## Introduction

Particulate settling in viscous fluids has been studied in a range of natural and industrial systems,1, 2, 3 with aggregation increasing the settling velocity of fine particles. Natural examples include the formation of river estuary systems when suspended silt is coagulated by the ions dissolved in the seawater,4 or droplet coalescence in clouds, ultimately producing rain.5 The action of soluble salts to coagulate and increase the settling of fine particles is exploited in drinking and wastewater treatment plants, typically by adding multivalent cation salts like alum.6–9 However, in mineral processing circuits coagulants have been largely superseded by high-molecular-weight (≈ 20 × 10^{6} g mol^{−1}) polymer flocculants.10, 11

Particle settling occurs in three distinctly different regimes depending on the effective solid volume fraction. When the solid volume fraction is low, the particles are well separated and settle independently from each other. The gravity sedimentation velocity of individual solid spheres settling in creeping flow (Re → 0) was first described theoretically by Stokes2, 12, 13

where *U* is settling velocity, m s^{−1}; *d* is particle diameter m; *g* is gravity, 9.8 m s^{−2}; ρ is density (s = solid, l = liquid), kg m^{−3}; and μ is fluid viscosity, N s m^{−2}.

In practice, mineral particles/aggregates are rarely spherical, and large aggregates may be highly porous. Nonspherical particles generally settle slightly slower (by a factor of ∼ 0.7–1.0) than spheres,12–16 although needle shaped particles settle slightly faster if the major axis remains vertical. Porous aggregates have larger hydrodynamic profiles than solid particles of the same mass, and the increased drag considerably reduces their settling velocity compared to solid particles of the same size.17 However, aggregates still settle more rapidly than individual fine particles. Highly porous aggregates may also allow some fluid flow through the structure.18

Large, dense particles may settle too rapidly to be accurately described by Stokes' law for viscous flow, and by a particle Reynolds number above about 0.1 inertial effects become significant.13, 19 Both empirical and theoretical functions have been proposed for the drag coefficient at higher Reynolds numbers,20–23 although in most cases aggregates settle sufficiently slowly to be described by Stokes' law.17

As the suspension solid fraction is increased there is a gradual transition to hindered-settling. The hindered settling regime is characterized by a distinct solid/liquid interface (mudline) that settles to leave a clear supernatant above.24–27 The settling velocity of the mudline decreases as a function of the solid fraction, due to the decreasing permeability of the settling layer and an increase in the upward velocity of the displaced fluid,23, 28–31 as described by Richardson and Zaki1

where *U*_{h} is hindered settling velocity, m s^{−1}; *U*_{o} is settling velocity at infinite dilution, m s^{−1}; *n* is the exponent, ∼ 4.65, and ϕ is the solid volume fraction, 0,1.

A wide variety of similar relationships have also been proposed in the literature,28, 30, 32 but usually as functions only of the volume fraction, ignoring the effects of flocculation. Of the various hindered settling functions proposed, Richardson and Zaki's equation (Eq. 2) is the most popular, with the exponent typically taken as 4.65.19, 23, 33–37

Aggregation increases the hindered-settling velocity by increasing the particle size, forming channels in the settling sediment that increase the suspension permeability. Settling behavior in the hindered settling regime is often described with Kynch29, 39 theory, but since the settling velocity is usually taken as a unique function of the solid fraction it does not account for the effect of flocculation7, 40. In the past this effect was difficult to quantify, because although the hindered settling velocity can be measured relatively easily (for example, by cylinder tests), the fragile aggregates needed to be subsampled for the traditional *ex situ* particle sizing instruments. However, in-stream aggregate sizing instruments are now available,41 allowing the effect of aggregate size on the hindered settling velocity to be studied directly.

As the solid loading is increased further, the particles will eventually form a continuous network and settling will also be restrained by mechanical support from below.22, 29, 36, 42, 43 The mechanical strength of the network is a function of the solid/packing fraction and the strength of the interparticle bonding. The sediment will compress when the weight of sediment overburden exceeds the compressive yield stress.31, 43–45 The excess overburden weight not supported mechanically is restrained hydrodynamically, and is the force required to squeeze the fluid back up through the collapsing sediment.46–50 In practice channel formation may occur spontaneously, or can be encouraged by rakes with vertical pickets.7, 29, 40, 51

The aim of this work is to develop a mathematical relationship between the aggregate size and the initial hindered settling velocity under a range of conditions likely in a mineral processing thickener. The influence of fluid shear, flocculant dosage, feed solid volume fraction, primary particle size and residence time are addressed. This work continues on from previous studies,52 where the aggregate size data was used to develop a population balance model describing the kinetics of aggregation and breakage. Hence, a direct link is made between the aggregate size and the corresponding hindered settling velocities of the same suspensions at each stage of the flocculation process.