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Influence of flotation cell hydrodynamics on the flotation kinetics and scale up, Part 2: Introducing turbulence parameters to improve predictions
Minerals Engineering 100 (2017) 31–39
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Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Influence of flotation cell hydrodynamics on the flotation kinetics and scale up, Part 2: Introducing turbulence parameters to improve predictions E. Amini ⇑, D.J. Bradshaw, W. Xie SMI/JKMRC, University of Queensland, Brisbane, Australia
a r t i c l e
i n f o
Article history: Received 7 July 2015 Revised 2 September 2016 Accepted 2 October 2016
Keywords: Flotation kinetic Cell hydrodynamic Dimensionless parameters Turbulence Ore property
a b s t r a c t The AMIRA P9 model has floatability (P) as the ore property which is considered to remain constant in different flotation cell sizes under different hydrodynamic conditions. However, in this study increasing the power input increased the P value, especially in finer particle size classes (below 75 lm). Acceptable explanations for the floatability variations, as a result of hydrodynamic condition variations in the flotation cells, have been sought by looking at the literature and the results obtained in this study were published in part one of this manuscript. To improve the accuracy of the AMIRA P9 flotation model in predicting flotation rate constant (k) and to improve the consistency of ore property, measurable and appropriate turbulence parameters were sought to be incorporated into the model. Therefore, two dimensionless turbulence parameters æ and EVF, derived from practical measurements, were formulated and introduced to the AMIRA P9 model. The modified ore floatability parameter, P00 , was demonstrated to be a more consistent characterisation of the ore property than P and both the accuracy and precision of the k prediction improved for a variety of hydrodynamic conditions of flotation cells. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The AMIRA P9 model has been applied in a number of flotation studies and has been successfully used to optimise several industrial flotation circuits (AMIRA P9 reports, 2000, 2004a,b). The model is explained in detail in part one of this paper. Other studies (Amini et al., 2009; Collins, 2007) showed that the ore property (P value) is not constant when scaling up from laboratory to pilot plant or industrial scales, and that the P value changes with varying hydrodynamic conditions in cells. It was shown in the first part of this manuscript that increasing impeller speed and turbulent kinetic energy dissipation rate (TKEDR) changed the P value, which is contrary to the proposed mechanism of Gorain (1997) and Gorain et al. (1998) who reported that impeller speed changed bubble surface area (Sb) which is a function of bubble size distribution, but not the P value in his experiments on industrial cells. This implies other factors such as impeller speed influence the P value in small scale flotation cells. Varying impellor speed is not a control variable available in industrial plant operation. It is however an important concept in relating laboratory flotation kinetic test data to industrial plant scale-up. As a result, a fitted scale-up number ⇑ Corresponding author. E-mail address: [email protected] (E. Amini). http://dx.doi.org/10.1016/j.mineng.2016.10.001 0892-6875/Ó 2016 Elsevier Ltd. All rights reserved.
needs to be applied to the P value predicted by the AMIRA P9 model. Hernandez-Aguilar (2011) and Hernandez-Aguilar et al. (2005) reported that the P value does not have a linear relationship with the flotation rate constant as claimed by Gorain (1997) and Gorain et al. (1998). The authors showed that incorporating a new bubble size factor into the model can improve the model predictions in column flotation. Their new model seems to work when the Sauter mean bubble diameter (d32) is lower than 1.2 mm but Gorain’s relationship is still valid in the column flotation cells where no agitation is required. The mean value of d32 in the mechanical flotation cells in the industry is equal to 1.7 mm (Schwarz and Alexander, 2006) showing that Hernandez-Aguilar (2011) is not valid for the mechanical cells. Moreover, the authors did not take into account the effect of bubble size distribution and bubble particle loading on froth stability and froth drop-back (Laskowski and Woodburn, 1998; Bulatovic, 2007) when the Rf was calculated in their studies. That is a crucial factor in the rate constant calculation in industrial test work. Putting the result of the studies together indicates experimental test work with minimum froth effect is required to analyse the relationship between flotation rate constant (k) and Sb in pulp zone. In contrast, the fundamental flotation models discussed in part one already contain hydrodynamic factors such as energy
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E. Amini et al. / Minerals Engineering 100 (2017) 31–39
Nomenclature
æ db EVF k P
dimensionless number that represent hydrodynamic condition of the cell diameter of bubble (mm) effective volume factor (dimensionless) flotation rate constant (1/min) floatability as defined in AMIRA P9 flotation model (dimensionless)
dissipation rate, bubble size, viscosity, and number of bubbles. Application of several parameters makes the fundamental models sensitive to the majority of influential parameters in the flotation process. However, calculation or measurement of each parameter has an associated error and more parameters simply mean a bigger uncertainty in the calculated/predicted flotation rate constant. Also, some of the parameters in the fundamental models cannot be directly measured, which increases the uncertainties in calculation even more. It was found that bubble size, energy dissipation rate and viscosity are the most important factors influencing bubble-particle collision rates (Schubert, 1999; Abrahamson, 1975). It was also assumed that the volume of the cell that contains higher TKEDR affects the flotation rate constant by increasing the overall collision rate in the cell (Savassi, 1998). Savassi (1998) divided the pulp in a flotation cell into two zones: (1) The ‘‘Collection” zone (near the impeller), which contains high turbulence intensity and where the majority of bubble particle attachment occurs and (2) The ‘‘Quiescent” zone (outside the impeller region), which contains very low turbulence intensity and where the interaction between bubbles and particles is very low. The existence of these zones was also observed by other researchers (Pyke, 2004; Newell and Grano, 2006, 2007) and is demonstrated in part two of this manuscript. The quiescent zone or disengagement zone indicates the zone that bubble-particle attachment or detachment rate is considerably lower compare to the high turbulent zone (Matis, 1995). It was demonstrated in part one of this manuscript that it is possible to make appropriate measurements to characterise the hydrodynamic conditions of flotation cells of any size. Turbulent kinetic energy dissipation rate (TKEDR), bubble size, shear and viscosity are the key influential parameters in a flotation process which can be measured with practical methodologies and used for modelling purposes. Introducing dimensionless turbulence parameters to the AMIRA P9 model (modified AMIRA P9 model) can reduce the error in calculating of ore property. Thus it enhances the flotation rate constant prediction for a variety of size and hydrodynamic conditions of flotation cells.
P0 and P00 floatability as defined in the modified P9 flotation model (dimensionless) Rf recovery of particles across the froth (%) Sb bubble surface area flux (1/s) TKEDR turbulent kinetic energy dissipation rate (m3/s2) m kinematic viscosity (m2/s) q density (m3/kg)
the dimensions of (L2T1), (L2T3), (L), (ML3) and (T1), respectively. Bubble surface area flux is already present in the AMIRA P9 model so the extension of the model could contain the other four parameters. Thus, a dimensionless number, which is called æ, can be written using db, TKEDR, m and q:
æ ¼ db :TKEDRa mb qc
ð1Þ
By introducing the dimensions we have
æ¼L
L2 T3
!a
L2 T
!b c M 3 L
ð2Þ
where a, b and c are the parameters which should be calculated by solving the equation for each fundamental quantity to eliminate the dimensions.
For length ðLÞ : 1 þ 2a þ 2b 3c ¼ 0
ð3Þ
For time ðTÞ : 3a þ b ¼ 0
ð4Þ
For mass ðMÞ : c ¼ 0
ð5Þ
Solving the Eqs. (3)–(5) returns values of a = 0.25, b = 0.75 and c = 0. Therefore, æ can be written as An expression to describe pulp hydrodynamic conditions incorporating pulp viscosity (m), arithmetic mean bubble size (db) and the turbulent kinetic energy dissipation rate (TKEDR) has been developed and is shown as Eq. (6). The derivation of the equation was published in detail (Amini, 2013).
æ¼
db TKEDR0:25
m0:75
!n ð6Þ
where n is estimated by the number of flotation tests over a range of operational conditions. It is suggested to conduct three lab scale flotation tests at three levels of impeller speed to fit n value where floatability remains constant for each floatability class (e.g. size by size P values) over the operational range. When only æ is incorporated into AMIRA P9 model ore property (P0 ) can be estimated from this model. The P0 becomes more of an ore property compared to the P in the original AMIRA P9 model (k = P Sb Rf).
2. Development of dimensionless numbers
k ¼ P 0 Sb æ R f
It was shown that the floatability (P value) is proportional to hydrodynamic conditions of a flotation cell (see part one). The hydrodynamic effect may be described using two expressions related to the pulp hydrodynamics (æ) and effective cell volume (EVF). The fundamental flotation models indicated that the flotation rate constant is directly proportional to bubble-particle collision frequency, which is a function of kinematic viscosity (m), turbulent kinetic energy dissipation rate (TKEDR), bubble diameter (db), density (q) and bubble surface area flux (Sb). These parameters have
From Savassi’s assumption, it can be concluded that increasing the energy dissipation rate increases the collision frequency. Thus, increasing the energy dissipation rate increases the flotation rate constant, but only up to a point where detachment overrides the collision frequency effect, especially for coarse particles. Increasing the volume of the ‘‘high” turbulent zone (TKEDR > 0.1 m2/s3 in this study) increases the collision frequency in a flotation cell and the flotation rate constant increases. Therefore, the proportion of the cell volume, which contains a high-energy dissipation rate (here called the Effective Volume Factor or EVF), is introduced here:
ð7Þ
E. Amini et al. / Minerals Engineering 100 (2017) 31–39
EVF ¼
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Volume of the cell that contains high energy dissipation rate Total v olume of the cell ð8Þ
Incorporating EVF factor into Eq. (6) gives;
k ¼ P00 Sb æ EVF Rf
ð9Þ
In this equation P00 represents floatability of the ore in a flotation process. 3. Review of experimental data Details on flotation test work which were carried out on the 5 L and 60 L flotation cells was explained in part one of the manuscript. Twelve different hydrodynamic conditions were studied. It was found that the flotation rate (k) - bubble surface area flux relationship (Sb) observed in commercial flotation cells reported previously (Gorain, 1997; Gorain et al., 1998) were not observed in laboratory (5 L) and pilot scale (60 L) flotation cells. Hydrodynamic test conditions did affect flotation rate, especially for laboratory scale flotation conditions. Thus, other hydrodynamic variables need to be included in the flotation scale-up models. 4. Results and discussion 4.1. Unsized ore floatability P, P0 , and P00 for all of the test results obtained in the 5 L and 60 L cells (calculated using, original AMIRA P9 model, Eqs. (7) and (9)) are presented in Fig. 1. The coefficient of variance (standard deviation to average) for P value calculated from three repeat tests conducted in 60 L cell at Jg = 0.6 cm/s and impeller tip speed (ITS) = 4.32 m/s. The P values are 0.00117, 0.00116, and 0.00127 so the coefficient of variance of 5.7% at 97.7% confidence interval was obtained. The figure shows, unlike the P value, the P0 value is almost independent of impeller speed and air rate in both cells. The P value obtained at the lowest impeller speed (3.22 m/s) and superficial gas velocity of 0.4 m/s in the 5 L cell is very close to the P value obtained at impeller speed of 3.99 m/s most likely due to the experimental error. A series of t-test analyses were conducted on the extracted floatability parameters to evaluate the improvements in the predictions accuracy. In the calculations, the null hypothesis was used, which claims there is no significant difference between the mean values of two sets of data. The confidence interval to reject the null hypothesis is calculated from the t-test analyses. The confidence interval values more than 90% (in industrial studies) shows that the null hypothesis is rejected and the mean values are not the same. Firstly, the t-test was applied to the total P values that are extracted from the 5 L tests. The test compares the values obtained from the flotation tests at ITS = 3.22 m/s with the flotation tests at ITS = 4.93 m/s. The total P values at the minimum and maximum ITS (relevant to the level of turbulence) in the tests are not significantly the same at 95.26% confidence interval. The same analysis was then applied for the 60 L cell at the minimum and maximum ITS, which are 2.99 m/s and 4.99 m/s, respectively. Again, the total P values were not significantly the same at 97.61% confidence interval. The same analysis on the total P0 values at the lowest and highest ITS values in the 5 L cell shows the values are significantly the same as the confidence interval is not high enough to reject the null hypothesis (73.49%). In the 60 L cell, confidence interval is at 78.51% for the total P0 values at the lowest and the highest ITS values. This shows that the total P0 is significantly more robust for flotation modelling compared to the P values when the impeller speed and level of turbulence changes in a cell.
Fig. 1. Unsized ore floatability vs. impeller tip speed in 5 L (open symbol) and 60 L (closed symbol) experiments.
Effective Volume Factor (EVF) was used to calculate P00 value as well. The results are shown at the bottom of Fig. 1. In order to calculate n value in Eq. (6), the model was fitted to all results obtained from the 5 L cell tests. An assumption was made that the floatability value (P0 ) should remain constant for each size fraction as the chemistry of pulp was maintained in the 5 L cell tests. The variable parameters are the hydrodynamic parameters in the 5 L cell tests which were measured and extracted from flotation rate constant. The n value is equal to 1.47 and was kept constant in the calculations of floatability values in the 60 L cell tests as well. Generally, P00 is higher than P0 and lower than P. A series of t-test analyses was also conducted on the total P00 values that are extracted from the 5 L tests comparing the values obtained from the flotation tests at ITS = 3.22 m/s with the flotation tests at ITS = 4.93 m/s. The total P00 values at the minimum and the maximum tip speed (relative to level of turbulence) in
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E. Amini et al. / Minerals Engineering 100 (2017) 31–39
cell still needs to be considered in the CV value calculation and this is discussed in the next section. Both statistical analyses showed that the total P0 and total P00 are more reliable than the total P value, and total P00 is the most reliable floatability number to predict the results in scale-up studies because increasing the level of turbulence (relative to TKEDR) improves the collision rate (Abrahamson, 1975; Pyke, 2004). It eventually increases the flotation rate constant; therefore, the effect of turbulence should be incorporated in the scale-up analysis or when a cell with variable impeller speed is used. The industrial cells usually built with fixed impeller speed but the impeller speed and the TKEDR is different in different size, type and brand of the flotation cells. Therefore, in scale-up analysis from batch or pilot scale to industrial flotation cells the TKEDR should be estimated and incorporated according to the size, type and brand of flotation cell being used in the circuit. Increasing impellor tip speed and hence mixing intensity in a stator-type flotation cell is shown to increase flotation rate up to a point where pulp shear results in a loss of attachment for coarse particles. The effect is greater in a laboratory size cell (5-L) than in a pilot cell (60-L). The effect is attributed to increasing bubbleparticle contact with more intensive mixing. The possibility of mineral surface effects contributing to the observed behaviour is not considered as the pulp chemistry and particle size distribution remained constant. The effect of increasing Jg is inconsistent, likely due to the effect of Jg on bubble size distribution and froth recovery.
the tests are significantly the same because the confidence level is low (73.49%). That is lower than the confidence level obtained for the total P value and exactly similar to the total value. Then the same analyses were applied for the 60 L cell at the minimum and the maximum ITS, which are 2.99 m/s and 4.99 m/s, respectively. Again, the mean of the total P00 values were the same at a 78.51% confidence level. The confidence values for the total P00 are similar to those for the total P0 values because the EVF is a constant value and does change when the flotation cell size is constant. That puts the total P00 and total P0 in the same position when predictions are needed to be done in one cell at different hydrodynamic conditions. But the confidence interval was much lower (31.07%) when the t-test analysis was applied to the total P00 values obtained from both cells at the maximum impeller speed. That indicates the P00 value is the more robust and reliable value for the scale-up from the 5 L to the 60 L. Flotation rate constant (k), P, P0 and P00 values for unsized results obtained from all flotation tests conducted in 5 L and 60 L cell are summarised in Table 1. The coefficient of variance for P, P0 and P00 values obtained in 5 L and 60 L cells are presented in Table 2. The coefficient of variance (CV) of P is much higher than the CV’s for P0 and P00 in both flotation cell test results, especially in the 60 L cell where the CV for P is more than two times larger for the unsized result. The CV increased 1% when EVF was introduced to the model (i.e., P00 ) suggesting that P0 is more appropriate than P00 to describe particle properties. However, the size of the flotation
Table 1 Summary of flotation rate constant (k), P, P0 and P00 values obtained from 5 L to 60 L tests. Condition ID
Cell volume
Impeller tip speed (m/s)
Jg (cm/s)
P
P0
P00
k (1/min)
1 2 3 4 5 6 7 8 9 10 11 12
5L
3.2 3.9 4.5 4.9 3.2 3.9 4.5 4.9 3.2 3.9 4.5 4.9
0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7
0.00165 0.00181 0.00229 0.00228 0.00111 0.00128 0.00221 0.00220 0.00096 0.00155 0.00196 0.00242
0.00023 0.00024 0.00024 0.00022 0.00017 0.00020 0.00030 0.00030 0.00017 0.00024 0.00026 0.00023
0.00114 0.00118 0.00120 0.00109 0.00085 0.00097 0.00148 0.00147 0.00084 0.00121 0.00128 0.00116
1.34 1.79 2.29 2.42 1.11 1.47 2.67 2.66 1.16 2.00 2.73 3.15
1 2 3 4 5 6 7 8 9 10 11 12
60 L
3.0 3.7 4.3 5.0 3.0 3.7 4.3 5.0 3.0 3.7 4.3 5.0
0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7
0.00090 0.00113 0.00128 0.00161 0.00092 0.00095 0.00117 0.00149 0.00083 0.00105 0.00121 0.00180
0.00017 0.00020 0.00018 0.00018 0.00017 0.00015 0.00016 0.00018 0.00018 0.00018 0.00018 0.00022
0.00104 0.00122 0.00107 0.00109 0.00101 0.00092 0.00098 0.00107 0.00107 0.00110 0.00107 0.00133
1.09 1.43 1.56 1.83 1.24 1.32 1.62 2.00 1.16 1.48 1.75 2.63
Table 2 Coefficient of variation (CV) for P, P0 and P00 values at 97.7% confidence interval. Particle size (lm)
+90 +75–90 +53–75 +38–53 +20–38 –20 Unsized
60 L cell
5 L cell
P (%)
P0 (%)
P00 (%)
P (%)
P0 (%)
P00 (%)
16.2 18.0 16.5 14.9 17.7 44.0 25.4
14.8 8.6 9.9 9.9 8.2 27.6 9.9
17.3 12.2 11.9 11.2 9.8 27.3 11.3
23.2 22.0 24.1 21.5 28.3 32.8 27.7
7.4 12.3 12.6 13.4 17.2 25.2 16.2
7.7 12.5 12.9 13.8 17.8 26.4 17.0
E. Amini et al. / Minerals Engineering 100 (2017) 31–39 Table 3 Coefficient of variation (CV) for flotation rate constant (k), P, P0 and P00 values at 97.7% confidence interval (both cells). Particle size (lm)
k (%)
P (%)
P0 (%)
P00 (%)
+90 +75–90 +53–75 +38–53 +20–38 –20 Unsized
29.7 30.6 29.0 29.3 34.7 43.0 33.5
32.0 32.1 31.1 32.0 34.9 40.3 33.9
17.1 18.4 16.6 19.4 19.4 26.5 17.4
12.6 12.6 12.1 13.6 14.1 25.4 13.2
4.2. Size-by-size ore floatability Coefficient of Variation (CV) for P, P0 and P00 values of all given size fractions are shown in Table 2 to indicate which parameter is more representative of particle properties at different-size fractions. Similar to the total (unsized) values, P0 generally has the lower CV in both cells. The CV is much higher in the finest 20 lm size fraction for P, P0 and P00 . In this case the 20 lm is the most important size fraction because it contains the majority of the Cu (about 45%) in the feed. The variation of P, P0 and P00 across size fractions is the largest in the 5 L cell in most cases. Calculating CV for floatability values in a given size fraction for both flotation cells together is reported in Table 3 together with the CV for the flotation rate constant k. This shows more readily which of P, P0 or P00 is statistically more reliable for scale-up studies. The CV values at different-size fractions for k, shown in Table 3, are very similar to the CV for P at the different-size fractions, which suggest that bubble surface area flux alone is not sufficient to reduce the error/bias in the flotation rate constant calculation when the cell is increased in size from 5 L to 60 L. But the CV values for P00 are lower than the CV value for P0 , implying that including EVF is important in scale-up calculations and will reduce the error in flotation rate constant predictions. Variation of both P0 and P00 values in the results obtained in both cells is significantly lower than P and k values. The error analysis on the ore floatability values in this study indicates Eq. (9) is reliable for use in the calculation of ore floatability properties in a scale-up study. The equation reduces the error that usually occurs in ore floatability predictions from small-scale flotation test work significantly. Pulp chemistry is an important factor in scale-up analysis which is usually is kept constant as possible in these kinds of studies. 4.3. Prediction of flotation rate constant The new models are evaluated for delivering better predictions of k for different flotation scales. The flotation rate constant (k) in the 60 L cell was predicted using floatability values obtained from 5 L tests. The flotation rate constant of each flotation test conducted in the 5 L cell was used to predict the rate constant of 12 different conditions in the 60 L cell at the given particle size fractions. The impeller speed and TKEDR variation range is different in the 5 L cell compared to the 60 L. That variations are taken into account when P0 and P00 values are obtained but not in the calculation of P using the original AMIRA P9 model as the model does not have any parameter to take them into account. It is a common practice in flotation modelling to apply a scale-up factor that is either a fitted parameter or a rule of thumb value (e.g., 2.5) to adjust the gap between the flotation rate constant values obtained from processing the same ore in two different-size flotation machines. In this study, no fitted values were applied to adjust the results for the predictions in order to evaluate the models. Bubble size distribution in the 5 L and 60 L at the 12 hydrodynamic conditions is different which was reported in detail by
35
Amini et al. (2013). Bubble size is about 13% coarser in the 5 L cell than the 60 L cell at the same level of TKE. Fig. 2 shows the predicted k values using the original AMIRA P9 model (which applies only Sb as the scale-up criteria) versus the experimental k values, which were calculated directly from recovery-time data obtained in the 60 L flotation tests at 12 different hydrodynamic conditions for each size fraction. In Fig. 2, results for each size fraction are shown with different symbols and each size has 12 data points that represent 12 different hydrodynamic conditions. The title of each graph represents the hydrodynamic condition (Jg and ITS) of the flotation tests in the 5 L cell where floatability values from the 5 L test were used to predict k values in the 60 L cell tests. In the majority of the graphs, data points in a size fraction follow a horizontal line instead of being aligned among the 45° line. For instance, in the size class of 20 lm in the top three graphs, the predicted k is almost constant at around 1 (min1), but the experimental k varies between 0.5 and 1.5. There is another point far away from the others on 2.6 (min1) that is not considered to complete the linear trend and assumed to be an outlier. The trend shows that the variation in the hydrodynamic condition with scale-up is not detectable by the AMIRA P9 model. Moving from the top graphs to the bottom shows that the accuracy of the predictions reduces dramatically as impeller tip speed (ITS) increases, which is because the AMIRA P9 model lacks a hydrodynamic factor that accounts for the intensity of turbulence in the process. At the lowest ITS, increasing the air rate improves the accuracy of the predictions, but the worst predictions are obtained at the highest ITS and air rate. The æ values for the hydrodynamic conditions in the cells are presented in Table 4 were used to calculate P0 using k values obtained from the 5 L cell flotation tests in 12 different hydrodynamic conditions. The P0 values then were used to predict k in the 60 L cell and the predicted versus experimental values are presented in Fig. 3. Application of the P0 to predict k values in the 60 L test results is similar to the procedure that was used for P value earlier, but the only difference is that æ was added to the model. In all graphs in Fig. 3, the experimental k for the flotation test that was conducted at Jg = 0.7 (cm/s) and ITS = 4.93 (m/s) is much higher than the predicted k and this point is far away from the other points (red1 squares) in the series. Thus, the point is suspected to be an outlier. In all of the conditions, the predictions are improved compared to the original AMIRA P9 model as the data is aligned more closely to the 45° line, implying that the P0 model is more successful in considering hydrodynamic conditions of the cell in flotation rate constant calculations. These results show that the new model provides a better prediction of flotation rate constants (k) than the AMIRA P9 model. This provides more evidence to support the results presented in Tables 2 and 3 which confirms that P0 is less influenced by the hydrodynamic condition of a cell compared to P. Still Eq. (7) over predicts the k values, especially predictions of the flotation test conducted at Jg = 0.6 (cm/s) and ITS = 4.93 (m/s) and Jg = 0.4 (cm/s) and ITS = 3.92 (m/s). This means one or more parameters are still missing in the equation that causes bias in the results. P00 values calculated for the 12 hydrodynamic conditions at different size fractions from the 5 L tests were applied to predict k values of 60 L cell tests at different conditions and the results are shown in Fig. 4. EVF was 16.68% and 20.16% in the range of ITS for 60 L and 5 L flotation cells, respectively which was explained in part one of this manuscript. Thus, it was expected that the over predictions in k should reduce by about 4%. As Fig. 4 demonstrates, the predicted k values reduced and are closer to the experimental k
1 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.
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E. Amini et al. / Minerals Engineering 100 (2017) 31–39
Fig. 2. Predicted flotation rate constant (k) vs. experimental values using P value from original AMIRA P9 model.
Table 4
æ values at different impeller tip speed and Jg in 5 L and 60 L cells. Jg (cm/s)
Impeller tip speed (m/s) 5 L cell
0.4 0.6 0.7
60 L cell
3.2
3.9
4.5
4.9
3.0
3.7
4.3
5.0
3.83 3.55 3.27
3.99 3.61 3.53
4.63 3.91 3.99
4.91 5.16 4.91
3.06 3.18 2.83
3.20 3.46 3.26
3.82 3.82 3.67
4.41 4.24 4.16
values in all of the 12 conditions when EVF is included in the model. In particular, the predictions obtained from the test conducted at highest ITS and Jg when P00 is used (the graph at the
bottom-right in Fig. 4) are much closer to the experimental results compared to the original AMIRA P9 model (the graph at the bottom-right in Fig. 2).
E. Amini et al. / Minerals Engineering 100 (2017) 31–39
37
Fig. 3. Predicted flotation rate constant (k) vs. experimental values using P0 value.
To assess the quality of the predictions, F-test was applied to compare the precision of the three sets of predictions using P, P0 and P00 values. To calculate the deviation of the predicted flotation rate constant values from their experimental values, the predicted flotation rate constants at different particle size classes and flotation test conditions were divided by the relative experimental flotation rate constant values. So, three large sets of data were created with 1008 values for each of Figs. 2–4. The values equal to unity indicates the predicted and the experimental values are equal, the values larger than 1 indicated the flotation rate constant is over predicted and the values lower than 1 shows the rate constant is under predicted. So the variance value of each of the three data sets, which is closer to 0, indicates that the data set is more precise. Also, the mean value closer to 1 indicates the data
set is more accurate. First F-test was conducted on the predictions obtained using P and P0 values (results from Figs. 2 and 3). Variance values are 0.3213 and 0.0885 for P and P0 data sets, respectively indicating that the P0 is not varying as much as the P value at the different flotation conditions. The F value obtained from the variance values is equal to 3.63 at a high degree of freedom (1007). This indicates using P0 values improved the precision significantly at 99.99% confidence interval. The same analysis between the P0 and P00 data sets provides F value equal to 1.46 with the variance equal to 0.0606 for P00 data set. This indicates that the P00 data set is significantly more precise than the P0 data set at a confidence interval of 99.99%. The mean values for the P, P0 and P00 data sets are 1.62, 1.34 and 1.10, respectively, which indicates the P00 data set is the most accurate data set. Therefore, using the hydrodynamic
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E. Amini et al. / Minerals Engineering 100 (2017) 31–39
Fig. 4. Predicted flotation rate constant (k) vs. experimental values using P00 value.
parameters improved the precision and accuracy of the predictions significantly at confidence interval of 99.99%. The mean value for the P00 data set is a bit higher than 1, which indicates there might be other parameters still hidden in the P00 value, which most likely are the parameters that influence the pulp-froth interface and needs further investigations. For example, the influence of froth recovery on overall flotation rate constant in batch flotation test has been addressed by Amelunxen et al. (2014). 5. Conclusions and recommendations In this paper, the AMIRA P9 floatation model was modified by application of two dimensionless numbers (æ and EVF) to improve the prediction of the flotation rate constant where the hydrodynamic conditions and the size of flotation cell changes. The coeffi-
cient of variance (CV) for the ore property values (P, P0 and P00 ) reduced by introducing æ and EVF, confirming the precision improvements in ore floatability calculation. The ore property values obtained from 5 L tests were used to predict the flotation rate constant in the 60 L cell under different hydrodynamic conditions. The precision and accuracy of the predictions were enhanced when P0 was used instead of P, and they were further improved when P00 was used. It was concluded that the fractional volume of the cell that contains a high turbulent energy dissipation rate and the intensity of turbulence in a cell is important in calculating the flotation rate constant, providing further evidence to support the findings mentioned above. In this study, no fitted value was applied to predict flotation rate constants in the 60 L cell. However, the predictions using Eq. (9) are accurate and precise enough to be used in simulations and
E. Amini et al. / Minerals Engineering 100 (2017) 31–39
no further adjustment is required. That means æ and EVF are two important factors that reduce the bias in simulation and scale-up of flotation test results. It is recommended that small scale tests be repeated at different air rates and impeller speeds to quantify the variability of the parameters in the reconciliation process. This improves the quality of the results due to providing more data at a wider range of hydrodynamic conditions when the methodology is used. The TKEDR effect on flotation rate constant is different for coarse particles compared to fines. So it is recommended to calculate n in Eq. (6) separately for each size fraction when dealing with a wide range of particle sizes in the flotation pulp to incorporate the effect of TKEDR on different particle size fractions separately. For instance, n value for very coarse particles could be negative due to particles detachment rate enhancement when TKEDR increases. Since the turbulence factors have been successfully validated in batch flotation mode, further test work is suggested to test these parameters in a continuous system. Industrial flotation cells are much larger than the relative volume of the 60 L cell compared to the 5 L cell, which were the two scales of cells used in this study. Thus, this methodology should be further validated in larger scales so as to be applicable for industrial design and optimisation. Also a robust equation for froth phase is required to calculate the recovery of the material through the froth phase that takes the hydrodynamic parameter variations into account for the industrial-scale studies. Mineral separation efficiency (relative mineral flotation kinetics) is affected by factors including the following: 1. Bubble size distribution change due to impellor wear; 2. Pulp viscosity change due to variations in gangue type, especially when the mineral feed includes clay minerals; 3. Particle size distribution changes due to effects including classifier wear and ore hardness variations; 4. Classifier wear can also lead to variations in feed density; 5. Variations in mineral texture (liberation effects) which can also include the presence of soluble minerals present in flotation plant feed. However, the modifications to the flotation model in the existing P9 flotation model as proposed in the present work merit inclusion in plant testing, design and scale-up studies. This work has provided the first step in separating the ore property P00 into a more consistent particle property. Further work is necessary for the AMIRA P9 model to be integrated into a particle-based circuit model. Introducing more components, such
39
as mineral liberation, to the model is also expected to increase the quality of the predictions. Acknowledgements The authors would like to acknowledge JKTech manager Dr. Dan Alexander for the generous financial support for this research. Also JKMRC management team and staff are acknowledged for providing unlimited access to their lab facilities and technical supports. References Abrahamson, J., 1975. Collision rates of small particles in a vigorously turbulent fluid. Chem. Eng. Sci. 30, 1371–1379. Amini, E., 2013. Influence of Flotation Cell Hydrodynamics on the Flotation Kinetics and Scale up of Flotation Recovery PhD Thesis. University of Queensland, JKMRC. Amini, E., Bradshaw, D.J., Finch, J.A., Brennan, M., 2013. Influence of turbulence kinetic energy on bubble size in different scale flotation cells. Miner. Eng. J. 45 (May), 146–150. Amini, E., Alexander, D.J., Wightman, E., 2009. Using laboratory scale flotation testing to predict pilot scale flotation performance. In: 41st Annual Meeting of the Canadian Mineral Processors. CIM, Ottawa, Canada. Amelunxen, P., Sandoval, G., Barrigab, D., Amelunxen, R., 2014. The implications of the froth recovery at the laboratory scale. Miner. Eng. 66-68 (November), 54– 61. Bulatovic, S.M., 2007. Handbook of Flotation Reagents: Chemistry, Theory and Practice. Elsevier Publications. Collins, D.A., 2007. Validation of the JK Floatability Index Test Master Thesis. JKMRC, University of Queensland, Australia. Gorain, B.K., 1997. The Effect of Bubble Size on The Kinetics Of Flotation And Its Relevance To Scale-Up Ph.D.. University of Queensland. Gorain, B.K., Harris, M.C., Franzidis, J.P., Manlapig, E.V., 1998. The effect of froth residence time on the kinetics of flotation. Miner. Eng. 11 (7), 627–638. Hernandez-aguilar, J.R., 2011. On the role of bubble size in column flotation. In: CMP Conference, Ottawa, Canada. Hernandez-aguilar, J.R., Rao, S.R., Finch, J.A., 2005. Testing the k–Sb relationship at the microscale. Miner. Eng. 18 (6), 591–598. Laskowski, J., Woodburn, E.T., 1998. Frothing in Flotation II: Recent Advances in Coal Processing, vol. 2. CRC Press, Taylor and Francis Group. Matis, K.A., 1995. Flotation Science and Engineering. Marcel Dekker Inc., Aristotle University. Newell, R., Grano, S., 2006. Hydrodynamics and scale up in Rushton turbine flotation cells: part 2. Flotation scale-up for laboratory and pilot cells. Int. J. Miner. Process. 81, 65–78. Newell, R., Grano, S., 2007. Hydrodynamics and scale up in Rushton turbine flotation cells: part 1 – cell hydrodynamics. Int. J. Miner. Process. 81, 224–236. P9L, A., 2000. AMIRA P9L Technical Report. Brisbane, JKMRC. P9M, A., 2004. AMIRA P9M Technical Report. Brisbane, JKMRC. P9N, A., 2004. 2nd P9 Technical Report. Brisbane, JKMRC. Pyke, B., 2004. Bubble-Particle Capture in Turbulent Flotation Systems. Savassi, O.N., 1998. Direct Estimation of the Degree of Entrainment and the Froth Recovery of Attached Particles in Industrial Flotation Cells Ph.D.. University of Queensland. Schwarz, S., Alexander, D., 2006. Gas dispersion measurements in industrial flotation cells. Miner. Eng. 19, 554–560. Schubert, H., 1999. On the turbulence-controlled microprocesses in flotation machines. Int. J. Miner. Process. 56, 257–276.
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Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Influence of flotation cell hydrodynamics on the flotation kinetics and scale up, Part 2: Introducing turbulence parameters to improve predictions E. Amini ⇑, D.J. Bradshaw, W. Xie SMI/JKMRC, University of Queensland, Brisbane, Australia
a r t i c l e
i n f o
Article history: Received 7 July 2015 Revised 2 September 2016 Accepted 2 October 2016
Keywords: Flotation kinetic Cell hydrodynamic Dimensionless parameters Turbulence Ore property
a b s t r a c t The AMIRA P9 model has floatability (P) as the ore property which is considered to remain constant in different flotation cell sizes under different hydrodynamic conditions. However, in this study increasing the power input increased the P value, especially in finer particle size classes (below 75 lm). Acceptable explanations for the floatability variations, as a result of hydrodynamic condition variations in the flotation cells, have been sought by looking at the literature and the results obtained in this study were published in part one of this manuscript. To improve the accuracy of the AMIRA P9 flotation model in predicting flotation rate constant (k) and to improve the consistency of ore property, measurable and appropriate turbulence parameters were sought to be incorporated into the model. Therefore, two dimensionless turbulence parameters æ and EVF, derived from practical measurements, were formulated and introduced to the AMIRA P9 model. The modified ore floatability parameter, P00 , was demonstrated to be a more consistent characterisation of the ore property than P and both the accuracy and precision of the k prediction improved for a variety of hydrodynamic conditions of flotation cells. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction The AMIRA P9 model has been applied in a number of flotation studies and has been successfully used to optimise several industrial flotation circuits (AMIRA P9 reports, 2000, 2004a,b). The model is explained in detail in part one of this paper. Other studies (Amini et al., 2009; Collins, 2007) showed that the ore property (P value) is not constant when scaling up from laboratory to pilot plant or industrial scales, and that the P value changes with varying hydrodynamic conditions in cells. It was shown in the first part of this manuscript that increasing impeller speed and turbulent kinetic energy dissipation rate (TKEDR) changed the P value, which is contrary to the proposed mechanism of Gorain (1997) and Gorain et al. (1998) who reported that impeller speed changed bubble surface area (Sb) which is a function of bubble size distribution, but not the P value in his experiments on industrial cells. This implies other factors such as impeller speed influence the P value in small scale flotation cells. Varying impellor speed is not a control variable available in industrial plant operation. It is however an important concept in relating laboratory flotation kinetic test data to industrial plant scale-up. As a result, a fitted scale-up number ⇑ Corresponding author. E-mail address: [email protected] (E. Amini). http://dx.doi.org/10.1016/j.mineng.2016.10.001 0892-6875/Ó 2016 Elsevier Ltd. All rights reserved.
needs to be applied to the P value predicted by the AMIRA P9 model. Hernandez-Aguilar (2011) and Hernandez-Aguilar et al. (2005) reported that the P value does not have a linear relationship with the flotation rate constant as claimed by Gorain (1997) and Gorain et al. (1998). The authors showed that incorporating a new bubble size factor into the model can improve the model predictions in column flotation. Their new model seems to work when the Sauter mean bubble diameter (d32) is lower than 1.2 mm but Gorain’s relationship is still valid in the column flotation cells where no agitation is required. The mean value of d32 in the mechanical flotation cells in the industry is equal to 1.7 mm (Schwarz and Alexander, 2006) showing that Hernandez-Aguilar (2011) is not valid for the mechanical cells. Moreover, the authors did not take into account the effect of bubble size distribution and bubble particle loading on froth stability and froth drop-back (Laskowski and Woodburn, 1998; Bulatovic, 2007) when the Rf was calculated in their studies. That is a crucial factor in the rate constant calculation in industrial test work. Putting the result of the studies together indicates experimental test work with minimum froth effect is required to analyse the relationship between flotation rate constant (k) and Sb in pulp zone. In contrast, the fundamental flotation models discussed in part one already contain hydrodynamic factors such as energy
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E. Amini et al. / Minerals Engineering 100 (2017) 31–39
Nomenclature
æ db EVF k P
dimensionless number that represent hydrodynamic condition of the cell diameter of bubble (mm) effective volume factor (dimensionless) flotation rate constant (1/min) floatability as defined in AMIRA P9 flotation model (dimensionless)
dissipation rate, bubble size, viscosity, and number of bubbles. Application of several parameters makes the fundamental models sensitive to the majority of influential parameters in the flotation process. However, calculation or measurement of each parameter has an associated error and more parameters simply mean a bigger uncertainty in the calculated/predicted flotation rate constant. Also, some of the parameters in the fundamental models cannot be directly measured, which increases the uncertainties in calculation even more. It was found that bubble size, energy dissipation rate and viscosity are the most important factors influencing bubble-particle collision rates (Schubert, 1999; Abrahamson, 1975). It was also assumed that the volume of the cell that contains higher TKEDR affects the flotation rate constant by increasing the overall collision rate in the cell (Savassi, 1998). Savassi (1998) divided the pulp in a flotation cell into two zones: (1) The ‘‘Collection” zone (near the impeller), which contains high turbulence intensity and where the majority of bubble particle attachment occurs and (2) The ‘‘Quiescent” zone (outside the impeller region), which contains very low turbulence intensity and where the interaction between bubbles and particles is very low. The existence of these zones was also observed by other researchers (Pyke, 2004; Newell and Grano, 2006, 2007) and is demonstrated in part two of this manuscript. The quiescent zone or disengagement zone indicates the zone that bubble-particle attachment or detachment rate is considerably lower compare to the high turbulent zone (Matis, 1995). It was demonstrated in part one of this manuscript that it is possible to make appropriate measurements to characterise the hydrodynamic conditions of flotation cells of any size. Turbulent kinetic energy dissipation rate (TKEDR), bubble size, shear and viscosity are the key influential parameters in a flotation process which can be measured with practical methodologies and used for modelling purposes. Introducing dimensionless turbulence parameters to the AMIRA P9 model (modified AMIRA P9 model) can reduce the error in calculating of ore property. Thus it enhances the flotation rate constant prediction for a variety of size and hydrodynamic conditions of flotation cells.
P0 and P00 floatability as defined in the modified P9 flotation model (dimensionless) Rf recovery of particles across the froth (%) Sb bubble surface area flux (1/s) TKEDR turbulent kinetic energy dissipation rate (m3/s2) m kinematic viscosity (m2/s) q density (m3/kg)
the dimensions of (L2T1), (L2T3), (L), (ML3) and (T1), respectively. Bubble surface area flux is already present in the AMIRA P9 model so the extension of the model could contain the other four parameters. Thus, a dimensionless number, which is called æ, can be written using db, TKEDR, m and q:
æ ¼ db :TKEDRa mb qc
ð1Þ
By introducing the dimensions we have
æ¼L
L2 T3
!a
L2 T
!b c M 3 L
ð2Þ
where a, b and c are the parameters which should be calculated by solving the equation for each fundamental quantity to eliminate the dimensions.
For length ðLÞ : 1 þ 2a þ 2b 3c ¼ 0
ð3Þ
For time ðTÞ : 3a þ b ¼ 0
ð4Þ
For mass ðMÞ : c ¼ 0
ð5Þ
Solving the Eqs. (3)–(5) returns values of a = 0.25, b = 0.75 and c = 0. Therefore, æ can be written as An expression to describe pulp hydrodynamic conditions incorporating pulp viscosity (m), arithmetic mean bubble size (db) and the turbulent kinetic energy dissipation rate (TKEDR) has been developed and is shown as Eq. (6). The derivation of the equation was published in detail (Amini, 2013).
æ¼
db TKEDR0:25
m0:75
!n ð6Þ
where n is estimated by the number of flotation tests over a range of operational conditions. It is suggested to conduct three lab scale flotation tests at three levels of impeller speed to fit n value where floatability remains constant for each floatability class (e.g. size by size P values) over the operational range. When only æ is incorporated into AMIRA P9 model ore property (P0 ) can be estimated from this model. The P0 becomes more of an ore property compared to the P in the original AMIRA P9 model (k = P Sb Rf).
2. Development of dimensionless numbers
k ¼ P 0 Sb æ R f
It was shown that the floatability (P value) is proportional to hydrodynamic conditions of a flotation cell (see part one). The hydrodynamic effect may be described using two expressions related to the pulp hydrodynamics (æ) and effective cell volume (EVF). The fundamental flotation models indicated that the flotation rate constant is directly proportional to bubble-particle collision frequency, which is a function of kinematic viscosity (m), turbulent kinetic energy dissipation rate (TKEDR), bubble diameter (db), density (q) and bubble surface area flux (Sb). These parameters have
From Savassi’s assumption, it can be concluded that increasing the energy dissipation rate increases the collision frequency. Thus, increasing the energy dissipation rate increases the flotation rate constant, but only up to a point where detachment overrides the collision frequency effect, especially for coarse particles. Increasing the volume of the ‘‘high” turbulent zone (TKEDR > 0.1 m2/s3 in this study) increases the collision frequency in a flotation cell and the flotation rate constant increases. Therefore, the proportion of the cell volume, which contains a high-energy dissipation rate (here called the Effective Volume Factor or EVF), is introduced here:
ð7Þ
E. Amini et al. / Minerals Engineering 100 (2017) 31–39
EVF ¼
33
Volume of the cell that contains high energy dissipation rate Total v olume of the cell ð8Þ
Incorporating EVF factor into Eq. (6) gives;
k ¼ P00 Sb æ EVF Rf
ð9Þ
In this equation P00 represents floatability of the ore in a flotation process. 3. Review of experimental data Details on flotation test work which were carried out on the 5 L and 60 L flotation cells was explained in part one of the manuscript. Twelve different hydrodynamic conditions were studied. It was found that the flotation rate (k) - bubble surface area flux relationship (Sb) observed in commercial flotation cells reported previously (Gorain, 1997; Gorain et al., 1998) were not observed in laboratory (5 L) and pilot scale (60 L) flotation cells. Hydrodynamic test conditions did affect flotation rate, especially for laboratory scale flotation conditions. Thus, other hydrodynamic variables need to be included in the flotation scale-up models. 4. Results and discussion 4.1. Unsized ore floatability P, P0 , and P00 for all of the test results obtained in the 5 L and 60 L cells (calculated using, original AMIRA P9 model, Eqs. (7) and (9)) are presented in Fig. 1. The coefficient of variance (standard deviation to average) for P value calculated from three repeat tests conducted in 60 L cell at Jg = 0.6 cm/s and impeller tip speed (ITS) = 4.32 m/s. The P values are 0.00117, 0.00116, and 0.00127 so the coefficient of variance of 5.7% at 97.7% confidence interval was obtained. The figure shows, unlike the P value, the P0 value is almost independent of impeller speed and air rate in both cells. The P value obtained at the lowest impeller speed (3.22 m/s) and superficial gas velocity of 0.4 m/s in the 5 L cell is very close to the P value obtained at impeller speed of 3.99 m/s most likely due to the experimental error. A series of t-test analyses were conducted on the extracted floatability parameters to evaluate the improvements in the predictions accuracy. In the calculations, the null hypothesis was used, which claims there is no significant difference between the mean values of two sets of data. The confidence interval to reject the null hypothesis is calculated from the t-test analyses. The confidence interval values more than 90% (in industrial studies) shows that the null hypothesis is rejected and the mean values are not the same. Firstly, the t-test was applied to the total P values that are extracted from the 5 L tests. The test compares the values obtained from the flotation tests at ITS = 3.22 m/s with the flotation tests at ITS = 4.93 m/s. The total P values at the minimum and maximum ITS (relevant to the level of turbulence) in the tests are not significantly the same at 95.26% confidence interval. The same analysis was then applied for the 60 L cell at the minimum and maximum ITS, which are 2.99 m/s and 4.99 m/s, respectively. Again, the total P values were not significantly the same at 97.61% confidence interval. The same analysis on the total P0 values at the lowest and highest ITS values in the 5 L cell shows the values are significantly the same as the confidence interval is not high enough to reject the null hypothesis (73.49%). In the 60 L cell, confidence interval is at 78.51% for the total P0 values at the lowest and the highest ITS values. This shows that the total P0 is significantly more robust for flotation modelling compared to the P values when the impeller speed and level of turbulence changes in a cell.
Fig. 1. Unsized ore floatability vs. impeller tip speed in 5 L (open symbol) and 60 L (closed symbol) experiments.
Effective Volume Factor (EVF) was used to calculate P00 value as well. The results are shown at the bottom of Fig. 1. In order to calculate n value in Eq. (6), the model was fitted to all results obtained from the 5 L cell tests. An assumption was made that the floatability value (P0 ) should remain constant for each size fraction as the chemistry of pulp was maintained in the 5 L cell tests. The variable parameters are the hydrodynamic parameters in the 5 L cell tests which were measured and extracted from flotation rate constant. The n value is equal to 1.47 and was kept constant in the calculations of floatability values in the 60 L cell tests as well. Generally, P00 is higher than P0 and lower than P. A series of t-test analyses was also conducted on the total P00 values that are extracted from the 5 L tests comparing the values obtained from the flotation tests at ITS = 3.22 m/s with the flotation tests at ITS = 4.93 m/s. The total P00 values at the minimum and the maximum tip speed (relative to level of turbulence) in
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E. Amini et al. / Minerals Engineering 100 (2017) 31–39
cell still needs to be considered in the CV value calculation and this is discussed in the next section. Both statistical analyses showed that the total P0 and total P00 are more reliable than the total P value, and total P00 is the most reliable floatability number to predict the results in scale-up studies because increasing the level of turbulence (relative to TKEDR) improves the collision rate (Abrahamson, 1975; Pyke, 2004). It eventually increases the flotation rate constant; therefore, the effect of turbulence should be incorporated in the scale-up analysis or when a cell with variable impeller speed is used. The industrial cells usually built with fixed impeller speed but the impeller speed and the TKEDR is different in different size, type and brand of the flotation cells. Therefore, in scale-up analysis from batch or pilot scale to industrial flotation cells the TKEDR should be estimated and incorporated according to the size, type and brand of flotation cell being used in the circuit. Increasing impellor tip speed and hence mixing intensity in a stator-type flotation cell is shown to increase flotation rate up to a point where pulp shear results in a loss of attachment for coarse particles. The effect is greater in a laboratory size cell (5-L) than in a pilot cell (60-L). The effect is attributed to increasing bubbleparticle contact with more intensive mixing. The possibility of mineral surface effects contributing to the observed behaviour is not considered as the pulp chemistry and particle size distribution remained constant. The effect of increasing Jg is inconsistent, likely due to the effect of Jg on bubble size distribution and froth recovery.
the tests are significantly the same because the confidence level is low (73.49%). That is lower than the confidence level obtained for the total P value and exactly similar to the total value. Then the same analyses were applied for the 60 L cell at the minimum and the maximum ITS, which are 2.99 m/s and 4.99 m/s, respectively. Again, the mean of the total P00 values were the same at a 78.51% confidence level. The confidence values for the total P00 are similar to those for the total P0 values because the EVF is a constant value and does change when the flotation cell size is constant. That puts the total P00 and total P0 in the same position when predictions are needed to be done in one cell at different hydrodynamic conditions. But the confidence interval was much lower (31.07%) when the t-test analysis was applied to the total P00 values obtained from both cells at the maximum impeller speed. That indicates the P00 value is the more robust and reliable value for the scale-up from the 5 L to the 60 L. Flotation rate constant (k), P, P0 and P00 values for unsized results obtained from all flotation tests conducted in 5 L and 60 L cell are summarised in Table 1. The coefficient of variance for P, P0 and P00 values obtained in 5 L and 60 L cells are presented in Table 2. The coefficient of variance (CV) of P is much higher than the CV’s for P0 and P00 in both flotation cell test results, especially in the 60 L cell where the CV for P is more than two times larger for the unsized result. The CV increased 1% when EVF was introduced to the model (i.e., P00 ) suggesting that P0 is more appropriate than P00 to describe particle properties. However, the size of the flotation
Table 1 Summary of flotation rate constant (k), P, P0 and P00 values obtained from 5 L to 60 L tests. Condition ID
Cell volume
Impeller tip speed (m/s)
Jg (cm/s)
P
P0
P00
k (1/min)
1 2 3 4 5 6 7 8 9 10 11 12
5L
3.2 3.9 4.5 4.9 3.2 3.9 4.5 4.9 3.2 3.9 4.5 4.9
0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7
0.00165 0.00181 0.00229 0.00228 0.00111 0.00128 0.00221 0.00220 0.00096 0.00155 0.00196 0.00242
0.00023 0.00024 0.00024 0.00022 0.00017 0.00020 0.00030 0.00030 0.00017 0.00024 0.00026 0.00023
0.00114 0.00118 0.00120 0.00109 0.00085 0.00097 0.00148 0.00147 0.00084 0.00121 0.00128 0.00116
1.34 1.79 2.29 2.42 1.11 1.47 2.67 2.66 1.16 2.00 2.73 3.15
1 2 3 4 5 6 7 8 9 10 11 12
60 L
3.0 3.7 4.3 5.0 3.0 3.7 4.3 5.0 3.0 3.7 4.3 5.0
0.4 0.4 0.4 0.4 0.6 0.6 0.6 0.6 0.7 0.7 0.7 0.7
0.00090 0.00113 0.00128 0.00161 0.00092 0.00095 0.00117 0.00149 0.00083 0.00105 0.00121 0.00180
0.00017 0.00020 0.00018 0.00018 0.00017 0.00015 0.00016 0.00018 0.00018 0.00018 0.00018 0.00022
0.00104 0.00122 0.00107 0.00109 0.00101 0.00092 0.00098 0.00107 0.00107 0.00110 0.00107 0.00133
1.09 1.43 1.56 1.83 1.24 1.32 1.62 2.00 1.16 1.48 1.75 2.63
Table 2 Coefficient of variation (CV) for P, P0 and P00 values at 97.7% confidence interval. Particle size (lm)
+90 +75–90 +53–75 +38–53 +20–38 –20 Unsized
60 L cell
5 L cell
P (%)
P0 (%)
P00 (%)
P (%)
P0 (%)
P00 (%)
16.2 18.0 16.5 14.9 17.7 44.0 25.4
14.8 8.6 9.9 9.9 8.2 27.6 9.9
17.3 12.2 11.9 11.2 9.8 27.3 11.3
23.2 22.0 24.1 21.5 28.3 32.8 27.7
7.4 12.3 12.6 13.4 17.2 25.2 16.2
7.7 12.5 12.9 13.8 17.8 26.4 17.0
E. Amini et al. / Minerals Engineering 100 (2017) 31–39 Table 3 Coefficient of variation (CV) for flotation rate constant (k), P, P0 and P00 values at 97.7% confidence interval (both cells). Particle size (lm)
k (%)
P (%)
P0 (%)
P00 (%)
+90 +75–90 +53–75 +38–53 +20–38 –20 Unsized
29.7 30.6 29.0 29.3 34.7 43.0 33.5
32.0 32.1 31.1 32.0 34.9 40.3 33.9
17.1 18.4 16.6 19.4 19.4 26.5 17.4
12.6 12.6 12.1 13.6 14.1 25.4 13.2
4.2. Size-by-size ore floatability Coefficient of Variation (CV) for P, P0 and P00 values of all given size fractions are shown in Table 2 to indicate which parameter is more representative of particle properties at different-size fractions. Similar to the total (unsized) values, P0 generally has the lower CV in both cells. The CV is much higher in the finest 20 lm size fraction for P, P0 and P00 . In this case the 20 lm is the most important size fraction because it contains the majority of the Cu (about 45%) in the feed. The variation of P, P0 and P00 across size fractions is the largest in the 5 L cell in most cases. Calculating CV for floatability values in a given size fraction for both flotation cells together is reported in Table 3 together with the CV for the flotation rate constant k. This shows more readily which of P, P0 or P00 is statistically more reliable for scale-up studies. The CV values at different-size fractions for k, shown in Table 3, are very similar to the CV for P at the different-size fractions, which suggest that bubble surface area flux alone is not sufficient to reduce the error/bias in the flotation rate constant calculation when the cell is increased in size from 5 L to 60 L. But the CV values for P00 are lower than the CV value for P0 , implying that including EVF is important in scale-up calculations and will reduce the error in flotation rate constant predictions. Variation of both P0 and P00 values in the results obtained in both cells is significantly lower than P and k values. The error analysis on the ore floatability values in this study indicates Eq. (9) is reliable for use in the calculation of ore floatability properties in a scale-up study. The equation reduces the error that usually occurs in ore floatability predictions from small-scale flotation test work significantly. Pulp chemistry is an important factor in scale-up analysis which is usually is kept constant as possible in these kinds of studies. 4.3. Prediction of flotation rate constant The new models are evaluated for delivering better predictions of k for different flotation scales. The flotation rate constant (k) in the 60 L cell was predicted using floatability values obtained from 5 L tests. The flotation rate constant of each flotation test conducted in the 5 L cell was used to predict the rate constant of 12 different conditions in the 60 L cell at the given particle size fractions. The impeller speed and TKEDR variation range is different in the 5 L cell compared to the 60 L. That variations are taken into account when P0 and P00 values are obtained but not in the calculation of P using the original AMIRA P9 model as the model does not have any parameter to take them into account. It is a common practice in flotation modelling to apply a scale-up factor that is either a fitted parameter or a rule of thumb value (e.g., 2.5) to adjust the gap between the flotation rate constant values obtained from processing the same ore in two different-size flotation machines. In this study, no fitted values were applied to adjust the results for the predictions in order to evaluate the models. Bubble size distribution in the 5 L and 60 L at the 12 hydrodynamic conditions is different which was reported in detail by
35
Amini et al. (2013). Bubble size is about 13% coarser in the 5 L cell than the 60 L cell at the same level of TKE. Fig. 2 shows the predicted k values using the original AMIRA P9 model (which applies only Sb as the scale-up criteria) versus the experimental k values, which were calculated directly from recovery-time data obtained in the 60 L flotation tests at 12 different hydrodynamic conditions for each size fraction. In Fig. 2, results for each size fraction are shown with different symbols and each size has 12 data points that represent 12 different hydrodynamic conditions. The title of each graph represents the hydrodynamic condition (Jg and ITS) of the flotation tests in the 5 L cell where floatability values from the 5 L test were used to predict k values in the 60 L cell tests. In the majority of the graphs, data points in a size fraction follow a horizontal line instead of being aligned among the 45° line. For instance, in the size class of 20 lm in the top three graphs, the predicted k is almost constant at around 1 (min1), but the experimental k varies between 0.5 and 1.5. There is another point far away from the others on 2.6 (min1) that is not considered to complete the linear trend and assumed to be an outlier. The trend shows that the variation in the hydrodynamic condition with scale-up is not detectable by the AMIRA P9 model. Moving from the top graphs to the bottom shows that the accuracy of the predictions reduces dramatically as impeller tip speed (ITS) increases, which is because the AMIRA P9 model lacks a hydrodynamic factor that accounts for the intensity of turbulence in the process. At the lowest ITS, increasing the air rate improves the accuracy of the predictions, but the worst predictions are obtained at the highest ITS and air rate. The æ values for the hydrodynamic conditions in the cells are presented in Table 4 were used to calculate P0 using k values obtained from the 5 L cell flotation tests in 12 different hydrodynamic conditions. The P0 values then were used to predict k in the 60 L cell and the predicted versus experimental values are presented in Fig. 3. Application of the P0 to predict k values in the 60 L test results is similar to the procedure that was used for P value earlier, but the only difference is that æ was added to the model. In all graphs in Fig. 3, the experimental k for the flotation test that was conducted at Jg = 0.7 (cm/s) and ITS = 4.93 (m/s) is much higher than the predicted k and this point is far away from the other points (red1 squares) in the series. Thus, the point is suspected to be an outlier. In all of the conditions, the predictions are improved compared to the original AMIRA P9 model as the data is aligned more closely to the 45° line, implying that the P0 model is more successful in considering hydrodynamic conditions of the cell in flotation rate constant calculations. These results show that the new model provides a better prediction of flotation rate constants (k) than the AMIRA P9 model. This provides more evidence to support the results presented in Tables 2 and 3 which confirms that P0 is less influenced by the hydrodynamic condition of a cell compared to P. Still Eq. (7) over predicts the k values, especially predictions of the flotation test conducted at Jg = 0.6 (cm/s) and ITS = 4.93 (m/s) and Jg = 0.4 (cm/s) and ITS = 3.92 (m/s). This means one or more parameters are still missing in the equation that causes bias in the results. P00 values calculated for the 12 hydrodynamic conditions at different size fractions from the 5 L tests were applied to predict k values of 60 L cell tests at different conditions and the results are shown in Fig. 4. EVF was 16.68% and 20.16% in the range of ITS for 60 L and 5 L flotation cells, respectively which was explained in part one of this manuscript. Thus, it was expected that the over predictions in k should reduce by about 4%. As Fig. 4 demonstrates, the predicted k values reduced and are closer to the experimental k
1 For interpretation of color in Fig. 3, the reader is referred to the web version of this article.
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E. Amini et al. / Minerals Engineering 100 (2017) 31–39
Fig. 2. Predicted flotation rate constant (k) vs. experimental values using P value from original AMIRA P9 model.
Table 4
æ values at different impeller tip speed and Jg in 5 L and 60 L cells. Jg (cm/s)
Impeller tip speed (m/s) 5 L cell
0.4 0.6 0.7
60 L cell
3.2
3.9
4.5
4.9
3.0
3.7
4.3
5.0
3.83 3.55 3.27
3.99 3.61 3.53
4.63 3.91 3.99
4.91 5.16 4.91
3.06 3.18 2.83
3.20 3.46 3.26
3.82 3.82 3.67
4.41 4.24 4.16
values in all of the 12 conditions when EVF is included in the model. In particular, the predictions obtained from the test conducted at highest ITS and Jg when P00 is used (the graph at the
bottom-right in Fig. 4) are much closer to the experimental results compared to the original AMIRA P9 model (the graph at the bottom-right in Fig. 2).
E. Amini et al. / Minerals Engineering 100 (2017) 31–39
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Fig. 3. Predicted flotation rate constant (k) vs. experimental values using P0 value.
To assess the quality of the predictions, F-test was applied to compare the precision of the three sets of predictions using P, P0 and P00 values. To calculate the deviation of the predicted flotation rate constant values from their experimental values, the predicted flotation rate constants at different particle size classes and flotation test conditions were divided by the relative experimental flotation rate constant values. So, three large sets of data were created with 1008 values for each of Figs. 2–4. The values equal to unity indicates the predicted and the experimental values are equal, the values larger than 1 indicated the flotation rate constant is over predicted and the values lower than 1 shows the rate constant is under predicted. So the variance value of each of the three data sets, which is closer to 0, indicates that the data set is more precise. Also, the mean value closer to 1 indicates the data
set is more accurate. First F-test was conducted on the predictions obtained using P and P0 values (results from Figs. 2 and 3). Variance values are 0.3213 and 0.0885 for P and P0 data sets, respectively indicating that the P0 is not varying as much as the P value at the different flotation conditions. The F value obtained from the variance values is equal to 3.63 at a high degree of freedom (1007). This indicates using P0 values improved the precision significantly at 99.99% confidence interval. The same analysis between the P0 and P00 data sets provides F value equal to 1.46 with the variance equal to 0.0606 for P00 data set. This indicates that the P00 data set is significantly more precise than the P0 data set at a confidence interval of 99.99%. The mean values for the P, P0 and P00 data sets are 1.62, 1.34 and 1.10, respectively, which indicates the P00 data set is the most accurate data set. Therefore, using the hydrodynamic
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E. Amini et al. / Minerals Engineering 100 (2017) 31–39
Fig. 4. Predicted flotation rate constant (k) vs. experimental values using P00 value.
parameters improved the precision and accuracy of the predictions significantly at confidence interval of 99.99%. The mean value for the P00 data set is a bit higher than 1, which indicates there might be other parameters still hidden in the P00 value, which most likely are the parameters that influence the pulp-froth interface and needs further investigations. For example, the influence of froth recovery on overall flotation rate constant in batch flotation test has been addressed by Amelunxen et al. (2014). 5. Conclusions and recommendations In this paper, the AMIRA P9 floatation model was modified by application of two dimensionless numbers (æ and EVF) to improve the prediction of the flotation rate constant where the hydrodynamic conditions and the size of flotation cell changes. The coeffi-
cient of variance (CV) for the ore property values (P, P0 and P00 ) reduced by introducing æ and EVF, confirming the precision improvements in ore floatability calculation. The ore property values obtained from 5 L tests were used to predict the flotation rate constant in the 60 L cell under different hydrodynamic conditions. The precision and accuracy of the predictions were enhanced when P0 was used instead of P, and they were further improved when P00 was used. It was concluded that the fractional volume of the cell that contains a high turbulent energy dissipation rate and the intensity of turbulence in a cell is important in calculating the flotation rate constant, providing further evidence to support the findings mentioned above. In this study, no fitted value was applied to predict flotation rate constants in the 60 L cell. However, the predictions using Eq. (9) are accurate and precise enough to be used in simulations and
E. Amini et al. / Minerals Engineering 100 (2017) 31–39
no further adjustment is required. That means æ and EVF are two important factors that reduce the bias in simulation and scale-up of flotation test results. It is recommended that small scale tests be repeated at different air rates and impeller speeds to quantify the variability of the parameters in the reconciliation process. This improves the quality of the results due to providing more data at a wider range of hydrodynamic conditions when the methodology is used. The TKEDR effect on flotation rate constant is different for coarse particles compared to fines. So it is recommended to calculate n in Eq. (6) separately for each size fraction when dealing with a wide range of particle sizes in the flotation pulp to incorporate the effect of TKEDR on different particle size fractions separately. For instance, n value for very coarse particles could be negative due to particles detachment rate enhancement when TKEDR increases. Since the turbulence factors have been successfully validated in batch flotation mode, further test work is suggested to test these parameters in a continuous system. Industrial flotation cells are much larger than the relative volume of the 60 L cell compared to the 5 L cell, which were the two scales of cells used in this study. Thus, this methodology should be further validated in larger scales so as to be applicable for industrial design and optimisation. Also a robust equation for froth phase is required to calculate the recovery of the material through the froth phase that takes the hydrodynamic parameter variations into account for the industrial-scale studies. Mineral separation efficiency (relative mineral flotation kinetics) is affected by factors including the following: 1. Bubble size distribution change due to impellor wear; 2. Pulp viscosity change due to variations in gangue type, especially when the mineral feed includes clay minerals; 3. Particle size distribution changes due to effects including classifier wear and ore hardness variations; 4. Classifier wear can also lead to variations in feed density; 5. Variations in mineral texture (liberation effects) which can also include the presence of soluble minerals present in flotation plant feed. However, the modifications to the flotation model in the existing P9 flotation model as proposed in the present work merit inclusion in plant testing, design and scale-up studies. This work has provided the first step in separating the ore property P00 into a more consistent particle property. Further work is necessary for the AMIRA P9 model to be integrated into a particle-based circuit model. Introducing more components, such
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as mineral liberation, to the model is also expected to increase the quality of the predictions. Acknowledgements The authors would like to acknowledge JKTech manager Dr. Dan Alexander for the generous financial support for this research. Also JKMRC management team and staff are acknowledged for providing unlimited access to their lab facilities and technical supports. References Abrahamson, J., 1975. Collision rates of small particles in a vigorously turbulent fluid. Chem. Eng. Sci. 30, 1371–1379. Amini, E., 2013. Influence of Flotation Cell Hydrodynamics on the Flotation Kinetics and Scale up of Flotation Recovery PhD Thesis. University of Queensland, JKMRC. Amini, E., Bradshaw, D.J., Finch, J.A., Brennan, M., 2013. Influence of turbulence kinetic energy on bubble size in different scale flotation cells. Miner. Eng. J. 45 (May), 146–150. Amini, E., Alexander, D.J., Wightman, E., 2009. Using laboratory scale flotation testing to predict pilot scale flotation performance. In: 41st Annual Meeting of the Canadian Mineral Processors. CIM, Ottawa, Canada. Amelunxen, P., Sandoval, G., Barrigab, D., Amelunxen, R., 2014. The implications of the froth recovery at the laboratory scale. Miner. Eng. 66-68 (November), 54– 61. Bulatovic, S.M., 2007. Handbook of Flotation Reagents: Chemistry, Theory and Practice. Elsevier Publications. Collins, D.A., 2007. Validation of the JK Floatability Index Test Master Thesis. JKMRC, University of Queensland, Australia. Gorain, B.K., 1997. The Effect of Bubble Size on The Kinetics Of Flotation And Its Relevance To Scale-Up Ph.D.. University of Queensland. Gorain, B.K., Harris, M.C., Franzidis, J.P., Manlapig, E.V., 1998. The effect of froth residence time on the kinetics of flotation. Miner. Eng. 11 (7), 627–638. Hernandez-aguilar, J.R., 2011. On the role of bubble size in column flotation. In: CMP Conference, Ottawa, Canada. Hernandez-aguilar, J.R., Rao, S.R., Finch, J.A., 2005. Testing the k–Sb relationship at the microscale. Miner. Eng. 18 (6), 591–598. Laskowski, J., Woodburn, E.T., 1998. Frothing in Flotation II: Recent Advances in Coal Processing, vol. 2. CRC Press, Taylor and Francis Group. Matis, K.A., 1995. Flotation Science and Engineering. Marcel Dekker Inc., Aristotle University. Newell, R., Grano, S., 2006. Hydrodynamics and scale up in Rushton turbine flotation cells: part 2. Flotation scale-up for laboratory and pilot cells. Int. J. Miner. Process. 81, 65–78. Newell, R., Grano, S., 2007. Hydrodynamics and scale up in Rushton turbine flotation cells: part 1 – cell hydrodynamics. Int. J. Miner. Process. 81, 224–236. P9L, A., 2000. AMIRA P9L Technical Report. Brisbane, JKMRC. P9M, A., 2004. AMIRA P9M Technical Report. Brisbane, JKMRC. P9N, A., 2004. 2nd P9 Technical Report. Brisbane, JKMRC. Pyke, B., 2004. Bubble-Particle Capture in Turbulent Flotation Systems. Savassi, O.N., 1998. Direct Estimation of the Degree of Entrainment and the Froth Recovery of Attached Particles in Industrial Flotation Cells Ph.D.. University of Queensland. Schwarz, S., Alexander, D., 2006. Gas dispersion measurements in industrial flotation cells. Miner. Eng. 19, 554–560. Schubert, H., 1999. On the turbulence-controlled microprocesses in flotation machines. Int. J. Miner. Process. 56, 257–276.