Chemical Engineering Science 62 (2007) 5736–5744www.elsevier.com/locate/ces

Convective–dispersive gangue transport in flotation froth

Paul Stevensonb,∗, Seher Ataa, Geoffrey M. Evansa

aCentre for Advanced Particle Processing, University of Newcastle, Callaghan, NSW 2308, AustraliabCentre for Multiphase Processes, University of Newcastle, Callaghan, NSW 2308, Australia

Received 14 March 2007; received in revised form 29 May 2007; accepted 29 May 2007Available online 14 June 2007

Abstract

The transport of gangue through flotation froth has been described by solving the convection–diffusion equation. Gangue recovery is predictedto be proportional to liquid recovery, which is consistent with experimental observation. In addition, it is seen that the dependency of ganguerecovery upon particle size is due to processes within the pulp phase rather than the froth, insofar as the transport of particles in a given froth isapproximately independent of size. The importance of maintenance of positive bias in column flotation, previously stressed by other workers,is reinforced. This model utilises a simplified representation of the froth and, as a consequence, it does not necessarily give accurate ganguerecovery estimates for practical flotation processes. However, the convective–diffusive model does illuminate the physical processes behindgangue recovery in the concentrate which will aid the development of automatic control strategies.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Flotation; Gangue; Foam; Convection; Dispersion

1. Introduction

It has been noted by Ata et al. (2002) that, no matter how hardthe froth phase is washed in column flotation, there is alwayssome unwanted recovery of gangue material in the concentratestream. This observation has a profound influence on the designof flotation circuits since a number of flotation operations arerequired in series to achieve the desired product recovery andselectivity.

Kirjavainen (1996) asserted that there are two principalmechanisms by which gangue particles are recovered in theconcentrate stream in flotation. Entrainment is caused by con-vection of liquid from the pulp to the froth, entrapment occurswhen particles become ‘wedged’ between bubbles. In thisarticle we will consider the former mechanism.

There have been a number of published models for theentrainment of gangue into the concentrate stream. It haslong been recognised that gangue entrainment rate approxi-mately scales with the rate of water recovery (Engelbrecht andWoodburn, 1975), and this observation has recently been

∗ Corresponding author. Tel.: +61 2 49616192.E-mail address: Paul.Stevenson@newcastle.edu.au (P. Stevenson).

0009-2509/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ces.2007.05.038

reaffirmed by Zheng et al. (2006). Kirjavainen (1992) presentedan empirical expression for this scaling factor as a functionof water recovery rate, particle density, slurry viscosity and aparticle shape factor. However, the expression is dimensionallyinconsistent and is therefore only strictly effective for the sys-tem on which the data was taken (Stevenson and Galvin, 2007).Neethling and Cilliers (2002a) developed a numerical modelfor two-dimensional gangue entrainment and presented resultsof simulations. Their model was dependent upon a dispersioncoefficient, but no indication of how this should be calculatedwas given, with an arbitrary selection made for the benefit ofthe simulations. A similar approach was taken by Neethlingand Cilliers (2002b). In addition, the channel-dominated foamdrainage equation of Verbist et al. (1996) was assumed; all pub-lished foam drainage data investigated by Stevenson (2007a)suggest that this model under-predicts liquid drainage rate by afactor of at least 10. Moreover, Neethling and Cilliers (2002a, b)did not explicitly give boundary conditions for their model mak-ing it difficult to be replicated by other researchers.

It is apparent that the transport of gangue through the frothdue to entrainment is as a result of two processes: (1) con-vection of particles due to net transport of liquid through thefroth, and (2) dispersion of particles within the froth. Thus,if simplifying assumptions are made about the state of the

P. Stevenson et al. / Chemical Engineering Science 62 (2007) 5736–5744 5737

froth, the transport of gangue can be predicted by solving theconvection–dispersion equation for the gangue particles. Suchanalyses are very common, especially in the modelling of en-vironmental transport of contaminants; for an example pleasesee Nguyen et al. (1999). However, in order to construct theappropriate convection–dispersion equation the liquid drainagerate and dispersion coefficient must be known, and these are es-timated using previously published empirical correlations. Byadopting such a semi-empirical approach we will show whatthe underlying physical factors govern gangue entrainment inflotation are. Such knowledge of the principal governing factorswill enable the development of more effective strategies for au-tomatic control of the flotation process. In addition, this modelis the first to provide estimates of the effect of washwater rateupon gangue entrainment. We are fully aware that our worklacks experimental verification. Previous work of Neethlingand Cilliers (2002b, 2003a) suffered similarly, and thisreflects the complexity of the flotation system. However, theliquid drainage rates and dispersion coefficients used hereinhave been experimentally verified.

Both Xu and Finch (1991) and Mavros (1993) have studiedaxial dispersion in the pulp phase in flotation (known as thecollection zone in column flotation), whereas we specificallystudy the froth phase (known as the cleaning zone in columnflotation) as it is the transport of gangue through the froth thatinterests us. The theory that we will present is general to bothcolumn flotation devices as well as mechanical cells. The rel-evant differences between the two types of flotation are:

1. Mechanical agitation is provided to the pulp phase of a me-chanical cell, whereas mixing of the gangue in the collectionzone must come from the turbulence of the bubbly mixturein column flotation.

2. The addition of washwater to the froth of mechanical cellsis rare, but washwater addition to the surface of the froth incolumn flotation is universal.

3. The depth of the froth in mechanical flotation is often small(i.e., a few centimetres in depth in rougher cells), whereascleaning zone depths in column flotation are typicallyaround one metre (Finch and Dobby, 1990).

The theory presented applies to the transport of gangue throughthe froth phase in new generation flotation cells, such as theJameson Cell, too.

A note on nomenclature: We will adhere where possible toconventional nomenclature, but it is appropriate to explicitlydefine the direction of fluxes. Gas and liquid flux in the froth, jg

and jf , are measured positive upwards. However, bias rate, jB ,is given as positive downwards by Finch and Dobby (1990), andwe maintain this convention herein. In addition we define theliquid drainage superficial velocity, jd , and added washwatersuperficial velocity, jW , as positive downwards.

2. Hydrodynamics of rising foam: the convection term

Before we can begin to consider the behaviour of parti-cles within a pneumatic froth, we must first understand its

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0 0.1 0.2 0.3

Liquid fraction, �

jf (

mm

/s)

Equilibrium liquidfraction

Enhanced liquidfraction

jW

jB

Fig. 1. jf versus � for a pneumatic foam showing the graphical calculationof equilibrium liquid fraction, enhanced liquid fraction due to washwater ratejW and the definition of bias rate, jB (after Stevenson, 2007b).

hydrodynamic condition. Stevenson (2006a) showed that theliquid superficial drainage rate from a stationary column offoam, jd , could be non-dimensionalised as a Stokes-type num-ber, Sk, where

Sk = �jd

�gr2b

(1)

(� is the liquid density, � is the liquid dynamic viscosity, g is theacceleration due to gravity and rb is the harmonic mean bubbleradius) and, for a temporally and spatially invariant froth, couldbe expressed as a power-law function of the volumetric liquidfraction, �, only:

Sk = m�n, (2)

where m and n are adjustable dimensionless constants specificto a certain surfactant system. Two adjustable constants arethe minimum required to describe this system since we cannotquantify the viscous losses at the nodes (Koehler et al., 1999)and we cannot measure the surface shear viscosity (Stevenson,2005). Stevenson et al. (2007) have shown that, for foam sta-bilised by 2.92 g l−1 SDS, m = 0.016 and n = 2.

Now, this simple equation for the liquid drainage rate froma stationary foam may be readily adapted to describe the hy-drodynamics of a rising foam. Stevenson (2006b) showed thatthe liquid superficial rate, jf , rising in the foam could beexpressed as

jf = �jg

(1 − �)− �gr2

b

�m�n, (3)

where jg is the superficial velocity of gas sparged to the flotationmachine. The dependency of jf upon � is shown in Fig. 1 as-suming rb =0.5 mm, �=1000 kg m−3, �=1 cP, jg =7 mm s−1

and using the drainage parameters, m and n, of Stevensonet al. (2007). Stevenson (2006b) showed that the maximum ofthe curve in Fig. 1 represented the equilibrium condition. Theimplication is that a pneumatic foam adjusts its liquid fractionto maximise the liquid rate. The equilibrium volume fractionmay be calculated through numerical solution of

�jg

mn�gr2b

= �n−1(1 − �)2 (4)

5738 P. Stevenson et al. / Chemical Engineering Science 62 (2007) 5736–5744

whence the liquid fraction, jmaxf , is obtained by substituting

the liquid fraction into Eq. (3). The efficacy of this method hasbeen proved experimentally by Stevenson (2006b). Note thatit is assumed that there is no internal coalescence nor is therecoalescence of bubbles on the surface of the froth. Stevenson(2007b) has given an analysis of the effect of coalescence uponthe froth condition.

In froth flotation, washwater is added at a superficial rate jW

to the surface of the froth to aid rejection of gangue from theconcentrate stream. Washwater enhances the liquid fraction ofthe foam, and can be readily calculated by reference to Fig. 1or by numerical solution of

jmaxf − jW = �jg

(1 − �)− �gr2

b

�m�n. (5)

The efficacy of this method of calculating the enhanced liq-uid fraction due to washwater addition has been confirmed byIreland and Jameson (2007). Stevenson (2006b) showed that,no matter what the rate of washwater, the liquid recovery fromthe top of the column remained constant at jmax

f .Finch and Dobby (1990) suggested that columns should be

operated in positive bias (i.e., jW > jmaxf ) in order to ensure

effective gangue rejection. In this work we define the bias rateas jB = jW − jmax

f . Note that jB = −jf (see the note onnomenclature above).

3. The dispersion term

Lee et al. (2005) measured the axial dispersion coeffi-cient, D, for particles and liquid tracer in a foam undergoingforced drainage. They found that the dispersion coefficients forparticles and liquid tracer were similar, thereby indicating in-dependence upon particle size. They gave a dimensionally in-consistent expression for a fit to their data. However, Stevenson(2006c) has, by invoking dimensional analysis, shown that thedimensionally consistent expression:

D

rbjd

= 2.84Sk−1/3 (6)

provides an excellent fit to the particle and liquid dispersiondata of Lee et al. (2005) and has positively demonstratedthat the axial dispersion coefficient is independent of particleself-dispersivity. The implication of this observation is thatdispersion is a mechanical process and does not involve Taylordispersion which is consistent with the theory of dispersionin porous media of Saffman (1959) and the experimental ob-servations of Stevenson et al. (2003) for tracer dispersion inrising foam. Sk is as defined in Eq. (1) and jd is calculatedby substituting Eq. (1) into Eq. (2). Note that Eq. (6) is of thesame functional form as correlations for dispersion in porousmedia in the ‘Pure Mechanical Dispersion’ regime (Fried andCombarnous, 1971) save for the appearance of the Stokesnumber that takes into account the variable volume fraction ofthe liquid phase as a function of the liquid rate; in a packedbed the liquid volume fraction is constant.

Meloy et al. (2007) have recently asserted that “The axialdispersion coefficient was found to change significantly as a

result of particle size and density.” They are quite correct. Leeet al. (2005) found that the dispersion coefficients were practi-cally identical for liquid dispersion and the dispersion of parti-cle of diameter 5 �m. This result indicates for “particles” in asize range of between a few Angstroms and 5 �m that the dis-persion coefficient is independent of particle size; there is noexperimental data for dispersion coefficients of larger particles.Since these two sets of data (i.e., those for liquid and parti-cle dispersion) are practically identical, it is indicated that thephysics that govern dispersion for particles of this size range ispurely mechanical since there is no dependency on ‘particle’size, and this physics will pertain to still larger particles. How-ever, a 5 �m particle has a negligible terminal velocity (in theorder of a few tens of microns per second). For larger particles(or more dense) particles that settle at a faster rate we mustmodify Eq. (6) since the particles no longer travel with the liq-uid but exhibit a greater velocity and are therefore exposed to agreater number of tortuous channels within the foam network.Thus, we suggest that Eq. (6) is strictly modified as follows:

D

rb(jd + �VS)= 2.84

(�(jd + �VS)

�gr2b

)−1/3

(7)

since the velocity experienced by the particles is modified fromjd to (jd +�VS). (Eq. (7) is identical to Eq. (6) with jd replacedby (jd + �VS). In fact, it is demonstrated below that ganguerecovery is approximately independent of the dispersion coef-ficient.)

4. The convection–dispersion of gangue in froth

4.1. Assumptions and definitions

Practical flotation froths are incredibly complex systems,with processes such as bubble coalescence, particle attachmentand liquid drainage all inter-related. In order to proceed withthe convective–dispersive model, we must first make nine sim-plifying assumptions, and therefore we do not claim to be ableto capture the entire physical reality of a flotation froth. Thevalidity of each of our assumptions will be discussed in Section5. Our assumptions are as follows:

1. There is no bubble coalescence.2. The volumetric liquid fraction is constant throughout the

froth.3. The froth exhibits only one-dimensional behaviour.4. Washwater is perfectly distributed over the froth.5. The top of the froth is a perfect sink for gas, liquid and

gangue.6. The pulp phase is perfectly mixed with respect to particle

distribution.7. The gangue concentration is low everywhere in the froth to

preclude consideration of hindered settling.8. Foam drainage parameters, m and n (explained below), are

constant throughout the froth.9. The interstitial rheology is not modified by the presence of

gangue.

P. Stevenson et al. / Chemical Engineering Science 62 (2007) 5736–5744 5739

Fig. 2. Schematic of the froth column showing the mass balance on thegangue particles.

Because our aim is to develop an accessible theory, a num-ber of variables, and their inter-relation, must be clearly de-fined. Let c be the volumetric fraction of gangue particles inthe entire froth (i.e., the gas and interstitial liquid). Therefore,the volumetric concentration of gangue in the interstitial liquidalone is c/�.

The bias rate, defined above, is jB measured positive down-wards, so the absolute velocity of the liquid phase (measuredpositive downwards) is jB/�. We assume that the particle set-tling velocity (relative to the absolute velocity of the interstitialliquid) is VS . Thus the absolute downwards mean velocity ofthe particles to the stationary observer is jB/� + VS . However,because only a fraction c of the entire froth consists of particles,the convective flux of gangue particles (superficial velocity),jP , measured positive downwards, is

jP =(

jB

�+ VS

)c. (8)

In addition we let the volumetric solids fraction of the feed be�. Because we assume that the pulp phase is perfectly mixed,the solids fraction of the pulp liquid is also approximately �.

4.2. Mass balance

By reference to Fig. 2, we may write a dynamic mass balanceon gangue particles across an elemental slice of the column ofthickness �x of the froth (i.e., in = out + accumulation) asfollows:

−Ddc

dx+ jP = − D

(dc

dx+ d

dx

dc

dx�x

)

−(

jP + djP

dx�x

)+ dc

dt�x. (9)

Simplifying

dc

dt= D

d2c

dx2+ djP

dx(10)

and, noting that we consider a steady-state system, and by in-voking Eq. (7), we obtain:

0 = Dd2c

dx2+(

jB

�+ VS

)dc

dx. (11)

4.3. General solution

Eq. (11) is a second order ordinary differential equation thatis standard in convection diffusion equations, and can readilybe solved by inspection. However, we proceed herein to solveby Laplace transformation to make the solution explicit and toautogenously engender boundary conditions at the bottom ofthe froth. For convenience we re-write Eq. (11) as

d2c

dx2= �

dc

dx, (12)

where

� = −1

D

(jB

�+ VS

). (13)

Transforming Eq. (12) into the frequency domain we obtain

c̄(s) = scx=0 + dc/dx|x=0 − �cx=0

s(s − �)(14)

and by taking partial fractions:

c̄(s) = �cx=0 − dc/dx|x=0

�s+ dc/dx|x=0

�(s − �). (15)

Expressing this in the spatial domain gives

c(x) = cx=0 + 1

�

dc

dx

∣∣∣∣x=0

(exp[�x] − 1) (16)

which is the general solution of the steady-state convection–dispersion equation.

4.4. Boundary conditions

Given that we invoke perfect mixedness in the pulp, we assertthat the solids fraction of the liquid just entering the froth isthe same as the solids fraction of the liquid in the pulp (�).However, since only a fraction � of the column is actually froth,the volumetric concentration of gangue at the bottom of thecolumn is

cx=0 = �� (17)

which is our first boundary condition. The second boundarycondition is obtained as follows. Consider an infinitely highfroth in which � is negative (i.e., the exponential term in Eq. (16)tends to zero). Because the froth is being effectively washed,the concentration of gangue at the top of the froth approacheszero. Thus, from Eq. (16) we obtain our second boundarycondition i.e.,:

dc

dx

∣∣∣∣x=0

= �cx=0 = ��� for � < 0. (18)

5740 P. Stevenson et al. / Chemical Engineering Science 62 (2007) 5736–5744

Thus, the solution for the concentration profile in the frothphase is

c(x) = �� exp[�x] = �� exp

[−1

D

(jB

�+ VS

)x

]

for

(jB

�+ VS

)> 0. (19)

However, if � > 0 (i.e., (jB/� + VS) < 0), no net washing ofgangue occurs. By reference to Eq. (16), we see that, for aninfinitely deep froth, c tends to either infinitely positive or in-finitely negative, dependent on the sign of the concentrationgradient at the bottom of the froth. These scenarios are clearlyunphysical, and this can only be reconciled if the concentrationgradient at the bottom of the froth is zero. As a consequencewe write

c(x) = �� for

(jB

�+ VS

)< 0. (20)

Thus, the universal solution for all bias rates is given by

c(x) = min

[�� exp

[−1

D

(jB

�+ VS

)x

], ��

]. (21)

4.5. Gangue recovery

The volumetric fraction of gangue in the column is given byEq. (21). However, since only a fraction � is liquid, then theconcentration of gangue within the liquid itself is c/�. Now,since the liquid recovered from the top of the column is jmax

f ,it follows that the volumetric flux of recovered gangue fromthe top of the column, R, is

R = cT jmaxf

�(22)

or

R = min

[jmaxf � exp

[−1

D

(jB

�+ VS

)xT

], jmax

f �

], (23)

where cT is the gangue concentration at the top of the froth,located height of xT above the froth–pulp interface.

Eq. (23), for the recovery of gangue from the top of the froth,is worthy of further consideration. For � > 0 (i.e., no net frothwashing) we find that gangue recovery is linear in liquid recov-ery, which is in consistent with the experimental observationsof Engelbrecht and Woodburn (1975) and Zheng et al. (2006).However, Eq. (23) suggests that gangue recovery, for � > 0 , isindependent of particle size, which is most certainly inconsis-tent with experimental observation. In the following section, itwill be argued that the dependency of gangue recovery uponparticle size must be governed by the physics of the pulp phase,rather than the physics of the froth phase.

For � < 0 Eq. (23) predicts that gangue recovery is linear inliquid recovery (since jmax

f is independent of jB). However,gangue recovery is dependent upon particle size as a conse-quence of the dependency of VS , although this dependency willbe seen to be inconsequential due to the rapid attenuation ingangue concentration with froth height.

5. Validity of the assumptions

Eq. (23) gives predictions of gangue recovery from the frothbased upon the nine assumptions presented in Section 4.1. Wedo not claim that Eq. (23) can give accurate a priori predictionsof gangue recovery in practical flotation processes. However,by careful consideration of the validity of each of the nine as-sumptions, the physics behind gangue entrainment from prac-tical flotation froths can be illuminated.

5.1. Bubble coalescence

It has been assumed in the above analysis that the bubblesize is constant throughout the column, so that bubble coales-cence (and indeed bubble coarsening due to Ostwald ripening)is zero. In fact, bubble coalescence can have a great effect uponthe hydrodynamic condition of the rising foam as is predictedby the hydrodynamic theory of rising foam (Stevenson, 2007b).At steady-state, the liquid flux within the column is spatially in-variant, and, since bubble coalescence increases the bubble size,the capacity for liquid transport through the froth diminishes.Bursting bubbles cause a release of liquid down the foam thatis like an autogenous source of washwater. However, no matterwhat the bubble coalescence rate, washwater that is sufficientto ‘create’ a positive bias cannot be released. Thus, for flota-tion cells with no externally added washwater, such as rougherand scavenger cells (which are preliminary stages in a typicalflotation circuit), the gangue recovery will be given by R =jmaxf �, but jmax

f will be diminished due to bubble coalescence.However, if bubbles burst in a column with external washwateraddition such that � < 0, because the gangue recovery is depen-dent upon bias rate, liquid fraction and dispersion coefficientwhich are all changed by bubble bursting, the gangue recov-ery changes accordingly. Neethling and Cilliers (2003b) havespeculated about possible mechanisms for bubble coalescencein rising foams, but no experimental data has been produced toconfirm, or otherwise, this speculation.

5.2. Constant liquid fraction

In the analysis given above, it has been assumed that theliquid fraction is constant throughout the froth. In fact, if theliquid fraction exhibits a relatively high value at the pulp–frothinterface before relaxing, over a distance of a few centimetres,to an equilibrium value as described by Stevenson (2007b). Indeep froths, such as those typically found in column flotation,this variable liquid fraction near the pulp interface may notsignificantly influence the gangue recovery rate. However, inshallower froths, such as those typically found in mechanicalcells, the variable liquid fraction may be a significant factor.

5.3. One-dimensional froth

It has been assumed that the froth travels up the column ina one dimensional fashion. In fact, gross convective flows inrelatively wet foam have been observed (Hutzler et al., 1998),

P. Stevenson et al. / Chemical Engineering Science 62 (2007) 5736–5744 5741

that have been explained recently by the theory of Embley andGrassia (2006). These convective structures may, or may not,have implications upon gangue recovery.

5.4. Distribution of washwater

The theory of Stevenson (2006b) for the liquid fractionenhancement due to added washwater that has been invokedherein assumes that externally added washwater is distributedevenly across the froth surface. However, Ireland et al. (2007)have measured the liquid fraction enhancement due to washwa-ter addition to froth by vertical and horizontal jets. Their resultssuggest that washwater distribution by discrete jets cannotstrictly be considered as uniform. They found that horizontaljets submerged in the froth were more effective at approach-ing a uniform distribution. Cilliers (2007) has attributed plantobservations of excess gangue recovery to poorly distributedwashwater causing gangue to bypass washed zones and travelto the concentrate.

5.5. Froth surface

The model assumes that the top of the froth, located at a dis-tance of xT above the pulp–froth interface, is a perfect sink i.e.,all material arriving at this position is instantaneously spiritedto the concentrate. This is, of course, not the physical reality ofthe top of a flotation froth. In practical froths the bubbles mustmove horizontally to reach a weir, over which they fall into alaunder, a process which Neethling and Cilliers (2003a,b) haveattempted to model. Such horizontal motion will increase resi-dence time and therefore the propensity for bubble coalescence.

5.6. Mixing of the pulp phase

The model assumes that the pulp phase is perfectly mixed sothe boundary condition of cx=0 =�� is valid. Trahar (1981) hasshown that the recovery of 5 �m particles is similar to that ofwater recovery, suggesting that, in the system he investigated,the pulp was almost perfectly mixed with respect to 5 �m par-ticles. However, the degree of mixedness in a mechanical cellis dependent upon the type and rate of mechanical agitation,as well as particle size. The critical agitation rate for the sus-pension of solid particles in two-phase (liquid–solid) systemsis predicted by the seminal correlation of Zwietering (1958).van der Westhuizen and Deglon (2007) have shown that a sim-ilar approach can be taken for agitation in the three-phase sys-tem that is present in the pulp of a mechanical cell. Of course,larger particles are more difficult to suspend so their concen-tration at the top of the pulp will be relatively smaller. Thishas the implication that, in reality, � at the top of the frothis a decreasing function of particle size, meaning that ganguerecovery will also be a decreasing function of particle size,which is consistent with the observations of Engelbrecht andWoodburn (1975) and Zheng et al. (2006). The important con-sequence is that, for flotation with no externally added washwa-ter, the dependency of gangue recovery on particle size is due to

agitation in the pulp (as investigated by Zheng et al., 2005)rather than any physical processes that occur in the froth. Itshould be noted that Kirjavainen suggests that gangue recov-ery is dependent upon particle shape. Thorpe and Stevenson(2003) drew analogy between the incipient condition forhydraulic conveying of particles down pipes with incipientparticle suspension in two-phase and three-phase agitators.Because they noticed that the threshold of hydraulic convey-ing is strongly dependent upon particle shape, it is likely thatparticle suspension in agitators, and therefore the concentra-tion of a particular size of particle at the pulp–froth interface,shares this dependency. Of course, in column flotation, gangueparticles are not suspended by mechanical agitation and mustreach the interface between collection and cleaning zones byconvection and turbulent diffusion.

5.7. Hindered settling

Particle settling velocity is required for Eqs. (7) and (23) tobe applied. This settling velocity is included in the model ofNeethling and Cilliers (2002a). Although they do not state ex-plicitly how they calculated the settling velocity, they probablyused Stokes law to calculate the terminal particle velocity inan infinite medium and used the Richardson and Zaki (1954)equation to take into account hindered settling, as was doneby Neethling and Cilliers (2002b). There is no evidence forthe efficacy of this approach; indeed, it was shown by Munroe(1888) that forces on particles were modified in the environsof a boundary. In addition, it is clear that as the gangue con-centration changes with height in the froth, then the hinderedsettling function of Richardson and Zaki will change, therebymaking VS a function of height and significantly complicat-ing the problem. However, if the gangue concentration is rel-atively low, particle–particle interaction can be assumed to below throughout the froth so VS can be considered to be unhin-dered. The reader should be cautioned against direct applica-tion of the Richardson and Zaki correlation for gangue particlesthat exhibit a poly-dispersive size distribution; Greenspan andUngarish (1982) have shown that the treatment of the hinderedsettling of poly-dispersive sizes of particles is far from trivial,although the simpler treatment of Masliyah (1979) has foundfavour amongst some researchers.

It should be noted that we assume that the particles settle ata constant velocity VS relative to the absolute liquid velocitybut there is no reason why this should be at all true. For exam-ple entrapment effects within the foam itself my retard particlesettling, whereas the Boycott effect which enhances sedimen-tation in inclined channels and has been exploited in the de-sign of the Reflux Classifier for size and density based particleseparations (Laskovski et al., 2006) may enhance settling rate.

5.8. Foam drainage

Estimation of the convection term in the model requires priorknowledge of the Stevenson’s (2006a) drainage parameter’s,m and n and assumes that the values are constant in the froth.

5742 P. Stevenson et al. / Chemical Engineering Science 62 (2007) 5736–5744

However, these are dependent upon the type and concentrationof the frother. In a column in which bubbles coalesce, theconcentration of frother will increase with height due to frac-tionation of the surfactant as described by Stevenson andJameson (2007). However, if fresh water is applied to the top,this could wash the frother away resulting in a decrease insurfactant concentration. It is therefore by no means obviousthat m and n should in fact be constant within the column.Moreover, the presence of hydrophobic particles at the bubblesurfaces is likely to change the stress state and therefore thevalues of the drainage parameters, although the form of thisdependency is not yet known.

5.9. Interstitial rheology

Because dilute concentrations of gangue have been con-sidered in this theory, modifications to the rheology of theinterstitial fluid due to the presence of particles has not beennecessary. However, if many gangue particles are present, thesewill increase the effective viscosity of the fluid (and thereforethe hydrodynamic condition of the froth). Such modificationcould be effected by employing the equation of Thomas (1964)who asserted that the actual viscosity, �a of a slurry of solidsfraction � could be estimated using the expression:

�a = �(1 + 2.5� + 10.05�2 + 0.00273 exp(16.6 �)) (24)

as suggested by Neethling and Cilliers (2003a). This approachassumes that the rheology of the interstitial slurry is Newto-nian, whereas, depending on concentration and surface proper-ties of the particles, non-Newtonian rheology may be exhibited(Johnson et al., 2000).

6. Calculation of some parameters for column flotation

At this juncture, it is appropriate to estimate � for columnflotation to illuminate what the effect of washwater addition hason gangue recovery, according to the theory presented herein.It is noted that Finch and Dobby suggest that a bias rate greaterthan 4 mm s−1 can be detrimental to the rejection of ganguebecause it engenders mixing, and that Clingan and McGregor(1987) has stated that the bias rate is generally unimportant solong as it is maintained at a positive value. Take the case ofa 20% wet froth (i.e., � = 0.2) and a bias rate of 1 mm s−1.Finch and Dobby suggest that the d80 of particles in columnflotation is typically 30– 74�m which corresponds to a termi-nal velocity of between 80 and 500 �m s−1 assuming a particledensity of 2650 kg m−1 (i.e., that of silica sand) and Stokes law,and the density and viscosity of the liquid as 1000 kg m−1 and1 cP, respectively. However, for the reasons indicated above, wecannot assume that these calculated velocities correspond withVS as described in the model. Thus, in the interests of conser-vative design, we will assume that VS is, in fact, zero. Now,Finch and Dobby suggest that bubble diameters of between 0.5and 1.5 mm are typical in column flotation, so will assume thatrb=0.5 mm in this example (i.e., d=1 mm). Given these param-eters, and assuming that the drainage parameters for the foam

are m = 0.016 and n = 2 (i.e., those measured by Stevensonet al., 2007) we calculate by Eq. (2) that jd = 1.57 mm s−1

and, by Eq. (6), that D = 26 mm2 s−1. Now, let us imagine thatour bias rate is set at 1 mm s−1 (i.e., greater than zero but lessthan the value of 4 mm s−1 cautioned by Finch and Dobby).We therefore calculate, by Eq. (13) that �=193 indicating veryrapid attenuation of gangue concentration with froth height byEq. (19). If a finite VS is assumed then this attenuation will bestill more rapid. This observation is consistent with Clingan andMcGregor’s assertion that bias rate is unimportant as long as itis positive, or, in the nomenclature of the current work, if � < 0In addition, the predicted rapid attenuation of gangue concen-tration is consistent with Clingan and McGregor’s observationthat gangue recovery is largely independent of froth height.

7. Conclusions

We have presented a convective–dispersive model for the re-covery of gangue from a simplified representation of a flotationfroth both with and without externally added washwater. Al-though we do not claim that the model can give accurate a pri-ori estimates of gangue recovery from practical flotation froths,it has illuminated the physical processes that govern ganguerecovery. This will ultimately help the development of effec-tive automatic control strategies for the flotation process, notleast because the assertion of the importance of positive bias incolumn flotation made by Clingan and McGregor (1987) andFinch and Dobby (1990).

Amongst the implications of the theory presented herein are:

1. The gangue recovery rate is linear in liquid recovery, asobserved by Engelbrecht and Woodburn (1975) and Zhenget al. (2006), inter alia.

2. The recovery of gangue particles from columns with noexternally added washwater, or if the washwater added isinsufficient to create positive bias, the transport of ganguethrough froth is independent of particle size. The fact thatsmaller particles have been seen to be more easily recoveredin the concentrate is due to physical processes in the pulprather than in the froth.

3. No gangue rejection benefit is achieved by adding wash-water if � > 0. Indeed such washwater addition will bedetrimental if it stabilises the froth by inhibiting bubblecoalescence.

4. If washwater is added such that � < 0, gangue transportthrough the froth phase is weakly dependent upon particlesize by changing the dispersion coefficient as well as theconvective term in the transport equation.

5. Ireland et al. (2007) attribute the observation that totalgangue rejection is impossible, no matter how hard the frothis washed, to back-mixing when washwater is added. Back-mixing may indeed complicate the model presented herein,but the exponential form of Eq. (23) indicates zero ganguerecovery is only asymptotically approached as froth heightincreases or � becomes more negative.

6. In fact, in column flotation, if washwater is provided suchthat � < 0, the concentration of gangue attenuates so rapidly

P. Stevenson et al. / Chemical Engineering Science 62 (2007) 5736–5744 5743

with respect to froth height that gangue entrainment islargely independent of the value of the bias rate and theheight of the froth, which is consistent with the observa-tions of Clingan and McGregor (1987). In a sense, ganguerecovery is independent of dispersion coefficient since, if� > 0 there is no dispersion effects by Eq. (20), whereas if�>0 (which it is for even moderately positive bias rates)the gangue concentration drops rapidly whatever the valueof dispersion coefficient.

Notation

c volumetric gangue fraction within the froth, dimen-sionless

cT volumetric gangue fraction in the froth at the top ofthe column, dimensionless

D coefficient of axial dispersion, m2 s−1

g acceleration due to gravity, m s−2

jB liquid superficial bias rate (positive downwards),m s−1

jd liquid superficial drainage velocity (positive down-wards), m s−1

jf liquid superficial velocity up the column (positive up-wards), m s−1

jmaxf maximum liquid superficial velocity up the column

(positive upwards), m s−1

jg gas superficial velocity (positive upwards), m s−1

jP convective flux of particles (positive upwards), m s−1

jW superficial washwater velocity (positive downwards),m s−1

m dimensionless number used in Eq. (2), dimensionlessn dimensionless index used in Eq. (2), dimensionlessR volumetric flux of recovered gangue, m s−1

rb bubble radius (≡ 0.5d), mSk Stokes number (≡ jd�/(r2

bg�)), dimensionlesst time, sVS particle settling velocity, m s−1

x vertical distance from pulp interface measured positiveupwards, m

xT height of the froth, m

Greek letters

� volumetric liquid fraction in the foam, dimensionless� a composite parameter defined in Eq. (13), m−1

� liquid dynamic viscosity, Pa s�a actual viscosity of a slurry, Pa s� liquid density, kg m−3

�s particle density, kg m−3

� solids fraction in the slurry, dimensionless

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