B

Ba

b

c

a

ARRAA

KFFHCA

1

cstrmtcm

fnotrfiatp

dm

0d

Chemical Engineering and Processing 49 (2010) 622–627

Contents lists available at ScienceDirect

Chemical Engineering and Processing:Process Intensification

journa l homepage: www.e lsev ier .com/ locate /cep

ubble–particle collision and attachment probability on fine particles flotation

. Shahbazia,∗, B. Rezaib, S.M. Javad Koleini c

Mining Department, Research and Science Branch, Islamic Azad University, Tehran, IranAmirkabir University of Technology, Tehran, IranTarbiat Modares University, Tehran, Iran

r t i c l e i n f o

rticle history:eceived 11 February 2009eceived in revised form 4 April 2010ccepted 23 April 2010

a b s t r a c t

Particle size is an important parameter in flotation and has been the focus of flotation research for decades.The difficulty in floating fine particles is attributed to the low probability of bubble–particle collision. Inthis research, the influence of hydrodynamic parameters on collision probability of fine particles wasinvestigated. Collision probability was obtained using Stokes, intermediate I and intermediate II and

vailable online 20 May 2010

eywords:lotationine particlesydrodynamic

potential equations. Maximum collision probability was 5.65% obtained with impeller speed of 1100 rpm,air flow rate of 30 l/h and particle size of 50 �m. Also, attachment probability under Stokes flow, turbulentand potential flow conditions was calculated 100, 99.49 and 81.87% respectively. Maximum attachmentprobability was obtained with impeller speed of 700 rpm, contact angle of 90◦, particle size of 20 �m andair flow rate of 15 l/h. Collision angles were obtained between 60.71◦ and 60.18◦ and attachment angles

.15◦ a

ollisionttachment

were obtained between 9

. Introduction

For flotation occurring under quiescent conditions, one canalculate the probability of collision using stream functions. Thetream functions used by earlier workers are applicable for bubbleshat are either too large or too small [1], while those developed inecent years are useful for flotation size bubbles [2,3]. However,ost of the flotation machines are operated under intensely agi-

ated conditions, which make it difficult to use the interceptionalollision models based on stream functions. Under such conditions,odels based on microturbulence may be more useful [4].Froth flotation is widely used for separating different minerals

rom each other. However, its influence is limited to a relativelyarrow particle size range of 10–100 �m [5–7]. Although the effectf particle size on flotation performance has been widely studiedo date [5,6,8–10], and many important physico-chemical factorselated to particle size have been identified, the net effect of theseactors are very difficult to predict. For example, in particle–bubblenteraction, particle size is known to play a critical role in the prob-bility of particles colliding with bubbles, attachment of particleso bubbles after collision, as well as remaining attached in the pulp

hase [11,12].

Fine particles typically show slow recovery rates, owing toecreased particle–bubble collisions, and are prone to entrain-ent. Moreover, very small particles tend to have large specific

∗ Corresponding author. Tel.: +98 2182883516; fax: +98 2182884324.E-mail address: bzshahbazi@yahoo.com (B. Shahbazi).

255-2701/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.cep.2010.04.009

nd 59.83◦.© 2010 Elsevier B.V. All rights reserved.

areas, which can lead to excessive adsorption of reagents, andother effects associated with chemically active particles. Thesefactors can have a considerable impact on grades and recoveries,depending on the dominant effects in operation [13,14]. Efficiencyof the bubble–particle stability depends on the particle size, particlehydrophobicity and external detaching forces. Even in the flotationof fine particles, the bubble–particle detachment can significantlyinfluence the kinetics of flotation taking place in mechanical cellsby intensive turbulent agitation [15]. For these fine particles, thebubble–particle detachment is often neglected.

In this research a theoretical analysis of fine particle flota-tion was investigated based on experimental measurement ofbubble size and raise velocity. Using Stokes equation collisionprobability was obtained very low while using potential equa-tion collision probability was exaggerated. According to this study,for fine particles, best equation for calculating collision probabil-ity are intermediate equations cause to collision probability bythese equations can be estimated between Stokes and potentialequations. Also, for fine particles with different air flow rates andimpeller speeds, collision angle was obtained between 63.18◦ and60.71◦.

Furthermore, attachment probability of fine particles was cal-culated under turbulent and Stokes conditions. Stokes equation ismore useful for column flotation and when the attachment proba-

bility was calculated under Stokes flow conditions, the probabilitywas exaggerated. So, attachment probability was calculated underturbulent flow conditions that is more suitable for mechanical flota-tion. Finally, after calculating collision probability by intermediateI equation and attachment probability under turbulent flow condi-

B. Shahbazi et al. / Chemical Engineering and Processing 49 (2010) 622–627 623

Fr

tc

2

mmCds7tec0

diawcitaboc

3

3

a

Table 1Values of A and n for different flow conditions [20].

Flow conditions A n

Stokes [1] 3/2 2

Intermediate I [25][

3 + 4Re0.72]

2

equation was 0.27–4.92%, intermediate II equation was 0.35–5.65%

ig. 1. Bubble diameter measurement for different impeller speeds and air flowates.

ions, the probability of collecting a particle by an air bubble wasalculated.

. Materials and methods

Bubble size distribution and rise velocity were measured in aechanical flotation cell. The frother used in the flotation cell wasethyl iso-butyl carbinol – MIBC with concentration of 22 ppm and

/CCC = 1.96 (CCC, critical coalescence concentration). An impelleriameter of 0.07 m was used for agitation in a cell with a squareection of 0.12 and 0.1 m height. The impeller rotating speeds were00, 900, 1100 and 1300 rpm. The type of impeller was a Rushtonurbine with 8 paddles and a stator was used around the rotor. Allxperiments were carried out without any baffling in the flotationell. With impeller speed of 700 rpm and superficial air rates of.14 cm/s, bubble surface area flux (Sb) was obtained 6.67 s−1.

The bubble size distribution and rise velocity were measured in aevice similar to the McGill bubble viewer [16]. According to Fig. 1,

t consisted of a sampling tube attached to a viewing chamber withwindow inclined 15◦ from vertical. The closed assembly was filledith water of a similar nature to that in the flotation cell (to limit

hanges in bubble environment during sampling) and the tube wasmmersed to the desired location below the froth. Bubbles rose intohe viewing chamber and were imaged by a digital video cameras they slid up the inclined window, which was illuminated fromehind. In this research, at first frother has been added to the waterf the cell then viewing chamber has been filled with water of theell to prevent bubble coalescence.

. Results and discussion

.1. Bubble size distribution and raise velocity

The mean bubble diameter adopted was the Sauter diameter,s calculated by Eq. (1). The bubble Reynolds number is calculated

2 15

Intermediate II [2] 32

[1 + (3/16)Re

1+0.249Re0.56

]2

Potential [26] 3 1

from Eq. (2):

d32 =∑

nid3i∑

nid2i

(1)

Reb = Vbd32�f

�(2)

Parameters influencing the bubble Reynolds number are n,number of bubbles, d, bubble diameter, Vb, bubble raise velocity,�, fluid dynamic viscosity and �f, fluid density. Air flow rate andimpeller speed are important operational variables in flotation andcontrolling of them is significant for improving the flotation per-formance. With increasing impeller speed, rotor will be more ableto disperse air bubbles but with increasing air flow rate, dispersingability of rotor will be decreased. So, Sauter mean bubble diameteris decreased by increasing impeller speed and decreasing air flowrate. The effect of impeller speed on Sauter mean bubble diameterat different air flow rates is shown in Fig. 2. According to Fig. 2, theminimum bubble diameter was 0.55 mm with air flow rate of 15 l/hand an impeller speed of 1300 rpm and the maximum bubble diam-eter was 1.52 mm with air flow rate of 75 l/h and an impeller speedof 1300 rpm. Furthermore, the minimum bubble Reynolds num-ber was 100 with bubble velocity of 16.26 cm/s, an air flow rate of15 l/h and an impeller speed of 1100 rpm and the maximum bubbleReynolds number was 327 with bubble velocity of 19.28 cm/s, anair flow rate of 75 l/h and an impeller speed of 1300 rpm.

3.2. Collision probability

Bubble–particle collision and attachment interaction in flotationhave been defined in Fig. 3. The probability (P) of a particle beingcollected by an air bubble in the pulp phase of a flotation cell canbe given by:

P = PcPa(1 − Pd) (3)

Pc = A

(dp

db

)n

(4)

in which Pc is the probability of bubble particle collision, Pa is theprobability of adhesion, Pd is the probability of detachment, dp isthe diameter of the particle, db is the diameter of the bubble and Aand n are the parameters that vary with Reynolds numbers. Table 1gives these values for the three different flow regimes considered,i.e., Stokes, intermediate and potential flows.

The probability of collision was calculated for different parti-cle sizes (20, 30, 40 and 50 �m), air flow rates (15, 30, 45 and75 l/h) and impeller speeds (700, 900, 1100 and 1300 rpm), usingthe different equations in Table 1. The minimum collision probabil-ity was obtained using Stokes equation and the maximum collisionprobability was obtained using the potential equation. The collisionprobability using Stokes equation was 0.01–0.41%; intermediate I

and potential equation was 4.48–23.44%.According to Fig. 4 and under intermediate I equation, the max-

imum collision probability obtained was 4.92% with an impellerspeed of 1300 rpm, an air flow rate of 15 l/h and a particle size of

624 B. Shahbazi et al. / Chemical Engineering and Processing 49 (2010) 622–627

5waflwos1[

3

d

ti is induction time, V is the bubble raise velocity. There is another

Fig. 2. Bubble size, raise velocity and Reynolds number.

0 �m and the minimum collision probability obtained was 0.27%ith an impeller speed of 700 rpm, an air flow rate of 15 l/h andparticle size of 20 �m. These results indicate that difficulty in

oating fine particles is due to low collision probability, whileith coarse particles it can be attributed to a high probability

f detachment. Under Stokes flow conditions, a maximum colli-ion probability of 48.35% was obtained with an impeller speed of100 rpm, an air flow rate of 15 l/h and a particle size of 545 �m17].

.3. Collision angles

Collision angles are shown in Fig. 5. The collision angle, �c,epends weakly on particle size, but strongly on the particle den-

Fig. 3. Bubble–particle collision and attachment interaction in flotation [19].

sity and the bubble Reynolds number and may be predicted by Eq.(5) [18] and used elsewhere [19]:

�c = arccos(D) (5)

where the dimensionless parameter D can be calculated using thefollowing equations [18]:

X = 32

+ 9Re

32 + 9.888Re0.694(6)

Y = 3Re

8 + 1.736Re0.518(7)

C = V

U

(Db

Dp

)2

(8)

D =√

(X + C)2 + 3Y2 − (X + C)3Y

(9)

In this research, for various air flow rates, impeller speeds andparticle size, the collision angle was calculated using Eq. (5) andshows that for different particle sizes, the collision angles are same.According to Fig. 5 with increasing air flow rate, the collision angledecreases. The maximum collision angle obtained was 63.18◦ withan air flow rate of 15 l/h and an impeller speed of 700 rpm and theminimum collision angle obtained was 60.71◦ with an air flow rateof 75 l/h and an impeller speed of 1300 rpm.

3.4. Attachment probability

The availability of stream function also makes it possible topredict Pa. For predicting attachment probability under turbulentconditions, can use below equation [20]:

Pa = sin2

[2 arctan exp

(−(45 + 8Re0.72)vti

15db(db/dp + 1)

)](10)

where dp is the diameter of particle, db is the diameter of bubble,

generalized equation for calculation Pa according to [19]:

Pa = sech2

(2UAti

dp + db

)(11)

B. Shahbazi et al. / Chemical Engineering and Processing 49 (2010) 622–627 625

e I equ

wiSfl

A

A

wo

F(

Fig. 4. (A–D) Collision probability was calculated by intermediat

here dp is the diameter of particle, db is the diameter of bubble, tis induction time, U is particle settling velocity (was calculated bytokes equation) and A is a dimensionless parameter under Stokesow (A1) or potential flow (A2) according below equations:

1 = V

U+ 1 − 3

4

(1 + dp

db

)−1

− 14

(1 + dp

db

)−3

(12)

( )−3

2 = V

U+ 1 + 1

21 + dp

db(13)

here V is the bubble raise velocity. The induction time is a functionf the particle size and contact angle which can be determined by

ig. 5. Collision angle of fine particles in various impeller speed and air flow ratesparticle density 2.65 g/cm3).

ation at different air flow rate, impeller speed and particle size.

experiment and correlated in the form of [21]:

ti = AdBp (14)

where parameters A and B are independent of particle size. It wasfound that parameter B is constant with a value of 0.6, and parame-ter A is inversely proportional to the particle contact angle �. Basedon these findings, the following equation was used [22]:

ti = 75�

d0.6p (15)

where ti is given in second, � in degrees and dp in meter. The proba-bility of adhesion can now be calculated for given values of bubblesize, particle size and contact angle.

When the attachment probability was calculated under tur-bulent conditions, with increasing air flow rate, impeller speedand particle size, the attachment probability decreased but withincreasing contact angle, the attachment probability increased.Attachment probabilities were obtained between 20.59 and 99.49%with the maximum probability at an impeller speed of 700 rpm,contact angle of 90◦, particle size of 20 �m and air flow rate of 15 l/h.

When the attachment probability was calculated under Stokesflow conditions (Eq. (11)), the probability was exaggerated 100%approximately. When the attachment probability was calculatedunder potential flow conditions, the probability was less than

the attachment probability calculated under turbulent condi-tions. Under potential flow conditions, the attachment probabilityobtained was between 3.09 and 81.87% with the maximum proba-bility at an impeller speed of 700 rpm, a contact angle of 90◦, particlesize of 20 �m and an air flow rate of 15 l/h.

626 B. Shahbazi et al. / Chemical Engineering and Processing 49 (2010) 622–627

flow r

3

r[

P

c6btiaaptw

3

i

P

ws

probability was obtained 2.13%.

Fig. 6. (A–D) Probability of collecting a particle by an air bubble at different air

.5. Attachment angles

The bubble–particle attachment probability, Pa, is defined as theatio of two specific numbers [19]. So, Pa was calculated by Eq. (16)23,24]:

a =(

sin �cr

sin �c

)2

(16)

Attachment angles, �cr, were calculated under potential flowonditions using Eq. (11). First, all of ti have been calculated for0◦, 70◦, 80◦ and 90◦ contact angles and attachment angles haveeen calculated using Eq. (16). With decreasing air flow rate, par-icle size and impeller speed, attachment angle increased and withncreasing contact angle, attachment angle increased. Attachmentngles were obtained between 9.15◦ and 59.83◦ and the maximumngle was at an impeller speed of 700 rpm, contact angle of 90◦,article size of 20 �m and air flow rate of 15 l/h. So, in these condi-ions for a particle with contact angle of 90◦, successful attachmentill occur only by 30.17◦ particle sliding.

.6. The probability of collecting a particle by an air bubble

For fine particles, Pd can be negligibly small because of the low

nertia, in which case Eq. (3) becomes:

= PcPa (17)

here Pc is determined by the hydrodynamic of system, which istrongly affected by the particle size, bubble size and turbulent of

ate, impeller speed and particle size (density 2.65 g/cm3 and contact angle 90◦).

the system. Pa is also affected by the hydrodynamic, but is largelya function of surface involved [20].

After calculating collision probability by intermediate I equationand attachment probability under turbulent flow conditions, theprobability of collecting a particle by an air bubble was calculatedby Eq. (17). According to Fig. 6, for fine particles with contact angleof 90◦, the probability of collecting a particle by an air bubble is verylow due to low probability of collision. Pmax was obtained 2.85%with air flow rate of 15 l/h, particle size of 50 �m and impeller speedof 1100 rpm.

4. Conclusions

• The effect of impeller speed on Sauter mean bubble diameterat different air flow rate was investigated. So, minimum bubblediameter was obtained 0.55 mm with air flow rate of 15 l/h andimpeller speed of 1300 rpm and maximum bubble diameter wasobtained 1.52 mm with air flow rate of 75 l/h and impeller speedof 1300 rpm. By intermediate I equation, when bubble diameterand particle size were 0.55 mm and 50 �m respectively, colli-sion probability was obtained 4.92% and when bubble diameterand particle size were 1.52 mm and 50 �m respectively, collision

• Collision probability was calculated using different equations.So, minimum collision probability was obtained Using Stokesequation and maximum collision probability was obtained usingpotential equation. Medium of collision probability was obtainedusing intermediate I and intermediate II equations.

ering

•

•

•

•

•

•

R

[

[

[

[

[

[

[

[

[

[

[

[

[

[

Processing 27 (1987) 241–263.

B. Shahbazi et al. / Chemical Engine

Collision probability was obtained using Stokes equation0.01–0.41%, intermediate I equation 0.27–4.92%, intermediateII equation 0.35–5.65% and potential equation 4.48–23.44%.Medium collision probability was obtained using intermediateI and intermediate II equations.Under intermediate I equation, maximum collision probabilitywas obtained 4.92% with impeller speed of 1300 rpm, air flowrate of 15 l/h and particle size of 50 �m and minimum collisionprobability was obtained 0.27% with impeller speed of 700 rpm,air flow rate of 15 l/h and particle size of 20 �m. So, this resultsshow that difficulty in floating fine particles is due to collisionprobability of fine particles being very low.Maximum collision angle was obtained 63.18◦ with 15 l/h air flowrate and impeller speed of 700 rpm and minimum collision anglewas obtained 60.71◦ with 75 l/h air flow rate and impeller velocityof 1300 rpm.When attachment probability was calculated under turbulentconditions, attachment probability was obtained between 20.59and 99.49% that the maximum was around impeller speed of700 rpm, contact angle of 90◦, particle size of 20 �m and air flowrate of 15 l/h.When attachment probability was calculated under Stokes flow,the amount of probability was exaggerated (100% approxi-mately). So, the attachment probability was calculated underpotential flow and the amount of probability was less than attach-ment probability calculated under turbulent conditions. Underpotential flow, attachment probability was obtained between3.09 and 81.87% that the maximum was around impeller speed of700 rpm, contact angle of 90◦, particle size of 20 �m and air flowrate of 15 l/h.Attachment angles were obtained between 9.15◦ and 59.83◦ thatthe maximum was around impeller speed of 700 rpm, contactangle of 90◦, particle size of 20 �m and air flow rate of 15 l/h.

eferences

[1] A.M. Gaudin, Flotation, 2nd ed., McGraw-Hill, New York, 1957.[2] M.E. Weber, D. Paddock, Interception al and gravitational collision efficiencies

for single collectors at intermediate Reynolds numbers, Journal of Colloid andInterface Science 94 (1983) 328–335.

[3] R.-H. Yoon, G.H. Luttrell, G.T. Adel, M.J. Mankosa, Recent advances in fine coalflotation, in: S. Chander (Ed.), Advances in Coal and Mineral Processing UsingFlotation, Society of Mining Engineers, Littleton, CO, 1989, pp. 211–218 (Chap-ter 23).

[[

[

and Processing 49 (2010) 622–627 627

[4] H. Schubert, C. Bischofberger, On the optimization of hydrodynamics in flota-tion processes, in: Proceedings of the 13th International Mineral ProcessingCongress, vol. 2, Warszawa, 1979, pp, 1261–1287.

[5] W.J. Trahar, The selective flotation of galena from sphalerite with special refer-ence to the effects of particle size, International Journal of Mineral Processing3 (1976) 151–166.

[6] W.J. Trahar, L.J. Warren, The floatability of very fine particles—a review, Inter-national Journal of Mineral Processing 3 (1976) 103–131.

[7] B.A. Wills, Mineral Processing Technology, 4th ed., Pergamon, New York, 1998.[8] A.M. Gaudin, J.O. Groh, H.B. Henderson, Effect of Particle Size on Flotation, vol.

414, American Institute of Mining and Metallurgical Engineering, Tech. Publ.,1931, pp. 3–23.

[9] R.M. Anthony, D.F. Kelsall, W.J. Trahar, The effect of particle size on the acti-vation and flotation of sphalerite, Proceedings of the Australian Institute ofMining and Metallurgy 254 (1975) 47–58.

10] W.J. Trahar, A rational interpretation of the role of particle size in flotation,International Journal of Mineral Processing 8 (1981) 289–327.

11] H.R. Spedden, W.S. Hannan, Attachment of mineral particles to air bubbles inflotation, Mineral Technology 12 (1984), T.P. 2354.

12] P.F. Whelan, D.S. Brown, Particle–bubble attachment in froth flotation, Trans-actions of the Institute of Mining and Metallurgy 65 (1956) 181–192.

13] T.M. Morris, Measurement of equilibrium forces between an air bubble and anattached solid in water, Transactions of the AIME 187 (1950) 91–95.

14] H.J. Schultze, New theoretical and experimental investigations on stability ofbubble/particle aggregates in flotation: a theory on the upper particle size offloatability, International Journal of Mineral Processing 4 (1977) 241–259.

15] D.A. Deglon, F. Sawyerr, C.T. O’Connor, A model to relate the flotation rate con-stant and the bubble surface area flux in mechanical flotation cells, MineralsEngineering 12 (1999) 599–608.

16] E.H. Girgin, S. Do, C.O. Gomez, J.A. Finch, Bubble size as a function of impellerspeed in a self-aeration laboratory flotation cell, Minerals Engineering 19(2006) 201–203.

17] B. Shahbazi, B. Rezai, S.M. Javad Koleini, The effect of hydrodynamic parameterson probability of bubble–particle collision and attachment, Minerals Engineer-ing 22 (2009) 57–63.

18] V.A. Nguyen, The collision between fine particle and single bubble in flotation,Journal of Colloid and Interface Science 162 (1994) 123–128.

19] A.V. Nguyen, J. Ralston, H.J. Schulze, On modeling of bubble–particle attachmentprobability in flotation, Minerals Engineering 53 (1998) 225–249.

20] R.H. Yoon, The role of hydrodynamic and surface forces in bubble–particleinteraction, International Journal of Mineral Processing 58 (1–4) (2000)129–143.

21] P.T.L. Koh, M.P. Schwarz, CFD modelling of bubble–particle attachments inflotation cells, Minerals Engineering 19 (2006) 619–626.

22] Z. Dai, D. Fornasiero, J. Ralston, Particle–bubble attachment in mineral flotation,Journal of Colloid and Interface Science 217 (1999) 70–76.

23] G.S. Dobby, J.A. Finch, Particle size dependence in flotation derived from afundamental model of the capture process, International Journal of Mineral

24] J.A. Finch, G.S. Dobby, Column Flotation, Pergamon Press, Oxford, 1990, p. 180.25] R.H. Yoon, G.H. Luttrell, The effect of bubble size on fine particle flotation,

Mineral Processing 5 (1989) 101–122.26] K.L. Sutherland, Physical chemistry of flotation. XI. Kinetics of the flotation

process, Journal of Physical Chemistry 52 (1948) 394–425.