Polymer flocculation of calcite: Relating the aggregate size to the settling rate


  • Alex R. Heath,

    Corresponding author
    1. A.J. Parker Cooperative Research Centre for Hydrometallurgy (CSIRO Minerals), Clayton South, Victoria, Australia
    • A.J. Parker Cooperative Research Centre for Hydrometallurgy (CSIRO Minerals), Clayton South, Victoria, Australia
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  • Parisa A. Bahri,

    1. A.J. Parker Cooperative Research Centre for Hydrometallurgy (Murdoch University), School of Engineering - Rockingham Campus, Western Australia
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  • Phillip D. Fawell,

    1. A.J. Parker Cooperative Research Centre for Hydrometallurgy (CSIRO Minerals), Waterford, Western Australia
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  • John B. Farrow

    1. A.J. Parker Cooperative Research Centre for Hydrometallurgy (CSIRO Minerals), Waterford, Western Australia
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The initial hindered settling velocities of flocculated calcite suspensions have been related to the mean aggregate sizes and solid fraction under a range of flocculation conditions that are similar to those occurring within a mineral processing thickener. Calcite particles were flocculated in turbulent pipe flow and the aggregates sized with an in situ probe (Lasentec FBRM). The flocculated suspension then flowed into vertical column used to measure the hindered settling velocity. This allowed the investigation of key flocculation parameters — the flocculant dosage, the primary particle size, the suspension solid fraction and the fluid shear rate. The hindered settling velocity was related to the mean aggregate size using a relationship based on Richardson and Zaki's extension of Stokes' law. The effect of aggregate porosity was incorporated using fractal geometry, allowing the estimation of the fractal dimension (2.4) from experimental data. It is an important step toward the development of a link between the size of aggregates produced by flocculation, and their settling and dewatering characteristics. This is also an essential step in the development of robust process models describing the performance of gravity thickeners which are widely used throughout the minerals and other industries. © 2006 American Institute of Chemical Engineers AIChE J, 2006


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 × 106 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

equation image(1)

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

equation image(2)

where Uh is hindered settling velocity, m s−1; Uo 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.


Aggregation of calcite (Omyacarb, Commercial Minerals) with a commercial 30% anionic high-molecular-weight polymer (Nalco 9902) was performed in a linear horizontal pipe reactor, as previously described.53 The flocculation conditions are summarized in Table 1. Size measurements were taken using the Lasentec FBRM probe,41 and hindered settling velocities taken in a graduated 0.5 m settling column (Figure 1).

Table 1. Matrix of Pipe Reactor Experimental Runs
Run No.Pipe ID (m)Flow Velocity (m s−1)Solid ϕ (m3 m−3)Mean dp (μm)Floc. Dose (g t−1)
  1. Bold face indicates where conditions differ from baseline.

Figure 1.

Valve arrangement to capture portion of flow in settling column.

The settling column was constructed from acrylic tube (38.1 mm ID), and the flow was isolatable (note valves in Figure 1) and removable for extended settling measurements. The valve arrangement allowed a portion of the flow to be captured in the settling column, without disrupting the flow through the pipe reactor and past the sizing probe, allowing continuous operation. The column was marked at regular intervals (1 and 5 cm) and lit from behind by a fluorescent tube, allowing the measurement of the hindered settling velocity by following the fall of the mudline through time using a stopwatch.

The pipe work and fittings were designed to keep the top of the settling column as close as possible to the particle-sizing probe, reducing the additional residence time. This was the primary reason for mounting the pipe reactor above floor level (note Figure 1). The fittings around the bend and valve were also constructed to produce a smooth inner surface in an effort to reduce turbulence. Despite this, the reaction time offset between the aggregate size measurement and the top of the settling column was still considerable, requiring a correction (see next section).

Results and Discussion

Effect of fluid shear

Figure 2 shows the development of the mean aggregate size, and corresponding hindered settling velocity as a function of the spatially averaged fluid shear rate and residence time in the pipe. A high turbulent shear rate initially increases the aggregation rate by increasing the mixing and particle collision rates. However, aggregate breakage is also dramatically increased at high shear rates, and larger aggregates are ultimately formed with a lower shear rate. The hindered settling velocity follows a similar trend, but there is a time offset due to the additional pipe-work and fittings between the sizing probe and top of the settling column (Figure 1).

Figure 2.

Effect of mean pipe shear rate on the development of the mean aggregate size and hindered settling velocity (other conditions as per baseline).

The reaction residence time in the main part of the pipe reactor was taken simply from the mean flow velocity (assuming plug flow), and the distance from the flocculant addition point and the aggregate sizing probe. The additional effective residence time to the top of the settling column was calculated by the velocity-head method,23 using the Bernoulli equation for incompressible fluids.

The velocity head method was used rather than the simple approach of dividing the volumetric flow rate by the combined volumes of the valve and bend, and so on, because the additional flocculation is a function of both the residence time and the turbulent shear rate. The velocity head is more appropriate and can be converted back to an equivalent number of pipe diameters in the main pipe reactor. Figure 1 shows that in this case several fittings are present between the particle sizing probe and the settling column. The valve on the settling tube was assumed to be fully open with K = 0.17 as suggested by Perry and Green23. The protruding sizing probe was also assumed to have K = 0.17, and the tee given as K = 1.0.23 In addition the expansion loss when using the smaller diameter pipe reactor (25.4 mm ID) was given by the Borda-Carnot equation.23

equation image(3)

where A is cross sectional area of the pipe, m2.

The contribution from the various fittings and pipe were combined, and converted to an equivalent time depending on the flow velocity and size of the main pipe. The additional equivalent time between the FBRM sizing probe and the top of the settling column varied from 0.44 s (25.4 mm pipe, 39.33 L min.−1) to 7.31 s (38.1 mm pipe, 14.16 L min.−1), and was added to the x-axis of Figures 2–5. In Figures 3, 4 and 5 below (all with 25.4 mm pipe, 14.01 L min.−1) the offset was a constant 1.86 s. The additional time offset is unfortunate, since the data is lost from low-residence times, where the aggregate size increases rapidly. However, it does serve to reinforce the observation that flocculation is rapid in well-mixed suspensions of high-solid fraction.

Figure 3.

Effect of flocculant dosage on the development of the mean aggregate size and hindered settling velocity (other conditions as per baseline).

Figure 4.

Effect of suspension solid fraction on the development of the mean aggregate size and hindered settling velocity (other conditions as per baseline).

Figure 5.

Effect of primary particle size (dp) on the development of the mean aggregate size and hindered settling velocity (other conditions as per baseline).

Effect of flocculant dosage

Figure 3 shows the increase in the aggregate size and hindered settling velocity with flocculant dosage. The increased size is attributed to a higher flocculant surface coverage and increased particle bridging,10 leading to stronger aggregates and a reduced breakage rate. However, repeated aggregation/breakage causes polymer chain scission and/or rearrangement, tending to flatten the polymer chains onto the particle surface and reducing their ability to form polymer bridges between particles. This results in a reduced flocculant activity and the reduction in the aggregate size at extended residence times. If the particles were coagulated, or if charge neutralization (patch model) was the dominant flocculation mechanism, a stable steady-state aggregate size would be produced,10 where the aggregation and breakage rates were balanced.

Higher flocculant dosages (40 and 80 g t−1) were also tried (not shown), increasing the aggregate size and settling velocity further, however, the increased settling velocity caused the solid to settle to the base of the pipe reactor, invalidating the measurements. Although the flow rate could have been increased, allowing the suspension of larger aggregates formed at a higher flocculant dosage, this would also result in a higher shear rate and increased aggregate breakage (Figure 2).

Effect of suspension solid fraction

Figure 4 shows the effect of the solid fraction on the mean aggregate size and corresponding hindered settling velocity. The decreased size at high-solid fraction is attributed to an increase in fluid viscosity and energy dissipation rate during flocculation, leading to an increased breakage rate and smaller aggregates. The hindered settling velocity is further reduced by greater hindrance at an increased solid fraction, as described by Eq. 2.

Effect of primary particle size

Figure 5 shows the effect of primary particle size on the aggregate size and subsequent hindered settling velocity. A larger primary particle has a lower surface area (per unit mass) and, hence, a higher flocculant surface coverage. This leads to a stronger aggregate and an increased size due to a reduction in the breakage rate. In addition, a larger primary particle also leads to a lower aggregate porosity (Eq. 6), further increasing the settling velocity.

Fractal geometry

Aggregate porosity is usually described by fractal geometry,54 where although the total enclosed volume of the aggregate increases as a cube of its diameter, its mass increases at some lower fractional (hence, fractal) power

equation image(4)

where m to aggregate mass kg; d to diameter (agg = aggregate, p = primary particle), m, and Df to mass-length fractal dimension.

The increase in porosity with aggregate size is usually rationalized on the basis of the increasing voidage produced later in the aggregation process where large aggregates result from aggregate-aggregate collision. Aggregate porosity has two major impacts on the hindered settling velocity. First, it dramatically reduces the aggregate density, reducing the driving force for the particle to settle under gravity (Eq. 1), giving55–58

equation image(5)

where ρ is density (s = solid, l = liquid, agg = aggregate), kg m−3.

Second, the effective suspension solid volume fraction is increased, increasing the inter-particle hindrance as described by Richardson and Zaki1, 59, 60 (Eq. 2)

equation image(6)

where ϕ is the solid volume fraction (s = solid, eff = effective) [0,1].

Combining Eqs. 1, 2, 5 and 6, and substituting mean sizes gives

equation image(7)

The settling velocity may be further reduced by the effect of particle shape as the particles deviate from the sphericity assumed by the earlier equations. However, the effect of shape is relatively minor at low-particle Reynolds number (∼ 0.7–1.1 times the rate12–16), and is partially offset by the permeability of porous aggregates, allowing some fluid flow through the structure, decreasing the drag slightly.17, 18 Hence, the effects of particle shape, permeability, and Reynolds' number have been ignored as unnecessary complications at this stage, and Eq. 7 incorporates the dominant effects (suspension solid fraction and aggregate density) into the Stokes' Equation to predict the hindered settling velocity.

In this case the fluid viscosity is taken to be a constant 0.00102 N s m−2 (water at 20 °C61), and is unaffected by the aggregation process. During hindered settling the particles are essentially joined together as a continuous, permeable network, with the displaced water passing through channels in the sediment. Equation 7 was used to estimate the fractal dimension (Df) from the experimental data via

equation image(8)

where Uh,Exp is the experimental hindered settling velocity, m s−1; Uh, (Eq. 7) is the predicted hindered settling velocity from Eq. 7, m s−1.

In this case the fractal dimension (Df) was found to produce the best fit with a value of 2.42 (Figure 6), which is a typical value for aggregates formed by flocculation at high shear rates.62–66

Figure 6.

Comparison between measured and predicted (Eq. 7) hindered settling velocities.

Aggregate settling data typically displays considerable scatter (perhaps due to shape effects), however, a sensitivity analysis was performed to determine if the fractal dimension (Df) was a function of any of the experimental variables (shear rate G, flocculant dosage θ, primary particle size dp and solid fraction ϕ). Both linear and nonlinear sensitivities were tried as per Eqs. 12 and 13, the linear according to

equation image(9)

With G to the mean shear rate, s−1; θ is the flocculant dosage, kg m−2; dp is the Primary particle size (m); ϕ is the solid fraction [0,1]; Exp is the experimental condition; BL is the experimental condition at baseline Dmath image = 2.43; A = 0.0244; B = 0.0842; C = -0.0060; D = 0.0786, and the nonlinear

equation image(10)

With Dmath image = 2.44; W = 0.01949; X = 0.0337; Y = 0.00314, and Z = -0.0337.

Figure 7 shows the fit using the linear method (Eq. 9), with the residual reduced from 19.4 (Figure 6) to 11.9. The nonlinear method (not shown, Eq. 10) reduced the residual fractionally further, to 10.5. The improvement of Figure 7 over Figure 6 is minor, considering the number of degrees of freedom increased from 1 to 5, indicating that Equation 7 holds reasonably well within the experimental scatter.

Figure 7.

Comparison between measured and predicted (Eqs. 7 and 9) hindered settling velocities.

The sensitivity analysis indicates some increase in the fractal dimension (less porous) with shear and flocculant dosage, but a decrease with solid fraction. However, caution may be required interpreting these observations, since none of the effects are very strong (that is, Df ≈ 2.4 ± 0.1), and do not change the overall observation that the initial hindered settling velocity is increased with flocculant dosage (Figure 3), and primary particle size (Figure 5), but decreased with solid fraction (Figure 4) and shear rate (Figure 2).

Figures 2, 3, 4, and 5 show a slight reduction in aggregate size on extended shearing, typical of aggregates formed by polymer flocculants.10, 67–69 This effect could be interpreted as either aggregate compaction, or partially irreversible breakage (or both).57, 64, 70 However, aggregate breakage appears to be the dominant mechanism for this system, because in all cases the hindered settling velocities also drop with aggregate size on extended shearing. If compaction was the dominant size reduction mechanism, the hindered settling velocity would be expected to rise due to the increased density (Eq. 5), and reduced effective solid fraction (Eq. 6).


The initial hindered settling velocities of flocculated calcite suspensions have been studied under various process conditions (fluid shear, flocculant dosage, primary particle size, solid fraction) and related to on-line aggregate size measurements. As would be expected a larger aggregate size leads to a higher hindered settling velocity, however, the effects of inter-particle hindrance must also be considered. Richardson and Zaki's1 relationship describing the hindered settling velocity of nonflocculated suspensions has been adapted to flocculated suspensions by incorporating fractal geometry to describe aggregate porosity. This has enabled the estimation of the fractal dimension (2.4) from the experimental data.



cross sectional area of the pipe, m2


subscript, experimental condition at baseline


particle dia., m


diameter of primary particle, m


diameter of aggregate, m


mass-length fractal dimension


subscript, experimental condition


gravity, m s −2


mean shear rate, s−1


aggregate mass, kg


exponent in Eq. 2, usually taken as 4.65


pressure drop, N m−2


settling velocity, m s−1


settling velocity at infinite dilution, m s−1


hindered settling velocity m s−1


predicted hindered settling velocity from Eq. 7, m s−1


experimental hindered settling velocity, m s−1


mean fluid velocity, m s−1


density of solid, kg m−3


density of liquid, kg m−3


density of aggregate, kg m−3


flocculant dosage, kg m−2


solid volume fraction, 0,1


solid volume fraction, excluding aggregate porosity, 0,1


effective solid volume fraction, including aggregate porosity, 0,1


fluid viscosity, N s m−2


This research has been supported by the Australian Government's Cooperative Research Centre (CRC) program, through the AJ Parker CRC for Hydrometallurgy. Considerable support was also provided by the industrial sponsors of the AMIRA P266C Improving Thickener Technology project. This support is gratefully acknowledged.