Review of froth modelling in steady state flotation systems

Minerals

Pergamon 0892687S(98)00020-X

Engineering.

Vol. 11, No. 5. pp. 397421, 1998 0 1998 Elstvier Science Ltd All rights reserved. Printed in Great Britain 0892-6875/98 $19.OOtO.00

REVIEW OF FROTH MODELLING IN STEADY STATE FLOTATION SYSTEMS

Z.T. MATHE”, M.C. HARRIS*, C.T. O’CONNOR” and J-P. FRANZIDIS+ $ Department of Chemical Engineering, University of Cape Town, Private Bag Rondebosch, 7701 South Africa. E-mail: [email protected] t Julius Kruttschnitt Mineral Research Centre, University of Queensland, Australia (Received 26 November 1997; accepted 3 January 1998)

ABSTRACT Froth models proposed in the literature are reviewed with the aim of identifying their significance and usefulness in the modelling and scale-up of thefroth phase in steady state flotation systems. Literature indicates thatfroth phase performance is better understood in terms of froth recovery, the fraction of material presented at the pulp-froth intelface that reports to the concentrate. This review suggests that froth recovery is a strong function of drainage rate of particles from the froth phase to the slurry phase. Drainage rate, in turn, is determined by physical factors, such as froth removal technique, geometry of the flotation cell, flux and distribution of air at the pulp-froth interface, the water content, particle size and solids content, and chemical factors, such as froth stability and froth loading. These factors infruence the froth residence time, which has been ident$ed as a key froth parameter. Finally, it is proposed that future work should focus largely on the development of a methodology to investigate froth performance based on the froth recovery in different flotation systems. This will enable generic relationships between the froth recovery and froth sub-processes and key froth parameters to be established, and make it possible to relate froth performance in different flotation systems. 0 1998 Elsevier Science Ltd. All rights reserved Keywords Froth flotation;

flotation froths; modelling;

flotation kinetics

INTRODUCTION The froth flotation process is one of the least well understood engineering unit operations today. It is believed that a poor understanding of the processes occurring in the froth phase contributes significantly to the incomplete understanding of the flotation process as a whole. This is evident in the inability of flotation correlations and kinetic models, which lump together the pulp and froth phases, to accurately model complex flotation systems. It is widely acknowledged, however, that froth flotation involves the recovery of minerals in two distinct phases (see Figure 1). The pulp phase is adequately described by first-order kinetics, whereas no appropriate model of a general nature exists for the froth phase. As a result, a firstorder kinetic equation is usually imposed on the overall flotation process.

397

398

2. T. Mathe et (11.

In the early days of froth modelling, the emphasis was on developing a mathematical description of the froth phase based on the assumption that both froth and pulp phases are completely mixed [ 1, 21. This approach was subsequently used by Ring [3] and Woodburn et al. [4] to estimate flowsheet parameters for circuit modelling, with limited success. Thereafter the emphasis shifted towards studying the phenomena which take place in the froth phase [5-81. From these investigations, it emerged that froth stability, froth mobility, entrainment and drainage of particles are the most important factors which affect mineral recovery and grade. However, the interactive nature of these factors makes it extremely difficult to quantify the effect of froth on the overall flotation process. Consequently, the froth recovery factor was seen.as a valuable practical parameter to describe the influence of the froth phase in flotation. Froth recovery refers to the recovery to the launders of floatable and entrained ore particles entering the froth zone from the collection zone. A typical example of the effect of froth recovery on the flotation process is illustrated by Figure 2.

Tails

Concentrate b % = fOL,RL) (entrained + floatable)

j

Feed

+-i-

Drainage water +

I

floatable and entrained particles

.___........___.._...... Fig. 1

R, = froth recovery;

Two-phase flotation process. R, = collection zone recovery; residence time in the pulp zone; 2, = froth residence time.

r,= mean

1

0

1

2

3

4

Mean Residence lime (mins)

Fig.2 The effect of froth recovery on the overall flotation process. Although several froth models have been used successfully to describe froth behaviour in specially designed flotation cells, the results have seldom been useful in predicting the froth recovery and the overall recovery

Review of froth modelling

399

in a cell, and hence in the modelling of flotation circuits. Studies carried out in these cells, however, have contributed significantly to the current understanding of frothing phenomena and the behaviour of mineral particles in froths. Due to the complex and diverse nature of froth modelling, the significance and usefulness of these investigations is either not known or confusing to most researchers. To review these studies, the present paper proposes that froth models, in general, may be divided into two broad categories, viz. steady and non-steady state models. Steady state models refer to froth models derived from systems where the froth characteristics do not change with time for fixed operating conditions, such as in equilibrium and continuous flotation cells. The non-steady state models, on the other hand, are models derived from systems such as batch cells, where the froth characteristics are changing with time. The aim of this paper, therefore, is to review the current understanding and modelling of the froth phase in steady-state systems. Subsequent review will deal with non-steady state systems. The material in this paper is organised according to the transfer processes involved in the transportation of particles out of the flotation cell. It further highlights the significance of froth investigations based on froth recovery. Future research areas which can significantly contribute to the current understanding of froth modelling are also proposed.

MODELLING

OF FROTHS

The steady-state models found in the literature are derived from equilibrium cells (defined below), column cells and conventional cells. Froth phase modelling, in general, involves a number of transfer processes: selective transfer of material from slurry to the froth by particle-bubble attachment; non-selective transfer of material from the slurry to the froth by entrainment; drop back (both selective and non-selective) of material from the froth to the slurry; and mechanical or hydraulic transfer of material from the froth into the concentrate (see Figure 3). The models presented here either treat some of these transfer processes individually, or lump them into a single equivalent process described by the froth recovery factor.

Froth phase

I

=Qfl

1,;: . .il . ..:

w

Feed

Pulp phase Fig.3

Froth transfer process (after Laplante et action; 4. froth transfer.

al.

Tails

[ 161). 1. true flotation; 2. entrainment; 3. drop-back

Quantification of the individual transfer processes is difficult, especially in industrial froths. As a result, the performance of the froth phase is sometimes inferred from the appearance of the froth at the surface. Image analysis has proved to be very useful in the analysis of the froth based on its appearance. In addition to the review of froth models, a very brief review of the work involving this technique in steady state systems is provided at the end of this section.

Z. T. Mathe et ul.

400

Material Transfer Processes Equilibrium

Cell

The equilibrium cell was developed by Warren Spring Laboratories [9] to study froth by bringing the froth stability under more direct experimental observation. The sides of an equilibrium cell are built up so as not to allow continuous froth removal. This allows effects due to the rate of froth collapse to be separated from the rates of pulp-froth transfer processes. Furthermore, it allows separation of froth and pulp zones by interposing slide(s) between the froth and the pulp for assay purposes. Transfer within the froth Moys [7] developed

dm,(h) -=

a plug-flow

model to describe equilibrium

froth behaviour

given by the equation:

mJh)+mei(h) mrJh)

(1)

+’

dh

v(h)

u(h) I

where ml(h) is the mass of a particular mineral at froth height h, m&h) is the entrained mass flow rate at froth height h, mei is the mass flowrate of species i attached to bubbles at froth height h, m,i(h) is the mass flowrate of particles diffusing from the upward-flowing stream to the downward-tlowing stream, u(h) is the velocity of downward-flowing streams, and v(h) is the velocity of the bubble films. This equation

is based on the following

postulates:

6)

The rate of detachment froth.

(ii)

Both floatable and entrained particles (class i) rise at the velocity of the bubble films v(h) towards the top of the froth column.

(iii)

The net upward flowrate of water is constant and equal to the concentrate

(iv>

The solid particles diffusing from the upward-flowing at velocity u and rate m,l(h).

of particles of component

i is proportional

to their concentration

within the

water flowrate.

stream enter the downward-flowing

stream

Although this model worked very well for a complex Zn-Cu-Fe sulphide ore (as seen in Figures 4 and S), it does not quantify froth structure in terms of typical froth properties, such as drainage velocity, viscosity and bubble film thickness. Furthermore, the possibility of re-attachment of particles was neglected. Therefore, the general applicability of this model to deep froths, where re-attachment of particles is most likely to occur, may be limited. Drainage

mechanism

Drainage of material horizontal direction. minimal. Cutting et equilibrium cell can

in equilibrium froths is mainly associated with mobility of froth in the vertical direction as opposed to the In equilibrium cells the significance of froth mobility in the horizontal direction is al. [lo] suggested that the effects of vertical froth mobility within the froth in the be classified into two modes:

(9

Film drainage,

(ii)

Column drainage, which refers to unstable conditions, such as solids accumulation resulting in collapse and downward movement of fluid.

which refers to water and solids draining

around the air bubbles. in the froth,

Review of froth modelling

401

I

20 GRADE

Fig.4

60 %

80

Froth structure showing inflation in the pyrite-grade curve. Batch float with pyrite depressed under steady state conditions (i.e. no removal of concentrate) (Moys [7]).

20 GRAOE

Fig.5

40 OR CONCENTRATION,

OR

4.0 CONCENTRATION,

%

60

Simulated froth structure showing inflation in the pyrite-grade curve. Based on the plug-flow model (Moys

[71).

They then proposed the following equation:

(2)

where C, is the material concentration of froth constituent n, y is the column drainage coefficient, j3 is the

402

Z. T. Mathe et d.

film drainage Integrating

coefficient,

and L2 is the froth removal coefficient.

the above equation,

assuming

the froth removal factor is negligible,

yields:

(3)

where C, is the concentration

of particles at the froth-pulp

Equation (3) can be used to estimate the probability the expression:

interface.

of transfer of froth/material

P(h) = C(h)/C[

to the top of the froth by

(4)

The above expression has a similar meaning to the froth recovery factor to be discussed later in this paper. Other coefficients and probabilities for interpreting the recovery of minerals within the froth have also been proposed [ 11, 121. The advantage of using eq. (4) is that the recovery of mineral/s within the froth can be studied as a function of froth height without independently varying the froth depth. Ross [ 131 estimated the separate contributions of water and mineral particles to the thickness of the bubble film above the interface and then used the resulting total thickness to determine the drainage rate constants to describe the rate at which entrained particles and water diffuse from the upward-flowing entrained stream of slurry. Based on the above approach, it was possible to calculate the drainage velocities of the various mineral species and water from the froth. Using Moys’s approach to entrainment of species i (i.e., rate of drainage of an entrained species i is proportional to its concentration in the froth), Ross [14] showed that the total mass flowrate of that species ascending to a height h above the pulp-froth interface is given by:

(5)

where kd, i is the drainage

rate constant for entrained

particles of species i.

It was further postulated that the total thickness of the bubble films at any height h, above the interface, is a function of both entrained water and particles immersed in the film. This model combines various factors that affect the behaviour of mineral particles and water in flotation froths, such as the size of the bubbles, the mass of particles and water floated and entrained, and the rates and velocities at which they drain from the froth. Based on small values of the fractional bubble-surface coverage (described in Ross [13]), the detachment of floating particles by displacement from the bubble surfaces was taken to be negligible. The observed decrease in concentration of floated particles with increasing height was therefore associated with washing of the bubble surfaces by the draining slurry. From Figure 6, it can be seen that the drainage of water decreases with an increase in air flowrate. Figure 7 indicates an initial increase in drainage rate constants (kd, i) with an increase in froth height during flotation of coarse particles of pyritic ore at different aeration rates. This illustrates the importance of the influence of water and air content and entrainment on the drainage mechanism within the froth which is not well documented in the literature. Entrainment

in an equilibrium

cell

Moys [7] developed an expression for entrainment of particles which he used successfully in developing a plug-flow model for studying flotation froth behaviour. The expression indicates that the mass of entrained particles decreases exponentially with increase in froth height.

Review of froth modelling

403

( J&y

mJh> - me,i(0)exp - Id

(6)

Equation (6) is expected to work well in froth regions where plug-flow rarely the case in most flotation systems.

t ii

1

3 0.5

conditions

exist. However,

this is

--a-4

1 0

1 --s-4

--A-6 -0-E

1

2

3

5

4

Froth Height (cm)

Fig.6

Variation of the drainage rate constant for water during flotation of the -150+75 urn fraction of the pyrite ore (Ross [13]). Airflow 4 Ilmin for Test 1 & 3; airfiow 8 Umin for Tests 4 and 6.

Airibw (litres/min)

0

1

2

3

4

5

Froth Height (cm) Fig.7

Variation of the drainage rate constant for the -150+75 pm fraction of the pyrite ore (Ross [13]). Airflow 4 Umin for Tests 1 & 3; airflow 8 llmin for Tests 4 & 6.

404

Z. T. Mathe et (II.

Continuous Mechanical Cells The froth phase found in mechanical cells is also influenced by a number of transfer processes, viz. selective transfer of material from slurry to the froth by true flotation; transfer of material from the slurry to the froth by entrainment; drop back (both selective and non-selective) of material from the froth to the slurry; and mechanical or hydraulic transfer of material from the froth into the concentrate. Likewise the froth models developed for this system can be classified in terms of these transfer processes. Transfer of material from slurry to froth By floating with almost no froth layer, it is generally assumed that the flotation rate constant obtained represents flotation of minerals in the collection zone. Alternatively, the collection zone flotation rate constant (k,) is determined by extrapolating , to zero froth height, a line or curve relating the overall flotation rate constant to froth height [12, 15-171. This rate constant (k,) is, by definition, highly influenced by processes occurring in the pulp or collection phase. Recently, Gorain et al. [ 181 proposed that a linear relationship exists between the overall flotation rate constant (obtained at very shallow froth depth) and the bubble surface area flux St,‘. This could be a very useful relationship for use in flotation modelling, optimisation and scale-up. The presence of the froth layer, however, has a significant influence on this relationship [19] since the overall rate constant, k, (i.e., in the presence of a froth phase), will not equal the collection zone rate constant, k,. To predict k, from St,, it is necessary to understand the sub-processes occurring within the froth zone, and how they relate to the overall froth performance. Entrainment Correlations between gangue and water recoveries have been studied both in batch and continuous systems [5, 20-231. Smith and Warren [24] reviewed the available work on entrainment in conventional flotation cells. They highlighted the effect of particle size on the degree of entrainment, which was studied by Engelbrecht and Woodbum [5], on a continuous pilot plant system of finely ground silica and finely ground pyrite. The latter authors developed an expression for gangue recovery by an equation of the form:

R’g=

eig

( R, - R,,,,o J

where R’, is the gangue recovery of species i, R, is the water recovery, R,,o is the recovery the absence of gangue material, and eig is the entrainment factor for gangue of size i. The major conclusion reached by Engelbrecht and Woodbum was gangue recovery and water recovery for fine particles. For coarse between gangue and water recovery. It is believed that the settling which creates a difference between the recovery of gangue versus and fine particles.

of water in

that a linear relationship exists between particles, a parabolic relationship exists velocity of coarser particles is the factor the recovery of water curves for coarse

Recently, Kirjavainen [25] proposed a method to evaluate the entrainment of gangue material in flotation circuits. The semi-empirical method relates the entrainment factor (P) to the variables in the Newtonian region (described in Kirjavainen [26]) given by the following equation: w 0.7

P=

(8)

w 0.7+ b,,,h - O.%,!mO.C%++

where w is the water recovery rate (kg/m%), b is a constant, the dynamic shape factor, and q is the slurry viscosity.

m is the average mass of the particles,

’S, = 6 Jdd, where J, is the superficial gas velocity and d, is the bubble size

w is

Review of froth modelling

405

It should be noted, however, that eq. (8) was derived for a system without any hydrophobic species present (only quartz and phlogopite were used), so the usefulness of this entrainment model in real ore systems may be questionable. Froth transfer within the froth The design of mechanical cells is such that not all material in the froth zone has a chance of freely flowing out of the cell to the concentrate launder. Under these circumstances, it is sometimes necessary to use mechanical scrapers to remove the froth. The physical removal of froth induces disturbances within the froth phase. This action also affects the behaviour of particles within the froth phase. In conventional cells, three flow regimes have been identified [lo, 27, 281, viz. the stagnant flow regime next to the cell backplate, where particles have a very low probability of being transferred to the launder, the centre zone where “plug-flow” behaviour of particles prevails, and the region just below the paddles, where there is a considerable mixing activity and which, according to Moys [27], is the main path followed by entrained material. The three regions within the froth volume are shown in Figure 8. The three zones were characterised from studies of froth behaviour at pilot-scale in a continuously operated flotation bank [ 81 and observations of froth mobility and structure in large scale flotation cells. Interesting results from this work indicated that no significant concentration gradient existed in the froth at low froth heights (O-4 cm for cleaner cells, and O-2.5 cm for rougher cells: Wemco No. 20, 80 1). From this it is therefore expected that the froth will behave like a completely mixed system with respect to solids for mechanical cells operating at shallow froths. This approach has been used with some success by both Ring [3] and Woodburn et al. [4] in circuit modelling studies.

h

1

i I 3

Froth

Fig,8

Froth volume regions (after Ross [28]). 1. Plug flow equilibrium; 2. Lateral movement in the direction of the concentrate weir; 3. Path followed by entrainment particles; 4. Plug-flow.

Moys [27] proposed a two-dimensional streamline behaviour of froths. This approach was based on the Laplace equation-similar to the approach taken by Murphy et al. [29] in a two-phase system. To use the Laplace equation, however, one has to assume that the froth is well drained with no frictional forces. The applicability of this assumption to three-phase froths is somewhat doubtful. More recently, Ross [28] studied the behaviour of particles and water in the froth phase of rougher and cleaner cells in a large-scale pyrite flotation plant. This work utilised the findings discussed earlier that three regions exist in froths found in mechanical cells, and that deep froths can be modelled as plug-flow systems. Ross improved Moys’ plug-flow model of drainage by considering variation of particle and water drainage rates with respect to froth height.

406

Z. T. Mathe et al.

Froth transfer out of the cell Upon arrival of the particles in the froth phase, froth removal techniques determine the fraction of particles successfully recovered. Froth removal induces drainage of particles by breaking bubbles. Froth removal is also strongly believed to be the rate controlling step in froth flotation in many instances, as has been demonstrated by Meyer and Klimpel [30]. With longer residence times the entrained particles drop back to the pulp phase thus yielding a high grade concentrate, whereas the opposite is expected for shorter residence times [31]. Laplante et al. [ 161 studied the effect of air flowrate and froth thickness on the froth transfer rate in a small continuous mechanical cell. This was carried out using an unclassified and classified (0-12pm; 12-22 pm and 22-4Opm) galena ore. This work showed that the froth transfer rate increases with an increase in gas rate, and decreases with an increase in particle size. It also showed a decrease in froth transfer rate with an increase in froth thickness and hence with an increase in froth residence time. A decrease in froth transfer rate would then result in a lower froth recovery and hence a lower overall recovery. Drop back of particles The recovery of material within the froth is determined by what fraction of material arriving at the interface returns to the pulp zone by drainage mechanisms. Particle drainage has been studied mostly in column and equilibrium flotation cells [32, 7, 131. Laplante et al. [16] studied dropback of particles in a small (5 I) mechanical cell using unclassified and classified galena ore. It was found that particles have a high probability of returning from the froth zone to the pulp zone when deep froths are used. Most of these studies, however, did not account for the contribution of entrained material to the overall drainage rate of particles. For a continuous flow in a two-phase system (i.e. inflow of feed, removal of concentrate, removal of tailings, and transfer of concentrate from the pulp to the froth and from froth to pulp), Arbiter and Harris [l] developed a dimensionless equation:

ICP V’, -SCf

Qc

(9)

kcVp+ k&

where r is the ratio of pulp volumes (with air/without air), s is the ratio of froth volumes (with air/without air), C, is the concentration of particles in the pulp phase, C, is the concentration of particles in the froth, V, is the pulp phase volume, V, is the volume of froth, Q, is the mass flow rate from pulp to the interface, and kd is the drainage rate constant of particles. By changing the feed to the cell, and measuring from the linear plot of (rC++) against QjVp . Allan [33] who determined material transfer rates the froth to the pulp zone in a single mechanical

C,, C, and Qc, the constants k, and k, can be obtained This method has been used successfully by Greaves and from the pulp to froth zone and dropback constants from flotation cell.

Woodbum [34] and later Lynch et al. 1231 showed that for a two-phase continuous system (assuming perfectly mixed froth region) the overall flotation rate constant (slurry and froth) can be expressed as:

a

(10)

where zf is the froth residence

time, the precise definition

of which will be discussed

later in this paper.

Equation (10) is probably only suitable for flotation at relatively shallow froth heights, as it is under these conditions that the froth can be regarded as perfectly mixed. For very shallow froths (i.e. negligible froth

Review of froth modelling

residence Column

time), the overall rate constant

will approach the collection

407

zone rate constant.

Cells

It is relatively easier to investigate processes occurring in column flotation cell froths compared to mechanical cell froths because of the well-defined flow patterns in column froths as a result of the absence of moving parts. However, the addition of wash water creates some turbulence within these froths which nullifies the assumption of plug-flow froth behaviour. In addition, there is a significant difference between concentrate quality collected when wash water is added below the cell lip and when wash water is added above the cell lip [35]. Nevertheless, it has been shown that the behaviour of particles within the froth zone can be adequately described by plug-flow models [36]. Detachment and Drainage of particles in column froths Finch and Dobby [38] postulated an expression to describe the overall recovery in a column flotation system by treating the drop back of particles from the froth to the pulp zone as a recycle stream which is mixed with the fresh feed (see Figure 9). This leads to the expression:

R,

=

RcRr

(11)

1- R,+ R$$

where R, is the collection

zone recovery and Rf is the froth recovery.

Froth Zone Wl-w 4 R,

1 v Feed Fig.9 Interaction

Collection Zone * (

between

froth and collection

I_-*

1-K

zone (after Falutsu and Dobby [32]).

Falutsu and Dobby [32] used a modified laboratory column, which isolates the froth zone from the collection zone, for direct measurement of froth drop back [32]. Pure silica with a dso particle size of approximately 35 pm was used for all tests. They then determined froth recovery, using eq. (1 l), at different flotation conditions. Results indicated that froth recovery is (i) dependent on particle size, increasing as particle size decreases, (ii) dependent on froth bias velocity, decreasing as froth bias velocity increases (i.e. more wash water added); (iii) not strongly froth height dependent, suggesting the possibility of particle re-attachment or negligible particle detachment within the froth. This is consistent with the view that detachment of particles from bubbles occurs primarily at the pulp-froth interface.

408

2. T. Mathe et al.

The problem with this technique of directly measuring froth drop back is that the two zones (i.e. collection and froth) are modelled separately. The drop back is recycled back to the feed line, or discarded. This allows for no interaction of the froth zone and collection zone. Also, the geometry of the column is different from that of the regular column flotation cell. It is, however, a useful approach with regard to understanding the role played by drop back in the froth phase. Falutsu [37] later presented a fundamental analysis of the stability of the bubble-particle aggregate in column froths. The approach involved identifying forces acting on the bubble-particle system at equilibrium, during detachment, and during reattachment. Forces involved at equilibrium include: (a) the gravity force; (b) the static buoyancy of the immersed particle; (c) the hydrostatic pressure of the liquid column of height Z on the contact area; (d) the capillary force at the three-phase interface; and (e) the capillary pressure. Destructive forces identified during detachment were: (i) liquid flow (drag force); (ii) slippage (combination of a centrifugal force and the gravity force); (iii) bubble deceleration; and (iv) bubble oscillation. To assess the importance of these forces, fixed conditions for flotation cell variables were assumed. It was found that estimations of liquid flow and slippage forces were significantly lower than the force of surface tension. These results indicated that the drag force exerted by the draining liquid or wash water addition, even at higher rates, cannot be responsible for particle detachment. Slippage of particles along the bubble surface was also considered unlikely to initiate detachment of particles in the froth. The results strongly suggested that a significant proportion of detachment was caused by deceleration and impact of particle-bubble aggregates upon arrival at the interface, which supports the finding by Falutsu and Dobby

WI. Based on this work, it can be inferred that instability of the froth system is caused by breakage of bubbles in the bed structure which causes deformation of the froth (bubble coalescence). The limitations of this work however, are the assumptions on which the calculations were based. Particle diameter, bubble diameter in froth, bubble velocity in the froth, froth density, and bias velocity were all assumed to be the same for all calculations. However, it has been shown in many studies that all these parameters vary with froth height. Entrainment

in column froths

Entrainment is usually assumed to be insignificant in a column froth when wash water is used. Yianatos et al. [39] suggested that entrained particles are rejected very close to the pulp-froth interface in column cells when froth washing is employed. However, a number of studies have been conducted on column froths in the absence of wash water to investigate entrainment. Tuteja et al. [40] looked at the influence of column parameters (i.e. collection zone height, superficial air rate, superficial feed rate, and feed grade) on entrainment in froth using a gold bearing sulphide ore. Their work showed that an increase in superficial feed rate and feed grade results in an increase in concentrate grade without significantly affecting recovery of gangue minerals. This suggests that entrainment in column froths is indeed negligible even in the absence of wash water. Froth Recovery Consideration of only the overall flotation rate constant, or individual transfer processes, in the determination of the kinetic response of an ore to flotation is inadequate and misleading, particularly when modelling flotation cells with deep froths. This is true for both laboratory and plant scale flotation cells. For very deep froths, the overall flotation rate constant is highly influenced by processes occurring within the froth phase. Presently, the poor understanding of the behaviour of the froth phase makes it difficult to interpret the overall flotation rate constant in terms of froth sub-processes. A more promising and practical approach is to represent the processes occurring within the froth-phase collectively in terms of froth recovery. Froth recovery refers to the recovery to the concentrate of floatable and entrained ore particles entering the froth zone from the collection zone. Although this concept has been explored extensively in column flotation cells, it has largely been neglected with respect to the analysis of froth performance in mechanical cells (particularly in batch flotation cells). This section reviews the different expressions for froth recovery found in the literature.

Review of froth modelling

409

Froth recovery modelling Mathematically, froth recovery is described by the ratio of the overall flotation rate constant to the collection zone rate constant: (12)

!f = k/k, where k, is the overall flotation rate constant

(i.e. in the presence of froth and pulp phases).

If certain assumptions are made, eq. (12) can be expressed in terms of the overall flotation recovery and the collection zone recovery. Furthermore, it can be shown that eq. (11) and eq. (12) are mathematically equivalent. Perfect Mixing If the collection zone is assumed zone recovery is given by:

to be perfectly

mixed with a mean residence

time t, then the collection

R, = I-(l+k,z)-’ and the overall recovery,

(13) including

the froth zone, is given by:

R, = 1 -(I +k#l Solving

(14)

for k, and k, in equations

(13) and (14), and substituting

them into eq. (12) leads to eq. (11).

Plug Flow Where transport of material in the collection collection zone recovery is given by:

zone is described

R, = I-exp(-k,z) and the overall recovery,

time 2, the

(15) including

the froth zone, is given by:

R, = 1-exp(-k,T) Solving for k, and k, in equations R, = 1- ( l-RJRf Whereas eq. systems, the to the kinetics and entrained

by plug flow with residence

(16) (15) and (16), and substituting

them into eq. (12) leads to the expression: (17)

(12) can be useful in terms of modelling recovery data from simple and complex flotation assumption on which it is based, viz. that the behaviour of the froth zone is directly related of the collection zone, is questionable. Furthermore, it does not distinguish between floatable ore particles.

The approach of Laplante et al. [ 161 treats the froth phase as a separate unit from the collection zone, with the pulp/froth interface as their common link. Using this approach leads to an alternative definition of the froth recovery factor, namely: l?f = k’l(K+kJ where k’ is the froth transfer rate.

(18)

Z. T. Mathe et ul.

410

The advantage of this approach is that one can study froth recovery with respect to processes occurring within the froth phase.

only

The other useful approach to froth flotation modelling uses air recovery or enhancement factors as opposed to particle recovery to account for froth effects in flotation models [41, 421. Since most of this work was carried out in non-steady state batch flotation systems, further results will be discussed in the subsequent review paper on non-steady state systems. Recent work by Gorain et al. [ 191 indicates that bubble surface area flux, or St.,, is a better parameter than air flowrate in this type of analysis. Given the present understanding of the k,-St, relationship, it might be necessary to look at the use of air recovery correction factors in flotation circuit modelling from this perspective. Froth recovery investigations Column cells Contini et al. [43] used ground silica (80% passing 35pm) in a laboratory column cell capable of operating either countercurrently or cocurrently to investigate froth recovery. The difference between the two operating modes was mainly that particles which dropped out of the froth in cocurrent mode were immediately lost to tailings, whereas in counter-current mode, these particles were subject to recollection. Each operating mode was divided into three zones: froth zone, secondary collection zone, and primary collection zone. Expressions for the rate constant and froth recovery were developed for each operating mode based on a first-order kinetic model describing the particle removal in the primary and secondary collection zones. The rate constant and froth recovery was then determined by simultaneous solutions of the two expressions. In this system Rf varied between 27% and 65%, increasing with decrease in particle size (Figure 10). This modelling approach was based on the following assumptions:

(9 (ii) (iii)

the rate constant for the primary collection zone was assumed to be constant for both operating modes, as the gas holdup in cocurrent flow was only slightly less than in countercurrent flow; the rate constant in the primary collection zone was assumed to be the same as in the secondary collection zone; and froth recovery in the counter-current mode was assumed to be the same as in the cocurrent mode, as the solids content of the froth was the same for both operating modes.

It is questionable

whether the third assumption

is likely to be satisfied.

0.8 t-

t

FmUwdose

tfmtherdose

= 40 mgtlii = 30 mgtlii

[:‘\r--.:-.

0.2 .-

0

5

10

15

20 Awage

Fig.10 Rate constant

and froth recovery

25 particle

30 size

35

40

45

50

@II)

versus particle size (Contini

et al. [43]).

Review of froth modelling

411

Falutsu and Dobby [32] using a specially designed laboratory column cell operating at 15% solids content (80% passing 35 pm of silica) also observed that Rf varied between 20 and 60%. Rf increased slightly with an increase in solids flux at the interface, and decreased slightly with increase in froth height. At low wash water bias velocities (co.3 cm/set), it was found that Rfremained fairly constant. However, Rf was observed to decrease at higher bias velocities. For the particle size range tested, froth recovery was observed not to be a strong function of particle size, except for very fine particles (Figure 11). The problem with this technique, however, is that the collection and froth zones do not interact because the drop back is recycled to the feed line or discarded. Therefore, extrapolation of these observations of Rf behaviour to regular column cells could be questionable.

20 --

0, 0

10

20

30

40

50

Particle size, d, (pm)

Fig.1 1 Froth recovery versus particle size (Falutsu and Dobby [32]). Test 1: froth height = 116 cm and feed slurry = 244 mllmin; Test 2: froth height = 56 cm and feed slurry = 153 mllmin. Burger [44] proposed a model which described the composition of the pulp, at a specific level in a pilotscale column cell, in terms of interstitial concentrations of species, bubble loading, overall concentration of species and gas hold-up. The interstitial concentration of species i is described as: Ci =( (Total mass)i - (Mass floating)i}/{ Mass of Overall S1t.1~ - Total MSS Floating}

(19)

This expression relates the bubble loading and the interstitial concentration of a species at a specific level in the column cell to the overall concentration of species at that level. Knowing the recovery in the pulp phase at the level directly below the pulp/froth interface, Finch and Dobby’s expression (eq. 11) was used to determine froth recovery. This work (conducted using Cr,O, ore and SiO, in a 15 cm diameter column cell) indicated that froth recovery varied between 40 and 80% for Cr,O,, and between 20 and 40% for SiO,. Contrary to a suggestion made by Falutsu and Dobby [32] that froth recovery increased with decreasing particle size, Burger’s results surprisingly showed an increase in froth recovery with increasing particle size. It is very unlikely, however, that the froth recovery will increase with an increase in particle size. Nevertheless, his work highlighted the interactive nature of flotation parameters such as rate of aeration, reagent dosages, bubble size, air hold-up, pulp viscosity, wash water addition, and froth depth. It also

412

Z. T. Mathe et al.

showed that the size, shape and density of a particle have a strong effect on the drainage rate and possible detachment of particles from air bubbles. Finally, the work highlighted the strong dependence of particle-bubble detachment on the floatability or hydrophobicity of the mineral species. Recently, Vera [45] used Contini’s approach to test the effect of operating conditions such as air flowrate and froth depth on the collection rate constant and froth recovery. A zinc cleaner feed stream (80% passing 14 to 20 pm) consisting of sphalerite (50% by mass) and pyrite (42% by mass) was used for all tests. Froth recovery was found to vary between 6 and 20%, and increase with an increase in solids flux at the interface. This work confirmed to some extent the independence of the collection zone flotation rate constant with respect to froth height, suggesting that for this system the processes occurring within the froth phase had no influence whatsoever on the collection zone processes. It also supported the conclusion by Falutsu and Dobby [32] that froth recovery is not a strong function of froth height. However, when the secondary collection zone length was varied, froth recovery was strongly affected by froth depth, indicating that there are still many unresolved issues in our understanding of the behaviour of this type of system. Mechanical cells Recently, Harris [46] following earlier work by Gorain et al. [ 191, used industrial data obtained at different froth depths to analyse the effect of the froth phase on the relationship between bubble surface area flux and the flotation rate constant. The results showed that the slope of a line relating the overall flotation rate constant and bubble surface area flux decreased with an increase in froth depth. Most significantly, it was observed that the relationship between k, and S, transforms into a non-linear relationship as froth depth increased. This suggests that k,-S, relationship depends strongly on the froth residence time. Consequently, the k,-S, relationship was modified to incorporate the froth recovery factor:

where P represents

the mineral

floatability.

The froth recovery factor by itself is not useful, unless it is correlated with key froth parameters such as froth residence time. However, it is difficult to measure the true residence time in flotation froths. Currently, froth residence time is inferred by the use of froth retention time (FRT) which can be defined in a number of ways. One definition is given by: (FRT),,

= NJ,

To account given by:

FW

for different

cell sizes, Gorain et al. [47] proposed

the use of a specific froth retention

time

specific = (h/Jg)/L

where L is the distance The other definition

from the center of a flotation cell to the launder.

(proposed

by Lynch et al. [23]) is based on the slurry flow through the froth:

(FRV,,,, = Vf/Qf where Qf is the concentrate

(23) flowrate.

Froth retention time based on the concentrate flowrate represents the consequence of the difference between the water flowrate into the froth phase and water flowrate back into the slurry phase. This parameter is not the froth residence time in the true sense of the classical residence time 7. However, eq. (23) is very useful for modelling the froth phase in flotation systems. This is supported by the recent work on the effect froth residence time on the kinetics of flotation [47] which indicated that froth recovery decreases exponentially with an increase in froth retention time (Figure 12). Expressed mathematically,

Review of froth modelling I$=

I-

(I

-exp (-h

413

* FRT)}

where h is the exponential

(24)

parameter

which could depend on the physical and chemical

froth factors.

??

0 0

loo0

zoo0

3ooo

4ooo

5ooo

WOJ

7ooo

axfl9oal

Froth retention time (set)

Fig.12 Variation

of the froth recovery factor with froth retention

time (Gorain et al. [47]).

Image Processing As discussed above, a useful fundamental mechanistic description of the behaviour of the froth phase has not yet been proposed. This is mainly due to the ill-defined and fragile nature of froth structure. The visual appearance of froth has been used to determine and possibly even control the performance of the froth phase. Moolman et al. [48], however, pointed out that different sets of metallurgical parameters may result in similar visual appearances of the froth. Nevertheless, research based on machine vision systems for the characterisation of froth is currently an active endeavour. Moolman et al. [49] classified the features of the froth structure by using spatial grey level dependence matrix (SGLDM) methods and neighbouring grey level dependence matrix methods. It is reported that a combination of the two methods can improve the accuracy of froth structure identification to approximately 90%. It has been shown that froth structures can also be classified accurately by using learning vector quantization neural networks for discrete-time classification of surface froth features extracted by means of image processing [48]. Attempts have also been made to relate image features to flotation control and output variables, and to identify the most important froth characteristics [50]. In froth structures found in industrial flotation cells, three distinct classes of froths were distinguished. The froths in class 1 had intermediate bubble sizes, mobility and stability (associated with a high degree of mineralization); in class 2 the froths were highly mobile and unstable with small bubbles (associated with poorly mineralised loosely packed bubbles); and in class 3 the froths were stable with large bubbles and low mobility (associated with overloaded minerals). This agrees well with results obtained in a batch pyrite system [51] which showed that froth mobility and stability have the most significant effect on the overall recovery. In addition, Hargrave et al. [52] recently demonstrated, also using a neural network approach, that froth instability in industrial flotation cells decreases along a cell bank-indicating less floatable material down the bank of cells.

414

Z. T. Mathe et ~1.

Furthermore, on-line and off-line image processing techniques have been used to link drainage and kinetics [41, 531. For instance, Woodbum et al. [53] proposed a kinetic specific surface of the bubbles estimated by off-line image processing of the froth. discussed further in the next review paper on non-steady systems, as most of the tests a batch flotation cell.

froth model This were

structure with based on the work will be carried out in

DISCUSSION Just as the process of froth flotation is often labelled as complex and difficult to understand, the froth investigations presented here are similarly complex and difficult. However, in spite of all these difficulties, a considerable amount of fundamental knowledge about the behaviour of mineralised froths has been established. The important aspects of froth modelling provided above can be categorised into three main research areas, viz. the investigation of froth transfer processes, froth characterisation and froth recovery modelling. Froth Transfer Processes The transfer of floatable material from the collection to the froth zone is largely governed by processes occurring in the pulp phase such as aeration, agitation and the chemical treatment of the ore. The kinetics of processes in the collection zone may be globally reflected by the collection zone rate constant which is estimated by extrapolation of a curve relating the overall flotation rate constant and froth depth to zero froth depth, or by floating with a very shallow froth. It has been found that this rate constant is directly proportional to the bubble surface area flux within the pulp phase. Although these studies do not distinguish between entrained and truly floatable material arriving at the pulp-froth interface, the k,-St, relationship could be very useful in flotation modelling, optimisation and scale-up. Furthermore, it has been shown that an increase in water content within the froth phase induces detachment and drainage of particles. The drainage rate constants for water and solids increase when the rate of aeration increases. Also, the drainage rate constant for coarse particles in particular, has been found to decrease only slightly with increasing froth height. This suggests that detachment of loosely attached particles occurs mainly at the pulp-froth interface. It should be kept in mind, however, that the drainage mechanism has been studied mostly in column and equilibrium flotation cells, mainly because these systems have well defined froth flow regions within the froth phase. Results obtained from the equilibrium cell, however, cannot be used to interpret the behaviour of particles in all regions found in industrial froths. This is due to the difference in froth flow patterns that exist in froths in conventional industrial cells. Although it is possible to study the behaviour of entrained and floatable particles in this type of system, the manner in which it is operated limits its potential for interpreting continuously overtlowing froth systems. Therefore the use of the models based on results obtained from equilibrium cells is probably very limited. The recovery of fine particles by entrainment is governed by the thickness of liquid lamella which strongly depends on the water content within the froth. It is therefore clear that by controlling the variables which affect the recovery of water-such as air flowrate and bubble size-improvements in the recovery and grade of desired particles could be achieved. Furthermore, a very important observation from Moys’ work, that entrained particles in mechanical flotation cells follow the path just below the paddles (Figure S), is an indication of the importance of froth residence time. Froth transfer and behaviour of particles within the froth is often determined by the design of the flotation cell. From the literature, it can be concluded that at high gas rates and low froth depth, the transfer of material from pulp phase to the pulp-froth interface is rate limiting. Under these conditions, the overall rate constant approaches the collection zone flotation rate constant. At low gas rates and high froth height, however, transfer of the material out of the cell is rate limiting, and the overall rate constant approaches the froth transfer constant. At industrial scale, froths are thicker and the horizontal transport distances that particles have to travel to reach the concentrate launder are longer. Higher froth residence times lead to high probability of particle drop-back within the froth which results in low froth recoveries.

Review of froth modelling

415

Froth Characterisation Significant progress has been made in froth characterisation and understanding of the frothing phenomena occurring within the froth phase. These advances are due to devices and tools (such as the equilibrium cell and image analysis) that have been developed to study froth structure and behaviour of floating and entrained mineral particles in the froth phase. The information obtained from these studies, however, has a very limited use in flotation circuit design, modelling and optimisation. In addition, these studies do not identify the appropriate froth parameters that can be used to relate froth performance in different flotation systems. Furthermore, literature indicates that the difficulty in sampling of the froth phase is one of the problems that hinders the successful understanding of this aspect of the flotation process. Image analysis methods may well provide a viable solution to this problem. This approach should be very useful if it proves possible to relate the visual images of froth to the sub-processes occurring within the froth and collection zones. The decrease in bubble-film thickness with increase in froth height, as shown by Ross [ 131, is an indication that the bubble size is a strong function of froth height. The decrease in bubble-film thickness leads to bubble coalescence. This process is believed to be governed by the entrainment and drainage of particles. Under these conditions, the frequently used approach of using the average bubble size to determine froth parameters can lead to significant errors and misleading conclusions. Froth Recovery Froth recovery studies have injected new hope into the modelling of the froth phase. The froth recovery factor globally represents the froth transfer processes and their interaction with each other. This parameter has been studied extensively in laboratory column flotation cells, but there is a need to investigate it further in mechanical cells. Moreover, no froth recovery study has decoupled Rf into the contributions from floatable and entrained material. Such a study can be very useful in flotation circuit design and modelling. Recent work done in mechanical flotation cells has shown a very strong dependence of the froth recovery on the froth residence time [47]. The measurement of the froth residence time, however, is not easy and as a result, froth retention time (eq. (23)) is currently being used to infer froth residence time. As shown earlier, an exponential decrease of the froth recovery factor with an increase in froth residence time has been observed. This Rt-FRT relationship is expected to be influenced by a number of process variables which can be classified into chemical and physical factors (Figure 13). Chemical factors Although the influence of chemical factors on flotation systems is a major topic on its own, it is imperative to mention here that chemical reagents (particularly frothers) play a major role in determining froth stability and flotation kinetics in general. Frothers are surface-active and usually non-ionic reagents which provide a stable air-water interface to ensure that floated particles do not drop back into the flotation pulp from the froth zone. The use of frothers is further complicated by differences in machine designs which lead to operational differences. No research has been performed which has incorporated the effect of chemical factors into froth models.

Furthermore, reagents used to condition pulp (particularly collectors) influence the bubble surface coverage by mineral particles. Loosely attached particles are susceptible to drop back action in the froth phase. Some useful information on reagents and how they may affect flotation kinetics, in general, has been reported by a number of workers [54-581, but as yet there is no practical way in which this information can be used in the modelling, and, in particular, the scale-up of flotation froths. Consequently more work is needed in this regard.

416

2. T. Mathe

et (11.

key froth parameter . . . .._.._._.._________ Froth residence time, of i

Macro level Transfer nrccesses: froth transfer (l?) drop back ob) pulp to froth transfer (kJ (selective and non-selective)

Fk’

T

Micro level

Sub-Droccsses: entrainment , drainage detachment reattachment 1

Physical factors

- hther compositionand Hydrodynamic factors

Operation components

bubble oscillation bubble deceleration cell bank configuration liquid flow (drag force)

Fig.13

cell bank control

Schematic diagram indicating the processes (macro and micro level) which influence the froth recovery, and how physical and chemical froth factors are linked to the froth recovery. Arrows indicate the various factors contributing to a particular classification of factors/processes involved in the froth phase.

Physical factors Physical factors that influence froth performance can be classified into operational and hydrodynamic components (Figure 13). Operational components include cell geometry, cell bank configuration, cell bank control and froth removal methods. Hydrodynamic factors, on the other hand, are factors that affect turbulence within the froth phase. For fixed chemical conditions and operation mode, froth hydrodynamics will affect the water recovery. Water recovery, in turn, affects froth sub-processes such as entrainment, drainage and detachment of particles. From the work on Rf found in the literature, one can generally conclude that Rt will decrease with an increase in particle size, and increase with an increase in solids and gas flux at the pulp-froth interface (Figure 14). However, no studies involving direct correlation of these physical factors with froth recovery versus froth residence time have been reported in the literature, and this represents a key area for future research efforts.

Review of froth modelling

Froth Residence Fig.14 Qualitative

417

Time

effect of froth parameters

on froth recovery.

CONCLUSIONS This review shows that froth phase has a very pronounced influence on the overall flotation kinetics in any steady state system. Because of the complexity and interactive nature of processes occurring within the froth phase, the key to understanding the influence of froth depends on determining those froth parameters, such as froth recovery factor, which can be correlated with measurable flotation parameters (e.g. froth residence time) irrespective of the flotation system. The froth recovery factor presents a useful and practical way of modelling froth performance. In general, column flotation studies report that froth recovery varies between 20 and 80% depending on the gas flux at the interface and that froth recovery is not a strong function of froth height. A relationship between froth recovery and froth retention time, which indicates an exponential decrease in froth recovery with increase in froth retention time, represents the latest development in the modelling of the froth phase in mechanical flotation cells. In the discussion of this relationship the interactive nature of chemical and physical factors that influence froth recovery was highlighted, based on their influence on the various froth sub-processes. The Rr--FRT relationship forms the basis upon which the investigations of froths found in bench-scale flotation cells can be related to the froth phase in large-scale flotation systems. Not much is known, however, about the contribution of entrained and truly floatable mineral particles to this froth recovery factor. As a result of the findings 1.

highlighted

by this review, the following

general observations

can be made:

Studies in laboratory-scale flotation cells (where most of the current flotation investigations are being carried out to understand and optimise flotation processes) should be conducted to establish the effect of the key froth parameters (such as froth retention time) on the froth recovery. The determination of froth recovery in bench-scale conventional cells will present a significant improvement in the interpretation of laboratory data and will possibly serve as a basis for scale-up with respect to the froth phase. In addition, froth recovery studies in laboratory mechanical cells can enhance the understanding of the effect of froth on the use of bubble surface area flux to predict the overall flotation rate constant.

418

Z. T. Mathe er al.

2.

Further research is needed to study the phenomena of the froth sub-processes such as the drainage rate of particles on a size-by-size basis, drainage rate of water, and the influence of entrained particles on drainage. This will help in establishing the key froth parameters and their influence on the froth recovery factor.

3.

It is also imperative that a quantitative analysis of the influence of chemical factors on froth recovery is established. Presently, there is no measurable parameter within the froth that can be associated with the influence of chemistry.

ACKNOWLEDGMENTS Financial

support from Impala Platinum

Ltd is gratefully

acknowledged.

NOMENCLATURE

cf COO CI cll CP db

eig FRT h J, k, kd kde i kc2 ’

k’ m mi(h) me&h) mr,i(h) mr,i(h) mt,i :c

Qf

r Rc Rf R, Ro %

R w,u :b u(h)

vf vc vP

concentration of particles in the froth concentration of particles at height h in the froth concentration of particles at the froth-pulp interface material concentration of froth constituent n concentration of particles in the pulp phase bubble diameter entrainment factor for gangue of size i froth retention time froth depth superficial gas velocity collection zone flotation rate constant of species i froth drainage rate constant froth drainage rate constant for entrained particles of class i overall flotation rate constant (slurry + froth) froth transfer rate constant average particle mass mass of mineral i at froth height h entrained mass flow rate at froth height h mass flowrate of species i attached to bubbles at froth height h mass flowrate of particles diffusing from the upward-flowing stream to the downward-flowing stream total mass flowrate of component i ascending at a height h above the pulp-froth interface floatability constant mass flow rate from pulp to the interface concentrate flowrate ratio of pulp volumes (with air/without air) recovery of particles in the collection zone froth zone recovery gangue recovery overall flotation recovery water recovery recovery of water in the absence of gangue material ratio of froth volumes (with air/without air) bubble surface area flux velocity of downward-flowing streams volume of froth volume of floatable material in the collection zone pulp phase volume

Review of froth modelling

v(h) W

419

velocity of the bubble films water recovery rate, kg/m%

Greek symbols z Y P R w rl h

mean residence time column drainage coefficient film drainage coefficient froth removal coefficient dynamic shape factor slurry viscosity, mPa s exponential parameter

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9. 10. 11. 12. 13. 14. 15.

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