Predicting the Settling Velocity of Lime Softening Flocs using Fractal Geometry
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Stokes’ law that is traditionally used for modeling the sedimentation of flocs, incorrectly assumes that the floc is solid and spherical. Consequently the settling rates of flocs cannot be estimated using the Stokes law. The application of fractal dimensions to study the internal structure and settling of flocs formed in lime softening process was investigated. An optical microscope with motorized stage was used to measure the fractal dimensions of lime softening flocs directly on their images in 2 and 3D space. The fractal dimensions of the lime softening flocs were 1.15-1.27 for floc boundary, 1.49-1.90 for cross-sectional area and 2.55-2.99 for floc volume. Free settling tests were used for indirect determination of 3D fractal dimension. The measured settling velocity of flocs ranged from 0.1 to 7.1 mm/s (average: 2.37 mm/s) for the flocs with equivalent diameters from 10µm to 260µm (average: 124 µm). Floc settling model incorporating variable floc fractal dimensions as well as variable primary particle size was found to describe the settling velocity of large (>60 µm) lime softening flocs better than Stokes’ law. Settling velocities of smaller flocs (<60 µm) could still be quite well predicted by the 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 (>60 µm) and diffusion-limited aggregation for large flocs (<60 µm). Therefore, the relationship between the floc fractal dimension and floc size appears to be determined by floc aggregation mechanisms. The settling velocity of lime softening flocs was also modeled by a general model that assumes multiple normally distributed fractal dimensions for each floc size. The settling velocities were in the range of 0-10mm/s and in good agreement with measured settling velocities (0.1-7.1mm/s). The Stokes’ law overestimates the settling velocity of lime flocs. It seems that the settling velocity of flocs is mainly controlled by aggregation mechanisms and forming large floc does not guarantee improved sedimentation. The multifractal analysis of lime softening flocs showed that these aggregates are multifractal and a spectrum of fractal dimensions is required to describe the structure of an individual floc.