*Corresponding author. Tel.: 001 303 492 7314; fax: 001 303 4924341; e-mail: robert.davis@colorado.edu.

Chemical Engineering Science 54 (1999) 149—157

The flotation rates of fine spherical particles under Brownian andconvective motion

Jose A. Ramirez!, Alexander Zinchenko!, Michael Loewenberg", Robert H. Davis!,*

! Department of Chemical Engineering, University of Colorado, Engineering Center-ECCH 111, Campus Box 424, Boulder, CO 80309-0424, U.S.A.;" Department of Chemical Engineering, Yale University, New Haven, CT 06520-2159, U.S.A.

Received 8 July 1997; accepted 26 June 1998

Abstract

The flotation of spherical colloidal particles by small spherical bubbles is considered. The model accounts for the effects ofbuoyancy motion, Brownian motion, van der Waals attractive forces, and hydrodynamic interactions. Conditions are such that thefluid has negligible inertia. The suspension is sufficiently dilute that the analysis is restricted to pairwise bubble—particle interactions.The quasi-steady formulation of the Fokker—Planck equation for the pair-distribution function is simplified for negligible transversaldiffusion and solved numerically. Allowance is made for bubbles with freely mobile or totally immobile interfaces. For size ratios ofthe captured particle to capturing bubble of 0.1 and higher, and for bubble Peclet numbers greater than approximately 105, convectivecapture dominates. For these conditions, the collision efficiencies calculated through the more complete Fokker—Planck formulationare in good agreement with those predicted by a particle trajectory analysis, both for free and rigid interfaces. For more extreme sizeratios of 0.01 and lower, and bubble Peclet numbers less than approximately 105, capture is dominated by diffusion of the smallparticles within the convective flow field created by the rising bubble; however, it is found that the classical mass-transport formula isnot entirely accurate, due to the effects of finite particle size and hydrodynamic interactions when the particles are large enough forboundary-layer mass transfer with high Peclet number to be dominant. A minimum flotation efficiency is observed for a givencollecting bubble size, while, for a fixed suspended particle diameter, it is always more effective to utilize smaller bubbles. Bubbles witha rigid interface exhibit lower collection efficiencies than those with mobile interfaces, especially in the regime of convective capture. Inall instances, the simple additivity approximation for diffusive and convective capture is shown to overpredict the collision efficiencies,in some cases by up to two-fold. ( 1998 Elsevier Science Ltd. All rights reserved.

Keywords: Microflotation; Flotation; Collision efficiency; Brownian motion

1. Introduction

Flotation is a separation method based on the adsorp-tion or attachment of material on the surfaces of gasbubbles passing through a solution or suspension. Itoriginated in the field of mineral processing in the late19th century, where it was used for the removal of metalparticulates from water (Kitchener, 1984). Currently, itplays a vital role in mineral processing, accounting for95% of the total amount of base metals recovered (Matisand Zouboulis, 1995). The flotation of ores consists of theinjection of air in the form of bubbles ranging in diameter

mainly from 0.1 to 5 mm to a vessel containing theore—water mixture with typical particle concentrations ofthe order of 25—40% by weight. The size range of par-ticles in the suspension typically lies between diameters of10—100 km, and a surface-active agent is sometimes usedto facilitate particle attachment (Kitchener, 1984).

Flotation technology is currently being employed ina variety of fields, such as wastewater treatment, biotech-nology, and food processing (Kitchener, 1984). For manyapplications, flotation is best carried out by dissolved-airor electrolytic methods, which generate small bubbleswith diameters in the range 20—100 km (Zabel, 1984).Such small bubbles are needed because the very smallcolloidal particles in these applications are not effectivelyfloated by large, millimeter-sized bubbles (Matis andZouboulis, 1995). The use of small bubbles, termed

0009-2509/99/$— see front matter ( 1998 Elsevier Science Ltd. All rights reserved.PII: S 0 0 0 9 - 2 5 0 9 ( 9 8 ) 0 0 2 1 5 - 2

microflotation, is also suitable for the treatment of emul-sified, oily wastewaters, as the suspended droplets arevery small and close to neutrally buoyant, thus makingconventional approaches of sedimentation or centrifu-gation expensive alternatives (Pushkarev et al., 1983).

The basic mathematical description of the flotationprocess was laid out by Smoluchowski (1916), who de-scribed the kinetics of Brownian particle aggregation inthe absence of hydrodynamic or interparticle forces.Fuchs (1934) later introduced the concept of a stabilityfactor in order to account for the effects of interparticleforces on the particle collision rate. Subsequent attemptsat describing theoretically the underlying processes inparticle aggregation and collection (of which flotationcan be considered to be a subset) focused mainly oneither of two extreme cases: (i) convective capture ofnon-Brownian particles, and (ii) diffusive capture ofBrownian particles.

Non-Brownian particle systems were analyzed bySutherland (1948), who assumed that the smaller particlewould follow potential flow stream lines around thelarger bubble and become captured if its center passeswithin one particle radius of the bubble surface. Flint andHowarth (1971) also studied the situation of negligiblefluid inertia, where the flow field around the capturingbubble is described by Stokes flow. Complete hy-drodynamic interactions for particle aggregation andmicroflotation for Stokes flow have been considered byDavis (1984) and Loewenberg and Davis (1994). Thecapture of Brownian particles has also been studied ex-tensively. Acrivos and Taylor (1962) considered diffusingpoint particles and a rising capturing sphere, and Spiel-man (1970) and Prieve and Ruckenstein (1974) developedexpressions for predicting capture rates of sphericalBrownian particles with hydrodynamic interactions.

Reay and Ratcliff (1973) calculated the collision ef-ficiencies for captured particles of varying diameters (fora fixed diameter of collecting bubble), and experimentallycorroborated the existence of two distinct capture re-gimes, in between which lies a transition zone where bothconvective and diffusive capture contribute substantially.Loewenberg and Davis (1994) supported this notionthrough a rigorous scaling analysis, and established ap-proximate bounding limits for the predominance ofa given capture mechanism in terms of the capturedparticle to capturing bubble (or drop) size ratio, thePeclet number based on the capturing bubble, and themobility of the interface of the capturing bubble.

Early attempts at a unified description of the problemof combined convective and Brownian aggregation in-clude those of Melik and Fogler (1984) and Wang andWen (1990). These workers considered the gravity-in-duced aggregation of particles, accounting for the exist-ence of van der Waals attraction, Brownian motion, andelectrostatic and hydrodynamic interactions. Their anal-ysis is based on the solution of the quasi-steady convec-

tion—diffusion equation in terms of the pair probabilitydensity (Batchelor, 1982); however, it is restricted to thecase where the gravity effects are weak in relation to theBrownian effects (small Peclet numbers). Conversely,Wen et al. (1991) solved the same problem, but for thecase where the influence of the Brownian-induced motionis small in comparison to the motion of the two particlesdue to gravity (large Peclet numbers). Recently, Zin-chenko and Davis (1994) numerically solved the quasi-steady convective—diffusion equation for particle ag-gregation or drop coalescence at arbitrary Peclet num-bers, thus covering the whole range of particle sizesencountered in typical hydrosols. They observed thattheir algorithm becomes computationally slow for higherPeclet numbers, and proposed a more practical approachbased on neglecting the effects of diffusion in the trans-versal direction along the surface of the capturingparticle. The resulting parabolized equation gives ag-gregation or coalescence rates which are typically within1—2% of those computed from the complete equation.

In this study, the parabolized quasi-steady convec-tive—diffusion equation resulting from the neglection oftransversal diffusion to the capturing bubble is solvednumerically for microflotation. Motion of both the cap-turing bubble and the captured particle is assumed to bethe result of gravitational, Brownian, and van der Waalsforces. Because of the small sizes involved, deformation ofthe capturing bubble is neglected, as is the inertia of thebubbles, particles, and surrounding fluid. Two cases, onefor which the interface of the capturing bubble is immobi-lized (by surfactants for example), and the other for whichthe bubble exhibits a freely mobile interface, are con-sidered. Computations are run for different particle-to-bubble size ratios at fixed bubble Peclet numbers, as wellas for different Peclet numbers at fixed size ratios. Thequalitative scaling results from Loewenberg and Davis(1994) are used in order to interpret the quantitativeresults of the present calculations. Finally, the results arecompared against those of a trajectory analysis for con-vective capture (Davis, 1984; Loewenberg and Davis,1994), a refined version of the Levich-Lighthill mass-transfer formula for diffusive capture (Acrivos and God-dard, 1965; Hirose, 1978), and the so-called additivityapproximation (Swift and Friendlander, 1964).

2. Formulation of the problem

Microflotation consists of the removal of very finesuspended particles or droplets (of micron or submicrondimensions) by bubbles or drops whose diameters aretypically less than 100 km. Under typical microflotationconditions, the Reynolds number for such small collect-ing bubbles is much less than unity, thus justifying theassumption of negligible fluid inertia (Loewenberg andDavis, 1994). The suspending fluid is Newtonian, and the

J.A. Ramirez et al./Chemical Engineering Science 54 (1999) 149—157150

Fig. 1. Schematic for semi-Brownian particle capture by a rising bubble.

collecting bubbles as well as the suspended particles areconsidered as spherical. Furthermore, it is supposed thatthere are no interfacial tension gradients on the surfacesof the bubbles and that electrostatic interactions arenegligible. The justification of the above is discussed byLoewenberg and Davis (1994).

Since the suspensions are fairly dilute and large gasholdups are not possible in practice with such smallbubbles (Matis and Zouboulis, 1995), the problem can beanalyzed by considering the interactions between a singlerising bubble and a single suspended particle. It is thusconvenient to formulate the problem in terms of thepair-distribution function, p(r), which represents theprobability density of finding a particle of radiusa2

whose center lies at r, provided that there is a bubbleof radius a

1centered at the origin (see Fig. 1). This

pair-distribution function satisfies the quasi-steadyFokker—Planck equation (Batchelor, 1982)

+ ) (p(r)V12

(r))"0 (1)

where V12

represents the relative velocity between thebubble and the particle. Assuming that, once a bubble anda suspended particle come into contact, a permanent doub-let is formed (i.e. there is no subsequent detachment), thefollowing boundary condition results (Batchelor, 1982):

p(r)"0 for r"a1#a

2. (2a)

Furthermore, if all bubble—particle encounters originateat very large separations, the remaining boundary condi-tion is that of uniform particle concentration:

p(r)P1 as rPR. (2b)

The relative velocity between the capturing bubble andthe suspended particle, V

12"V

1!V

2may be written,

based on a spherical polar coordinate system (Zhang andDavis, 1991), as

V12

(r)

»(o)12

"!¸ cos heLr#M sin heL h

!

1

Q12

Gd/

dseLr!

1

Pe12CG

L (ln p)

LseLr

#

H

s

L (ln p)

LheL hD (3)

where eLrand eL h are the unit vectors along and normal to

the line of centers, respectively. The relative velocity isnormalized by the magnitude of the far-field relativevelocity due to gravity, »(0)

12"DV(o)

1!V(o)

2D, and is given

by the well-known Hadamard—Rybczynski formula forsedimenting spheres:

V(o)12"V(o)

1!V(o)

2"

2

9

a21

ke

(kL1#1)(o

1!o

e)

(kL1#2/3)

][1!cj2a]g (4)

with

a"(kL

1#2/3)(kL

2#1)

(kL2#2/3)(kL

1#1)

, c"o2!o

eo1!o

e

,

j"a2

a1

. (5)

Here, kLirepresents the bubble (i"1) or particle (i"2) to

suspending fluid viscosity ratio, keis the viscosity of the

suspension fluid, oiis the density of the bubbles or par-

ticles, oeis the density of the suspension fluid, and g is the

gravitational acceleration vector.The relative mobility functions along the line of centers

(¸ and G) and normal to the line of centers (M and H) inEq. (3) account for the effects of hydrodynamic interac-tions between the bubble and particle. These functionsdepend on the viscosity ratios, kL

1and kL

2, the reduced

density ratio, c, the particle to bubble size ratio, j, and thedimensionless interparticle distance, s"2r/(a

1#a

2).

They were calculated by modifying previously developedalgorithms (Zinchenko, 1978, 1980, 1982). The para-meters Pe

12and Q

12in Eq. (3) provide a measure of the

relative importance between gravitational forces andBrownian diffusion forces, and between gravitational for-ces and interparticle forces, respectively, and are given by

Pe12"

(a1#a

2)»(o)

122D(o)

12

and Q12"

(a1#a

2)»(o)

122AD(o)

12/k¹

. (6)

J.A. Ramirez et al./Chemical Engineering Science 54 (1999) 149—157 151

The combined far-field diffusivity, D(o)12

, is the sum of theindividual bubble and particle diffusivities given by thewell-known Stokes—Einstein expression

D(o)12"D(o)

1#D(o)

2"

k¹

6nkea1

(kL1!1)

(kL1#2/3) C1#

ajD (7)

where k"1.381]10~23 J/K is the Boltzmann constantand ¹ is the absolute temperature.

The dimensionless interparticle force potential,/"'

12/A (where '

12is the force potential and A is the

composite Hamaker constant), describes the bubble—par-ticle force due to retarded van der Waals interactions.For successful flotation or capture, this force must beattractive. A detailed treatment of the interparticle forcepotential employed is given elsewhere (Zinchenko andDavis, 1994).

Substitution of Eq. (3) into Eq. (1) yields the dimen-sionless Fokker—Planck equation describing the com-bined gravitational convection and Brownian diffusionof the particles toward a collecting bubble:

1

s2

LLs C!s2pA¸ cos h#

G

Q12

d/

ds#

G

Pe12

L(ln p)

Ls BD#

M

s sin hL (p sin h)

Lh!

H

Pe12

s2 sin h

]LLh A sin h

Lp

LhB"0. (8)

The first term in Eq. (8) accounts for the relative motionbetween the bubble and the particle in the directionalong their line of centers, and includes a convectivecontribution due to gravity (or buoyancy), a convectivecontribution due to van der Waals attraction, and a dif-fusive contribution due to Brownian effects, in that order.The second term in Eq. (8) represents transversal (normalto the line of centers) convective motion due to gravity,while the third term represents transversal diffusion dueto Brownian motion. Note that no term accounting fortransversal motion due to van der Waals attraction ispresent, since these forces act only along the line ofcenters.

3. Calculation procedures

For small relative Peclet numbers (Pe12

@1), Browniandiffusion is dominant and the pair distribution is radiallysymmetric (depending only on s and not on h). Then thetransverse diffusion in the angular direction is negligible.Perhaps surprisingly, the opposite limit of Pe

12A1, for

which gravitational motion is dominant, also givesa pair-distribution function which is radially symmetric(Batchelor, 1982). Although this is not the case forintermediate Peclet numbers, the error resulting from

neglecting the transverse diffusion is expected to be small.Indeed, as shown by Zinchenko and Davis (1994) forcoalescence of drops of the same fluid dispersed in a sec-ond fluid, the contribution of the transversal diffusionterm is always insignificant, with errors in the coales-cence rate not exceeding 2—3% when the whole range ofrelative Peclet numbers is considered.

Substantial simplification of the mathematics of theproblem is gained by neglecting the transverse diffusionterm, as the original elliptic partial differential equationof Eq. (8) then becomes a parabolic partial differentialequation, whose solution can be pursued by a relativelysimple non-iterative scheme. The simplified equationthen becomes, after the transformation u"cos h andintroducing the dimensionless Hamaker constantAK "A/k¹,

LLs C s2p¸u#

Gs2pAKPe

12

d/

ds#

Gs2

Pe12

Lp

Ls D"Ms

L[p (u2!1)]

Lu. (9)

This simplification is referred to as the parabolic approx-imation (PA). Eq. (9) has been solved previously by Zin-chenko and Davis (1994) for the case of spheres of equaldensity and viscosity. For this study, their algorithm wasextended for particles of different densities and viscosi-ties. In essence, the method consists of expanding thesolution for p (s, u) in powers of u!1 (assuming regular-ity at u"1) for a given value of Pe

12, instead of finite-

difference marching from u"1, and successively findingthe coefficients of a sufficient number of terms by substi-tuting this power expansion into Eq. (9). In particular,the first term of the expansion, p

o, which is the initial

distribution of p (s, u) at u"1, is directly determinedfrom Eq. (9) as the solution of an ordinary differentialequation along u"1, when the right hand side of Eq. (9)is replaced by 2Msp, and boundary conditions (2a) and(2b) are used. In the radial direction, the same finite-difference scheme as in Zinchenko and Davis (1994) isused. The cutoff distance was typically set at s

="25,

which allows us to cover the whole range from Pe12

moderate to large Pe12

. For Pe12

A1, only a thin bound-ary-layer contribution to the probability flux is essential;however, considering the efficiency of the parabolic ap-proximation, we did not try to adjust the position of thecutoff distance s

=to the boundary-layer thickness, but

kept it fixed and simply used a large number of radialnodes.

The comparison of the exact and PA-based local prob-ability flux, as a function cos h, was made in Zinchenkoand Davis (1994) for drops with and without van derWaals attractions (see their Figs. 8 and 10). Althoughthere was a noticeable difference in the local flux for smalland moderate Peclet numbers, the integral error in thetotal probability flux was always small. For high Peclet

J.A. Ramirez et al./Chemical Engineering Science 54 (1999) 149—157152

Fig. 3. Collection efficiency as a function of the size ratio for Pe"103

(upper, thin lines) and Pe"105 (lower, thick lines) for a rigid interfacebubble. The solid lines represent the prediction of the parabolizedFokker—Planck equation, the dashed line is the result of the boundarylayer diffusion equation, the dotted line is from the trajectory analysis,and the dashed-dotted line is the additivity approximation.

Fig. 2. Collection efficiency as a function of the size ratio for Pe"103

(upper, thin lines) and Pe"105 (lower, thick lines) for a mobile inter-face bubble. The solid lines represent the prediction of the parabolizedFokker—Planck equation, the dashed line is the result of the boundarylayer diffusion equation, the dotted line is from the trajectory analysis,and the dashed-dotted line is the additivity approximation.

numbers, the exact and PA solutions were found tocoincide in the whole range of h. Of course, the parabolicapproximation cannot properly describe the asymptoticstructure of the wake at cos h"!1 (where tangentialdiffusion is essential) in higher orders of approximation,but this region has a small contribution to the total flux,and the solutions were found again to give practicallyidentical capture rates. As follows from Zinchenko andDavis (1994), the local probability flux vs cos h at h"nhas a finite slope that tends to be zero for large Pe

12. This

implies, in particular, a zero slope of flux vs h.The rate of particle capture per unit volume is given by

the flux of pairs into the contact surface r"a1#a

2:

J12"!n

1n2 P

r/a1`a2

pV12

)r

rdA. (10)

Here, n1

and n2

are the number of drops and particles ofradius a

1and a

2per unit volume of the dispersion. The

requirement that the pair density function be null at thecontact surface [see Eq. (2a)] and the necessity of satisfy-ing Eq. (1) reveal an indeterminancy in the integrand ofEq. (10). To overcome this problem, the Gauss diver-gence theorem is applied in conjunction with the steadyFokker—Planck equation to evaluate the above integralon any surface enclosing the original contact surface.

A dimensionless flotation rate, E12"J

12/J(o)

12(also

known as the collision efficiency or particle capture effi-ciency (Loewenberg and Davis, 1994)) is defined by tak-ing the ratio of the rate of particle capture calculatedfrom Eq. (10) to the rate of particle capture withoutallowance for hydrodynamic or interparticle interactions,as well as without the effects of Brownian diffusion:

J(o)12"n

1n2»(0)

12n(a

1#a

2)2. (11)

In summary, the procedure consists of calculating thepairwise distribution function p(r) by solving Eq. (9) forthe radial dependence using a series expansion for theangular dependence, calculating the bubble-particle rela-tive velocities V

12(s) through Eq. (3), and utilizing

Eq. (10) for calculating the particle capture rate J12

. Thisprovides the necessary information on the kinetics of therate of particle removal for designing or predicting theperformance of the microflotation cell.

4. Numerical results and discussion

Since, even in dimensionless form, there is a large num-ber of parameters governing the microflotation process,the practical case of an aqueous suspension at ambienttemperature using air bubbles as collectors (o

e!o

1"

1.0 g/cm3) to flotate neutrally buoyant particles(o

e"o

2) was considered. The value of the Hamaker

constant was taken as the typical A"k¹"4]10~21 J(Loewenberg and Davis, 1994), and the common retarda-

tion London wavelength of jL"0.1 km was assumed

(Loewenberg and Davis, 1994). Calculations were carriedout for a range of bubble radii, 3 km(a

1(32 km, and

a range of size ratios 0.003(j(1. The cases of a bubblewith a freely mobile interface or with a completely rigidinterface were treated by choosing kL

1"0 or kL

1PR,

respectively. The suspended particles were treated asrigid (kL

2PR); as shown by Loewenberg and Davis

(1994), however, flotation rates are not largely affected bythe physical properties of the suspended particles ordroplets.

4.1. Effect of particle size

Figs. 2 and 3 were generated for the case of bubbleswith completely mobile interfaces (kL

1"0) and rigid

interfaces (kL2PR), respectively, at two fixed Peclet

numbers of Pe"103 and Pe"105, corresponding to

J.A. Ramirez et al./Chemical Engineering Science 54 (1999) 149—157 153

bubble diameters of 6 and 20 km, respectively. Here,Pe"a

1»(o)

1/D(o)

1"2Pe

12(2#3j)/(3(j#j2 )) is the Pec-

let number of the collector bubble. The dimensionlessflotation rates calculated from the parabolized Fok-ker—Planck equation are compared to the results of a tra-jectory analysis (Davis, 1984; Loewenberg and Davis,1994) and to the mass-transport formulas derived fromthe convection—diffusion equation for point particlesand a thin convective boundary layer surrounding therising bubble (Hirose, 1978; Acrivos and Goddard,1965). In terms of the variables used herein, the leadingterms of the mass-transport formulas give E

12"1.503

(jPe)~1@2#2.205 (jPe)~1 or E12

"2.498 (jPe)~2@3

#1.824 (jPe)~1 for mobile or rigid bubbles, respectively.The former approach does not take into account theeffects of Brownian forces on either the bubble or par-ticle, while the latter neglects direct hydrodynamic andinterparticle interactions (except that the velocity fieldcreated by the rising bubble is considered with respect toits convection effects on the diffusing point particles). Thedimensionless flotation rates obtained from an additivityapproximation, representing the algebraic addition of thetrajectory results and the mass-transport results, are alsoincluded for comparison; the additivity approximationoverestimates the collection efficiency in almost all cases.

The dimensionless flotation rates are well described bythe trajectory analysis for suspended particles of compa-rable dimensions to the collecting bubble, implying thatthe assumption of dominant convective capture is accu-rate for these conditions with Pe*O (103). Furthermore,it is apparent that, at the higher bubble Peclet number,the convective capture mechanism dominates overa wider range of size ratios, as expected. It is noted that,for the case of bubbles with mobile interfaces at highPeclet numbers and with particle-to-bubble size ratiosclose to unity, the integration of Eq. (9) does not converge(see Zinchenko and Davis, 1994). Thus, the solution forthe Fokker—Planck equation is discontinued in the rangewhere these conditions occur (see Fig. 2); fortunately, thetrajectory analysis is accurate in this range.

As smaller size ratios are considered, diffusion effectsbecome increasingly important. Loewenberg and Davis(1994) predicted that a minimum value of the collectionefficiency results when convective and diffusive captureare comparable; the critical size ratio where this occursfor bubbles with mobile interfaces is j"O(Pe~1@3) andthe corresponding minimum collision efficiency isE12"O(Pe~1@3). These scaling estimates compare fa-

vorably with the results obtained from the parabolizedFokker—Planck equation, for which the minimum collec-tion efficiencies are E

12"0.15 and E

12"0.035, at

j"0.15 and j"0.022 for Pe"103 and Pe"105,respectively.

For the case of bubbles with completely rigid interfaces(see Fig. 3), the results exhibit similar qualitative behav-ior. While the collection efficiencies in the regime of

diffusive capture are not markedly different from thosefor bubbles with freely mobile interfaces (see Fig. 2), theyare substantially lower when convective capture domin-ates, due to the fact that a mobile bubble interface aids inthe displacement of fluid from the lubrication gap be-tween the bubble and particle. The minimum flotationefficiencies are E

12"0.05 at Pe"103 and E

12"0.007

at Pe"105, which occur at j"0.20 and j"0.055, re-spectively. These values lie within the order of magni-tudes predicted by the scaling analysis of Loewenbergand Davis (1994) of E

12"O(Pe~1@2) and j"O (Pe~1@4)

for bubbles with rigid interfaces.The existence of a minimum in the collection efficiency

can be explained on physical grounds. For size ratiosclose to unity, particles are captured by convective ef-fects; the velocity of the particle relative to the bubble ismostly a result of gravitational forces, and they willfollow deterministic trajectories resulting from their in-teraction. Whenever these trajectories are such that phys-ical contact occurs, it is assumed that the particle wascaptured. As smaller suspended particles are considered,the chance of physical contact is lessened; the smallerparticles will tend to follow the flow streamlines and beconvected around the larger rising bubble without theoccurrence of physical contact. This will result in lowervalues of the capture efficiency. If, however, the size of thesuspended particles is such that their Brownian diffus-ivity becomes important, a change in the underlyingmechanism for capture occurs. In this case, motion of thesuspended particles is primarily due to Brownian forces,and, since the magnitude of the Brownian diffusivityincreases inversely with the diameter of the suspendedparticles, the rate at which particles move towards thecollecting bubble will then increase with decreasing sizeratio.

An exact agreement is not met between the mass-transport formulae (when Brownian diffusion within theconvective flow field of the rising bubble is dominant)and the Fokker—Planck results, even for the case ofextremely small size ratios, although the general trend forthe collision efficiencies is observed. Since the mass-trans-port formulae were developed under conditions forwhich the combined Peclet number based on bubbleconvection and the particle diffusivity is large(Pe*"a

1»(o)

1/D(o)

2A1), and the dimensions of such par-

ticles are small in relation to the thickness of the mass-transfer boundary layer around the rising bubble, it isuseful to inquire if these conditions are met for theparticular cases treated.

For small size ratios (jP0), it has been shown byLoewenberg and Davis (1994) that the combined Pecletnumber is essentially equivalent to Pe*"jPe. On theother hand, it is also shown by Loewenberg and Davis(1994) that the boundary layer thickness is large com-pared to the suspended particle diameter whenj@Pe~1@3 or j@Pe~1@4, for mobile or rigid interface

J.A. Ramirez et al./Chemical Engineering Science 54 (1999) 149—157154

Fig. 4. Collection efficiency as a function of the bubble Peclet numberfor j"0.1 (thin lines), j"0.033 (thicker lines) and j"0.01 (thickestlines) for a mobile interface bubble. The solid lines represent theprediction of the parabolized Fokker—Planck equation, the dashed linesare the result of the boundary layer diffusion equation, and the dottedlines are the results of a trajectory analysis.

Fig. 5. Collection efficiency as a function of the bubble Peclet numberfor j"0.1 (thin lines), j"0.033 (thicker lines) and j"0.01 (thickestlines) for a rigid interface bubble. The solid lines represent the predic-tion of the parabolized Fokker—Planck equation, the dashed lines arethe result of the boundary layer diffusion equation, and the dotted linesare the results of a trajectory analysis.

bubbles, respectively. The high Peclet number conditionevaluated at a size ratio of j"0.01 yields values ofjPe"10 and jPe"103 for Pe"103 and Pe"105,respectively. Maintaining Pe"103 and consideringmore extreme size ratios results in the product jPe notbeing much larger than unity, thus violating the require-ment of high Pe*. This means that the results of themass-transport formula will not be accurate for eithercase of mobile or rigid interfaces at Pe)103, as long asj(0.01.

For the case of Pe"105, the product jPe is still atleast two orders of magnitude higher than unity in thelower range of size ratios considered in Figs. 2 and3 (0.0033(j(0.01), and thus, on these grounds, themass-transport formulae would be applicable. However,at this higher value of the Peclet number, the boundarylayer thickness is comparable to the dimensions ofthe suspended particles, since h (j)+O (Pe~1@3) andO(j)+O (Pe~1@4) for mobile or rigid interfaces, respec-tively. This means that, for practical purposes, the mass-transport results of Figs. 2 and 3 are not completelyaccurate.

4.2. Effect of bubble size

Calculations of the collection efficiency as a function ofthe bubble size or Peclet number for three particle-to-bubble size ratios were also performed, for each case ofcompletely mobile and completely rigid bubble interfaces(see Figs. 4 and 5). For bubbles with a

1*5 km and

completely mobile interfaces, floated particles which areten times smaller in diameter are captured predomi-nantly by convective effects, which is typical of microflo-tation. In this range, the collision efficiency is describedvery well by the trajectory analysis. As the particle sizedecreases with respect to the bubble size, the range ofbubble sizes for purely convective capture narrows, untildiffusive capture begins playing an important role as theparticles become Brownian. Fig. 4 shows that convectivecapture effects are dominant when a

1*10 km for a size

ratio of j"0.033, and that diffusive effects have gainedimportance in the lower range of Pe. For the moreextreme size ratio of j"0.01, diffusive capture nowplays a more important role for a

1)20 km, as is evident

from the poor agreement with the trajectory solutionand the improved agreement with the mass transportformula.

As discussed previously, the collection efficienciesare lower when the bubble possesses a completely rigidinterface (see Fig. 5), especially where convectivecapture dominates. Furthermore, the effects of Browniandiffusion gain relative importance at a larger Pecletnumber when rigid bubbles are involved, in comparisonto those with mobile interfaces. This is in accordancewith the scaling results of Loewenberg and Davis (1994),who found that the diffusive effects will dominate

capture if O (jPe)~1@3@1 for rigid interface bubbles, or ifO(jPe)~1@2@1 for mobile interface bubbles.

Conversely to what occurs when the effect of the sus-pended particle size is examined, the capture efficiencycurves do not exhibit minima when plotted as a functionof the bubble radius or Peclet number (Figs. 4 and 5).This is consistent with the fact that, as the size of thecollecting bubble is increased, both the contributionsfrom Brownian motion and from convective motion toparticle capture diminish. The former is due to the largersuspended particle (recalling that the size ratio is keptconstant for each curve of Figs. 4 and 5) possessinga smaller Brownian diffusivity, which reduces the drivingforce for collection by Brownian motion. The latter effectstems from the larger relative velocities between the

J.A. Ramirez et al./Chemical Engineering Science 54 (1999) 149—157 155

rising bubbles and particles that result from larger Pecletnumbers; these in turn translate into larger interparticlehydrodynamic forces relative to attractive van der Waalsforces. These two effects will contribute to lower captureefficiencies with increasing bubble size, independently ofwhether the dominating mechanism for capture is con-vective or diffusive in nature.

5. Concluding remarks

The flotation rates of fine spherical particles by spheri-cal bubbles have been calculated using a parabolizedform of the quasi-steady-state Fokker—Planck convec-tive—diffusion equation. Allowance was made in themodel for detailed pairwise hydrodynamic interactionsand attractive van der Waals forces between the bubbleand particle. Bubble and particle motions were assumedto be induced by both gravitational and Brownian ther-modynamic forces under conditions of negligible fluidinertia. The range of bubble Peclet numbers studiedcorresponds to that frequently encountered in microflo-tation practice, where the bubbles generated have dia-meters in between approximately 5 and 100 km. The sizeratios employed in the calculations provide results rel-evant to the microflotation of Brownian, semi-Brownianand non-Brownian particles.

For typical microflotation conditions, convectivemechanisms dominate particle capture when the bubblePeclet number is Pe*106 and the size ratio of particle tobubble is j*0.1, approximately. This corresponds to thesituation of bubbles of diameter greater than approxim-ately 35 km floating particles larger than approximatelyone-half of a micron in diameter. In this case, the ob-served floating efficiency will not be greater than tenpercent. On the other hand, flotation occurs from purelydiffusive mechanisms when the bubble Peclet numberand the particle to bubble size ratio are such thatPe)105 and j)0.033, approximately. For these condi-tions, the greatest capture efficiencies occur, with valuesas high as 100%.

Flotation performance is degraded significantly whenthe bubble interface is made rigid, for example due to thepresence of insoluble surfactants. This effect is not asimportant when Brownian particles are involved, but isespecially large for the case of flotation of non-Brownianparticles. As shown in earlier studies, the calculationsdemonstrate that the performance of the flotation pro-cess is greatly enhanced by using smaller collecting bub-bles. It is also observed, for a fixed collecting bubble size,that a minimum flotation rate will exist for suspendedparticles of a semi-Brownian size. This observation isconsistent with the results from the scaling analysis ofLoewenberg and Davis (1994).

It is seen that the additivity approximation signifi-cantly overestimates the actual collision efficiencies over

most of the range of the parameters considered, indicat-ing that the linear superposition of convective captureand diffusive capture results is not recommended in gen-eral. On the other hand, a trajectory analysis that takesinto account the complete hydrodynamic interactionsbetween bubble and particle will result in an adequatedescription of the process, at least for those conditionsunder which both bubble and particle Brownian diffus-ivity are negligible.

The flotation rates predicted by the mass-transportsolution of the convection-diffusion equation for pointparticles to a translating sphere do not coincide with thecomplete solution, at least for the conditions relevant tomicroflotation. Under the conditions of bubble Pecletnumber and particle-bubble size ratio for which diffusiveeffects dominate capture, one or both conditions for thevalidity of the simple mass-transport formulae are viol-ated. For the cases when the Peclet number is highenough for boundary-layer mass transfer to be dominant,the thickness of the boundary layer is comparable to thedimensions of the suspended particle, resulting in inac-curacies in the predicted capture efficiencies. In thesecases, the parabolized Fokker—Planck equation resultscan be used in lieu of the mass-transport formulae, whenmore accurate estimations of the particle capture efficiencyare desired.

Acknowledgements

This work was supported by the National ScienceFoundation through grant CTS-9416702

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