Dev. Chem. Eng. Mineral Process., S ( l / 2 ) , pp.21-42, 1997. Development of a Particle-Bubble Collision Model for Flotation Columns N.O. Egiebor', Z.A. Zhou and K. Nyavor Environmental Engineering Program Chemical Engineering Department Tuskegee University, Tuskegee, Alabama 36088, USA Particle-bubble collisions in bubble swarm systems have been modelled based on previous work on single bubble systems and bubbly flow characterization. This approach relates the overall collision probability to bubble and particle sizes, gas holdup, column height, and other operating parameters, thus providing a theoretical basisfor column design and analysis ofparticle collection inflotation columns. The simulated results indicate that lhe existing collision models based on single bubble systems cannot be accurately applied to actual flotation columnsfor analysis of the particle-bubble collision process. This is mainly due to the effect of gas holdup, bubble interaction,and multi-layers of bubbles in a swarm. For column flotation of fine particles under normal operating conditions, a recovery zone of about 10 meters is suficient to ensure that the collision probability approaches unity. For bigger particles, higher gas holdups, and smaller bubble sizes, much shorter recovery zone heights can be used to achieve the same high collisionprobability. *Authorfor correspondence. 21 N.O.Egiebor, ZA. Zhou and K. Nyavor Introduction Particle collection by air bubbles is regarded as the heart [l] of froth flotation operation. It basically consists of two sub-prwsses, viz: particle collision with rising bubbles, followed by adherence or attachment to the bubble surface. In column flotation of fine particles. particle detachment from the bubble is minimized or eliminated due to the quiescent nature of the bubbly flow in the column[l, 21. Consequently, particle-bubblecollision is the prerequisite step for recovering valuable minerals from gangues. If a particle does not collide with bubbles, the valuables cannot be enriched in the froth. Therefore, it is not surprising that much attention has been paid to understanding the mechanism of particle-bubble collision in flotation. Since Sutherland [3] first proposed a collision model in 1948, numerous models have been developed to describe this process[l, 4-10]. However, most of these models are based on single bubble systems. As pointed out by Flint and Howarth [q.the relatively high bubble voidages encountered in (column) flotation raises the question of how closely these predicted collision probabilities represent the actual situation. In bubble swarm systems, like those in the recovery mne of flotation columns, the gas holdup and column height, or the multi-layers of bubbles will affect the overall collision probability. Since column flotation is accomplished by the interaction of descending particles with the rise of a swarm of bubbles, it is the collision model in bubble swarm systems, rather than in single bubble systems, that could provide more reliable information for flotation column design and process optimization. The effect of gas holdup on particle collection has been considered by Dobby 113. and Finch and Dobby [ll], but they assumed that collision probability was not affected by bubble swarms, except by the lowering of the bubble rise velocity. Rulev et al. 1121 also observed the gas holdup effect in their analysis of flotation kinetics. However, this was based on an inadequate definition of gas holdup. Recently Sanchez-Pinoand Moys [13J proposed a simple probability model relating gas holdup effect to the collision process, but the bubble and particle sizes and other operating parameters were not taken into account, and hence could not describe the actual 22 Development of a particle-bubble collision modelfor flotation columns collision process in the column. Until now it appears that little attention has been given to a consideration of the effect of multi-layers of bubbles and/or the column height on collision probability, although the column height is an important parameter in column design and scale-up. The difficulties in such a study may be attributed to the problems related to the determinationof the changes in the streamline functions around a bubble in bubble swarm systems, resulting from the bubble interaction. In addition, modelling particle-bubble collision in column flotation is further complicated by the fact that the bubble size, and therefore, gas holdup and superficial gas velocity, change with the column height [14-161, due to the drop in the hydrostatic head. It is the objective of this paper to model and discuss the effects of gas holdup, column height, particle and bubble sizes on the particle-bubble collision in bubble swarm systems. The analysis is based on an extension of previous studies on single bubble systems and bubbly flow characterization in the recovery zone of a flotation column. Single Bubble System The most important factors governing particle-bubble collision in flotation systems [17]can be summarized as: (i ) interceptional effect; (ii) gravitational and inertial effect; (iii) collision in the turbulent region behind the bubble; (iv) diffusion effect; and (v) "cloud" effect in the case of a bubble swarm system. In column flotation of particles bigger than 3 pm,the effects (iii) and (iv) can be ignored. Thus, for a single bubble system, the collision probability is mainly influenced by the interceptionaland gravitational effects. Whether a particle collides with a bubble depends on the balance of viscous, inertia, and gravitational forces acting on it, and on the form of the liquid streamlines around the bubble. For particles colliding with a single bubble, the collision probability (psc) is defined by: 23 N.O. Egiebor,ZA.Zhou and K. Nyavor Only those particles within the limiting radius, b.will collide with the bubble. Those particles lying outside this area will sweep past the bubble without contact. Previous investigations [5] have shown that the collision probability in single-bubble systemscan be generally expressed as: where the parameters a and n vary with Reynolds number. The values of a and n for the different flow regimes of interest in column flotation are: ~ n How Conditions Stokes [4] 2 1.5 (3116)Reb Intermediate [S] 1.5 [1+ ]+(vp/Ub)(RdRp+1)2Sin2e,* 2 1+0.249RebO.56 Potential p3 3 1 * €Ic is the angle (measured from the front stagnation point of the bubble) where the fluid streamlines come closest to the bubble [1 1,191, and can be written as: Experimental verifications for these models [lo, 181 have shown that Weber and Paddock's equation [8] provides a reasonable prediction of the collision probability in single bubble systems because it considers both the interceptional and gravitational effects, making it applicable for wide ranges of bubble and particle sizes. Gaudin's 24 Development of a particle-bubble collision model forfloration columns equation [4] for Stokes regime tends to underestimate, and Sutherland’sequation [3] for potential flow overestimatesPsc for flotation sized bubbles. Bubble Swarm System The differences between particle-bubble collisions in bubble swarm systems from those in single bubble systems are due to several factors: (1) the individual bubble rise velocities are reduced, due to the gas holdup effect, which tends to increase the collision probability; (2) the shape of the liquid streamlines around each bubble is less divergent near the bubble surface because of the interaction between neighboring bubbles, which will also increase the chances of collision; (3) multi-layers of bubbles in the column provide repeated opportunitiesfor a particle to collide with bubbles at different column heights, thus contributing to the increase in overall collision probability. Because the collision models derived from single bubble systems, and other existing models have not considered these effects, it is obvious that they m o t be satisfactorily used to describe the particle-bubble collision process in actual flotation columns. 1. Assumptions In order to investigate the particle-bubble collision in a flotation column more realistically, the following assumptions were made. (1) Bubble diameters range from 0.05 cm to 0.2 cm,and the bubble can be regarded as approximately spherical and rise in a linear fashion within the column in the presence of frothers [20]. (2) Bubbles are regularly arranged in layers in the column, and the bubbles at a given column height (or a layer) are assumed to have the same average bubble size and rise velocity. (3) Gas holdup <Eg)is defined as the volumetxic ratio of a spherical bubble with the radius (Rb) to a cubic liquid envelope with a lateral length of 2b, thefefore: 25 N.O.Egiebor, ZA. Zhou and K. Nyavor (5) b =0 . W ' g 113 = Constant Therefore, the distance Q between the neighboring two layers of bubbles (2b) given by: If the height of the recovery zone is Hr, the total layers of bubbles (N) in the column can be written as: A bubbly flow regime is maintained in the column, which is characterized (4) by the ascent of discrete bubbles in the column,and approximately a linear relationship between gas flowrate and gas holdup [ll, 211. Under this condition, the average bubble rise velocity is relatively constant along the column height [16]. (5) When a bubble is rising, it expands slightly as a result of the reduction in the hydrostatic head. In the presence of frothers, bubble coalescence is minimized and cau be ignored. Thus the numbex of bubbles can be assumed to be constant at the different layers of bubbles in the column. Therefore, the gas holdup also increases with the column height, and the following relationship holds [161: 3 3 'gi/(Rbi) =Egj/<Rbj) =constant (6) 26 (9) There is no severe back mixing or liquid circulation in the recovery zone. Development of a particle-bubble collision modelfor flotation columns 2. Model Development Consider a particle dropped into a bubbly column with a square cross section. Because the bubbles in the column are arranged in layers, if the particle does not collide with bubbles on the first layer, it may collide with the second or third layer of bubbles, and so on. Therefore, the probability (p(Ci)) of the particle dropping from the (i-l)th layer, and colliding with the ith layer of bubbles, will be the ratio of the total projectd areas of bubbles in the ith layer to the total sectional areas of the liquid cubes, or the column cross sectional area, multiplied by KbPsc. Therefore: where Kb is a coefficient related to the bubble interactions, and should be a function of gas holdup and bubble size. If Kb = 1, then there is no interaction between bubbles, which might be the case for very low gas holdups, and the increase of the collision probability can only be attributed to the repeated opportunities for particle bubble contact due to the multi-layer of bubbles in the column. If Kb > 1, then bubble interaction occurs,and this interaction should be stronger at the higher value of 6. SubstitutingEquation (6) into Equation (10) gives: Thus the collection probability of the particle by the ith layer of bubbles, P(Ki), cau be expressed as: 27 N.O. Egiebor, Z.A. Zhou and K. Nyavor In order to consider the particle-bubble collision, the attachment probability, P(Ai), is assumed to be unity. Therefore, the particle collection probability is equal to the particle collision probability: P s i ) = P(Ci) (13) Hence, the probability of this particle being caphued by the first layer of bubbles is: a 1 = P(K1) =1-2mb(Rp/Rbl? ‘gl 213 (15) and the escaping probability of the particle from the first, or the ith layer bubbles is: 213 P S I e ) = 1 - P S I ) =1 -1.208aKb(RplRblrEgl (16) The collision probability of the particle with the second layer of bubbles can then be written as: pK2 = P(K:-K2) = P(K;)P(Kz I Kle) 213 = [ l - l . x b @ # b l ) n E g l 1pw2) = [1- 1.20&&@p/&l? ad 28 ‘g 121311.208aKb(Rp/Rb2)n‘g;l3 Development of a particle-bubble collision model for flotation columns Hence it can be concluded that the probability of this particle colliding with the Nth layer bubbles will be: ad Therefore, the overall probability for the particle colliding with bubbles in the column is: Now, consider two particular situations: &el: n=2 Equation (21) becomes: i =1 N = 1- n[l-1.20-p i =I 2 113 2 IRbi) J N.O. Egiebor, ZA. Zhou and K. Nyavor Thedore, by comb- Equation (8) and (9).Equation (22)can be written: 1/3 2N pKT= 1 - [1-1.208&b@pEgl IRbl) ] Case2: n = l Equation (21) becomes: Then: 30 Development of a particle-bubble collision model forflotation columns 113 =1- [1-1.208aKbRjfgN213/%Np62HrEgN '%N Therefore, the collision probabilities in flotation columns under different flow regimes can be summarizedas follows: I. For Stokes flow: II. For intmediate Reynolds number ranges: Where (3116)Rebs a1 = 1.5 [l + 1 1+0.249Rebs0.56 III. For potential flow: 213 PCT 2 1 - [l- 3.624K,,RpEg~ /RbNp62HregN 113 lRbN (30) Equations (25). (26) and (30) show that the overall collision probability in a flotation oolumn is related to the particle size, the bubble size of the fmt or last layer, the first or last layer gas holdup, and column height. While EgN (or Egl)can be measured 31 N.O. Egiebor, ZA. Zhou and K. Nyavor directly, &N (or Rbl) can be estimated from the established models [22]. The only unknown parameter is Kb. However, it can be determined experimentally by conducting a systematic study on the effect if column height andor gas holdup on the recovery, if the following conditions apply: (a) low solid concentration; (b) particles made completely hydrophobic to ensure approximately 100% attachment efficiency; high dosages of frothers to prevent bubble coalescence. (c) Whichever of Equations (25), (26) and (30) accurately describe the actual collision process in bubble swarm systems can only be determinedexperimentally, because the flow regimes defined above (to which the corresponding models are applicable) are based on single bubble systems. Discussion Collision probability depends on the physical or hydrodynamic conditions of the column and bubbly flow characterization, (which can be easily adjusted) and are less complicated than the chemical interactions taking place in the column. In order to obtain a higher recovery, it is necessary to f i t ensure that particles do collide with bubbles. If a 100% collision efficiency can be obtained in the column, the problems encountered in practical operations can be easily analyzed and solved. Whether the collision models for single bubble systems are suitable for describing bubble swam systems can be examined by setting Kb = 1 in Equations (25), (26)and (30)and comparison with experimental results. The following examples employ literature data and expressions derived from other models for single bubble systems. Bubble Interaction Effect Example 1. Zn cleaning by a flotation column with a diameter of 120 cm [23]. The feed has )&c particle size of 38 p ; Jg = 1.6 cmls; Eg = 15%. It was found that there was almost no difference in recovery (approx. 62%) for different height of recovery zone (Hr = 50,180 and 490 cm)used. 32 Development of a particle-bubble collision model for flotation columns Assuming that Rp = 10 pm, Pp = 4.0 g/cm3. Kb = 1, the predicted collision probabilities vs. the recovery zone height from different models are shown in Figure 1. It indicates that for a column height of about 50 cm,all the predicted collision probabilities are less than 60%, except under the potential flow condition. Therefore, even when the attachment probability is 100%. the collection probability in the column,or the theoretical recovery, will be less than 60%. This implies that bubble interaction OCCUTS in the column, i.e., > 1. Because the comprehensive model of Weber and Paddock [8] considers both the interceptional and gravitational effects on the collisions, all the following analyses are based on the model derived from Equation (26).i.e. for intermediateReynolds number range. 1.0 - . --.--.-stows now ( ~ g25) . Intermdhte (Eq.22. & Yoon & Luttrell, 1989) Intermediate (Eq.26) -- 1 10 100 1000 Recovery Zone Height & cm) Figure 1. Prediction of collision probabilityfor different models. (Kb = 1, Rb = 0.075 ~ m Rp , = O.oCl1 ~ m Eg , = 15%. pP = 4.0 g / d , Jg = 1.6 d s ) 33 N.O. Egiebor, Z.A. Zhou and K. Nyavor Figure 2 shows the bubble interaction effect on the collision probabilities. It can be seen that for a fixed column height, a larger & value gives higher collision probability. When Kb is greater than 3, the collision probability exceeds 60% if the recovery zone is higher than 50 an.When Q is 7 or larger, the collision probability approaches unity at a column height of 50 cm or above. Under this condition,the phenomenon listed in Example 1 can be reasonably explained. All the changes in the recovery can be attributed to the attachment probability. The bubbles reach their relative maximum capturing capacity under a given chemical condition soon after leaving the sparger. This can be justified by the observation that there was little if any concentrationgradient in the pulp along the recovery zone [U]. 1.o oa 0.6 0.4 0.2 0.0 1 10 100 Recovery Zone Height (Hr;cm) Figure 2. Bubble interactioneffect on collisionprobability. = 0.075 cm,Rp = 0.001 an.~g = 15%, pp = 4.0 g k d , Jg = 1.6 d s ) 34 1000 Development of a particle-bubble collision model forflotation columns For a high concentration of floatable particles in the feed, as is the case with most flotation columns, then bubbles almost stop collecting particles soon after they leave the sparger. In this case, no matter how long the recovery zone may be, both the collision and attachment probabilities are unchanged, thus keeping the recovery almost the same, as reported by Ounpuu and Tremblay [U]. The measured axial concentration profides and reverse flotation conducted by Sandvik et al. [24] in a flotation column, clearly showed that the collection process takes place over the entire height of the column only when the floatable particles in the feed were in low concentrations, and the fast capturing rate was observed around the area near the sparger. Therefore, the possible solutions to the problem in Example 1 are either to increase the collector dosages, or to reduce the feed rate and Zn concentration in the feed, as shown by the results [U]. Alternatively and most importantly, by increasing the total bubble surface area either through generating smaller bubbles, increasing the gas holdup, or by using multiple spargers along the column height [a]. Partick Size EfJect Example 2. Column flotation of coal [w. A column with inside diameter 22 cm and height 252 cm (the height of recovery zone is 122 cm,i.e. from the feed point to the bottom) was employed to treat a coal sample. The average particle size was dp = 2 5 0 ~Jg;= 1.417 a d s ; the highest recovery reported was 90.4%. Assuming that Rb = 0.075 cm and Eg= 15% the collision probability vs. the column height is presented in Figure 3, using our model represented by EQuation (26). Figure 3 shows that for the particle size used in this example, the collision probability reaches unity when column height is about 50 cm,even when not considering the bubble interaction. This may result from the dominant function of gravitational effects for bigger particles in which case the trajectory of particles deviates from the liquid streamlines, and fall virtually in a straight line. These results indicate that it is not necessary to use a long recovery-zone column to obtain a high recovery for coarser particles. It has been the conventional trend that for column flotation of coal,much shorter columns than those in mineral flotation are adopted 35 N.O. Egiebor, Z.A. Zhou and K. Nyavor [a, due to the high hydrophobicity and bigger feed sizes of coal relative to those in mineral flotation. From Figure 3, particle size has a great effect on collision probability. For an average particle size of 10p (radius),only when the recovery zone is near 10 meters can the collision probability approach unity for the case where the bubble interaction effect is not considered. Therefore a 10-meterrecovery mne should be long enough to ensure a 100% collision probability in the column. Figure 3 also indicates that, for fine particle flotation,the particle-bubble collision is an important factor for particle caphuing processes, as reported by other researchers [l], and the column height may play a significant role in improving the recovery if the bubbles are not fully loaded. -0.6 I I/ 1 / / 10 100 1000 Recovery Zone Height (Hr; cm) Figure 3. Particle size effect on collisionprobability. (Kb = 1,Rb = 0.075Cm. &g= 15%. pp = 1.4 g/Cm3. Jg = 1.417 d 36 S ) Development of a particle-bubble collision model forflotation columns Gas Holdup and Bubble Size EfJect Figure 4 shows the gas holdup effect on the collision probability based on our model in Equation (26). For a fixed recovery zone height, increasing gas holdup increases the collision probability significantly. This is expected because for a given bubble size, a higher gas holdup means a larger total bubble surface area in the column,thus increasing the chance for particle-bubblecollision. Comparing Figure 4 with the experimental conditions used by Sanchez-pino and Moys [13], their collision model underestimated the overall collision probability. 1 10 100 1000 Recovery Zone Height (Hr; m) Figure 4. Gas holdup efect on collisionprobability. (Kb = 1, Uba = M S , Rb = O.O75Cm, Rp = O . O o l ~ m ,pp = 4.OglCm3, Jg = 1.sCmlS) 37 N.O.Egiebor, ZA. Zhou and K. Nyavor Similarly, smaller bubbles will give a higher collision probability for a fixed gas holdup, as indicated in Figure 5. In practice, higher gas holdup and smaller bubbles should be adopted to achieve high recovery in flotation columns. This may be one of the reasons why Jameson cells are reported to give better performance than conventional columns, even though much shorter columns are employed. 1-0 oa 0.6 0.4 0.2 0.0 1 10 100 Recovery Zone Height (Hr; cm) Figure 5. Bubble size effixt on collisionprobability. (Kb = 1, Rb = 0.01 cm,Eg = 20%.pp = 4.0 g/cm3, Jg = 1.5 cmls) 38 1000 Development of a particle-bubblecollision model forflotation columns Conclusions Based on the results of the analysis presented here, the following conclusions can be drawn. The existing collision models dealing mainly with single bubble systems cannot be accurately applied to real flotation columns for the analysis of the particle-bubblecollision processes in a swarm. In bubble swarm systems, the reduced individual bubble rise velocities due to the gas holdup effect, the interaction between neighbowing bubbles, and the effect of multi-layers of bubbles contribute to the increase in overall collision probability. From a practical point of view, the particle-bubble collision probability should be maintained close to unity in the column to ensure high recoveries. This can be done by increasing the recovery zone height, gas holdup, and reducing the bubble size. A suitable recovery-zoneheight is dependent on gas holdups, and particleand bubble sizes. For bigger particles, higher gas holdups, smaller bubble sizes,and much shorter recovery zone can be used to achieve a high recovery. Acknowledgements Financial support for this work was provided by the Natural Sciences & Eugineenng Research Council (NSERC), and Energy Mines & Resources (Eh4.R) of Canada.and is grate€dly acknowledged. Nomenclature a Coefficient related to the flow regime b Constant related to the lateral length of a cubic liquid envelope c#, Particlediameter & Recovery zone height Jg Superficialgas velocity Q Coefficientrelated to the bubble interactions 39 N.O. Egiebor, Z.A. Zhou and K. Nyavor L Distance between two layers of bubbles constant Total number of bubbles in each layer Total number of layers of bubbles in a COiumn Probability of particles attaching to the ith layer of bubbles probability of particles colliding with the ith layer of bubbles Probability of particles captured by the ith layer of bubbles Robability of particles escapingfrom the ith layer of bubbles Overall attachment probability from the first to the ith layer of bubbles Overall collision probability from the first to the ith layer of bubbles Overall collision probability in the column Overall collection probability from the first to the ith layer of bubbles Overall collectionprobability in the column Collision probability in single bubble systems Bubble radius Limiting radius at an infinite distance from the bubble (Equation 1) Particleradius Bubble Reynolds number in single bubble systems Bubble Reynolds number in bubble swam systems Terminal bubble rise velocity Average bubble rise velocity Particle settling velocity Angle (measured from the front stagnationpoint of the bubble) where the fluid streamlines come closest to the bubble 4 Liquiddensity Pp Particledensity pl Liquidviscosity &g Gas holdup 40 Development of a particle-bubble collision modelfor flotation columns References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 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Received: 21 June 1995;Accepted afim revision: 30 October 1996. 42

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