It is widely accepted that the implementation of model-based predictive controllers (MPC) ensures optimal operation if an accurate model of the process is available. In the case of froth flotation, modelling for control purposes is a challenging task due to inherent process instabilities. Most models for control have only focused on the pulp phase rather than the froth phase, which is usually oversimplified or even neglected. Despite the fact that froth stability can significantly affect the overall performance of flotation cells, there is still a gap in literature regarding flotation models for control purposes that properly include froth physics.

In this paper we describe the development of a dynamic flotation model suitable for model predictive control, incorporating equations that describe the physics of flotation froths. Unlike other flotation models for control in the literature, the model proposed here includes important variables related to froth stability, such as bursting rate and air recovery, as well as simplified equations to calculate froth recovery and entrainment. These model equations allow estimating the amount of valuable material reporting to the concentrate, which can be used as a proxy to estimate grade and recovery. Additionally, pulp-froth interface physics was also included in our model, which enables a more accurate prediction of relevant flotation variables.

A sensitivity analysis of the parameters showed that two out of seven parameters were highly sensitive. The highly sensitive parameters are the exponential factor n of the equation for the overflowing bubble size, and the constant value a of the equation for the bursting rate. Although the other parameters showed a reasonably lower sensitivity than n and a, the results also revealed that there is a significant difference in the prediction accuracy if the parameters are poorly estimated. Further simulations of important variables for control exhibited a good adaptability to changes in typical variables, such as air and feed flowrates.

An analysis of degrees of freedom of the model established that two variables need to be fixed to have a completely determined system. This means that two variables are available for control purposes, which can be air and tailings flowrates (through the manipulation of the respective control valves). This study therefore paves the way for the implementation of a robust dynamic model for flotation predictive control, incorporating important froth phenomena.

人们普遍认为，如果过程的准确模型可用，则基于模型的预测控制器 (MPC) 的实施可确保最佳运行。在泡沫浮选的情况下，由于固有的过程不稳定性，用于控制目的的建模是一项具有挑战性的任务。大多数控制模型只关注纸浆阶段而不是泡沫阶段，这通常被过度简化甚至被忽视。尽管泡沫稳定性会显着影响浮选槽的整体性能，但有关用于控制目的的浮选模型的文献中仍然存在空白，其中适当地包括了泡沫物理。

在本文中，我们描述了适用于模型预测控制的动态浮选模型的开发，结合​​了描述浮选泡沫物理学的方程。与文献中用于控制的其他浮选模型不同，此处提出的模型包括与泡沫稳定性相关的重要变量，例如爆破率和空气回收率，以及用于计算泡沫回收率和夹带的简化方程。这些模型方程允许估计报告给精矿的有价值材料的数量，这可以用作估计品位和回收率的代理。此外，我们的模型中还包括纸浆-泡沫界面物理，这可以更准确地预测相关浮选变量。

参数的敏感性分析表明，七个参数中有两个是高度敏感的。高度敏感的参数是溢出气泡大小方程的指数因子n和爆裂率方程的常数值a。尽管其他参数的灵敏度比n和a 低，但结果还表明，如果参数估计不当，则预测精度会有显着差异。用于控制的重要变量的进一步模拟显示出对典型变量（例如空气和进料流速）变化的良好适应性。

对模型自由度的分析表明，需要固定两个变量才能拥有一个完全确定的系统。这意味着有两个变量可用于控制目的，它们可以是空气和尾矿流量（通过操纵相应的控制阀）。因此，这项研究为实施用于浮选预测控制的稳健动态模型铺平了道路，其中包含重要的泡沫现象。