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Development of a Particle-Bubble Collision Model for Flotation columns.

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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.
N.O.Egiebor, ZA. Zhou and K. Nyavor
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
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:
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:
How Conditions
Stokes [4]
Intermediate [S] 1.5 [1+
Potential p3
* €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
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.
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].
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.
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,
N.O.Egiebor, ZA. Zhou and K. Nyavor
b =0 . W ' g
= Constant
Therefore, the distance Q between the neighboring two layers of bubbles (2b) given
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
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].
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:
'gi/(Rbi) =Egj/<Rbj) =constant
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:
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)
Hence, the probability of this particle being caphued by the first layer of bubbles is:
a 1 = P(K1) =1-2mb(Rp/Rbl? ‘gl
and the escaping probability of the particle from the first, or the ith layer bubbles is:
P S I e ) = 1 - P S I ) =1 -1.208aKb(RplRblrEgl
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)
= [ l - l . x b @ # b l ) n E g l 1pw2)
= [1- 1.20&&@p/&l?
‘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:
Therefore, the overall probability for the particle colliding with bubbles in the
column is:
Now, consider two particular situations:
Equation (21) becomes:
i =1
= 1-
i =I
IRbi) J
N.O. Egiebor, ZA. Zhou and K. Nyavor
Thedore, by comb-
Equation (8) and (9).Equation (22)can be written:
pKT= 1 - [1-1.208&b@pEgl IRbl) ]
Case2: n = l
Equation (21) becomes:
Development of a particle-bubble collision model forflotation columns
=1- [1-1.208aKbRjfgN213/%Np62HrEgN '%N
Therefore, the collision probabilities in flotation columns under different flow
regimes can be summarizedas follows:
For Stokes flow:
For intmediate Reynolds number ranges:
a1 = 1.5 [l +
For potential flow:
PCT 2 1 - [l- 3.624K,,RpEg~ /RbNp62HregN
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
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:
low solid concentration;
particles made completely hydrophobic to ensure approximately 100%
attachment efficiency;
high dosages of frothers to prevent bubble coalescence.
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.
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.
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.
. --.--.-stows now ( ~ g25)
Intermdhte (Eq.22. &
Yoon & Luttrell, 1989)
Intermediate (Eq.26)
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 )
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].
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 )
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
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
N.O. Egiebor, Z.A. Zhou and K. Nyavor
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.
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
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.
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)
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.
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)
Development of a particle-bubblecollision model forflotation columns
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.
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.
Coefficient related to the flow regime
Constant related to the lateral length of a cubic liquid envelope
Recovery zone height
Superficialgas velocity
Coefficientrelated to the bubble interactions
N.O. Egiebor, Z.A. Zhou and K. Nyavor
Distance between two layers of bubbles
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)
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
Gas holdup
Development of a particle-bubble collision modelfor flotation columns
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Received: 21 June 1995;Accepted afim revision: 30 October 1996.
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