The increased sophistication of mathematical models depicting industrial processes requires advancement both in terms of expressing the state of participating materials and of describing their spatial distributions. In case of dispersed systems, the appropriate mathematical tools for such an advanced representation are the population balance equations and the computational fluid dynamics algorithms. In the presence of turbulence the situation is even more difficult because turbulence affects all size scales of the problem. Since the direct simulation of turbulence is impracticable with the present computational means, a large effort in modeling turbulent interactions is required. The present work is the follow up of a series of two recent works on the development of expressions for turbulent collisions frequency between bubbles and particles. Here, the whole route for transferring the localized collision information to the real-world device scale is presented, accounting for the dynamics of bubbles and particles populations. A particular procedure to (i) select the state and independent variables, (ii) select the occurring process, and (iii) mathematically reduce the multivariate population balance system after careful inspection of the problem in each stage of development, is proposed. The procedure is applied to a relatively complicated specific flotation process with the aim of demonstrating its details. A spatially homogeneous system of ordinary differential equations is derived and several results are illustrated in order to expose its usefulness. Finally, possible ways to transfer the spatially homogeneous model to a computational fluid dynamics environment are discussed.

描述工业过程的数学模型越来越复杂，需要在表达参与材料的状态和描述它们的空间分布方面取得进步。在分散系统的情况下，这种高级表示的适当数学工具是人口平衡方程和计算流体动力学算法。在存在湍流的情况下，情况更加困难，因为湍流会影响问题的所有大小尺度。由于目前的计算手段无法直接模拟湍流，因此需要在模拟湍流相互作用方面付出大量努力。目前的工作是最近两篇关于气泡和粒子之间湍流碰撞频率表达式发展的一系列工作的后续工作。这里，介绍了将局部碰撞信息传输到真实设备规模的整个路线，考虑了气泡和粒子群的动力学。提出了一个特定的程序，以 (i) 选择状态和自变量，(ii) 选择发生的过程，以及 (iii) 在仔细检查每个发展阶段的问题后，在数学上减少多元人口平衡系统。该程序应用于相对复杂的特定浮选过程，目的是展示其细节。导出了常微分方程的空间齐次系统，并说明了几个结果，以展示其实用性。最后，讨论了将空间均匀模型转移到计算流体动力学环境的可能方法。