A combined experimental and numerical approach to the assessment of floc settling velocity using fractal geometry

Sedimentation processes are fundamental to solids/liquid separation in water and wastewater treatment, and therefore a robust understanding of the settlement characteristics of mass fractal aggregates (flocs) formed in the flocculation stage is fundamental to optimized settlement tank design and operation. However, the use of settling as a technique to determine aggregates' traits is limited by current understanding of permeability. In this paper, we combine experimental and numerical approaches to assess settling velocities of fractal aggregates. Using a non-intrusive in situ digital image-based method, three- and two-dimensional fractal dimensions were calculated for kaolin-based flocs. By considering shape and fractal dimension, the porosity, density and settling velocities of the flocs were calculated individually, and settling velocities compared with those of spheres of the same density using Stokes' law. Shape analysis shows that the settling velocities for fractal aggregates may be greater or less than those for perfect spheres. For example, fractal aggregates with floc fractal dimension, D_f = 2.61, floc size, d_f > 320 μm and d_p = 7.5 μm settle with lower velocities than those predicted by Stokes' law; whilst, for D_f = 2.33, all aggregates of d_f > 70 μm and d_p = 7.5 μm settled below the velocity calculated by Stokes' law for spheres. Conversely, fractal settling velocities were higher than spheres for all the range of sizes, when D_f of 2.83 was simulated. The ratio of fractal aggregate to sphere settling velocity (the former being obtained from fractal porosity and density considerations), varied from 0.16 to 4.11 for aggregates in the range of 10 and 1,000 μm, primary particle size of 7.5 μm and a three-dimensional fractal dimension between 2.33 and 2.83. However, the ratio decreases to the range of 0.04-2.92 when primary particle size changes to 1.0 μm for the same fractal dimensions. Using the floc analysis technique developed here, the results demonstrate the difference in settlement behaviour between the approach developed here and the traditional Stokes' law approach using solid spheres. The technique and results demonstrate the improvements in understanding, and hence value to be derived, from an analysis based on fractal, rather than Euclidean, geometry when considering flocculation and subsequent clarification performance.