass tube, and a launder. It can be used to assess the response of a mineral for a specific flotation condition, such as pH or reagent dosage (McGill University, 2020).
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The schematic of a Hallimond tube can be seen in Figure 10. It contains a frit at the bottom to allow the air to flow. It uses similar conditions to the modified Partridge- Smith cell and also assesses the mineral response to a specific flotation condition. The difference is that it reduces the quantity of floated particles that fall back to the pulp through a bending shape (McGill University, 2020).
Figure 10 Schematic of a Hallimond tube (extracted from Wills & Napier-Munn, 2006).
Mechanical bench-scale laboratory cells (Figure 11) are broadly applied machines that demonstrate to be effective for flotation tests in terms of deciding the reagents to be used and determining the kinetic parameters for modelling (Mesa and Brito-Parada, 2019a).
Figure 11 Schematic of a laboratory bench-scale mechanical flotation equipment (extracted from Mesa and Brito-Parada, 2019).
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Nonetheless, the reduced size indicates that the majority of these cells have significant differences in impeller size, amount of stator blades, and in other physical proportions when contrasted to industrial cells. Also, in these laboratory equipment, the steady-state cannot be achieved, as water is constantly added while the froth overflows in order to keep up the pulp level, resulting in a variety of mineral grade, solid, and reagents concentration over time (Mesa and Brito-Parada, 2019a).
In mechanical batch flotation tests, the sample size usually ranges around 500 g, 1 kg, or 2 kg sample. These are mechanically agitated and simulate a large-scale flotation process (Figure 12). Air is introduced mainly through a hollow standpipe around the impeller shaft. The impeller pushes the air down the standpipe, being controlled using a valve and through the speed of the impeller. This generates bubbles that rise through the pulp, which are after collected in the froth zone (Napier-Munn and Wills, 2005).
Figure 12 Laboratory mechanical flotation cell (extracted from Wills & Napier-Munn, 2006).
These bench-scale flotation tests are commonly done using a Denver Lab Cell D-1 machine. It contains different cells with different capacities, usually three cells with capacities of 500 ml, 2500 ml, and 5000 ml (Figure 13) (McGill University, 2020).
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Figure 13 Denver Lab Cell D-1 flotation machine (extracted from McGill University, 2020).
Another example of a laboratory bench-scale flotation machine is the Outotec GTK LabCell, which has adjustable air feed and impeller speed, and automatic froth scraping mechanism. It also comes with different scale cells with its respective impellers, stators, and scrapers (Outotec, 2018).
Continuous laboratory systems have been introduced, together with laboratory cells that can perform at a steady state. Whereas laboratory tests permit assessing the impact of different variables in a single unit, pilot-scale testing is crucial for plant circuit design.
These are small (ranging from 60 to 150 litres) industrial flotation tanks applied for comparison of equipment and circuit efficiency in terms of costs and concentrate sample sizes (Mesa and Brito-Parada, 2019a).
Nowadays, the decrease in grades and higher mining capacities has led to an increase in the throughput of material being treated at modern industrial treatment plants.
Rather than the increase in the number of cells and banks, flotation equipment has increased its dimensions in the interest of handling more material (Mesa and Brito- Parada, 2019a).
Figure 14 shows the evolution in flotation tank sizes over the last century, alluding to the maximum tank volume commercially accessible. This has been beneficial in terms of
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decreasing the overall capital and operating expenses. On the other hand, these bigger and increasingly complex tanks created new challenges in its design, performance, and operation especially in respect of pulp hydrodynamics and froth transportation (Mesa and Brito-Parada, 2019a). With this increase in flotation cell sizes, it is important to understand the mechanisms governing the scale-up process.
Figure 14 Trend in the flotation tank size over the last century (y-axis is on a logarithmic scale) (extracted from Mesa and Brito-Parada, 2019a).
2.3 Scale-up of flotation process
Some mineral processing plants, when facing the issue of treating a higher throughput, apply a different technique called process intensification. This means studying and designing smaller reactors in order to improve transport and processing rates, providing better control of kinetics. This enhances energy efficiency and decreases capital costs.
This process intensification approach has been taken slow steps in terms of application in the industrial scale. Although it seems to play a vital part within the
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future of mineral processing operations, it is improbable that it will be applied in froth flotation, in terms of having reduced tanks within the near future. Based on that, scale- up studies will most likely continue to be crucial in terms of designing larger flotation equipment. As such, a stronger understanding of the hydrodynamics factors at variable scales and their influence on performance is required, especially for the pulp and froth zones in flotation tanks (Mesa and Brito-Parada, 2019a).
Previous flotation tanks were considerably small in volume (smaller than 1 m3) (Arbiter, 2000). Today, these can be larger than 300 m3. This increase results in technical and financial benefits, as fewer machines are required leading to reduced plant footprint, simpler operational control, and less energy consumption. Anyhow, fluid dynamic properties also influence the efficacy of flotation machines. The size, shape, speed, and location of the agitating mechanism directly influence in pulp dynamics.
Also, the higher gap between the bulk of the froth and the discharge lip alter the froth stability (Mesa and Brito-Parada, 2019a). These aforementioned factors can lead to new challenges when it comes to scale-up to the industrial scale.
The scale-up approach in flotation studies is divided into two distinctive groups, the kinetic scale-up and the machine design scale-up (Mesa and Brito-Parada, 2019a). The kinetic scale-up consists of diverse methods of scaling-up the flotation model such as kinetic parameters acquired over laboratory investigations, to anticipate the plant performance (Gorain et al., 1998). The machine design scale-up is referred to as the study field that analyses, in different scales, the behaviour and performance of flotation equipment. This is done by centralizing on air injection technologies and impeller speed and layout, taking into account the influences of hydrodynamic phenomena in the pulp zone, and the geometrical and dynamic resemblances (Gorain et al., 1994).
A large majority of studies related to machine design scale-up focuses on pulp zones.
Therefore, there is an information gap when it comes to scientific studies and
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manufacturers guidelines on the scale-up of flotation tanks. The few works on this topic centralize exclusively on hydrodynamics assessment of the pulp zone, by comparing the Computational Fluid Dynamics (CFD) model estimations versus tests operated solely with water. When it comes to modelling flotation tanks, this tool is sometimes applied to evaluate flotation efficiency without integrating the froth and pulp zones (Mesa and Brito-Parada, 2019a).
2.3.1 Computational Fluid Dynamics (CFD)
Recently, researchers have started applying CFD for modelling mechanically agitated flotation cells for assessing the flow complexity regarding the different phases in the interior of the cells (Koh et al., 2003). The design of flotation cells is normally done based on empirically derived relations. When applying CFD modelling, individual finite volumes categorizes the flotation cell in order to estimate local values of flow properties.
This provides a more detailed comprehension of the flow permitting the equipment adjustment and operation and enhancing flotation performance (Koh and Schwarz, 2006).
Three flotation sub-processes are modelled using the collision, attachment, and detachment approach. The particle-bubble collisions rate is estimated using a turbulent collision model, using the local turbulent speed, the size, and the number of bubbles and particles present in the different regions of the cell. Additionally, the collision, adhesion, and stabilization probabilities are determined allowing the estimation of the attachment rates. Likewise, the fluid turbulence allows the estimation of the detachment rates. These attachment and detachment rates are applied in the CFD kinetic modelling containing simulations of the transient population-balance eliminating the froth bubble–particle aggregates (Koh and Schwarz, 2006).
Although CFD has been extensively studied for modelling of the flotation process, the literature regarding its practice for the scale-up of flotation cells is still scarce (Mesa
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and Brito-Parada, 2019a). One of the first examples of the use of CFD in flotation equipment design was made in 2000 in the work of Koh et al. (2000). In this work, a standard Rushton turbine tank and a CSIRO flotation cell were contrasted regarding its number of bubble-particle collisions per time per unit volume. The number of bubble- particle collisions was predicted using the computed flow properties acquired from the flow variables derivative from an Eulerian-Eulerian multiphase concept associated with the standard k–e turbulence model (Koh et al., 2000).
Afterward, a Denver-type flotation cell was simulated using a combination of the bubble–particle collision and attachment rates that have been introduced in a CFD model (Koh and Schwarz, 2003). For the estimation of the bubble–particle collision number, the Saffman, and Turner equation was used (Saffman and Turner, 1956), and for accounting to the particles following the fluid streamlines, the Yoon and Luttrell (1989) (Hassanzadeh et al., 2019) model was utilized. Similarly to the beforehand mentioned work, the Eulerian–Eulerian method was applied in order to model the multiphase in combination with the Multiphase Reference Frames (MRF) method for the rotation of the impeller (Koh and Schwarz, 2003).
Subsequently, in 2006, the detachment rate and the attachment probability were inserted in the CFD model. They implemented a first-order kinetic model including different sub-processes equations. The CFX4.4 was used for determining the flotation kinetic and the gas-liquid governing equations for a Rushton turbine tank and a CSIRO Denver flotation cell. The flotation rate constant was estimated based on the particles that remained within the cell. The same process was also applied in subsequent work, in a self-aerated flotation cell, in which the gravitational force was included in the dispersed phase equation, which leads to an increase in the detachment frequency (Koh and Schwarz, 2006; Koh and Schwarz, 2007).
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Another work was produced in a modified Denver batch flotation cell using a CFD model for predicting the flotation rate constant and for assessing the influence of the impeller speed on the flotation performance. This work proved to be in good qualitatively and quantitatively agreement based on the contrast among the measurements obtained and the predicted flotation rate constants. In this case, an Eulerian-Eulerian and a Lagrangian-Eulerian method were applied for modelling, using a higher-order coupling manner among the different phases. This allowed researchers to employ a full momentum among the particulate phases, using the Lagrangian-Eulerian approach, providing better results (Koh and Smith, 2011).
Overall, incorporating numerical systems in fundamental models has proved to be limited, which is attributed to the struggle of incorporating complex flotation sub- processes models with the numerical modelling of high turbulent flow in the interior of a mechanically agitated flotation cell. Therefore, inserting a partial differential equation to the model, to estimate the free particles in the system, maximize the computational demands.
In order to improve that, Karimi et al. (2014) developed a new methodology for the estimation of the flotation performance using CFD modelling, considering the sub- processes taking place through the separation, excluding the insertion of a new equation for the number of particles. For that, the flotation rate constant is predicted using the fundamental flotation model of Pyke et al. (2003) into the Eulerian-Eulerian framework.
It was proven that the new CFD model improved the flotation rate constant estimations and allowed to assess the influence of the particles' hydrophobicity, impeller speed, and gas flow rate on the flotation rate constant (Karimi et al., 2014).
The main literature focusing on a CFD model for the scale-up of flotation presents the effect of machine scale-up for comparing the pulp behaviour in different Metso tanks using CFD and Discrete Element Modelling (DEM). However, the air injection was not
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considered and not many details are presented, therefore, the whole complexity of the system was not considered (Mesa and Brito-Parada, 2019a).
Similar industries have, however, applied CFD for equipment scale-up. An example is the scale-up of fluidized-bed hydrodynamics in which the flow turbulence and particle size affect the selection for the appropriate size of the cell (Knowlton et al., 2005).
Another work is regarding the scale-up of binder agglomeration processes, in which the operating conditions are influenced in more than one process, therefore when reaching a determined scale, it was recommended that the different processes should be split into different staged unit operations (Mort, 2005).
Improving the predicting competencies of current models requires additional investigations on CFD methods to model the collection zone, addressing the three phases and 3D flotation systems. Although kinetic models have been useful for assessing flotation performance, addressing the physical interactions among the different phases would improve the actual limitations regarding scale-up methodologies. Therefore, theoretical and experimental research regarding the scale-up methods is vital in order to fill the gaps in knowledge that needs to be addressed (Mesa and Brito-Parada, 2019a).
2.3.2 Kinetic scale-up
The fundamental goal of kinetic scale-up is to apply mathematical methods to anticipate the performance of an industrial-scale plant in terms of concentrate grade and recovery, through the evaluation of laboratory-scale data acquired in flotation tests (Mesa and Brito-Parada, 2019a).
Researches related to kinetic flotation models are plenty. These are based on simplification, as in chemical reactions, in which flotation is considered as a kinetic rate operation (Mesa and Brito-Parada, 2019a).
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The subsequent ordinary differential equation is based on this assumption (Equation 2), in which C is the concentration of particles, k is the kinetic constant for the flotation rate, and n is the reaction order (Bahrami et al., 2019).
(1)
Introducing the compartmental model for continuous flotation tanks in the kinetic models enabled the deviation of its focus on the froth zone, redirecting it to the collection zone. Based on that, the model splits the flotation process into a pair of independent yet interlinked parts containing its recovery. These are the froth and the collection zone. This is presented in Equation 3, in which the recoveries are represented as overall recovery (R), collection zone recovery (Rc), and froth zone recovery (Rf). The last can be represented by Equation 4, where the overall flotation rate constant is represented by k, and the collection zone rate constant by kc (Mesa and Brito-Parada, 2019a).
(2)
(3)
In the following Equation 5, the flotation rate (k) can be estimated assuming that the flotation of particles of diameter i follows a first-order rate reaction. R is the recovery at a flotation time t, and R∞ is the recovery towards an infinite time (Duan et al., 2003).
(4)
The following Equations 6-9 shows other flotation kinetic models that can be used for estimation of the flotation rate. These can be achieved considering different simplifications and can be applied to describe the collection recovery component and the overall recovery as in Equation 5 (Mesa and Brito-Parada, 2019a).
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(5)
(6)
(7)
(8)
Equation 6 is a variation of the first-order model, called the first-order model with a rectangular distribution of flotabilities. Equation 7 is the second-order kinetic model and Equation 8 is the Klimpel model, or also called, the second-order model with a rectangular distribution of flotabilities (Bahrami et al., 2019). Finally, Equation 9 is called the non-integral order equation (Merna et al., 2015).
The main objective of these models is to explain the flotation process using the kinetic parameters, which can be, for example, the variables k and the theoretical maximum recovery reachable considering the machine efficiency and mineral liberation, R∞. These variables can be achieved through applying the models in the experimental data, and accordingly, are reliant on the mineral properties, for example, its composition, particle size distribution, liberation, operational settings, and the flotation machinery (Mesa and Brito-Parada, 2019a).
The currently applicable kinetic models consider the froth as a simple zone in which an experimentally determined fraction of the solid particles are rejected and returned to the pulp zones. Therefore, performance may be considerably different among laboratory-, pilot- and industrial-scale units, depending on froth stability, mixing systems, and residence-times (Flint, 2001).
Finding the kinetic parameters at an industrial scale based on laboratory experiments is not a simple operation for the different combinations of ore and machinery. The flotation kinetics variables acquired from industrial-scale and experimental data are
25
different. The amount of time that a particle stays in the system and the hydrodynamic settings of the flow are not the same. This process is behind the theory of kinetic scale-up and it has not been solved, as typical results in laboratory-scale data overpredict industrial rates (Mesa and Brito-Parada, 2019a).
Many existing empirical methods for scaling-up kinetic variables are reliant on a scaling factor. This factor commonly ranges from 1.5 to 3 and it can be achieved based on the ratio between industrial and laboratory residence times. The estimation of the plant flotation rate can be assumed as the division of the experimental flotation rate to the scaling factor. The purpose is to accomplish an identical recovery by means of linking the residence time required in a batch test and a continuous flotation circuit (Mesa and Brito-Parada, 2019a).
There are many published studies related to the scaling factor application. Yianatos et al. (2003) applied separability curves, which are the ratio between mineral recovery and yield, for determining the comparison recovery. The scale-up factor then established was kPlant τ = kLab t, based on the assumption of an ideal separability point for the comparison recovery. This means having the concentrate incremental grade matching the feed grade. For further studies, a non-dimensional scaling variable (φ) was added to the equation in order to split the impacts of mixing and kinetic variations on the time scale-up factor, as presented in Equation 10 (Yianatos et al., 2006).
(9)
Followed by these studies, Yianatos et al. (2010) integrated some impacts of tank dimensions in the scale-up factor, which is now represented as ξ = Kac/KLab, in which the real value from the plant (Kac) can be assumed from Equation 11. In this equation, the influences of froth zone (ζ = kapp/kc), variances in cell mixing (ŋ), and solids segregation (Ψ) are now inserted in the calculation of the apparent flotation rate constant, Kapp.
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Yet, none of these studies considers the influences of entrainment, the detachment of particles during sampling, and further influences of the froth zone, for example, liquid drainage and transport phenomena.
(10)
Gorain et al. (1998) suggested the following Equation 12 for the representation of k. In this equation, kc = P.Sb, where P is a non-dimensional variable dependent on the ore properties, is called the floatability index. And the bubble surface area is represented by Sb (Sb = 6 Jg/d32), in which Jg is the gas superficial velocity in cm/s, while d32 is the bubble Sauter mean diameter in mm. This scale-up process model was developed with the intention of decoupling the ore properties (P) from the operational parameters and the flotation machinery design (Sb), so it is only dependent on Sb.
(11)
Later researches suggested what is shown in Equation 13. That is an empirical equation for Sb that considers the hydrodynamic effects associated with the impeller design and operational settings, ignoring some aspects such as tank size and shape. In this equation, the constants a = 123, b = 0.44, c = 0.75, d = − 0.10 and e = − 0.42 come from the analysis of experimental data, while the peripheral speed of the impeller is represented by Ns, the aspect ratio among the impeller’s diameter and height is represented by As and the particle size of the feed is represented by P80 (Mesa and Brito- Parada, 2019a).
(12)
To add the entrainment mechanisms in these studies, Welsby et al. (2010) proposed the following Equation 14. In this equation, the degree of entrainment is represented by
27
ENT, Rw represents the concentrates water recovery, i is the size class and j the liberation class.
(13)
Lately, two non-dimensional variables (EVF and æ) were added by Amini et al. (2016) to improve the scale-up model competences. These are shown in Equation 15 and Equation 16. EVF is the Effective Volume in Flotation and it is the portion among the volume of the cell in which ε (the turbulent kinetic energy dissipation rate) is bigger than 0.1 m2/s3 and the entire volume of the cell. For Equation 16, æ represents the hydrodynamic factor. The fluid kinematic viscosity in cm2/s is represented by ν, and n is predicted based on various flotation experiments ranging in the operational settings.
(14)
(15)
The previously mentioned kinetic scale-up models rely on deterministic models.
These have received many critiques over time, due to its industrial application and estimation capacity. Consequently, probabilistic models have been suggested and investigated. These states that k is the effect of merging the bubble-particle collision (Pc), attachment (Pa), and detachment (Pd) probabilities, as presented in Equation 17, in which Z is the collision rate (Schuhmann, 1942).
(16)
Pyke et al. (2003) presented a model that has been applied in the design of a CFD kinetic model. This is shown in Equation 18, where the gas flow rate is Qg, the volume of reference is Vr, the density of the solids is given by ρs (g/cm3 ), the density of the liquid is given by ρl (g/cm3) and the fluid turbulent speed by ui (cm/s).
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(17)
Although many studies related to the mechanisms of flotation, multiphase flotation is a complex system that is still not entirely explained by those models. Kinetic models are primarily focused on the pulp zone events because the froth zone is not a kinetic process (Mesa and Brito-Parada, 2019a).
2.3.3 Machine design scale-up
The objective of machine scale-up in flotation is to allow the conversion of laboratory- scale to industrial-scale with the smallest interference on its efficiency. That can be accomplished by describing the influence of machinery design, shape, and size on flotation performance. However, this is a challenging procedure as flotation phenomena comprise many micro-, meso- and macro-scale unrelated processes. Therefore, similitude factors and non-dimensional analysis have been applied for simplification of the process (Mesa and Brito-Parada, 2019a).
For the process of stirred tank scale-up, the distinct phases are combined in a turbulent zone, and the focus is to scale-up just the pulp zone. This process includes the development of a larger system that is expected to accomplish a mixing quality that is equal to the experimental one (Mesa and Brito-Parada, 2019a).
As seen in Figure 15, the power spent by the diverse impellers can be calculated based on the Reynolds number and Power number. The following Equation 19 and Equation 20 were suggested by Arbiter (2000) and it considers the Power number and the power per volume (P/V) as fixed. It is done by changing the rotor diameter (D) and rotational velocity (N).
(18)
(19)
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Figure 15 Power number for the diverse impellers Reynolds number (extracted from Mesa and Brito-Parada, 2019a).
Equation 21 shows the nominal shear rate, in which δ is a rotor-stator scale- independent variable that represents the shear gap width. Based on some studies, for a scale-up process, the nominal shear rate should be kept as constant. However, this statement just addresses the liquid phase agitation phenomena, ignoring the existence of solids and air bubbles in a flotation process (Mesa and Brito-Parada, 2019a).
(20)
Some examples of nondimensional variables suggested for the development of froth flotation tanks are: the Reynolds number (Re) that assess to the turbulence in the system, and it is represented by Equation 22; the Power number (Np), which is associated to the torque and inertial forces required to spin the impeller at a certain rate, and it is represented by Equation 23; the Frode number (Fr) which is the ratio between the inertial and gravitational forces, and it is represented by Equation 24; the Zwietering constant (S) that is related to the impeller nature and geometry, and it is represented by Equation 25; and the airflow number (Na, also called air capacity number - Ca) represented by
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Equation 26. In these equations, ρ is the density of the pulp, µ and ν are the dynamic and kinematic viscosity, respectively.
The power spent by the impeller is given as P and the gravitational speed is given as g.
Reimp is the Reynolds number produced by the different impeller types, ρl is the liquid phase average density and ρs is the solid phase average density. Njs is the lowest agitation velocity where all the particles achieve the total suspension. The particle size mean is given by dp, the mass ratio between suspended solids and liquid is represented as X, and the gas inflow rate by Qg (Mesa and Brito-Parada, 2019a).
(21)
(22)
(23)
(24)
(25)
A typical flotation scale-up practice, in terms of examining the gas injection and bubble creation, is to maintain constant the connections among gas and liquid flow rate, and tank diameter, as presented in Equation 27 (Arbiter et al., 1976).
(26)
The previously examined researches focus on different strategies for equipment design scale-up. The different procedures have been implemented each for a specific tank and at different operating settings. Based on that, there is a lack of additional investigations for comparing the different scale-up systems, in both theoretical and experimental aspects (Mesa and Brito-Parada, 2019a).
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It is also important to mention that all the previously mentioned methods related to nondimensional variables can only be applied to get a comparable particle suspension and agitation, meaning that there is still a lack of proof that these are related to achieving similar metallurgical effectiveness. Also, the froth zone presents several complexities that must be analysed for anticipating its performance (Mesa and Brito- Parada, 2019a).
According to Newell (2006), if the chemical conditions within the cells of different scale are constant, and geometric similitude applies, the scale-up of the flotation process should be governed by hydrodynamic factors. These are the volumetric flow rate of gas and the impeller speed.
2.4 Influence of the impeller speed and design in flotation
Mechanical cells are highly turbulent vessels in which a significant amount of energy is present for permitting the collision of small particles with the rising bubbles. The main function of the impeller is to keep particles in suspension, create and disperse bubbles, and to promote bubble-particle collision. Additionally, it can produce a more turbulent environment, affecting froth stability (Wang et al., 2015). These are usually installed in a rotor-stator system, in order to produce high shear rates and turbulence. A turbulent system can be responsible for either a good collection of fines but also for an increase in detachment of particles from the bubble at larger particle sizes (Flint, 2001).
The impeller provides air to the flotation system producing bubbles at the bottom of the cell, mixing the pulp, and avoiding particle sedimentation. It breaks the air bubbles producing smaller bubbles offering an environment that permits the bubble-particle collision within the slurry (Silva et al., 2018). Although it is a crucial parameter in flotation, it also has a negative impact as it can generate excessive turbulence within the cell. The turbulence of the pulp is strongly dependent on the impeller size and shape.
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Impellers are usually classified according to its mixing properties into axial, radial, or mixed flow. Axial impellers are commonly applied in solids suspension and radial impellers for dispersion of gas (Rushton turbine, for example). Generally, the impeller design is associated with the flotation tank design. In industrial-scale flotation systems, most impellers have a flat disc design associated with distinct blade shapes, commonly adopting a half-spherical rotor design. These are generally installed as a rotor-stator system, which is a high-speed rotor attached closely to a stator (which is fixed) allowing the production of high shear rates and a turbulent system (Kauppila, 2019).
The following Equation 28 can be used to calculate the impeller tip speed, in which ω is the impeller tip speed (m/s), D the is rotor diameter (m) and N is the rotation speed (1/s) (Kauppila, 2019):
(27)
Additionally, the power input can be used for addressing the impact of the impeller speed on flotation kinetics, independent from the shape of the impeller. Practically, an improve in flotation can be observed when moving from lower power input to higher power input. However, this could lead to an unstable flotation system as a result of increasing turbulence. The following Equation 29 shows the relation between the rotor power input (P, in W), the full mass of the fluid (m, in Kg), and the energy dissipation rate (ε), which is a parameter that defines the real energy input to the slurry mass. An increase in the impeller speed leads to an increase in the energy dissipation among the flotation cell, increasing the bubble-particle probability of collision. This is advantageous for finer particles, as these are less likely to attach to the air bubble due to its size (Kauppila, 2019).
(28)
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The different mineral types and size distributions require specific hydrodynamic conditions for flotation. This regulates the particles recovered and consequently, the flotation rate of recovery. This highlights the importance of particle size analysis prior and posterior to flotation. Usually, the flotation rate increases for finer particles and decreases for coarser particles with an increase in the impeller speed. This can also be associated with the energy input, as the flotation rate is more likely to improve for particles in between the fines and coarse size range, with an increase in the energy input (Kauppila, 2019).
There are 3 main zones associated to flow conditions in a flotation cell: the turbulent zone, the quiescent zone, and the froth zone. The froth zone is responsible for the final recovery achieved. Its recovery is directly connected to the energy input, as an increase in the energy input leads to an unstable froth zone thus reporting a lower recovery (Kauppila, 2019).
Another important variable is the Power number (Np), which is described as the ratio among dissipated energy as shear and the energy applied for bulk flow production. As an example, it can be improved through applying a lower cell aspect ratio, lower impeller speed, and a larger impeller-stator arrangement leading to an increase in the area with greater turbulence zone (Kauppila, 2019). This is presented in Equation 24.
According to Amini et al. (2016) in a laboratory-scale flotation cell (5 litres, for example), increasing the impeller speed leads to a decrease in bubble size up to a critical speed, in which the impeller speed will no longer be able to reduce the bubble size.
Consequently, an increase in the surface area of the bubble can be observed. This allows a better bubble-particle collision probability, increasing the flotation rate. However, for bigger cells (60 litres, for example), the same pattern might not be observed. In this case, bubble size keeps constant independent from the impeller speed. This can be explained by better contact between the bubble and particle due to the increase in turbulence.
