Settling Velocities of Fractal Aggregates

Abstract

Aggregates generated in water and wastewater treatment systems and those found in natural systems are fractal and therefore have different scaling properties than assumed in settling velocity calculations using Stokes' law. In order to demonstrate that settling velocity models based on impermeable spheres do not accurately relate aggregate size, porosity and settling velocity for highly porous fractal aggregates, we generated fractal aggregates by coagulation of latex microspheres in paddle mixers and analyzed each aggregate individually for its size, porosity, and settling velocity. Settling velocities of these aggregates were on average 4−8.3 times higher than those predicted using either an impermeable sphere model (Stokes' law) or a permeable sphere model that specified aggregate permeability for a homogeneous distribution of particles within an aggregate. Fractal dimensions ( D ) derived from size−porosity relationships for the three batches of aggregates were 1.78 ± 0.10, 2.19 ± 0.12 and 2.25 ± 0.10. These fractal dimensions were used to predict power law relationships between aggregate size and settling velocity based on Stokes' law. When it was assumed that the the drag coefficient, C D, was constant and fixed at its value of C D = 24/ Re for the creeping flow region ( Re ≪ 1), predicted slopes of size and settling velocity were in agreement with only the data sets where D > 2. As a result, when D < 2, aggregate porosities will be overestimated and fractal dimensions will be calculated incorrectly from settling velocity data and Stokes' law.