**Gonzalo R. Quezada 1, Luís Ayala 2, Williams H. Leiva 3, Norman Toro 4,5,\*, Pedro G. Toledo 6, Pedro Robles <sup>7</sup> and Ricardo I. Jeldres 3,\***


Received: 7 January 2020; Accepted: 29 January 2020; Published: 11 February 2020

**Abstract:** A population balance model (PBM) is used to describe flocculation of particle tailings in seawater at pH 8 for a range of mixing intensities. The size of the aggregates is represented by the mean chord length, determined by the focused beam reflectance measurement (FBRM) technique. The PBM follows the dynamics of aggregation and breakage processes underlying flocculation and provides a good approximation to the temporal evolution of aggregate size. The structure of the aggregates during flocculation is described by a constant or time-dependent fractal dimension. The results revealed that the compensations between the aggregation and breakage rates lead to a correct representation of the flocculation kinetics of the tailings of particles in seawater and, in addition, that the representation of the flocculation kinetics in optimal conditions is equally good with a constant or variable fractal dimension. The aggregation and breakage functions and their corresponding parameters are sensitive to the choice of the fractal dimension of the aggregates, whether constant or time dependent, however, under optimal conditions, a constant fractal dimension is sufficient. The model is robust and predictive with a few parameters and can be used to find the optimal flocculation conditions at different mixing intensities, and the optimal flocculation time can be used for a cost-effective evaluation of the quality of the flocculant used.

**Keywords:** clay-based copper tailings; fractal dimension; mixing intensity; population balance model; seawater flocculation
