# Sedimentation of Fractal Aggregates in Shear-Thinning Fluids
###### Abstract  
Solid-liquid separation is a key unit operation in the wastewater treatment, generally consisting of coagulation and flocculation steps to promote aggregation and increase the particle size, followed by sedimentation, where the particles settle due to the effect of gravity. The sedimentation efficiency is related to the hydrodynamic behavior of the suspended particles that, in turn, depends on the aggregate morphology. In addition, the non-Newtonian rheology of sludges strongly affects the drag coefficient of the suspended particles, leading to deviations from the known settling behavior in Newtonian fluids. In this work, we use direct numerical simulations to study the hydrodynamic drag of fractal-shaped particles suspended in a shear-thinning fluid modeled by the power-law constitutive equation. The fluid dynamics governing equations are solved for an applied force with different orientations uniformly distributed over the unit sphere. The resulting particle velocities are interpolated to compute the aggregate dynamics and the drag correction coefficient. A remarkable effect of the detailed microstructure of the aggregate on the sedimentation process is observed. The orientational dynamics shows a rich behavior characterized by steady-state, bistable, and periodic regimes. In qualitative agreement with spherical particles, shear-thinning increases the drag correction coefficient. Elongated aggregates sediment more slowly than sphere-like particles, with a lower terminal velocity as the aspect ratio increases. s +
Footnote †: journal: Applied Physics Letters  
_Applied Physics Letters_  
where \(N_{\rm p}\) is the number of primary particles forming the aggregate, \(R_{\rm g}\) is the radius of gyration, \(D_{\rm f}\) is the fractal dimension, \(k_{\rm f}\) is the fractal pre-factor, and \(a\) is the radius of the primary particles. Aggregates with a shape satisfying Equation (1) are referred to as fractal-like or quasi-fractal aggregates, as the scaling law relation is independent of whether the particle has a real scale-invariant (self-similar) morphology [8]. The fractal dimension provides a scaling law between the number of primary particles composing the aggregate and a characteristic cluster size (e.g., the gyration radius). It assumes values between 1 and 3 corresponding to rod-like and spherical-like particles, respectively. The fractal prefactor is a descriptor of the aggregate local structure and is related to the packing factor. Finally, the radius of gyration is a geometric measure of the spatial mass distribution about the aggregate center of mass. The particle morphology has a relevant influence on the settling velocity, and thus it must be accounted for when dealing with the sedimentation process of flocculated particles [9; 10]. Therefore, it is not surprising that the hydrodynamic drag of particles with complex shapes has been thoroughly studied in the literature. Several methodologies have been proposed to compute the hydrodynamic drag of a set of spherical particles in contact, based on expansions of analytical solutions for Stokes flows [11; 12; 13] or direct numerical simulations [14; 15; 16]. Due to the linearity of the creeping flow equations, the dynamics of a particle with arbitrary shape can be determined by the mobility tensor that univocally relates translational and angular velocities to the forces and torques acting on the particle [17]. The knowledge of the mobility tensor allows to completely predict the translational and orientational motion of the aggregate. In this regards, it is well known that the settling velocity depends on the aggregate orientation. The average velocity over all possible orientations can be linearly related to the applied force, with a proportionality constant given by the arithmetic mean of the three eigenvalues of the translational mobility tensor divided by the fluid viscosity [13; 18; 19]. A hydrodynamic radius can be defined as the radius of a sphere that gives the same drag force acting on the aggregate in a uniform flow [13].