Coupling Flotation Rate Constant and Viscosity Models
Abstract
:1. Introduction
Researcher | Reported Suspension Viscosity | Flotation Results (Rate Constant or Recovery/Grade) | References |
---|---|---|---|
Farrokhpay et al., 2011 | The viscosity of 50 vol.% glycerol–water mixture, used in their study, was 0.0076 Pa s. | The recovery of coarse composite copper-bearing particles (+210 μm) of porphyry copper ore, recovered in the tailings of rougher, at a grind size of d80 = 250 μm, increased from 83 to 90% with the increase in viscosity from 0.001 to 0.0076 Pa s. | [24] |
Shabalala et al., 2011 | The viscosity of kaolin ore slurry increased exponentially to the maximum values of between 0.03 and 0.08 Pa s with the increase in solid concentration from 15 to 40 wt.% of kaolin. | Bubble size generated within kaolin ore suspension decreased from 1 to 0.65 mm with the increase in solid concentration from 30 to 40 wt.% at an impeller speed of 650 rpm. | [26] |
Forbes et al., 2014 | The viscosity of pulp containing chalcopyrite and clay minerals (kaolinite and quartz) was found between 0.001 and 0.15 Pa s. | The recovery of chalcopyrite (copper) was 92% at quartz/kaolinite content of 100/0; however, with the change in quartz/kaolinite ratio (i.e., 70/30 and 30/70), the chalcopyrite recovery reduced to 87% and 82%, respectively. | [28] |
Cruz et al., 2015 | The viscosity of copper–gold ore slurry increased from 0.0035 to 0.014 Pa s with the increase of solid concentration of bentonite from 0 to 15 wt.% at a shear rate of 100 s−1. | The baseline flotation of their ore (i.e., 100 wt.% ore without bentonite/kaolinite) resulted in a copper recovery of 92% at a grade of 10% copper, and 81% gold recovery at a grade of 7 ppm gold. The addition of 15 wt.% bentonite to the ore (100 wt.%) decreased the recovery (i.e., copper recovery from 92 to 83% and gold recovery from 81 to 64%) and slightly decreased in copper and gold grades from 10 to 8% and 7 ppm to 5 ppm, respectively. The addition of 30 wt.% kaolinite to the ore (100 wt.%) did not decrease copper and gold recoveries but did decrease copper and gold grades from 10 to 2% and 7 ppm to 1 ppm, respectively. | [20] |
Wang et al., 2015 | The apparent viscosity of Telfer clean ore increased from 0.001 to 0.008 Pa s with the increase in solid concentration of bentonite from 5 to 25 wt.% at a shear rate of 100 s−1. | The copper recovery decreased from 76 to 25% with the increase in solid concentration of bentonite from 0 to 20 wt.%; however, it slightly decreased the copper grade from 5.1 to 5%. The copper recovery decreased from 80 to 67% with the increase in solid concentration of kaolin from 0 to 20 wt.%; however, it decreased the copper grade from 5 to 4%. | [16] |
Zhang & Peng, 2015 | The apparent viscosity of a copper–gold ore increased from 0.0018 to 0.0076 Pa s with the increase in solid concentration of bentonite from 0 to 15 wt.% at a shear rate of 100 s−1. | The copper recovery decreased from 82 to 60% with the increase in solid concentration of bentonite from 0 to 15 wt.%. The gold recovery decreased from 78 to 65% with the increase in solid concentration of bentonite from 0 to 15%. | [14] |
Farrokhpay et al., 2016 | The apparent viscosity of their copper ore slurry in the presence of 15 wt.% montmorillonite, 30 wt.% of kaolinite, and 30 wt.% illite at a shear rate of 100 s−1 was 0.17, 0.03, and 0.02 Pa s, respectively. | The copper recovery decreased (90 to 80%) in the presence of 15 wt.% swelling clay (montmorillonite); however, in the presence of 15 wt.% non-swelling clays (illite and kaolinite), the copper recovery decreased slightly to 87% and 88%, respectively. The copper grade decreased from 18% to about 1% in the presence of both 30 wt.% of kaolinite and 15 wt.% of montmorillonite, respectively; however, it decreased to about 5% in the presence of 30% illite. The copper ore flotation rate constants were 0.51 s−1 and 0.49 s−1 in the presence of 15 wt.% of kaolinite and 15 wt.% of illite, respectively, and 0.33 s−1 in the presence 15 wt.% of montmorillonite, as compared with the copper ore flotation rate constant of 0.70 s−1 in the absence of clay minerals. | [25] |
Basnayaka et al., 2017 | The viscosity of gold ore increased from 0.0018 to 0.0035 Pa s by the addition of 10 wt.% kaolin at pH 7, a shear rate of 100 s−1, and polyacrylate depressant concentration of 0 and 200 g/t, respectively; however, by the addition of 5 wt.% bentonite, the viscosity increased to 0.0060 Pa s, under the same conditions. | The flotation rate constant of their gold-bearing pyrite ore was decreased from 13.71 to 3.37 s−1 (822.6 to 202.2 min−1) without and with the presence of 10 wt.% kaolin at pH 7, air rate of 5 L/min, and polyacrylate depressant concentration of 0 and 200 g/t, respectively. The presence of bentonite under same conditions reduced the flotation rate constant to 4.14 s−1 (248.4 min−1). | [33] |
Chen et al., 2017 | The apparent viscosity of amorphous silica and quartz suspension increased from 0.109 to 0.147 Pa s with the increase in the concentration of amorphous silica (in amorphous silica and quartz suspension) from 30 to 50 vol.%, at a shear rate of 100 s−1. | The copper recovery dropped sharply from 95 to 63.6% after the percentage of amorphous silica increased from 30 to 50 vol.%. The copper grade decreased slightly from 3.9 to 3.8% with the increase in the concentration of amorphous silica (in amorphous silica and quartz suspension) from 30 to 50 vol.%. | [18] |
Farrokhpay et al., 2018 | The viscosity of their copper ore slurry increased from 0.010 to 0.038 Pa s with the increase in solid concentration of muscovite from 0 to 30 wt.% at a shear rate of 100 s−1; however, with the increase in solid concentration of talc from 0 to 7.5 wt.%, there was a slight increase in the slurry viscosity from 0.010 to 0.012 Pa s, at a shear rate of 100 s−1. | The flotation grade decreased (19 to 2%) with the increase in solid concentration of muscovite from 0 to 30 wt.%; however, the change in the recovery was reported negligible. The flotation recovery (90 to 83%) and grade (19 to 2%) decreased with the increase in solid concentration of talc from 0 to 7.5 wt.%. | [17] |
2. Theory
2.1. Collection Efficiency ()
2.1.1. Collision Efficiency ()
2.1.2. Attachment Efficiency
2.1.3. Stability Efficiency
2.2. Flotation Rate Constant under Turbulent Flow Condition
2.3. Viscosity Modeling and Factors Affecting Viscosity
2.3.1. Hard Sphere Suspensions
2.3.2. Effect of Shear Rate
2.3.3. Colloidal Suspensions
3. Results and Discussion
3.1. Calculated Flotation Collection Efficiencies , , , and Rate Constant
3.2. Effect of Suspension Viscosity on , , , and Rate Constant
3.3. Suspension Viscosity Calculation
3.3.1. Modified Krieger and Dougherty Model—Hard Sphere Suspensions
3.3.2. Our Predictive Model—Hard Sphere and Interacting Colloidal Particle Suspensions
Homogeneous Case (i.e., Same Particle with Same Sizes)
Homogeneous Case (i.e., Same Particles with Different Particle Sizes)
3.4. Flotation Efficiencies and Rate Constant Calculations
3.4.1. Incorporation of the Modified Krieger and Dougherty Model—Hard Sphere Suspensions
3.4.2. Incorporation of Our Predictive Model—Hard Sphere and Interacting Colloidal Particle Suspensions
Homogeneous Case—Same Particles with the Same Sizes
Homogeneous Case—Same Particle with Different Particle Sizes
4. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Nomenclature
Hamaker constant | |
Radius of hard spheres | |
Bond number | |
Concentration of ions | |
Bubble diameter | |
Particle diameter | |
Elementary charge | |
Collection efficiency | |
Collision efficiency | |
Attachment efficiency | |
Stability efficiency | |
Attractive forces | |
Buoyancy force | |
Capillary force | |
Machine acceleration force | |
Detachment forces | |
Gravitational force | |
Hydrostatic force | |
Capillary pressure force | |
Gas flow rate | |
Interparticle separation distance | |
Flotation rate constant | |
Boltzman’s constant | |
Avogadro’s number | |
Number concentration of ions | |
Peclet number | |
Characteristic Peclet number | |
Bubble radius | |
Reynolds number | |
Particle radius | |
Absolute temperature | |
Induction time | |
Volume of flotation cell | |
Kinematic viscosity | |
Bubble velocity | |
Particle velocity | |
Ionic valence | |
Collision angle | |
Maximum collision angle | |
Contact angle | |
Particle density | |
Fluid density | |
Dynamic viscosity | |
Suspension viscosity | |
Liquid’s viscosity | |
Relative viscosity | |
Intrinsic viscosity of the particles | |
Solid volume fraction | |
Maximum packing fraction | |
Shear rate | |
Reduced surface potential | |
Debye–Huckel reciprocal length | |
Dielectric constant of the medium | |
Permittivity of free space | |
Zeta potential |
Appendix A
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ϕ/a | 8 (nm) | 12 (nm) | 16 (nm) | 20 (nm) | 24 (nm) | 28 (nm) | 32 (nm) | 36 (nm) | 40 (nm) | 60 (nm) | 120 (nm) | 500 (nm) | 1000 (nm) | 10,000 (nm) | 50,000 (nm) | 100,000 (nm) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 3.36 | 1.91 | 1.31 | 1.07 | 9.70 × 10−1 | 9.26 × 10−1 | 9.06 × 10−1 | 8.98 × 10−1 | 8.94 × 10−1 | 8.92 × 10−1 | 8.92 × 10−1 | 8.92 × 10−1 | 8.92 × 10−1 | 8.92 × 10−1 | 8.92 × 10−1 | 8.92 × 10−1 |
0.176 | 1.51 × 101 | 9.63 | 5.24 | 3.04 | 1.99 | 1.47 | 1.20 | 1.05 | 9.57 × 10−1 | 8.35 × 10−1 | 8.24 × 10−1 | 8.24 × 10−1 | 8.24 × 10−1 | 8.24 × 10−1 | 8.24 × 10−1 | 8.24 × 10−1 |
0.2 | 2.40 × 101 | 1.80 × 101 | 1.01 × 101 | 5.53 | 3.30 | 2.19 | 1.62 | 1.30 | 1.11 | 8.33 × 10−1 | 7.99 × 10−1 | 7.99 × 10−1 | 7.99 × 10−1 | 7.99 × 10−1 | 7.99 × 10−1 | 7.99 × 10−1 |
0.3 | 1.42 × 102 | 3.19 × 102 | 3.69 × 102 | 2.85 × 102 | 1.76 × 102 | 9.81 × 101 | 5.30 × 101 | 2.91 × 101 | 1.67 × 101 | 2.39 | 6.91 × 10−1 | 6.59 × 10−1 | 6.59 × 10−1 | 6.59 × 10−1 | 6.59 × 10−1 | 6.59 × 10−1 |
0.4 | 5.20 × 102 | 5.55 × 103 | 3.16 × 104 | 1.07 × 105 | 2.39 × 105 | 3.78 × 105 | 4.54 × 105 | 4.38 × 105 | 3.57 × 105 | 3.12 × 104 | 1.63 × 101 | 3.99 × 10−1 | 3.99 × 10−1 | 3.99 × 10−1 | 3.99 × 10−1 | 3.99 × 10−1 |
0.5 | 1.13 × 101 | 7.41 × 102 | 4.18 × 104 | 2.05 × 106 | 8.72 × 107 | 3.25 × 109 | 1.06 × 1011 | 3.05 × 1012 | 7.73 × 1013 | 1.40 × 1020 | 3.80 × 1032 | 4.60 × 1025 | 5.73 × 103 | 1.69 × 10−3 | 1.69 × 10−3 | 1.69 × 10−3 |
ϕ/a | 8 (nm) | 12 (nm) | 16 (nm) | 20 (nm) | 24 (nm) | 28 (nm) | 32 (nm) | 36 (nm) | 40 (nm) | 60 (nm) | 120 (nm) | 500 (nm) | 1000 (nm) | 10,000 (nm) | 50,000 (nm) | 100,000 (nm) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0.1 | 9.98 × 10−1 | 9.97 × 10−1 | 9.97 × 10−1 | 9.96 × 10−1 | 9.96 × 10−1 | 9.96 × 10−1 | 9.95 × 10−1 | 9.95 × 10−1 | 9.95 × 10−1 | 9.94 × 10−1 | 9.91 × 10−1 | 9.82 × 10−1 | 9.75 × 10−1 | 9.30 × 10−1 | 8.94 × 10−1 | 8.92 × 10−1 |
0.176 | 9.96 × 10−1 | 9.95 × 10−1 | 9.95 × 10−1 | 9.94 × 10−1 | 9.93 × 10−1 | 9.93 × 10−1 | 9.92 × 10−1 | 9.92 × 10−1 | 9.91 × 10−1 | 9.89 × 10−1 | 9.85 × 10−1 | 9.70 × 10−1 | 9.58 × 10−1 | 8.86 × 10−1 | 8.29 × 10−1 | 8.25 × 10−1 |
0.2 | 9.96 × 10−1 | 9.95 × 10−1 | 9.94 × 10−1 | 9.93 × 10−1 | 9.92 × 10−1 | 9.92 × 10−1 | 9.91 × 10−1 | 9.91 × 10−1 | 9.90 × 10−1 | 9.88 × 10−1 | 9.83 × 10−1 | 9.65 × 10−1 | 9.52 × 10−1 | 8.69 × 10−1 | 8.04 × 10−1 | 8.00 × 10−1 |
0.3 | 9.92 × 10−1 | 9.90 × 10−1 | 9.88 × 10−1 | 9.87 × 10−1 | 9.86 × 10−1 | 9.85 × 10−1 | 9.83 × 10−1 | 9.82 × 10−1 | 9.82 × 10−1 | 9.77 × 10−1 | 9.68 × 10−1 | 9.37 × 10−1 | 9.12 × 10−1 | 7.69 × 10−1 | 6.66 × 10−1 | 6.60 × 10−1 |
0.4 | 9.82 × 10−1 | 9.78 × 10−1 | 9.74 × 10−1 | 9.71 × 10−1 | 9.69 × 10−1 | 9.66 × 10−1 | 9.64 × 10−1 | 9.62 × 10−1 | 9.60 × 10−1 | 9.51 × 10−1 | 9.31 × 10−1 | 8.66 × 10−1 | 8.16 × 10−1 | 5.61 × 10−1 | 4.09 × 10−1 | 4.01 × 10−1 |
0.5 | 9.27 × 10−1 | 8.87 × 10−1 | 8.56 × 10−1 | 8.32 × 10−1 | 8.12 × 10−1 | 7.95 × 10−1 | 7.80 × 10−1 | 7.67 × 10−1 | 7.54 × 10−1 | 7.06 × 10−1 | 6.10 × 10−1 | 3.67 × 10−1 | 2.44 × 10−1 | 1.81 × 10−2 | 2.01 × 10−3 | 1.76 × 10−3 |
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Sajjad, M.; Otsuki, A. Coupling Flotation Rate Constant and Viscosity Models. Metals 2022, 12, 409. https://doi.org/10.3390/met12030409
Sajjad M, Otsuki A. Coupling Flotation Rate Constant and Viscosity Models. Metals. 2022; 12(3):409. https://doi.org/10.3390/met12030409
Chicago/Turabian StyleSajjad, Mohsin, and Akira Otsuki. 2022. "Coupling Flotation Rate Constant and Viscosity Models" Metals 12, no. 3: 409. https://doi.org/10.3390/met12030409