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June  2018, 13(2): 339-371. doi: 10.3934/nhm.2018015

A conservation law with multiply discontinuous flux modelling a flotation column

Raimund Bürger 1, , Stefan Diehl 2, and  María Carmen Martí 3,,

1. 

CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile
2. 

Centre for Mathematical Sciences, Lund University, P.O. Box 118, S-221 00 Lund, Sweden
3. 

Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

* Corresponding author: M.C. Martí

Received  November 2017 Revised  January 2018 Published  May 2018

Flotation is a unit operation extensively used in the recovery of valuable minerals in mineral processing and related applications. Essential insight to the hydrodynamics of a flotation column can be obtained by studying just two phases: gas and fluid. To this end, the approach based on the drift-flux theory, proposed in similar form by several authors, is reformulated as a one-dimensional non-linear conservation law with a multiply discontinuous flux. The unknown is the gas volume fraction as a function of height and time, and the flux function depends discontinuously on spatial position due to several feed inlets. The resulting model is similar, but not equivalent, to previously studied clarifier-thickener models for solid-liquid separation and therefore adds a new real-world application to the field of conservation laws with discontinuous flux. Steady-state solutions are studied in detail, including their construction by applying an appropriate entropy condition across each flux discontinuity. This analysis leads to operating charts and tables collecting all possible steady states along with some necessary conditions for their feasibility in each case. Numerical experiments show that the transient model recovers the steady states, depending on the feed rates of the different inlets.

Keywords: Kinematic flow models, flotation, steady state, conservation law, discontinuous flux.
Mathematics Subject Classification: Primary: 35L65, 35R05; Secondary: 76T10.
Citation: Raimund Bürger, Stefan Diehl, María Carmen Martí. A conservation law with multiply discontinuous flux modelling a flotation column. Networks & Heterogeneous Media, 2018, 13 (2) : 339-371. doi: 10.3934/nhm.2018015
References:
[1]

B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Rational Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.  Google Scholar

[2]

O. A. Bascur, Example of a dynamic flotation framework, In: Centenary of Flotation Symposium, Brisbane, QLD, 6-9 June 2005, Australasian Institute of Mining and Metallurgy Publication Series, 2005, 85-91. Google Scholar

[3]

R. Bürger, S. Diehl, S. Farås, I. Nopens and E. Torfs, A consistent modelling methodology for secondary settling tanks: A reliable numerical method, Water Sci. Technol., 68 (2013), 192-208.   Google Scholar

[4]

R. Bürger, S. Diehl and C. Mejías, A difference scheme for a degenerating convection-diffusion-reaction system modelling continuous sedimentation, ESAIM: Math. Model. Numer. Anal., to appear. Google Scholar

[5]

R. Bürger and K. H. Karlsen, Conservation laws with discontinuous flux: A short introduction, J. Eng. Math., 60 (2008), 241-247.  doi: 10.1007/s10665-008-9213-7.  Google Scholar

[6]

R. Bürger, A. García, K. H. Karlsen and J. D. Towers, On an extended clarifier-thickener model with singular source and sink terms, Eur. J. Appl. Math., 17 (2006), 257-292.  doi: 10.1017/S0956792506006619.  Google Scholar

[7]

R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Eng. Math., 60 (2008), 387-425.  doi: 10.1007/s10665-007-9148-4.  Google Scholar

[8]

R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Well-posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math., 97 (2004), 25-65.  doi: 10.1007/s00211-003-0503-8.  Google Scholar

[9]

R. Bürger, K. H. Karlsen, H. Torres and J. D. Towers, Second-order schemes for conservation laws with discontinuous flux modelling clarifier-thickener units, Numer. Math., 116 (2010), 579-617.  doi: 10.1007/s00211-010-0325-4.  Google Scholar

[10]

R. Bürger, K. H. Karlsen and J. D. Towers, A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units, SIAM J. Appl. Math., 65 (2005), 882-940.  doi: 10.1137/04060620X.  Google Scholar

[11]

R. Bürger, K. H. Karlsen and J. D. Towers, On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux, Netw. Heterog. Media, 5 (2010), 461-485.  doi: 10.3934/nhm.2010.5.461.  Google Scholar

[12]

J. M. Coulson, J. F. Richardson, J. R. Backhurst and J. H. Harker, Coulson and Richardson's Chemical Engineering. Volume 2: Particle Technology and Separation Processes, Fourth Ed., Butterworth-Heinemann, Oxford, 2000. Google Scholar

[13]

J. E. Dickinson and K. P. Galvin, Fluidized bed desliming in fine particle flotation, Part Ⅰ, Chem. Eng. Sci., 108 (2014), 283-298.  doi: 10.1016/j.ces.2013.11.006.  Google Scholar

[14]

S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), 1425-1451.  doi: 10.1137/S0036141093242533.  Google Scholar

[15]

S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math., 56 (1996), 388-419.  doi: 10.1137/S0036139994242425.  Google Scholar

[16]

S. Diehl, Operating charts for continuous sedimentation Ⅰ: Control of steady states, J. Eng. Math., 41 (2001), 117-144.  doi: 10.1023/A:1011959425670.  Google Scholar

[17]

S. Diehl, Operating charts for continuous sedimentation Ⅱ: Step responses, J. Eng. Math., 53 (2005), 139-185.  doi: 10.1007/s10665-005-6430-1.  Google Scholar

[18]

S. Diehl, A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Differential Equations, 6 (2009), 127-159.  doi: 10.1142/S0219891609001794.  Google Scholar

[19]

S. Diehl, Numerical identification of constitutive functions in scalar nonlinear convection-diffusion equations with application to batch sedimentation, Appl. Numer. Math., 95 (2015), 154-172.  doi: 10.1016/j.apnum.2014.04.002.  Google Scholar

[20]

S. Diehl, S. Farås and G. Mauritsson, Fast reliable simulations of secondary settling tanks in wastewater treatment with semi-implicit time discretization, Comput. Math. Appl., 70 (2015), 459-477.  doi: 10.1016/j.camwa.2015.05.005.  Google Scholar

[21]

J. A. Finch and G. S. Dobby, Column Flotation, Pergamon Press, London, 1990. Google Scholar

[22]

K. P. Galvin and J. E. Dickinson, bed desliming in fine particle flotation Part Ⅱ: Flotation of a model feed, Chem. Eng. Sci., 108 (2014), 299-309.  doi: 10.1016/j.ces.2013.11.027.  Google Scholar

[23]

S. K. Godunov, Finite difference methods for numerical computation of discontinuous solutions of equations of fluid dynamics, Mat. Sb., 47 (1959), 271-306.   Google Scholar

[24]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Second Edition, Springer Verlag, Berlin, 2015. doi: 10.1007/978-3-662-47507-2.  Google Scholar

[25]

P. M. Ireland and G. J. Jameson, Liquid transport in multi-layer froths, J. Colloid Interf. Sci., 314 (2007), 207-213.   Google Scholar

[26]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243.   Google Scholar

[27]

D. E. Langberg and G. J. Jameson, The coexistence of the froth and liquid phases in a flotation column, Chem. Eng. Sci., 47 (1992), 4345-4355.  doi: 10.1016/0009-2509(92)85113-P.  Google Scholar

[28]

S. Mishra, Numerical methods for conservation laws with discontinuous coefficients, Chapter 18 in R. Abgrall and C.-W. Shu (eds.), Handbook of Numerical Methods for Hyperbolic Problems: Applied and Modern Issues, North Holland, 18 (2017), 479-506.  Google Scholar

[29]

G. Narsimhan, Analysis of creaming and formation of foam layer in aerated liquid, J. Colloid Interface Sci., 345 (2010), 566-572.  doi: 10.1016/j.jcis.2010.02.003.  Google Scholar

[30]

O. A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation Uspekhi Mat. Nauk., 14 (1959), 165-170. Amer. Math. Soc. Trans. Ser. 2, 33 (1964), 285-290.  Google Scholar

[31]

R. Pal and J. H. Masliyah, Flow characterization of a flotation column, Canad. J. Chem. Eng., 67 (1989), 916-923.  doi: 10.1002/cjce.5450670608.  Google Scholar

[32]

R. Pal and J. H. Masliyah, Oil recovery from oil in water emulsions using a flotation column, Canad. J. Chem. Eng., 68 (1990), 959-967.  doi: 10.1002/cjce.5450680611.  Google Scholar

[33]

C.-H. Park, N. Subasinghe and O.-J. Han, Amenability testing of fine coal beneficiation using laboratory flotation column, Materials Transactions, 56 (2015), 766-773.   Google Scholar

[34]

J. F. Richardson and W. N. Zaki, Sedimentation and fluidisation: Part Ⅰ, Chemical Engineering Research and Design, 75 (1997), 82-100.  doi: 10.1016/S0263-8762(97)80006-8.  Google Scholar

[35]

K. Rietema, Science and technology of dispersed two-phase systems—Ⅰ and Ⅱ, Chem. Eng. Sci., 37 (1982), 1125-1150.  doi: 10.1016/0009-2509(82)85058-6.  Google Scholar

[36]

A. Rushton, A. S. Ward and R. G. Holdich, Solid-Liquid Filtration and Separation Technology, Wiley Online Library, 2008. doi: 10.1002/9783527614974.  Google Scholar

[37]

P. Stevenson, S. Ata and G. M. Evans, Convective-dispersive gangue transport in flotation froth, Chem. Eng. Sci., 62 (2007), 5736-5744.   Google Scholar

[38]

P. Stevenson, P. S. Fennell and K. P. Galvin, On the drift-flux analysis of flotation and foam fractionation processes, Canad. J. Chem. Eng., 86 (2008), 635-642.  doi: 10.1002/cjce.20076.  Google Scholar

[39]

J. Vandenberghe, J. Chung, Z. Xu and J. Masliyah, Drift flux modelling for a two-phase system in a flotation column, Canad. J. Chem. Eng., 83 (2005), 169-176.   Google Scholar

[40]

G. B. Wallis, One-Dimensional two-Phase Flow, McGraw-Hill, New York, 1969. Google Scholar

[41]

G. B. Wallis, The terminal speed of single drops or bubbled in an infinite medium, Int. J. Multiphase Flow, 1 (1974), 491-511.   Google Scholar

[42]

B. A. Wills and T. J. Napier-Munn, Wills' Mineral Processing Technology, Seventh Edition, Butterworth-Heinemann, Oxford, 2006. Google Scholar

[43]

J. B. Yianatos, J. A. Finch, G. S. Dobby and M. Xiu, Bubble size estimation in a bubble swarm, J. Colloid Interf. Sci., 126 (1988), 37-44.  doi: 10.1016/0021-9797(88)90096-3.  Google Scholar

[44]

Z. A. Zhou and N. O. Egiebor, A gas-liquid drift-flux model for flotation columns, Minerals Eng., 6 (1993), 199-205.   Google Scholar

show all references

References:
[1]

B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Rational Mech. Anal., 201 (2011), 27-86.  doi: 10.1007/s00205-010-0389-4.  Google Scholar

[2]

O. A. Bascur, Example of a dynamic flotation framework, In: Centenary of Flotation Symposium, Brisbane, QLD, 6-9 June 2005, Australasian Institute of Mining and Metallurgy Publication Series, 2005, 85-91. Google Scholar

[3]

R. Bürger, S. Diehl, S. Farås, I. Nopens and E. Torfs, A consistent modelling methodology for secondary settling tanks: A reliable numerical method, Water Sci. Technol., 68 (2013), 192-208.   Google Scholar

[4]

R. Bürger, S. Diehl and C. Mejías, A difference scheme for a degenerating convection-diffusion-reaction system modelling continuous sedimentation, ESAIM: Math. Model. Numer. Anal., to appear. Google Scholar

[5]

R. Bürger and K. H. Karlsen, Conservation laws with discontinuous flux: A short introduction, J. Eng. Math., 60 (2008), 241-247.  doi: 10.1007/s10665-008-9213-7.  Google Scholar

[6]

R. Bürger, A. García, K. H. Karlsen and J. D. Towers, On an extended clarifier-thickener model with singular source and sink terms, Eur. J. Appl. Math., 17 (2006), 257-292.  doi: 10.1017/S0956792506006619.  Google Scholar

[7]

R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Eng. Math., 60 (2008), 387-425.  doi: 10.1007/s10665-007-9148-4.  Google Scholar

[8]

R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Well-posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math., 97 (2004), 25-65.  doi: 10.1007/s00211-003-0503-8.  Google Scholar

[9]

R. Bürger, K. H. Karlsen, H. Torres and J. D. Towers, Second-order schemes for conservation laws with discontinuous flux modelling clarifier-thickener units, Numer. Math., 116 (2010), 579-617.  doi: 10.1007/s00211-010-0325-4.  Google Scholar

[10]

R. Bürger, K. H. Karlsen and J. D. Towers, A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units, SIAM J. Appl. Math., 65 (2005), 882-940.  doi: 10.1137/04060620X.  Google Scholar

[11]

R. Bürger, K. H. Karlsen and J. D. Towers, On some difference schemes and entropy conditions for a class of multi-species kinematic flow models with discontinuous flux, Netw. Heterog. Media, 5 (2010), 461-485.  doi: 10.3934/nhm.2010.5.461.  Google Scholar

[12]

J. M. Coulson, J. F. Richardson, J. R. Backhurst and J. H. Harker, Coulson and Richardson's Chemical Engineering. Volume 2: Particle Technology and Separation Processes, Fourth Ed., Butterworth-Heinemann, Oxford, 2000. Google Scholar

[13]

J. E. Dickinson and K. P. Galvin, Fluidized bed desliming in fine particle flotation, Part Ⅰ, Chem. Eng. Sci., 108 (2014), 283-298.  doi: 10.1016/j.ces.2013.11.006.  Google Scholar

[14]

S. Diehl, On scalar conservation laws with point source and discontinuous flux function, SIAM J. Math. Anal., 26 (1995), 1425-1451.  doi: 10.1137/S0036141093242533.  Google Scholar

[15]

S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math., 56 (1996), 388-419.  doi: 10.1137/S0036139994242425.  Google Scholar

[16]

S. Diehl, Operating charts for continuous sedimentation Ⅰ: Control of steady states, J. Eng. Math., 41 (2001), 117-144.  doi: 10.1023/A:1011959425670.  Google Scholar

[17]

S. Diehl, Operating charts for continuous sedimentation Ⅱ: Step responses, J. Eng. Math., 53 (2005), 139-185.  doi: 10.1007/s10665-005-6430-1.  Google Scholar

[18]

S. Diehl, A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Differential Equations, 6 (2009), 127-159.  doi: 10.1142/S0219891609001794.  Google Scholar

[19]

S. Diehl, Numerical identification of constitutive functions in scalar nonlinear convection-diffusion equations with application to batch sedimentation, Appl. Numer. Math., 95 (2015), 154-172.  doi: 10.1016/j.apnum.2014.04.002.  Google Scholar

[20]

S. Diehl, S. Farås and G. Mauritsson, Fast reliable simulations of secondary settling tanks in wastewater treatment with semi-implicit time discretization, Comput. Math. Appl., 70 (2015), 459-477.  doi: 10.1016/j.camwa.2015.05.005.  Google Scholar

[21]

J. A. Finch and G. S. Dobby, Column Flotation, Pergamon Press, London, 1990. Google Scholar

[22]

K. P. Galvin and J. E. Dickinson, bed desliming in fine particle flotation Part Ⅱ: Flotation of a model feed, Chem. Eng. Sci., 108 (2014), 299-309.  doi: 10.1016/j.ces.2013.11.027.  Google Scholar

[23]

S. K. Godunov, Finite difference methods for numerical computation of discontinuous solutions of equations of fluid dynamics, Mat. Sb., 47 (1959), 271-306.   Google Scholar

[24]

H. Holden and N. H. Risebro, Front Tracking for Hyperbolic Conservation Laws, Second Edition, Springer Verlag, Berlin, 2015. doi: 10.1007/978-3-662-47507-2.  Google Scholar

[25]

P. M. Ireland and G. J. Jameson, Liquid transport in multi-layer froths, J. Colloid Interf. Sci., 314 (2007), 207-213.   Google Scholar

[26]

S. N. Kružkov, First order quasilinear equations in several independent variables, Math. USSR-Sb., 10 (1970), 217-243.   Google Scholar

[27]

D. E. Langberg and G. J. Jameson, The coexistence of the froth and liquid phases in a flotation column, Chem. Eng. Sci., 47 (1992), 4345-4355.  doi: 10.1016/0009-2509(92)85113-P.  Google Scholar

[28]

S. Mishra, Numerical methods for conservation laws with discontinuous coefficients, Chapter 18 in R. Abgrall and C.-W. Shu (eds.), Handbook of Numerical Methods for Hyperbolic Problems: Applied and Modern Issues, North Holland, 18 (2017), 479-506.  Google Scholar

[29]

G. Narsimhan, Analysis of creaming and formation of foam layer in aerated liquid, J. Colloid Interface Sci., 345 (2010), 566-572.  doi: 10.1016/j.jcis.2010.02.003.  Google Scholar

[30]

O. A. Oleinik, Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation Uspekhi Mat. Nauk., 14 (1959), 165-170. Amer. Math. Soc. Trans. Ser. 2, 33 (1964), 285-290.  Google Scholar

[31]

R. Pal and J. H. Masliyah, Flow characterization of a flotation column, Canad. J. Chem. Eng., 67 (1989), 916-923.  doi: 10.1002/cjce.5450670608.  Google Scholar

[32]

R. Pal and J. H. Masliyah, Oil recovery from oil in water emulsions using a flotation column, Canad. J. Chem. Eng., 68 (1990), 959-967.  doi: 10.1002/cjce.5450680611.  Google Scholar

[33]

C.-H. Park, N. Subasinghe and O.-J. Han, Amenability testing of fine coal beneficiation using laboratory flotation column, Materials Transactions, 56 (2015), 766-773.   Google Scholar

[34]

J. F. Richardson and W. N. Zaki, Sedimentation and fluidisation: Part Ⅰ, Chemical Engineering Research and Design, 75 (1997), 82-100.  doi: 10.1016/S0263-8762(97)80006-8.  Google Scholar

[35]

K. Rietema, Science and technology of dispersed two-phase systems—Ⅰ and Ⅱ, Chem. Eng. Sci., 37 (1982), 1125-1150.  doi: 10.1016/0009-2509(82)85058-6.  Google Scholar

[36]

A. Rushton, A. S. Ward and R. G. Holdich, Solid-Liquid Filtration and Separation Technology, Wiley Online Library, 2008. doi: 10.1002/9783527614974.  Google Scholar

[37]

P. Stevenson, S. Ata and G. M. Evans, Convective-dispersive gangue transport in flotation froth, Chem. Eng. Sci., 62 (2007), 5736-5744.   Google Scholar

[38]

P. Stevenson, P. S. Fennell and K. P. Galvin, On the drift-flux analysis of flotation and foam fractionation processes, Canad. J. Chem. Eng., 86 (2008), 635-642.  doi: 10.1002/cjce.20076.  Google Scholar

[39]

J. Vandenberghe, J. Chung, Z. Xu and J. Masliyah, Drift flux modelling for a two-phase system in a flotation column, Canad. J. Chem. Eng., 83 (2005), 169-176.   Google Scholar

[40]

G. B. Wallis, One-Dimensional two-Phase Flow, McGraw-Hill, New York, 1969. Google Scholar

[41]

G. B. Wallis, The terminal speed of single drops or bubbled in an infinite medium, Int. J. Multiphase Flow, 1 (1974), 491-511.   Google Scholar

[42]

B. A. Wills and T. J. Napier-Munn, Wills' Mineral Processing Technology, Seventh Edition, Butterworth-Heinemann, Oxford, 2006. Google Scholar

[43]

J. B. Yianatos, J. A. Finch, G. S. Dobby and M. Xiu, Bubble size estimation in a bubble swarm, J. Colloid Interf. Sci., 126 (1988), 37-44.  doi: 10.1016/0021-9797(88)90096-3.  Google Scholar

[44]

Z. A. Zhou and N. O. Egiebor, A gas-liquid drift-flux model for flotation columns, Minerals Eng., 6 (1993), 199-205.   Google Scholar

Figure 1.  Left: Schematic of a typical flotation column (after [21,38]), including heights of singular sources $z_{{\rm{G}}}$, $z_{{\rm{F}}}$ and $z_{{\rm{W}}}$, the underflow level $z_{\rm{U}}$, and the effluent level $z_{\rm{E}}$. Right: Corresponding conceptual model of the flotation column used in this work, indicating the volumetric feed flows $Q_{{\rm{G}}}$, $Q_{{\rm{F}}}$ and $Q_{{\rm{W}}}$, the underflow rate $Q_{\rm{U}}$, the effluent rate $Q_{\rm{E}}$, and the spatially piecewise constant bulk velocity $q = q(z, t)$.
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Figure 2.  Flux functions' properties and specific volume fraction. Left: Drift-flux function $j_{\rm{g}}$ and flux curves for zones $1$ and $4$. Right: The local minimum $\phi_{\rm{M}}$ and appurtenant $\phi_{\rm{m}}$ for a zone flux $j$ with positive $q$, and flux curves with zero derivatives at $\phi_{\rm{max}} = 1$ and $\phi_{\rm{infl}}$. In these and other plots, we have used the expression (2.4) with $n_{\rm{RZ}} = 3.2$ in the drift-flux function $j_{\rm{g}}$. The unit on the vertical axis is ${\rm{cm/s}}$.
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Figure 3.  The decreasing function $\check{\jmath}_{\rm{U}}(\cdot;\phi_{\rm{U}}) = j_{\rm{U}}$ and three possible cases of graphs of ${\hat{\jmath}}(\cdot;\phi_1)$ depending on $\phi_1$. The intersection of ${\hat{\jmath}}(\cdot;\phi_1)$ and $j_{\rm{U}}$ defines the possible values in a steady-state solution.
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Figure 4.  Case G1: $q_1\leq 0\leq q_2$. Possible steady-state values for zones 1 and 2. The gas injection velocity is set to $q_{\rm{G}} = 0.2 \, {\rm{cm/s}}$ in all subplots except for (d) where it is $0.35 \, {\rm{cm/s}}$.
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Figure 5.  Case G2: $q_1\leq q_2 \leq 0$. Possible steady-state values for zones 1 and 2. The value of $q_{\rm{G}}$ is set to $q_{\rm{G}} = 0.2 \; {\rm{cm/s}}$ except for plot (b2), where it is $0 \; {\rm{cm/s}}$.
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Figure 6.  Case F1: $0\leq q_2\leq q_3$. Possible steady-state values for zones 2 and 3. In the special case $q_2 = q_3$, the diagonal plots (a), (e) and (i) are the only ones where $\phi_2 = \phi_3$ occurs.
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Figure 7.  Case F2, $q_2\leq0\leq q_3$: Possible intersections and steady states for zones 2 and 3.
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Figure 8.  Case F3, $q_2\leq q_3\leq0$. Possible intersections and steady-state values for zones 2 and 3. (d1) and (d2) correspond to positive and negative intersection flux values in subcase (d), respectively.
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Figure 9.  Case W1: $0\leq q_3\leq q_4$. Possible steady-state values for zones 3 and 4. Subcases (b) and (c) are not plotted since they are empty cases.
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Figure 10.  Operating charts in which condition (G) is satisfied in (a) $(q_2, q_{\rm{G}})$-plane and (b) $(q_{\rm{U}}, q_{\rm{G}})$-plane. As usual, the unit of $q$-fluxes is ${\rm{cm/s}}$.
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Figure 11.  Operating charts in which conditions (FⅠ)-(FⅢ) are satisfied in (a) $(q_2, q_3)$-plane (where $q_3\geq q_2$ holds) and (b) $(q_{\rm{G}}, q_{\rm{F}})$-plane. The value $q_{\rm{neg}} = -2.6941\, {\rm{cm/s}}$ is not shown in these and further plots. In the latter plot, the scale of the horizontal axis is adjusted with respect to the previously chosen fixed value $q_{\rm{U}}^{\rm{SS}}$.
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Figure 12.  Operating charts in which conditions (WⅠ)-(WⅡ) are satisfied in (a) $(q_3, q_4)$-plane and (b) $(q_{\rm{F}}, q_{\rm{W}})$-plane. In the latter plot, the scale of the horizontal axis is adjusted with respect to the previously chosen fixed values $q_{\rm{U}}^{\rm{SS}}$ and $q_{\rm{G}}^{\rm{SS}}$.
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Figure 13.  Examples 1 and 2: possible steady states with $\phi_1 = 0$ for initial data corresponding to Figures 15(a)-(d).
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Figure 14.  Examples 1 and 2: Possible steady states with $\phi_1\in[\phi_{1 {\rm{Z}}}, 1]$ for initial data corresponding to Figures 15(e)-(h).
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Figure 15.  Examples 1 and 2: possible steady states with gas (red) and fluid (blue) fluxes.
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Figure 16.  Example 1: time evolution of gas concentration from different angles.
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Figure 17.  Example 1: gas concentration profiles for (a) $t = 150$, (b) $t = 300$, (c) $t = 500$ and (d) $t = 2000$.
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Figure 18.  Example 2: time evolution of gas concentration from different angles.
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Figure 19.  Example 2: gas concentration profiles for (a) $t = 150$, (b) $t = 400$, (c) $t = 500$ and (d) $t = 2000$.
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Figure 20.  Example 3: possible steady states for initial data in Example 3.
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Figure 21.  Example 3: gas concentration profiles for (a) $t = 150$, (b) $t = 500$, (c) $t = 1000$ and (d) $t = 2000$.
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Figure 22.  Example 4: Steady state with gas (red) and fluid (blue) fluxes for a desliming test.
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Table 1.  Collection of possible steady states for the flotation column when $q_2 = q_{\rm{G}}-q_{\rm{U}}\geq0$. $^{(\ast)}$When $\phi_1\in[\phi_{1{\rm{Z}}}, 1]$ then $\phi_2 = \phi_{2}^{\rm{M}}$.
$\phi_{\rm{U}}$ $\phi_1$ $\phi_2$ $\phi_3$ $\phi_4$ $\phi_{\rm{E}}$
$\displaystyle -\frac{j_1(\phi_1)}{q_{\rm{U}}}$ 0 (G) $[0, \phi_{2}^{\rm{M}}]^*$ $[0, \phi_{3}^{\rm{M}}]$ $[0, \phi_{4{\rm{m}}}]$ $\displaystyle \frac{j_4(\phi_4)}{q_{\rm{E}}}$
$\phi_{4{\rm{M}}}\;{\rm{(WI)}}$
$[\phi_{1{\rm{Z}}}, 1]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]\;{\rm{FII}}$
0 (G) $(\phi_{2}^{\rm{M}}, \phi_{2{\rm{M}}}]$
$[\phi_{1{\rm{Z}}}, 1]$
0 $(q_1=0)$ 1 $[0, \phi_{3}^{\rm{M}}]$ (FⅠ) $[0, \phi_{4{\rm{m}}}]$
$\phi_{4{\rm{M}}}$ (WⅠ)
$(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]$ (FⅢ)
1
$[\phi_{3{\rm{M}}}, 1]$ (FⅢ) $\phi_{4{\rm{M}}}$ (WⅡ)
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$\phi_{\rm{U}}$ $\phi_1$ $\phi_2$ $\phi_3$ $\phi_4$ $\phi_{\rm{E}}$
$\displaystyle -\frac{j_1(\phi_1)}{q_{\rm{U}}}$ 0 (G) $[0, \phi_{2}^{\rm{M}}]^*$ $[0, \phi_{3}^{\rm{M}}]$ $[0, \phi_{4{\rm{m}}}]$ $\displaystyle \frac{j_4(\phi_4)}{q_{\rm{E}}}$
$\phi_{4{\rm{M}}}\;{\rm{(WI)}}$
$[\phi_{1{\rm{Z}}}, 1]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]\;{\rm{FII}}$
0 (G) $(\phi_{2}^{\rm{M}}, \phi_{2{\rm{M}}}]$
$[\phi_{1{\rm{Z}}}, 1]$
0 $(q_1=0)$ 1 $[0, \phi_{3}^{\rm{M}}]$ (FⅠ) $[0, \phi_{4{\rm{m}}}]$
$\phi_{4{\rm{M}}}$ (WⅠ)
$(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]$ (FⅢ)
1
$[\phi_{3{\rm{M}}}, 1]$ (FⅢ) $\phi_{4{\rm{M}}}$ (WⅡ)
Table 3.  Collection of possible steady-states for the flotation column when $q_2 = q_{\rm{G}}-q_{\rm{U}}<0$. $^{(\ast)}$When $\phi_1\in[\phi_{1{\rm{Z}}}, 1]$ then $\phi_2 = \phi_{2}^{\rm{M}}$.
$\phi_{\rm{U}}$ $\phi_1$ $\phi_2$ $\phi_3$ $\phi_4$ $\phi_{\rm{E}}$
$\displaystyle -\frac{j_1(\phi_1)}{q_{\rm{U}}}$ 0 (G) $[0, \phi_{2}^{\rm{M}}]^*$ $[0, \phi_{3}^{\rm{M}}]$ $[0, \phi_{4{\rm{m}}}]$ $\displaystyle \frac{j_4(\phi_4)}{q_{\rm{E}}}$
$\phi_{4{\rm{M}}}$ (WⅠ)
$(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]$ (FⅡ) $q_3\geq 0$
$[\phi_{1{\rm{Z}}}, 1]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{Z}}}]$ ${q_3} \le 0$
0 (G) $(\phi_{2}^{\rm{M}}, \phi_{2{\rm{Z}}}]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]$ (FⅡ) $q_3\geq 0$
$[\phi_{1{\rm{Z}}}, 1]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{Z}}}]$ ${q_3} \le 0$
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$\phi_{\rm{U}}$ $\phi_1$ $\phi_2$ $\phi_3$ $\phi_4$ $\phi_{\rm{E}}$
$\displaystyle -\frac{j_1(\phi_1)}{q_{\rm{U}}}$ 0 (G) $[0, \phi_{2}^{\rm{M}}]^*$ $[0, \phi_{3}^{\rm{M}}]$ $[0, \phi_{4{\rm{m}}}]$ $\displaystyle \frac{j_4(\phi_4)}{q_{\rm{E}}}$
$\phi_{4{\rm{M}}}$ (WⅠ)
$(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]$ (FⅡ) $q_3\geq 0$
$[\phi_{1{\rm{Z}}}, 1]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{Z}}}]$ ${q_3} \le 0$
0 (G) $(\phi_{2}^{\rm{M}}, \phi_{2{\rm{Z}}}]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]$ (FⅡ) $q_3\geq 0$
$[\phi_{1{\rm{Z}}}, 1]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{Z}}}]$ ${q_3} \le 0$
Table 2.  Admissible (green) and inadmissible (red) paths for steady-state construction in Table 1.
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Table 4.  Examples 1 and 2: admissible paths for steady states with $\phi_1 = 0$, corresponding to steady states (a)-(d) in Figure 13.
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