一个模拟带排液泡沫浮选的退化对流-扩散系统
A degenerating convection-diffusion system modelling froth flotation with drainage
AI总结：
本文将毛细排液效应纳入泡沫浮选的双曲型偏微分方程模型，构造稳态解并给出操作图可行性条件，同时提出单调数值格式模拟浮选柱动态行为。
摘要：
泡沫浮选是矿物加工中常用的一种单元操作，用于从细磨矿石中分离有价值的矿物颗粒与无价值的脉石颗粒。有价值的矿物颗粒具有疏水性，会附着在注入矿浆中的空气气泡上。这形成了气泡-颗粒聚集体，它们上升到浮选柱顶部，在那里积聚成泡沫层，并通过溢流槽移出以进行进一步处理。同时，亲水性的脉石颗粒沉降并被连续移除。由于毛细作用引起的液体排液对于形成稳定的泡沫层至关重要。这一效应被纳入先前建立的一个双曲型偏微分方程组中，该方程组模拟了浮升聚集体和沉降亲水性固体的体积分数 [R. Bürger, S. Diehl and M.C. Martí, IMA J. Appl. Math. 84 (2019) 930--973]。文中详细阐述了具有泡沫层的期望稳态解的构造，并在所谓操作图中可视化了进料体积分数以及进料、底流和冲洗水体积流量的可行性条件。推导了一个单调数值格式，并将其用于模拟浮选柱的动态行为。还证明了在适当的Courant-Friedrichs-Lewy（CFL）条件下，当初始数据介于零和一之间时，近似体积分数也保持在零和一之间。
英文摘要：
Froth flotation is a common unit operation used in mineral processing. It serves to separate valuable mineral particles from worthless gangue particles in finely ground ores. The valuable mineral particles are hydrophobic and attach to bubbles of air injected into the pulp. This creates bubble-particle aggregates that rise to the top of the flotation column where they accumulate to a froth or foam layer that is removed through a launder for further processing. At the same time, the hydrophilic gangue particles settle and are removed continuously. The drainage of liquid due to capillarity is essential for the formation of a stable froth layer. This effect is included into a previously formulated hyperbolic system of partial differential equations that models the volume fractions of floating aggregates and settling hydrophilic solids [R. Bürger, S. Diehl and M.C. Martí, {\it IMA J. Appl. Math.} {\bf 84} (2019) 930--973]. The construction of desired steady-state solutions with a froth layer is detailed and feasibility conditions on the feed volume fractions and the volumetric flows of feed, underflow and wash water are visualized in so-called operating charts. A monotone numerical scheme is derived and employed to simulate the dynamic behaviour of a flotation column. It is also proven that, under a suitable Courant-Friedrichs-Lewy (CFL) condition, the approximate volume fractions are bounded between zero and one when the initial data are.
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