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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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Figure 1.
Left: Schematic of a typical flotation column (after [21,38]), including heights of singular sources
Figure 2.
Flux functions' properties and specific volume fraction. Left: Drift-flux function
Figure 3.
The decreasing function
Figure 11.
Operating charts in which conditions (FⅠ)-(FⅢ) are satisfied in (a)
Figure 12.
Operating charts in which conditions (WⅠ)-(WⅡ) are satisfied in (a)
Figure 13.
Examples 1 and 2: possible steady states with
Figure 14.
Examples 1 and 2: Possible steady states with
Table 1.
Collection of possible steady states for the flotation column when
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Table 3.
Collection of possible steady-states for the flotation column when
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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
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