Minerals Engineering 22 (2009) 176–181

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Minerals Engineering

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

Validation of the generalised Sutherland equation for bubble–particleencounter efficiency in flotation: Effect of particle density q

Phong T. Nguyen, Anh V. Nguyen *

Division of Chemical Engineering, School of Engineering, The University of Queensland, Brisbane, Queensland 4072, Australia

a r t i c l e i n f o a b s t r a c t

Article history:Received 1 April 2008Accepted 12 June 2008Available online 24 July 2008

Keywords:Flotation bubbleFlotation kineticsFroth flotation

0892-6875/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.mineng.2008.06.004

q Presented at the Fifth International Conference onMelbourne, Australia.

* Corresponding author. Tel.: +61 7 336 53665; faxE-mail address: Anh.Nguyen@eng.uq.edu.au (A.V.

Bubble–particle encounter interaction is the first step of the particle collection by rising air bubbles inflotation and has been predicted based on the potential flow condition by Sutherland and others, leadingto the approximate generalised Sutherland equation (GSE). In this paper, the bubble–particle encounterinteraction with the potential flow condition has been analysed by solving the full motion equation forthe particle employing a numerical computational approach. Together with other inertial forces, the grav-itational forces were fully included in the motion equation. The numerical results were compared withthe GSE models for the encounter efficiency. The effect of particle density on the encounter interactionwas very significant and could counterbalance the ‘‘negative” effect of the inertial forces on the particleinteraction with air bubbles with a mobile surface under the potential flow condition. For the typical par-ticle size range used in flotation, significant deviation of the Sutherland approximate models from thenumerical results due to the particle density effect and gravitational forces was observed.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Bubble–particle interaction is central to froth flotation which iswidely used in the mineral industry. It is controlled by a number ofimportant forces such as the hydrodynamic resistance, the inertialforces, the gravitational forces, the short-range (surface) forces,and the capillary forces. Since the governing forces are not signifi-cantly interdependent, the particle–bubble interaction has conve-niently been quantified by considering collision, attachment anddetachment independently (Dai et al., 2000; Finch and Dobby,1990; Gontijo et al., 2007; Heindel and Bloom, 1999; Hewittet al., 1994; Jameson et al., 1977; King, 2001; Nguyen and Schulze,2004; Phan et al., 2003; Pyke et al., 2003; Ralston et al., 1999; Schu-lze, 1989; Yoon, 2000; Yoon and Luttrell, 1989). Collision is the ap-proach of a particle to encounter a bubble and is governed by thebubble and particle mechanics, and the physico-chemical aspectsof water flow around the air bubble surface in presence of flotationcollectors and frothers which dynamically interact with the risingbubbles. The limit of the collision is determined by the zonalboundary between the long-range hydrodynamic and interfacialforce interactions (Deryaguin and Dukhin, 1960–1961; Ralstonet al., 1999). The inter-surface separation distance at the zonalboundary is of the sub-micrometer order. Once the particle ap-

ll rights reserved.

CFD in the Process Industries,

: +61 7 366 54199.Nguyen).

proaches the bubble at a shorter separation distance, intermolecu-lar and colloidal forces are critical and the attachment begins. Thedetachment is governed by the capillary force, the particle weightand the turbulent eddies.

Air bubbles with a rigid (immobile) surface have traditionallybeen considered in predicting bubble–particle collision (Finchand Dobby, 1990; Flint and Howarth, 1971; Gaudin, 1957; Nguyen,1994; Schulze, 1989; Yoon and Luttrell, 1989). The argument for arigid bubble surface has been based on the strong surface retarda-tion caused by adsorption of surfactants, particles and other impu-rities from the solution The bubble–particle collision with a mobilebubble surface has attracted intensive research in recent years (Daiet al., 1998; Dai et al., 2000; Nguyen, 1998; Sarrot et al., 2007). Re-search into the collision of bubbles with a mobile surface is drivenby the observation that the adsorption of the surface contamina-tion at the surface of rising air bubbles is a dynamic process andas a result the surface contaminants are swept to the rear surfaceof the bubble by adjacent water during bubble rise in a flotationcell. Thus, the forward part of bubble surfaces can remain mobileeven when the bubbles reach terminal velocity, i.e. even whenequilibrium of the mass transfer of surfactants is established.While the forward surface of bubbles in solutions is mobile, thesurface contamination often forms an immobile (stagnant) cap atthe rear (Clift et al., 1978; Sam et al., 1996; Zhang and Finch,2001). It now can be established that not only the surface of freshbubbles is mobile, but the forward surface of aged bubbles onwhich bubble–particle interaction in actual flotation takes placecan also be mobile. At the mobile surface of air bubbles, the

Nomenclature

E bubble–particle encounter efficiencyf drag correction factor (f = 1 in this paper)g acceleration due to gravityK0 dimensionless number, K0 = (1 + 0.5d/q)StK00 dimensionless number, K00 = 1.5(d/q)StK000 dimensionless number, K 000 ¼ 2ðq� dÞUR2

p=ð9lRbÞR interception number, R = Rp/Rb

Rb bubble radiusRc radius of grazing trajectoryRp particle radiusr radial coordinate (Fig. 2)St particle Stokes number, St ¼ 2R2

pUq=ð9lRbÞt timeU bubble slip velocityvr scaled particle radial velocity (divided by U)

vg scaled particle settling velocity (divided by U)vu scaled particle tangential velocity (divided by U)wr scaled water radial velocity (divided by U)wu scaled water tangential velocity (divided by U)

Greek symbolsb dimensionless number, b = 2Rf/(3K000)d water densityl liquid viscosityq particle densitys dimensionless time, s = tU/Rb

u polar coordinate (Fig. 2)ut angle of tangency

40

50

60

70

80

90

0 10 20 30 40 50Particle radius (micron)

Ang

le o

f tan

genc

y (d

eg)

Numericalsolution

R b = 0.385 mm, U = 196 mm/sρ = 2500 kg/m3, δ = 1000 kg/m3

Eq. (2) with f = 2Eq. (2) with f = 1

Fig. 1. Comparison between the numerical solution to Eqs. (5) and (2) for the angleof tangency, ut, versus the particle radius, Rp (Nguyen et al., 2006).

P.T. Nguyen, A.V. Nguyen / Minerals Engineering 22 (2009) 176–181 177

tangential component of the liquid velocity is non-zero, but thecondition of zero tangential stress can be applied. However, bothof the velocity components and stress at the immobile surfaceare zero.

The first prediction for the bubble–particle interaction with amobile bubble surface was published by Sutherland (Sutherland,1948), who used the potential flow for water around air bubblesin a flotation cell and employed a simplified particle motion equa-tion without inertia. The Sutherland prediction for the bubble–par-ticle encounter efficiency, E, is described as E = 3R, where R is theinterception number, defined as the ratio of the particle to bubbleradius, R = Rp/Rb. The Sutherland model has been improved byincorporating inertial forces which are amplified at the mobile sur-face with a non-zero tangential velocity component of the liquidphase. For example, the Sutherland equation was extended to ac-count for the influence of the inertial forces by Dukhin (1983).Nowadays, this extension, referred to as the generalised Sutherlandequation (GSE), is described by Dai et al. (1998)

E ¼ 3R sin2 ut expf� cos ut½3K 000ðln Rþ 1:8Þ þ ð8� 12 cos ut

þ 4 cos3 utÞ=ð3 sin4 utÞ�g ð1Þ

where the angle of tangency, ut, is described as

ut ¼ arccosffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ b2

q� b

� �ð2Þ

Parameters b and K000 in these equations are defined by

K 000 ¼2ðq� dÞUR2

p

9lRbð3Þ

b ¼ 2Rf3K 000

ð4Þ

where U is the bubble (slip) velocity relative to the liquid phase, qand d are the particle and liquid densities, l is the liquid viscosity.The drag correction, f, in Eq. (4) was not considered by Dukhin(1983) (f = 1), but it was later changed to f = 2 (Dai et al., 1998).

Eq. (1) was obtained using the following equation for the parti-cle inertial motion around an air bubble (Dai et al., 1998; Dukhin,1983):

K 0d~vds� K 00

d~wds¼ ~w�~v ð5Þ

where v and w describe the dimensionless particle and water veloc-ities around the bubble surface, and the arrows over the symbolsare used to describe the vectors. The dimensionless velocities areobtained by dividing the appropriate dimensional velocities by

the bubble slip velocity, U. The dimensionless time in Eq. (5) is de-fined as s = tU/Rb, where t is the dimensional time. The dimension-less numbers describing the inertial effects are defined as K0 = (1 +0.5d/q)St and K00 = 1.5(d/q)St, where the particle Stokes number isdefined as St ¼ 2R2

pUq=ð9lRbÞ.The GSE model described by Eq. (1) was recently validated for

fine particles employing the numerical solution to Eq. (5) (Nguyenet al., 2006). It was also shown that the correction factor f in Eq. (4)should be equal to 1. Otherwise, the deviation of Eq. (2) from thenumerical results for the angle of tangency is significant (seeFig. 1). The angle of tangency is the polar angle u (defined relativeto the rising bubble as shown in Fig. 2), where the particle trajec-tory just meets the bubble surface at a single point and then de-parts from the surface. The angle of tangency is critical to thedetermination of the encounter efficiency as described by Eq. (1).

It is interesting to note that the numerical results in Fig. 1 indi-cate that with increasing the particle radius, the angle of tangencyfirst decreases from 90�, reaching a minimum value and then in-creases towards 90�. At the minimum angle, the inertial force ofthe water flow at a mobile bubble surface, as described by the sec-ond term on the left hand side of Eq. (5), and the particle inertia, asdescribed by the first term on the left hand side of Eq. (5), are bal-anced. For coarse particles, the particle inertia overcomes thewater inertial force, leading to an increase in the encounter effi-ciency as expected. Importantly, for coarse particles Eq. (5) has tobe extended to include the gravitational forces which are propor-tional to the particle volume and density. The aim of this paperis to examine the effect of the particle density on the GSEprediction.

2Rp

2Rb

Rc

r

ϕ

Particle

Bubble

eϕ

re

Fig. 2. Schematic of the bubble–particle encounter interaction. The collision radius,Rc, is measured by the distance between the parallel lines of the path of the risingbubble and the grazing trajectory of settling particles when being far apart from thebubble surface.~er and ~eu describe the unit vectors of the rotationally symmetricalcoordinate system (r, u).

178 P.T. Nguyen, A.V. Nguyen / Minerals Engineering 22 (2009) 176–181

2. Equations for particle motion around an air bubble

The bubble–particle encounter interaction in flotation is sche-matically shown in Fig. 2. Eq. (5) for the particle motion is ex-tended to include the particle weight and buoyancy, giving

K 0d~vds� K 00

d~wds¼ ~w�~vþ~vg ð6Þ

where vg accounts for the effect of the gravitation forces on the par-ticle motion around the bubble surface and is described as

vg ¼2R2

pðq� dÞg9lU

which is identical to the Stokes velocity for the

particle terminal settling, divided by the bubble slip velocity. Thedirection of gravity is the direction of vector ~vg .

Under the condition of the steady flow of water, the partial timederivative of the water velocity is zero and Eq. (6) gives (Nguyenet al., 2006)

K 0dvr

ds¼ K 0

v2u

rþ K 00 wr

owr

orþwu

rowr

ou�

w2u

r

!þwr � vr � vg

� cos u ð7Þ

K 0dvu

ds¼ �K 0

vuvr

rþ K 00 wr

owu

orþwu

rowu

ouþwuwr

r

� �þwu

� vu þ vg sin u ð8Þ

where wr and wu are the radial and tangential components of thewater velocity, and vr and vu are the radial and tangential compo-nents of the particle velocity. The particle velocity componentscan be related to the particle radial, r, and polar, u, positions as

vr ¼drds

ð9Þ

vu ¼ rduds

ð10Þ

The liquid velocity components contained in the above motionequation can be determined from the potential flow used in deriv-ing the generalised Sutherland equation (Dai et al., 1998; Dukhin,1983), giving

wr ¼ � 1� 1r3

� �cos u ð11Þ

wu ¼ 1þ 12r3

� �sin u ð12Þ

The partial derivatives of the velocity components are describedas

@wr

or¼ �3 cos u

r4 ð13Þ

owr

ou¼ 1� 1

r3

� �sinu ð14Þ

owu

or¼ � 3

2r4 sinu ð15Þ

owu

ou¼ 1þ 3

2r3

� �cos u ð16Þ

The minus sign before the brackets on the right hand side of Eq. (11)indicates that the radial liquid velocity is in the negative directiontowards the bubble surface as shown in Fig. 2.

3. Computational methodology

The equations for particle motion as described by Eqs. (7)–(10)can be numerically solved to obtain the particle position relative tothe bubble surface, using the fourth-order Runge–Kutta method.The particle position is obtained as the parametric solution interms of the dimensionless time, s. The numerical parametric solu-tion is very stable but small changes in the time can generate largechanges in the particle polar coordinate, u, in particular, in thebubble surface region close to the equator. Therefore, the paramet-ric solution is not convenient for the determination of the tangencyposition between the particle grazing trajectory and the bubblesurface which is normally close to the bubble equator. In orderto overcome this numerical difficulty, the governing equationscan be integrated with respect to the particle polar coordinate, u,as used in our previous paper (Nguyen et al., 2006). Eliminatingtime from Eqs. (7)–(10) gives

dvr

du¼

v2u

rþ K 00

K 0wr

owr

orþwu

rowr

ou�

w2u

r

!þwr � vr � vg cos u

K 0

( )

� rvu

ð17Þ

dvu

du¼ �vuvr

rþK 00

K 0wr

owu

orþwu

rowu

ouþwuwr

r

� �þwu�vuþvg sinu

K 0

� �

� rvu

ð18Þ

drdu¼ rvr

vuð19Þ

Eqs. (17)–(19) were discretised and integrated following thestandard procedure of the Runge–Kutta method for a system of dif-ferential equations of the first-order (Rice and Do, 1995). The par-ticle coordinates r and u were updated simultaneously with thevelocity components, vr and vu. The integration started at an initialposition (r0, u0) for the particle. This initial position was deter-mined based on the condition that the particle velocity far awayfrom the bubble surface was constant and was equal to the watervelocity plus the particle terminal settling velocity, giving

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100Particle radius (micron)

Ang

le o

f tan

genc

y (d

eg)

R b = 0.385 mm, U = 196 mm/sρ = 7000 kg/m3, δ = 1000 kg/m3

Fig. 4. Comparison between the computational results (points) and Eq. (2) (thickline) for the angle of tangency, ut, versus radius of a heavy (galena) particle withdensity q = 7000 kg/m3.

40

50

60

70

80

90

0 20 40 60 80 100Particle radius (micron)

Ang

le o

f tan

genc

y (d

eg)

ρ = 1300 kg/m3

ρ = 7000 kg/m3

Fig. 5. Numerical results for the angle of tangency versus the particle size anddensity. The bubble size and rise velocity are the same as those described in thecaption to Fig. 3. The particle density increment is 1300, 2000, 3000, 4000, 5000,6000 and 7000 kg/m3.

P.T. Nguyen, A.V. Nguyen / Minerals Engineering 22 (2009) 176–181 179

vrðr ¼ r0Þ ¼ �ð1þ vgÞ cos u0 ð20Þvuðr ¼ r0Þ ¼ ð1þ vgÞ sin u0 ð21Þ

The particle trajectories far from the bubble surface were notinfluenced by the bubble and were parallel (with the bubble axisof symmetry). The grazing trajectory for particle motion was com-putationally determined using the condition that there existedonly one contact point between the bubble surface and the particleon the grazing trajectory. This condition was computationally sat-isfied by trial-and-error, varying the initial particle position (r0, u0).A particle position very close to the symmetric axis of the bubblerise was initially chosen to start the numerical integration forobtaining a particle trajectory. If the trajectory reached a pointaway from the bubble surface by r = 1 + R, the integration wasstopped and the initially chosen particle position was shifted awayfrom the axis of symmetry by a very small increment, which wastypically about 1 � 10�3. The new particle position was then usedto calculate the corresponding trajectory for the particle. The con-tact between the bubble and particle surfaces was checked and thecomputation procedure was repeated until no contact was compu-tationally found within relative errors of 1 � 10�6 in the dimen-sionless collision radius, Rc/Rb (see Fig. 1). The last trajectory wasconsidered as the grazing trajectory. The polar position of the bub-ble–particle contact obtained with the last trajectory was deter-mined to give the angle of tangency, ut. The distance betweenthe parallel part of the last trajectory and the axis of symmetrywas determined to give the collision radius, Rc, as shown inFig. 2. The encounter efficiency was calculated using the followingequation:

E ¼ Rc

Rb þ Rp

� �2

ð22Þ

The relative error in the calculated encounter efficiency was1 � 10�12. The error in the calculated angle of tangency was 0.001�.

4. Results and discussion

Comparison between the computational solutions and analyti-cal solutions for the angle of tangency, ut, described by Eq. (2) isshown in Figs. 3 and 4 for a light particle such as coal and a heavyparticle such as galena, respectively. Good agreement between thecomputational results and the analytical solutions is observed forparticle radius up to about 25 lm for the light particle and up toabout 8 lm for the heavy particle. In all cases, the angle of tan-gency first decreases from 90� to a minimum and then increasesback to 90�. The minimum angle of tangency decreases withincreasing particle density (Fig. 5). This special dependence of the

0

10

20

30

40

50

60

70

80

90

0 20 40 60 80 100Particle radius (micron)

Ang

le o

f tan

genc

y (d

eg)

R b = 0.385 mm, U = 196 mm/sρ = 1300 kg/m3, δ = 1000 kg/m3

Fig. 3. Comparison between the computational results (points) and Eq. (2) (thickline) for the angle of tangency, ut, versus radius of a light (coal) particle with densityq = 1300 kg/m3.

angle of tangency on particle radius and density cannot be ob-tained by the analytical solution of the GSE theory. This significantdifference in the angle of tangency is important for flotation theorybecause in the modelling and predicting of bubble–particle attach-ment interaction in flotation we need the angle of tangency (Daiet al., 1999; Dobby and Finch, 1987; Duan et al., 2003; Nguyenet al., 1998; Pyke et al., 2003; Ralston et al., 2002).

The numerical results shown in Fig. 5 can be approximated bythe following equation:

ut ¼ acosabn

1þ bbm

� �ð23Þ

where the model parameters, a, b, n and m, can be found by the bestfit of Eq. (23) with the numerical results. The best fit with thenumerical data shown in Fig. 5 gives

a ¼ 2:665qd� 5:937

qd

� �2þ 4:288

qd

� �3

� 1:002qd

� �4þ 0:107

qd

� �5� 4:389� 10�3 q

d

� �6

b ¼ 7:668qd� 17:305

qd

� �2þ 12:276

qd

� �3

� 2:875qd

� �4þ 3:077� 10�1 q

d

� �5� 1:266� 10�2 q

d

� �6

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100Particle radius (micron)

Enco

unte

r effi

cien

cy, E

R b = 0.385 mm, U = 196 mm/sρ = 7000 kg/m3, = 1000 kg/m3

0

0.05

0.1

0.15

0 10 155

δ

Fig. 7. Encounter efficiency as a function of particle size calculated with the GSEdescribed by Eq. (1) (solid line) and with the numerical computation withoutgravitational forces (Nguyen et al., 2006) (unfilled circles) and with gravitationalforces included in Eq. (6) (filled circles).

180 P.T. Nguyen, A.V. Nguyen / Minerals Engineering 22 (2009) 176–181

n ¼ 2:258 tanh 0:722qd

� �1:108� �

m ¼ 2:946� 0:3958qdþ 0:4673

qd

� �2� 0:2047

qd

� �3þ 4:270

� 10�2 qd

� �4� 4:305� 10�3 q

d

� �5þ 1:683� 10�4 q

d

� �6

Eq. (23) with the best fitted expressions for the model parameters iswithin 4% relative error compared with the numerical solutionsshown in Fig. 5.

The numerical results for the bubble–particle encounter effi-ciency are shown in Fig. 6. The effect of the particle density onthe encounter efficiency is also very significant. For fine particles(with diameter smaller than 20 lm), the encounter efficiency de-creases with increasing the particle density. This decrease is dueto the ‘‘negative” effect of the inertia of the water flow at the slipsurface of air bubbles. For coarse particles, the encounter efficiencyincreases with increasing the particle density as expected. Forcoarse particles, the negative effect of the inertia of the water flowat the slip surface is overcome by the particle inertia. This featureof our computational modelling is beyond the capability of thegeneralised Sutherland theory. Finally, the efficiency for both fineand coarse particles increases with increasing the particle size aswe expect.

Shown in Fig. 7 is a plot of encounter efficiency versus particleradius calculated with the GSE described by Eq. (1) and with thenumerical computation without gravitational forces (Nguyenet al., 2006) and with gravitational forces included in Eq. (6). Theinfluence of gravitational forces is significant for coarse particles.Agreement among the three models can be seen only with ultrafineparticles. Validation of the three models against experiment is dif-ficult at present because direct measurements of encounter effi-ciency with bubbles with a mobile surface are not available. The

0

0.2

0.4

0.6

0.8

1

0 20 40 60 80 100Particleradius (micron)

Enco

unte

r effi

cien

cy, E

= 1300 kg/m3

= 7000 kg/m3

0

0.1

0.2

0 10 20 30

ρ

ρ

Fig. 6. Numerical results for the encounter efficiency versus particle size anddensity. The bubble size and rise velocity are the same as those described in thecaption to Fig. 3. The particle density increment is the same as that described in thecaption to Fig. 5.

data are only available for particle capture efficiency. Such datacould be used to calculate the encounter efficiency if the attach-ment and detachment efficiencies would be known but such acomparison between the models and the model results is not trulyan experimental validation of the theories. Our experience is thatthe model results calculated for encounter efficiency from the cap-ture, attachment and detachment efficiencies can exceeds unity inmany cases, in particular for fine and ultrafine model particles,which are physically inconsistent (see, for example, Figs. 6 and 7in (Hewitt et al., 1994)). This complication and the lack of directmeasurements of encounter efficiency urge further research in thisarea of flotation theory. Perhaps, prediction by improved numericalcomputation capable of solving many complicated mathematicaltasks is a useful approach.

5. Conclusion

The particle encounter interaction with an air bubble with amobile (slip) surface was computationally analysed. The computa-tional results were used to validate Sutherland’s theory on thebubble–particle collision interaction and its generalisation withthe inclusion of the centrifugal (inertial) effect. It was shown thatthe generalised Sutherland theory only agreed with the computa-tional simulation for ultrafine particles (<10 lm in diameter). Fornon-ultrafine particles, a significant deviation of the GSE modelfrom the numerical data has been observed for the angle of tan-gency and the encounter efficiency. The computational modellingwas extended to include gravitational forces. The computationalsimulation shows the very significant effect of the particle densityon the angle of tangency as well as on the encounter efficiency. Fi-nally, the computational results show that there exists a particlesize where the angle of tangency has a minimum. For bigger par-ticles, the other inertial effects on particle collision are more sig-nificant than the centrifugal effect and, consequently, the angleof tangency increases with increasing the particle size towards90�. The deviation of the angle of tangency shown in this paperstrongly depends on the particle density and will be significantfor modelling and predicting the particle attachment by air bub-bles in flotation.

P.T. Nguyen, A.V. Nguyen / Minerals Engineering 22 (2009) 176–181 181

Acknowledgements

PTN is grateful to the School of Engineering, The University ofQueensland for a fee-waiver postgraduate scholarship. Discussionswith Professor Jan D. Miller (University of Utah, Salt Lake City, USA)and Professor John Ralston (University of South Australia, Adelaide,Australia) are warmly acknowledged.

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