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A conservation law with multiply discontinuous flux modelling a flotation column

  • * Corresponding author: M.C. Martí

    * Corresponding author: M.C. Martí
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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.

    Mathematics Subject Classification: Primary: 35L65, 35R05; Secondary: 76T10.


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  • 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 $q = q(z, t)$.

    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 ${\rm{cm/s}}$.

    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.

    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 $0.35 \, {\rm{cm/s}}$.

    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 $0 \; {\rm{cm/s}}$.

    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.

    Figure 7.  Case F2, $q_2\leq0\leq q_3$: Possible intersections and steady states for zones 2 and 3.

    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.

    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.

    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 ${\rm{cm/s}}$.

    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 $q_{\rm{U}}^{\rm{SS}}$.

    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 $q_{\rm{G}}^{\rm{SS}}$.

    Figure 13.  Examples 1 and 2: possible steady states with $\phi_1 = 0$ for initial data corresponding to Figures 15(a)-(d).

    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).

    Figure 15.  Examples 1 and 2: possible steady states with gas (red) and fluid (blue) fluxes.

    Figure 16.  Example 1: time evolution of gas concentration from different angles.

    Figure 17.  Example 1: gas concentration profiles for (a) $t = 150$, (b) $t = 300$, (c) $t = 500$ and (d) $t = 2000$.

    Figure 18.  Example 2: time evolution of gas concentration from different angles.

    Figure 19.  Example 2: gas concentration profiles for (a) $t = 150$, (b) $t = 400$, (c) $t = 500$ and (d) $t = 2000$.

    Figure 20.  Example 3: possible steady states for initial data in Example 3.

    Figure 21.  Example 3: gas concentration profiles for (a) $t = 150$, (b) $t = 500$, (c) $t = 1000$ and (d) $t = 2000$.

    Figure 22.  Example 4: Steady state with gas (red) and fluid (blue) fluxes for a desliming test.

    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) $[0, \phi_{2}^{\rm{M}}]^*$ $[0, \phi_{3}^{\rm{M}}]$ $[0, \phi_{4{\rm{m}}}]$ $\displaystyle \frac{j_4(\phi_4)}{q_{\rm{E}}}$
    $[\phi_{1{\rm{Z}}}, 1]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{M}}}]\;{\rm{FII}}$
    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)
    $[\phi_{3{\rm{M}}}, 1]$ (FⅢ) $\phi_{4{\rm{M}}}$ (WⅡ)
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    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) $q_3\geq 0$
    $[\phi_{1{\rm{Z}}}, 1]$ $(\phi_{3}^{\rm{M}}, \phi_{3{\rm{Z}}}]$ ${q_3} \le 0$
     | Show Table
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    Table 2.  Admissible (green) and inadmissible (red) paths for steady-state construction in Table 1.

     | Show Table
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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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