Application of fractal dimensions to study the structure of flocs formed in lime softening process

Abstract

The use of fractal dimensions to study the internal structure and settling of flocs formed in lime softening process was investigated. Fractal dimensions of flocs were measured directly on floc images and indirectly from their settling velocity. An optical microscope with a motorized stage was used to measure the fractal dimensions of lime softening flocs directly on their images in 2 and 3D space. The directly determined fractal dimensions of the lime softening flocs were 1.11–1.25 for floc boundary, 1.82–1.99 for cross-sectional area and 2.6–2.99 for floc volume. The fractal dimension determined indirectly from the flocs settling rates was 1.87 that was different from the 3D fractal dimension determined directly on floc images. This discrepancy is due to the following incorrect assumptions used for fractal dimensions determined from floc settling rates: linear relationship between square settling velocity and floc size (Stokes’ Law), Euclidean relationship between floc size and volume, constant fractal dimensions and one primary particle size describing entire population of flocs. Floc settling model incorporating variable floc fractal dimensions as well as variable primary particle size was found to describe the settling velocity of large (>50 μm) lime softening flocs better than Stokes’ Law. Settling velocities of smaller flocs (<50 μm) could still be quite well predicted by Stokes’ Law. The variation of fractal dimensions with lime floc size in this study indicated that two mechanisms are involved in the formation of these flocs: cluster–cluster aggregation for small flocs (<50 μm) and diffusion-limited aggregation for large flocs (>50 μm). Therefore, the relationship between the floc fractal dimension and floc size appears to be determined by floc formation mechanisms.