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Closed-form and finite difference solutions to a population balance model of grinding mills

机译：研磨机种群平衡模型的封闭形式和有限差分解

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摘要

The wear of steel balls in continuously operated grinding mills, used in mineral processing to comminute metalliferous rocks, can be described by a simple population balance model. This model gives rise to a scalar transport equation with a singular source term for the number density of balls as a function of size and time. Exact solutions to this equation are determined under the assumption of a simple power-law type wear law. It is shown that a particular term proposed in the engineering literature that describes the removal of used balls from the mill leads to negative solutions (Model 1). An alternative, more realistic term for the sieve action, which admits nonnegative solutions only, is introduced (Model 2). A working first-order finite difference scheme for Model 2 and a second-order TVD variant are introduced and applied for numerical simulations along with an error study. A weak solution concept for Model 2 is proposed, uniqueness of weak solutions is shown and convergence of the first-order scheme to a weak solution is established. These results hold for a general class of wear laws, not just power-law type.
机译：连续运行的磨机中钢球的磨损（可用于矿物加工以粉碎含金属的岩石）可通过简单的人口平衡模型来描述。该模型产生了一个标量输运方程，该方程具有一个奇异的源项，即球的数量密度与大小和时间的函数关系。在简单的幂律型磨损定律的假设下，可以确定该方程式的精确解。结果表明，工程文献中提出的描述从磨机中取出用过的钢球的特定术语会导致产生负溶液（模型1）。引入了筛分作用的一个更现实的替代术语，即仅接受非负解（模型2）。介绍了适用于模型2的有效一阶有限差分方案和二阶TVD变量，并将其应用于数值模拟以及误差研究。提出了模型2的弱解概念，显示了弱解的唯一性，并建立了一阶格式到弱解的收敛性。这些结果适用于一般的磨损定律类别，而不仅仅是幂律类型。

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