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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5.2 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.
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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.
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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.
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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.
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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:
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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
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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.
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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
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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]
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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
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
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:
while the distribution of solids returned to the pulp from the froth per unit time was:
Consequently, the net rates of solids being carried with to the froth by entrainment were:
Finally, the rates of solids in the tailings stream, in terms of the different size and composition classes were:
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:
while the rates of particles entrained and returned to the pulp were:
239
The distribution of solids lost in the tailings per unit time was the following:
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:
The final (steady-state) mass rate of material in each size-floatability combination that entered the froth through flotation was:
247
Likewise, the mass rates of particles entering the froth by entrainment and returning to the pulp simultaneously were:
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:
Adding the amounts floated, entrained and discarded in the tailings per unit time yields the following:
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
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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.
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• 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

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Three-Phase Dynamic Model of the Column Flotation Process