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The development of a cavern model for mechanical flotation cells
Minerals Engineering 23 (2010) 968–972
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
The development of a cavern model for mechanical flotation cells C.W. Bakker a,*, C.J. Meyer b, D.A. Deglon a a b
Centre for Minerals Research, Dept. of Chemical Engineering, University of Cape Town, South Africa Centre for Research in Computational and Applied Mechanics, University of Cape Town, South Africa
a r t i c l e
i n f o
Article history: Received 3 December 2009 Accepted 17 March 2010 Available online 14 April 2010 Keywords: Modelling Computational fluid dynamics Froth flotation Fine particle processing Agitation
a b s t r a c t Certain fine particle and high solid concentration mineral slurries used in the froth flotation process have been shown to exhibit non-Newtonian rheologies, including a yield stress. The mixing characteristics of these fluids are often problematic as a cavern of yielded fluid forms around the impeller whilst the rest of the fluid remains stagnant and therefore unmixed. This paper aims to develop a semi-empirical model to calculate the height of caverns forming in non-Newtonian mineral slurries in a mechanical flotation cell. Cavern shapes in a pilot-scale Batequip flotation cell were numerically determined for a range of mineral slurries using an experimentally validated Computational Fluid Dynamics (CFD) model. Development of the cavern height model was based on the assumption that the cavern boundary was formed where the shear stress imposed on the slurry equaled the fluid yield stress and also that the flow along the cell walls could be represented by an annular wall jet. It was found that the cavern height was directly proportional to the product of the slurry density and the square of the impeller tip speed, and inversely proportional to the slurry yield stress. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Mining operations, such as lead–zinc and platinum, are increasingly having to grind ores to ever finer particle sizes in order to liberate the valuable minerals (Pease et al., 2006). These operations typically run flotation circuits at high solids concentrations, mainly in order to maximise residence time, but also to limit water consumption. This may cause mineral slurries to exhibit non-Newtonian pseudoplastic rheological behaviour, including a yield stress (Prestidge, 1997; Tseng and Chen, 2003; He et al., 2006). This behaviour is exacerbated when processing ores containing minerals known to adversely affect rheology, such as certain phyllosilicates. Fluids with similar rheological behaviour are encountered in a variety of other industries, including those involving food, pharmaceutical, cosmetic and paper pulp processing, and in many cases have been found to be problematic. Mixing of these yield stress fluids in mechanically stirred tanks has been found to cause the formation of a yielded region of fluid around the impeller, with the rest of the fluid in the bulk of the tank remaining unyielded, and therefore stagnant (Elson and Cheeseman, 1986; Moore et al., 1995; Wilkens et al., 2005). The formation of a cavern in these stirred tanks resulted in diminished mixing of the fluid in the tank, and is thus undesirable. The formation of such a cavern in a mechanical flotation cell has been found to form in certain slurries and operating conditions (Bakker et al., 2009), effectively reducing * Corresponding author. Tel.: +27 78 619 2835. E-mail address: [email protected] (C.W. Bakker). 0892-6875/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2010.03.016
the active volume of the cell where flotation occurs, since a high rate of turbulent energy dissipation, required for the flotation of fine particles (Deglon, 2005; Schubert, 2008), would be absent in the stagnant regions of slurry. Since non-Newtonian fluids encountered in industry are often solid suspensions and therefore opaque, many experimental techniques traditionally used to measure local fluid flow are unsuitable. In an attempt to be able to predict the shape and size of the cavern formed in these fluids, various researchers have developed empirical and mathematical models to calculate the cavern dimensions (Solomon et al., 1981; Elson and Cheeseman, 1986; Elson, 1990; Moore et al., 1995; Amanullah et al., 1998; Wilkens et al., 2005). Solomon et al. (1981) developed a mathematical model assuming that the cavern could best be represented by a sphere centered around the impeller. The shape of a sphere was chosen under the assumption that all the power dissipated by the impeller was transmitted to the cavern boundary, meaning that the shear rate at the boundary was equal to the yield stress. It also assumed that the fluid velocity was predominantly tangential. The model was only valid until the cavern reached the tank wall. This model was modified by Elson and Cheeseman (1986) to become a right angled cylinder, again only valid until the cavern reached the walls. Amanullah et al. (1998) developed a new model for highly shear thinning fluids by making the total momentum from the impeller equal to the sum of both the tangential and axial components from an axial flow impeller. The shape of the cavern was defined as a torus. Since the fluid did not exhibit a yield stress, a velocity of 1 103 m s1 was used to define the cavern boundary. Wilkens
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C.W. Bakker et al. / Minerals Engineering 23 (2010) 968–972
et al. (2005) used a novel experimental method to measure the cavern size in tomato sauce, or ketchup, a fluid known to exhibit a yield stress. The fluid was agitated by both a Rushton radial flow impeller and a pitch blade turbine. The cavern shape was found to be best approximated by an elliptical torus. The agitation method in mechanical flotation cells is similar to that in mechanically stirred tanks, however due to the ring of stator blades around the impeller, the flow has a predominantly radial velocity, leaving the impeller region in a form of free stream radial jet (Fuerstenau et al., 2007). This radial jet also always reaches the tank wall before forming two recirculation loops or vortices that return to the impeller region (Xia et al., 2009). For this reason, although the same reasoning can be applied to the development of a cavern model for a mechanical flotation cell, the models themselves cannot be used. A cavern model has however been developed for a different mixing device having similar flow characteristics to those in flotation cells, namely jet mixers (Meyer and Etchells, 2007). These devices consist of cylindrical tanks with either one or more pulse tubes inserted into the fluid to be mixed. The tubes either vent the fluid contained inside them, forming a jet, or refill, sucking in the fluid. In the venting mode this jet strikes the base of the tank and then follows the tank wall up the side of the tank. When mixing yield stress fluids, the fluid only follows the wall until the shear stress in the fluid cannot overcome the fluids yield stress and so forms a recirculation loop, returning to the jet. The shear stress imposed on the fluid was assumed to be from wall friction stresses as it flowed along the tank walls. Due to the difficulty in obtaining cavern measurements in opaque particle suspensions such as mineral slurries, this paper aims to use a experimentally validated Computational Fluid Dynamics (CFD) model, developed in previous research (Bakker et al., 2009), to provide a useful engineering prediction, rather than a physically measurement of the effect of the rheology of mineral slurries on the hydrodynamics in the flotation cell and thus the shape and size of caverns formed. This data will then be used to develop a model to predict whether a cavern will form and if so, its height. 2. Model description
Fig. 1. Flotation cell setup.
which cause repulsive forces between particles (Burdukova, 2007), causing the slurry to represent the extreme rheologies as found in industrial slurries. Three different solid concentrations were modelled for each type of slurry. For the Bindura nickel slurry concentrations of 40, 50 and 60 wt.% were chosen since slurry solid concentrations of approximately 40 wt.% are typical of many mineral processing operations, whereas slurries at 50 wt.% represent the upper bounds of current flotation practice, although it is anticipated that higher solid concentrations will be more common in the future due to pressures on reducing water consumption. The kaolin was found to be too viscous to pour after a solid concentration of 40 wt.%, and so values of 25, 30 and 40 wt.% were instead modelled. Similar findings were made by Loginov et al. (2008). The rheology of the slurries were experimentally measured using a Rheolab MCR 300 rheometer over a shear rate range of 0.1– 1000 s1. It was found that all slurries exhibited a yield stress and had a shear thinning behaviour at low shear rates, which was best described using the Herschel–Bulkley non-Newtonian model, defined as
This section describes the numerical model used to simulate the non-Newtonian mineral slurries in a mechanical flotation cell. The experimental validation of the modelling methodology used is described in Bakker et al. (2009).
s ¼ sy þ kc_ n
2.1. Geometry
s ¼ sy þ kc_
This study was conducted on a 100 l pilot-scale Bateman flotation cell (Fig. 1) as used in previous research (Bakker et al., 2009), with diameter, T ¼ 540 mm, and liquid depth, Z ¼ 445 mm, which consists of a 6 bladed impeller of maximum diameter, D ¼ 150 mm, tapering down to 70 mm at the lower edge, and height W ¼ 100 mm. This was surrounded by a ring of 16 stator blades of 70 mm by 36 mm attached to a 250 mm diameter stator disc. The impeller bottom clearance was C b ¼ 83 mm.
This has also been found in other research using limestone slurries (He et al., 2006), and by Tseng and Chen (2003) in nickel powders dispersed in terpineol solvent. The shear rate below which the Herschel–Bulkley model is used, and above which the Bingham plastic model is used is equal to the intercept of the two model curves. Constants used in each non-Newtonian model are shown in Table 1.
ð1Þ
At higher shear rates the gradient of the rheogram was constant, and therefore the slurry was modelled as a Bingham plastic fluid.
ð2Þ
2.3. CFD model 2.2. Fluid rheology Two mineral slurries were used in this research, namely a kaolin slurry and Bindura nickel ore slurry. Both slurries are also known to have complex mineralogies, and thus problematic rheologies. The ores were also ground to extremely fine particle sizes, the d50 for the kaolin being 2.5 lm, and for the Bindura ore 15 lm, and contacted with ‘plant water’, which contains high dissolved solids. This was done in order to get rid of any surface charge
The CFD simulations were implemented using the commercial CFD software, ANSYS Fluent 6.3 (ANSYS, 2007). A single phase model was used with a UDF to represent the slurries non-Newtonian rheology. Only half of the flotation cell was modelled, using a mesh of 778,700 hexahedral cells as shown in Fig. 2. The full 360° mesh is shown to improve clarity. Grid refinement tests were conducted and it was found that there was no appreciable difference in results predicted with finer meshes.
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Table 1 Densities and rheology model parameters of slurries.
q (kg/m3)
sy HB (Pa)
kHB (kg/m s)
nHB
sy Bing (Pa)
kBing (kg/m s)
Bindura 40 wt.%
1364
0.456
0.439
0.353
1.945
6:79 103
Bindura 50 wt.%
1500
1.742
1.020
0.427
8.189
1:47 102
Bindura 60 wt.%
1667
4.741
3.774
0.565
Kaolin 25 wt.%
1182
1.838
1.756
0.254
Kaolin 30 wt.%
1226
2.740
3.987
0.231
14.74
8:64 103
Kaolin 40 wt.%
1327
8.67
11.700
0.266
57.46
1:81 102
52.18 7.038
7:42 102 6:55 103
(P/V), of approximately 0.6–5.6 kW/m3, which straddles and exceeds the range of 1–3 kW/m3 found in typical industrial mechanical flotation cells (Deglon et al., 2000). The combined Herschel–Bulkley and Bingham plastic non-Newtonian rheology models were input into Fluent via a user-defined function. Since Fluent requires an equation defining value of the apparent viscosity, g ¼ s=c_ , as a function of the strain rate, the rheology model equations are undefined at a strain rate of zero. To avoid this, a very high constant yielding viscosity, ly is thus used to define the apparent viscosity near where c_ ¼ 0 until c_ ¼ sy =ly . The slurry was therefore defined to be stagnant when the strain rate was lower than sy =ly . 3. Results of numerical model Depending on the impeller speed and solid concentrations of the slurry used, the numerical model predicted caverns of yielded slurry formed in both types of slurries. In general, the flow closely resembled that inside a jet mixer, however the radial jet did not flow along the base of the tank, but was rather a free stream jet, as seen in Fig. 3. The slurry left the ring of stator blades in a predominantly radial jet, forming an upper and lower recirculation loop, or vortex. The lower, stronger vortex returned to the center of the impeller, while the upper vortex flowed up the cell wall until it could no longer overcome the yield stress, and thus turned inwards to return to the radial jet. The slurry above where the slurry left the cell wall remained stagnant. In the Bindura nickel slurries, caverns only formed at the lowest speeds and highest solid concentrations, while in the kaolin slurry, at the highest solid concentration caverns formed at all impeller speeds. 4. Cavern model As mentioned in the introduction, Meyer and Etchells (2007) developed a cavern model in a jet mixed vessel by assuming, like Solomon et al. (1981), that at the cavern boundary, the shear stress at the cavern boundary must balance the yield stress of the nonNewtonian fluid. These shear stresses are formed when the fluid moves along solid boundaries, like the tank wall, and can be expressed according to Blasius by
Fig. 2. Mesh used to model flotation cell (778,700 hexahedral cells).
There was found to be a strong time dependent interaction between the flow generated by the impeller and the stator blades due to their proximity, making the geometry unsuitable for the steadystate MRF model (Tabor et al., 1996). It was also found that the MRF model underpredicted the power draw of the cell, since it is dependent on the relative positioning between the impeller and stator blades. The unsteady sliding mesh method of modelling the impeller rotation was therefore used. The MRF model was used however to provide initial conditions for the sliding mesh model in order to reduce computational time. A time step corresponding to 0.5° of rotation was found to achieve adequate convergence during the first revolution. The SST k x turbulence mode was used to account for turbulent fluctuations in the flow. Simulations took approximately 10 h per revolution on a Intel Core 2 Quad processor with 8GB RAM. It was also found that simulations of the higher solid concentration slurries converged more rapidly due to the higher apparent viscosity and therefore higher viscous damping. Three impeller speeds, namely 300, 450 and 600 rpm, resulting in impeller tip speeds ranging between 2.36 and 4.71 ms1, were modelled for each slurry. These speeds were chosen as they produce a range of power intensity, defined as the power input per unit volume
sf ¼
C f qu2 2
ð3Þ
The friction coefficient, C f , is a weak function of the viscous Reynolds number in the turbulent regime, expressed as
Fig. 3. Contours of velocity magnitude in 60 wt.% Bindura nickel slurry at 450 rpm showing formation of a cavern.
C.W. Bakker et al. / Minerals Engineering 23 (2010) 968–972
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linearly proportional to the impeller tip velocity, V tip . When Eqs. (5) and (4) are substituted into Eq. (3), and equated to sy . The following equation for the cavern height normalized by the tank diameter is developed 1
2
Hc C 3 Re b qV tip C4 ¼ T sy
Fig. 4. Description of cavern height as used in model. 1
C f ¼ C R Reb
ð4Þ
where b has been empirically found to be between approximately 4 and 5 for typical turbulent boundary layers over flat plates and circular pipes (Douglas et al., 2001). The assumption that the same value of b is valid in this situation is thought to be valid since once reaching the tank wall the fluid flows axially up and down the wall, as is found in flow in a circular pipe. In order to use the cavern model of Meyer and Etchells (2007) one needs to know the velocity at the cavern boundary to calculate the shear stress due to wall friction using Eq. (3). The method developed by Kresta et al. (2001) to calculate this velocity was used, whereby the flow up and down the wall in a tank stirred by a radial flow impeller was represented by an annular wall jet, with the maximum velocity in this jet dependent on the height above or below the impeller. Kresta et al. (2001) determined that the maximum velocity in the wall jet had the following relationship to the height above the impeller:
C 1 U core U ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z þ jC 2 j T
ð5Þ
where z is the height above the impeller, T is the diameter of the tank, and U core is the jet core velocity due to the impeller discharge. C 1 and C 2 are constants Therefore the velocity at the cavern boundary would be when z ¼ Hc ; Hc being the cavern height above the impeller. The assumption is also made that U core is
ð6Þ
where C 3 and C 4 are constants. As Eq. (6) is a very weak function of 1 Reb , since b is approximately 4–5, the term is therefore neglected, in a similar way to Meyer and Etchells (2007). This also means that the only rheological variable that effects the cavern height is the yield stress. Initially the cavern height was defined as height of the cavern apex, but it was found that using a height equal to that where the cavern leaves the tank wall was more consistent with the model. This also is more consistent with the physical assumption that Hc is defined where the fluid has travelled along the wall a distance so that its shear stress is equal to the fluids yield stress, and therefore detaches from the wall to recirculate. The height of the stator disc was also added to the equation, so that Hc equaled the total height of the cavern, from the tank floor to the height where the cavern meets the tank outer wall, as can be seen in Fig. 4. Therefore 2
Hc C 3 qV tip C 4 þ Hdisc ¼ T sy
ð7Þ
In order to generate additional data points at conditions which could not be measured experimentally, the height of the tank was increased to H ¼ 0:745 m in the validated CFD model. Cavern heights were then predicted at conditions where caverns would form with heights previously above the free surface. A plot of qV 2tip =sy to the predicted cavern heights was made in order to find the constants a and b, as shown in Fig. 5. A close correlation was shown, indicating that the cavern height in a Bateman flotation cell containing a yield stress slurry can be predicted using the following equation: 5 2 Hc 4:250 10 qV tip þ 0:4837 ¼ T sy
ð8Þ
The maximum cavern height is always higher than the height at which the cavern meets the tank wall, and so there could therefore be certain conditions where the apex of the cavern reaches the free surface, but Hc is less than the slurry depth, a region near the outer tank wall remains stagnant. Any region of stagnant fluid is undesirable for efficient flotation however,
Fig. 5. Correlation of normalized cavern height, including cavern predictions in tall cell, to qV 2tip =sy in order to find constants. = numerically predicted cavern heights.
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and so conditions in a flotation cell should always be maintained so that Hc either equals or exceeds the slurry depth, meaning slurry movement throughout the cell occurs. This method of cavern height calculation can be used to get an estimation of the extent of the mixed slurry in flotation cells of similar geometry, however the constants will need to be obtained for different geometries using either computational cavern prediction, or experimental measurement. This could provide a useful tool to obtain an engineering approximation of whether a cavern will form or not in the first stages of cell design. To the authors’ knowledge, this is also the first time that such a correlation has been developed for a flotation cell.
5. Conclusion A validated CFD model was used to investigate the effects of non-Newtonian mineral slurry rheology on the hydrodynamics inside a mechanical flotation cell, specifically with relation to the formation of a yielded cavern of slurry around the impeller. The predicted cavern sizes were used to develop a semi-empirical mathematical cavern height model. The slurry flow up the cells upper wall was represented by an annular wall jet it was assumed that the jet separates from the wall to flow inwards and form a vortex when its shear stress is equal to the slurries yield stress. The cavern height was therefore found to be directly proportional to the product of the slurry density and the square of the impeller tip speed, and inversely proportional to the slurry yield stress. This method of cavern height calculation can be used to get an estimation of the extent of the mixed slurry in flotation cells of similar geometry, however the constants will need to be obtained for different geometries using either computational cavern prediction, or experimental measurement. This could provide a useful tool to obtain an engineering approximation of whether a cavern will form or not in the first stages of cell design. To the authors knowledge, this is also the first time such a correlation has been developed for a flotation cell.
Acknowledgement The authors would like to thank the SA Research Chair in Minerals Beneficiation for funding this research.
References Amanullah, A., Hjorth, S.A., Nienow, A.W., 1998. A new mathematical model to predict cavern diameters in highly shear thinning, power law liquids using axial flow impellers. Chemical Engineering Science 53, 455–469. ANSYS, 2007. Fluent 6.3. Bakker, C.W., Meyer, C.J., Deglon, D.A., 2009. Numerical modelling of nonNewtonian slurry in a mechanical flotation cell. Minerals Engineering 22, 944–950. Burdukova, E., 2007. Surface properties of new york talc as a function of pH, polymer adsorption and electrolyte concentration. Ph.D. Thesis, University of Cape Town. Deglon, D.A., 2005. The effect of agitation on platinum ores. Minerals Engineering 18, 839–844. Deglon, D.A., Egya-Mensah, D., Franzidis, J., 2000. Review of hydrodynamics and gas dispersion in flotation cells on South African platinum concentrators. Minerals Engineering 13 (3), 235–244. Douglas, J.F., Gasiorek, J.M., Swaffield, J.A., 2001. Fluid Mechanics. Prentice Hill. Elson, T.P., 1990. The growth of caverns formed around rotating impellers during the mixing of a yield stress fluid. Chemical Engineering Communications 96, 303–319. Elson, T.P., Cheeseman, D.J., 1986. X-ray studies of cavern sizes and mixing performance with fluids possessing a yield stress. Chemical Engineering Science 41, 255–2562. Fuerstenau, M.C., Jameson, G.J., Yoon, R.-H. (Eds.), 2007. Froth Flotation: A Century of Innovation. Society for Mining, Metallurgy, and Exploration. He, M., Wang, Y., Forssberg, E., 2006. Parameter studies on the rheology of limestone slurries. International Journal of Mineral Processing 78, 63–77. Kresta, S.M., Bittorf, K.J., Wilson, D.J., 2001. Internal annular wall jets: radial flow in a stirred tank. AIChE 47, 2390–2400. Loginov, M., Larue, O., Lebovka, N., Vorobiev, E., 2008. Fluidity of highly concentrated kaolin suspensions: influence of particle concentration and presence of dispersant. Colloids and Surfaces A: Physicochemical and Engineering Aspects 325, 64–71. Meyer, P.A., Etchells, A.W., 2007. Mixing with intermittent jets with application in handling radioactive sludge wastes. Transactions IChemE 85 (A5), 691–696. Moore, I.P.T., Cossor, G., Baker, M.R., 1995. Velocity distributions in a stirred tank containing a yield stress fluid. Chemical Engineering Science 50, 2467–2481. Pease, J.D., Curry, D.C., Young, M.F., 2006. Designing flotation circuits for high fines recovery. Minerals Engineering 19, 831–840. Prestidge, C.A., 1997. Rheological investigations of ultrafine galena particle slurries under flotation-related conditions. International Journal of Mineral Processing 51, 241–254. Schubert, H., 2008. On the optimization of hydrodynamics in fine particle flotation. Minerals Engineering 21 (12–14), 930–936. Solomon, J., Elson, T.P., Nienow, A.W., 1981. Cavern sizes in agitated fluids with a yield stress. Chemical Engineering Communications 11, 143–164. Tabor, G., Gosman, A.D., Issa, R.I., 1996. Numerical simulation of the flow in a mixing vessel stirred by a rushton turbine. In: Benkreira, H. (Ed.), Fluid Mixing 5, Institution of Chemical Engineers. ICHEM Symposium Series, pp. 25–34. Tseng, W.J., Chen, C.N., 2003. Effect of polymeric dispersant on rheological behavior of nickel–terpineol suspensions. Materials Science and Engineering A347, 145– 153. Wilkens, R.J., Miller, J.D., Plummer, J.R., Dietz, D.C., Myers, K.J., 2005. New techniques for measuring and modeling cavern dimensions in a Bingham plastic fluid. Chemical Engineering Science 60, 5269–5275. Xia, J., Rinne, A., Grönstrand, S., 2009. Effect of turbulence models on prediction of fluid flow in an Outotec flotation cell. Minerals Engineering 22 (11), 880–885.
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
The development of a cavern model for mechanical flotation cells C.W. Bakker a,*, C.J. Meyer b, D.A. Deglon a a b
Centre for Minerals Research, Dept. of Chemical Engineering, University of Cape Town, South Africa Centre for Research in Computational and Applied Mechanics, University of Cape Town, South Africa
a r t i c l e
i n f o
Article history: Received 3 December 2009 Accepted 17 March 2010 Available online 14 April 2010 Keywords: Modelling Computational fluid dynamics Froth flotation Fine particle processing Agitation
a b s t r a c t Certain fine particle and high solid concentration mineral slurries used in the froth flotation process have been shown to exhibit non-Newtonian rheologies, including a yield stress. The mixing characteristics of these fluids are often problematic as a cavern of yielded fluid forms around the impeller whilst the rest of the fluid remains stagnant and therefore unmixed. This paper aims to develop a semi-empirical model to calculate the height of caverns forming in non-Newtonian mineral slurries in a mechanical flotation cell. Cavern shapes in a pilot-scale Batequip flotation cell were numerically determined for a range of mineral slurries using an experimentally validated Computational Fluid Dynamics (CFD) model. Development of the cavern height model was based on the assumption that the cavern boundary was formed where the shear stress imposed on the slurry equaled the fluid yield stress and also that the flow along the cell walls could be represented by an annular wall jet. It was found that the cavern height was directly proportional to the product of the slurry density and the square of the impeller tip speed, and inversely proportional to the slurry yield stress. Ó 2010 Elsevier Ltd. All rights reserved.
1. Introduction Mining operations, such as lead–zinc and platinum, are increasingly having to grind ores to ever finer particle sizes in order to liberate the valuable minerals (Pease et al., 2006). These operations typically run flotation circuits at high solids concentrations, mainly in order to maximise residence time, but also to limit water consumption. This may cause mineral slurries to exhibit non-Newtonian pseudoplastic rheological behaviour, including a yield stress (Prestidge, 1997; Tseng and Chen, 2003; He et al., 2006). This behaviour is exacerbated when processing ores containing minerals known to adversely affect rheology, such as certain phyllosilicates. Fluids with similar rheological behaviour are encountered in a variety of other industries, including those involving food, pharmaceutical, cosmetic and paper pulp processing, and in many cases have been found to be problematic. Mixing of these yield stress fluids in mechanically stirred tanks has been found to cause the formation of a yielded region of fluid around the impeller, with the rest of the fluid in the bulk of the tank remaining unyielded, and therefore stagnant (Elson and Cheeseman, 1986; Moore et al., 1995; Wilkens et al., 2005). The formation of a cavern in these stirred tanks resulted in diminished mixing of the fluid in the tank, and is thus undesirable. The formation of such a cavern in a mechanical flotation cell has been found to form in certain slurries and operating conditions (Bakker et al., 2009), effectively reducing * Corresponding author. Tel.: +27 78 619 2835. E-mail address: [email protected] (C.W. Bakker). 0892-6875/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mineng.2010.03.016
the active volume of the cell where flotation occurs, since a high rate of turbulent energy dissipation, required for the flotation of fine particles (Deglon, 2005; Schubert, 2008), would be absent in the stagnant regions of slurry. Since non-Newtonian fluids encountered in industry are often solid suspensions and therefore opaque, many experimental techniques traditionally used to measure local fluid flow are unsuitable. In an attempt to be able to predict the shape and size of the cavern formed in these fluids, various researchers have developed empirical and mathematical models to calculate the cavern dimensions (Solomon et al., 1981; Elson and Cheeseman, 1986; Elson, 1990; Moore et al., 1995; Amanullah et al., 1998; Wilkens et al., 2005). Solomon et al. (1981) developed a mathematical model assuming that the cavern could best be represented by a sphere centered around the impeller. The shape of a sphere was chosen under the assumption that all the power dissipated by the impeller was transmitted to the cavern boundary, meaning that the shear rate at the boundary was equal to the yield stress. It also assumed that the fluid velocity was predominantly tangential. The model was only valid until the cavern reached the tank wall. This model was modified by Elson and Cheeseman (1986) to become a right angled cylinder, again only valid until the cavern reached the walls. Amanullah et al. (1998) developed a new model for highly shear thinning fluids by making the total momentum from the impeller equal to the sum of both the tangential and axial components from an axial flow impeller. The shape of the cavern was defined as a torus. Since the fluid did not exhibit a yield stress, a velocity of 1 103 m s1 was used to define the cavern boundary. Wilkens
969
C.W. Bakker et al. / Minerals Engineering 23 (2010) 968–972
et al. (2005) used a novel experimental method to measure the cavern size in tomato sauce, or ketchup, a fluid known to exhibit a yield stress. The fluid was agitated by both a Rushton radial flow impeller and a pitch blade turbine. The cavern shape was found to be best approximated by an elliptical torus. The agitation method in mechanical flotation cells is similar to that in mechanically stirred tanks, however due to the ring of stator blades around the impeller, the flow has a predominantly radial velocity, leaving the impeller region in a form of free stream radial jet (Fuerstenau et al., 2007). This radial jet also always reaches the tank wall before forming two recirculation loops or vortices that return to the impeller region (Xia et al., 2009). For this reason, although the same reasoning can be applied to the development of a cavern model for a mechanical flotation cell, the models themselves cannot be used. A cavern model has however been developed for a different mixing device having similar flow characteristics to those in flotation cells, namely jet mixers (Meyer and Etchells, 2007). These devices consist of cylindrical tanks with either one or more pulse tubes inserted into the fluid to be mixed. The tubes either vent the fluid contained inside them, forming a jet, or refill, sucking in the fluid. In the venting mode this jet strikes the base of the tank and then follows the tank wall up the side of the tank. When mixing yield stress fluids, the fluid only follows the wall until the shear stress in the fluid cannot overcome the fluids yield stress and so forms a recirculation loop, returning to the jet. The shear stress imposed on the fluid was assumed to be from wall friction stresses as it flowed along the tank walls. Due to the difficulty in obtaining cavern measurements in opaque particle suspensions such as mineral slurries, this paper aims to use a experimentally validated Computational Fluid Dynamics (CFD) model, developed in previous research (Bakker et al., 2009), to provide a useful engineering prediction, rather than a physically measurement of the effect of the rheology of mineral slurries on the hydrodynamics in the flotation cell and thus the shape and size of caverns formed. This data will then be used to develop a model to predict whether a cavern will form and if so, its height. 2. Model description
Fig. 1. Flotation cell setup.
which cause repulsive forces between particles (Burdukova, 2007), causing the slurry to represent the extreme rheologies as found in industrial slurries. Three different solid concentrations were modelled for each type of slurry. For the Bindura nickel slurry concentrations of 40, 50 and 60 wt.% were chosen since slurry solid concentrations of approximately 40 wt.% are typical of many mineral processing operations, whereas slurries at 50 wt.% represent the upper bounds of current flotation practice, although it is anticipated that higher solid concentrations will be more common in the future due to pressures on reducing water consumption. The kaolin was found to be too viscous to pour after a solid concentration of 40 wt.%, and so values of 25, 30 and 40 wt.% were instead modelled. Similar findings were made by Loginov et al. (2008). The rheology of the slurries were experimentally measured using a Rheolab MCR 300 rheometer over a shear rate range of 0.1– 1000 s1. It was found that all slurries exhibited a yield stress and had a shear thinning behaviour at low shear rates, which was best described using the Herschel–Bulkley non-Newtonian model, defined as
This section describes the numerical model used to simulate the non-Newtonian mineral slurries in a mechanical flotation cell. The experimental validation of the modelling methodology used is described in Bakker et al. (2009).
s ¼ sy þ kc_ n
2.1. Geometry
s ¼ sy þ kc_
This study was conducted on a 100 l pilot-scale Bateman flotation cell (Fig. 1) as used in previous research (Bakker et al., 2009), with diameter, T ¼ 540 mm, and liquid depth, Z ¼ 445 mm, which consists of a 6 bladed impeller of maximum diameter, D ¼ 150 mm, tapering down to 70 mm at the lower edge, and height W ¼ 100 mm. This was surrounded by a ring of 16 stator blades of 70 mm by 36 mm attached to a 250 mm diameter stator disc. The impeller bottom clearance was C b ¼ 83 mm.
This has also been found in other research using limestone slurries (He et al., 2006), and by Tseng and Chen (2003) in nickel powders dispersed in terpineol solvent. The shear rate below which the Herschel–Bulkley model is used, and above which the Bingham plastic model is used is equal to the intercept of the two model curves. Constants used in each non-Newtonian model are shown in Table 1.
ð1Þ
At higher shear rates the gradient of the rheogram was constant, and therefore the slurry was modelled as a Bingham plastic fluid.
ð2Þ
2.3. CFD model 2.2. Fluid rheology Two mineral slurries were used in this research, namely a kaolin slurry and Bindura nickel ore slurry. Both slurries are also known to have complex mineralogies, and thus problematic rheologies. The ores were also ground to extremely fine particle sizes, the d50 for the kaolin being 2.5 lm, and for the Bindura ore 15 lm, and contacted with ‘plant water’, which contains high dissolved solids. This was done in order to get rid of any surface charge
The CFD simulations were implemented using the commercial CFD software, ANSYS Fluent 6.3 (ANSYS, 2007). A single phase model was used with a UDF to represent the slurries non-Newtonian rheology. Only half of the flotation cell was modelled, using a mesh of 778,700 hexahedral cells as shown in Fig. 2. The full 360° mesh is shown to improve clarity. Grid refinement tests were conducted and it was found that there was no appreciable difference in results predicted with finer meshes.
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Table 1 Densities and rheology model parameters of slurries.
q (kg/m3)
sy HB (Pa)
kHB (kg/m s)
nHB
sy Bing (Pa)
kBing (kg/m s)
Bindura 40 wt.%
1364
0.456
0.439
0.353
1.945
6:79 103
Bindura 50 wt.%
1500
1.742
1.020
0.427
8.189
1:47 102
Bindura 60 wt.%
1667
4.741
3.774
0.565
Kaolin 25 wt.%
1182
1.838
1.756
0.254
Kaolin 30 wt.%
1226
2.740
3.987
0.231
14.74
8:64 103
Kaolin 40 wt.%
1327
8.67
11.700
0.266
57.46
1:81 102
52.18 7.038
7:42 102 6:55 103
(P/V), of approximately 0.6–5.6 kW/m3, which straddles and exceeds the range of 1–3 kW/m3 found in typical industrial mechanical flotation cells (Deglon et al., 2000). The combined Herschel–Bulkley and Bingham plastic non-Newtonian rheology models were input into Fluent via a user-defined function. Since Fluent requires an equation defining value of the apparent viscosity, g ¼ s=c_ , as a function of the strain rate, the rheology model equations are undefined at a strain rate of zero. To avoid this, a very high constant yielding viscosity, ly is thus used to define the apparent viscosity near where c_ ¼ 0 until c_ ¼ sy =ly . The slurry was therefore defined to be stagnant when the strain rate was lower than sy =ly . 3. Results of numerical model Depending on the impeller speed and solid concentrations of the slurry used, the numerical model predicted caverns of yielded slurry formed in both types of slurries. In general, the flow closely resembled that inside a jet mixer, however the radial jet did not flow along the base of the tank, but was rather a free stream jet, as seen in Fig. 3. The slurry left the ring of stator blades in a predominantly radial jet, forming an upper and lower recirculation loop, or vortex. The lower, stronger vortex returned to the center of the impeller, while the upper vortex flowed up the cell wall until it could no longer overcome the yield stress, and thus turned inwards to return to the radial jet. The slurry above where the slurry left the cell wall remained stagnant. In the Bindura nickel slurries, caverns only formed at the lowest speeds and highest solid concentrations, while in the kaolin slurry, at the highest solid concentration caverns formed at all impeller speeds. 4. Cavern model As mentioned in the introduction, Meyer and Etchells (2007) developed a cavern model in a jet mixed vessel by assuming, like Solomon et al. (1981), that at the cavern boundary, the shear stress at the cavern boundary must balance the yield stress of the nonNewtonian fluid. These shear stresses are formed when the fluid moves along solid boundaries, like the tank wall, and can be expressed according to Blasius by
Fig. 2. Mesh used to model flotation cell (778,700 hexahedral cells).
There was found to be a strong time dependent interaction between the flow generated by the impeller and the stator blades due to their proximity, making the geometry unsuitable for the steadystate MRF model (Tabor et al., 1996). It was also found that the MRF model underpredicted the power draw of the cell, since it is dependent on the relative positioning between the impeller and stator blades. The unsteady sliding mesh method of modelling the impeller rotation was therefore used. The MRF model was used however to provide initial conditions for the sliding mesh model in order to reduce computational time. A time step corresponding to 0.5° of rotation was found to achieve adequate convergence during the first revolution. The SST k x turbulence mode was used to account for turbulent fluctuations in the flow. Simulations took approximately 10 h per revolution on a Intel Core 2 Quad processor with 8GB RAM. It was also found that simulations of the higher solid concentration slurries converged more rapidly due to the higher apparent viscosity and therefore higher viscous damping. Three impeller speeds, namely 300, 450 and 600 rpm, resulting in impeller tip speeds ranging between 2.36 and 4.71 ms1, were modelled for each slurry. These speeds were chosen as they produce a range of power intensity, defined as the power input per unit volume
sf ¼
C f qu2 2
ð3Þ
The friction coefficient, C f , is a weak function of the viscous Reynolds number in the turbulent regime, expressed as
Fig. 3. Contours of velocity magnitude in 60 wt.% Bindura nickel slurry at 450 rpm showing formation of a cavern.
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linearly proportional to the impeller tip velocity, V tip . When Eqs. (5) and (4) are substituted into Eq. (3), and equated to sy . The following equation for the cavern height normalized by the tank diameter is developed 1
2
Hc C 3 Re b qV tip C4 ¼ T sy
Fig. 4. Description of cavern height as used in model. 1
C f ¼ C R Reb
ð4Þ
where b has been empirically found to be between approximately 4 and 5 for typical turbulent boundary layers over flat plates and circular pipes (Douglas et al., 2001). The assumption that the same value of b is valid in this situation is thought to be valid since once reaching the tank wall the fluid flows axially up and down the wall, as is found in flow in a circular pipe. In order to use the cavern model of Meyer and Etchells (2007) one needs to know the velocity at the cavern boundary to calculate the shear stress due to wall friction using Eq. (3). The method developed by Kresta et al. (2001) to calculate this velocity was used, whereby the flow up and down the wall in a tank stirred by a radial flow impeller was represented by an annular wall jet, with the maximum velocity in this jet dependent on the height above or below the impeller. Kresta et al. (2001) determined that the maximum velocity in the wall jet had the following relationship to the height above the impeller:
C 1 U core U ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z þ jC 2 j T
ð5Þ
where z is the height above the impeller, T is the diameter of the tank, and U core is the jet core velocity due to the impeller discharge. C 1 and C 2 are constants Therefore the velocity at the cavern boundary would be when z ¼ Hc ; Hc being the cavern height above the impeller. The assumption is also made that U core is
ð6Þ
where C 3 and C 4 are constants. As Eq. (6) is a very weak function of 1 Reb , since b is approximately 4–5, the term is therefore neglected, in a similar way to Meyer and Etchells (2007). This also means that the only rheological variable that effects the cavern height is the yield stress. Initially the cavern height was defined as height of the cavern apex, but it was found that using a height equal to that where the cavern leaves the tank wall was more consistent with the model. This also is more consistent with the physical assumption that Hc is defined where the fluid has travelled along the wall a distance so that its shear stress is equal to the fluids yield stress, and therefore detaches from the wall to recirculate. The height of the stator disc was also added to the equation, so that Hc equaled the total height of the cavern, from the tank floor to the height where the cavern meets the tank outer wall, as can be seen in Fig. 4. Therefore 2
Hc C 3 qV tip C 4 þ Hdisc ¼ T sy
ð7Þ
In order to generate additional data points at conditions which could not be measured experimentally, the height of the tank was increased to H ¼ 0:745 m in the validated CFD model. Cavern heights were then predicted at conditions where caverns would form with heights previously above the free surface. A plot of qV 2tip =sy to the predicted cavern heights was made in order to find the constants a and b, as shown in Fig. 5. A close correlation was shown, indicating that the cavern height in a Bateman flotation cell containing a yield stress slurry can be predicted using the following equation: 5 2 Hc 4:250 10 qV tip þ 0:4837 ¼ T sy
ð8Þ
The maximum cavern height is always higher than the height at which the cavern meets the tank wall, and so there could therefore be certain conditions where the apex of the cavern reaches the free surface, but Hc is less than the slurry depth, a region near the outer tank wall remains stagnant. Any region of stagnant fluid is undesirable for efficient flotation however,
Fig. 5. Correlation of normalized cavern height, including cavern predictions in tall cell, to qV 2tip =sy in order to find constants. = numerically predicted cavern heights.
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and so conditions in a flotation cell should always be maintained so that Hc either equals or exceeds the slurry depth, meaning slurry movement throughout the cell occurs. This method of cavern height calculation can be used to get an estimation of the extent of the mixed slurry in flotation cells of similar geometry, however the constants will need to be obtained for different geometries using either computational cavern prediction, or experimental measurement. This could provide a useful tool to obtain an engineering approximation of whether a cavern will form or not in the first stages of cell design. To the authors’ knowledge, this is also the first time that such a correlation has been developed for a flotation cell.
5. Conclusion A validated CFD model was used to investigate the effects of non-Newtonian mineral slurry rheology on the hydrodynamics inside a mechanical flotation cell, specifically with relation to the formation of a yielded cavern of slurry around the impeller. The predicted cavern sizes were used to develop a semi-empirical mathematical cavern height model. The slurry flow up the cells upper wall was represented by an annular wall jet it was assumed that the jet separates from the wall to flow inwards and form a vortex when its shear stress is equal to the slurries yield stress. The cavern height was therefore found to be directly proportional to the product of the slurry density and the square of the impeller tip speed, and inversely proportional to the slurry yield stress. This method of cavern height calculation can be used to get an estimation of the extent of the mixed slurry in flotation cells of similar geometry, however the constants will need to be obtained for different geometries using either computational cavern prediction, or experimental measurement. This could provide a useful tool to obtain an engineering approximation of whether a cavern will form or not in the first stages of cell design. To the authors knowledge, this is also the first time such a correlation has been developed for a flotation cell.
Acknowledgement The authors would like to thank the SA Research Chair in Minerals Beneficiation for funding this research.
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