A conservation law with multiply discontinuous flux modelling a flotation column

Raimund Bürger; Stefan Diehl; María Carmen Martí; Raimund Bürger; Stefan Diehl; María Carmen Martí

doi:10.3934/nhm.2018015

Networks and Heterogeneous Media

2018, Volume 13, Issue 2: 339-371. doi: 10.3934/nhm.2018015

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A conservation law with multiply discontinuous flux modelling a flotation column

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

Received: 01 November 2017 Revised: 01 January 2018
Primary: 35L65, 35R05; Secondary: 76T10

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.
- Kinematic flow models,
- flotation,
- steady state,
- conservation law,
- discontinuous flux
Citation: Raimund Bürger, Stefan Diehl, María Carmen Martí. A conservation law with multiply discontinuous flux modelling a flotation column[J]. Networks and Heterogeneous Media, 2018, 13(2): 339-371. doi: 10.3934/nhm.2018015

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Abstract

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.

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