The main objective of batch grinding modeling is the estimation of the product particle size distribution over time or specific energy input to the mill. So far, the developed analytical methods require often complicated calculations which are time-consuming. Thus, more simple approaches that allow the reliable prediction of the complete product size distribution need to be developed. In the present paper, an approach, based on the population balance model (PBM), that reliably predicts the product size distribution is proposed. In order to enable this, the simplified form of the fundamental batch grinding equation was transformed into the well-known Rosin–Rammler (R-R) distribution, thus allowing the determination of the breakage rate for each grinding period. A time-dependent breakage rate framework was developed where the traditional linear theory of the PBM is considered a partial case. This approach allows the deviations from the linear theory to be categorized and the degree of the acceleration-deceleration of the breakage rate to be predicted. Modeling results were validated by laboratory grinding studies using two homogeneous materials, quartz and marble, and one heterogeneous, a limonitic laterite. The experimental data revealed that grinding exhibits non-first-order behavior and the degree of deviation from the linear theory depends on the tested material. The reliability of the proposed model was validated with the use of the distribution modulus n, the a value of the breakage rate parameter, and the optimum n values that minimize the sum of the differences between the experimental and estimated particle size distributions.

批量研磨建模的主要目标是估算随时间变化的产品粒度分布或输入到磨机的比能。到目前为止，已开发的分析方法通常需要复杂的计算，这很耗时。因此，需要开发允许可靠地预测整个产品尺寸分布的更简单的方法。在本文中，提出了一种基于人口平衡模型（PBM）的可靠预测产品尺寸分布的方法。为了实现这一点，将基本批磨方程的简化形式转换为众所周知的Rosin-Rammler（RR）分布，从而可以确定每个磨削周期的破损率。建立了随时间变化的破损率框架，其中传统的PBM线性理论被认为是部分情况。这种方法可以对与线性理论的偏差进行分类，并可以预测破损率的加减速程度。通过实验室研磨研究，使用两种均质材料（石英和大理石）和一种非均质的褐铁质红土对模型结果进行了验证。实验数据表明，研磨表现出非一阶行为，并且偏离线性理论的程度取决于所测试的材料。利用分布模量验证了所提出模型的可靠性 这种方法可以对与线性理论的偏差进行分类，并可以预测破损率的加减速程度。通过实验室研磨研究，使用两种均质材料（石英和大理石）和一种非均质的褐铁质红土对模型结果进行了验证。实验数据表明，研磨表现出非一阶行为，并且偏离线性理论的程度取决于所测试的材料。利用分布模量验证了所提出模型的可靠性 这种方法可以对与线性理论的偏差进行分类，并可以预测破损率的加减速程度。通过实验室研磨研究，使用两种均质材料（石英和大理石）和一种非均质的褐铁质红土对模型化结果进行了验证。实验数据表明，研磨表现出非一阶行为，并且偏离线性理论的程度取决于所测试的材料。利用分布模量验证了所提出模型的可靠性 实验数据表明，研磨表现出非一阶行为，并且偏离线性理论的程度取决于所测试的材料。利用分布模量验证了所提出模型的可靠性 实验数据表明，研磨表现出非一阶行为，并且偏离线性理论的程度取决于所测试的材料。利用分布模量验证了所提出模型的可靠性n，破损率参数的a值和最佳n值，它们使实验和估计的粒度分布之间的差异之和最小。