A Model of Froth Flotation with Drainage: Simulations and Comparison with Experiments
Abstract
:1. Introduction
2. Materials and Methods
2.1. Pilot Flotation Column and Experimental Setup
2.2. Experimental Determination of Stability Regions
3. Theory
3.1. Mathematical Model
3.2. Reduced Model for Two-Phase Flow of Bubbles in Liquid
3.3. Numerical Method
3.4. Desired Steady States for the Two-Phase System
4. Results
4.1. Choice of Parameters
4.2. Comparison between the Model and Experimental Stationary Data
4.3. Simulation of Dynamic Behaviour and a Case with a Solids Feed
4.3.1. A Dynamic Simulation of Two-Phase Bubble–Liquid Flow
4.3.2. A Dynamic Simulation of Three-Phase Bubble–Solids–Liquid Flow
5. Discussion
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
Glossary
List of Symbols | |
The following symbols are used in this manuscript: | |
Symbol | Significance and Unit |
A | interior cross-sectional area of column |
integrated capillarity function | |
convective flux function of bubbles | |
convective flux function of solids | |
convective flux function of solids | |
Heaviside step function | |
N | no. of numerical subintervals for numerical method |
Q | volumetric flow |
function giving height of pulp–froth interface | |
capillarity function | |
capillarity constant | |
bubble batch flux function | |
solids batch sedimentation flux function | |
constant exponent in bubble batch flux | |
constant exponent related to Plateau borders in foam | |
Richardson–Zaki exponent | |
q | bulk velocity, flow rate |
t | time |
drift–flux velocity function | |
hindered–settling velocity function | |
terminal velocity of single bubble | |
terminal velocity of single particle | |
z | height |
height of pulp–froth interface | |
Dirac delta distribution | |
temporal step size of numerical method | |
spatial step size of numerical method | |
characteristic function; inside column; outside | |
volume fraction of bubbles (aggregates) | |
steady-state solution | |
critical volume fraction | |
volume fraction of fluid | |
volume fraction of bubbles of numerical method | |
volume fraction of solids in suspension outside bubbles | |
volume fraction of solids | |
Subscripts and Superscript | |
The following sub- and superscripts are used in this manuscript: | |
Sub-/Superscript | Significance |
, , | zone 1, zone 2, zone 3 |
effluent | |
feed | |
gas | |
(local) minimum point | |
steady state | |
underflow | |
wash water | |
zero (of a function) | |
critical | |
batch | |
fluid | |
froth | |
parabolic | |
(local) maximum point | |
Abbreviations | |
The following abbreviations are used in this manuscript: | |
CFD | computational fluid dynamics |
CFL | Courant–Friedrichs–Lewy (condition) |
MIBC | methyl isobutyl carbinol |
ODE | ordinary differential equation |
PDE | partial differential equation |
References
- Finch, J.A.; Dobby, G.S. Column Flotation; Pergamon Press: London, UK, 1990. [Google Scholar]
- Wills, B.A.; Napier-Munn, T.J. Wills’ Mineral Processing Technology, 7th ed.; Butterworth-Heinemann: Oxford, UK, 2006. [Google Scholar]
- Dunne, R.C.; Kawatra, S.K.; Young, C.A. (Eds.) SME Mineral Processing & Extractive Metallurgy Handbook; Society for Mining, Metallurgy, and Exploration: Englewood, CO, USA, 2019. [Google Scholar]
- Pal, R.; Masliyah, J.H. Flow characterization of a flotation column. Can. J. Chem. Eng. 1989, 67, 916–923. [Google Scholar] [CrossRef]
- Vandenberghe, J.; Chung, J.; Xu, Z.; Masliyah, J. Drift flux modelling for a two-phase system in a flotation column. Can. J. Chem. Eng. 2005, 83, 169–176. [Google Scholar] [CrossRef]
- Cruz, E.B. A Comprehensive Dynamic Model of the Column Flotation Unit Operation. Ph.D. Thesis, Virginia Tech, Blacksburg, VA, USA, 1997. [Google Scholar]
- Maldonado, M.; Desbiens, A.; del Villar, R. Potential use of model predictive control for optimizing the column flotation process. Int. J. Miner. Process. 2009, 93, 26–33. [Google Scholar] [CrossRef]
- Bergh, L.G.; Yianatos, J.B. The long way to multivariate predictive control of flotation processes. J. Process Control 2022, 21, 226–234. [Google Scholar] [CrossRef]
- Tian, Y.; Azhin, M.; Luan, X.; Liu, F.; Dubljevic, S. Three-phases dynamic modelling of column flotation process. IFAC-PapersOnLine 2018, 51, 99–104. [Google Scholar] [CrossRef]
- Tian, Y.; Luan, X.; Liu, F.; Dubljevic, S. Model predictive control of mineral column flotation process. Mathematics 2018, 6, 100. [Google Scholar] [CrossRef] [Green Version]
- Azhin, M.; Popli, K.; Afacan, A.; Liu, Q.; Prasad, V. A dynamic framework for a three phase hybrid flotation column. Miner. Eng. 2021, 170, 107028. [Google Scholar] [CrossRef]
- Azhin, M.; Popli, K.; Prasad, V. Modelling and boundary optimal control design of hybrid column flotation. Can. J. Chem. Eng. 2021, 99 (Suppl. 1), S369–S388. [Google Scholar] [CrossRef]
- Quintanilla, P.; Neethling, S.J.; Brito-Parada, P.R. Modelling for froth flotation control: A review. Miner. Eng. 2021, 162, 106718. [Google Scholar] [CrossRef]
- Quintanilla, P.; Neethling, S.J.; Navia, D.; Brito-Parada, P.R. A dynamic flotation model for predictive control incorporating froth physics. Part I: Model development. Miner. Eng. 2021, 173, 107192. [Google Scholar] [CrossRef]
- Quintanilla, P.; Neethling, S.J.; Mesa, D.; Navia, D.; Brito-Parada, P.R. A dynamic flotation model for predictive control incorporating froth physics. Part II: Model calibration and validation. Miner. Eng. 2021, 173, 107190. [Google Scholar] [CrossRef]
- Wang, G.; Ge, L.; Mitra, S.; Evans, G.M.; Joshi, J.B.; Chen, S. A review of CFD modelling studies on the flotation process. Miner. Eng. 2018, 127, 153–177. [Google Scholar] [CrossRef]
- Wallis, G.B. One-Dimensional Two-Phase Flow; McGraw-Hill: New York, NY, USA, 1969. [Google Scholar]
- Narsimhan, G. Analysis of creaming and formation of foam layer in aerated liquid. J. Colloid Interface Sci. 2010, 345, 566–572. [Google Scholar] [CrossRef]
- Bürger, R.; Diehl, S.; Martí, M.C.; Vásquez, Y. A degenerating convection-diffusion system modelling froth flotation with drainage. IMA J. Appl. Math. 2022, 87, 1151–1190. [Google Scholar] [CrossRef]
- Vásquez, Y. Conservation Laws with Discontinuous Flux Modeling Flotation Columns. Doctoral Thesis, Universidad de Concepción, Concepción, Chile, 2022. [Google Scholar]
- Bürger, R.; Wendland, W.L.; Concha, F. Model equations for gravitational sedimentation-consolidation processes. Z. Angew. Math. Mech. 2000, 80, 79–92. [Google Scholar] [CrossRef]
- Bascur, O.A. A unified solid/liquid separation framework. Fluid/Part. Sep. J. 1991, 4, 117–122. [Google Scholar]
- Stevenson, P.; Fennell, P.S.; Galvin, K.P. On the drift-flux analysis of flotation and foam fractionation processes. Can. J. Chem. Eng. 2008, 86, 635–642. [Google Scholar] [CrossRef]
- Dickinson, J.E.; Galvin, K.P. Fluidized bed desliming in fine particle flotation, Part I. Chem. Eng. Sci. 2014, 108, 283–298. [Google Scholar] [CrossRef]
- Galvin, K.P.; Dickinson, J.E. Fluidized bed desliming in fine particle flotation Part II: Flotation of a model feed. Chem. Eng. Sci. 2014, 108, 299–309. [Google Scholar] [CrossRef]
- Galvin, K.P.; Harvey, N.G.; Dickinson, J.E. Fluidized bed desliming in fine particle flotation – Part III flotation of difficult to clean coal. Miner. Eng. 2014, 66–68, 94–101. [Google Scholar] [CrossRef]
- Bürger, R.; Diehl, S.; Martí, M.C. A conservation law with multiply discontinuous flux modelling a flotation column. Netw. Heterog. Media 2018, 13, 339–371. [Google Scholar] [CrossRef] [Green Version]
- Bürger, R.; Diehl, S.; Martí, M.C. A system of conservation laws with discontinuous flux modelling flotation with sedimentation. IMA J. Appl. Math. 2019, 84, 930–973. [Google Scholar] [CrossRef]
- Bürger, R.; Diehl, S.; Martí, M.C.; Vásquez, Y. Flotation with sedimentation: Steady states and numerical simulation of transient operation. Miner. Eng. 2020, 157, 106419. [Google Scholar] [CrossRef]
- Bürger, R.; Diehl, S.; Martí, M.C.; Vásquez, Y. Simulation and control of dissolved air flotation and column froth flotation with simultaneous sedimentation. Water Sci. Technol. 2020, 81, 1723–1732. [Google Scholar] [CrossRef] [PubMed]
- Kynch, G.J. A theory of sedimentation. Trans. Faraday Soc. 1952, 48, 166–176. [Google Scholar] [CrossRef]
- Diehl, S. Operating charts for continuous sedimentation I: Control of steady states. J. Eng. Math. 2001, 41, 117–144. [Google Scholar] [CrossRef]
- Diehl, S. The solids-flux theory—Confirmation and extension by using partial differential equations. Water Res. 2008, 42, 4976–4988. [Google Scholar] [CrossRef]
- Bürger, R.; Karlsen, K.H.; Risebro, N.H.; Towers, J.D. Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units. Numer. Math. 2004, 97, 25–65. [Google Scholar] [CrossRef] [Green Version]
- Bürger, R.; Karlsen, K.H.; Towers, J.D. A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units. SIAM J. Appl. Math. 2005, 65, 882–940. [Google Scholar] [CrossRef] [Green Version]
- Diehl, S. On scalar conservation laws with point source and discontinuous flux function. SIAM J. Math. Anal. 1995, 26, 1425–1451. [Google Scholar] [CrossRef] [Green Version]
- Diehl, S. A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM J. Appl. Math. 1996, 56, 388–419. [Google Scholar] [CrossRef] [Green Version]
- Neethling, S.J.; Lee, H.T.; Cilliers, J.J. A foam drainage equation generalized for all liquid contents. J. Phys. Condens. Matter 2002, 14, 331–342. [Google Scholar] [CrossRef]
- Neethling, S.J.; Cilliers, J.J. Modelling flotation froths. Int. J. Miner. Process. 2003, 72, 267–287. [Google Scholar] [CrossRef]
- Neethling, S.J.; Brito-Parada, P.R. Predicting flotation behaviour – The interaction between froth stability and performance. Miner. Eng. 2018, 120, 60–65. [Google Scholar] [CrossRef]
- Neethling, S.J.; Cilliers, J.J. Solids motion in flowing froths. Chem. Eng. Sci. 2002, 57, 607–615. [Google Scholar] [CrossRef]
- Richardson, J.F.; Zaki, W.N. Sedimentation and fluidisation: Part I. Trans. Inst. Chem. Eng. 1954, 32, 35–53. [Google Scholar] [CrossRef]
- Xu, M.; Finch, J.A.; Uribe-Salas, A. Maximum gas and bubble surface rates in flotation columns. Int. J. Miner. Process. 1991, 32, 233–250. [Google Scholar] [CrossRef]
- Bergh, L.G.; Yianatos, J.B. Experimental studies on flotation column dynamics. Miner. Eng. 1994, 7, 345–355. [Google Scholar] [CrossRef]
- Bergh, L.G.; Yianatos, J.B. Flotation column automation: State of the art. Control Eng. Pract. 2003, 11, 67–72. [Google Scholar] [CrossRef]
- Yianatos, J.B.; Bucarey, R.; Larenas, J.; Henríquez, F.; Torres, L. Collection zone kinetic model for industrial flotation columns. Miner. Eng. 2005, 18, 1373–1377. [Google Scholar] [CrossRef]
- Yianatos, J.B.; Henríquez, F.H.; Oroz, A.G. Characterization of large size flotation cells. Miner. Eng. 2006, 19, 531–538. [Google Scholar] [CrossRef]
- Bürger, R.; Diehl, S.; Martí, M.C.; Vásquez, Y. A difference scheme for a triangular system of conservation laws with discontinuous flux modeling three-phase flows. Netw. Heterog. Media 2023, 18, 140–190. [Google Scholar] [CrossRef]
- Diehl, S. A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients. J. Hyperbolic Differ. Equations 2009, 18, 127–159. [Google Scholar] [CrossRef] [Green Version]
- Wallis, G.B. The terminal speed of single drops or bubbles in an infinite medium. Int. J. Multiph. Flow 1974, 1, 491–511. [Google Scholar] [CrossRef]
- Diehl, S. Operating charts for continuous sedimentation III: Control of step inputs. J. Eng. Math. 2006, 54, 225–259. [Google Scholar] [CrossRef] [Green Version]
- Diehl, S. Operating charts for continuous sedimentation IV: Limitations for control of dynamic behaviour. J. Eng. Math. 2008, 60, 249–264. [Google Scholar] [CrossRef] [Green Version]
- Diehl, S. A regulator for continuous sedimentation in ideal clarifier-thickener units. J. Eng. Math. 2008, 60, 265–291. [Google Scholar] [CrossRef]
- Diehl, S.; Farås, S. Control of an ideal activated sludge process in wastewater treatment via an ODE-PDE model. J. Process Control 2013, 23, 359–381. [Google Scholar] [CrossRef] [Green Version]
- Betancourt, F.; Bürger, R.; Diehl, S.; Farås, S. Modelling and controlling clarifier-thickeners fed by suspensions with time-dependent properties. Miner. Eng. 2014, 62, 91–101. [Google Scholar] [CrossRef]
- Torfs, E.; Maere, T.; Bürger, R.; Diehl, S.; Nopens, I. Impact on sludge inventory and control strategies using the benchmark simulation model no. 1 with the Bürger-Diehl settler model. Water Sci. Technol. 2015, 71, 1524–1535. [Google Scholar] [CrossRef] [Green Version]
Symbol | Significance | Value |
---|---|---|
underflow level | ||
gas feed level | ||
pulp feed level | ||
wash water feed level | ||
overflow level | ||
A | interior cross-sectional area |
Instrument | Tag | Quantity Measured | Connected to PLC? |
---|---|---|---|
Mass flowmeter transmitter | FIT-01/02 | feed/discharge flowrate | yes |
Mass flowmeter controller | FIC-03 | air flowrate | yes |
Magnetic flowmeter | FT-01 | wash water flowrate | yes |
Variable frequency drive | SV-01/02/03 | pump velocity | yes |
Differential pressure transmitter | PT-01 | holdup | yes |
Feed manual valve | V-01 | —— | no |
Discharge manual valve | V-02 | —— | no |
Air manual valve | V-03 | —— | no |
Pressure taps valve | V-04 | —— | no |
Wash water manual valve | V-05 | —— | no |
Equipment | Tag | Type | Range/Dimensions and Unit |
Feed pump | P-01 | centrifuge | 20–110 L/min |
Discharge pump | P-02 | peristaltic | 0–18 L/min |
Wash water pump | P-03 | peristaltic | 0–12 L/min |
Regulator filter with water decanter | FLR | manual | 0–16 |
Pulp tank | T-01 | plastic cylinder | 200 L |
Flotation column | T-02 | acrylic tube | 55 L |
Wash water tank | T-03 | plastic cylinder | 200 L |
Experiment No. | 1 | 2 | 3 | 4 | 5 |
---|---|---|---|---|---|
[−] | 1 | 1 | 1 | 1 | 1 |
[−] | 0 | 0 | 0 | 0 | 0 |
[cm/s] | 1.3 | 1.8 | 2.3 | 1.8 | 1.8 |
[cm/s] | 0 | 0 | 0 | 0.3 | 0.5 |
[cm/s] | |||||
[cm/s] | see Figure 3 | see Figure 3 | see Figure 3 | see Figure 4 | see Figure 4 |
Parameter | Symbol | Working Range (Literature) | Range in Present Work |
---|---|---|---|
Froth height | 0.5–2.0 | 0.5–1.5 | |
Bubble diameter | 0.5–2.0 | 0.5–1.3 | |
Hold-up in zone 2 | 0.05–0.30 | 0.09–0.20 | |
Gas feed rate | 0.5–3.0 | 1.3–2.3 | |
Pulp feed rate | 0.2–2.0 | 0.8–1.5 | |
Discharge rate | 0.2–2.0 | 1.0–1.4 | |
Wash water rate | 0.2–1.0 | 0.3–0.5 |
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Betancourt, F.; Bürger, R.; Diehl, S.; Gutiérrez, L.; Martí, M.C.; Vásquez, Y. A Model of Froth Flotation with Drainage: Simulations and Comparison with Experiments. Minerals 2023, 13, 344. https://doi.org/10.3390/min13030344
Betancourt F, Bürger R, Diehl S, Gutiérrez L, Martí MC, Vásquez Y. A Model of Froth Flotation with Drainage: Simulations and Comparison with Experiments. Minerals. 2023; 13(3):344. https://doi.org/10.3390/min13030344
Chicago/Turabian StyleBetancourt, Fernando, Raimund Bürger, Stefan Diehl, Leopoldo Gutiérrez, M. Carmen Martí, and Yolanda Vásquez. 2023. "A Model of Froth Flotation with Drainage: Simulations and Comparison with Experiments" Minerals 13, no. 3: 344. https://doi.org/10.3390/min13030344