FLUID FLOW AND KINETIC MODELLING IN

Prévia do material em texto

REVIEW PAPER
FLUID FLOW AND KINETIC MODELLING IN
FLOTATION RELATED PROCESSES
Columns and Mechanically Agitated
Cells---A Review
J. B. Yianatos�
Department of Chemical, Biotechnological and Environmental Processes, Santa Maria University,
Valparaı́so, Chile.
Abstract: In this paper, fluid flow and kinetic models related to minerals flotation process are
presented and the advantages and limitations of using this type of models are discussed. The
modelling of such processes was firstly developed assuming perfect mixing for the whole
system as a black box. Then, a more realistic approach was developed recognizing the inter-
action between two zones, the particle–bubble collection zone and the froth transport zone.
From a hydrodynamic point of view, experimental data showed that single large mechanical
flotation cells can deviate significantly from perfect mixing, while the mixing conditions in a flo-
tation bank of mechanical cells (three to nine cells in series) can be well described as a
series of continuous perfectly mixed reactors. From plant experience, it was observed that per-
formance of large industrial pneumatic flotation columns, originally regarded as a counter-current
operation, also operate closer to a single perfectly mixed reactor.
Advances in the field of modelling and design of flotation cells and columns, have been achieved
because the fluid flow regime, the mass transport conditions at the pulp/froth interface and the
froth transport mechanisms are better known and understood. Key parameters such as the
bubble surface area flux, related to the bubble generation and the rate of particle collection,
bubble loading related to the mass transport across the pulp-froth interface and froth recovery,
which is mainly related to the gas residence time in the froth, are relevant for a deeper understand-
ing of this type of equipment.
Keywords: modelling; froth flotation; flotation machines; residence time distribution.
INTRODUCTION
Flotation is a widely used process within min-
erals processing industry in the last century, as
well as being used for water and waste water
treatment, and more recently for de-inking
of recycled paper and electrolyte cleaning (oil
separation) among other less conventional
applications.
In the case of water and waste water treat-
ment the aim of the process is to remove very
fine particles of few microns, in very low con-
centrations, which are collected by small
bubbles of 50–100 mm. In mineral processing,
however, particles from few microns to several
hundred of microns, in 10–40% solid suspen-
sions, are selectively collected by bubbles of
0.5–2 mm. This makes a difference in terms
of the type of flotation used, i.e., dissolved air
flotation DAF for the water treatment and dis-
persed air flotation for minerals separation.
Studies on DAF modelling are rather scarce
and mainly related to hydrodynamics (i.e.,
Kwon et al., 2006). Recently, Emmanouil
et al. (2006) presented the basis for a three
phase modelling of an industrial scale DAF
tank, using flotation kinetic concepts.
In the last decade, flotation equipment
related to mineral processing industry has
shown a dramatic increase in size, reaching
values of 250 m3 in unitary mechanical flo-
tation cells (Weber et al., 2005) and more
than 250 m3 in pneumatic columns (De
Aquino et al., 1998). The general feeling,
however, is that despite the great advances
observed in terms of process knowledge,
the mechanisms and principles as well as
the design and scale-up of industrial flotation
1591 Vol 85 (A12) 1591–1603
�Correspondence to:
Professor J.B. Yianatos,
Department of Chemical,
Biotechnological and
Environmental Processes,
Santa Maria University,
P.O. Box 110-V, Valaparaiso,
Chile.
E-mail: juan.yianatos@usm.cl
DOI: 10.1205/cherd07068
0263–8762/07/
$30.00þ 0.00
Chemical Engineering
Research and Design
Trans IChemE,
Part A, December 2007
# 2007 Institution
of Chemical Engineers
cells, are still not fully understood. Thus, the scope of this
paper is mainly addressed to mineral processing flotation
applications, particularly columns and mechanical cells.
From a unit operation point of view, flotation is a solid–solid
separation, where fine solid particles, suspended in water,
contact an air bubble swarm, in a well-mixed air–pulp dis-
persion. Particles are classified as floatable if they
adhere to the bubbles and are transported up to a froth
layer. Flotation separation is based on different mineral
surface properties. Hydrophobic particles can attach to air
bubbles, while hydrophilic particles do not. In this form
hydrophobic particles can be selectively separated by levita-
tion against gravity in the aqueous medium. Thus, it can be
seen that flotation is a multi-component, multiphase and
heterogeneous separation process.
From a conceptual point of view flotation can be observed
as a sequence of two operations, ‘reaction’ and ‘separation’
(Finch, 1995), as it is shown in Figure 1.
The reactor is fed with the slurry containing the solids to be
separated. Chemical reagents (collector, frother, depressant,
and so on.) are added to induce differences in particle sur-
face properties in order to promote the selective aggregation
of particles with air bubbles. Energy is required to keep the
solids in suspension, typical particle size around 50–150
microns, and to disperse the air into fine bubbles, typically
of 1–2 mm. The specific power used at present in large
size industrial flotation cells is about 1 kW m23.
The probability of collection and transfer of a mineral par-
ticle from the pulp to the froth can be described as a product
of probabilities of occurrence of several sub-processes (Ek,
1992). However, for practical purposes it is very difficult to
quantify most of these probabilities. Alternatively, the collec-
tion process has been represented similar to a chemical reac-
tion. In this approach the ‘reactants’ are hydrophobic mineral
particles that collide with and adhere to air bubbles. The reac-
tion ‘product’ is a particle-bubble aggregate that is less dense
than the medium and moves upwards against gravity while
hydrophilic particles are reported down to the tails.
A necessary condition for mineral separation in a flotation
process is the existence of a froth zone with a distinctive
pulp-froth interface. Conditions for the co-existence of the
froth and fluid (pulp) phases in a flotation column has been
theoretically derived from hydrodynamics for two phase sys-
tems (air–water), i.e., Pal and Masliyah (1990), Xu et al.
(1991a), Langberg and Jameson (1992). The critical bound-
ary conditions for industrial flotation equipment in terms of
bubble size and superficial gas rate, regarding the loss of
the pulp–froth interface, froth stability and limiting carrying
capacity has been reported by Yianatos and Henrı́quez
(2007). Thus, it was found that for typical superficial gas
rates, JG ¼ 1–2 cm s
21, the optimal range of bubble diameter
at the pulp/froth interface level was dB ¼ 1.0–1.5 mm, in order
to maximize the bubble surface area flux, SB ¼ 50–100 s
21.
Otherwise several constraints, i.e., loss of pulp/froth interface
(flooding) or greater disturbance at the interface level (boiling),
limit the flotation process (Finch et al., 2007).
Mechanically Agitated Cells
Figure 2(a) shows the schematic view of the forced-air cell,
with the rotor located near the bottom, and Figure 2(b) shows
the self-aerated cell, with the rotor located near the top. In
both designs, pulp circulation through the rotor is required
to enhance the particle collection performance. Recently, a
mid-rotor cell has been developed where the rotor is located
at the center of the cell (Lelinski et al., 2005). Also, froth crow-
ders and internal radial launders have been incorporated in
order to enhance the froth transport.
Pneumatic Cells
Pneumatic flotation columns are devices of simple construc-
tion in which a gas is distributed at the bottom and rises up in
the form of a dispersed phase of bubbles in a continuousfluid
phase which also contains suspended fine particles.
Counter-current column
Figure 3(a) shows a scheme of the classical flotation
column design, considering counter-current contact between
Figure 1. Conceptual flotation design. Figure 3. Counter-current and co-current pneumatic flotation.
Figure 2. Mechanical flotation cells.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
1592 YIANATOS
the pulp feed and a bubble swarm, generated by the gas
sparger near the bottom of the column (Finch and Dobby,
1990). The feed pulp descends by gravity allowing that
hydrophobic particles collide and adhere to the bubbles.
The froth zone consists of an approximately 1 m froth bed.
At the top of the froth wash water is distributed over the
column cross-section. The water plays an important role in
eliminating fine particles entrainment (Yianatos et al., 1987).
Industrial columns are circular in cross-section with diameter
up to 4.5 (m), square up to 4 (m) side and rectangular, i.e.,
2 � 8 or 4 � 6 (m) in side. The total column height varies
typically between 10–15 (m) height (De Aquino et al., 1998).
Co-current contact cells
An interesting point of discussion has been related to the
inefficient collision mechanism for attaching mineral particles
to a large number of small bubbles in the pulp zone of flo-
tation columns. In this sense, alternative designs of pneu-
matic cells considering intensive contact of pulp and
bubbles in a downflow co-current column have been devel-
oped. Figure 3(b) shows a typical co-current flotation
device (Jameson, 1988). In this device, pulp and air bubbles
flow co-currently downwards against the buoyancy of
the bubbles which creates high gas holdups. Here, the
active collection zone is located in the downcomer tube,
which provides an efficient particle–bubble interaction.
FLOTATION MODELLING
Fundamental Collection Models
Historically, a large number of fundamental models have
been developed to describe the flotation process, mainly
related to the collection zone in terms of hydrodynamics.
A comprehensive description on fundamentals of the flotation
process modeling of mechanical cells has been presented by
King (2001), and for column flotation by Finch and Dobby
(1990), Rubinstein (1995) and Finch et al. (1995). A general
flotation model for bubble–particle capture occurring within a
turbulent environment was reported by Pyke et al. (2003).
Recently, Kostoglou et al. (2006) presented a critical review
on the state of the art of flotation fundamental modelling,
and also introduces a new generalized particle–bubble
aggregation model, developed for a combined gravitational
and turbulent flow field. Also, particle collision and
detachment frequencies in flotation have been related to
the turbulent energy density (Bloom and Heindel, 2002). In
addition, the number of bubble–particle collisions has been
predicted by CFD simulation of flotation cells (Koh et al.,
2000; Koh and Schwarz, 2003). Another complex problem
is to link the model parameters with the surface-chemical
properties such as the degree of hydrophobicity. In this
sense, the surface forces measurement has been used to
model pulp flotation rates, including particle–bubble attach-
ment and detachment, in terms of surface chemistry par-
ameters (contact angle, zeta potential, Hamaker constant
and surface tension), by relating the energy barriers to the
kinetic energies of the bubble–particle interaction (Yoon,
2000). Recent evidence indicates that the surface properties
of the solid, the type and concentration of dissolved gas, and
surface microbubbles all influence the stability of the thin,
aqueous film that forms between a particle and a gas
bubble during the final stages of the capture process (Ralston
et al., 2001). A different approach to model the flotation pro-
cess considers the specific floatability of particles as a func-
tion of particle size, degree of liberation and chemical
adsorption of reagents (Niemi, 1995). However, quantitative
models of flotation cell performance do not at present make
any significant use of quantitative chemical parameters
such as the pH of the slurry and the concentration of chemi-
cal collectors or frothers to define overall process behaviour
(King, 2001).
Flotation Rate Models
From a practical point of view, the overall flotation process
was generally represented by a first order system with
lumped parameters, assuming the collection zone was per-
fectly mixed. Equation (1) shows the first order model for a
perfect mixed batch flotation process (Ek, 1992),
R
R1
¼ 1� e�kt (1)
where R represents the mineral recovery at time t, R1 rep-
resents the maximum flotation recovery at infinite time, and
k is the kinetic rate constant which involves all the micro-
scopic sub-processes. A similar first order model can also
be used to describe the operation, closer to plug flow, in lab-
oratory and pilot flotation columns (Finch and Dobby, 1990).
On the other hand, the mineral recovery of a continuous
flotation process can be described by the general equation
(Polat and Chander, 2000),
R
R1
¼
ð1
0
ð1
0
(1� e�kt)F(k)E(t)dkdt (2)
where the term (12 e 2kt) represents the mineral recovery of
a first order process with invariant kinetic constant k, as a
time function. F(k) is the kinetic constant distribution function
for mineral species with different flotation rates, and E(t) is the
residence time distribution function for continuous processes
with different mixing characteristics. According to equation
(2), the mineral recovery depends on the mixing regime in
the collection zone and the actual mean residence time that
is related to the effective pulp volume in flotation equipment.
Equation (1) can be derived from equation (2) considering the
batch condition E(t) ¼ d(t), equivalent to plug flow, and a
single rate constant k for the whole operation, assuming
F(k) ¼ d(k).
Industrial Flotation: Model Structure
Industrial flotation processes are continuous and multi-
stage. Also, the presence of different mineral species and
the critical effect of different particle sizes, among other con-
ditions, make it necessary to develop flexible and different
approaches to describe the flotation performance. For
example, assuming that the gas and pulp phases are com-
pletely mixed, or partially mixed, different model approaches
can be developed. In all cases, the residence time of the
phases as well as the mechanisms for particle–bubble
aggregate formation and separation must be known. Math-
ematical models for flotation cells and columns are based
on the law of conservation and can have various structures.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
COLUMNS AND MECHANICALLY AGITATED CELLS 1593
The single phase model
The flotation process was originally characterized as a
well-mixed cell, where the froth zone was neglected. Accord-
ing to Harris (1978), problems which may be encountered
are: (1) inadequate solids suspension and dispersion; (2)
particle, density and size distribution gradients in pulp;
(3) inadequate dispersion of air throughout the pulp.
Multiphase models
A two-phase model was introduced in order to account for
the presence of two distinct zones, pulp and froth, in flotation
machines. Assumptions made in the original model are:
(1) the pulp and froth phases are each ideally mixed and
(2) material transport occurs between the phases in one or
both directions, and first order flotation rates in both direc-
tions was most favoured. Harris (1978) pointed out that divid-
ing the cell contents into froth and pulp is not an assumption
of the model but the recognition of a fact of flotation cell
operation.
Flint (1974) refers to equation (1) as the apparent cell per-
formance where the net result of all processes occurring
within a cell to be first-order with respect to concentration.
In this case an apparent first-order rate constant can be
obtained, but the problem is to relatethis apparent recovery
rate to process conditions which affect the various transport
rates within a cell. With this in aim, Flint (1974) developed
a multiphase model (Figure 4) to include material transfer
steps in the sub-processes occurring in the cell. Thus, the
internal transport rates such as bubble transport, entrain-
ment, drainage, particle attachment and detachment, were
identified. In formulating the model, first order rate equations
were used to describe particle collection by bubbles. The
contribution of this work lies in the general structure of the
model and the application of it to experimental systems, as
a means of gaining insight into flotation processes, rather
than in the particular forms of the model equations
themselves.
Harris et al. (1975) derived a recycle flow model for the col-
lection zone of mechanical cells assuming that the pulp circu-
lates between two perfectly mixed regions: an intensely
stirred impeller region and the remainder of the cell volume.
The model also considered the effective liquid residence
time in the cell. Multiphase models including more than two
phases have been described, i.e., two froth phases plus
one pulp phase, or two pulp phases plus one froth phase
(Harris, 1978).
Laplante et al. (1983a, b) developed a two phase model for
transient studies on the mass transfer from the pulp to the
froth and from the froth over the cell lip. This allowed the
identification of the flotation rate from the slurry to the froth
using a specially designed batch flotation cell.
A different approach was presented by Deglon et al. (1999)
and Deglon (2003) who introduced a bubble population
balance methodology applied to a mechanical flotation cell.
This approach together with an attachment–detachment
model for the collection process allowed for description of
the non-linear dependence among the flotation rate constant
(s21), the specific power input (W kg21) and the bubble sur-
face area flux SB (s
21) for mechanical flotation cells of
different volume. Here, the new relevant parameter was the
gas residence time in the collection zone.
Vera et al. (1999), Alexander et al. (2003) and Seaman
et al. (2004) also considered a two phase model (pulp and
froth) to characterize the flotation process, but they still kept
the idea of using an apparent flotation rate constant k to
account for the overall cell performance. The apparent rate
constant k was related to the collection zone rate constant
kC and the froth recovery Rf, by equation (3).
k ¼ kC � Rf (3)
Equation (3) is strictly valid only if the maximum collection
zone recovery R1 is equal to 100%, which is an ideal
condition.
Recently, Savassi (2005) described a compartment model
to account for the mass transfer inside a conventional flo-
tation cell. In this model the total volume of the cell was
divided into three compartments: pulp collection zone, pulp
quiescent zone and froth region. The model also takes into
account the simultaneous mechanisms of true flotation and
entrainment, and considers a first order process for particle
collection. The principal mass transfer factors are identified
as: the flotation rate constant, the mean residence time in
the collection zone, the froth recovery of attached particles,
the degree of entrainment through the froth and the water
recovery from the feed to the concentrate. In this approach
the overall recovery was obtained as the interaction of
three independent zones (collection, quiescent and froth).
Despite the effort to separate both phases, a single overall
flotation rate constant has been typically used to represent
the combined effect of particle collection and froth transport.
This approach has been used for design, simulation
and optimization of flotation cells and circuits. However, it is
widely recognized that the knowledge and characterization
of the sub-processes involved in flotation are more powerful
regarding the effect that multiple variables have on flotation
performance. Unfortunately, most of the more complex
models are not practical, because they involve a large
number of parameters, which are generally difficult to
measure.
Alternatively, a good compromise to characterize industrial
flotation equipment has been described by Finch and Dobby
(1990), where the collection zone and froth recoveries are
identified and considered independently in order to estimate
the overall flotation performance, according to Figure 5.Figure 4. Concept of the multipath model (Flint, 1974).
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
1594 YIANATOS
From a mass balance,
R ¼
RCRf
1� RC(1� Rf )
(4)
where R is the overall flotation recovery, RC is the collection
zone recovery and Rf is the froth zone recovery. Bubble load-
ing, a key parameter in developing the model further, needs
to be evaluated to calculate froth recovery.
Bubble loading BL corresponds to the apparent density
(ton m23) of the particle–bubble aggregates entering the
froth. The bubble load can be estimated from the bubble sur-
face coverage, assuming that the bubble surface area occu-
pied by a particle of diameter dp was equal to dp
2 (Szatkowski
and Freiberger, 1985; Li et al., 2004),
BL ¼
K1prpdp
dB
(5)
where K1 is the fraction of bubble surface coverage with a
monolayer of particles, rp is the mineral density, dp is the
mean particle diameter and dB is the bubble diameter. Other
estimates upon the bubble surface coverage with particles
have been derived assuming different particle projection
areas or using shape factors (Patwardhan and Honaker,
2000; Gallegos et al., 2006).
The bubble loading, BL, and the superficial gas velocity, JG,
allow for the calculation of the superficial mass transport Mi
(ton h21m22) of floatable mineral (true flotation) across the
pulp–froth interfacse, according to the following relationship
Mi ¼ BL � JG (6)
Then, the froth recovery of the floatable mineral RF can be
estimated by
Rf ¼
MC
Mi
(7)
where MC is the superficial mass overflow (ton h
21m22) of
floatable mineral recovered into the concentrate by true flo-
tation. Direct measurement of bubble loading in a range of
30–60 g L21 has been reported (Alexander et al., 2003;
Seaman et al., 2004; Hidalgo, 2006).
In order to characterize the collection zone recovery, RC,
the general equation (2) can be used. To solve equation (2),
two functions must be known: the flotation rate constant
distribution F(k) and the residence time distribution E(t).
The interest of using a rate constant distributed function F(k)
is to account for minerals of different characteristics and sizes
which float at different rates. The F(k) function depends on par-
ticle size and chemical conditioning. Different approaches
havebeendescribed to characterize this function, i.e., using tri-
angular, sinusoidal, rectangular, normal or gamma functions
(Polat and Chander, 2000), or using the sum of two normal dis-
tributions (Ferreira and Loveday, 2000), but the most success-
ful has been the rectangular distribution model considered by
Klimpel (1980). The rectangular distribution has the advantage
of keeping the parsimony principle in line with using the
minimum number of parameters.
RTD of Industrial Mechanical Cells
The residence time distribution function E(t) depends on
the hydrodynamic regime and is related to cell design and cir-
cuit arrangement. A common assumption for process model-
ling has been the perfect mixing condition in the collection
zone of flotation cells (King, 2001). Equation (8) represents
the RTD function E(t) of a perfectly mixed cell,
E(t) ¼
e�(t=tF)
tF
(8)
where tF is the fluid mean residence time in the flotation cell
(¼V/Q), where V is the cell volume and Q is the volumetric
flowrate through it. However, since early times, observations
and discussions about the non-perfect mixing behaviour in
flotation cells has been reported, i.e., due to back-mixing
flow, dead zones and short-circuiting (Harris, 1978; Mehrotra
and Saxena, 1983; Mavros, 1992a). Thus, a morerealistic
approach to describe the mixing condition in the pulp of
industrial cells is the use of the actual RTD function (Niemi,
1995; Polat and Chander, 2000).
RTD of single industrial flotation cells
Mixing data on single large industrial flotation cells is
scarce. Some preliminary testing has been reported by
Burgess (1997), who showed that the RTD of a 100 m3
OK100 tank cell, provided with forced air, was close to well-
mixed. Also, Lelinski et al. (2002) reported the comparison
between the RTD of three single flotation cells, 148 to
160 m3, tested in parallel. Here it was observed that cells
operating with forced air plus the rotor located near the
bottom, Figure 2(a), showed a RTD close to well-mixed,
while the self-aerated cells, Figure 2(b), showed a more
significant deviation from perfect mixing. Neither model was
reported to correlate this data.
New experimental data on RTD of single large cells has
been recently reported by Yianatos et al. (2007). Here the
model given by equation (9), consisting of one large perfect
mixer and one small perfect mixer in series, gave the best
fitting to describe the mixing conditions in a 130 m3 Wemco
SmartCell.
E(t) ¼
e�(t�L)=tS � e�(t�L)=tL
(tS � tL)
(9)
Figure 5. Two-phase model for fluid transport in a flotation device.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
COLUMNS AND MECHANICALLY AGITATED CELLS 1595
where tL is the residence time of the large mixer, tS is the
residence time of the small mixer and L is the lag time.
Figure 6 shows the good agreement of equation (9) for mod-
elling the RTD of liquid, fine solid (245 microns), medium
solid (2150þ 45 microns) and coarse solid (þ150 microns),
in a large mechanical flotation cell. The experimental
work was developed using liquid (Br-82 in solution) and
solid (mineral) radioactive tracers (Tello, 2006).
RTD of industrial flotation banks
Because of the large short circuit in single continuous cells,
the industrial flotation operation considers the arrangement of
cells in banks. Thus, banks of 5–10 cells in series are com-
monly used.
In order to characterize the hydrodynamic performance of
an industrial flotation bank, the following equation, describing
the continuous operation of ‘N ’ perfectly mixed tanks in
series, has shown to be a realistic and efficient model.
E(t) ¼
tN�1 � e�(t�N=t)
(t=N)N � G(N)
(10)
where t is the mean residence time of the bank, G(N) is the
Gamma function, which allows to account for the non integer
solutions of N.
Other more complex models can include a large number of
parameters, such as dead zones, pure delay or back-mixing
(Wilson and Frew, 1986; Ando et al., 1990). Recent trends in
flotation circuit designs are towards the use of flotation banks
consisting of a low number (4–6) of large cells, where neither
open flow nor back-mixing flow is present between adjacent
cells (Bourke, 2002).
Industrial banks of large cells have been characterized as
perfect mixers in series (Yianatos et al., 2000, 2001; Arbiter,
2000). Using the radioactive tracer technique, it was shown
that the RTD of flotation banks consisting of 4, 8 and 9
cells (42.5 m3) can be well represented by a tank in series
model (Yianatos et al., 2005a).
Figure 7 shows the good agreement between experimental
data, from banks of 3, 5 and 7 cells of 130 m3 in series, and
the classical N tank-in-series model, equation (10), while for
the first (single) cell the best fit corresponds to the model
described by equation (9), (Tello, 2006).
Flotation Columns
The fluid mechanical behaviour in a flotation column is
complex, since two fluid phases, characterized by very differ-
ent masses and with one far more compressible than the
other, are in contact with each other. The multiphase flow in
Figure 6. RTD in a 130 m3 industrial flotation cell.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
1596 YIANATOS
the column is ‘self-adjusting’, i.e., there are no driven parts,
for the attainment of uniform phase distribution. Therefore,
in formulating a model for a flotation column, a number of
simplifying assumptions need to be posed, as e.g., homo-
geneous bubble-flow regime, uniform gas holdup, mineral
properties constant during the process, there is no sedimen-
tation, so the liquid phase containing the suspended solid can
be assumed to be a pseudo continuum. Also, the residence
time for each phase must be known. Thus, the flotation pro-
cess can be described as a one or multi-component model,
and the component balance is typically formulated for the
fluid (pulp) phase.
A further problem, independent of the exact structure of the
model equations consists of the evaluation of the parameters
required for these equations. Particularly important are the
volume fraction (holdup) of the phases, the collection rate
constant and the mass transport across the pulp/froth inter-
face. The fluid mechanical quantities depend on a complex
manner of the operating conditions, the geometrical dimen-
sions, the way the phases are distributed, and the mineral
properties. So experimentation is necessary for model
parameters setting.
Column hydrodynamic
Flotation columns typically operate at low gas velocities
(1–2 cm s21), low viscosity media, and small bubbles (0.5–
2 mm in diameter) generated with an approximately normal
size distribution, which moves upwards in a rather homo-
geneous bubbly flow condition. In this flow regime, a size dis-
tribution of bubbles occurs. Hence the bubbles rise with
different velocities and their residence times in the system
are different. If no interaction in the form of bubble coalesc-
ence and breakup occurs, bubble flow is segregated. Fluid
is entrained upwards, in the wake of bubbles, and because
of continuity an equal amount of fluid must therefore flow
downwards. Thus a fluid circulation develops.
The bubbly flow regime in flotation columns has been
characterized using the concept of drift flux in order to
relate phase flow rates, holdup and physical properties.
The drift flux, JGF, is defined as
JGF ¼ JG(1� 1G)+ JF1G ¼ JG � 1G(JG + JF) (11)
where 1G is the fractional gas holdup and JG and JF are super-
ficial velocities (flowrate per unit area) and are positive
upwards. The+ sign indicates counter-current and co-current
gas–fluid flow, respectively.
Plotting JGF versus 1G reveals at which gas velocity the
bubble flow switches over into the churn turbulent region,
since at that point the drift flux increases steeply. Bholes
and Joshi (2003) derived a criterion for the prediction of criti-
cal gas holdup at which the transition in the flow regime occur
in the collection zone of a flotation column. From drift flux
analysis an estimate of terminal bubble rise velocity, UT, is
obtained which can be used to calculate bubble diameter in
flotation columns (Dobby et al., 1988).
For a two-phase (gas-fluid) system the velocity of gas
relative to the fluid, called slip velocity, USl, is given by
USl ¼
JG
1G
+
JF
(1� 1G)
(12)
Shah et al. (1982) suggested that for a gas holdup of less
than 30%, the most suitable expression for relating slip vel-
ocity to terminal rise velocity is given by
USl ¼ UT(1� 1G)
m�1 (13)
where m is the Richardson–Zaki index and typical values of
m ¼ 2.39 (Pal and Masliyah, 1990) and m ¼ 3 (Banisi and
Finch, 1993) have been reported for the range of flotation
column operation. More recently, Vanderberghe et al.
(2005) reported a new correlation for the m parameter,
m ¼
20:26þ 1:89� ReB
4:38þ ReB
(14)
where ReB is the bubble Reynolds number for the pulp zone
defined as
ReB ¼
dBUTrF
mF
(15)
Several correlations between the bubble diameter dB and
terminal velocity UT have been reported in literature related
to flotation processes, i.e., Dobby et al. (1988) and Yianatos
et al. (1988). With the help of such hydrodynamic models,
relationships have been derived for slip velocity, gas holdup
or bubble size. Recently a comprehensive review and discus-
sion on column flotationfundamentals has been reported by
Finch et al. (2007).
Extensive discussions on the transport phenomena at the
pulp-froth interface in a flotation column has been presented
by Ross (1991) and Van Deventer et al. (2004a, b), who
pointed out that a great deal of ignorance still exists with
regard to particle detachment at the pulp–froth interface
and the factors that play a role in such action.
Modelling of industrial flotation columns
Mineral recovery in a flotation column depends upon flo-
tation rates and the flow regime in the collection zone and
the froth zone behaviour. The fundamental description of
the internal flow regimes in a flotation column is complex
and difficult to link with fundamental relationships for minerals
recovery. A review of column flotation models was reported
Figure 7. Model fitting for liquid RTD of industrial flotation cells in
series (N ¼ 1, 3, 5, 7).
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
COLUMNS AND MECHANICALLY AGITATED CELLS 1597
by Tuteja et al. (1994) where the associated shortcomings
were discussed.
The first approach to model the column flotation process
was developed by Sastry and Fuerstenau (1970), who
considered the axial dispersion model ADM in a counter-
current process with a first order flotation rate. Later
this approach become the most popular for column flotation
modelling and scale-up (Dobby and Finch, 1986; Finch and
Dobby, 1990; Luttrell et al., 1993). All these models are
one-dimensional.
In general, two kinds of problems have been addressed
using the axial dispersion model. The first one is to predict
the mineral recovery, which is a primary objective of model-
ling column flotation, and consists of using the ADM model
combined with the first order flotation rate equation. Finch
and Dobby (1990), adopted this approach to describe the col-
lection zone recovery RC related to the collection rate con-
stant kC, particle residence time tP and vessel dispersion
number Nd, according to the following relationship
RC ¼ 1�
4a � e(1=2Nd)
(1þ a)2e(a=2Nd) � (1� a)2e�(a=2Nd)
(16)
where
a ¼ (1þ 4 kCtPNd)
1=2 (17)
Equation (16) defaults to equation (1) for plug flow transport
(Nd ¼ 0) and it corresponds to the perfect mixing operation
for Nd ¼ 1.
The second type of problem is the transient solution often
used to fit the residence time distribution (RTD) data in order
to obtain Nd, which describes the intensity of dispersion in
a column and the mean residence time, t. In this case a
suitable technique for flotation columns is the pulse injection
of a liquid or solid tracer at the top of the collection zone,
followed by sampling and tracer analysis of the discharge.
RTD of Industrial Flotation Columns
In modelling the RTD of flotation columns using the axial
dispersion approach, the choice of boundary conditions and
fitting routines is important and has been thoroughly dis-
cussed (Ityokumbul et al., 1988; Xu et al., 1991b; Tokoro
and Okano, 2001). The usual assumption in plant work is
to consider both boundaries as either open or closed. Also,
because the use of wash water prevents most tracers from
exiting within the froth product (Dobby and Finch, 1985;
Yianatos et al., 1987; Yianatos and Bergh, 1992) and the
detection point in the underflow line is a good approximation
of a closed boundary compared with inside the column, most
of the experimental conditions correspond to the ‘closed–
closed’ boundary conditions. Even so, sometimes the
‘open–open’ solution is used because a relatively simple
analytical solution exists and for small dispersion numbers
(Nd , 0.25) the analytical solution is adequate.
For closed–closed boundaries in a flotation column, the
analytical solution was described by (Xu et al., 1991b;
Mavros, 1992b). According to Xu et al. (1991b), the most suit-
able combination is the numerical solution to the closed
vessel case with a least squares fitting routine. Also,
Mavros (1992b) has shown that the numerical solution, expli-
citly developed for a Dirac delta impulse, is applicable for
vessel dispersion numbers larger than 0.14 which is normally
the case for industrial flotation columns.
Experimental observations in pilot columns (50 mm in
diameter) with H/D ratio about 100–200 have shown Nd
values of 0.03–0.08, which are closer to a plug flow operation
(Laplante et al., 1988; Mills et al., 1992). For industrial
columns with H/D ratio about 4–12, the Nd values are
0.4–2.7 (Laplante et al., 1988; Mills et al., 1992; Yianatos
et al., 2005a).
Deckwer and Schumpe (1987) presented an extensive dis-
cussion on phenomenological and empirical studies on
bubble columns, and they also claimed that some postulates
of the ADM model are questionable, i.e., such as the use of
only one lumped parameter to account for the macroscopic
circulatory flows and the axial and radial mass flows, and
the capability to describe the gas phase residence time distri-
bution due to the occurrence of different bubble classes.
However, despite the ADM model not being a good physical
description of the industrial column flotation process, it has
been generally considered to provide a good data fitting.
Ityokumbul (1992) presented an alternative approach to
describe the flotation column operation using the analogy of
interface mass transfer. The main assumptions are the rate
of particle attachment to air bubbles is proportional to the
concentration of floatable solids and uncovered bubble sur-
face, while the rate of particle detachment is proportional to
the surface concentration of solid particles on the air bubbles
(bubble load), the same as suggested by Sastry and Fuerste-
nau (1970), and the solid behaviour in the collection zone
may be described using the sedimentation convection
model. Despite the use of some simplifying conditions (i.e.,
plug flow and constant parameters), this approach empha-
sizes the fact that in some cases column height can be
much shorter than usually built.
More sophisticated hydrodynamic models to describe the
collection zone in flotation columns, i.e., two-dimensional
models (axial and radial dispersion) such as proposed by
Deng et al. (1996), cell models with back-flow such as the cir-
culation cells model of Joshi and Sharma (1979), or the eddy-
cell model of Zehner (1986), as suggested by Deckwer and
Schumpe (1987) and Mavros (1992a), have not yet been
considered in modelling of flotation columns, i.e., in the calcu-
lation of recovery and performance by means of solving the
material balance equations.
Alternatively, other researchers have proposed that it
may be more appropriate to use the tanks-in-series model
with back-mixing flow (Mavros et al., 1989) or the tank in
series model (Mills and O’Connor, 1990; Goodall and
O’Connor, 1991) for describing the fluid flow regime (RTD)
of laboratory flotation columns. Also, because the operation
of industrial columns is much closer to perfect mixing, the
RTD of large industrial flotation columns can be described
by the tank-in-series model, equation (10), considering a
non-integer number, between one and two, of perfect
mixers in series.
More recently, Yianatos et al. (2005a) presented a better fit
using a model consisting of one large perfect mixer (resi-
dence time tL) and two small perfect mixers in series (resi-
dence time tS). Figure 8 shows the model structure where
the column local mixing conditions, described by the two
small mixers, can be related to the feed input and the
bubble generation zones, while the single large mixing
stage was related to the volume of the baffled section of
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
1598 YIANATOS
the column, typically extended from below the feed entrance
to the bubble generation level.
The analytical solution for this model is given by the
following equation
E(t) ¼
�½(t � L)=(ts � a)� � e�(t�L)=ts þ a � e�(t�L)=tL
(tL � tS)
(18)
where
a ¼
tL
(tL � tS)
(19)
Figure 9 shows good agreement between the model and
experimentaldata for liquid RTD (Br-82 radioactive liquid
tracer) and solid RTD (radioactive mineral tracer), in an
industrial flotation column of dimensions 2 � 6 � 13 m.
Following the good results of the RTD model, equation
(18), a new model for predicting the mineral recovery in the
collection zone of industrial flotation columns was derived.
The new analytical solution, equation (20), was obtained
from equation (2), with E(t) given by equation (18). The
model also has the advantage of considering a rectangular
rate constant distribution F(k). Equation (20) allows the esti-
mation of the maximum flotation rate constant kM and it
was satisfactorily tested for industrial flotation column data
(Yianatos et al., 2005b).
RC
R1
¼ 1�
1
kM(tL � tS)
1
kMtS þ 1
� 1þ a ln
kMtL þ 1
kMtS þ 1
� �� �
(20)
If tS is equal to zero, equation (20) reduces to the solution of
a single perfect mixer with rectangular flotation rate
distribution as it is shown in equation (21).
RC
R1
¼ 1�
1
kMtL
� ln (kMtL þ 1)
� �
(21)
Equation (21) represents the lower recovery condition for the
column flotation model, which can also be derived from
equation (2) considering the RTD of a perfect mixer, given
in equation (8).
On the other hand, the maximum recovery condition for the
column flotation model corresponds to the operation near
plug flow observed in a pilot size column, and it is shown in
equation (22):
RC
R1
¼ 1�
1
kMt
� (1� e�kMt)
� �
(22)
Equation (22) was derived from equation (2) considering
E(t) ¼ d(t), for the batch flotation operation, and a rectangular
rate constant distribution F(k).
Effect of baffles on fluid flow in flotation columns
Moys et al. (1991) showed that vertical baffles contained
entirely within the liquid phase and which divided the
column into several parallel and independent flow chambers
actually reduced the quality of mixing in the column. This
effect was related to pulp circulation due to mal-distribution
of gas flux at the bottom of the column. Kawatra and Eisele
(1995) studied the use of horizontal baffles at laboratory
Figure 8. Conceptual model for mixing in big flotation columns
(Yianatos et al., 2005a).
Figure 9. Model fitting for liquid and solid in industrial flotation column.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
COLUMNS AND MECHANICALLY AGITATED CELLS 1599
scale, and they claimed that back mixing and short-circuiting
can be reduced in a much shorter column, provided that the
flow in the column approaches plug-flow. The use of baffles in
large size columns seems not affect the overall mixing but pre-
vents the macro-circulation between baffled zones (Yianatos,
2003). Yigang et al. (2001) reported the liquid RTD in a labora-
tory packed flotation column which showed a significant
mixing, despite the presence of ripple packing.
Froth Transport Modelling
Early works showing the distributed nature of the fluid
transport through the froth have been reported, i.e., Bishop
and White (1976) developed empirical studies on hydrophilic
particle entrainment into the froth and the rate of recovery as
a function of water recovery and pulp density. They con-
cluded that the most important factor in the rate of particle
recovery is the residence time in the froth. Recently, Johnson
(2005) presented a comprehensive review of the entrainment
mechanisms and its modelling in industrial flotation
processes.
Moys (1978) developed a plug flow model to describe the
mass transport along the froth, in a steady state batch oper-
ation. Moys (1984) developed a theoretical model, based on
the Laplace equation, to provide a two-dimensional descrip-
tion of the streamlines in the froth as it moves towards the
concentrate weir. The froth model was used for estimating
mass transport in continuous and steady state batch flotation
froths. More recently, a model considering only vertical and
horizontal paths was presented by Zheng et al. (2004),
from which an analytical relationship was derived to estimate
the froth residence time distribution. Other studies have been
focused on characterizing the froth structure and transport
mechanisms (i.e., Ross and Van Deventer, 1988; Falutsu
and Dobby, 1992).
An extensive review on froth modelling in steady and non-
steady state flotation systems was reported by Mathe et al.
(1998, 2000). It was found that despite the significant pro-
gress made in froth characterization and understanding of
the frothing phenomena, the information obtained from
these studies has a very limited use in flotation circuit
design, modelling and optimization. The problem is that
these studies have been mainly developed in environments
with ideal characteristics, such as the equilibrium cell,
because of the difficulty of sampling the froth phase.
Thus, the froth parameters that can be used to relate froth
performance in different flotation systems are still not fully
identified. Also, the authors claimed that no research has
been performed considering the effect of chemical factors
into froth transport models.
In recent years, a significant number of fundamental
studies on froth modeling have been developed describing
the gas, liquid and solid motion in flowing froths (Neethling
et al., 2000; Neethling and Cilliers, 2002, 2003; Finch et al.,
2007). Stevenson et al. (2003) studied the dispersion in a
two-dimensional rising froth, and the numerical solutions of
foam drainage in rising systems were also presented at
values of background liquid holdup relevant to the flotation
process. Zheng et al. (2006) reported the evaluation of differ-
ent models of water recovery in flotation froths, where the
fundamental approach adopted by Neethling and Cilliers
(2003) was useful to predict the water recovery. However,
the interactive effects of particle size, particle shape,
hydrophobicity and turbulence on froth recovery remains as
a challenge to froth phase modelling.
In the middle of 1980s, studies on column flotation froths
introduced the froth recovery concept. The froth recovery is
a practical parameter describing the recovery to the concen-
trate launders of floatable minerals entering the froth from the
collection zone (Van Deventer et al., 2001). The froth recov-
ery has been directly measured using modified pilot flotation
columns (Falutsu and Dobby, 1989; Vera, 1995). Also, froth
recovery estimates have been indirectly derived by extrapol-
ation to zero froth depth (Feteris et al., 1987; Yu and Finch,
1990; Vera et al., 1999) and by fitting grade profiles from
industrial column froths (Yianatos et al., 1998).
A new alternative approach for froth recovery estimation is
based on direct measurement of the bubble load near the
pulp–froth interface, in flotation cells and columns (i.e.,
Alexander et al., 2003; Seaman and Franzidis, 2004;
Hidalgo, 2006). Thus, the froth recovery Rf can be directly
obtained from equations (6) and (7) provided that superficial
gas rate and concentrate mass flowrate of floatable minerals
are known. These studies have shown that operating vari-
ables affect the pulp and froth zones performance in a differ-
ent and often opposite way. Experimentally, it was found that
a significant fraction of particles entering the froth return to
the collection zone.
So far, refined hydrodynamic models have not been taken
into account in the actual modeling of flotation equipment,
i.e., in the calculation of recovery and performance of the flo-
tation equipment by means of solving the material balance
equations, because still most of the simplifying assumptions,
parameters and boundary conditions need to be validated.
CONCLUSIONS
Industrial flotation is a complex separation process involving
the interaction of physico-chemical and hydrodynamic
phenomena. The modeling of such processes was firstly
developed by analogy with a chemical reaction, where a
single overall rate constant was considered to describe the
whole process, assuming perfect mixing for the whole
system as a black box. Then, a more realistic approach hasincluded the interaction between two zones, the collection
zone and the froth transport zone (separation).
From plant experience it has been observed that the flow
regime in the collection zone of single self-aerated mechan-
ical cells was not perfectly mixed. Thus, the mixing condition
in the collection zone of a single cell, with internal recircula-
tion, was better characterized by the tank in series approach.
The fluid flow in industrial flotation banks, consisting of
three to nine mechanical cells, has been well characterized
using the N tanks in series model, where N corresponds to
the actual number of cells in the bank.
The collection zone of industrial pneumatic flotation columns
operates in the bubbly flow regime and has been typically
characterized by the axial dispersion model. Even considering
that this approach was not realistic the model structure allows
for a good data fitting. Alternatively, a new approach for indus-
trial flotation columns allowed a better fitting considering a
simple model, based on large and small tanks in series.
The study of fluid transport in the froth zone of flotation
equipment has been of great attention, specifically in recent
years. Fundamental and empirical correlations on fluid and
gas transport in the froth allowed the estimation of the solid
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
1600 YIANATOS
mass transport by true flotation, entrainment of liquid and
solids, as well as liquid and particles drainage.
Theoretical estimation or direct measurement of bubble
loading, at the pulp froth interface, are matters of continuing
research interest because they allow for froth recovery calcu-
lation and then the independent identification of the collection
zone performance.
A deeper understanding of flotation related processes has
been achieved because the flow regime, the mass transport
conditions at the pulp/froth interface and the froth transport
characteristics, are better known and understood. Key par-
ameters are the bubble surface area flux, related to the
bubble generation and collection rate processes, bubble
loading related to the mass transport across the pulp–froth
interface and froth recovery which is mainly related to the
gas residence time in the froth.
In summary, despite the significant advances in fundamen-
tal process knowledge as well as in industrial flotation
process modelling, further work has to be done on metallur-
gical scale-up laws combining surface chemistry, hydrodyn-
amics and flotation rates, for the pulp and froth zones.
NOMENCLATURE
BL bubble loading, ton /m
23
dB bubble diameter, mm
dP particle diameter, mm
E(t) residence time distribution function
F cell feed flowrate, m3 h21
F(k) rate constant distributed function
JF superficial fluid velocity, cm s
21
JG superficial gas velocity, cm s
21
JGF drift flux, cm s
21
k overall kinetic rate constant, min21
K1 fraction of bubble surface coverage with a monolayer of
particles
kC collection rate constant, min
21
kM maximum flotation rate constant, min
21
L lag time, min
m Richardson–Zaki index
MC superficial concentrate mass overflow, ton h
21 m22
Mi superficial mass transport at the interface, ton h
21 m22
N number of mixer in series
Nd vessel dispersion number
Q volumetric flowrate, m3 h21
R overall flotation recovery
R1 maximum flotation recovery at infinite time
RC collection zone recovery
ReB bubble Reynolds number
Rf froth zone recovery
SB bubble surface area flux, s
21
USl slip velocity, cm s
21
UT terminal bubble rise velocity, cm s
21
V cell volume, m3
Greek symbols
G gamma function
d Dirac delta function
e fractional holdup
r density, ton m23
t mean residence time, min
m viscosity, g cm21 s21
Subscripts
F fluid
G gas
L large mixer
M maximum
P particle
S small mixer
REFERENCES
Alexander, D.J., Franzidis, J.P. and Manlapig, E.V., 2003, Froth
recovery measurement in plant scale flotation cells, Miner Eng,
16: 1197–1203.
Ando, K., Obata, E., Ikeda, K. and Fukuda, T., 1990, Mixing time of
liquid in horizontal stirred vessels with multiple impellers, Can
J Chem Eng, 68: 278–283.
Arbiter, N., 2000, Development and scale-up of large flotation cells,
Mining Eng, 52(3): 28–33.
Banisi, S. and Finch, J.A., 1993, Reconciliation of bubble size esti-
mation methods using drift flux analysis,Miner Eng, 7: 1555–1559.
Bholes, M.R. and Joshi, J.B., 2003, Hydrodynamics of column flo-
tation: Revisited, in International Symposium on the Role of Chemi-
cal Engineering in Processing Minerals and Materials, Indian
Chemical Engineering Congress, Mohanti, J.N., Biswal, S.K.,
Reddy, P.S.R. and Misra, V.N. (eds). 17–28 (Bhubaneswar, India).
Bishop, J.P. and White, ME., 1976, Study of particle entrainment in
flotation froths, Trans Inst Min Metall, 85: 191–194.
Bloom, F. and Heindel, T.J., 2002, On the structure of collision and
detachment frequencies in flotation models, Chem Eng Sci, 57:
2467–2473.
Bourke, P., 2002, Selecting flotation cells: How many and what size?
MEI on-line, www.min-eng.com.
Burgess, F.L., 1997, OK-100 tank cell operation at Pasminco-Broken
Hill, Miner Eng, 10(7): 723–741.
De Aquino, J.A., Oliveira, M.L. and Dias, M., 1998, Flotaçao em
coluna, in da Luz, A.B., Valente, M. and de Almeida, S.L. (eds).
Tratamento de Minerios, 435–476 (CETEM/CNPq, Rı́o de
Janeiro, Brasil).
Deckwer, W.D. and Schumpe, A., 1987, Bubble columns—the state
of the art and current trends, Int Chem Eng, 27(3): 405–422.
Deglon, D.A., 2003, A novel attachment-detachment kinetic model, in
Proc of the Strategic Conference, Flotation and Flocculation: From
Fundamentals to Applications, Ralston, J., Miller, J. and Rubio, J.
(eds). 109–116, (Snap Printing, Australia).
Deglon, D.A., Sawyerr, F. and O’Connor, C., 1999, A model to relate
the flotation rate constant and the bubble surface area flux in
mechanical flotation cells, Miner Eng, 12(6): 599–608.
Deng, H., Mehta, R.K. and Warren, G.W., 1996, Numerical modelling
of flows in flotation column, Int J of Miner Process, 48(1–2):
61–72.
Dobby, G.S., Yianatos, J.B. and Finch, J.A., 1988, Estimation of
bubble diameter in flotation columns from drift flux analysis, Can
Metall Quart, 27(2): 85–90.
Dobby, G.S. and Finch, J.A., 1986, Flotation column scale-up and
modeling, CIM Bulletin, 89–96.
Dobby, G.S. and Finch, J.A., 1985, Mixing characteristics of industrial
flotation columns, Chem Eng Sci, 40(7): 1061–1068.
Ek, C., 1992, Flotation kinetics, Mavros, P. and Matis, K.A. (eds). in
Innovations in Flotation Technology, Vol. 208, 183–209 (Nato
ASI Series, Kluwer Academic Publishers, Dordrecht, The
Netherlands).
Emmanouil, V., Skaperdas, E., Karapantsios, T.D. and Matis, K.A.,
2006, Towards three-phase modeling of a dissolved air flotation
tank, in Proceedings of the XXIII International Mineral Processing
Congress, Istanbul, Turkey, 3–8 September.
Falutsu, M. and Dobby. G.S., 1992, Froth performance in commercial
sized flotation columns, Miner Eng, 5(10–12): 1207–1223.
Falutsu, M. and Dobby, G.S., 1989, Direct measurement of froth
drop-back and collection zone recovery in a laboratory flotation
column, Miner Eng, 2(3): 377–386.
Ferreira, J.P. and Loveday, B.K., 2000, An improved model for simu-
lation of flotation circuits, Miner Eng, 13(14–15): 1441–1453.
Feteris, S., Frew, J. and Jowet, A., 1987, Modelling the effect of froth
depth in flotation, Int J Miner Process, 20: 131–135.
Finch, J.A., Cilliers, J. and Yianatos, J., 2007, Column flotation, in
Fuerstenau, M.C., Jameson, G. and Yoon, R.H. (eds). Froth
Flotation: A Century of Innovation, 681–737. (Society for Mining,
Metallurgy and Exploration, Littleton, USA).
Finch, J.A., 1995, Column flotation: a selected review—Part IV: novel
flotation devices, Miner Eng, 8(6): 587–602.
Finch, J.A., Uribe-Salas, A. and Xu, M., 1995, Column flotation, in
Matis, K.A. and Marcel, D. (eds). Flotation Sciences and Engineer-
ing, 291–330 (New York, USA).
Finch, J.A. and Dobby, G.S., 1990, Column Flotation, 1st edition,
(PergamonPress, London, UK).
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
COLUMNS AND MECHANICALLY AGITATED CELLS 1601
Flint, L.R., 1974, A mechanistic approach to flotation kinetics, Trans
Inst Min Metall, 83: C90–95.
Gallegos, P.M., Pérez, R. and Uribe, A., 2006, Maximum bubble
loads: Experimental measurements vs. analytical estimation,
Miner Eng, 19: 12–18.
Goodall, C.M. and O’Connor, C.T., 1991, Residence time distribution
studies in a flotation column. Part 1: the modelling of residence
time distribution in a laboratory column flotation cell, Int J Miner
Process, 31: 97–113.
Harris, C.C., 1978, Multiphase models of flotation machines beha-
viour, Int J Miner Process, 5: 107–129.
Harris, C.C., Chakravarti, A. and Degaleesan, S.N., 1975, A recycle
flow flotation machine model, Int J Miner Process, 2(1): 39–58.
Hidalgo, L., 2006, Determinación de recuperación de espuma en
celdas de flotación. Chem Eng thesis, Santa Maria University,
Chile.
Ityokumbul, M.T., 1992, A mass transfer approach to flotation column
design, Chem Eng Sci, 47(13–14): 3605–3612.
Ityokumbul, M.T., Kosaric, N. and Bulani, W., 1988, Parameter esti-
mation with simplified boundary conditions, Chem Eng Sci, 43(9):
2457–2462.
Jameson, G.J., 1988, New concept in flotation column design, Miner
Metall Process, 5(1): 44–47.
Johnson, N.W., 2005, A review of the entrainment mechanism and its
modelling in industrial flotation processes, in Centenary of Flotation
Symposium, 487–496 (Brisbane, Australia).
Joshi, J.B. and Sharma, M.M., 1979, A circulation cell model for
bubble columns, Trans Instn Chem Engrs, 57: 244–251.
Kawatra, S.K. and Eisele, T.C., 1995, Laboratory baffled-column
flotation of mixer lower/middle Kittanning seam bituminous coal,
Miner Metall Process, 12(2):103–107.
King, R.P., 2001, Modeling and Simulation of Mineral Processing
Systems, 1st edition, (Butterworth-Heinemann, Oxford, UK).
Klimpel, R., 1980, Selection of chemical reagents for flotation, in
Mular, A.L. and Bhappu, R. (eds). Mineral Processing Plant
Design, 2nd edition 907–934 (SME, Littleton, USA).
Koh, P.T.L. and Schwarz, M.P., 2003, CFD modeling of bubble-
particle collision rates and efficiencies in a flotation cell, Miner
Eng, 16: 1055–1059.
Koh, P.T.L., Manickam, M. and Schwarz, M.P., 2000, CFD simulation
of bubble-particle collisions in mineral flotation cells, Miner Eng,
13(14–15): 1455–1463.
Kostoglou, M., Karapantsios, T.D. and Matis, K.A., 2006, Modeling
local flotation frequency in a turbulent flow field, Advances in
Colloid and Interface Science, 122: 79–91.
Kwon, S.B., Park, N.S., Lee, S.J., Ahn, H.W. and Wang, C.K., 2006,
Examining the effect of length/width ratio on the hydro-dynamic
behaviour in a DAF system using CFD and ADV techniques,
Water Science and Technology, 53(7): 141–149.
Langberg, D.E. and Jameson, G.J., 1992, The coexistence of the
froth and liquid phases in a flotation column, Chem Eng Sci, 47:
4345–4355.
Laplante, A.R., Yianatos, J.B. and Finch, J.A., 1988, On the mixing
characteristics of the collection zone in flotation columns, in,
Sastry, K.V.S. (ed.). Column Flotation’88, 69–80 (SME, Littleton,
USA).
Laplante, A.R., Toguri, J.M. and Smith, H.W., 1983a, The effect of air
flowrate on the kinetics of flotation. (I): The transfer of material from
the slurry to the froth, Int J Miner Process, 11: 203–219.
Laplante, A.R., Toguri, J.M. and Smith, H.W., 1983b, The effect of air
flowrate on the kinetics of flotation. (II) The transfer of material from
the froth over the cell lip, Int J Miner Process, 11: 221–234.
Lelinski, D., Allen, J., Redden, L. and Weber, A., 2002, Analysis of
the residence time distribution in large flotation machines, Miner
Eng, 15(7): 499–505.
Lelinski, D., Redden, L.D. and Nelson, M.G., 2005, Important con-
siderations in the design of mechanical flotation machines, in
Centenary of Flotation Symposium, 217–223 (Brisbane, Australia).
Li, H., Del Villar, R. and Gomez, C.O., 2004, Reviewing the exper-
imental procedure to determine the carrying capacity in flotation
columns, Can Met Quart, 43(4): 513–520.
Luttrell, G.H., Mankosa, M.J. and Yoon, R.H., 1993, Design and
scale-up criteria for column flotation, in XVIII IMPC, 785–791
(Sydney, Australia).
Mathe, Z., Harris, M. and O’Connor, C., 2000, A review of methods to
model the froth phase in non-steady state flotation systems, Miner
Eng, 13(2): 127–140.
Mathe, Z., Harris, M., O’Connor, C. and Franzidis, J.P., 1998, Review
of froth modelling in steady state flotation systems, Miner Eng,
11(5): 397–421.
Mavros, P., 1992a, Mixing and hydrodynamics in flotation cells, in
Mavros, P. and Matis, K.A. (eds). Innovation in Flotation Technol-
ogy, Vol. E208, 211–234 (NATO Series, Kluwer Academic Publish-
ers, The Netherlands).
Mavros, P., 1992b, Validity and limitations of the closed-vessel
analytical solution to the axial dispersion model, Miner Eng, 5(9):
1053–1060.
Mavros, P., Lazaridis, N.K. and Matis, K.A., 1989, A study and
modelling of liquid phase mixing in a flotation column, Int J Miner
Process, 26(1–2): 1–16.
Mehrotra, S.P. and Saxena, A.K., 1983, Effects of process
variables on the residence time distribution of a solid in a
continuously operated flotation cell, Int J Miner Process, 10:
255–277.
Mills, P.J.T. and O’Connor, C.T. 1990, The modeling of liquid and
solids mixing in a flotation column, Miner Eng, 3(6): 567–576.
Mills, P.J.T., Yianatos, J.B. and O’Connor, C.T., 1992, The effect of
particle size on the mixing characteristics of a flotation column,
Miner Eng, 3(6): 567–576.
Moys, M.H., 1984, Residence time distributions and mass transport
in the froth phase of the flotation process, Int J Miner Process,
13: 117–142.
Moys, M.H., 1978, A study of a plug-flow model for flotation froth
behavior, Int J Miner Process, 5: 21–38.
Moys, M.H., Engelbrecht, J. and Terblanche, N., 1991, The design of
baffles to reduce axial mixing in flotation columns, in Agar, G.E.,
Huls, B.J. and Hyma, D.B. (eds). Proc International Conference
Column’91, Vol. 1, 275–288 (CIM, Canada).
Neethling, S.J. and Cilliers, J.J., 2003, Modelling flotation froths, Int J
Miner Process, 72: 267–287.
Neethling, S.J. and Cilliers, J.J., 2002, Solids motion in flowing froths,
Chem Eng Sci, 57: 607–615.
Neethling, S.J., Cilliers, J.J. and Woodburn, E.T., 2000, Prediction
of the water distribution in a flowing foam, Chem Eng Sci, 55:
4021–4028.
Niemi, A.J., 1995, Role of kinetics in modelling and control of flotation
plants, Powder Technol, 82: 69–77.
Pal, R. and Masliyah, J., 1990, Flow characteristics of a flotation
column, Can J Chem Eng, 29(2): 97–103.
Patwardhan, A. and Honaker, R.Q., 2000, Development of a carrying-
capacity model for column froth flotation, Int J Miner Process, 59:
275–293.
Polat, M. and Chander, S., 2000, First order flotation kinetics models
and methods for estimation of the true distribution of flotation rate
constant, Int J Miner Process, 58: 145–166.
Pyke, B., Duan, J., Fornasiero, D. and Ralston, J., 2003, From
turbulence and collision to attachment and detachment: A
general flotation model, in Proc of the Strategic Conference
Flotation and Flocculation: From Fundamentals to Applications,
Ralston, J., Miller, J. and Rubio, J. (eds). 77–89 (Snap Printing,
Australia).
Ralston, J., Fornasiero, D. and Mishchuk, N., 2001, The hydrophobic
force in flotation: a critique, Colloids and Surfaces A: Physico-
chemical and Engineering Aspects, 192(1–3): 39–51.
Ross, V.E., 1991, The behavior of particles in flotation froths, Int J
Miner Process, 4(7–11): 959–974.
Ross, V.E. and Van Deventer, J.S.J., 1988, Mass transport in flotation
column froths, in Sastry, K.V.S. (ed.). Proc Int Symposium
Column’88, 129–139 (SME, Littleton, USA).
Rubinstein, J.B., 1995, Column Flotation: Processes, Designs and
Practices, 1st edition (Gordon and Breach Science Publishers,
Basilea, Switzerland).
Sastry, K.V.S. and Fuerstenau, D.W., 1970, Theoretical analysis
of a counter-current flotation column, Transactions,SME of
AIME, 247: 46–52.
Savassi, O.N., 2005, A compartment model for the mass transfer
inside a conventional cell, Int J Miner Process, 77: 65–79.
Seaman, D.R., Franzidis, J.P. and Manlapig, E.V., 2004, Bubble load
measurement in the pulp zone of industrial flotation machines-a
new device for determining the froth recovery of attached particles,
Int J Miner Process, 74: 1–13.
Shah, Y.T., Kelkar, B.G., Goldbole, S.P. and Deckwer, W.D., 1982,
Design parameters estimation for bubble column reactors,
AICHE J, 28(3): 353–379.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
1602 YIANATOS
Stevenson, P., Sullivan, S.P. and Jameson, G.J., 2003, Short-time
tracer dispersión in a two-dimensional rising froth, Chem Eng
Sci, 56: 5025–5043.
Szatkowski, M. and Freiberger, W.L., 1985, Kinetics of flotation with
fine bubbles, Trans Instn Min Metal, 94: C61–70.
Tello, K., 2006, Caracterización hidrodinámica y metalúrgica de un
banco de flotación industrial, Met Eng thesis, Santa Maria Univer-
sity, Chile.
Tokoro, C. and Okano, Y., 2001, Boundary conditions for column
flotation—A study by transfer function representation of an axial
diffusion model, Miner Eng, 14(1): 49–64.
Tuteja, R.K., Spottiswood, D.J. and Misra, V.N., 1994, Mathematical
models of the column flotation process: a review, Miner Eng, 7(12):
1459–1472.
Vandenberghe, J., Choung, J., Xu, Z. and Masliyah, J., 2005, Drift
flux modelling for a two-phase system in a flotation column, Can
J Chem Eng, 83: 169–176.
Van Deventer, J.S.J., Feng, D. and Burger, A.J., 2004a, Transport
phenomena at the pulp-froth interface in a flotation column: 1.
Recovery profiles, Int J Miner Process, 74: 201–215.
Van Deventer, J.S.J., Feng, D. and Burger, A.J., 2004b, Transport
phenomena at the pulp-froth interface in a flotation column: 2.
Detachment, Int J Miner Process, 74: 201–215.
Van Deventer, J.S.J., Feng, D. and Burger, A.J., 2001, The use of
bubble loads to interpret transport phenomena at the pulp-froth
interface in a flotation column, Chem Eng Sci, 56: 6313–6319.
Vera, M.A., 1995, The determination of the collection zone rate con-
stant and froth zone recovery by column flotation, Master thesis,
Queensland University, Australia.
Vera, M.A., Franzidis, J.P. and Manlapig, E.V., 1999, Simultaneous
determination of collection zone rate constant and froth recovery
in a mechanical flotation environment, Miner Eng, 12(10): 1163–
1176.
Weber, A., Meadows, D., Villanueva, F., Palomo, R. and Prado, S.,
2005, Development of the world’s largest flotation machine, in
Centenary of Flotation Symposium, 285–291 (Brisbane, Australia).
Wilson, S.G. and Frew, J.A., 1986, Effect of operating variables
upon liquid backmixing in a laboratory flotation bank, Int J Miner
Process, 16: 281–298.
Xu, M., Finch, J.A. and Uribe-Salas, A., 1991a, Maximum gas and
bubble surface rates in flotation columns, Int J Miner Process,
32: 233–250.
Xu, M., Finch, J.A. and Laplante, A.R., 1991b, Numerical solution to
axial dispersion model in flotation column studies, Can Metall
Quart, 30(2): 71–77.
Yianatos, J.B. and Henrı́quez, F., 2007, Boundary conditions for gas
rate and bubble size at the pulp-froth interface in flotation equip-
ment, Miner Eng, 20: 625–628.
Yianatos, J.B., Bergh, L.G., Tello, K., Dı́az, F. and Villanueva, A.,
2007, Residence time distribution in single big industrial flotation
cells, Miner Metall Process J, accepted for publication.
Yianatos, J.B., Bergh, L., Diaz, F. and Rodriguez, J., 2005a, Mixing
characteristics of industrial flotation equipment, Chem Eng Sci,
60: 2273–2282.
Yianatos, J.B., Bucarey, R., Larenas, J., Henrı́quez, F. and Torres, L.,
2005b, Collection zone kinetic model for industrial flotation
columns, Miner Eng, 18: 1373–1377.
Yianatos, J.B., 2003, Current status of column flotation, in Ralston, J.,
Miller, J. and Rubio, J., (eds). Proceedings Strategic Conference,
Flotation and Flocculation: From Fundamentals to Applications,
213–220 (Snap Printing, Australia).
Yianatos, J.B., Bergh, L.G. and Aguilera, J., 2000, The effect of grind-
ing on mill performance at División Salvador, Codelco-Chile, Miner
Eng, 13(5): 485–495.
Yianatos, J.B., Bergh, L.G., Condori, P. and Aguilera, J., 2001,
Hydrodynamic and metallurgical characterization of industrial
flotation banks for control purposes,Miner Eng, 14(9): 1033–1046.
Yianatos, J.B., Bergh, L.G. and Cortés, G.A., 1998, Froth zone mod-
eling of an industrial flotation column, Miner Eng,11(5): 423–435.
Yianatos, J.B. and Bergh, L.G., 1992, RTD studies in an industrial
flotation column: use of the radioactive tracer technique, Int J
Miner Process, 36: 81–91.
Yianatos, J.B., Finch, J.A., Dobby, G.S. and Xu, M., 1988, Bubble
size estimation in a bubble swarm, J Colloid Inter Sci, 126(1):
37–44.
Yianatos, J.B., Finch, J.A. and Laplante, A.R., 1987, The cleaning
action in column flotation froths, Trans Inst Min Metall, Section
C(96): C199–C205.
Yigang, D., Yuanxin, W., Dinghuo, L. and Jiashen, Z., 2001, A study
on the mixing characteristics of a packed flotation column, Miner
Eng, 14(9): 1101–1105.
Yoon, R.-H., 2000, The role of hydrodynamic and surface forces
in bubble-particle interaction, Int J Miner Process, 58(1–4):
129–143.
Yu, S. and Finch, J.A., 1990, Froth zone recovery in a flotation
column, Can Metall Quart, 29(3): 237–238.
Zehner, P., 1986, Momentum, mass and heat transfer in bubble
columns. Part I. Flow models of the bubble column and liquid
velocities, Int Chem Eng, 26: 22–28.
Zheng, X., Franzidis, J.P. and Johnson, N.W., 2006, An evaluation
of different models of water recovery in flotation, Miner Eng, 19:
871–882.
Zhang, X., Franzidis, J.P. and Manlapig, E., 2004, Modelling of froth
transportation in industrial flotation cells Part 1. Development of
froth transportation models for attached particles, Miner Eng, 17:
981–988.
ACKNOWLEDGEMENTS
Funding for process modelling and control research is provided by
CONICYT, project Fondecyt 1070106, and Santa Marı́a University,
project 270522.
The manuscript was received 3 May 2007 and accepted for
publication after revision 23 August 2007.
Trans IChemE, Part A, Chemical Engineering Research and Design, 2007, 85(A12): 1591–1603
COLUMNS AND MECHANICALLY AGITATED CELLS 1603
	FLUID FLOW AND KINETIC MODELLING INFLOTATION RELATED PROCESSESColumns and Mechanically AgitatedCells---A Review
	INTRODUCTION
	Mechanically Agitated Cells
	Pneumatic Cells
	FLOTATION MODELLINGFundamental Collection Models
	Flotation Rate Models
	Industrial Flotation: Model Structure
	RTD of Industrial Mechanical Cells
	Flotation Columns
	RTD of Industrial Flotation Columns
	Froth Transport Modelling
	CONCLUSIONS
	NOMENCLATURE
	REFERENCES
	ACKNOWLEDGEMENTS