Minerals Engineering 58 (2014) 80–89

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Minerals Engineering

journal homepage: www.elsevier .com/ locate/mineng

Particle shape effects in flotation. Part 1: Microscale experimentalobservations q

http://dx.doi.org/10.1016/j.mineng.2014.01.0040892-6875/Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved.

q Extension of paper presented at IMPC conference, New Delhi, India, 24–28September 2012.⇑ Corresponding author. Address: ASAM, 2 Technology Place, Macquarie Univer-

sity, NSW 2109, Australia. Tel.: +61 2 9850 2755; fax: +61 2 9850 2701.E-mail addresses: David.Verrelli@mq.edu.au (D.I. Verrelli), Warren.Bruckard@

csiro.au (W.J. Bruckard), Peter.Koh@csiro.au (P.T.L. Koh), Phil.Schwarz@csiro.au(M.P. Schwarz), Bart.Follink@unisa.edu.au (B. Follink).

1 Currently at Macquarie University.2 Currently at CSIRO Process Science and Engineering.

David I. Verrelli a,⇑,1, Warren J. Bruckard a, Peter T.L. Koh b, M. Philip Schwarz b,2, Bart Follink c

a CSIRO Process Science and Engineering, Bayview Avenue, Clayton, VIC 3168, Australiab CSIRO Computational Informatics, Bayview Avenue, Clayton, VIC 3168, Australiac Ian Wark Research Institute, University of South Australia, Mawson Lakes, SA 5095, Australia

a r t i c l e i n f o

Article history:Received 19 August 2013Accepted 5 January 2014Available online 13 February 2014

Keywords:Flotation bubblesFlotation kineticsFroth flotationGrindingInduction timeMineral processing

a b s t r a c t

There has long been speculation as to whether some particle shapes are more ‘floatable’ than others,which might be due to differences in the induction period required to achieve attachment betweenthe particles and the air bubbles in the pulp. To resolve this, we used the Milli-Timer apparatus to directlyobserve the process of particle–bubble interaction and attachment by means of a magnified, high-speedvideo recording, thus providing a direct measure of the induction period for attachment.

To assess the influence of particle shape on induction time we used two varieties of methylated boro-silicate glass particles — spheres and angular ‘frit’ — in a range of tightly-sized fractions. Other factorsthat could affect the induction time, such as the polar angle of sliding commencement, and approachvelocity, are accounted for using multiple nonlinear regression.

Our results illustrate the importance of particle shape on induction period, with angular particlesexhibiting induction periods that were an order of magnitude lower than those of spheres. Furthermore,the induction period was seen to decrease with increasing particle velocity, or kinetic energy onapproach, but increased as the trajectory approached the limit of just grazing the bubble. These resultsindicate that attention should be paid to the shape of particles obtained from the grinding operation,besides particle size.

Crown Copyright � 2014 Published by Elsevier Ltd. All rights reserved.

1. Introduction ticles are easier to float than rounder particles. However, such

Flotation is a key unit operation employed in mineral process-ing and a range of other industries. Successful flotation hinges onthe attachment of (certain) particles to bubbles. In real industrialsystems the particles fed into a flotation cell will exhibit a varietyof shapes, varying from approximately spherical (e.g. zircon, slags)(see Wotruba et al., 1991), to cuboidal (e.g. galena) (Dippenaar,1982), to platy (e.g. talc, chlorite) or acicular (e.g. tremolite)(Kursun and Ulusoy, 2006). How important is this shape in deter-mining attachment?

Anecdotally it is expected that particle shape can have a signif-icant effect on ‘floatability’: the common view is that angular par-

overall tendencies leave it unclear as to which flotation sub-processes are most affected by particle shape. For example, in thepulp we could consider collision rate, attachment efficiency, stabil-ity against detachment, and entrainment, besides a number offroth characteristics. Traditionally the efforts to model the interac-tion of particles and bubbles have typically assumed that bothobjects are perfect spheres (see Nguyen and Schulze, 2004). Someexceptions are studies on the final moments prior to breaching ofthe liquid gap, in which bubble deformation is modelled (Chanet al., 2011), and studies that examine the equilibrium configura-tion of a non-spherical particle after attachment has occurred inthe pulp (Huh and Mason, 1974; Schulze, 1984: 199ff) (cf. Coghilland Anderson, 1923: 44ff) or the froth (Dippenaar, 1982; Morriset al., 2011). Hence, modelling is not yet at a stage to fully explainthe relative importance of particle shape or roughness.

1.1. Measurement of particle shape and roughness

In flotation the particles of interest are typically sized at around10–150 lm (Shergold, 1984: 231). If the deviations occur on a scalecomparable to the particle size, then we shall refer to this as a

D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89 81

difference in shape. If the deviations occur on a scale much smallerthan the particle size, then we shall refer to this as a difference insurface roughness. An equivalent distinction is between ‘‘struc-ture’’ and ‘‘texture’’ (Ahmed, 2010).

An important element of working with real particles is to char-acterise their shape and surface roughness. Unfortunately, there isno single unambiguous parameter to fully characterise either ofthese properties, which has hindered research into this importantarea (Holt, 1981; Yekeler et al., 2004; Kursun and Ulusoy, 2006;Ahmed, 2010). The topic of roughness is scarcely more tractable:the topography of a solid surface can exhibit different characteris-tics on different lengthscales, such as ‘rough’ or ‘smooth’ asperities.Attempts to deal with these problems through use of fractaldimensions (e.g. Filippov et al., 1999; Ahmed, 2010) are subjectto question, in general, as the features of a given particle are unli-kely to conform to a true fractal rule over a broad enough range oflengthscales to be physically useful — unlike the features of aggre-gates (Verrelli, 2008).

1.2. Influences on particle shape, and roughness

Different views exist on the dominant reason why particleshave the shapes that they do, which can be summarised as theeffect of the inherent material character, and the effect of theprocessing (Holt, 1981). For example, malleable materials suchas some metals can deform plastically (e.g. Ofori-Sarpong andAmankwah, 2011), while brittle materials such as quartz will frac-ture (e.g. Holt, 1981). Laminated materials preferentially fail alongcertain orientations, while amorphous materials do not; on theother hand, the easy slip plane in the layered material may yielda smoother failure surface. The tendency to fail along grain bound-aries or through grains also varies between materials, dependinginter alia upon the structural homogeneity (Gaudin, 1926).

In the industrial context, working with a given material, thetype of grinding takes on increased importance. A number of stud-ies have been reported in the literature in which the effects of dif-ferent grinding processes on a given sample are reported. A typicalresult is that more angular particles are produced from rod mills(attributed to impact processes), compared to rounder particlesobtained from ball mills (attributed to abrasion and chipping)(Yekeler et al., 2004), although this depends on the details of oper-ation (Gaudin, 1926). Yet at the same time the particles ground in arod mill have been reported to be smoother than particles from aball mill (Yekeler et al., 2004).3 Autogenous milling produced inter-mediate results in each parameter in the work of Yekeler et al.(2004), while Forssberg and Zhai (1985) reported that autogenouslyground ore particles were less elongated than those from either ballor rod milling.

Hiçyılmaz et al. (2006) characterised pyrite particles of a givensize fraction obtained from dry autogenous grinding as beingrougher (by BET surface area) and more acute (by aerodynamicresistance) than those obtained from dry ball milling. This differedfrom their earlier results for barite (Hiçyılmaz et al., 2005), inwhich dry ball milling produced rougher and more acute particlesthan dry autogenous milling, for a given size fraction. This empha-sises the effect of the feed material on the outcome (Hiçyılmazet al., 2006). For the barite samples the smaller size fractions werereported to be rougher and more acute in shape (Hiçyılmaz et al.,2005), while the opposite trend was reported for the pyritesamples (Hiçyılmaz et al., 2006).

3 The rod-milled particles were also claimed to be more hydrophobic than the ball-milled product (Yekeler et al., 2004). However, that conclusion rests on comparingextrapolations of surface tension that are subject to uncertainties larger than theirdifferences, besides being founded on Zisman’s theory, which is itself subject tocriticism (Siboni et al., 2004).

Dry and wet milling were carefully studied by Feng and Aldrich(2000), who found that the dry-ground particles had relativelyrough surfaces containing microstructural defects, whereas thewet-ground particles had smoother, cleaner surfaces.

It should be clear that comparison between studies on differentmaterials is fraught with difficulty; yet even when the material isidentical, the situation may not be straightforward. Besides thetype of material and type of machine, the resulting particle shapesdepend inter alia on: size of feed material; speed of operation ofequipment; geometry of equipment; product particle size, and par-ticle size distribution (Gaudin, 1926; Holt, 1981).

1.3. Effect on flotation

1.3.1. ShapeOne of the key resistances to attachment is the hydrodynamic

resistance arising as liquid drains out of the gap between a bubbleand an approaching particle. Aspherical particles could experiencea lower resistance (depending upon their shape and orientation),and hence require less time for the intervening gap to thin suffi-ciently to be breached, and attachment occur.

There have been few systematic studies on the effect of particleshape or roughness upon flotation performance.

One of the earliest notable studies on this topic was carried outby Anfruns and Kitchener (1977). They concluded that angular par-ticle shapes have much higher attachment efficiencies thanspheres, which they attributed to easier rupture of the ‘‘wettingfilm’’. In that work the angular particles were composed of quartz(Brazilian rock crystal), while the ‘ballotini’ spheres were of leadglass. Despite care taken to ensure the same size (31 lm equivalentStokes diameter), similar density, and similar surface properties, aneffect of the difference in materials cannot be entirely ruled out (cf.Verrelli et al., 2012). In distilled water, Anfruns & Kitchener foundcollection efficiencies about four times greater for angular quartzthan for spherical glass particles. According to their experimentaldescription, the collision rates would be the same in each case, en-trained particles were eliminated, and detachment was implied tobe negligible; hence, the differences should be due to variation inthe ease of attachment.

Studies on talc have suggested that rounder (but rougher) par-ticles produced by ball milling are less easily recovered in a micro-flotation cell (Yekeler et al., 2004) and in column flotation (Kursunand Ulusoy, 2006), compared to more elongated (but smoother)particles produced by rod milling. These experiments used a rela-tively broad class of particle sizes (40–250 lm). Despite the ‘‘sim-ilar’’ particle size distributions from each mill (Yekeler et al., 2004),narrower fractions would avoid the potential issue associated withdependence of shape upon particle size. Even within a given nar-row size fraction, the predominant shape may depend uponwhether it comes from the lower end of a coarse grind, or theupper end of a fine grind.

Wotruba et al. (1991) reported that prismatic zircon particles ofa given size floated better than those with more spherical or ellip-soidal shapes in a 1.5 L cell. (Confirmed independently by Gül(2006).) The particles all had essentially the same composition byXRF. The prismatic particles tended to have smoother faces andsharper edges than the other types of particle in the flotation feed.However, even after roughening the faces and rounding down thesharp edges, the prismatic particles remained more floatable. Wot-ruba et al. proceeded to characterise the ease of detachment of theoriginal particles from a bubble, and found that the energy neces-sary for detachment was significantly greater for prismatic parti-cles attached parallel to their long axes (‘flat-attached’) than foreither rounded particles or ‘end-attached’ prismatic particles. Thiscan be related to the length of the three-phase contact line (TPCL)formed. Wotruba et al. accepted that rounded particles may also be

Fig. 1. Schematic of the progress of a particle interacting with and attaching to abubble (see also Albijanic et al., 2010).

82 D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89

disadvantaged in forming an attachment to a bubble, althoughthey did not isolate this step in their own experiments.

Schmidt & Berg have described an opposite tendency: theyfound that spherical printer toner particles floated better than discor platy particles (Ahmed, 2010).

Ahmed’s work showed increases in flotation recovery — andgreater resistance to detachment — with increases in either parti-cle asphericity or particle roughness (Ahmed, 2010). As the twoparameters were highly correlated, it is difficult or impossible totell from his data which was the more important factor. Further-more, as with Anfruns & Kitchener, Ahmed used particles of some-what different composition, although it was nominally accountedfor by methylation of each sample to an equal degree ofhydrophobicity.

Finally, Hiçyılmaz et al. probed the effect of particle shape onthe attachment process alone by testing recoveries in a modifiedHallimond tube across a series of size fractions. The barite (Hiçyıl-maz et al., 2005) and pyrite (Hiçyılmaz et al., 2006) particles thatwere characterised as rougher and more acute in shape showedlower recoveries. However, the results may have been influencedby differing degrees of surface oxidation, and also skewed in favourof rounder particles by the use of low collector dosages (Hiçyılmazet al., 2006).

1.3.2. RoughnessParticles with rough surfaces composed of microscale projec-

tions might achieve attachment more easily, if those bumps or jagsprotrude through the gap, so that the separation between particleand bubble is smaller than it might otherwise seem. A three-phasecontact would then be expected to occur on one of the projections.Nanoscale topological features seem less likely to have much effecton the ease of breaching the interjacent liquid; however, they maystill affect the expansion of the three-phase contact line in dewett-ing (Oliver and Mason, 1977). Moreover, chemical composition andsurface properties can vary on the atomic scale in layered crystal-line materials (Yekeler et al., 2004; Franks and Gan, 2007), whichwould be accentuated on rough surfaces. A further feature of roughsurfaces is that they may more readily harbour small air bubbles(‘microbubbles’ or ‘nanobubbles’), which promote attachmentby some mechanism — possibly including ‘hydrophobic forces’(Krasowska et al., 2007).

Feng and Aldrich (2000) found a large number of differences inthe particles prepared by wet and dry milling, with the dry-groundsamples exhibiting faster dissolution, faster reagent adsorption,more stable and higher-loaded froths, and faster flotation kinetics.These researchers were very careful to attain comparable degreesof comminution in the two milling operations. Nevertheless, it isnotable that no consideration of the effect of particle shape wasmade. The flotation kinetics were evaluated from recovery timesin a Leeds flotation cell under typical operating conditions; hence,effects such as entrainment and detachment will have influencedthe results.

As alluded to above, Wotruba et al. (1991) roughened the sur-faces of a mixture of near-spherical, ellipsoidal, and prismatic zir-con particles through attrition in a planetary mill. The particles’size and basic shapes were ‘‘nearly unaffected’’ by this processing.Their results indicated a much better floatability of the roughenedparticles in comparison to the original flotation feed, for a givencollector dose.

1.3.3. Complicating factorsDifferent grinding techniques do not just affect the physical

form of the minerals, but can also affect the chemistry of thesurface, for example through more or less reducing conditions(Forssberg et al., 1988; Bruckard et al., 2011), and ‘‘activation’’of the mineral surface through creation of material defects or

disorder (Feng and Aldrich, 2000; Yekeler et al., 2004). Under somecircumstances microscopic air pockets may persist in crevices atthe surface of the particles (Harvey et al., 1944), making subse-quent attachment to an air bubble easier (Coghill and Anderson,1923: 48); this would be more relevant in the case of dry grinding.

1.4. Induction period

The attachment between a bubble and a particle (see Fig. 1) iscommonly described as requiring a minimum, ‘critical’ time to oc-cur, once the two bodies are brought into proximity; this is theinduction period, s (Sven-Nilsson, 1934). In flotation, the conven-tional theory compares s against the time available for a particleto slide over the bubble’s surface (Nguyen and Schulze, 2004:265ff). Based on hydrodynamics, the sliding time is known to de-pend upon factors such as particle size, and the approach trajec-tory. The induction period is expected to depend on the surfacechemistry, and perhaps other factors such as particle shape. For agiven particle and bubble pair, the induction period is tacitly pre-sumed to be constant; however, the latest research indicates thats actually depends upon the particle’s approach trajectory (Verrelliet al., 2012).

We are interested in using our technique of direct observationof the induction period in order to assess the role of the attachmentprocess in the previously reported changes in floatability. This willestablish whether the altered floatability is due to changes in theease of attachment, following from changes in the time requiredfor thinning of the interposed liquid, or whether it is due to otherfactors, such as detachment or entrainment.

In order to assess the importance of particle shape and rough-ness on the induction period, s, it is necessary to account for asmany other factors as possible. Hence, we conduct our experimentson narrowly sized borosilicate glass particles that differ only intheir shape and surface roughness. We are unable to precisely con-trol the particle approach trajectories or speeds, so instead we in-clude corrections for these parameters in the analysis.

2. Methodology

2.1. Milli-Timer

CSIRO’s Milli-Timer (Fig. 2) has been described in detail previ-ously (Verrelli et al., 2011). Briefly, it consists of a stationary bubble

Fig. 2. CSIRO’s Milli-Timer. Adapted from Verrelli et al. (2011). The bulb of the pipette is not squeezed.

D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89 83

that is held underwater at the end of a capillary, and onto whichparticles are dropped. The interactions are recorded on a high-speed video camera.

In the present set-up the camera used was a Phantom v210(Vision Research, Wayne, U.S.A), fitted with a Zoom 6000 lensassembly (Navitar, Rochester, U.S.A) comprising a 2� magnifyingF-mount extension tube, a zoom lens with focussing capability,and a 0.75� lens attachment, for a nominal magnification of6.75� through the lens assembly. The actual magnification of thecamera–lens system is approximately 7.1�. The camera was oper-ated at a capture rate of 1000 frames per second, and exposuretime of 10 ls.

A KL 1500 LCD 150 W halogen cold light source (Schott, Mainz,Germany) provides intense white light. To minimise imageblur due to differential refraction of different wavelengths oflight, a BP635 colour filter (Midwest Optical Systems, Palatine,U.S.A.) was interposed, which passes wavelengths in the range590–670 nm.

Induction periods are defined as the time that elapses from themoment at which sliding commences, until the moment thatattachment is initiated (see Fig. 1). The latter moment is clearlyevidenced by a ‘jump in’ event, as described previously (Verrelliand Koh, 2010; Verrelli et al., 2011). The former requires somejudgement. In the ideal case, the particle will approach the bubbleand then undergo sliding within the plane perpendicular tothe camera’s viewing axis and passing through the bubble’scentre (i.e. approximately the middle of the focal ‘plane’). In realitythe particles are commonly observed to travel along slightlydeviating azimuthal angles (Verrelli et al., 2012), so that it is morepractical to determine sliding commencement by identifying asubstantial reduction in the radial velocity of the particle withrespect to the bubble’s centre.

In each ‘run’, a dilute swarm of particles settles through thewater column toward a freshly blown bubble. Depending uponthe attachment efficiency, any given run might result in multipleopportunities to estimate induction time, a single opportunity, orno opportunities. Only interactions that could be clearly seen, thatresulted in attachment, and that were not subject to any apparentinterferences, were included in the data analysis.

The polar angle of particle position at the moment that slidingbegins is given the symbol4 /, and is obtained from the videorecordings as above (schematically depicted in Fig. 1). The initial

4 The symbol /1 was used in our previous publication (Verrelli et al., 2012). Thesubscript is omitted here for clarity.

approach velocity of a given particle toward the bubble, v0, isobtained by measuring displacement of the particle from frame toframe as it first enters the field of view, when the effect of thebubble’s presence is minimal.

2.2. Materials

2.2.1. BubblesBubbles of 2.0 lL volume were blown at the end of a blunt

20 gauge capillary shortly before each run using ambient air, givingdiameters of 1.56 ± 0.04 mm.

2.2.2. ParticlesThe particles used were composed of borosilicate glass, in two

forms: spheres, and angular ‘frit’ (Mo-Sci Specialty Products, Rolla,U.S.A). The spheres were supplied in a 75–90 lm size fraction, des-ignated as ‘‘Precision Grade’’, which are specified as being at least90% in the nominated size range and at least 90% classed as ‘‘spher-ical’’. The scanning electron microscope (SEM) images we have ta-ken in Fig. 3(A) confirm a narrow size distribution, and highsphericity, with rather smooth surfaces. The frit is manufacturedby comminution of large chunks of recycled borosilicate glass, firstin a hammer mill, and then in a disk mill (e.g. Bico plate pulveriser).SEM images of the frit in Fig. 3(B) show the variety of shapesencountered. Using new screens, we sieved the as-received frit intonarrow size fractions, of which the 75–90 lm fraction was used inthe present work.

Due to the origin of the frit, its composition is practically thesame as that of the borosilicate spheres (see Verrelli et al., 2012).This is confirmed in assays reported in the online SupplementaryMaterial.

All particles were washed using progressive stages of waterrinsing, alkaline washing and acid washing (Verrelli et al., 2012).The clean particles were then methylated using chloro(tri-methyl)silane (CTMS) to achieve a surface coverage equal to 50%of the maximum achievable coverage (Verrelli et al., 2012), desig-nated as ‘‘50% methylated’’. This process increases particle hydro-phobicity by adsorption of a stable coating on the particle. Fulldetails of the methylation procedure are provided in the Supple-mentary Material.

2.2.3. Characterisation of particlesThe specific surface area of the frit was characterised using the

approach suggested by Brunauer et al. (1938). Measurementswere attempted first using a TriStar 3000 surface area analyser

Fig. 3. Micrographs of the particles at various levels of magnification. (A) Spherical particles. (B) Angular frit. Magnifications are (i) 100� (ii) 1000� and (iii) 15000�.

84 D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89

(Micromeritics, Atlanta, U.S.A) operating with nitrogen gas, withsample masses between 0.6 and 0.7 g. However, this proved inad-equate for the small surface areas involved. Hence, final character-isation was carried out on a TriStar II 3020 (Micromeritics),operating with argon gas for improved sensitivity, with samplemasses between 1 and 2 g. Confidence intervals on each measure-ment were evaluated based on uncertainty in the regression(including covariance (Montgomery et al., 2001: 32,74,82f.,103f));the intervals do not include errors from other sources.

Micrographs were taken with a field emission, environmentalSEM (FEI, Oregon, U.S.A) operating at 10 kV and a probe currentof approximately 300 pA. Additionally, energy-dispersive spectro-scopic (EDS) measurements were made on selected regions of theparticles with this instrument, as a check of their composition.

2.3. Multiple nonlinear regression

In order to systematically investigate the influence of three keymechanisms on induction period, rigorous statistical analysis wasperformed in the form of multiple nonlinear regression. The threekey issues explored are the effect of approach velocity, approachposition, and particle shape or roughness. These are described bythe approach velocity, v0, the polar angle of sliding commence-ment, /, and a binary ‘‘indicator variable’’, f, that takes a value ofeither 1 for frit or 0 for spheres.

We performed the regression using the following generalequation:

s¼ b11

v100

þb21v2

0

þb31v0þb4v0þb5v2

0þb6/þb7/2þb8 cosð/Þþb9

� �ðb10f þð1�f ÞÞ; ð1Þ

in which the bi are constant coefficients. We evaluated Eq. (1) forthe set of all possible regressions (Montgomery et al., 2001:302ff). The ‘best’ regression for a given number of parameters wasthat which minimised the mean squared error (‘‘MSE’’). Further de-tails are provided in the Appendix.

3. Results and discussion

3.1. Particle shape, and roughness

The nominally spherical particles are seen in Fig. 3(A) to mostlyapproximate ideal spheres. They do, however, exhibit some rough-ness due to ‘debris’ melded on their surfaces. This debris is of irreg-ular shape, but of size approximately 0.1–5 lm, and is composed ofthe parent material, as confirmed by EDS measurements. A smallnumber of angular particles are seen intermingled in Fig. 3(A)(i);those angular particles are readily discerned on the video record-ing, and are excluded from further analysis.

The frit used in the present experiments was produced by com-minution in a hammer mill and disk mill in sequence. It would beexpected that the final particle shape and roughness would beinfluenced more by the later stage. Particle size reduction in diskmills is due principally to shear, with smaller contributions fromparticle compression, impact and other mechanisms, which is notso different from the hammer mill (Austin and Rogers, 1985).The few previous reports on disk mills in the literature (Husemannet al., 1976; Ofori-Sarpong and Amankwah, 2011) are not directlyrelevant, due to the different materials used.

Arguably the most straightforward means of shape classifica-tion is still semi-qualitative descriptions obtained from visualinspection (cf. Ahmed, 2010). Micrographs of the frit are presented

0.000

0.020

0.040

0.060

0.080

0.100

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0.160

45 to 53µm 53 to 63µm 63 to 75µm 75 to 90µm 90 to 106µm

Mea

sure

d su

rfac

e ar

ea [

m²/g

]

Size fraction

Spheres : Predicted

Spheres (Argon)

Frit (Nitrogen)

Frit (Argon)

Frit : Predicted

Fig. 4. Specific BET surface area of spherical and angular borosilicate glass. The gas in parentheses indicates the species used for characterisation. The prediction for spheresuses the average of surface areas computed for the two end-points of the respective size class; the prediction for frit is double the value for spheres. The error bars enclose 95%confidence intervals based on the linear regression used in the BET technique. The source and preparation of all particles were the same.

D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89 85

in Fig. 3(B). It is apparent that the most common shapes are platyfragments, although other shapes are also observed, such as tetra-hedra and cuboids. As with the spherical particles, the frit exhibitsroughness due to ‘debris’ of size approximately 0.1–5 lm stuck onthe surface. In addition to this, step-like features, cavities, ridges,grooves and so forth can be seen in the substrate, of a similardimension.

3.1.1. Surface areaThe specific surface area provides a quantitative indication of

particle shape and roughness (Hiçyılmaz et al., 2006). Experimen-tal results are given in Fig. 4. For the spheres used in theMilli-Timer, the BET surface area was found to be approximately0.0365 m2/g, which is close to the expected value for a perfectsphere of 75–90 lm diameter. For frit of the same nominal sizefraction, the surface area was 0.0614 m2/g. This represents a ratioof 1/1.68 = 0.59, which can be interpreted as a sphericity-like mea-sure (Koh et al., 2009), although conventionally the sphericity isdefined as relating particles of equal volume (Wadell, 1935: 264),rather than equal size fraction. Recognising the uncertainty in esti-mating the small surface area of the spherical particles, and recal-culating the ratio using the lower limit of the confidence interval orthe theoretically expected value for ideal spheres (which happen tobe approximately equal here) the ratio becomes 1/1.85 = 0.54.Although in the present work only particles in a single size fractionwere used, namely 75–90 lm, the BET data indicates that particleasphericity increases as the particle size decreases; this is consis-tent with previous reports (Gaudin, 1926; Holt, 1981).

Measured reagent consumption for the borosilicate spheres at100% methylation indicates chemisorption of 10 mg/m2 of pureCTMS, assumed independent of particle size5, after correcting forreagent degradation and losses (Verrelli et al., 2012). The same con-sumption is obtained for borosilicate frit when its specific surfacearea is taken as 2.0 times the specific surface area of spherical parti-cles in the same size fraction, implying a ratio of specific surfaceareas equal to 0.50.

(For the soda–lime glass Ballotini used previously, 18.5 mg/m2

of CTMS was consumed for full methylation (Verrelli et al.,

5 In this calculation the surface area is estimated using the mid-range particle size

.

2012). The same level of consumption applied also to groundBallotini (Verrelli et al., 2012), when its surface area was taken asbeing 1/0.41 = 2.44 times greater than that of an ideal sphere inthe same size fraction, as suggested by BET surface area measure-ments (Koh et al., 2009). These revised estimates differ from earlierestimates made on the basis of pH measurement (Koh et al.,2009).)

3.2. Induction period for attachment

In Figs. 5 and 6 induction period results are presented forspheres and frit, respectively. We have previously described a the-oretical rationale for the induction period to depend upon the polarangle at which the particle impinges on the bubble (Verrelli et al.,2012). Furthermore, models of the attachment process commonlyposit that the attachment will be easier for particles approachingat higher velocities (i.e. greater kinetic energy) (Yoon and Luttrell,1989; Yoon and Mao, 1996; Nguyen and Schulze, 2004: 265,268).Both of these influences are depicted in the graphs.

It should be noted that in the case of angular particles muchgreater variation in the results can be expected as a consequenceof the anticipated dependence on the orientation of the impingingparticle. For example, a cubic particle could impinge point-first orface-on. The hydrodynamic resistances will assuredly be different,the relationship between gap and surface forces will be different,and even the surface chemistry may be different, so a differentinduction period could also be expected. However, as seen fromthe SEM images, the frit exists in a wide range of shapes, so thata simple quantification of the impinging particle’s orientation foreach event is very difficult. Hence, the approach has been to gathermore data for the angular particles, in order to provide a more rep-resentative selection of the possible combinations of particle shapeand orientation.

In short, the experimental data suggest that (on average) theinduction period for attachment is reduced in the case of angularparticles, and is also reduced at greater approach velocities. The di-rect dependence on polar angle seems weaker, but still suggests anincrease in induction period for particles that impinge further fromthe apex (i.e. at larger polar angles), in accordance with predictionsfrom numerical modelling (Verrelli et al., 2012).

0

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0 10 20 30 40

Indu

ctio

n tim

e, τ

[ms]

Polar angle of sliding commencement, φ [ ° ]

Run ARun CRun DLowest v0

Highest v 0

0

20

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0 10 20 30

Indu

ctio

n tim

e, τ

[ms]

Approach velocity, v0 [mm/s]

Run A

Run C

Run D

Highest φLowest φ

(a)

(b)

Fig. 5. Induction period for attachment of 75–90 lm borosilicate glass spheres to a1.56 mm air bubble, as a function of (a) polar angle (Verrelli et al., 2012) and (b)approach velocity. Symbols are experimental observations; curves represent therange of the proposed correlation.

(a)

(b)

0

10

20

30

40

50

60

0 10 20 30 40

Indu

ctio

n tim

e, τ

[ms]

Polar angle of sliding commencement, φ [ ° ]

Run aRun bRun cRun dLowest v0

Highest v 0

0

10

20

30

40

50

60

0 10 20 30 40 50

Indu

ctio

n tim

e, τ

[ms]

Approach velocity, v0 [mm/s]

Run aRun bRun cRun dHighest φLowest φ

Fig. 6. Induction period for attachment of 75–90 lm borosilicate glass frit to a1.56 mm air bubble, as a function of (a) polar angle and (b) approach velocity.Symbols are experimental observations; curves represent the range of the proposedcorrelation.

86 D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89

It should be noted that the apparent dependence of inductionperiod on approach velocity is not linked to the particle size, asmight have been thought. In the experiments, the particles fall un-der the action of gravity, and it is known that larger particles (ofthe same shape and density), will fall faster. However, analysis ofthe data indicates that in the present set-up, variations in the ap-proach velocity are predominantly due to the influence of otherparticles — i.e. swarm effects (Verrelli et al., 2012).

Viewing individual scatterplots in isolation may not give a com-plete picture of the correlation if the parameter of interest dependson multiple variables (Montgomery et al., 2001: 84f). Hence, to as-sess the various proposed dependencies simultaneously we under-take multiple nonlinear regression, which permits a clearerevaluation of the importance of each explanatory variable. For thisanalysis data from all runs are combined.

3.2.1. RegressionRegressions for each combination of terms produced varying

levels of fit, as indicated by the MSE in Fig. 7(a). The lowest valuesof MSE for each number of coefficients is shown in Fig. 7(b). Table 1

gives the best choice of coefficients in Eq. (1) for increasing num-bers of included terms. The information in Table 1 correspondsto the points in Fig. 7(b). There is little benefit in introducing morethan 4 coefficients, and indeed 3 coefficients is sufficient to capturethe influence of each of the three main effects under consideration.The estimated coefficients are generally found to be statisticallysignificant; the only notable exception is for the case of 3 retainedterms (when b1 is included).

The most important influence on induction time, statisticallyspeaking, is the polar angle. If only one coefficient were allowedto be retained in Eq. (1), then the best fit would be proportionalto approximately 0.0222 /2; however, this is quite a crude approx-imation (seen by the large value of MSE in Table 1), and so it isdesirable to retain more coefficients in the fitting equation. Whenmore coefficients are retained, the trend for larger induction timesto be correlated with greater /2 is still consistently found; b7 isconsistently estimated equal to approximately 0.1. This depen-dence accords with our previous numerical modelling (Verrelliet al., 2012). It does not appear to be described elsewhere in theliterature. The difference between /2 and cos(/) is small, as

(a)

(b)

0 2 4 6 8 100

100

200

300

400

500

600

700

800

900

Number of parameters

Mea

n sq

uare

d er

ror [

ms²

]

0 2 4 6 8 10101

102

103

Number of parameters

Mea

n sq

uare

d er

ror [

ms²

]

Any combinationWithout 1/v0

10

Fig. 7. MSE for multiple nonlinear regression according to Eq. (1). (a) Allcombinations of coefficients. (b) Optimal combinations of coefficients, eitherallowing or disallowing inclusion of a 1=v10

0 term.

D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89 87

cos(/) � 1 is directly proportional to /2 for small /. The /2 termdoes not have an obvious physical interpretation, while the cos(/) term does.

The next-most important influence is the particle shape. In allcases the frit is estimated to have induction periods that are onlya few percent of the equivalent spheres, although the precise

Table 1Optimal regressions of s on combinations of the candidate variables, using Eq. (1) with 1–indicate the 95% confidence intervals. Entries in italic denote high uncertainty. Entries in

Coefficient Corresponding parameter Number of coefficients

1 2

b1 1=v100

[s10/mm10]

b2 1=v20

[s2/mm2]

b3 1/v0 [s/mm]b4 Approach velocity, v0 [mm/s]b5 v2

0[mm2/s2]

b6 Polar angle, / [�]b7 /2 [�2] 0.0222 ± 0.0117 0.124 ± 0.013b8 cos(/) [–]b9 Constant [ms]b10 Frit flag, f [–] 0.0569 ± 0.041

MSE [ms2] 717.8 116.8

values fluctuate. This is in agreement with more general studiescomparing ‘floatability’ of angular and round particles, as discussedearlier.

Finally the influence of approach velocity is introduced.Although the 1=v10

0 term is a good fit for the frit data, there is noobvious physical reason for such an unusual dependence. If thisterm is disallowed, the penalty is not too great, and the regressionis then optimised by including a 1=v2

0 term, which is related to theinitial kinetic energy of the falling particle. (The term does not fullyaccount for the energy, as this would require the particle mass,which is especially difficult to determine for the angular frit parti-cles). Although separate models exist relating probability of attach-ment to the induction period (Yoon and Luttrell, 1989; Nguyen andSchulze, 2004: 265,268) and kinetic energy (Luttrell and Yoon,1992), it is not clear from these models how the induction periodshould depend upon kinetic energy.

In all cases the regression predicts increase in s with decreasingv0: the slower the particles approach, the longer the induction per-iod. This is conceptually consistent with models of particle–bubbleattachment in which the two bodies need to overcome an ‘energybarrier’ in order for attachment to occur (Luttrell and Yoon, 1992).The hydrodynamically-based models of induction period(see Nguyen and Schulze, 2004: 507ff) tend to suggest dependenceon 1/v0; this is still compatible with the overall trend observedpresently — the data do not allow a precise judgement of the bestexponent. Similarly, the experimental data of Gu et al. (2003) for aparticle-pickup device is best fit by an equation of the form

t50% attach ¼ a11

v1:14aþ a2: ð2Þ

In this device the bubble is squashed against a bed of particles,approaching at a speed va, and the kinetics may be quite differentto the more natural motions in the present system.

The constant term, b9, is not needed to describe our data.The suggested fit to the data is:

s ¼ 73:8mm=sv0

� �2

þ /

3:11�

� �2" #

ð0:0792f þ ð1� f ÞÞ; ð3Þ

with s in milliseconds. Eq. (3) indicates that the induction period isreduced for angular particles, high-velocity particles, and particlesimpinging nearest the apex of the bubble.

The correlation has been plotted in Figs. 5 and 6. Given thateach individual graph includes only one of either / or v0 on thehorizontal axis, the span of the other parameter is accounted forby plotting two curves: one for the largest value and one for thesmallest value of that ‘missing’ parameter. Considering the scatterof the data, the fits are satisfactory. The correlation provides a rea-sonable prediction of the average induction period under specified

4 coefficients. (For 30 , the 1=v100 term is disallowed: b1 is set to 0). The uncertainties

bold are the adopted data set.

3 30 4

7.63 � 1011 ± 8.82 � 1011

5.45 � 103 ± 3.04 � 103 84.3 � 103 ± 6.3 � 103

23.9 ± 3.5

0.123 ± 0.010 0.103 ± 0.016�698 ± 76

2 0.0303 ± 0.0311 0.0792 ± 0.0362 0.0299 ± 0.0071

64.2 97.0 39.4

88 D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89

interaction conditions. The correlation does not describe the sto-chastic variability, although this may be significant.

4. Conclusions

In the literature a number of papers have suggested increasingfloatability of more angular (or rougher) particles. A few studieshave judged this by flotation recovery, but have not been able toquantify the induction period directly. In the present work we havebeen able to observe particle–bubble interactions, and thus di-rectly obtain estimates of induction period.

Statistical analysis indicated that the induction period, s,depended on three parameters. In order of decreasing statisticalsignificance, these are: the polar angle of sliding commencement,/, an ‘‘indicator variable’’, f, (equal to 1 for frit and 0 for spheres),and the approach velocity, v0. As particles become moreangular, approach at higher velocity, or impinge closer to thebubble’s apex, the induction period is reduced.

Particle shape depends in part on the milling process. The re-sults presented herein indicate the level of attention that shouldbe paid to the shape of particles obtained from the grinding oper-ation, besides particle size. It may be anticipated that the moreangular the particle, the lower the induction period. An extensionof the present work would be to measure the effect of different de-grees of angularity on the induction period.

Acknowledgements

We gratefully acknowledge the assistance of numerous col-leagues at CSIRO who assisted with sample preparation (Cathy Ed-wards), chemical analysis (Cheryl McHugh & team) andmicroscopy (Matthew Glenn), or advised on other aspects. Fundingfor this project was provided from CSIRO’s Process Science andEngineering Capability Development Fund.

Appendix A. Regression formulæ

The simplest regression equation to apply would be

s ¼ b1v0 þ b2/þ b3f þ b4: ð4Þ

The true induction period may take on arbitrarily small values— fractions of a millisecond — but should not become negative.This suggests perhaps an equation of the form

logðsÞ ¼ b1v0 þ b2/þ b3f þ b4: ð5Þ

The variances in s were assumed to be of similar absolute value,irrespective of the magnitude of s. Hence Eq. (5) would produce aninappropriate minimisation of the residuals (Montgomery et al.,2001: 420). The scheme could be returned to homoskedasticityby rearranging the formula to

s ¼ exp b1v0 þ b2/þ b3f þ b4½ �; ð6Þ

but this has the disadvantage of requiring nonlinear regression.The main concern associated with prediction of negative values

of s stems from the (unbounded) approach velocity, for which largevalues may be correlated with small values of s. Therefore an alter-native tactic is regress s on, say, 1/v0, according to

s ¼ b11v0þ b2/þ b3f þ b4: ð7Þ

Note that this is still a linear correlation, because 1/v0 is treatedas an explanatory variable, rather than needing to be decomposedinto the reciprocal of a variable. We are less concerned about this

issue for /, because previous work (Verrelli et al., 2012) indicatesthat although s decreases with decreasing /, there is a natural low-er limit of /, namely / = 0.

A further concern is that it seems more likely for the inductionperiod for the frit to be a multiple or fraction of the correspondingvalue for spheres, whereas the foregoing equations all propose thatthey are interrelated by an offset, namely b3f. A multiplier can beimplemented like so:

s ¼ b11v0þ b2/þ b3

� �b4f þ ð1� f Þð Þ: ð8Þ

The neat thing about Eq. (8) is that b4 has a clear interpretationas the factor by which one needs to multiply s for spheres to obtainthe corresponding value for frit.

It is not possible to be sure in advance what relationship willbest describe the influence of v0, / and f upon s. To leave a varietyof options open, we perform the regression using the followinggeneral equation:

s ¼ b11

v100

þ b21v2

0

þ b31v0þ b4v0 þ b5v2

0 þ b6/þ b7/2

�

þb8 cosð/Þ þ b9Þðb10f þ ð1� f Þ�: ð9Þ

The first term, involving 1=v100 , was obtained by optimising n in

a linear regression in the general form s ¼ b=vn0 + constant, for the

frit data only. The optimum n does not change much when a /2

term is added. There is less data for the spherical particles fromwhich to obtain a reliable estimate of the optimal exponent. Fors ¼ b=vn

0 + constant, n seemed to be optimal at about 3; however,inclusion of a /2 term brought the optimum exponent back up toabout 10, for the sphere data alone.

Nonlinear regression was performed with MATLAB (The Math-Works), making use of the Statistics Toolbox.

Initially it was expected that a stepwise-type regression proce-dure would be most practicable (Montgomery et al., 2001: 310ff).However, given the discordant results seen from different kindsof stepwise regression in preliminary investigations, the complica-tion introduced by nonlinearity, and the power of the computa-tional environment selected, more helpful information wasderived from considering the set of all possible regressions(Montgomery et al., 2001: 302ff). In the case of a linear regressionequation with 10 coefficients, that would imply 210 = 1024 combi-nations. The nonlinearity of Eq. (9) means that when the effect of fwas to be neglected, instead of setting b10 to zero it was set tounity; also the case of only the coefficient b10 being retained wasconsidered physically meaningless, so that only 1023 combina-tions were computed.

The ‘best’ regression for a given number of parameters was cho-sen based on the objective of minimising the mean squared error(‘‘MSE’’), which is similar to — but preferable to — maximisingthe relevant coefficient of determination, R2 (Montgomery et al.,2001: 47f.,90,296ff). Furthermore, the principle of parsimony isemployed, in which extra terms are only included in the model ifthey lead to a substantial reduction in MSE. One other helpful toolto ensure that an excessive number of terms has not been retainedin the model is to ensure all retained coefficients are statisticallysignificant.

Statistical significance can be assessed by checking whether theconfidence interval on the estimated coefficient excludes zero.Here the 95% confidence interval is computed from the diagonalsof the asymptotic covariance matrix and the relevant Student’st-statistic (Montgomery et al., 2001: 428ff.,434ff).

Finally, for completeness it may be noted that an alternative,linear equation,

D.I. Verrelli et al. / Minerals Engineering 58 (2014) 80–89 89

s ¼ b11

v100

þ b21v2

0

þ b31v0þ b4v0 þ b5v2

0 þ b6/þ b7/2 þ b8

� cosð/Þ þ b9 þ b10f ; ð10Þ

which replaces the multiplier effect with a bias effect, was alsoinvestigated. This produced less favourable results, including coeffi-cients with fluctuating signs (see Montgomery et al., 2001: 120ff),and is not considered further herein.

Appendix B. Supplementary material

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.mineng.2014.01.004.

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