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Simulation of the effect of froth washing on flotation performance
Chemical Engineering Science 56 (2001) 6303–6311
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Simulation of the e ect of froth washing on !otation performance S. J. Neethling , J. J. Cilliers ∗ Froth and Foam Research Group Chemical Engineering Department, UMIST PO Box 88, Manchester M60 1QD, UK
Abstract The froth phase is extremely important in the operation of a !otation cell, seeing that -it is critical in determining the amount of unwanted gangue collected to the concentrate and thus the purity of the product. This paper uses a fundamentally based model of !owing froths to simulate the performance of a !otation cell. The study concentrates speci,cally on the e ect of wash water addition on the overall performance, i.e. the grade and the recovery. The froth model that is used within this work includes a large number of the e ects seen within a !otation froth and approaches their description from a fundamental point of view. Some of the phenomena that are included are bubble coalescence, liquid drainage including the e ects of gravity, surface tension and viscous dissipation, particle settling and particle dispersion. The results show the advantages and disadvantages of di erent water addition strategies on the performance of !otation vessels. Since most recent !otation work has been concerned with improving recoveries, rather than grade, the addition of water into !otation froths has been largely limited to column cells. This work demonstrates how water addition can be optimised in terms of both water addition point and quantity in order to produce the desired performance. ? 2001 Published by Elsevier Science Ltd. Keywords: Flotation; Simulation; Wash water
1. Introduction The froth phase of the !otation system, to a great extent, determines the separation performance, as the grade (ratio of desired to total solids recovered) of the product (concentrate) depends primarily on its structure and stability. The froth also contributes to the recovery (fraction of valuable solids recovered from the pulp to the concentrate) achieved, since the amount of desired material drop-back from the froth, together with the kinetics of the pulp phase, determines the recovery. This paper uses a fundamentally-based model of !owing froths to simulate the performance of a !otation cell. The study will concentrate speci,cally on the e ect of wash water addition on the overall performance, i.e. the grade and the recovery. Wash water is traditionally added to column !otation cells, but it has found some, though more limited, use in the operation of more conventional !otation cells. This paper will examine not only the e ect of wash water rate on the performance of !otation cells, but also the e ect of the wash water addition point (see Fig. 1). ∗
Corresponding author. E-mail address: [email protected](J. J. Cilliers).
The ,rst section gives a brief description of the assumptions inherent in the model and the mathematical formulations and boundary conditions resulting from these assumptions. A fuller description of the mathematical modelling of the liquid and gas motion can be found in previously published work (Neethling & Cilliers, 1998; Neethling, Cilliers, & Woodburn, 2000).
2. Mathematical modelling The mathematical model described below is solvable in both 2 and 3 dimensions, but since the computer code that has been written to solve it takes the best part of a day to solve a two-dimensional simulation, no three-dimensional simulations have yet been attempted. This calculation time is likely to improve dramatically as the code is further optimised for the non-linear set of equations that are being solved. In 2-D, the model has been implemented using cylindrical and Cartesian co-ordinates, but the examples show later are of the Cartesian type. All the equations are expressed relative to a stationary reference frame.
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S. J. Neethling, J. J. Cilliers / Chemical Engineering Science 56 (2001) 6303–6311
Fig. 1. Schematic diagram of !otation froth.
2.1. Gas motion and coalescence
2.2. Liquid motion and content
In order to model the motion of solids within a !otation cell, the motion of both the gas and liquid phases are required, since the motion of all three phases are interdependent. The froth motion is modelled using Laplace’s equation, and assumes the !ow to be incompressible and irrotational. Since !otation froths have a relatively low liquid content and !ow quite slowly, the assumptions inherent in Laplace’s equation produce accurate descriptions of the froth motion (Murphy, Zimmerman, & Woodburn, 1996; Neethling & Cilliers, 1998). This yields a !ow velocity vector, v, ˆ for each position within the system. The boundary condition for Laplace’s equation is the gas !ux through each of the boundaries. The !ux through the pulp-froth interface is the amount of air being added to cell. Gas can leave the froth in the cell by escaping from bursting bubbles at the surface, or within the froth that !ows over the weir. The permeability of the bursting surface to air (i.e. the super,cial !ux of air released by bursting) is a direct function of froth stability and is required model parameter. The !owrate of the gas in the bubble over!owing the weir is obtained by di erence. The diameter of the bubbles in the moving froth is required as a function of position, because, as we will see later, a number of froth phenomena are dependent on the local bubble diameter. The model incorporates a coalescence algorithm based on two criteria: the drainage time of the lamellae, and the stability of the lamellae at the equilibrium thickness. The equilibrium ,lm thickness is a function of liquid content, since it is a balance between the pressure exerted on the ,lm by the curvature of the Plateau border, van der Waals forces and the electric double layer repulsion. Both criteria are a function of bubble size and therefore, the stability of individual lamellae varies within a single !otation froth.
The motion of the liquid in the froth is described using an extension of the model of Verbist, Weaire, and Kraynik (1996), and is described brie!y here (Neethling et al., 2000). The liquid velocity, u, ˆ is estimated from a force balance along a Plateau border incorporating gravity, capillary and viscous dissipation forces. Since the froth is slow !owing, inertial forces are ignored, and the three forces sum to zero. This yields a liquid velocity equation, with the cross-sectional Plateau border area, A, as the only dependent variable.
@A + vx @x ; uˆ = k2 @A + vy −k1 A − √ × A @y √ 3 − =2 × k2 = 300
k2 A
−√ ×
where k1 =
g ; 150
(1)
This must be combined with a continuity equation · (uˆ × A × ) = Q(x; y)
(2)
to yield a boundary value problem that is solved numerically. The length of Plateau borders per volume of froth, , determines the e ect of bubble size on the behaviour of the liquid within the system and is of the form ˙ (bubble radius)−2 : Q(x; y) is the volume of water being added per volume of froth per time as a function of position and allows wash water addition. The solution of Eq. (2) gives the liquid content and liquid velocity vectors at each point through the froth.
S. J. Neethling, J. J. Cilliers / Chemical Engineering Science 56 (2001) 6303–6311
2.3. Solids motion 2.3.1. General In !otation there are, based on surface chemistry, two di erent types of solids, hydrophilic mineral that will not attach to bubble lamellae and hydrophobic mineral, able to attach. In the froth phase two classes of material exist: particles attached to the liquid=gas interfaces, and the unattached material that is found within the liquid in the Plateau borders. While the attached particles are assumed to be hydrophobic, the unattached material can be either hydrophilic or hydrophobic. Each class of solids can be subdivided according to size and density, and varying degrees of hydrophobicity. 2.3.2. Attached material This class is relatively simple to model, as it follows the bubbles. Coalescence in the froth or bursting on the surface is due to the rupture of the lamellae to which particles are attached. It is assumed that when a ,lm ruptures all the particles that were attached to it become detached. This is an initial modelling assumption that will need to be improved by means of experimental studies. 2.3.3. Unattached material The unattached material, while predominantly following the liquid in the Plateau borders, also moves relative to that liquid. As noted previously, attached particles are added to the unattached class as they detach from bursting or coalescing bubbles. Three di erent ways in which the particles move relative to the net !uid motion have been identi,ed; geometric dispersion, Plateau border dispersion and hindered particle settling. These mechanisms disperse the particles throughout the froth. 2.3.3.1. Geometric dispersion. Geometric dispersion is caused by the geometric layout of the Plateau borders through which the particles move. The Plateau borders form a three-dimensional network of interconnected channels, with four Plateau borders meeting at a vertex. When a particle enters a vertex from a Plateau border, there are three other Plateau borders by which it could leave. Geometric dispersion provides components of particle velocity that are perpendicular to the direction of net liquid motion. The net !ux is described by means of the following equation. @Csi − D d A | v − v − u | G b y Settling;i y @x ; (3) Fˆ G Net = @Csi −DG db A |vx − ux | @y where Fˆ G Net is the net !ux due to Geometric dispersion, Csi the mass of particle type i per volume of liquid, v the liquid velocity, u the gas velocity, vSettling; i the hindered
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settling velocity of particle type i; DG the dimensionless Geometric dispersion coeJcient (≈ 0:25); db the bubble diameter, A the cross-sectional Plateau border area and the length of Plateau borders per volume of froth. The value of DG of 0.25 was estimated by considering a two-dimensional grid of hexagons. 2.3.3.2. Plateau border dispersion. Plateau border dispersion is caused by the velocity pro,le of the !uid in the Plateau borders, which results in a particle velocity pro,le. Plateau border dispersion is in the same direction as the liquid motion and its !ux is described by the following equation. @Csi −Dp × dp × A × × |vx − ux | × @x ; Fˆ P Net = @Csi −Dp × dp × A × × |vy − uy | × @y (4) where Fˆ P Net is the net !ux due to Plateau border dispersion, dp the characteristic Plateau border diameter (2× radius of curvature) and DP the dimensionless Plateau border dispersion coeJcient (≈ 0:3). The Plateau border dispersion coeJcient (Dp ≈ 0:3) is estimated from the equivalent axial dispersion coeJcient (1=Peclet Number) for !ow in a cylindrical tube. 2.3.3.3. Hindered settling of particles. The rate at which the particles settle under gravity is a function of their size, density and concentration. Fine particles will predominantly follow the liquid, while coarse particles will tend to settle out of the froth. This leads to a strati,cation of particle sizes within the froth and hence particle size selective transport of particles from the pulp, through the froth, and across the weir. The settling velocity of each particle size class can be estimated from the following hindered settling formulae (Coulson & Richardson, 1993): g × ( s − ) × d2i 18 vTerm; i × ( 4:65 ) vSettling; i = 3
vTerm; i =
(5)
where s is the solid density, the liquid density, di the particle size, the voidage (volume fraction of liquid within the system), vTerm the terminal settling velocity in the absence of other particles and vSettling the apparent settling velocity of particles within the froth. The !ux of particles due to the !ux of liquid and their settling is given by vx × A × × Csi : (6) Fˆ Bulk = (vy − vSettling; i ) × A × × Csi
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2.3.4. Continuity equations A continuity equation is required for the !ow of solids within the Plateau borders. · (Fˆ Bulk + Fˆ G Net + Fˆ P Net ) = Rate of addition per volume:
(7)
for, at most, 3% of the volumetric feed to the cell. The solids kinetics were taken from an industrial operation !oating a 1% chalcopyrite ore. Three size classes were simulated: −38 m; +38–106 m and +106 m. The feed concentrations were as follows: −38 m
+38–106 m
+106 m
The rate of addition of solids to the Plateau borders is the same as the rate at which the particles become detached from the lamellae. If it is assumed that all bubbles are equally loaded, then the rate of particle addition to the Plateau borders is proportional to rate of loss of bubble surface area due to coalescence and bursting. Eq. (7) must be solved for each class of solids, i. The i equations are only indirectly interdependent in that the settling velocity depends on the total concentration of solids within the Plateau borders. The addition of wash water does not appear directly in the solids motion equations, as water addition does not directly a ect solids mass continuity. However, it can dramatically a ect the liquid motion, which a ects the solids motion through enhanced liquid transport and dispersion.
The system was simulated using cartesian co-ordinates. The simulated cell was 0:5 m long, 2 m wide and the froth depth was 0:275 m for all the simulations given in this paper. The permeability of the surface in every simulation was such that 80% of the gas entering the froth leaves due to bubble rupture at the surface. This is typical of industrial rougher !otation cells. It is recognised that water addition has, potentially, a signi,cant e ect on froth stability. However, without a suitable model or experimental data, this e ect was neglected.
2.4. The e
3.2. The e
The particles a ect the !uid by changing the density, viscosity and the volume entering the Plateau borders.
These simulations examine the e ect of wash water addition rate on cell performance, when added uniformly across the upper surface of the froth. The water rates explored cover both net positive and negative bias. Neutral bias, when there is zero net water !ux across the pulp-froth interface, is at a wash water addition rate of 108 l=h. The overall performance trends (Figs. 2 and 3) are as would be expected; an increase in water addition results in an increase in grade, but at the expense of recovery. It should be noted that the purest product theoretically obtainable is 31.5%, taking into account the copper content of chalcopyrite and imperfect liberation. Thus, at the highest water rate simulated, the product is very pure. It can be seen that the curves do not show a discontinuity at neutral bias. What is slightly unexpected from
• The density of the !uid is a ected by the solid concen-
tration:
n Csi + × ; f =
(8)
0
where f is the !uid density and the liquid density.
• The viscosity of the !uid is a ected by the solid con-
centration [5]:
f ≈ × (1 + 2:5 × (1 − ) + 14:1 × (1 − )2 + 0:00273 × e16:6(1− ) );
(9)
where f is the !uid viscosity and the liquid viscosity.
Desired material Gangue Rate constant
kg=m3
4:2 224 kg=m3 0:85 min−1
kg=m3
3:5 288 kg=m3 0:75 min−1
3. Simulation results 3.1. System details For all simulations, the pulp phase was considered to have a volume of 1:5 m3 and to be well mixed. The !otation kinetics in the pulp, and the pulp feed rate, were taken to be the same for each simulation. The feed rate to the cell was set at 150 l=min of a 45 wt% solids slurry, while the air rate was kept at 900 l=min for all simulations. The e ect of water addition on the pulp residence time was neglected, as it accounts
Fig. 2. Grade as a function of water rate.
1:7 kg=m3 128 kg=m3 0:34 min−1
S. J. Neethling, J. J. Cilliers / Chemical Engineering Science 56 (2001) 6303–6311
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Fig. 3. Recovery as a function of water rate. Fig. 5. Plateau border area (m2 ) for uniform wash water addition of 108 l=h.
Fig. 4. Grade-Recovery curve for variations in wash water rate.
these curves and the grade=recovery curve (Fig. 4) is that the position of the point of zero net bias is not as obvious from the grade and recovery curves as it would be, for instance, in the results from a column cell !oating very ,ne particles. The reasons for this are twofold. Firstly, the critical water rate is not that at neutral bias, but the rate at which the upward water rate equals the particle settling rate. At wash water rates exceeding this, particles can only move upwards by dispersion. For very ,ne particles, these rates are approximately equal, but as particle size increases, this di erence becomes more signi,cant. Since these simulations have a range of particle sizes, there is also a range of water rates over which the dominant particle transport mechanism changes from net liquid !ux to dispersion. Secondly, in column !otation cells with deep froths, there is no horizontal gas or liquid motion through most of the froth height and capillary e ects will cause the liquid !ow to become virtually uniform over the width of the cell. The comparatively shallow froths simulated here have a signi,cant horizontal component, which allows local regions of upward water !ow even when the net water bias is downwards. This non-uniform water !ow also obscures the location of the zero net water bias on the grade-recovery curve.
Fig. 6. Grade (%Cu) for uniform wash water addition of 108 l=h.
Fig. 7. Gangue concentration (kg=m3 ) for uniform wash water addition of 108 l=h.
Figs. 5 –8 are plots of some of the data produced for the uniform wash water simulation at neutral bias. The gangue and desired material concentrations shown here (and elsewhere in this paper) are for the intermediate
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Fig. 9. Grade as a function of water position. Fig. 8. Desired concentration (kg=m3 ) for uniform wash water addition of 108 l=h.
sized particle class (+38–106 m). The x and y axes are the grid numbers for the ,nite di erence solution. Since the cell is 0:5 m wide, the grid numbers represent 0:25 cm. It may appear anomalous that the largest Plateau borders are not found where the water is being added at the top, but rather in a band lower down (see Fig. 5). The main reason for this is that the wash water removes virtually all the gangue and a large portion of the desired material away from the upper reaches of the froth. Not only does this remove the volume of the solids from these Plateau borders, but also lowers the viscosity and allows the water to drain away more easily. The reason for the band of slightly wider Plateau borders across the middle of the froth is due, in part, to an accumulation of water where the entrained water meets the wash water. This band also corresponds to the region in which a sizeable amount of solids is again found and where, therefore, the viscosity of the slurry increases dramatically. The fact that this band, while being quite broad in this region, intercepts the front wall at roughly the height of the weir is due to the water bias being about neutral. The reason why the band of high desired material (Fig. 8) content also corresponds to the region of high water content is partly due to this being the lowest point at which wash water has a direct e ect, but is also because the desired material is hindered from settling through the region of high gangue concentration below it. These simulations con,rm that the froth model produces plausible results when the wash water rate is varied. The e ect of varying the position at which water is added is, however, less well understood and will be explored in the following section. 3.3. Wash water addition position In conventional !otation cells the wash water is generally either added uniformly by sprays over the entire froth
Fig. 10. Recovery as a function of water position.
surface, or by a single row of sprays near the weir over!ow. In these simulations these strategies will be compared, and the e ect of moving the single addition point will be examined. For these simulations, a single water addition rate of 108 l=h was used, the neutral bias point in the uniform water addition simulations. Figs. 9 and 10 show the grade and recovery, respectively, as the water addition position is changed. What is immediately evident is that the highest grade is not achieved when the water is added near the weir at the front of the cell, as is done conventionally, but occurs when the water is added towards the back of the froth. Further, the highest grade does not correspond to the lowest recovery, as observed when the wash water rate was changed (Fig. 4). Two e ects cause a decrease in grade when water is added near the front of the cell. Firstly, the majority of the wash water !ows directly over the weir, observed by a decrease in the concentrate solids concentration. Further, since most of the wash water !ows straight over the weir, the net bias is compromised and, as observed at low wash water rates, the grade decreases (see Fig. 16, as compared to Fig. 12 for Plateau border size near the weir over!ow). Secondly, a very large fraction of the bubbles burst at the top surface rather than over!ow the weir and release
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Fig. 12. Plateau border area (m2 ) for wash water addition 30% in from back wall.
Fig. 11. Grade Recovery curve for changing water position superimposed on curve for varying water rate.
the attached, desired material into the Plateau borders. Typical values, as in these simulations, are approximately 80%. Desired material released into the Plateau borders at the top surface has diJculty reaching the weir if released behind the wash water addition point. This explains the observed reduction in recovery as the wash water addition point is moved closer to the weir. This decrease does not occur close to the weir as most of the wash water !ows over the weir and a ords the desired material an opportunity to pass under it. Fig. 11 shows the grade-recovery behaviour observed when the water addition position is varied, superimposed on the grade recovery curve obtained when the water rate is changed but added uniformly. The results obtained when the wash water addition point is changed lie on a similar curve to the one obtained when the water rate is changed. Note in particular that water addition at the neutral bias rate at a single position 30% from the rear of the cell yields an equivalent recovery but a higher grade than when the wash water is added uniformly across the surface. The point for water addition near the weir lies virtually on (though very slightly below) the grade-recovery curve obtained for uniform water addition. It corresponds to an e ective uniform water rate that is much lower than was added as a sizeable portion of the water added in this region goes straight over the weir without in!uencing the performance of the cell. The behaviour of the cell in which water is added 30% of the distance from the back wall (Figs. 12–15) shows
Fig. 13. Grade (%Cu) for wash water addition 30% in from back wall.
Fig. 14. Gangue concentration (kg=m3 ) for wash water addition 30% in from back wall.
similar internal characteristics to the system in which the water is added 60% of the distance in from the back wall. The region directly below the water addition point is, as would be expected, characterised by very large Plateau borders (see Fig. 12). What is also observed, though, is that the Plateau borders on either side of this water addition point are not only much smaller than those near where the water is added, but also smaller than the Plateau borders elsewhere in the froth. The reason for this is
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Fig. 15. Desired concentration (kg=m3 ) for wash water addition 30% in from back wall.
Fig. 16. Plateau border area (m2 ) for wash water addition near front wall.
that the wash water (as it is intended to do) removes material from the Plateau borders in this vicinity. Since the solids content of the Plateau borders and the viscosity of the slurry within them are both strongly and directly related, the lower solids content in this region results in a lower viscosity. This, in turn means that the slurry in this vicinity is able to !ow away more easily. As the wash water moves away from the addition point, its strong downward motion becomes di used over a wider region. This gives an opportunity for some of the material released behind the wash water addition point to settle and=or disperse to a position from which it can move towards the weir below the region of strongest wash water e ect. This is characterised by the horizontal position where the desired solids concentration increases and then decreases with height (see Fig. 15). The concentration of gangue shows no similar saddle point, seeing that no gangue is release into the Plateau borders by due to coalescence and bursting (see Figs. 16 –19). 4. Conclusions This paper shows that the behaviour of !otation froths and the particles within them are not straightforward. This
Fig. 17. Grade (%Cu) for wash water addition near front wall.
Fig. 18. Gangue concentration (kg=m3 ) for wash water addition near front wall.
Fig. 19. Desired concentration (kg=m3 ) for wash water addition near front wall.
is especially true of the wash water addition position, where it was found that the simulations indicate that the best performance is achieved when the wash water is added slightly further back in the cell. This is due to these wash water addition positions being able remove a similar fraction of the gangue, while at the same time washing out less desired material. The ine ectiveness of water addition right at the front of the cell is because a sizeable fraction of the water !ows straight over the weir without having any signi,cant in!uence on the cell performance.
S. J. Neethling, J. J. Cilliers / Chemical Engineering Science 56 (2001) 6303–6311
Even the uniform wash water addition, while showing the totally expected trends, has a slightly unexpected water distribution, with the wettest portion of the froth being found in the middle. While the simulation study cannot eliminate the need for experimental studies, these studies can be carried out far more quickly and inexpensively than a set of equivalent experiments and could be used to restrict the experimental studies to a set of promising looking alternatives. References Coulson, J. M., & Richardson, J. F. (1993). Chemical Engineering, Vol. 2, Oxford: Pergamon Press.
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Murphy, D. G., Zimmerman, W., & Woodburn, E. T. (1996). Kinematic model of bubble motion in a !owing froth. Powder Technology, 87, 3–12. Neethling, S. J., & Cilliers, J. J. (1998). A visual kinematic model of !owing foams incorporating coalescence. Powder Technology, 101, 249–256. Neethling, S. J., Cilliers, J. J., & Woodburn, E. T. (2000). The distribution of liquid in !owing foams. Chemical Engineering Science, 55, 4021–4028. Verbist, G., Weaire, D., & Kraynik, A. M. (1996). The foam drainage equation. Journal of Physics Condensed Matter, 8, 3715–3731.
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Simulation of the e ect of froth washing on !otation performance S. J. Neethling , J. J. Cilliers ∗ Froth and Foam Research Group Chemical Engineering Department, UMIST PO Box 88, Manchester M60 1QD, UK
Abstract The froth phase is extremely important in the operation of a !otation cell, seeing that -it is critical in determining the amount of unwanted gangue collected to the concentrate and thus the purity of the product. This paper uses a fundamentally based model of !owing froths to simulate the performance of a !otation cell. The study concentrates speci,cally on the e ect of wash water addition on the overall performance, i.e. the grade and the recovery. The froth model that is used within this work includes a large number of the e ects seen within a !otation froth and approaches their description from a fundamental point of view. Some of the phenomena that are included are bubble coalescence, liquid drainage including the e ects of gravity, surface tension and viscous dissipation, particle settling and particle dispersion. The results show the advantages and disadvantages of di erent water addition strategies on the performance of !otation vessels. Since most recent !otation work has been concerned with improving recoveries, rather than grade, the addition of water into !otation froths has been largely limited to column cells. This work demonstrates how water addition can be optimised in terms of both water addition point and quantity in order to produce the desired performance. ? 2001 Published by Elsevier Science Ltd. Keywords: Flotation; Simulation; Wash water
1. Introduction The froth phase of the !otation system, to a great extent, determines the separation performance, as the grade (ratio of desired to total solids recovered) of the product (concentrate) depends primarily on its structure and stability. The froth also contributes to the recovery (fraction of valuable solids recovered from the pulp to the concentrate) achieved, since the amount of desired material drop-back from the froth, together with the kinetics of the pulp phase, determines the recovery. This paper uses a fundamentally-based model of !owing froths to simulate the performance of a !otation cell. The study will concentrate speci,cally on the e ect of wash water addition on the overall performance, i.e. the grade and the recovery. Wash water is traditionally added to column !otation cells, but it has found some, though more limited, use in the operation of more conventional !otation cells. This paper will examine not only the e ect of wash water rate on the performance of !otation cells, but also the e ect of the wash water addition point (see Fig. 1). ∗
Corresponding author. E-mail address: [email protected](J. J. Cilliers).
The ,rst section gives a brief description of the assumptions inherent in the model and the mathematical formulations and boundary conditions resulting from these assumptions. A fuller description of the mathematical modelling of the liquid and gas motion can be found in previously published work (Neethling & Cilliers, 1998; Neethling, Cilliers, & Woodburn, 2000).
2. Mathematical modelling The mathematical model described below is solvable in both 2 and 3 dimensions, but since the computer code that has been written to solve it takes the best part of a day to solve a two-dimensional simulation, no three-dimensional simulations have yet been attempted. This calculation time is likely to improve dramatically as the code is further optimised for the non-linear set of equations that are being solved. In 2-D, the model has been implemented using cylindrical and Cartesian co-ordinates, but the examples show later are of the Cartesian type. All the equations are expressed relative to a stationary reference frame.
0009-2509/01/$ - see front matter ? 2001 Published by Elsevier Science Ltd. PII: S 0 0 0 9 - 2 5 0 9 ( 0 1 ) 0 0 2 4 8 - 2
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S. J. Neethling, J. J. Cilliers / Chemical Engineering Science 56 (2001) 6303–6311
Fig. 1. Schematic diagram of !otation froth.
2.1. Gas motion and coalescence
2.2. Liquid motion and content
In order to model the motion of solids within a !otation cell, the motion of both the gas and liquid phases are required, since the motion of all three phases are interdependent. The froth motion is modelled using Laplace’s equation, and assumes the !ow to be incompressible and irrotational. Since !otation froths have a relatively low liquid content and !ow quite slowly, the assumptions inherent in Laplace’s equation produce accurate descriptions of the froth motion (Murphy, Zimmerman, & Woodburn, 1996; Neethling & Cilliers, 1998). This yields a !ow velocity vector, v, ˆ for each position within the system. The boundary condition for Laplace’s equation is the gas !ux through each of the boundaries. The !ux through the pulp-froth interface is the amount of air being added to cell. Gas can leave the froth in the cell by escaping from bursting bubbles at the surface, or within the froth that !ows over the weir. The permeability of the bursting surface to air (i.e. the super,cial !ux of air released by bursting) is a direct function of froth stability and is required model parameter. The !owrate of the gas in the bubble over!owing the weir is obtained by di erence. The diameter of the bubbles in the moving froth is required as a function of position, because, as we will see later, a number of froth phenomena are dependent on the local bubble diameter. The model incorporates a coalescence algorithm based on two criteria: the drainage time of the lamellae, and the stability of the lamellae at the equilibrium thickness. The equilibrium ,lm thickness is a function of liquid content, since it is a balance between the pressure exerted on the ,lm by the curvature of the Plateau border, van der Waals forces and the electric double layer repulsion. Both criteria are a function of bubble size and therefore, the stability of individual lamellae varies within a single !otation froth.
The motion of the liquid in the froth is described using an extension of the model of Verbist, Weaire, and Kraynik (1996), and is described brie!y here (Neethling et al., 2000). The liquid velocity, u, ˆ is estimated from a force balance along a Plateau border incorporating gravity, capillary and viscous dissipation forces. Since the froth is slow !owing, inertial forces are ignored, and the three forces sum to zero. This yields a liquid velocity equation, with the cross-sectional Plateau border area, A, as the only dependent variable.
@A + vx @x ; uˆ = k2 @A + vy −k1 A − √ × A @y √ 3 − =2 × k2 = 300
k2 A
−√ ×
where k1 =
g ; 150
(1)
This must be combined with a continuity equation · (uˆ × A × ) = Q(x; y)
(2)
to yield a boundary value problem that is solved numerically. The length of Plateau borders per volume of froth, , determines the e ect of bubble size on the behaviour of the liquid within the system and is of the form ˙ (bubble radius)−2 : Q(x; y) is the volume of water being added per volume of froth per time as a function of position and allows wash water addition. The solution of Eq. (2) gives the liquid content and liquid velocity vectors at each point through the froth.
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2.3. Solids motion 2.3.1. General In !otation there are, based on surface chemistry, two di erent types of solids, hydrophilic mineral that will not attach to bubble lamellae and hydrophobic mineral, able to attach. In the froth phase two classes of material exist: particles attached to the liquid=gas interfaces, and the unattached material that is found within the liquid in the Plateau borders. While the attached particles are assumed to be hydrophobic, the unattached material can be either hydrophilic or hydrophobic. Each class of solids can be subdivided according to size and density, and varying degrees of hydrophobicity. 2.3.2. Attached material This class is relatively simple to model, as it follows the bubbles. Coalescence in the froth or bursting on the surface is due to the rupture of the lamellae to which particles are attached. It is assumed that when a ,lm ruptures all the particles that were attached to it become detached. This is an initial modelling assumption that will need to be improved by means of experimental studies. 2.3.3. Unattached material The unattached material, while predominantly following the liquid in the Plateau borders, also moves relative to that liquid. As noted previously, attached particles are added to the unattached class as they detach from bursting or coalescing bubbles. Three di erent ways in which the particles move relative to the net !uid motion have been identi,ed; geometric dispersion, Plateau border dispersion and hindered particle settling. These mechanisms disperse the particles throughout the froth. 2.3.3.1. Geometric dispersion. Geometric dispersion is caused by the geometric layout of the Plateau borders through which the particles move. The Plateau borders form a three-dimensional network of interconnected channels, with four Plateau borders meeting at a vertex. When a particle enters a vertex from a Plateau border, there are three other Plateau borders by which it could leave. Geometric dispersion provides components of particle velocity that are perpendicular to the direction of net liquid motion. The net !ux is described by means of the following equation. @Csi − D d A | v − v − u | G b y Settling;i y @x ; (3) Fˆ G Net = @Csi −DG db A |vx − ux | @y where Fˆ G Net is the net !ux due to Geometric dispersion, Csi the mass of particle type i per volume of liquid, v the liquid velocity, u the gas velocity, vSettling; i the hindered
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settling velocity of particle type i; DG the dimensionless Geometric dispersion coeJcient (≈ 0:25); db the bubble diameter, A the cross-sectional Plateau border area and the length of Plateau borders per volume of froth. The value of DG of 0.25 was estimated by considering a two-dimensional grid of hexagons. 2.3.3.2. Plateau border dispersion. Plateau border dispersion is caused by the velocity pro,le of the !uid in the Plateau borders, which results in a particle velocity pro,le. Plateau border dispersion is in the same direction as the liquid motion and its !ux is described by the following equation. @Csi −Dp × dp × A × × |vx − ux | × @x ; Fˆ P Net = @Csi −Dp × dp × A × × |vy − uy | × @y (4) where Fˆ P Net is the net !ux due to Plateau border dispersion, dp the characteristic Plateau border diameter (2× radius of curvature) and DP the dimensionless Plateau border dispersion coeJcient (≈ 0:3). The Plateau border dispersion coeJcient (Dp ≈ 0:3) is estimated from the equivalent axial dispersion coeJcient (1=Peclet Number) for !ow in a cylindrical tube. 2.3.3.3. Hindered settling of particles. The rate at which the particles settle under gravity is a function of their size, density and concentration. Fine particles will predominantly follow the liquid, while coarse particles will tend to settle out of the froth. This leads to a strati,cation of particle sizes within the froth and hence particle size selective transport of particles from the pulp, through the froth, and across the weir. The settling velocity of each particle size class can be estimated from the following hindered settling formulae (Coulson & Richardson, 1993): g × ( s − ) × d2i 18 vTerm; i × ( 4:65 ) vSettling; i = 3
vTerm; i =
(5)
where s is the solid density, the liquid density, di the particle size, the voidage (volume fraction of liquid within the system), vTerm the terminal settling velocity in the absence of other particles and vSettling the apparent settling velocity of particles within the froth. The !ux of particles due to the !ux of liquid and their settling is given by vx × A × × Csi : (6) Fˆ Bulk = (vy − vSettling; i ) × A × × Csi
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2.3.4. Continuity equations A continuity equation is required for the !ow of solids within the Plateau borders. · (Fˆ Bulk + Fˆ G Net + Fˆ P Net ) = Rate of addition per volume:
(7)
for, at most, 3% of the volumetric feed to the cell. The solids kinetics were taken from an industrial operation !oating a 1% chalcopyrite ore. Three size classes were simulated: −38 m; +38–106 m and +106 m. The feed concentrations were as follows: −38 m
+38–106 m
+106 m
The rate of addition of solids to the Plateau borders is the same as the rate at which the particles become detached from the lamellae. If it is assumed that all bubbles are equally loaded, then the rate of particle addition to the Plateau borders is proportional to rate of loss of bubble surface area due to coalescence and bursting. Eq. (7) must be solved for each class of solids, i. The i equations are only indirectly interdependent in that the settling velocity depends on the total concentration of solids within the Plateau borders. The addition of wash water does not appear directly in the solids motion equations, as water addition does not directly a ect solids mass continuity. However, it can dramatically a ect the liquid motion, which a ects the solids motion through enhanced liquid transport and dispersion.
The system was simulated using cartesian co-ordinates. The simulated cell was 0:5 m long, 2 m wide and the froth depth was 0:275 m for all the simulations given in this paper. The permeability of the surface in every simulation was such that 80% of the gas entering the froth leaves due to bubble rupture at the surface. This is typical of industrial rougher !otation cells. It is recognised that water addition has, potentially, a signi,cant e ect on froth stability. However, without a suitable model or experimental data, this e ect was neglected.
2.4. The e
3.2. The e
The particles a ect the !uid by changing the density, viscosity and the volume entering the Plateau borders.
These simulations examine the e ect of wash water addition rate on cell performance, when added uniformly across the upper surface of the froth. The water rates explored cover both net positive and negative bias. Neutral bias, when there is zero net water !ux across the pulp-froth interface, is at a wash water addition rate of 108 l=h. The overall performance trends (Figs. 2 and 3) are as would be expected; an increase in water addition results in an increase in grade, but at the expense of recovery. It should be noted that the purest product theoretically obtainable is 31.5%, taking into account the copper content of chalcopyrite and imperfect liberation. Thus, at the highest water rate simulated, the product is very pure. It can be seen that the curves do not show a discontinuity at neutral bias. What is slightly unexpected from
• The density of the !uid is a ected by the solid concen-
tration:
n Csi + × ; f =
(8)
0
where f is the !uid density and the liquid density.
• The viscosity of the !uid is a ected by the solid con-
centration [5]:
f ≈ × (1 + 2:5 × (1 − ) + 14:1 × (1 − )2 + 0:00273 × e16:6(1− ) );
(9)
where f is the !uid viscosity and the liquid viscosity.
Desired material Gangue Rate constant
kg=m3
4:2 224 kg=m3 0:85 min−1
kg=m3
3:5 288 kg=m3 0:75 min−1
3. Simulation results 3.1. System details For all simulations, the pulp phase was considered to have a volume of 1:5 m3 and to be well mixed. The !otation kinetics in the pulp, and the pulp feed rate, were taken to be the same for each simulation. The feed rate to the cell was set at 150 l=min of a 45 wt% solids slurry, while the air rate was kept at 900 l=min for all simulations. The e ect of water addition on the pulp residence time was neglected, as it accounts
Fig. 2. Grade as a function of water rate.
1:7 kg=m3 128 kg=m3 0:34 min−1
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Fig. 3. Recovery as a function of water rate. Fig. 5. Plateau border area (m2 ) for uniform wash water addition of 108 l=h.
Fig. 4. Grade-Recovery curve for variations in wash water rate.
these curves and the grade=recovery curve (Fig. 4) is that the position of the point of zero net bias is not as obvious from the grade and recovery curves as it would be, for instance, in the results from a column cell !oating very ,ne particles. The reasons for this are twofold. Firstly, the critical water rate is not that at neutral bias, but the rate at which the upward water rate equals the particle settling rate. At wash water rates exceeding this, particles can only move upwards by dispersion. For very ,ne particles, these rates are approximately equal, but as particle size increases, this di erence becomes more signi,cant. Since these simulations have a range of particle sizes, there is also a range of water rates over which the dominant particle transport mechanism changes from net liquid !ux to dispersion. Secondly, in column !otation cells with deep froths, there is no horizontal gas or liquid motion through most of the froth height and capillary e ects will cause the liquid !ow to become virtually uniform over the width of the cell. The comparatively shallow froths simulated here have a signi,cant horizontal component, which allows local regions of upward water !ow even when the net water bias is downwards. This non-uniform water !ow also obscures the location of the zero net water bias on the grade-recovery curve.
Fig. 6. Grade (%Cu) for uniform wash water addition of 108 l=h.
Fig. 7. Gangue concentration (kg=m3 ) for uniform wash water addition of 108 l=h.
Figs. 5 –8 are plots of some of the data produced for the uniform wash water simulation at neutral bias. The gangue and desired material concentrations shown here (and elsewhere in this paper) are for the intermediate
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Fig. 9. Grade as a function of water position. Fig. 8. Desired concentration (kg=m3 ) for uniform wash water addition of 108 l=h.
sized particle class (+38–106 m). The x and y axes are the grid numbers for the ,nite di erence solution. Since the cell is 0:5 m wide, the grid numbers represent 0:25 cm. It may appear anomalous that the largest Plateau borders are not found where the water is being added at the top, but rather in a band lower down (see Fig. 5). The main reason for this is that the wash water removes virtually all the gangue and a large portion of the desired material away from the upper reaches of the froth. Not only does this remove the volume of the solids from these Plateau borders, but also lowers the viscosity and allows the water to drain away more easily. The reason for the band of slightly wider Plateau borders across the middle of the froth is due, in part, to an accumulation of water where the entrained water meets the wash water. This band also corresponds to the region in which a sizeable amount of solids is again found and where, therefore, the viscosity of the slurry increases dramatically. The fact that this band, while being quite broad in this region, intercepts the front wall at roughly the height of the weir is due to the water bias being about neutral. The reason why the band of high desired material (Fig. 8) content also corresponds to the region of high water content is partly due to this being the lowest point at which wash water has a direct e ect, but is also because the desired material is hindered from settling through the region of high gangue concentration below it. These simulations con,rm that the froth model produces plausible results when the wash water rate is varied. The e ect of varying the position at which water is added is, however, less well understood and will be explored in the following section. 3.3. Wash water addition position In conventional !otation cells the wash water is generally either added uniformly by sprays over the entire froth
Fig. 10. Recovery as a function of water position.
surface, or by a single row of sprays near the weir over!ow. In these simulations these strategies will be compared, and the e ect of moving the single addition point will be examined. For these simulations, a single water addition rate of 108 l=h was used, the neutral bias point in the uniform water addition simulations. Figs. 9 and 10 show the grade and recovery, respectively, as the water addition position is changed. What is immediately evident is that the highest grade is not achieved when the water is added near the weir at the front of the cell, as is done conventionally, but occurs when the water is added towards the back of the froth. Further, the highest grade does not correspond to the lowest recovery, as observed when the wash water rate was changed (Fig. 4). Two e ects cause a decrease in grade when water is added near the front of the cell. Firstly, the majority of the wash water !ows directly over the weir, observed by a decrease in the concentrate solids concentration. Further, since most of the wash water !ows straight over the weir, the net bias is compromised and, as observed at low wash water rates, the grade decreases (see Fig. 16, as compared to Fig. 12 for Plateau border size near the weir over!ow). Secondly, a very large fraction of the bubbles burst at the top surface rather than over!ow the weir and release
S. J. Neethling, J. J. Cilliers / Chemical Engineering Science 56 (2001) 6303–6311
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Fig. 12. Plateau border area (m2 ) for wash water addition 30% in from back wall.
Fig. 11. Grade Recovery curve for changing water position superimposed on curve for varying water rate.
the attached, desired material into the Plateau borders. Typical values, as in these simulations, are approximately 80%. Desired material released into the Plateau borders at the top surface has diJculty reaching the weir if released behind the wash water addition point. This explains the observed reduction in recovery as the wash water addition point is moved closer to the weir. This decrease does not occur close to the weir as most of the wash water !ows over the weir and a ords the desired material an opportunity to pass under it. Fig. 11 shows the grade-recovery behaviour observed when the water addition position is varied, superimposed on the grade recovery curve obtained when the water rate is changed but added uniformly. The results obtained when the wash water addition point is changed lie on a similar curve to the one obtained when the water rate is changed. Note in particular that water addition at the neutral bias rate at a single position 30% from the rear of the cell yields an equivalent recovery but a higher grade than when the wash water is added uniformly across the surface. The point for water addition near the weir lies virtually on (though very slightly below) the grade-recovery curve obtained for uniform water addition. It corresponds to an e ective uniform water rate that is much lower than was added as a sizeable portion of the water added in this region goes straight over the weir without in!uencing the performance of the cell. The behaviour of the cell in which water is added 30% of the distance from the back wall (Figs. 12–15) shows
Fig. 13. Grade (%Cu) for wash water addition 30% in from back wall.
Fig. 14. Gangue concentration (kg=m3 ) for wash water addition 30% in from back wall.
similar internal characteristics to the system in which the water is added 60% of the distance in from the back wall. The region directly below the water addition point is, as would be expected, characterised by very large Plateau borders (see Fig. 12). What is also observed, though, is that the Plateau borders on either side of this water addition point are not only much smaller than those near where the water is added, but also smaller than the Plateau borders elsewhere in the froth. The reason for this is
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Fig. 15. Desired concentration (kg=m3 ) for wash water addition 30% in from back wall.
Fig. 16. Plateau border area (m2 ) for wash water addition near front wall.
that the wash water (as it is intended to do) removes material from the Plateau borders in this vicinity. Since the solids content of the Plateau borders and the viscosity of the slurry within them are both strongly and directly related, the lower solids content in this region results in a lower viscosity. This, in turn means that the slurry in this vicinity is able to !ow away more easily. As the wash water moves away from the addition point, its strong downward motion becomes di used over a wider region. This gives an opportunity for some of the material released behind the wash water addition point to settle and=or disperse to a position from which it can move towards the weir below the region of strongest wash water e ect. This is characterised by the horizontal position where the desired solids concentration increases and then decreases with height (see Fig. 15). The concentration of gangue shows no similar saddle point, seeing that no gangue is release into the Plateau borders by due to coalescence and bursting (see Figs. 16 –19). 4. Conclusions This paper shows that the behaviour of !otation froths and the particles within them are not straightforward. This
Fig. 17. Grade (%Cu) for wash water addition near front wall.
Fig. 18. Gangue concentration (kg=m3 ) for wash water addition near front wall.
Fig. 19. Desired concentration (kg=m3 ) for wash water addition near front wall.
is especially true of the wash water addition position, where it was found that the simulations indicate that the best performance is achieved when the wash water is added slightly further back in the cell. This is due to these wash water addition positions being able remove a similar fraction of the gangue, while at the same time washing out less desired material. The ine ectiveness of water addition right at the front of the cell is because a sizeable fraction of the water !ows straight over the weir without having any signi,cant in!uence on the cell performance.
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Even the uniform wash water addition, while showing the totally expected trends, has a slightly unexpected water distribution, with the wettest portion of the froth being found in the middle. While the simulation study cannot eliminate the need for experimental studies, these studies can be carried out far more quickly and inexpensively than a set of equivalent experiments and could be used to restrict the experimental studies to a set of promising looking alternatives. References Coulson, J. M., & Richardson, J. F. (1993). Chemical Engineering, Vol. 2, Oxford: Pergamon Press.
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Murphy, D. G., Zimmerman, W., & Woodburn, E. T. (1996). Kinematic model of bubble motion in a !owing froth. Powder Technology, 87, 3–12. Neethling, S. J., & Cilliers, J. J. (1998). A visual kinematic model of !owing foams incorporating coalescence. Powder Technology, 101, 249–256. Neethling, S. J., Cilliers, J. J., & Woodburn, E. T. (2000). The distribution of liquid in !owing foams. Chemical Engineering Science, 55, 4021–4028. Verbist, G., Weaire, D., & Kraynik, A. M. (1996). The foam drainage equation. Journal of Physics Condensed Matter, 8, 3715–3731.