178 Chapter 5 Three-Phase Dynamic Model of the Column Flotation Process 5.1 Introduction An evaluation of the industrial performance of flotation columns would reveal that there is still room for improvement in column control and optimization. One of the advantages of column flotation over the conventional flotation cell technology appears to be the improved suitability for modeling and automation. This feature has encouraged several attempts at developing column representations for incorporation into advanced control and optimization routines. Most of those control techniques require a dynamic model, that is, a set of differential equations that account for the state of the process between steady-state operations. Unfortunately, the complexities of some of the subprocesses integrated in the operation of column flotation have made the task of finding an appropriate dynamic model very difficult. This has lead to the adoption of alternative techniques, that do not demand any detailed knowledge of the process. They are based on either empirical responses over a limited range of operation, or generalized verbal rules from a human operator. The empirical techniques are limited by the lack of capability to generalize, but they can be successfully employed for stabilizing control loops, when the process is operating around a set point. The heuristic approach has been regarded as a relatively easy method to apply for column optimization, and, undoubtedly, it can perform well as a diagnostics tool and as an expert advisor. Nevertheless, it is limited by the depth of knowledge extracted from the operators, and it does not provide any insight into the internal structure of the process. Some determinants of column flotation performance, especially concerning to froth behavior, cannot be easily defined in terms of a few other variables in simple rules. It is not only because there are significant correlations among many parameters, but also because these relationships may vary on a case-by-case basis. Besides, the temporal behavior should not be ignored since the speed of response of the main process variables can be critical in determining the proper control actions. A reasonable description of the column dynamics would therefore be very useful for experimenting with a diversity of control techniques that have been successfully applied in other industries. With that direction in mind, the task of developing such model was undertaken. The model was intended to provide a good representation of the behavior of column flotation units in both the pulp and froth regions. Since these regions are in fact interdependent, due to the exchange of material between them, both have to be adequately represented in order to attain a functional column flotation model. In addition, it was proposed that the model should integrate the available understanding on the various subprocesses that take place during operation, such as particle collection, detachment and bubble coalescence.
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5.1 Introduction
An evaluation of the industrial performance of flotation columns
would reveal that there is still room for improvement in column
control and optimization. One of the advantages of column flotation
over the conventional flotation cell technology appears to be the
improved suitability for modeling and automation. This feature has
encouraged several attempts at developing column representations
for incorporation into advanced control and optimization routines.
Most of those control techniques require a dynamic model, that is,
a set of differential equations that account for the state of the
process between steady-state operations. Unfortunately, the
complexities of some of the subprocesses integrated in the
operation of column flotation have made the task of finding an
appropriate dynamic model very difficult. This has lead to the
adoption of alternative techniques, that do not demand any detailed
knowledge of the process. They are based on either empirical
responses over a limited range of operation, or generalized verbal
rules from a human operator. The empirical techniques are limited
by the lack of capability to generalize, but they can be
successfully employed for stabilizing control loops, when the
process is operating around a set point. The heuristic approach has
been regarded as a relatively easy method to apply for column
optimization, and, undoubtedly, it can perform well as a
diagnostics tool and as an expert advisor. Nevertheless, it is
limited by the depth of knowledge extracted from the operators, and
it does not provide any insight into the internal structure of the
process. Some determinants of column flotation performance,
especially concerning to froth behavior, cannot be easily defined
in terms of a few other variables in simple rules. It is not only
because there are significant correlations among many parameters,
but also because these relationships may vary on a case-by-case
basis. Besides, the temporal behavior should not be ignored since
the speed of response of the main process variables can be critical
in determining the proper control actions.
A reasonable description of the column dynamics would therefore be
very useful for experimenting with a diversity of control
techniques that have been successfully applied in other industries.
With that direction in mind, the task of developing such model was
undertaken. The model was intended to provide a good representation
of the behavior of column flotation units in both the pulp and
froth regions. Since these regions are in fact interdependent, due
to the exchange of material between them, both have to be
adequately represented in order to attain a functional column
flotation model. In addition, it was proposed that the model should
integrate the available understanding on the various subprocesses
that take place during operation, such as particle collection,
detachment and bubble coalescence.
179
5.2 Background
The first published report on column flotation modeling is
attributed to Sastry and Fuerstenau (1970), who derived and solved
the steady-state equations describing the concentration profiles of
free and attached solids along the collection region. Afterwards,
there has been numerous publications on parametric studies, which
establish a link between operating conditions such as gas rate,
bubble and particle sizes to froth depth, liquid content, and solid
concentration profiles. Recent publications on the modeling of
column flotation units include scale-up models, like the ones
developed by Dobby and Finch (1986) and Mankosa et al.(1990),
steady-state simulators (Luttrell and Yoon, 1991; Alford, 1992),
and coarse-particle flotation models (Oteyaka and Soto, 1995).
Characteristics such as froth cleaning, recovery, selectivity,
column carrying capacity, and entrainment have also been
mathematically interpreted (Flynn and Woodburn, 1987; Szatkowski,
1988; Espinosa-Gomez et al., 1988; Tuteja et al., 1995). However,
as to the existence of column dynamic models, progress has been
more modest. Sastry and Lofftus (1988) extended the concepts first
introduced by Sastry and Fuerstenau (op.cit.) to obtain a
mechanistic representation. Luttrell (1986) also presented a set of
fundamental dynamic equations that describe the process, and Bascur
and Herbst (1982) developed a flotation cell dynamic model which
has been used as the foundation for a column simulator (Lee, Pate,
Oblad, Herbst, 1991). However, these representations have not been
successful in integrating the dynamic behavior of both gas bubbles
and solid particles. For instance, the approach of using a series
of perfectly mixed tanks for the entire column length fails to
reproduce the transition in flow regime that occurs at the
interface. In all cases, a solution for the collection region can
only be obtained for a very limited number of situations, involving
a series of simplifying assumptions. As to the froth equations, a
simultaneous solution for the air phase and the solid phase has not
yet been presented. Each subprocess (detachment, froth washing,
entrainment) is usually characterized by an unknown first- order
rate constant, while consideration for bubble growth throughout the
froth is normally absent from these dynamic models. The problem is
that if the number of unknown parameters is too large, the model is
transformed into a merely theoretical exercise, particularly
because determination of such rate constants is a difficult
task.
5.3 Model Development
As a separation process, a flotation column operates with three
particulate phases: air bubbles, solids in the continuous slurry
phase, and solids attached to air bubbles. The column can be
regarded as a series of regions characterized by the function they
play. The collection region, located below the interface down to
the zone where the bubbles are produced, is mainly where the
interception of bubbles and particles lead to particle attachment.
Because the feed enters the column in this region, it can be
subdivided into a lower and an upper part, with the feed entry port
defining the transition. Above the interface, the region located
below the wash water distributor is the stabilized froth, which is
also called the cleaning region because the bias water washes down
entrained material. Above the wash water addition zone, the froth
behaves like in a conventional flotation cell.
180
Its function is to carry the floated material to the overflow
launder so that it can be recovered. Since there is no
countercurrent flow to keep the froth stable, this region is also
known as the draining froth in a column. The diagram on Figure 5.1
illustrates how the flotation column is viewed in terms of
operating regions.
The methodology followed in the development of a column flotation
dynamic model was the application of a population balance; that is,
a number balance of the particulate species in the system. As a
general rule, a population balance model is based on the following
conservation principle:
Accumulation = Input - Output + Generation [1]
There are two general forms of a population balance model. One is
the macroscopic model, where the species quantity has been averaged
over the reactor volume. The other one is the microscopic model,
whose solution provides information on the changes in the species
concentration along any of the three spatial directions. The
general form of the macroscopic number balance equation is:
( ) ( ) ( )1 1
∂ ψ ∂
∑ , [2]
where ψ is the number concentration of the species under
consideration, which is defined by
ψ ζ= nfn, . [3]
In the equation above, n is the total number of particles in the
reactor volume and fn,ζ is the number distribution with respect to
property ζ. The second term in Equation [2] represents the
continuous generation of the population characterized by property
ζ, with vj as the rate of change in property ζ with time. The third
and fourth terms account for the net discrete generation of the
same species. The right-hand side of the equation represents the
net flowrate of the species into the system volume being
considered.
The microscopic population balance model, on the other hand, has
the following general form:
( ) ( ) ( ) ( )∂ψ ∂
∂ ψ ∂
∂ ψ ∂
∂ ψ ∂
0 [4]
The quantities vx, vy and vz stand for the average interstitial
velocities of the population
species in the x-, y-, and z-directions inside the reactor volume.
The variable ψ is the number of particulates of a given species per
unit volume and characteristic ζ, at a location [x,y,z] inside the
reactor. The terms D and A are also defined on the basis of
position; that is, they are not average quantities.
181
Feed Zone
Interface
182
5.3.1 Model Assumptions
In order to substitute in Equations [2] and [4] for the particular
transport and rate terms for each particulate species, several
assumptions have to be made from previous knowledge about the
process. The assumptions involved in the development of the column
flotation dynamic equations are listed next:
• Collection Region
a) The portion of the column located between the feed entry port
and the pulp-froth interface is called the upper collection region,
while the volume below the feed port down to the gas port is termed
the lower collection region. Each of these regions can be
represented by an integer number of perfectly-mixed tanks. The
number of tanks corresponding to each of these regions is specified
separately as a model input.
b) A set of transition regions represented by perfectly mixed zones
are also defined. They include the gas entry region (around the gas
entry port) and the feed entry region (around the feed port). Other
transition regions are the wash water addition zone and the
interface, which are considered hereinafter with the froth
zones.
c) There is no bubble coalescence taking place in the entire
collection zone. Also, the increase in bubble size which occurs in
large columns due to head pressure effects is not incorporated into
the model. Therefore, the bubble size distribution stays the same
up to the interface. This distribution is expressed as a discrete
set of number fractions fn,d,k , a number average bubble size and a
number Nb of size classes.
d) All the air entering the column leaves with the concentrate (air
in the tailings is negligible in most situations).
e) Particle detachment in the collection region, due to their
inertia, turbulence, or bubble oscillations, is not
significant.
( )( )
0.687 [5]
g) Bubble loading has the effect of reducing the bubble rise
velocity. Such effect is quantified through the previous equation
as the bubble density changes with the extent of loading.
183
h) The average slip velocity in each perfectly-mixed region is
given by
Ugs Ugs
ε . [6]
i) The feed solids are categorized into size classes and
composition classes. A feed size distribution and composition
distribution have to be specified. The total number of solid
species is given by the product Np*Nc, where Np is the number of
discrete particle size classes and Nc is the number of discrete
solid composition classes. Therefore, the model includes Np*Nc
equations for free particles and Np*Nc equations for attached
particles, for each column zone.
• Stabilized Froth
a) In the stabilized froth region, the flow behavior is assumed to
be plug-flow. Such premise is in agreement with the froth models
derived by Moys (1978) for conventional froths, and by Yianatos et
al. (1988) for column froths. The mean bubble velocity is given
by
Ub Vg
= ∑ε . [7]
b) The bubbles are assumed to remain spherical due to the downward
bias flow which helps stabilize the region and prevents bubble
deformation.
c) Coalescence is considered to be proportional to the number of
interactions or collisions between bubbles of size classes i and j,
for i=1...Nb and j=1...Nb, and to a coalescence efficiency rate
parameter. This parameter is a measure of the number of
interactions that result in coalescence. From the published studies
on the stability of mineralized froths (Subrahmanyam and Forssberg,
1988; Ross, 1991; Johansson and Pugh, 1992; Falutsu, 1994;
Szatkowski, 1995), it is expected that the coalescence efficiency
rate parameters are somehow dependent on the presence of solids.
However, at the present time, a correlation between solid
properties and froth stability is not feasible, partly because the
observed effects vary with the type of mineral system. According to
Szatkowski (1995) the more loaded the bubbles are, the less likely
it is that they will coalesce. On the other hand, Dippenaar (1982)
has suggested that hydrophobic particles destabilize the froth. In
another study, Johansson and Pugh (1992) suggested that there is a
critical degree of hydrophobicity. Particles that are at this level
of hydrophobicity, or higher, would be capable of bridging and
rupturing the film separating the bubbles. Given the degree of
uncertainty on the kind of relationship between the presence of
solids and coalescence, only the dependence on bubble size was
explicitly considered throughout this work.
184
d) The detachment of particles from the bubble surfaces occur when
two loaded bubbles coalesce. Two different scenarios were
considered. In the first situation, it was assumed that the
particles attached to coalescing bubbles try to accomodate on the
newly created bubble. However, since the available surface area has
been reduced, some particles may not be able to find enough free
space. The concentration of particles detached is then a function
of the difference between the available surface area on the new
bubble and the occupied surface area of the original bubbles.
Larger particles and hydrophilic particles are assumed to detach
first than the small hydrophobic ones. A second approach was to
assume that froth detachment is non-selective, so that all
particles on the surface of two coalescing bubbles become detached
as a result of the oscillations that take place. In this case, no
reattachment ensues.
The drag force exerted by the downward liquid flow is sometimes
considered to be a source of detachment. However, it was assumed
that the main role of the bias flow is to wash down entrained free
particles, and it does not affect the attached solid material. In a
related study, Falutsu (1994) concluded that this drag force is not
of the same order of magnitude of the detachment force, but
smaller.
e) Collection of particles due to bubble-particle collision is
considered to be negligible. Besides the lack of reported evidence
that would support the existence of significant particle collection
in the froth, several factors work against the formation of bubble-
particle aggregates. They include the low relative velocities
between bubbles and particles and the occurrence of bubble
oscillations. Falutsu (op. cit.) made an analysis of the various
conditions that promote or disfavor particle collection in the
froth regions. Among the features that could increase the
likelihood of attachment, he included the large contact times and
the reduced thickness of the liquid films. However, another factor
which also works against bubble-particle collection is the
reduction in available bubble surface area in comparison with the
collection region.
• Draining Froth
a) The bubbles in the draining froth are considered to maintain a
spherical shape. This assumption eliminates the geometrical
parameters characteristic of a cellular foam model. After all, the
objective is not to describe accurately the structure of this
region, but to represent adequately the transport conditions and
the particle transfer from the air to the water phase.
b) The air equations describing the draining froth have the same
form as those representing the stabilized froth. Since the froth is
considered to overflow evenly along the circular path of the column
lip, the premise of plug-flow bubble movement is still appropriate.
The differences between the air fraction profiles can be
mathematically explained by the choice of values for the
coalescence rate parameters and their relationship to conditions
such as bubble sizes. The rate of coalescence is expected to
increase, in comparison to that in the stabilized froth, due to the
drainage of the liquid films without a dowward liquid flow to
replenish them.
185
c) The rate of bubble-particle collection is insignificant. As in
the stabilized froth, particles on coalescing bubbles may all
become detached, or they may rearrange themselves on the available
surface area until the bubble coverage reaches its maximum. Then,
the remaining particles detach and either leave with the overflow
or move downwards due to settling.
d) The product slurry flowrate is given by the following
expression:
( ) Qp
1 ε ε , [8]
and, at steady state, an overall volume balance equation has to be
met so that
Qt Qp Qw Qf+ = + [9]
since it is assumed that the air content of the tailings flow is
zero. From the general mass balance, the slurry flowrate in the
concentrate is also given by the following relationship:
Qp Qw Qb= − [10]
Consequently, the model has to be solved iteratively in order for
Equation [9] to hold true. The tailings flowrate has to be adjusted
after the model equations reach a stable solution; then, the model
has to be solved again. This process should be repeated until the
difference between the concentrate flowrate calculated with
Equation [8] and that calculated with Equation [10] is sufficiently
small.
5.3.2 Collection Zone Modeling
• Air Phase Equations
A macroscopic population balance equation was written for each of
the perfectly- mixed zones of the collection region. The
microscopic balance model could not be applied since, for a
perfectly mixed reactor,
( ) ( ) ( )∂ ψ ∂
∂ ψ ∂
∂ ψ ∂
v
x
v
y
v
z x y z= = = 0 [11]
Since it was assumed that there is no coalescence in this region
and that the reduced pressure with height is not a concern, the
discrete and continuous generation terms were
186
dropped from Equation [2]. The general macroscopic population
balance model can then be written as follows:
( ) ( )d n f
Vz
d t n d t d in d t n d t in d out d t n d t, , , , , , , , ,
=
− [12]
A transformation from a number-based population balance to a
volume-based balance yields the dynamic model equations for the air
holdup in the perfectly-mixed regions. The number-based bubble
distribution is converted to a volume-based one using the following
expression:
f D f
6
3
3
[13]
The dependence of the average bubble size on gas rate is taken into
account by using the empirical relationship reported by Dobby and
Finch (1986), which suggests that
D CJb g= 0 25. [14]
In this way, changes in the gas rate during the simulations will
have an effect on the number-average bubble size and, therefore, in
the bubble size distribution. The quantities Qdin and Qdout are
calculated from the following drift-flux relationship:
Qg Qsl Ugs
1 [15]
From this relationship, the interstitial gas flowrate is found to
be given by the next expression:
( )Qg Qg Qsl AUgs
total ave totalε
ε= − + −1 [16]
Substituting Equation [6] in Equation [16], an expression in terms
of the individual air fraction components and hindered rise
velocities is obtained:
Qg Qg Qsl A Ugs A Ugsk k k
k k
+∑ ∑ ∑ε ε ε [17]
Drift flux theory also indicates that the interstitial slurry
flowrate is given by:
187
ε [18]
For each of the air fractions components εk , which pertain to each
bubble size class k, the following expression applies at steady
state:
f Qg Qg Qsl AUgs AUgsv d k ave total k k k, , ( )= − − +ε ε ε
[19]
Application of the preceding drift-flux relationships results in
the following air- phase model equations for the collection
zone:
⇒ In the gas entry zone (or aeration zone):
d
v kε = , [21]
k z
+∑ [22]
⇒ in each of the zones in the lower collection region (between the
gas entry zone and the feed zone):
d
ave z
k z
Q Qg Qt AUgs AUgsout k ave z
k z
ave z
k z
Q Qg Qb AUgs AUgsout k ave z
k z
+∑ [28]
⇒ in each of the zones in the upper collection region (above the
feed zone and below the interface):
d
ave z
k z
Q Qg Qb AUgs AUgsout k ave z
k z
+∑ [31]
In these equations, Qg, Qt and Qb stand for the gas, tailings and
bias flowrates, and A symbolizes the column cross sectional
area.
• Free Solids Equations
The general number balance equation for the solid particles of size
class s and composition class c in a perfectly-mixed zone is the
following:
( )∂ ψ ∂
s c s c s c
s in s cin out s cout
, , , , ,− + = −
1 [32]
ψf s,c is the volume-averaged number concentration of free solids
in size class s and
composition class c. The detachment term, TDs c, , is assumed to
equal zero in the collection region, and the attachment term is
given by:
TA k fs c s c k s c k
k , , , ,
max
β β
1 , [33]
where ks,c,k is the attachment rate constant for particles of size
class s and composition c,
which collide with bubbles of size class k. The parameters βk and
βmax are the fractional surface coverage of bubbles in size class k
and the maximum surface coverage,
189
respectively. With the exceptions of the aeration zone and the feed
zone, the flowrates into and out of each perfectly-mixed tank are
given by:
( )( )Q AUgs AUps Qg Qsl AUgsin z
ave z
( )( )Q AUgs AUps Qg Qsl AUgsout z
ave z
total z= + + − + −, 1 ε [35]
The first term in the right-hand side of each of these equations
represents the drainage of liquid due to gravity, while the second
term accounts for the particle settling. The expression in
parentheses represents the interstitial gas flowrate (Equation
[16]), which is linked to the entrainment of free particles. To
provide a clearer picture of the various flows transporting free
solids, they are graphically depicted in Figure 5.2.
( )∂ ∂
s c
k s in s c
in out s c
[36]
The set of population balance equations, along with the
corresponding expressions for the transport terms QinCfin and
QoutCf, for free solids of size class s and floatability class c in
the collection zone are provided next.
⇒ In the aeration zone:
k
ave z
, , , ,= ++ + + +1 1 1 1 [38]
( )( )Q Cf AUps Cf Qg Qt AUgs Cf QtCfout s c s c z
s c z
⇒ in each of the zones in the lower collection region:
dCf
k
190
( )( )Q Cf AUgs Cf AUps Cf Qg Qt AUgs Cfin s c in
ave z
, , , , ,= + + − + −+ + + + − − −1 1 1 1 1 1 11 ε [41]
( )( )Q Cf AUgs Cf AUps Cf Qg Qt AUgs Cfout s c ave z
s c z
s c z
s c z
Vz k Cf
out s c
k
( )1 β β [43]
( )( )Q Cf AUgs Cf AUps Cf Qg Qt AUgs Cfin s c in
ave z
, , , , ,= + + − + −+ + + + − − −1 1 1 1 1 1 11 ε [44]
( )( )Q Cf AUgs Cf AUps Cf Qg Qb AUgs Cfout s c ave z
s c z
s c z
s c z
⇒ in each of the zones in the upper collection region:
dCf
k
−∑ 1 β β [46]
( )( )Q Cf AUgs Cf AUps Cf Qg Qb AUgs Cfin s c in
ave z
, , , , ,= + + − + −+ + + + − − −1 1 1 1 1 1 11 ε [47]
( )( )Q Cf AUgs Cf AUps Cf Qg Qb AUgs Cfout s c ave z
s c z
s c z
s c z
s c z
, , , , ,= + + − + −1 ε [48]
In order to estimate the fractional bubble surface coverage, it is
assumed that each
particle of size Dps and density ρc occupies an area equal to π
4
2Dps on the bubble
surface. The bubble surface area is equal to πDbk 2, the total
number of bubbles of size
class k per unit volume is 6
3
k
kD b , and the mass of particles in classes s and c attached
to
bubbles of size k per unit volume is Cas,c,k. The following
expression then provides the fractional surface coverage of bubbles
in size class k at each time unit:
β ρ εk
191
In the literature, Dobby and Finch (1986) and Luttrell and Yoon
(1991) used a maximum
surface coverage (βmax ) equal to 0.8 in their simulations. Others
have suggested that
βmax should be 0.5 since the particles seem to slide and fill only
the bottom half of the
bubble. The model being introduced, however, allows that βmax be
set to any value between 0 and 1.
Perfectly Mixed Zone Vz
Drainage: UgsACf z+1
(Qg-Qsl+UgsA(1- ))Cfεz-1 z-1
(Qg-Qsl+UgsA(1- ))Cfεz z
Figure 5.2: Flows of Free Particles Around a Perfectly Mixed Zone
in the Collection Region
192
• Attached Solids Equations
The solids attached to air bubbles are classified not only
according to the particle size and particle floatability, but also
based on the size of the accompanying bubble. This classification
is necessary because the rate of attachment is dependant on bubble
size. The general population conservation equation is provided
below.
( )∂ ψ ∂
s c k s c k s c k
s in s c kin out s c kout
, , , , , , , , , ,+ − = −D
1 [50]
The relationship between the number concentration and the mass
concentration of attached particles is given by the following
equation:
C a D p as c k c s s c k, , , ,= π
ρ ψ 6
3 [51]
The flows of attached solids leaving and entering each of of
perfectly-mixed tanks are illustraded in Figure 5.3. The mass of
solids per unit volume which is entering the zone with volume Vs
per unit time is found to be:
Q Ca Qg Qsl AUgs Ca AUgs Cain s c k in z
ave z
k z
k z
+− − − − − −∑1 1 1 1 1 1ε , [52]
while the mass of solids per unit volume which leaves the zone z
per unit time is:
Q Ca Qg Qsl AUgs Ca AUgs Caout s c k z
ave z
k z
k z
+∑ε [53]
The slurry flowrate Qsl equals the tailings rate in the region
below the feed port and it is equal to the bias flowrate above it.
The terms Qslz and Qslz-1 are evaluated according to such
definition.
The rate of collection for particles of size class s and
floatability class c attaching to bubbles of size class k is given
by:
A k fs c k s c k s c k
, , , , , max
β β1 [54]
For a system with Nb bubble size classes, Np particle size classes
and Nc different floatabilities, substitution of Equations [52],
[53] and [54] in Equation [50] yields a set of N N Nb p c x x
ordinary differential equations.
193
U g sA C az-1
U g sA C az(Q g -Q sl-U gsA )C aεz z
(Q g -Q sl-U g sA )C aεz-1 z-1
Figure 5.3: Flows of Attached Particles Around a Perfectly Mixed
Zone in the Collection Region
These equations represent the dynamic changes in the concentration
of attached solid particles in each of the collection zone tanks.
The general form of the dynamic equations for each of the
collection region tanks are:
⇒ In the aeration zone:
out s c k
, , , max
Q Cain s c k in , , = 0 [56]
Q Ca Qg Qt AUgs Ca AUgs Caout s c k ave z
k z
k z
⇒ in each of the zones in the lower collection region:
dCa
out s c k
, , , max
−1 β β [58]
Q Ca Qg Qt AUgs Ca AUgs Cain s c k in
ave z
k z
k z
+− − − − −∑1 1 1 1 1ε [59]
Q Ca Qg Qt AUgs Ca AUgs Caout s c k ave z
k z
k z
out s c k
, , , max
−1 β β [61]
Q Ca Qg Qt AUgs Ca AUgs Cain s c k in
ave z
k z
k z
+− − − − −∑1 1 1 1 1ε [62]
Q Ca Qg Qb AUgs Ca AUgs Caout s c k ave z
k z
k z
⇒ in each of the zones in the upper collection region:
dCa
out s c k
, , , max
195
Q Ca Qg Qb AUgs Ca AUgs Cain s c k in
ave z
k z
k z
+− − − − −∑1 1 1 1 1ε [65]
Q Ca Qg Qb AUgs Ca AUgs Caout s c k ave z
k z
k z
5.3.3 Stabilized Froth Modeling
From the general form of the microscopic population balance model,
with the assumption of plug-flow movement along the froth height,
the dynamic equation for the volumetric fraction of kth-class
bubbles in the stabilized froth is:
( )∂ε ∂
∂ ε ∂
z D A+ + − = 0 [67]
where the subscript k refers to the bubble size class, v is the
average bubble rise velocity, and the appearance and disappearance
terms (Dk and Ak) are defined according to the following
equations:
( )D k j t Db
Db j
Db Db
ε , , , [69]
The parameter λ in Equations [68] and [69] is the coalescence
efficiency rate that corresponds to the pair of interacting
bubbles.
Since the average bubble rise velocity is given by Equation [7],
the space derivative can be expanded in the following manner:
( )d v
dz v
∑ ∑ [70]
Substituting in the general equation, the changes in air fraction
with time and position along the froth are represented by:
196
d
dt
∑ 2 0 [71]
Immediately above the interface, the air fraction at each time
interval is provided by the other possible solution to the
nonlinear equation
( ) ( )Vg Vl Ugs1 1 0− + − − =ε ε ε ε [72]
for the same conditions existing right below the interface, that
is, same average bubble size and phase velocities.
The number of solids carried by each bubble across the interface is
considered to be the same as that at the highest section of the
collection zone. Therefore, the concentration of attached solids at
the interface zone is given by the following relationship:
Ca Cas c k z
s c k z k
z
ε ε [73]
( )∂ ∂ ∂
Ca
t
vCa
z s c k s c k, , , ,+ − +A D = 0s,c,k s,c,k [74]
The attachment term, As,c,k, was assumed to be zero in the froth
regions, while the detachment term, Ds,c,k, is determined from a
calculation of the reduction in available surface area for each
bubble size class due to coalescence.
When analyzing the circumstances in which particles detach from the
bubble surfaces, two different situations were considered. First,
it was assumed that when a bubble of size class k coalesces with
another bubble, the particles that were attached to it are
rearranged on the newly created larger bubble. If the particles on
the disappearing bubbles cannot all be accomodated on the new
bubble surface, the excess particles become detached. Under this
assumption, the net loss in utilizable surface area per unit time
is given by:
197
∑ , , maxβ β [75]
If Nb is the number of bubble size classes, k takes integer values
from 1 to Nb-1, and j ranges from 1 to Nb-k. In the previous
equation, Dk,j and Ak,j stand for the volume of bubbles in size
class k that disappears and appears due to coalescence with bubbles
in size class j. The function f is defined so that
f x x
x0 [76]
If evaluation of Equation [75] yields zero, no detachment takes
place (all particles have found space where to reattach).
Therefore,
D = Ds,c,k k = 0 [77]
Otherwise, the rate of particles that becomes detached from bubble
size class k per unit region volume is calculated as follows:
a) Starting with the least hydrophobic species, if the excess
surface area Sk is less than the total area occupied by the solid
species of size s and composition c on the k-th class bubbles prior
to coalescence, the mass concentration of particles of size s and
composition c that becomes detached from bubbles of class k per
unit time is:
Ds,c,k = 4S Dpk c sρ , [78] and Sk reaches zero. b) if Sk is
greater than the area that was covered by the solid species
under
consideration, all particles of that type and composition become
detached. The mass concentration of particles of size s and
composition c that becomes detached from bubble size class k per
unit time is thus given by:
Ds,c,k = 24D Dp
k
S S D
. [80]
198
d) The steps a)-c) are repeated with all particles sizes (from the
larger size class to the smallest one) of the most hydrophilic
particles. If Sk is still greater than zero, the whole process
continues with the next least hydrophobic species.
e) Finally, all the detachment rates corresponding to each bubble
size class are added to
determine the overall mass rate of particles of size s and
composition c that go from the attached phase to the free phase in
a unit volume:
D Ds,c s,c,k= ∑ k
[81]
The other picture of particle detachment in the froth is based on
the premise that
when two bubbles coalesce, the oscillations caused all particles
attached to their surfaces to become free. Without consideration of
rearrangement on the surface of other bubbles, the equations are
the following:
Ds,c,k = 24D Dp
k
k
Nb
s,c = −
[83]
After developing each of the terms in Equation [74] and using, for
example, Equations [82] to replace the detachment rate term, the
resulting model equation for an attached solid species is:
∂ ∂ ε
k
24 =0 [84]
The general mass balance equation for the free mineral particle in
each solid species is:
( )∂ ∂ ∂ Cf
t
+ + −A D = 0s,c s,c [85]
The parameter Uf represents the interstitial free particle
velocity, given by the following relationship:
U Qsl
199
where Ups is the particle hindered settling velocity calculated
using the expression provided by Masliyah (1979) for multispecies
systems.
The first term on the right-hand side of Equation [86] represents
the net liquid flow resulting from the algebraic sum of two types
of flow, the entrainment of slurry by the rising bubbles and the
drainage of slurry through the films between bubbles. The drainage
velocity, Uds, in a countercurrent system is given by:
Uds Ugs= − , [87]
while particle entraiment can be assumed to be directly
proportional to water entrainment. Entrained water is considered to
be transported at the average bubble rise velocity. Therefore, for
a countercurrent process, the net interstitial slurry flowing
downwards is given by the difference between the average hindered
rise velocity of the bubbles with respect to the slurry, Ugs, and
the average rise velocity with respect to a stationary
reference.
Qsl Ugs
ε ε [88]
Substituting the detachment term with Equation [83], the equation
which represents the dynamic behavior of the free solid species is
finally determined to be:
∂ ∂ ε ε
β ρCf
D Dp
k s c k c s
k
Nb ,
Model Equations:
In summary, the model equations for the interface and stabilized
froth zones are:
• Interface:
( ) ( )Vg Vl Ugs1 1 0− + − − =ε ε ε ε [90]
200
z
s c s c
z s c
, , , , , , ,+
− −
− + −
k
D Dp
k s c k c s
k
Nb ,
5.3.4 Wash-Water Zone and Draining Froth Equations
At the wash water addition point, another transition occurs. The
wash water flowrate is split into two parts: the bias water, which
flows down the stabilized froth, and the concentrate water, which
leaves with the overflow material. This water partition is
influenced by a series of factors such as the gas rate and bubble
size, which have a direct effect on water entrainment. However, due
to the complexity of these interactions and the lack of sufficient
knowledge, it is not yet possible to mathematically predict which
fraction of the wash water flow will flow downwards as bias flow,
unless the froth air fraction is known beforehand.
The bias flow used in all the model equations is an estimated
value. When the model reaches steady-state, the concentrate water
is calculated using the following relationship:
( ) Q
= −1 ε
ε [98]
and compared to the difference between the wash water rate and the
assumed bias flowrate. If the concentrate water obtained from
equation [98] is higher than the one calculated based on the
estimated bias rate, a new bias flowrate is determined according to
the following expression:
( )( ) Q Q
cw w b = −
− − 2
[99]
The model is then solved again with the new bias flowrate.
Convergence is reached when
( )Q Q Qcw w b− − ≤ tolerance. Alternatively, if the bias rate is
considered to be known, a new interface position can be calculated
until the sum of the concentrate water (Equation [98]) and the bias
rate are approximately equal to the wash water added to the system.
If the froth is too wet (concentrate water too high), the interface
position would be lowered. The whole process is illustrated more
clearly in Figure 5.4, which shows the sequence of steps followed
for the solution of the model equations.
202
Last dt?
Calculate Qp
Qp-(Qw-Qb) <= tol?
No
Yes
Yes
No
Figure 5.4: Flowchart of the Procedure Followed for Solving the
Model Equations
203
The equations for the attached and free solid species are derived
in the same manner as those used to model the stabilized
froth.
a) Air Phase:
k
D Dp
k s c k c s
k
Nb ,
ε ε [103]
These equations apply to both the wash water zone and the draining
froth region, which extends from the wash water zone up to the
overflow lip. The coalescence rate parameters in the wash water
zone and the draining froth zones can have different values in the
model.
5.3.5 Recovery Calculations
The overall recovery for each solid species j is given by:
R RC RF
RC RC RFj
= − +1
[104]
The equation above is derived based on the block diagram in Figure
5.5.
204
( ) ( )
, [105]
where Qfeed stands for the feed flowrate, and Cfeed j is the mass
concentration of particles of class j in a unit feed volume. The
term Yj(t) is the mass rate of particles in class j that cross from
the collection region to the froth by either flotation or
entrainment, and Yrj(t) is the mass rate of particles in class j
that return to the collection region.
The froth recoveries RFj for each solid composition class were
obtained from the following relationship:
( ) ( )RF t QpCp
j
j
= , [106]
where Qp is the concentrate flowrate and Cpj is the concentration
of species j in the concentrate.
The mass rate of attached jth-class particles leaving a given zone
is given by:
( ) ( ) ( )Yf t Ca t Qbu tj j j= * , [107]
where Caj is the concentration of attached solids belonging to
class j, and Qbuj is the average bubble rate provided by:
( )[ ]Qbu Qg Qsl Ugsj ave= − + −1 ε [108]
The mass rate of entrained solids is then calculated using the
following relationship:
( ) ( )Ye t Cf t Quj j= * [109]
In Equation [109], Cfj is the concentration of free solids in class
j, and Qu is the flowrate moving upward from the collection region
to the zone above the interface. The return rate is then given
by:
( ) ( )Yr t Cf t Qdj j z =
+1 * [110]
The term Qd stands for the downward flowrate, which is determined
by drainage rate and settling. (See Figure 5.2).
205
The overall recovery of species j in the concentrate is also given
by the relationship:
( )R t QpCp
206
5.4 Simulations
A number of simulations were carried out to determine the type of
predictions provided by the model and how they compare to
established knowledge about column flotation behavior. The input
parameters which have to be provided in order to solve the dynamic
equations include:
♦ number of zones in the upper collection region and in the lower
collection region; ♦ column dimensions, such as, cross sectional
area, position of the wash water
distributor from the top, position of the feed port from the top; ♦
position of the pulp-froth interface from the column top or,
alternatively, the known
tailings rate and an estimate of the interface position. In the
latter case, the iteration would proceed by adjusting the interface
position until the flow balance is satisfied.
♦ discrete number size distribution and average size of the bubbles
produced; ♦ combined discrete mass size distribution and mass
floatability distribution of the feed
particles; ♦ feed percent solids; ♦ particle sizes (corresponding
to the discrete size distribution); ♦ particle densities
(corresponding to the various composition classes); ♦ superficial
gas velocity; ♦ superficial feed slurry velocity; ♦ estimated
superficial tailings velocity; ♦ probabilities of attachment for
all the particle species; ♦ coalescence efficiency rate parameters
for the stabilized froth; ♦ coalescence efficiency rate parameters
for the wash water transition region; ♦ coalescence efficiency rate
parameters for the draining froth; ♦ maximum bubble surface
coverage.
Almost all of these parameters are set at the start of operation,
can be measured, or are found in the literature. The exception is
the newly introduced coalescence efficiency rate parameter, which
has been assigned values so that the air fraction solution at the
top of the froth remain in the range between zero and one. In
addition, the coalescence parameters were given higher values in
the draining froth than in the stabilized froth.
5.4.1 Simulation No. 1:
This simulation was performed to determine the type of profiles
predicted by the model, examine the response time constants, and
evaluate these results on the basis of a priori knowledge about
column flotation behavior. The parameter values were selected based
on typical operating conditions. The probabilities of attachment
were assigned arbitrarily, but with the condition that the
resulting flotation rate constant were not unrealistic.
207
Operating Conditions and Parameters:
Number of Bubble Size Classes: Nb=4; Number of Particle Size
Classes: Np=2; Number of Particle Composition Classes: Nc=3; Column
Diameter: Cd=5 cm; Cross Sectional Area: A=π*Cd*Cd/4; Number of
Perfectly Mixed Zones in the Lower Collection Region: nZlp=6;
Number of Perfectly Mixed Zones in the Upper Collection Region:
nZup=3; Number of Height Intervals in the Stabilized Froth: nZsf=6;
Number of Height Intervals in the Draining Froth: nZdf=2; Column
Length above Gas Entry Level: L=200 cm; Collection Region Height:
Lp=150 cm; Total Froth Height: Lf=L-Lp; Distance between Wash Water
Addition Port and Overflow Lip: Lww=10 cm; Distance between Gas
Entry Port and Column Bottom: Lg=10 cm; Distance from the Overflow
Lip to the Feed Port: FP=70 cm; Height of the Transition Regions:
Lt=2 cm;
Superficial Gas Velocity: Vg=1.0 cm/sec; Superficial Wash Water
Velocity: Vw=0.3 cm/sec; Superficial Feed Slurry Velocity:
Vfeed=0.4 cm/sec; Initial Estimated Tailings Flowrate:
Qt=1.2418*Vfeed*A; Bias Flowrate: Qb=Qt-Qfeed; Product Slurry
Flowrate: Qp=Qw-Qb; Number-Average Bubble Diameter at Gas Inlet:
Dbave=Cg*Vg^(1/4); Constant relating Bubble Size and Gas Flowrate:
Cg=0.0947; Number Size Distribution of Generated Bubbles:
fnd=[0.5471;0.4529];
Bubble Size Classes:Db Db
Volume Size Distribution of Bubbles: f k f k Db
f i Db vd
3
3
Particle Size Classes: Dp=[0.0056 0.008]'; Particle Species
Densities: SGp=[1.2 2.0 3.0]; Initial Bubble Densities:
SGb=zeros(Nb,1); Feed Percent Solids: Fs=20%; Feed Slurry Specific
Gravity: Fsg=1.8; Total Feed Solid Concentration: TCf=Fs*Fsg/100;
Feed Solids Size Distribution fsd=[0.7;0.3]; Feed Solids
Composition Distribution: fcd=[0.6;0.2;0.2]; Concentration of Each
Solids Species in the Feed: C TCf fsd fcdFeed = * *
Probabilities of Bubble-Particle Collision:
Probabilities of Attachment:
0 9 0 5 4 0
0 9 0 5 4 0
0 8 0 4 8 0
0 8 0 4 8 0
0 7 0 4 2 0
0 7 0 4 2 0
0 6 0 3 6 0
. .
. .
. .
. .
. .
. .
. .
. .
Probabilities of Particle Collection: P=Pc*Pa;
Selection Function for the Detachment of Particle Species in the
Froth:
S =
( )λi j
i j
( )λi j i jKw Db Db, exp= − +
Maximum Bubble Surface Coverage: βmax=0.5;
Total Simulation Time: Tf=960 sec; Time Steps: dt=0.08 sec;
Air Fraction Initial Values: in collection region, εk=0.10/Nb; in
and above the interface, εk=0.70/Nb;
209
Free Solids Initial Values: Cfs,c = 0 in all column regions;
Attached Solids Initial Values: Cas,c,k = 0 in all column regions;
The detachment term is calculated based on the following expression
for the net loss of bubble surface area:
S f D
∑ β β
The final product velocity, after 4 iterations was 0.138 cm/sec,
while the estimated bias velocity was 0.162 cm/sec. Figure 5.6
shows the changes of air fraction with time in a few zones along
the column length during the last iteration, while Figure 5.7
illustrates the air fraction profile along the full column length
at t=900 secs (15 mins).
0 5 10 15 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1 - In a Zone in Collection Region 2 - At Interface
3 - In Middle of Stabilized Froth 4 - Top Column Zone
4
3
2
1
Figure 5.6: Dynamic Changes in Air Fraction in Several Zones Along
the Column (Simulation No. 1)
210
50
100
150
200
250
)
Figure 5.7: Final Air Fraction Profile Along Column Height
(Simulation No. 1)
The predicted attached solids concentrations in several column
zones, at each time step, are provided in Figure 5.8. The total
attached solids profile, provided in Figure 5.9, shows that the
concentration of attached solids increases steadily along the
collection region for the conditions of this simulation. The jump
at the interface corresponds to the sudden increase in the number
of bubbles. Throughout the froth, the total concentration of
attached species decreases with height.
The dynamic behavior of the free solid species in this case is more
sluggish than the responses of the air phase and attached solid
species (Figure 5.10). The slow reaction is probably due to the
transport of the small hydrophilic species by settling and
drainage. The concentration of free solids is highest at the feed
zone (Figure 5.11); it decreases down the collection region mainly
due to flotation, while in the froth, it is significantly reduced
as a result of drainage and particle settling.
According to these results, the overall solids concentration
decreases along the froth, while it increases with height in the
collection region (Figure 5.12). A similar behavior was reported by
Ross and van Deventer (1988), and by Falutsu and Dobby
211
(1992), who measured the concentration of attached and free solid
species in two flotation columns. They suggested that coalescence
and drainage were responsible for such response. The experimental
froth profiles are shown in Figure 5.13 along with the section of
the calculated profile depicted in Figure 5.12 that corresponds the
froth. The trends followed by the profiles shown match fairly well.
The reason why they do not overlap is that no attempt was made to
adjust the model parameters so that the calculated profile fits the
literature data. The purpose of comparing the three curves is
mainly to establish the capability of the model for representing
the internal structure of an operating flotation column.
As to the collection region, the data obtained by Dobby and Finch,
(1986) in a full- scale column (shown in Figure 5.14) indicates
that the total solid concentration increases with height in the
collection region, which is similar to the behavior indicated by
the simulation results.
0 5 10 15 0
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
1 - In a Zone in Collection Region 2 - At Interface
3 - In Middle of Stabilized Froth 4 - Top Column Zone
4 3
2
1
Figure 5.8: Dynamic Changes in the Concentration of Attached Solids
in Several Zones Along the Column (Simulation No. 1)
212
50
100
150
200
250
D is
ta nc
e fro
m B
ot to
m o
)
Figure 5.9: Final Mass Concentration of Attached Solids Along
Column Height (Simulation No. 1)
213
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
2 - At Interface
3 - In Middle of Stabilized Froth 4 - Top Column Zone
4
1
2
3
Figure 5.10: Dynamic Changes in the Concentration of Free Solids in
Several Zones Along the Column (Simulation No. 1)
214
50
100
150
200
250
D is
ta nc
e fro
m B
ot to
m o
)
Figure 5.11: Final Mass Concentration of Free Solids Along Column
Height (Simulation No. 1)
215
50
100
150
200
250
)
Figure 5.12: Total Concentration of Solids Along Column Height at
Steady-State (Simulation No. 1)
216
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
_ From Simulation No.1
Figure 5.13: Total Solid Concentration Profiles Along the Froth of
a Flotation Column (Data After Ross and vanDeventer (1988), and
Falutsu and Dobby (1992))
217
200
300
400
500
600
700
800
900
1000
ce (
cm )
Figure 5.14: Total Solid Concentration Profiles Along the
Collection Region of a Flotation Column, after Dobby and Finch
(1986).
The mass rates of feed particles, according to their size class and
floatability class, were:
Fast-Floating Slower-Floating Nonfloating Dp1 1.1875 0.3958 0.3958
Dp2 0.5089 0.1696 0.1696
where Dp1 and Dp2 represent the two particle size classes. The
upper row corresponds to the smallest particle size class (Dp1),
while the leftmost column was assigned to the most floatable
class.
At steady-state, the distribution of the rate of material floated
at t = 15 mins was:
Fast-Floating Slower-Floating Nonfloating Dp1 1.1401 0.3626 0 Dp2
0.5002 0.1628 0
218
The distribution of solids carried to the froth by entrainment per
unit time was:
Fast-Floating Slower-Floating Nonfloating Dp1 2.3879 0.9800 2.9824
Dp2 0.7184 0.2916 0.8966
while the distribution of solids returned to the pulp from the
froth per unit time was:
Fast-Floating Slower-Floating Nonfloating Dp1 2.3736 0.9762 2.9752
Dp2 0.7145 0.2909 0.8958
Consequently, the net rates of solids being carried with to the
froth by entrainment were:
Fast-Floating Slower-Floating Nonfloating Dp1 0.0143 0.0038 0.0072
Dp2 0.0039 0.0007 0.0008
Finally, the rates of solids in the tailings stream, in terms of
the different size and composition classes were:
Fast-Floating Slower-Floating Nonfloating Dp1 0.0332 0.0294 0.3868
Dp2 0.0049 0.0062 0.1687
The final recoveries in the concentrate for each of the three
different types of materials were:
Fast-Floating Slower-Floating Nonfloating 0.9772 0.9360
0.0138
The feed rates of each solid composition class during the
simulation run are illustrated in Figure 5.15, while Figure 5.16
and 5.17 show the net rates of each composition class that cross
the interface by flotation (Figure 5.16) and by entrainment (Figure
5.17). The proportion of material that is entrained into the froth,
according to the simulation results, is very small for all
materials. In a flotation column operating with positive bias, a
negligible amount of entrained material is expected due to the
action of the bias water. A better look of Figure 5.17 can be
appreciated in Figure 5.19, at a different scale. The mass rates in
the tailings can be seen in Figure 5.19.
219
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Slower-Floating Material Nonfloating Material
Figure 5.15: Rates of Each Composition Class in the Feed Stream at
Each Time Step (Simulation No. 1)
220
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Fast-Floating Material
Slower-Floating Material
Nonfloating Material
Figure 5.16: Rates of Each Composition Class Entering the Froth by
Flotation at Each Time Step (Simulation No. 1)
221
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
ec )
Figure 5.17: Rates of Each Composition Class Entering the Froth by
Entrainment at Each Time Step (Simulation No. 1)
222
0.02
0.04
0.06
0.08
0.1
0.12
1
2
3
Figure 5.18: Rates of Each Composition Class Entering the Froth by
Entrainment at Each Time Step - Bigger Scale (Simulation No.
1)
223
0.1
0.2
0.3
0.4
0.5
0.6
1 - Fast-Floating Material
2 - Slower-Floating Material
3 - Nonfloating Material
Figure 5.19: Rates of Each Composition Class Leaving with the
Tailings Stream at Each Time Step (Simulation No. 1)
224
The predicted dynamic changes in the solid recoveries for each
composition class are plotted in Figure 5.20, while the
corresponding recoveries in the tailings are shown in Figure 5.21.
The calculated recoveries for the floating species were very high
(over 90%), but this was just a function of the flotation rate
values used in the simulation. The model predicted a very good
rejection of the nonfloatable species. The change in the
concentrate grade with time is provided in Figure 5.22. The final
fractional content of each composition species in the concentrate
were:
Fast-Floating Slower-Floating Nonfloating 0.7552 0.2412
0.0036
0 5 10 15 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3 - Nonfloating Material
Figure 5.20: Fractional Recoveries in the Concentrate of Each
Composition Class at Each Time Step (Simulation No. 1)
225
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3
Figure 5.21: Fractional Recoveries in the Tailings of Each
Composition Class at Each Time Step (Simulation No. 1)
226
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
on te
nt in
C on
ce nt
ra te
Fast-Floating Material
Nonfloating Material
Slower-Floating Material
Figure 5.22: Fractional Content of Each Composition Class in the
Concentrate at Each Time Step (Simulation No. 1)
In order to determine how well the response times predicted by the
model approximate the times observed during column operation, two
experiments were carried out at two different feed rates (retention
times). Samples were collected during timed intervals from the
moment the feed material first entered the column until the system
achieved steady state. In the first experiment, the feed
superficial velocity was around 0.4 cm/sec, as in the past
simulation. Figure 5.23 presents the measured and simulated solid
rates in the concentrate, both normalized by dividing over their
maximum value since the purpose is to compare the times needed for
reaching steady state. A comparison of the changes in solid rate
with time in the tailings stream is provided in Figure 5.24. Both
plots indicate that the model can provide adequate predictions
about the process time constant. Also, the model is capable of
representing the fact that the tailings response is slower than the
concentrate dynamic reaction. As mentioned before, this is
attributed to the settling and drainage rates.
227
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
1
Figure 5.23: Predicted and Measured Dynamic Responses in
Concentrate for a Feed Velocity of 0.4 cm/sec
228
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
1
Figure 5.24: Predicted and Measured Dynamic Responses in Tailings
for a Feed Velocity of 0.4 cm/sec
For the second test, the feed superficial velocity was increased to
1.0 cm/sec. The model equations were solved for this new feed rate,
and the normalized rates for the concentrate and tailings flows are
contrasted in Figures 5.25 and 5.26, respectively. Since the
retention time was reduced when the feed rate was increased, the
time required to reach steady state is significantly less than in
the previous case. Once again, the model results were found to be
quite reasonable, in terms of the response speeds.
229
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 1
Figure 5.25: Predicted and Measured Dynamic Responses in
Concentrate for a Feed Velocity of 1.0 cm/sec
230
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2
1
Figure 5.26: Predicted and Measured Dynamic Responses in Tailings
for a Feed Velocity of 1.0 cm/sec
231
5.4.2 Simulation 2:
The purpose of this simulation was to determine the type of
response predicted by the model after an increase in aeration rate
while keeping the other operating conditions constant (as in
Simulation No.1). From experience, it is known that a higher air
rate would probably produce a wetter froth and would reduce the
positive bias. The air fraction in the collection region, as well
as the flotation rate constant, should become higher. The amount of
entrained solid particles is also expected to increase.
New Operating Conditions and Parameters:
Vg=2.0 cm/sec; Number-Average Bubble Diameter at Gas Inlet:
Dbave=Cg*Vg^(1/4); The rate constants k also increase, since they
are directly related to Vg.
The coalescence efficiency rate parameters had to be increased;
otherwise, the predicted froth liquid content was unreasonably
high. This event appears to indicate that as the predicted bias
rate moves into negative direction, the model requires that the
froth stability decreases. It is likely that the increase in gas
rate affects other parameters that determine the magnitude of the
coalescence rate parameters, such as frother concentration and
bubble loading.
Proportionality Constant: Kw=5e-4; Proportionality Constant:
Kw=1e-1; Proportionality Constant: Kd=8e-2;
The predicted air fraction at the top of the froth was e=0.85, and
the calculated product velocity was 0.352 cm/sec. The predicted
bias, obtained as Vw-Vp, was -0.052 cm/sec, while the bias velocity
used at the start of the last iteration was Vb=-0.055 cm/sec. The
increase in gas velocity in the equations caused, therefore, a
large decrease in bias rate so that the predicted column operation
did not have the countercurrent washing action.
Figure 5.27 illustrates the air fraction profile along the full
column length at t=840 secs (14 mins). The simulator keeps track of
the number of bubbles of each size class in each zone and,
therefore, the air content corresponding to each bubble class. The
final profiles of the volumetric fraction of bubbles in each size
class are compared in Figure 5.28. This plot shows how coalescence
in the froth caused the air fraction of the larger bubble size
class to increase at the expense of the smaller bubbles. The degree
of the increase in overall air fraction is determined by the values
of the coalescence rate parameters.
The calculated concentrations of attached solids along several
regions of the column, at each time step, are represented in Figure
5.29. The total attached solids profile, provided in Figure 5.30,
shows that the total concentration of attached species in the froth
decreased slightly with height, although the net detachment in the
froth due to
232
coalescence was zero. This effect is due to the fact that the
larger bubbles are considered to be less loaded than the smaller
ones, but they rise faster.
0 0.2 0.4 0.6 0.8 1 0
50
100
150
200
250
)
Figure 5.27: Steady-State Air Fraction Profile at t=14 mins
(Simulation No.2)
233
50
100
150
200
250
e1 e2e3 e4
Figure 5.28: Steady-State Profiles of All Air Fraction Component at
t=14 mins (e1:Smallest Size Class, e4: Largest Size Class)
(Simulation No.2)
234
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
Time (min)
M as
s C
on ce
nt ra
tio n
A tta
ch e
d S
ol id
s (g
/m l)
1 - In a Zone in Collection Region 2 - At Interface 3 - In Middle
of Stabilized Froth 4 - In Top Column Zone
1
2
3
4
Figure 5.29: Dynamic Responses of the Total Attached Solid
Concentration in Various Column Zones (Simulation No.2)
235
50
100
150
200
250
D is
ta nc
e fro
m B
ot to
m o
)
Figure 5.30: Steady-State Profile of the Concentration of Attached
Solids at t=14 mins (Simulation No.2)
236
As to the solids in the slurry phase, some of the dynamic responses
are shown in Figure 5.31. The predicted amount of free solids in
the froth is higher than in the previous simulation, as can be seen
in the profile in Figure 5.32. This can be attributed to the
negative direction of the bias flow, which results in higher
entrainment flows. Given that both the concentrations of solids in
the slurry and on the bubbles decrease with froth height, the total
solid concentration profile also gets lower (seen in Figure 5.33).
This reduction, however, is of a slightly lesser magnitude than in
the profile obtained in the previous simulation. This is explained
by the fact that the drainage effect is smaller and there is enough
surface area for the rearrangement of particles released through
bubble coalescence.
0 2 4 6 8 10 12 14 0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
0.045
0.05
2 - At Interface 3 - In Middle of Stabilized Froth
4 - In Top Column Zone
4
3
2
1
Figure 5.31: Dynamic Responses of the Total Free Solid
Concentration in Various Column Zones (Simulation No.2)
237
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0
50
100
150
200
250
D is
ta nc
e fro
m B
ot to
m o
)
Figure 5.32: Steady-State Profile of the Concentration of Free
Solids at t=14 mins (Simulation No.2)
238
50
100
150
200
250
)
Figure 5.33: Steady-State Profile of the Total Solid Concentration
at t=14 mins (Simulation No.2)
The distribution of the rate of material floated at t = 14 mins
was:
Fast-Floating Slower-Floating Nonfloating Dp1 1.1417 0.3694 0 Dp2
0.4990 0.1644 0
while the rates of particles entrained and returned to the pulp
were:
Fast-Floating Slower-Floating Nonfloating Dp1 1.4889 0.6524 2.4969
Dp2 0.4102 0.1847 0.7900
Fast-Floating Slower-Floating Nonfloating Dp1 1.4578 0.6412 2.4629
Dp2 0.4020 0.1822 0.7833
239
Fast-Floating Slower-Floating Nonfloating Dp1 0.0311 0.0112 0.0340
Dp2 0.0082 0.0025 0.0067
The distribution of solids lost in the tailings per unit time was
the following:
Fast-Floating Slower-Floating Nonfloating Dp1 0.0146 0.0153 0.3610
Dp2 0.0017 0.0027 0.1629
The fractional bubble coverage, for each size class, at the top of
the froth at t= 14 mins was:
Blk Db1 0.3729 Db2 0.1715 Db3 0.0003 Db4 0.0002
The predicted mass rates in each composition class which are
collected by bubbles in the collection region, at each time step,
are represented in Figure 5.34. The entrainment rates are plotted
in Figure 5.35, while the tailings rates are provided in Figure
5.36. It is again observed that the tailings stream requires a
longer time to stabilize, as previously reported.
Variations with time in the recoveries in the concentrate and
tailings are described in Figure 5.37 and 5.38 respectively, and
the changes in the concentrate fractional content of each
composition class along the simulation run are illustrated in
Figure 5.39. As expected, the calculated amount of entrained
material increased with respect to the conditions in Simulation
No.1 due to the negative bias flow. Consequently, more nonfloating
solids are recovered in the concentrate, as indicated by a
comparison between Figures 5.37 and 5.20. In addition, the fraction
of the concentrate solids belonging to the nonfloating species
increased (see Figure 5.39) with respect to the value in Figure
5.22.
240
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Slower-Floating Species
Nonfloating Species
Figure 5.34: Dynamic Changes in the Rate of Solids Carried by the
Bubbles from the Pulp to the Froth for Each Solid Species
(Simulation No.2)
241
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
3 21
Figure 5.35: Dynamic Changes in the Net Rate of Solids Transported
by the Slurry from the Pulp to the Froth for Each Solid Species
(Simulation No.2)
242
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
3
2 1
Figure 5.36: Dynamic Changes in Tailings Solid Rates for Each Solid
Species (Simulation No.2)
243
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
2 - Slower-Floating Species
3 - Nonfloating Species
Figure 5.37: Dynamic Changes in the Fractional Recovery in the
Concentrate for Each Solid Species (Simulation No.2)
244
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
3 - Nonfloating Species
3
Figure 5.38: Dynamic Changes in the Fractional Recovery in the
Tailings Stream for Each Solid Species (Simulation No.2)
245
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
on te
nt in
C on
ce nt
ra te
Fast-Floating Species
Slower-Floating Species
Nonfloating Species
Figure 5.39: Fractional Content of Each Composition Class in the
Concentrate at Each Time Step (Simulation No. 2)
246
5.4.3 Simulation 3:
In this simulation, particle detachment was assumed to be
unselective, that is, all particles on the surface of two
coalescing bubbles would become detached, regardless of their
characteristics. The remaining parameters were the ones used in
Simulation No.1. The objective was to compare the profiles obtained
with both detachment equations.
New Operating Conditions and Parameters:
Vg=1.0 cm/sec; Proportionality Constant: Kw=2e-4; Proportionality
Constant: Kw=4e-2; Proportionality Constant: Kd=3e-2; The
detachment term is calculated using the following expression:
detachs,c,k = 24D Dp
k
β ρ, ,
The final fractional coverage for each of the four bubble classes
(from smallest to largest) was:
Bubble Size Class
Db1 0.3323 Db2 0.2054 Db3 0.1080 Db4 0.0512
The distribution of particles in the feed was again the
following:
Fast-Floating Slower-Floating Nonfloating Dp1 1.1875 0.3958 0.3958
Dp2 0.5089 0.1696 0.1696
The final (steady-state) mass rate of material in each
size-floatability combination that entered the froth through
flotation was:
Fast-Floating Slower-Floating Nonfloating Dp1 1.4331 0.4418 0 Dp2
0.6535 0.2176 0
247
Likewise, the mass rates of particles entering the froth by
entrainment and returning to the pulp simultaneously were:
Fast-Floating Slower-Floating Nonfloating Dp1 4.7908 1.7829 2.3516
Dp2 1.7071 0.6636 0.7647
and
The net entrainment rates were therefore:
Fast-Floating Slower-Floating Nonfloating Dp1 -0.3132 -0.0998
0.0565 Dp2 -0.1560 -0.0613 0.0133
The corresponding steady-state mass rates in the tailings
were:
Fast-Floating Slower-Floating Nonfloating Dp1 0.0676 0.0539 0.3387
Dp2 0.0114 0.0134 0.1563
Adding the amounts floated, entrained and discarded in the tailings
per unit time yields the following:
Fast-Floating Slower-Floating Nonfloating Dp1 1.1875 0.3958 0.3952
Dp2 0.5089 0.1696 0.1696
which is approximately equal to the feed solid rates.
By ignoring particle reattachment after coalescence, the decrease
in the concentration of attached solids along the froth was very
large, as indicated by Figure 5.40. Correspondingly, the
concentration of free solids in the froth was very high (Figure
5.41). The total solid concentration in the froth, shown in Figure
5.42, was significantly higher than the concentration calculated in
Simulation No.1 (Figure 5.12). A probable cause for this occurrence
is that the settling rates for all species were reduced as the
volume concentration of free solids increased. Most of the floating
material that went from the attached to the free state remained in
the froth and left with the concentrate. In
248
addition, the movement of nonfloating species down the column was
slowed down as the slurry viscosity rose. Since it was assumed that
the particles detached from the bubble surfaces in the same
proportion they were initially on the bubbles (unselective
detachment), the fast-floating species constituted the major
fraction of detached particles. Under the assumptions of this
model, the large bubbles created in the froth would remain
unloaded. Given that, in practice, the bubbles at the top of the
froth are normally loaded with solids, reattachment should be taken
into consideration.
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0
50
100
150
200
250
D is
ta nc
e fro
m B
ot to
m o
)
Figure 5.40: Steady-State Profile of the Concentration of Attached
Solids at t=15 mins (Simulation No.3)
249
50
100
150
200
250
D is
ta nc
e fro
m B
ot to
m o
)
Figure 5.41: Steady-State Profile of the Concentration of Free
Solids at t=15 mins (Simulation No.3)
250
50
100
150
200
250
)
Figure 5.42: Steady-State Profile of the Total Solid Concentration
at t=15 mins (Simulation No.3)
5.5 Model Validation
In order to validate the model, the results of dynamic studies
performed previously (Cruz, 1994) with coal in a laboratory 2-inch
flotation column were compared to the model solutions under
equivalent conditions. Individual step changes in each of a series
of manipulated variables were carried out, while the dynamic
responses in product ash content and ash recovery were determined
by collecting samples of the concentrate and feed streams at timed
intervals. The experimental conditions, as well as the magnitude of
the step changes are specified in Table 5.1. During these
experiments, the tailings flowrate was set through a peristaltic
pump, while the interface position was allowed to change in
response to the variations in operating conditions. Likewise, a
constant tailings velocity was used in the simulations, while the
pulp level varied.
251
Table 5.1: Experimental Conditions During Study of Column Dynamic
Responses
Test No.
Frother Addition Rate Superficial Feed Velocity
1 1.3 ---> 1.4 cm/sec 0.007 µl/min 0.1 cm/sec 2 1.3 cm/sec 0.007
---> 0.01 µl/min 0.1 cm/sec 3 1.3 cm/sec 0.007 µl/min 0.1
---> 0.12 cm/sec
The values given above were used in the simulator equations, and
two discrete classes of feed material were defined: fully liberated
coal and liberated ash particles. The overall ash content in the
feed was around 8%. The model was first solved using the 'normal'
conditions in Table 5.1 (before the step changes). The
probabilities of attachment were then adjusted in order to achieve
an initial concentrate ash content similar to that in the sample
collected before each change. The changes in gas rate, frother
(estimated bubble size) and feed rate were then simulated
separately.
After the actual step change in aeration rate, the ash percentage
in the concentrate, as well as the ash recovery, increased.
Increases in pulp level and air fraction were also recorded.
However, due to the small magnitude of the change in the
steady-state response and the scattering of the data points, the
comparison between the empirical and simulated responses in Figures
5.43 and 5.44 is not very effective. In adddition, the nature of
the experimental transient response cannot be clearly appreciated.
Nevertheless, it seems that the model is capable of predicting the
correct type of steady-state reaction.
The response to the increase in frother addition rate showed a
small increase in ash percentage, as pictured in Figures 5.45. The
recovery curve is not provided due to the impossibility of drawing
a significant conclusion from the data available. The simulated
response appear to follow a path similar to the experimental
values, but, once again, the data is very noisy.
Finally, the step change in feed rate resulted in a reduction in
the ash content in the product, which agrees with the direction of
the simulated dynamic response (Figure 5.46). The errors involved
in the recovery calculation were very significant in comparison to
the magnitude of the change.
252
0 1 2 3 4 5 6 7 8 2
2.5
3
3.5
4
4.5
5
5.5
6
* - Experimental Values
_ - Model Solution
Figure 5.43: Predicted and Experimental Concentrate Ash Values
After an Increase in Aeration Rate
253
0 1 2 3 4 5 6 7 8 0.2
0.3
0.4
0.5
0.6
0.7
0.8
Figure 5.44: Predicted and Experimental Concentrate Ash Recoveries
After an Increase in Aeration Rate
254
0 1 2 3 4 5 6 7 8 2
3
4
5
6
7
8
* - Experimental Values
_ - Model Solution
Figure 5.45: Predicted and Experimental Concentrate Ash Values
After an Increase in Frother Rate (Smaller Average Bubble
Size)
255
0 1 2 3 4 5 6 7 8 2
3
4
5
6
7
8
Figure 5.46: Predicted and Experimental Concentrate Ash Content
After an Increase in Feed Rate
256
5.6 Summary
• A three-phase dynamic model of the column flotation process has
been developed and solved numerically. The model allows to work
with any number of discrete bubble size classes, particle size
classes and particle floatability types. Constraints are only
imposed by the speed of the simulation and the memory capacity of
the computer system. The number of regions in a flotation column
were identified as a collection region, a stabilized froth and a
conventional froth. The feed entry zone was defined as a transition
volume that divides the collection region into a lower section and
an upper one. The lower part of the collection region is
represented as a series of Nl stirred taks in series, while the
upper region is modelled using Nu stacked tanks. The stabilized
froth and the draining froth are, on the other hand, described as
plug-flow volumes. Other column sections defined as transition
zones are the aeration zone, the interface, and the wash-water
addition zone.
• The solids exist in one of two different states at any time:
either free in the slurry
phase, or attached to air bubbles. The attached solids rise at the
speed of the bubbles, and it is assumed that air and consequently,
attached solids, are not transported to the tailings flow. Only
upward flow of attached solids and gas bubbles is thus considered.
In the case of the free solids, slurry is carried upward by
entrainment, while free particles are transported down the column
through drainage and gravity. Both flow directions are represented
in the model equations.
• The drift-flux equation was utilized to describe the relationship
between the bubble rise
velocities and the gas and liquid rates. The model keeps track of
the air fraction components corresponding to each bubble size
class. Through application of the drift- flux equation at both
sides of the interface, the model can theoretically predict loss of
the interface due to high gas rate or very small bubble size.
• A distribution of flotation rate constants in the collection
region was calculated based
on the combined probabilities of collision and attachment for each
particle species, the bubble sizes, and the volume-based size
distribution of the bubbles introduced at the foot of the
column.
• An important feature in the dynamic model is that the change in
bubble surface
coverage is calculated at each time step so that a maximum bubble
loading is not exceeded. The degree of bubble loading, as well as
the slurry density and viscosity, are also incorporated into the
calculations of bubble rise velocity. Therefore, the effects of
those parameters on air fraction are taken into account during the
simulations.
• A mechanistic representation of bubble coalescence has been
utilized to describe the increase in air fraction along the froth.
This representation is based on the high number of bubble
collisions in the froth due to packing, and it assumes spherical
particles. Film drainage and bubble deformation were not considered
as coalescence mechanisms.
257
• The coalescence equations contain a set of coalescence efficiency
rate parameters, which quantify the fraction of collision events
between a pair of bubbles that result in coalescence, per unit
time. Determination of the coalescence rate parameters in a
three-phase system must take into consideration the presence of
solids in the froth liquid films and the surfactant concentration.
Since the air holdup in a column stabilized froth has been observed
to stabilize with height, the values utilized in the simulations
were inversely related to the bubble sizes, looking to duplicate
the profile shape. A similar relationship was employed for the
upper froth region since it has been observed that, in the presence
of solids, drainage is impaired and air fraction becomes almost
constant. Given the contrasting evidence on the effect of solids
upon the froth stability, no attempt was made to explicitly
correlate the particle properties with the coalescence parameters.
It was however inferred from the simulation conditions that these
parameters may also be influenced by other operating conditions
such as bias rate and, on that account, gas rate.
• The detachment of particles from the bubble surfaces was regarded
as a consequence
of the coalescence process. No detachment term was therefore
incorporated into the collection region equations, since only froth
coalescence was represented. Depending on the assumptions made in
relation to the occurrence of particle reattachment, two different
expressions for detachment rate were obtained. Evaluation of each
of these detachment models within the dynamic equations suggested
that particle reattachment should be taken into
consideration.
• Most of the parameters required for solving the model are known
operating conditions
or can be calculated using established relationships. The
probabilities of attachment for a number of systems can be
determined from the literature. The only values which have to be
estimated are the coalescence efficiency rate parameters.
• The steady-state solid concentrations profiles resulting from the
simulations resemble
experimental profiles found in the literature. The model predicts a
decrease in solid concentration along the froth as well as an
increase in concentration in the pulp up to the interface. Such
kind of behavior has been observed by other workers during studies
of solid mass tranfer in flotation columns. The times to reach
steady state in the simulation plots appear to be reasonable, based
on a comparison with actual dynamic responses collected during
laboratory tests with coal samples.
• Available dynamic data which had been collected in a laboratory
flotation column were