University of Manitoba, Department of Civil Engineering, 15 Gillson Street, Winnipeg, MB, Canada.
Water Res. 2012 Sep 1;46(13):4188-94. doi: 10.1016/j.watres.2012.04.031. Epub 2012 May 14.
Here we introduce a distribution of floc fractal dimensions as opposed to a single fractal dimension value into the floc settling velocity model developed in earlier studies. The distribution of fractal dimensions for a single floc size was assumed to cover a range from 1.9 to 3.0. This range was selected based on the theoretically determined fractal dimensions for diffusion-limited and cluster-cluster aggregation. These two aggregation mechanisms are involved in the formation of the lime softening flocs analyzed in this study. Fractal dimensions were generated under the assumption that a floc can have any value of normally distributed fractal dimensions ranging from 1.9-3.0. A range of settling velocities for a single floc size was calculated based on the distribution of fractal dimensions. The assumption of multiple fractal dimensions for a single floc size resulted in a non-unique relationship between the floc size and the floc settling velocity, i.e., several different settling velocities were calculated for one floc size. The settling velocities calculated according to the model ranged from 0 to 10 mm/s (average 2.22 mm/s) for the majority of flocs in the size range of 1-250 μm (average 125 μm). The experimentally measured settling velocities of flocs ranged from 0.1 to 7.1 mm/s (average 2.37 mm/s) for the flocs with equivalent diameters from 10 μm to 260 μm (average 124 μm). Experimentally determined floc settling velocities were predicted well by the floc settling model incorporating distributions of floc fractal dimensions calculated based on the knowledge of the mechanisms of aggregation, i.e., cluster-cluster aggregation and diffusion-limited aggregation.
在这里，我们引入了絮体分形维数的分布，而不是早期研究中开发的絮体沉降速度模型中的单个分形维数值。假设单个絮体大小的分形维数分布范围为 1.9 到 3.0。这个范围是根据扩散限制和团聚体团聚的理论确定的分形维数来选择的。这两种聚合机制都参与了本研究中分析的石灰软化絮体的形成。分形维数是在假设絮体可以具有任何值的正态分布分形维数（范围为 1.9-3.0）的情况下生成的。根据分形维数的分布，计算了单个絮体大小的一系列沉降速度。对于单个絮体大小的多个分形维度的假设导致絮体大小和絮体沉降速度之间的关系不唯一，即对于一个絮体大小计算了几个不同的沉降速度。根据该模型计算的沉降速度范围为 0 至 10mm/s（平均为 2.22mm/s），适用于 1 至 250μm 范围内的大多数絮体（平均为 125μm）。实验测量的絮体沉降速度范围为 0.1 至 7.1mm/s（平均为 2.37mm/s），适用于等效直径为 10μm 至 260μm 的絮体（平均为 124μm）。通过将基于团聚机制（即团聚体团聚和扩散限制团聚）的知识计算的絮体分形维数分布纳入絮体沉降模型，可以很好地预测实验确定的絮体沉降速度。
