| Cite this article as: |
Hua Han, An Liu, Caili Wang, Runquan Yang, Shuai Li, and Huaifa Wang, Flotation kinetics performance of different coal size fractions with nanobubbles, Int. J. Miner. Metall. Mater., 29(2022), No. 8, pp.1502-1510. https://dx.doi.org/10.1007/s12613-021-2280-8
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Flotation is the most efficient separation method in fine particle processing, and particle size plays an important role in the flotation procedure. Many researchers have suggested that the highest combustible recovery and flotation rate can be obtained in the intermediate size range, whereas the low capture probability of particle–bubble (low collision and attachment probability) results in poor flotation performance on fine particles. Meanwhile, the high detachment probability between particles and bubbles is the major barrier for the flotation of coarse particles [1–3].
Flotation kinetic describes the recovery rate of the froth product against the variation of flotation time. Flotation rate constant and theoretical maximum recovery are the main parameters that represent flotation speed and potential recovery. Flotation kinetics is affected by many factors, including particle size, flotation reagent, slurry density, and flotation equipment [4–6]. The flotation kinetic model was primitively built on the basis of chemical reaction process kinetics, and then many flotation kinetics models were proposed by researchers [7–11]. Abkhoshk et al. [12] studied the flotation kinetics of various particle sizes and reported that the difference of particle sizes has a more remarkable effect on kinetics constant than theoretical maximum flotation recovery.
Nanobubbles (NBs) are tiny bubbles, which are smaller than hundreds of nanometers. NBs can be produced by pressure reduction, temperature change, and hydrodynamic or ultrasonic cavitation methods [13]. Compared with regular bubbles, NBs have larger specific surface area and stronger stability [14]. In accordance with the position of bubble nucleation, NBs can be divided into two types: bulk and surface NBs, which are produced in the solution and on the mineral surface, respectively. Given the lower energy barrier of heterogeneous nucleation compared with homogeneous nucleation, NBs are more easily grown on hydrophobic surfaces than on hydrophilic surfaces and solutions. Consequently, NBs provide a natural selectivity to particles with different hydrophobicity. Compared with surface NBs, bulk NBs work poorer than surface NBs due to their low quality and kinetic energy in the solution; thus, the energy barrier between particles and bulk NBs is difficult to break [15–17].
Recent studies have proven that NBs are beneficial to the flotation process [18–20]. The existence of NBs contributes to increasing the recovery of flotation and deducing the consumption of flotation reagents; the contact angle of particle surfaces is also improved by the cover of NBs [21]. Furthermore, once NBs are adsorbed on the fine hydrophobic particle surface selectively, these fine particles tend to aggregate and form flocs due to the effect of bridging of NBs between fine particles [22]. For the flotation performance of different particle size, Fan et al. [23] studied different size fraction flotation in columns with NBs; 8%–46% increments of combustible recovery were obtained at –0.045+0.038 mm, –0.038+0.020 mm, and −0.020 mm, and the combustible recovery of the coarse size fraction (–2+0.3 mm) also increased by 22%–26%. Other researchers reported that NBs have a remarkable effect on coarse and fine particle sizes with the assistance of NBs [24–25].
However, only a few researchers focused on the flotation kinetics performance of NBs on various size fractions of particles. An atomic force microscope (AFM) was used to detect NBs in the solid–liquid interface. Six flotation kinetics models were selected to explore coal flotation behavior on different particle size ranges with NBs. Moreover, the flotation performances of conventional and NB flotation were compared. The effect of particle size with NBs on the flotation speed contributes to a further understanding of the role of NBs on coal flotation.
The coal sample was obtained from the Bailong coal preparation plant in Shanxi Province, China. The proximate and size distribution analysis are shown in Tables 1 and 2, respectively. The XRD (X-ray diffraction) result of the coal sample is given in Fig. 1.
| Mad | Vad | FCad | Aad | |||
| 1.27 | 27.21 | 46.91 | 24.61 | |||
| Notes: Mad—Moisture content; Vad—Volatile content; FCad—Fixed carbon content; Aad—Ash content. | ||||||
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| Size fraction / mm | Weight ratio / wt% | Ash content / wt% |
| −0.5+0.25 | 24.48 | 23.71 |
| −0.25+0.125 | 21.88 | 22.04 |
| −0.125+0.074 | 11.46 | 22.27 |
| −0.074+0.045 | 8.85 | 22.06 |
| −0.045 | 33.33 | 28.46 |
| Total | 100.00 | 24.61 |
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The total ash content of the coal sample is 24.61wt%, and the ash content of all the size fractions is close, except for the highest value of −0.045 mm at 28.46wt%. The main gangue minerals are kaolinite, quartz, and calcite.
Conventional flotation was conducted in a 1 L flotation machine (type: XFDIV-1 L, China). Kerosene and sec-octyl alcohol were selected as the collector and frother, respectively. First, 1 L tap water and 60 g coal sample were added into the flotation cell and mixed for 2 min at 1800 r/min. Then, kerosene of 125 g/t was added into the suspension and stirred for 1 min, and sec-octyl alcohol of 100 g/t was added into the pulp and conditioned for 10 s. Lastly, the air was introduced at 0.2 m3/h aeration rate, and a scraper blade was activated at 30 r/min.
The NB flotation experiment system is presented in Fig. 2. The coal sample and tap water were poured into the conditioning tank. After 1 min of agitation, the suspension with air was sent to a Venturi tube (parameters given in Table 3) by a pump (type: ASP5540, China). The flow rate of the pump was 3 L/min, and the throat velocity and aeration rate were 16 m/s and 75 mL/min, respectively. After a pretreatment time of 1 min, the slurry was transferred to the flotation cell. The rest of the flotation procedure was similar to the conventional one, except for the mixing time, which was reduced to 30 s before adding the flotation reagent.
| Diameter of inlet / mm | Diameter of outlet / mm | Diameter of throat / mm | Length of throat / mm | Angle of inlet / (°) | Angle of outlet / (°) |
| 8 | 8 | 2 | 4 | 22 | 11 |
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AFM imaging was conducted to explore the NB nucleation of particle surfaces under hydrodynamic cavitation. A highly polished quartz plate, which was modified by dodecyltriethoxysilane, was simulated as the coal surface, and the contact angle was measured to be approximately 60°. The prepared sample was placed into the conditioning tank under hydrodynamic cavitation conditions (operating parameters presented in Section 2.2.2) for 5 min and then transferred into the sample stage of AFM.
NBs were characterized on a MultiMode 8 SPM (Bruker, Karlsruhe, Germany) with a NanoScope V controller by ScanAsyst in fluid mode, and a silicon probe (SNL, 0.35 N/m, Bruker) was used for the test. The scan size and rate were 1000 nm and 0.977 Hz, respectively. During the measurement, the scanning parameters were automatically optimized. NanoScope Analysis 1.8 software was used to analyze the final image.
The coal sample was divided into five size ranges: −0.5+0.25, −0.25+0.125, −0.125+0.074, −0.074+0.045, and −0.045 mm. Each of the size fractions underwent a flotation kinetics test separately.
The flotation products were divided into several parts according to the following periods: 0−15, 15−30, 30−45, 45−60, 60−90, 90−120, 120−180, and 180−300 s. The flotation products were filtered, dried, weighed, and analyzed. The ash content and combustible recovery were used to evaluate the separation performance of conventional and NB flotation.
Six flotation kinetics models were selected to fit the flotation result to calculate the flotation rate constant k and correlation coefficient R2, their formulations and remarks were given in Table 4. The fittings were done by nonlinear regression analysis [7–11].
| Model No. | Name | Formulation | Remarks |
| 1 | Classical first-order model | ε=ε∞[1−exp(−kt)] | The model is applicable to the observed data with low recovery. |
| 2 | First-order rectangular distribution model | ε=ε∞{1−1kt[(1−exp(−kt)]} | The model is reported to be a better form of the first-order model because of the flexibility of the rectangular distribution of floatability. |
| 3 | Fully mixed reactor model | ε=ε∞(1−11+t/k) | The model has more flexibility than the classical first-order model by assuming that flotation components are exponentially distributed. |
| 4 | Improved gas/solid adsorption model | ε=ε∞(kt1+kt) | The model can be considered to be a transformation from model 3 by adjusting parameter k. |
| 5 | Second-order model | ε=ε2∞kt1+ε∞kt | The model can describe the flotation of the monodisperse feed, which has a constant floatability by the two parameters expression. |
| 6 | Second-order rectangular distribution model | ε=ε∞{1−1kt[ln(1+kt)]} | The model has an added parameter dilution in the confidence intervals; however, the fitting and confidence intervals decrease when recovery approaches 1.0. |
| Notes: ɛ—Combustible recovery at flotation time t; ɛ∞—Maximum theoretical combustible recovery; k—Flotation rate constant. | |||
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The scanning micrograph images of the hydrophobic surface before and after hydrodynamic cavitation are shown in Fig. 3. The original surface is flat, whereas several bright spots appear on the surface after hydrodynamic cavitation pretreatment. These spots have a few nanometers in height and about a hundred nanometers in diameter. Considering the disk shape of the bright spots, we assume that NB nucleation is generated on the hydrophobic surface. In addition, to investigate the characteristic of NBs further, one of the large spots in Fig. 3(b) was selected and amplified as shown in Fig. 3(c); meanwhile, the height through this bright spot was measured (Fig. 3(d)), as one can see that the height and diameter of the bright spot are 6.06 and 115.20 nm, respectively. The images conform to the characteristics of NBs in previous studies [26–29], and the result demonstrates that hydrodynamic cavitation could promote heterogeneous nucleation and NB generation on hydrophobic surfaces.
Fig. 4(a) and (b) indicates the combustible recovery of different size fractions with and without NBs, respectively. The combustible recovery of the medium particle size range (−0.125+0.074 mm) is the highest of all size ranges either in conventional and NB flotation, whereas the finest size fraction (−0.045 mm) is the hardest part to be floated. The floatability of the coarsest size range (−0.5+0.25 mm) is better than the finest size range but much lower than the other size ranges. Generally, the flotation process is determined by probabilities of the collision, attachment, and detachment between particle and bubble. Fine particles have low collision and attachment probabilities due to the small size and kinetic energy, whereas coarse particles have high gravity, resulting in high detachment probabilities between bubbles and particles. Compared with conventional flotation, NBs increase the combustible recovery of the raw sample (−0.5+0.25 and −0.045 mm) but have minimal effect on the other size fractions.
The cumulative ash content of products with and without NBs are given in Fig. 4(c) and (d), respectively. The results show that the difference of the ash content of −0.074+0.045 mm and all size fractions between conventional and NB flotation is small; however, the introduction of NBs increases the ash content of −0.25+0.074 mm. The existence of NBs is believed to increase the floatability of particles that contain gangue minerals. By contrast, the ash content of the finest part (−0.045 mm) is reduced by NBs.
The nonlinear regression results of the combustible recovery of different size fractions in conventional flotation are shown in Fig. 5 and Table 5, respectively. The results demonstrate that the multitude correlation coefficients R2 > 0.985 in raw sample and −0.045 mm fraction, indicating that all the flotation kinetics models fit these flotation data well. For the other size fractions, the similar theoretical maximum yield and multitude correlation coefficients were obtained by using models 3, 4, and 5, but models 1 and 2 are more suitable. Once the flotation sample contains the fine particles, which are difficult to float (−0.045 mm), the flotation kinetics result does not conform to most of the models, except for models 1 and 2. The order of theoretical maximum combustible recovery of various size fractions is –0.125+0.074 mm ˃ –0.074+0.045 mm ˃ –0.25+0.125 mm ˃ −0.045 mm ˃ –0.5+0.25 mm ˃ raw sample. Meanwhile, the order of flotation rate constant of different size fractions is –0.25+0.125 mm ˃ –0.125+0.074 mm ˃ –0.074+0.045 mm ˃ –0.5+0.25 mm ˃ raw sample ˃ −0.045 mm. Except for −0.5+0.25 mm, the maximum theoretical combustible recovery of models 3, 4, 5, and 6 is more than one hundred percent because the actual combustible recovery is less than one hundred percent; the inaccurate prediction resulted from the low convergence of the models.
| Model No. | Raw sample | −0.5+0.25 mm | −0.25+0.125 mm | −0.125+0.074 mm | −0.074+0.045 mm | −0.045 mm | |||||||||||||||||
| ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ||||||
| 1 | 69.67 | 2.0831 | 0.9956 | 81.35 | 2.7034 | 0.9808 | 91.01 | 4.1606 | 0.9997 | 94.75 | 3.7832 | 0.9999 | 93.64 | 3.0388 | 0.9984 | 81.24 | 0.5884 | 0.9992 | |||||
| 2 | 75.99 | 4.4893 | 0.9971 | 87.87 | 5.9569 | 0.9624 | 96.09 | 10.4362 | 0.9946 | 100.41 | 9.2762 | 0.9938 | 100.31 | 7.0117 | 0.9894 | 95.83 | 1.0459 | 0.9996 | |||||
| 3 | 79.43 | 0.3676 | 0.9912 | 90.56 | 0.2596 | 0.9452 | 97.47 | 0.1319 | 0.9894 | 102.13 | 0.1521 | 0.9876 | 102.85 | 0.2135 | 0.9795 | 110.15 | 2.0151 | 0.9989 | |||||
| 4 | 79.44 | 2.7197 | 0.9912 | 90.56 | 3.8516 | 0.9452 | 97.47 | 7.5801 | 0.9894 | 102.14 | 6.5735 | 0.9876 | 102.86 | 4.6827 | 0.9795 | 110.16 | 0.4962 | 0.9989 | |||||
| 5 | 79.44 | 0.0342 | 0.9958 | 90.56 | 0.0425 | 0.9452 | 97.47 | 0.0778 | 0.9894 | 102.14 | 0.0644 | 0.9876 | 102.86 | 0.0455 | 0.9754 | 110.16 | 0.0045 | 0.9989 | |||||
| 6 | 84.54 | 6.1165 | 0.9855 | 95.04 | 9.2194 | 0.9325 | 100.25 | 20.5574 | 0.9841 | 105.42 | 17.5522 | 0.9815 | 107.24 | 11.6535 | 0.9710 | 126.94 | 0.9048 | 0.9980 | |||||
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All conventional flotation data can be described by the classical first-order model and the first-order rectangular distribution model, although the classical first-order model is more suitable. The fully mixed reactor model, improved gas/solid adsorption model, second-order model, and second-order rectangular distribution model only fit in all size and finest size fractions, and the second-order rectangular distribution model has the lowest coefficient of determination.
The fitting results of six flotation models on different coal size fractions with NBs are shown in Fig. 6 and Table 6, respectively. Similar to conventional flotation, models 3, 4, and 5 have almost the same R2 and ɛ∞ in NB flotation, and six flotation kinetics models all fit the flotation data of the raw sample and −0.045 mm, which correspond with regular flotation. The maximum theoretical combustible recovery of models 3, 4, 5, and 6 all exceed 100 percent in the size fraction of −0.25+0.074 mm. Combined with the low R2 at approximately 0.98, models 3, 4, 5, and 6 cannot describe the flotation data well. A comparison of the six models reveals that the classical first-order model is the most suitable model for all the size fractions, and the raw coal sample has the highest R2 in NB flotation. Only the raw sample and −0.045 mm can be estimated by fitting all six kinetics models. In addition, in regular flotation, the combustible recovery basically increases to the maximum value at 90 s for size fractions of −0.5+0.25, −0.25+0.125, −0.125+0.074, and −0.074+0.045 mm and barely grows with the increase in flotation time in the after period; meanwhile, the time at which combustible recovery almost reaches the maximum is reduced to 60 s with the introduction of NBs. The order of ɛ∞ of different size fractions in NB flotation is the same as conventional flotation. Meanwhile, −0.5+0.25 mm has the highest flotation rate constant; −0.25+0.125, −0.125+0.074, and −0.074+0.045 mm have similar flotation speed; the raw sample and −0.045 mm have the lowest flotation rate constant.
| Model No. | Raw sample | −0.5+0.25 mm | −0.25+0.125 mm | −0.125+0.074 mm | −0.074+0.045 mm | −0.045 mm | |||||||||||||||||
| ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ||||||
| 1 | 79.01 | 2.3824 | 0.9901 | 85.19 | 4.3342 | 0.9972 | 93.09 | 3.5180 | 0.9958 | 96.66 | 3.4585 | 0.9945 | 94.15 | 3.6091 | 0.9997 | 82.79 | 0.6698 | 0.9972 | |||||
| 2 | 85.40 | 5.3442 | 0.9996 | 89.82 | 10.9245 | 0.9854 | 99.13 | 8.3406 | 0.9835 | 102.97 | 8.1731 | 0.9787 | 100.04 | 8.7000 | 0.9923 | 96.63 | 1.2155 | 0.9982 | |||||
| 3 | 88.69 | 0.2996 | 0.9990 | 90.87 | 0.1230 | 0.9782 | 101.24 | 0.1721 | 0.9732 | 104.92 | 0.1754 | 0.9670 | 101.91 | 0.1643 | 0.9850 | 109.60 | 1.6901 | 0.9975 | |||||
| 4 | 88.69 | 3.3378 | 0.9990 | 90.88 | 8.1251 | 0.9782 | 101.03 | 5.8093 | 0.9732 | 104.93 | 5.5973 | 0.9670 | 101.92 | 6.0836 | 0.9850 | 109.61 | 0.5916 | 0.9975 | |||||
| 5 | 88.70 | 0.0376 | 0.9990 | 90.88 | 0.0894 | 0.9782 | 101.03 | 0.0575 | 0.9732 | 104.93 | 0.0543 | 0.9670 | 101.92 | 0.0597 | 0.9850 | 109.61 | 0.0054 | 0.9975 | |||||
| 6 | 93.66 | 7.7745 | 0.9963 | 93.24 | 22.8384 | 0.9716 | 104.60 | 15.1296 | 0.9646 | 108.66 | 14.8193 | 0.9573 | 105.42 | 15.9762 | 0.9782 | 125.17 | 1.0973 | 0.9963 | |||||
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According to the comparison of six flotation kinetics models in conventional and NB flotation, the classical first-order model was selected as the appropriate model in all size fractions. Table 7 indicates the difference of various coal size fractions between conventional and NB flotation. On the one hand, compared with regular flotation, almost all the maximum theoretical combustible recoveries of various size fractions in NB flotation have been improved at different degrees. In contrast to a single-size fraction, NBs display a better flotation enhancement in the presence of all size fractions. On the other hand, NBs have a remarkable improvement in the flotation speed of the coarse coal size fraction (−0.5+0.25 mm), the flotation rate constant increases from 2.7034 to 4.3342, and the maximum theoretical combustible recovery increases from 81.35% to 85.19%. Nevertheless, the positive effect of NBs on flotation rate is not shown in medium size fraction; with the introduction of NBs, the flotation rate constant of −0.25+0.125 and −0.125+0.074 mm is decreased by 15.45% and 8.59%, respectively. However, compared with coarser and finer size fractions, the flotation speed of medium size is still fast enough even through a reduction due to NBs. For the fine coal particles, NBs improve the flotation rate constants of −0.074+0.045 and −0.045 mm by 18.77% and 13.91%, respectively. As the most difficult and slowest coal size fraction to be floated, fine coal particles greatly affect the total flotation performance. The problem is the incomplete flotation process of fine particles at the general time, and the increase in the flotation rate of fine particles with NBs results in a more effective separation performance at a limited flotation time. For this reason, NBs show the best improvement of maximum theoretical combustible recovery in raw coal samples.
| Size fraction / mm | Conventional flotation | NBs flotation | |||
| ɛ∞ / % | k / s−1 | ɛ∞ / % | k / s−1 | ||
| −0.5+0.25 | 81.35 | 2.7034 | 85.19 | 4.3342 | |
| −0.25+0.125 | 91.01 | 4.1606 | 93.09 | 3.5180 | |
| −0.125+0.074 | 94.75 | 3.7832 | 96.66 | 3.4585 | |
| −0.074+0.045 | 93.64 | 3.0388 | 93.09 | 3.6091 | |
| −0.045 | 81.24 | 0.5880 | 82.79 | 0.6698 | |
| Raw sample | 69.67 | 2.0831 | 79.01 | 2.3824 | |
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Researchers have proposed that the coarse coal particle covered by NBs has large hydrophobicity, and the increase in true flotation (particle–bubble attachment) is the main reason for improving flotation recovery [30]. The medium-sized coal particle with the best floatability has high separation performance and flotation rate even in conventional flotation, thus the improvement of flotation performance of NBs is not obvious. Whereas the additional floated medium–ash particles increase the ash content of concentrate and decrease the flotation rate. Furthermore, the occupation of NBs on coal particle surface leads to the obstruction of adsorption of flotation reagent [31], that is an another possible reason for the reduction of flotation rate. As for the fine particles, on the one hand, NBs on the particle surface increase the particle’s contact angle and reduce the requirement of flotation reagent; on the other hand, the existence of “NB–Bridge” promotes the aggregation of fine particles, thus increasing the collision probability between coal particles and bubbles; these factors may be the reasons for the improvement of flotation rate and recovery on fine particles [32].
The following conclusions can be drawn on the basis of the above results.
(1) AFM scanning proves that NB nucleation appears on hydrophobic surfaces under hydrodynamic cavitation. The characteristics of the flotation performance of different sample sizes in conventional and NB flotation are similar that medium-size fraction (−0.25+0.074 mm) has the highest combustible recovery. The combustible recoveries of almost all the size fractions are improved with the assistance of NBs. The product ash content of −0.045 mm is decreased, whereas the product ash content of −0.25+0.074 mm is increased by NBs.
(2) Six flotation modes are used to fit the flotation data. All flotation models can describe the flotation data of the −0.045 mm size fraction and raw sample both in conventional and NB flotation. However, for the single size fraction (except for −0.045 mm), the correlation coefficients of the fully mixed reactor model, the improved gas/solid adsorption model, the second-order kinetics model, and the second-order model with rectangular distribution are relatively low. The classical first-order model is the most suitable model by comparison.
(3) Compared with the maximum theoretical recovery and flotation rate constant between conventional and NB flotation, the combustible recovery of coal particles within almost all size fractions can be increased with addition of NBs. However, compared with the improvement of flotation rate of coal particles within −0.5+0.25 mm and −0.074 mm size fractions, the flotation rate of −0.25+0.125 mm is reduced in NBs flotation.
This work was financially supported by the National Natural Science Foundation of China (No. 51704208).
The authors declare no conflicts of interest.
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Figures(6) / Tables(7)
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| Mad | Vad | FCad | Aad | |||
| 1.27 | 27.21 | 46.91 | 24.61 | |||
| Notes: Mad—Moisture content; Vad—Volatile content; FCad—Fixed carbon content; Aad—Ash content. | ||||||
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| Size fraction / mm | Weight ratio / wt% | Ash content / wt% |
| −0.5+0.25 | 24.48 | 23.71 |
| −0.25+0.125 | 21.88 | 22.04 |
| −0.125+0.074 | 11.46 | 22.27 |
| −0.074+0.045 | 8.85 | 22.06 |
| −0.045 | 33.33 | 28.46 |
| Total | 100.00 | 24.61 |
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| Diameter of inlet / mm | Diameter of outlet / mm | Diameter of throat / mm | Length of throat / mm | Angle of inlet / (°) | Angle of outlet / (°) |
| 8 | 8 | 2 | 4 | 22 | 11 |
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| Model No. | Name | Formulation | Remarks |
| 1 | Classical first-order model | ε=ε∞[1−exp(−kt)] | The model is applicable to the observed data with low recovery. |
| 2 | First-order rectangular distribution model | ε=ε∞{1−1kt[(1−exp(−kt)]} | The model is reported to be a better form of the first-order model because of the flexibility of the rectangular distribution of floatability. |
| 3 | Fully mixed reactor model | ε=ε∞(1−11+t/k) | The model has more flexibility than the classical first-order model by assuming that flotation components are exponentially distributed. |
| 4 | Improved gas/solid adsorption model | ε=ε∞(kt1+kt) | The model can be considered to be a transformation from model 3 by adjusting parameter k. |
| 5 | Second-order model | ε=ε2∞kt1+ε∞kt | The model can describe the flotation of the monodisperse feed, which has a constant floatability by the two parameters expression. |
| 6 | Second-order rectangular distribution model | ε=ε∞{1−1kt[ln(1+kt)]} | The model has an added parameter dilution in the confidence intervals; however, the fitting and confidence intervals decrease when recovery approaches 1.0. |
| Notes: ɛ—Combustible recovery at flotation time t; ɛ∞—Maximum theoretical combustible recovery; k—Flotation rate constant. | |||
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| Model No. | Raw sample | −0.5+0.25 mm | −0.25+0.125 mm | −0.125+0.074 mm | −0.074+0.045 mm | −0.045 mm | |||||||||||||||||
| ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ||||||
| 1 | 69.67 | 2.0831 | 0.9956 | 81.35 | 2.7034 | 0.9808 | 91.01 | 4.1606 | 0.9997 | 94.75 | 3.7832 | 0.9999 | 93.64 | 3.0388 | 0.9984 | 81.24 | 0.5884 | 0.9992 | |||||
| 2 | 75.99 | 4.4893 | 0.9971 | 87.87 | 5.9569 | 0.9624 | 96.09 | 10.4362 | 0.9946 | 100.41 | 9.2762 | 0.9938 | 100.31 | 7.0117 | 0.9894 | 95.83 | 1.0459 | 0.9996 | |||||
| 3 | 79.43 | 0.3676 | 0.9912 | 90.56 | 0.2596 | 0.9452 | 97.47 | 0.1319 | 0.9894 | 102.13 | 0.1521 | 0.9876 | 102.85 | 0.2135 | 0.9795 | 110.15 | 2.0151 | 0.9989 | |||||
| 4 | 79.44 | 2.7197 | 0.9912 | 90.56 | 3.8516 | 0.9452 | 97.47 | 7.5801 | 0.9894 | 102.14 | 6.5735 | 0.9876 | 102.86 | 4.6827 | 0.9795 | 110.16 | 0.4962 | 0.9989 | |||||
| 5 | 79.44 | 0.0342 | 0.9958 | 90.56 | 0.0425 | 0.9452 | 97.47 | 0.0778 | 0.9894 | 102.14 | 0.0644 | 0.9876 | 102.86 | 0.0455 | 0.9754 | 110.16 | 0.0045 | 0.9989 | |||||
| 6 | 84.54 | 6.1165 | 0.9855 | 95.04 | 9.2194 | 0.9325 | 100.25 | 20.5574 | 0.9841 | 105.42 | 17.5522 | 0.9815 | 107.24 | 11.6535 | 0.9710 | 126.94 | 0.9048 | 0.9980 | |||||
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| Model No. | Raw sample | −0.5+0.25 mm | −0.25+0.125 mm | −0.125+0.074 mm | −0.074+0.045 mm | −0.045 mm | |||||||||||||||||
| ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ɛ∞ / % | k / s−1 | R2 | ||||||
| 1 | 79.01 | 2.3824 | 0.9901 | 85.19 | 4.3342 | 0.9972 | 93.09 | 3.5180 | 0.9958 | 96.66 | 3.4585 | 0.9945 | 94.15 | 3.6091 | 0.9997 | 82.79 | 0.6698 | 0.9972 | |||||
| 2 | 85.40 | 5.3442 | 0.9996 | 89.82 | 10.9245 | 0.9854 | 99.13 | 8.3406 | 0.9835 | 102.97 | 8.1731 | 0.9787 | 100.04 | 8.7000 | 0.9923 | 96.63 | 1.2155 | 0.9982 | |||||
| 3 | 88.69 | 0.2996 | 0.9990 | 90.87 | 0.1230 | 0.9782 | 101.24 | 0.1721 | 0.9732 | 104.92 | 0.1754 | 0.9670 | 101.91 | 0.1643 | 0.9850 | 109.60 | 1.6901 | 0.9975 | |||||
| 4 | 88.69 | 3.3378 | 0.9990 | 90.88 | 8.1251 | 0.9782 | 101.03 | 5.8093 | 0.9732 | 104.93 | 5.5973 | 0.9670 | 101.92 | 6.0836 | 0.9850 | 109.61 | 0.5916 | 0.9975 | |||||
| 5 | 88.70 | 0.0376 | 0.9990 | 90.88 | 0.0894 | 0.9782 | 101.03 | 0.0575 | 0.9732 | 104.93 | 0.0543 | 0.9670 | 101.92 | 0.0597 | 0.9850 | 109.61 | 0.0054 | 0.9975 | |||||
| 6 | 93.66 | 7.7745 | 0.9963 | 93.24 | 22.8384 | 0.9716 | 104.60 | 15.1296 | 0.9646 | 108.66 | 14.8193 | 0.9573 | 105.42 | 15.9762 | 0.9782 | 125.17 | 1.0973 | 0.9963 | |||||
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| Size fraction / mm | Conventional flotation | NBs flotation | |||
| ɛ∞ / % | k / s−1 | ɛ∞ / % | k / s−1 | ||
| −0.5+0.25 | 81.35 | 2.7034 | 85.19 | 4.3342 | |
| −0.25+0.125 | 91.01 | 4.1606 | 93.09 | 3.5180 | |
| −0.125+0.074 | 94.75 | 3.7832 | 96.66 | 3.4585 | |
| −0.074+0.045 | 93.64 | 3.0388 | 93.09 | 3.6091 | |
| −0.045 | 81.24 | 0.5880 | 82.79 | 0.6698 | |
| Raw sample | 69.67 | 2.0831 | 79.01 | 2.3824 | |
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