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Experimental validation of a statistical reliability method for the liberation distribution measurement of ore particles
Minerals Engineering 140 (2019) 105880
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Experimental validation of a statistical reliability method for the liberation distribution measurement of ore particles
T
Takao Ueda Environmental Management Research Institute, Department of Energy and Environment, National Institute of Advanced Industrial Science and Technology (AIST), 16-1 Onogawa, Tsukuba, Ibaraki 305-8569, Japan
ARTICLE INFO
ABSTRACT
Keywords: Statistical reliability Mineral liberation Liberation distribution X-ray computed tomography Binary particle
Accurate assessment of mineral liberation is important in mineral processing. Recently popularized automated liberation analyzers have enabled the rapid measurement of the liberation state of particles. However, the statistical reliability of liberation distribution assessment has not been thoroughly investigated. Recently, a statistical method which correlates particle measurements with the confidence interval of the measured liberation distribution has been proposed. In the present study, this statistical method was validated by a series of experiments using binary particles and X-ray computed tomography (X-ray CT) analysis. Specifically, three types of binary particles composed of a matrix phase (epoxy resin) and grain phase (glass beads or silica sand) were created; the liberation distribution of all the particles was obtained as the mother population, through X-ray CT and post-processing; a certain number of particles determined by the statistical method were selected from the mother population, and the percentage in which the sampled liberation distribution satisfied the designated confidence interval was determined. The percentages ranged from 94% to 97%, which agreed with the required reliability of 95%, demonstrating the effectiveness of the statistical model.
1. Introduction Liberation distribution (the cumulative distribution of the particles with respect to their content of the mineral phase of interest ( x )) is one of the most important parameters for mineral liberation assessment of crushed ore particles. It is commonly used for engineering purposes, such as simulating the recovery rate of the mineral of interest by adjusting the separation operating conditions. By nature, x can be a continuum function, but it is traditionally treated as twelve bins for ease of experimental analysis (when the bins are designated as i , i = 1 12 corresponds to x = 0, 0 0.1, , 0.9 1, 1, respectively), and this was adopted in the present study. The liberation distribution is generally measured through analysis of ore particle sections or surfaces. Recently popularized automated analyzers employing a combination of scanning electron microscopy and energy dispersive X-ray analysis (Fandrich et al., 2007; Gottlieb et al., 2000; Hrstka et al., 2018; TESCAN, 2012) are effective tools that enable the analysis of a great number of particle sections in a reasonable time. However, the correlation between the confidence interval of the liberation distribution and the number of measured particle sections has hitherto not been fully understood. Studies investigating the liberation distribution confidence interval have included the development of a dispersion model using the arcsine
transformation and comparison with the bootstrapping technique (Efron, 1979; Leigh et al., 1993; Lotter et al., 2018); an estimation of the variation coefficient with respect to the number of particle section measurements (with the equation y = ax b ) using the bootstrapping technique (Evans and Napier-Munn, 2013); a measurement of the liberation distribution confidence interval for bins of copper ore (Mariano and Evans, 2015). Recently, using a statistical approach, Ueda et al. investigated the relationship between the liberation distribution confidence interval and the number of particle section measurements; and developed a statistical reliability method for liberation distribution assessment (Ueda et al., 2016a, 2018a). A remarkable feature of the method is that the confidence interval for each bin can be independently calculated; and conversely, the required number of particle section measurements can be calculated from the required confidence interval for each bin. In the present study, this method was experimentally validated using artificial binary particles and X-ray CT. 2. Methodology This section briefly describes the statistical reliability method for liberation distribution assessment; the detailed information is in Ueda et al. (2018a).
E-mail address: [email protected]. https://doi.org/10.1016/j.mineng.2019.105880 Received 11 April 2019; Received in revised form 17 July 2019; Accepted 22 July 2019 Available online 31 July 2019 0892-6875/ © 2019 Elsevier Ltd. All rights reserved.
Minerals Engineering 140 (2019) 105880
T. Ueda
Table 1 Examples of statistic reliability requirements for bins (after (Ueda et al., 2018a)). Case
Statistical reliability requirements
1
The confidence interval ( i ) for all bins should be less than 0.02
2 3
Specified
for bins with x 0.8 (i 10) should be less than 0.02 , because we are concerned with the liberation distribution range with high phase A content i for bins with x = 0 and 1 (i = 1 and 12) should be less than 0.02 , because we are concerned with the respective degrees of liberation of phases A and B i
The liberation distributions of binary particles (composed of phases A and B) are considered. Actual ore particles are generally composed of multiple mineral phases, but if the mineral phases are grouped into the mineral of interest and others, the particles can be simply treated as binary particles. Therefore, the binary particle methodology may be applied to the assessment of actual ore particles, without loss of generality. As noted in the Introduction, mineral liberation assessment is based on particle section measurement; that is, 2D assessment. Therefore, 2D liberation distribution will be focused on in this paper. However, if 3D particle information can somehow be obtained [such as by X-ray CT (Gay and Morrison, 2006; Lin and Miller, 1996; Miller et al., 2003, 2009; Videla et al., 2007), serial particle sectioning (Lätti and Adair, 2001; Miller and Lin, 1988; Schneider et al., 1991), or stereological correction (Gay, 1994; Gay and Morrison, 2006; King and Schneider, 1998; Leigh et al., 1996; Miller and Lin, 1988; Ueda et al., 2017, 2016b)], the following methodology can be equally applied in the case of 3D liberation distribution.
i i
i
i
= 0.02 where i = 1
{ 0.02 ={
0.02 where i = 10 = where i = 1
12 12 9
where i = 1 and 12 where i = 2 11
interval is obtained. Given this implementation method, in this study, Nr is adjusted to 100 when the calculatedNr is less than 100. 3. Experimental validation The experimental validation employed binary particles composed of grain and matrix phases (corresponding to phase A and phase B), respectively. Artificial binary particles were created (Section 3.1) and analyzed by X-ray CT and post-processing (Sections 3.2 and 3.3); after which, the proposed method was validated (Section 3.4). Sections 3.1–3.3 are based on the methodology in Ueda et al. (2018b), and briefly described here. 3.1. Preparation of binary particle samples A key feature of the experiment was the assignment of different specific gravities (SGs) to the grain and matrix phases, in order to later distinguish them by X-ray CT. The binary material was crushed by a double shaft shredder, and screened by a Ro-tap sieve shaker. Table 2 lists the three types of artificial binary particle samples, with the different materials, grain sizes (initial size of the grain phase for binary materials), and particle sizes after crushing and screening.
2.1. Statistical reliability method for liberation distribution assessment As aforementioned, the liberation distribution was treated in terms of twelve bins (i = 1 12 corresponds to x = 0, 0 0.1, , 0.9 1, 1, respectively). For the mother population, the probability of a particle section being classified into the i -th bin (Pi ) is Pi = Si Sall , where Si denotes the sum of the sectional areas entering the i -th bin, and Sall denotes the total sectional area. Similarly, for a given sample (a limited number of randomly selected particles from the mother population), the probability of a particle section being classified into the i -th bin is Pi . When the confidence interval is arbitrarily determined as i (i.e., Pi is expected to be in the range of Pi ± i ), the required number of particle sections in the i -th bin (Ni ) is determined as: Ni = (KP i ) 2Pi (1 Pi ) , where KP is a statistical coefficient obtained from the table of normal distribution, and KP = 1.96 when the required reliability (Rr ) is 95%, according to the estimation of the confidence interval for the population proportion. Then, the required number of particle section measurements (Nr ) is calculated by Nr = max Ni .
3.2. X-ray CT analysis 3.2.1. X-ray CT scanning The binary particles were packed into cylindrical polyethylene containers (7.5 mm in diameter and 20.0 mm in height), with 19 containers prepared for each sample type. The samples were analyzed by an X-ray CT system (inspeXio SMX-100CT, Shimadzu Corporation) with an X-ray tube output of 60 kV and 40 μA, a resolution of 0.017 mm/ pixel, and an image size of 512 × 512. Fig. 1 shows examples of the Xray transmission images. 3.2.2. Post-processing for identification of particles and phases In order to identify the particles and phases, further post-processing was conducted as follows: (1) the particles and background were binarized using Otsu’s method (Otsu, 1979); (2) the particles were detached and individually identified using the watershed method; (3) the grain and matrix phases were identified by binarization using Otsu’s method; and finally, (4) three-valued voxel data (grain phase, matrix phase, background), with dimensions 512 × 512 × 460 , were obtained. The particle identification in Step 2 (above) was not flawless; for example, some aggregates were misidentified as particles. However, the misidentified particles were manually eliminated, and the total number of successfully identified particles are listed in the rightmost column of Table 2. Then, six cylindrical sample sections (parallel to the bottom), with heights of 20, 105, 190, 275, 360, and 445 voxels, were analyzed. A sectional interval of 85 voxels (=1.45 mm), larger than the maximum particle diameter (1.40 mm), was specified, to avoid multiple sectioning of a single particle. The particle sections emerging in the six sample sections were treated as identified particle sections. The total number of particle sections are also listed (in parentheses) in the rightmost column of Table 2.
i = 1 12
A remarkable feature of the method is that i can be independently determined for each bin, which makes it possible to flexibly meet measurement needs. Table 1 presents three possible examples (after (Ueda et al., 2018a)). 2.2. An implementation method Detailed information about the method of implementation for the statistical reliability method is in (Ueda et al., 2018a), but may be summarized as follows. First, a preliminary measurement is conducted for a small number of particles (e.g., 100) to temporally determine Pi . Second, the measurement and Nr estimation are repeated until the number of particle measurements reaches Nr . When Nr is reached, the measurements are complete, and the liberation distribution with the required confidential 2
Minerals Engineering 140 (2019) 105880
Fig. 2 shows 3D particle structures obtained by the post-processing, and Fig. 3 shows sectional images of Fig. 2 at a height of 275. 3.3. Liberation analysis 3.3.1. 3D liberation distribution The particle volume (Vp ) was calculated by multiplying the number of voxels in a given particle by the voxel volume (0.0173 mm3), which was determined by the X-ray CT scanning resolution (0.017 mm/pixel). The phase A volume in a given particle (VA ) was similarly calculated, and the phase A volume fraction ( x ) was calculated as x = VA VP . After completing this calculation for all the particles, the number of particles entering the i -th bin was counted with respect to x , for all 12 bins, and the fraction of each bin (Pi ) was determined. Pi corresponds to the 3D liberation distribution. 13,093 (8549) 19,384 (11,499) 2950 (2994)
3.3.2. 2D liberation distribution The subsequent analysis was similar to that of the 3D calculation above. The particle sectional area (SP ) was calculated by multiplying the number of voxels in a particle section by the pixel area (0.0172mm2 ), the phase A area in a given particle section (SA ) was similarly calculated, and the phase A areal fraction (x ) was calculated as x = SA SP . The number of sections entering the i -th bin was counted with respect to x , for all 12 bins, and the fraction of each bin (Pi ), which corresponds to the 2D liberation distribution, was determined. 3.4. Validation
0.5–0.85 0.5–0.85 0.85–1.4
The total data for all the 3D particles and 2D particle sections was considered as the parent population. Three cases (required statistical reliabilities) in Table 1 were considered, with Nr in each case calculated by the statistical reliability method (Section 2). Then, Nr particles were randomly selected from the parent population, allowing overlap, and Pi was calculated (as in Section 3.3). This random sampling was repeated 1000 times, and the probability of Pi being within the confidence interval of Pi ± i was determined. This probability is defined as the measurement reliability (Rm ), and is expected to satisfy Rm Rr . 4. Results and discussion
0.2 0.2 0.2
4.1. Sample 1 Table 3 shows the Rm values for Sample 1, Cases 1–3, calculated with 1000 sampling repetitions of Nr particles. The shaded values in Table 3 indicate the bins with i = 0.02 , namely the bins of interest (see Table 2). The bottom row shows the minimums of the shaded values. The minimum Rm s for the 2D and 3D liberation distributions (roughly 94–97.5%) are comparable to Rr (95%), except for the 3D distribution in Case 3, where Rm = 100 , because in this latter case the number of preliminary measurements (1 0 0) exceeds Nr , which results in excessive measurement. Fig. 4 compares the 2D liberation distributions of the parent population with those of the 2.5–97.5 percentile range of 1000 samples for Sample 1, Cases 1–3. The respective designated confidence interval ranges (with i = 0.02 ) are indicated by the double arrows. It is confirmed that the dispersion ranges of the sample cases successfully fell within the designated confidence interval ranges.
Glass beads (2.5) Glass beads (2.5) Silica sand (2.2) Epoxy resin (1.1) Epoxy resin (1.1) Epoxy resin (1.1) 1 2 3
0.105–0.125 0.35–0.5 0.16–0.25
Grain phase (SG) Matrix phase (SG) Sample
Table 2 Binary particle samples.
Grain size (mm)
Grain phase volume fraction
Particle size after crushing and screening (mm)
Number of particles in post-processing after X-ray CT (number of sections)
T. Ueda
4.2. Sample 2 Table 4 shows the Rm values for Sample 2, Cases 1–3 (corresponding to Table 3). The results show the same pattern as in Table 3: the minimum Rm s of the 2D and 3D liberation distributions (roughly 95–96%) are comparable to Rr (95%). 3
Minerals Engineering 140 (2019) 105880
T. Ueda
Fig. 1. X-ray transmission images of (a) Sample 1 (epoxy resin and 0.105–0.125 mm glass beads), (b) Sample 2 (epoxy resin and 0.35–0.5 mm glass beads), and (c) Sample 3 (epoxy resin and 0.16–0.25 mm silica sand); grain phase (glass beads or silica sand) in white, matrix phase (epoxy resin) in gray.
Fig. 2. 3D cylindrical particle structures created by X-ray CT post-processing of (a) Sample 1 (epoxy resin and 0.105–0.125 mm glass beads), (b) Sample 2 (epoxy resin and 0.35–0.5 mm glass beads), and (c) Sample 3 (epoxy resin and 0.16–0.25 mm silica sand); grain phase (glass beads or silica sand) in black, matrix phase (epoxy resin) in gray.
Fig. 3. 2D cylindrical particle section patterns created by X-ray CT post-processing of (a) Sample 1 (epoxy resin and 0.105–0.125 mm glass beads), (b) Sample 2 (epoxy resin and 0.35–0.5 mm glass beads), and (c) Sample 3 (epoxy resin and 0.16–0.25 mm silica sand); grain phase (glass beads or silica sand) in black, matrix phase (epoxy resin) in gray.
Fig. 5 shows the 2D liberation distributions for Sample 2, Cases 1–3 (corresponding to Fig. 4). It is again confirmed that the dispersion ranges of the sample cases successfully fell within the designated confidence interval ranges ( i = 0.02 ).
dispersion ranges of the sample cases successfully fell within the designated confidence interval ranges ( i = 0.02 ).
4.3. Sample 3
As remarked in the Introduction, a notable feature of the statistical reliability method for liberation distribution assessment is that the required confidential interval of each bin can be independently designated. As only this method has this capability, comparison with previous studies (Evans and Napier-Munn, 2013; Leigh et al., 1993) was not possible. The main result, experimental validation of the statistical reliability method, is free from experimental error. The validation consisted of two stages, the first described in Sections 3.1 and 3.2, and the second in Sections 3.3 and 3.4. Vertical binary particle data was created from
4.4. Discussion
Table 5 shows the Rm values for Sample 3, Cases 1–3 (corresponding to Tables 3 and 4). The results show the same pattern as in the two tables: the minimum Rm s of the 2D and 3D liberation distribution (roughly 94–97%) are comparable to Rr (95%), except for the 3D distribution in Case 3, where Nr was less than in the preliminary measurement (1 0 0), which results in excessive measurement. Fig. 6 shows the 2D liberation distributions for Sample 3, Cases 1–3 (corresponding to Figs. 4 and 5). It is again confirmed that the 4
Minerals Engineering 140 (2019) 105880
T. Ueda
Table 3 Probability of the calculated particle number samples satisfying the designated confidence interval, for Sample 1. Shaded values indicate the designated confidence interval bins. The bottom row shows the minimums of the shaded values.
Nr A
Nr A
x
x
Nr A
x
Fig. 4. Comparison of the 2D liberation distribution of the parent population (8549 particle sections), with the 2.5–97.5 percentile range of 1000 samples with Nr particle sections, for Sample 1, Cases 1–3. 5
Minerals Engineering 140 (2019) 105880
T. Ueda
Table 4 Probability of the calculated particle number samples satisfying the designated confidence interval, for Sample 2, Cases 1–3. Shaded values indicate designated confidence interval bins, and the bottom row shows the minimums of the shaded values.
Nr A
Nr x
A
x
Nr A
x
Fig. 5. Comparison of the 2D liberation distribution of the parent population (11,499 particle sections), with the 2.5–97.5 percentile range of 1000 samples with Nr particle sections, for Sample 2, Cases 1–3. 6
Minerals Engineering 140 (2019) 105880
T. Ueda
Table 5 Probability of the calculated particle number samples satisfying the designated confidence interval, for Sample 3, Cases 1–3. Shaded values indicate designated confidence interval bins, and the bottom row shows the minimums of the shaded values.
Nr A
Nr x
A
x
Nr A
x
Fig. 6. Comparison of the 2D liberation distribution of the parent population (2994 particle sections), with the 2.5–97.5 percentile range of 1000 samples with Nr particle sections, for Sample 3, Cases 1–3. 7
Minerals Engineering 140 (2019) 105880
T. Ueda
actual binary particles in the first stage, and the statistical reliability method was validated in the second stage, using this vertical binary particle data. In the first stage, experimental error was possible in the Xray CT scanning and post-processing; however, in the second stage, since the liberation analysis and validation was computed automatically, experimental error was not possible.
petrology – applications of TESCAN integrated mineral analyzer (TIMA). J. Geosci. (Czech Republic) 63, 47–63. https://doi.org/10.3190/jgeosci.250. King, R.P., Schneider, C.L., 1998. Stereological correction of linear grade distributions for mineral liberation. Powder Technol. 98, 21–37. https://doi.org/10.1016/S00325910(98)00013-8. Lätti, D., Adair, B.J.I., 2001. An assessment of stereological adjustment procedures. Miner. Eng. 14, 1579–1587. https://doi.org/10.1016/S0892-6875(01)00176-5. Leigh, G.M., Lyman, G.J., Gottlieb, P., 1996. Stereological estimates of liberation from mineral section measurements: a rederivation of Barbery’s formulae with extensions. Powder Technol. 87, 141–152. https://doi.org/10.1016/0032-5910(95)03080-8. Leigh, G.M., Sutherland, D.N., Gottlieb, P., 1993. Confidence limits for liberation measurements. Miner. Eng. 6, 155–161. https://doi.org/10.1016/0892-6875(93) 90129-B. Lin, C.L., Miller, J.D., 1996. Cone beam X-ray microtomography for three-dimensional liberation analysis in the 21st century. Int. J. Miner. Process. 47, 61–73. https://doi. org/10.1016/0301-7516(96)00005-1. Lotter, N.O., Evans, C.L., Engstrőm, K., 2018. Sampling – a key tool in modern process mineralogy. Miner. Eng. 116, 196–202. https://doi.org/10.1016/j.mineng.2017.07. 013. Mariano, R.A., Evans, C.L., 2015. Error analysis in ore particle composition distribution measurements. Miner. Eng. 82, 36–44. https://doi.org/10.1016/j.mineng.2015.06. 001. Miller, J.D., Lin, C.L., 1988. Treatment of polished section data for detailed liberation analysis. Int. J. Miner. Process. 22, 41–58. https://doi.org/10.1016/0301-7516(88) 90055-5. Miller, J.D., Lin, C.L., Garcia, C., Arias, H., 2003. Ultimate recovery in heap leaching operations as established from mineral exposure analysis by X-ray microtomography. Int. J. Miner. Process. 72, 331–340. https://doi.org/10.1016/S0301-7516(03) 00091-7. Miller, J.D., Lin, C.L., Hupka, L., Al-Wakeel, M.I., 2009. Liberation-limited grade/recovery curves from X-ray micro CT analysis of feed material for the evaluation of separation efficiency. Int. J. Miner. Process. 93, 48–53. https://doi.org/10.1016/j. minpro.2009.05.009. Otsu, N., 1979. A threshold selection method from gray-level histograms. IEEE Trans. Syst. Man. Cybern. 9, 62–66. https://doi.org/10.1109/TSMC.1979.4310076. Schneider, C.L., Lin, C.L., King, R.P., Miller, J.D., 1991. Improved transformation technique for the prediction of liberation by a random fracture model. Powder Technol. 67, 103–111. https://doi.org/10.1016/0032-5910(91)80032-E. TESCAN, 2012. TESCAN Introduces the TIMA Mineralogy Solution. [WWW Document]. URL https://www.tescan.com/en-us/about-tescan/news/tescan-introduces-the-timamineralogy-solution (accessed 5 July 2019). Ueda, T., Oki, T., Koyanaka, S., 2018a. Statistical reliability of the liberation distribution of ore particles with respect to number of particle measurements. Miner. Eng. Miner. Eng. 126, 82–88. Ueda, T., Oki, T., Koyanaka, S., 2018b. Experimental analysis of mineral liberation and stereological bias based on X-ray computed tomography and artificial binary particles. Adv. Powder Technol. 29, 462–470. https://doi.org/10.1016/j.apt.2017.11. 004. Ueda, T., Oki, T., Koyanaka, S., 2017. Stereological correction method based on sectional texture analysis for the liberation distribution of binary particle systems. Adv. Powder Technol. 28, 1391–1398. https://doi.org/10.1016/j.apt.2017.03.007. Ueda, T., Oki, T., Koyanaka, S., 2016a. Statistical effect of sampling particle number on mineral liberation assessment. Miner. Eng. 98, 204–212. https://doi.org/10.1016/j. mineng.2016.08.026. Ueda, T., Oki, T., Koyanaka, S., 2016b. Stereological bias for spherical particles with various particle compositions. Adv. Powder Technol. 27, 1828–1838. https://doi. org/10.1016/j.apt.2016.06.016. Videla, A.R., Lin, C.L., Miller, J.D., 2007. 3D characterization of individual multiphase particles in packed particle beds by X-ray microtomography (XMT). Int. J. Miner. Process. 84, 321–326. https://doi.org/10.1016/j.minpro.2006.07.009.
5. Conclusion A series of experiments were conducted to validate the proposed statistical method for liberation distribution assessment, which correlates the number of particle measurements with the confidence interval range. Three types of binary particles composed of a matrix phase (epoxy resin) and grain phase (glass beads or silica sand) were created, and 3D particle data (as well as 2D particle sectional data), which included the volumetric (and areal) fraction of the grain phase, were analyzed by X-ray CT. The liberation distribution calculated based on the total number of particles (or particle sections) was determined as the true data of the mother population. Then, a certain number of particles (or particle sections) determined by the statistical model were randomly selected from the mother population, and the percentage in which the sample liberation distribution satisfied the designated confidence interval range was determined. The percentage ranged from 94% to 97%, demonstrating agreement with the required reliability of 95%, and validating the effectiveness of the statistical model. Acknowledgment This work was supported by Japan Oil, Gas and Metals National Corporation. References Efron, B., 1979. Bootstrap methods: another look at the jackknife. Ann. Stat. 7, 1–26. https://doi.org/10.1214/aos/1176344552. Evans, C.L., Napier-Munn, T.J., 2013. Estimating error in measurements of mineral grain size distribution. Miner. Eng. 52, 198–203. https://doi.org/10.1016/j.mineng.2013. 09.005. Fandrich, R., Gu, Y., Burrows, D., Moeller, K., 2007. Modern SEM-based mineral liberation analysis. Int. J. Miner. Process. 84, 310–320. https://doi.org/10.1016/j.minpro. 2006.07.018. Gay, S.L., 1994. Liberation Modelling Using Particle Sections. The University of Queensland, Australia (accessed 5 July 2019). Gay, S.L., Morrison, R.D., 2006. Using two dimensional sectional distributions to infer three dimensional volumetric distributions – validation using tomography. Part. Part. Syst. Charact. 23, 246–253. https://doi.org/10.1002/ppsc.200601056. Gottlieb, P., Wilkie, G., Sutherland, D., Ho-Tun, E., Suthers, S., Perera, K., Jenkins, B., Spencer, S., Butcher, A., Rayner, J., 2000. Using quantitative electron microscopy for process mineralogy applications. J. Miner. Met. Mater. Soc. 52, 24–25. https://doi. org/10.1007/s11837-000-0126-9. Hrstka, T., Gottlieb, P., Skála, R., Breiter, K., Motl, D., 2018. Automated mineralogy and
8
Contents lists available at ScienceDirect
Minerals Engineering journal homepage: www.elsevier.com/locate/mineng
Experimental validation of a statistical reliability method for the liberation distribution measurement of ore particles
T
Takao Ueda Environmental Management Research Institute, Department of Energy and Environment, National Institute of Advanced Industrial Science and Technology (AIST), 16-1 Onogawa, Tsukuba, Ibaraki 305-8569, Japan
ARTICLE INFO
ABSTRACT
Keywords: Statistical reliability Mineral liberation Liberation distribution X-ray computed tomography Binary particle
Accurate assessment of mineral liberation is important in mineral processing. Recently popularized automated liberation analyzers have enabled the rapid measurement of the liberation state of particles. However, the statistical reliability of liberation distribution assessment has not been thoroughly investigated. Recently, a statistical method which correlates particle measurements with the confidence interval of the measured liberation distribution has been proposed. In the present study, this statistical method was validated by a series of experiments using binary particles and X-ray computed tomography (X-ray CT) analysis. Specifically, three types of binary particles composed of a matrix phase (epoxy resin) and grain phase (glass beads or silica sand) were created; the liberation distribution of all the particles was obtained as the mother population, through X-ray CT and post-processing; a certain number of particles determined by the statistical method were selected from the mother population, and the percentage in which the sampled liberation distribution satisfied the designated confidence interval was determined. The percentages ranged from 94% to 97%, which agreed with the required reliability of 95%, demonstrating the effectiveness of the statistical model.
1. Introduction Liberation distribution (the cumulative distribution of the particles with respect to their content of the mineral phase of interest ( x )) is one of the most important parameters for mineral liberation assessment of crushed ore particles. It is commonly used for engineering purposes, such as simulating the recovery rate of the mineral of interest by adjusting the separation operating conditions. By nature, x can be a continuum function, but it is traditionally treated as twelve bins for ease of experimental analysis (when the bins are designated as i , i = 1 12 corresponds to x = 0, 0 0.1, , 0.9 1, 1, respectively), and this was adopted in the present study. The liberation distribution is generally measured through analysis of ore particle sections or surfaces. Recently popularized automated analyzers employing a combination of scanning electron microscopy and energy dispersive X-ray analysis (Fandrich et al., 2007; Gottlieb et al., 2000; Hrstka et al., 2018; TESCAN, 2012) are effective tools that enable the analysis of a great number of particle sections in a reasonable time. However, the correlation between the confidence interval of the liberation distribution and the number of measured particle sections has hitherto not been fully understood. Studies investigating the liberation distribution confidence interval have included the development of a dispersion model using the arcsine
transformation and comparison with the bootstrapping technique (Efron, 1979; Leigh et al., 1993; Lotter et al., 2018); an estimation of the variation coefficient with respect to the number of particle section measurements (with the equation y = ax b ) using the bootstrapping technique (Evans and Napier-Munn, 2013); a measurement of the liberation distribution confidence interval for bins of copper ore (Mariano and Evans, 2015). Recently, using a statistical approach, Ueda et al. investigated the relationship between the liberation distribution confidence interval and the number of particle section measurements; and developed a statistical reliability method for liberation distribution assessment (Ueda et al., 2016a, 2018a). A remarkable feature of the method is that the confidence interval for each bin can be independently calculated; and conversely, the required number of particle section measurements can be calculated from the required confidence interval for each bin. In the present study, this method was experimentally validated using artificial binary particles and X-ray CT. 2. Methodology This section briefly describes the statistical reliability method for liberation distribution assessment; the detailed information is in Ueda et al. (2018a).
E-mail address: [email protected]. https://doi.org/10.1016/j.mineng.2019.105880 Received 11 April 2019; Received in revised form 17 July 2019; Accepted 22 July 2019 Available online 31 July 2019 0892-6875/ © 2019 Elsevier Ltd. All rights reserved.
Minerals Engineering 140 (2019) 105880
T. Ueda
Table 1 Examples of statistic reliability requirements for bins (after (Ueda et al., 2018a)). Case
Statistical reliability requirements
1
The confidence interval ( i ) for all bins should be less than 0.02
2 3
Specified
for bins with x 0.8 (i 10) should be less than 0.02 , because we are concerned with the liberation distribution range with high phase A content i for bins with x = 0 and 1 (i = 1 and 12) should be less than 0.02 , because we are concerned with the respective degrees of liberation of phases A and B i
The liberation distributions of binary particles (composed of phases A and B) are considered. Actual ore particles are generally composed of multiple mineral phases, but if the mineral phases are grouped into the mineral of interest and others, the particles can be simply treated as binary particles. Therefore, the binary particle methodology may be applied to the assessment of actual ore particles, without loss of generality. As noted in the Introduction, mineral liberation assessment is based on particle section measurement; that is, 2D assessment. Therefore, 2D liberation distribution will be focused on in this paper. However, if 3D particle information can somehow be obtained [such as by X-ray CT (Gay and Morrison, 2006; Lin and Miller, 1996; Miller et al., 2003, 2009; Videla et al., 2007), serial particle sectioning (Lätti and Adair, 2001; Miller and Lin, 1988; Schneider et al., 1991), or stereological correction (Gay, 1994; Gay and Morrison, 2006; King and Schneider, 1998; Leigh et al., 1996; Miller and Lin, 1988; Ueda et al., 2017, 2016b)], the following methodology can be equally applied in the case of 3D liberation distribution.
i i
i
i
= 0.02 where i = 1
{ 0.02 ={
0.02 where i = 10 = where i = 1
12 12 9
where i = 1 and 12 where i = 2 11
interval is obtained. Given this implementation method, in this study, Nr is adjusted to 100 when the calculatedNr is less than 100. 3. Experimental validation The experimental validation employed binary particles composed of grain and matrix phases (corresponding to phase A and phase B), respectively. Artificial binary particles were created (Section 3.1) and analyzed by X-ray CT and post-processing (Sections 3.2 and 3.3); after which, the proposed method was validated (Section 3.4). Sections 3.1–3.3 are based on the methodology in Ueda et al. (2018b), and briefly described here. 3.1. Preparation of binary particle samples A key feature of the experiment was the assignment of different specific gravities (SGs) to the grain and matrix phases, in order to later distinguish them by X-ray CT. The binary material was crushed by a double shaft shredder, and screened by a Ro-tap sieve shaker. Table 2 lists the three types of artificial binary particle samples, with the different materials, grain sizes (initial size of the grain phase for binary materials), and particle sizes after crushing and screening.
2.1. Statistical reliability method for liberation distribution assessment As aforementioned, the liberation distribution was treated in terms of twelve bins (i = 1 12 corresponds to x = 0, 0 0.1, , 0.9 1, 1, respectively). For the mother population, the probability of a particle section being classified into the i -th bin (Pi ) is Pi = Si Sall , where Si denotes the sum of the sectional areas entering the i -th bin, and Sall denotes the total sectional area. Similarly, for a given sample (a limited number of randomly selected particles from the mother population), the probability of a particle section being classified into the i -th bin is Pi . When the confidence interval is arbitrarily determined as i (i.e., Pi is expected to be in the range of Pi ± i ), the required number of particle sections in the i -th bin (Ni ) is determined as: Ni = (KP i ) 2Pi (1 Pi ) , where KP is a statistical coefficient obtained from the table of normal distribution, and KP = 1.96 when the required reliability (Rr ) is 95%, according to the estimation of the confidence interval for the population proportion. Then, the required number of particle section measurements (Nr ) is calculated by Nr = max Ni .
3.2. X-ray CT analysis 3.2.1. X-ray CT scanning The binary particles were packed into cylindrical polyethylene containers (7.5 mm in diameter and 20.0 mm in height), with 19 containers prepared for each sample type. The samples were analyzed by an X-ray CT system (inspeXio SMX-100CT, Shimadzu Corporation) with an X-ray tube output of 60 kV and 40 μA, a resolution of 0.017 mm/ pixel, and an image size of 512 × 512. Fig. 1 shows examples of the Xray transmission images. 3.2.2. Post-processing for identification of particles and phases In order to identify the particles and phases, further post-processing was conducted as follows: (1) the particles and background were binarized using Otsu’s method (Otsu, 1979); (2) the particles were detached and individually identified using the watershed method; (3) the grain and matrix phases were identified by binarization using Otsu’s method; and finally, (4) three-valued voxel data (grain phase, matrix phase, background), with dimensions 512 × 512 × 460 , were obtained. The particle identification in Step 2 (above) was not flawless; for example, some aggregates were misidentified as particles. However, the misidentified particles were manually eliminated, and the total number of successfully identified particles are listed in the rightmost column of Table 2. Then, six cylindrical sample sections (parallel to the bottom), with heights of 20, 105, 190, 275, 360, and 445 voxels, were analyzed. A sectional interval of 85 voxels (=1.45 mm), larger than the maximum particle diameter (1.40 mm), was specified, to avoid multiple sectioning of a single particle. The particle sections emerging in the six sample sections were treated as identified particle sections. The total number of particle sections are also listed (in parentheses) in the rightmost column of Table 2.
i = 1 12
A remarkable feature of the method is that i can be independently determined for each bin, which makes it possible to flexibly meet measurement needs. Table 1 presents three possible examples (after (Ueda et al., 2018a)). 2.2. An implementation method Detailed information about the method of implementation for the statistical reliability method is in (Ueda et al., 2018a), but may be summarized as follows. First, a preliminary measurement is conducted for a small number of particles (e.g., 100) to temporally determine Pi . Second, the measurement and Nr estimation are repeated until the number of particle measurements reaches Nr . When Nr is reached, the measurements are complete, and the liberation distribution with the required confidential 2
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Fig. 2 shows 3D particle structures obtained by the post-processing, and Fig. 3 shows sectional images of Fig. 2 at a height of 275. 3.3. Liberation analysis 3.3.1. 3D liberation distribution The particle volume (Vp ) was calculated by multiplying the number of voxels in a given particle by the voxel volume (0.0173 mm3), which was determined by the X-ray CT scanning resolution (0.017 mm/pixel). The phase A volume in a given particle (VA ) was similarly calculated, and the phase A volume fraction ( x ) was calculated as x = VA VP . After completing this calculation for all the particles, the number of particles entering the i -th bin was counted with respect to x , for all 12 bins, and the fraction of each bin (Pi ) was determined. Pi corresponds to the 3D liberation distribution. 13,093 (8549) 19,384 (11,499) 2950 (2994)
3.3.2. 2D liberation distribution The subsequent analysis was similar to that of the 3D calculation above. The particle sectional area (SP ) was calculated by multiplying the number of voxels in a particle section by the pixel area (0.0172mm2 ), the phase A area in a given particle section (SA ) was similarly calculated, and the phase A areal fraction (x ) was calculated as x = SA SP . The number of sections entering the i -th bin was counted with respect to x , for all 12 bins, and the fraction of each bin (Pi ), which corresponds to the 2D liberation distribution, was determined. 3.4. Validation
0.5–0.85 0.5–0.85 0.85–1.4
The total data for all the 3D particles and 2D particle sections was considered as the parent population. Three cases (required statistical reliabilities) in Table 1 were considered, with Nr in each case calculated by the statistical reliability method (Section 2). Then, Nr particles were randomly selected from the parent population, allowing overlap, and Pi was calculated (as in Section 3.3). This random sampling was repeated 1000 times, and the probability of Pi being within the confidence interval of Pi ± i was determined. This probability is defined as the measurement reliability (Rm ), and is expected to satisfy Rm Rr . 4. Results and discussion
0.2 0.2 0.2
4.1. Sample 1 Table 3 shows the Rm values for Sample 1, Cases 1–3, calculated with 1000 sampling repetitions of Nr particles. The shaded values in Table 3 indicate the bins with i = 0.02 , namely the bins of interest (see Table 2). The bottom row shows the minimums of the shaded values. The minimum Rm s for the 2D and 3D liberation distributions (roughly 94–97.5%) are comparable to Rr (95%), except for the 3D distribution in Case 3, where Rm = 100 , because in this latter case the number of preliminary measurements (1 0 0) exceeds Nr , which results in excessive measurement. Fig. 4 compares the 2D liberation distributions of the parent population with those of the 2.5–97.5 percentile range of 1000 samples for Sample 1, Cases 1–3. The respective designated confidence interval ranges (with i = 0.02 ) are indicated by the double arrows. It is confirmed that the dispersion ranges of the sample cases successfully fell within the designated confidence interval ranges.
Glass beads (2.5) Glass beads (2.5) Silica sand (2.2) Epoxy resin (1.1) Epoxy resin (1.1) Epoxy resin (1.1) 1 2 3
0.105–0.125 0.35–0.5 0.16–0.25
Grain phase (SG) Matrix phase (SG) Sample
Table 2 Binary particle samples.
Grain size (mm)
Grain phase volume fraction
Particle size after crushing and screening (mm)
Number of particles in post-processing after X-ray CT (number of sections)
T. Ueda
4.2. Sample 2 Table 4 shows the Rm values for Sample 2, Cases 1–3 (corresponding to Table 3). The results show the same pattern as in Table 3: the minimum Rm s of the 2D and 3D liberation distributions (roughly 95–96%) are comparable to Rr (95%). 3
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Fig. 1. X-ray transmission images of (a) Sample 1 (epoxy resin and 0.105–0.125 mm glass beads), (b) Sample 2 (epoxy resin and 0.35–0.5 mm glass beads), and (c) Sample 3 (epoxy resin and 0.16–0.25 mm silica sand); grain phase (glass beads or silica sand) in white, matrix phase (epoxy resin) in gray.
Fig. 2. 3D cylindrical particle structures created by X-ray CT post-processing of (a) Sample 1 (epoxy resin and 0.105–0.125 mm glass beads), (b) Sample 2 (epoxy resin and 0.35–0.5 mm glass beads), and (c) Sample 3 (epoxy resin and 0.16–0.25 mm silica sand); grain phase (glass beads or silica sand) in black, matrix phase (epoxy resin) in gray.
Fig. 3. 2D cylindrical particle section patterns created by X-ray CT post-processing of (a) Sample 1 (epoxy resin and 0.105–0.125 mm glass beads), (b) Sample 2 (epoxy resin and 0.35–0.5 mm glass beads), and (c) Sample 3 (epoxy resin and 0.16–0.25 mm silica sand); grain phase (glass beads or silica sand) in black, matrix phase (epoxy resin) in gray.
Fig. 5 shows the 2D liberation distributions for Sample 2, Cases 1–3 (corresponding to Fig. 4). It is again confirmed that the dispersion ranges of the sample cases successfully fell within the designated confidence interval ranges ( i = 0.02 ).
dispersion ranges of the sample cases successfully fell within the designated confidence interval ranges ( i = 0.02 ).
4.3. Sample 3
As remarked in the Introduction, a notable feature of the statistical reliability method for liberation distribution assessment is that the required confidential interval of each bin can be independently designated. As only this method has this capability, comparison with previous studies (Evans and Napier-Munn, 2013; Leigh et al., 1993) was not possible. The main result, experimental validation of the statistical reliability method, is free from experimental error. The validation consisted of two stages, the first described in Sections 3.1 and 3.2, and the second in Sections 3.3 and 3.4. Vertical binary particle data was created from
4.4. Discussion
Table 5 shows the Rm values for Sample 3, Cases 1–3 (corresponding to Tables 3 and 4). The results show the same pattern as in the two tables: the minimum Rm s of the 2D and 3D liberation distribution (roughly 94–97%) are comparable to Rr (95%), except for the 3D distribution in Case 3, where Nr was less than in the preliminary measurement (1 0 0), which results in excessive measurement. Fig. 6 shows the 2D liberation distributions for Sample 3, Cases 1–3 (corresponding to Figs. 4 and 5). It is again confirmed that the 4
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Table 3 Probability of the calculated particle number samples satisfying the designated confidence interval, for Sample 1. Shaded values indicate the designated confidence interval bins. The bottom row shows the minimums of the shaded values.
Nr A
Nr A
x
x
Nr A
x
Fig. 4. Comparison of the 2D liberation distribution of the parent population (8549 particle sections), with the 2.5–97.5 percentile range of 1000 samples with Nr particle sections, for Sample 1, Cases 1–3. 5
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Table 4 Probability of the calculated particle number samples satisfying the designated confidence interval, for Sample 2, Cases 1–3. Shaded values indicate designated confidence interval bins, and the bottom row shows the minimums of the shaded values.
Nr A
Nr x
A
x
Nr A
x
Fig. 5. Comparison of the 2D liberation distribution of the parent population (11,499 particle sections), with the 2.5–97.5 percentile range of 1000 samples with Nr particle sections, for Sample 2, Cases 1–3. 6
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Table 5 Probability of the calculated particle number samples satisfying the designated confidence interval, for Sample 3, Cases 1–3. Shaded values indicate designated confidence interval bins, and the bottom row shows the minimums of the shaded values.
Nr A
Nr x
A
x
Nr A
x
Fig. 6. Comparison of the 2D liberation distribution of the parent population (2994 particle sections), with the 2.5–97.5 percentile range of 1000 samples with Nr particle sections, for Sample 3, Cases 1–3. 7
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actual binary particles in the first stage, and the statistical reliability method was validated in the second stage, using this vertical binary particle data. In the first stage, experimental error was possible in the Xray CT scanning and post-processing; however, in the second stage, since the liberation analysis and validation was computed automatically, experimental error was not possible.
petrology – applications of TESCAN integrated mineral analyzer (TIMA). J. Geosci. (Czech Republic) 63, 47–63. https://doi.org/10.3190/jgeosci.250. King, R.P., Schneider, C.L., 1998. Stereological correction of linear grade distributions for mineral liberation. Powder Technol. 98, 21–37. https://doi.org/10.1016/S00325910(98)00013-8. Lätti, D., Adair, B.J.I., 2001. An assessment of stereological adjustment procedures. Miner. Eng. 14, 1579–1587. https://doi.org/10.1016/S0892-6875(01)00176-5. Leigh, G.M., Lyman, G.J., Gottlieb, P., 1996. Stereological estimates of liberation from mineral section measurements: a rederivation of Barbery’s formulae with extensions. Powder Technol. 87, 141–152. https://doi.org/10.1016/0032-5910(95)03080-8. Leigh, G.M., Sutherland, D.N., Gottlieb, P., 1993. Confidence limits for liberation measurements. Miner. Eng. 6, 155–161. https://doi.org/10.1016/0892-6875(93) 90129-B. Lin, C.L., Miller, J.D., 1996. Cone beam X-ray microtomography for three-dimensional liberation analysis in the 21st century. Int. J. Miner. Process. 47, 61–73. https://doi. org/10.1016/0301-7516(96)00005-1. Lotter, N.O., Evans, C.L., Engstrőm, K., 2018. Sampling – a key tool in modern process mineralogy. Miner. Eng. 116, 196–202. https://doi.org/10.1016/j.mineng.2017.07. 013. Mariano, R.A., Evans, C.L., 2015. Error analysis in ore particle composition distribution measurements. Miner. Eng. 82, 36–44. https://doi.org/10.1016/j.mineng.2015.06. 001. Miller, J.D., Lin, C.L., 1988. Treatment of polished section data for detailed liberation analysis. Int. J. Miner. Process. 22, 41–58. https://doi.org/10.1016/0301-7516(88) 90055-5. Miller, J.D., Lin, C.L., Garcia, C., Arias, H., 2003. Ultimate recovery in heap leaching operations as established from mineral exposure analysis by X-ray microtomography. Int. J. Miner. Process. 72, 331–340. https://doi.org/10.1016/S0301-7516(03) 00091-7. Miller, J.D., Lin, C.L., Hupka, L., Al-Wakeel, M.I., 2009. Liberation-limited grade/recovery curves from X-ray micro CT analysis of feed material for the evaluation of separation efficiency. Int. J. Miner. Process. 93, 48–53. https://doi.org/10.1016/j. minpro.2009.05.009. Otsu, N., 1979. A threshold selection method from gray-level histograms. IEEE Trans. Syst. Man. Cybern. 9, 62–66. https://doi.org/10.1109/TSMC.1979.4310076. Schneider, C.L., Lin, C.L., King, R.P., Miller, J.D., 1991. Improved transformation technique for the prediction of liberation by a random fracture model. Powder Technol. 67, 103–111. https://doi.org/10.1016/0032-5910(91)80032-E. TESCAN, 2012. TESCAN Introduces the TIMA Mineralogy Solution. [WWW Document]. URL https://www.tescan.com/en-us/about-tescan/news/tescan-introduces-the-timamineralogy-solution (accessed 5 July 2019). Ueda, T., Oki, T., Koyanaka, S., 2018a. Statistical reliability of the liberation distribution of ore particles with respect to number of particle measurements. Miner. Eng. Miner. Eng. 126, 82–88. Ueda, T., Oki, T., Koyanaka, S., 2018b. Experimental analysis of mineral liberation and stereological bias based on X-ray computed tomography and artificial binary particles. Adv. Powder Technol. 29, 462–470. https://doi.org/10.1016/j.apt.2017.11. 004. Ueda, T., Oki, T., Koyanaka, S., 2017. Stereological correction method based on sectional texture analysis for the liberation distribution of binary particle systems. Adv. Powder Technol. 28, 1391–1398. https://doi.org/10.1016/j.apt.2017.03.007. Ueda, T., Oki, T., Koyanaka, S., 2016a. Statistical effect of sampling particle number on mineral liberation assessment. Miner. Eng. 98, 204–212. https://doi.org/10.1016/j. mineng.2016.08.026. Ueda, T., Oki, T., Koyanaka, S., 2016b. Stereological bias for spherical particles with various particle compositions. Adv. Powder Technol. 27, 1828–1838. https://doi. org/10.1016/j.apt.2016.06.016. Videla, A.R., Lin, C.L., Miller, J.D., 2007. 3D characterization of individual multiphase particles in packed particle beds by X-ray microtomography (XMT). Int. J. Miner. Process. 84, 321–326. https://doi.org/10.1016/j.minpro.2006.07.009.
5. Conclusion A series of experiments were conducted to validate the proposed statistical method for liberation distribution assessment, which correlates the number of particle measurements with the confidence interval range. Three types of binary particles composed of a matrix phase (epoxy resin) and grain phase (glass beads or silica sand) were created, and 3D particle data (as well as 2D particle sectional data), which included the volumetric (and areal) fraction of the grain phase, were analyzed by X-ray CT. The liberation distribution calculated based on the total number of particles (or particle sections) was determined as the true data of the mother population. Then, a certain number of particles (or particle sections) determined by the statistical model were randomly selected from the mother population, and the percentage in which the sample liberation distribution satisfied the designated confidence interval range was determined. The percentage ranged from 94% to 97%, demonstrating agreement with the required reliability of 95%, and validating the effectiveness of the statistical model. Acknowledgment This work was supported by Japan Oil, Gas and Metals National Corporation. References Efron, B., 1979. Bootstrap methods: another look at the jackknife. Ann. Stat. 7, 1–26. https://doi.org/10.1214/aos/1176344552. Evans, C.L., Napier-Munn, T.J., 2013. Estimating error in measurements of mineral grain size distribution. Miner. Eng. 52, 198–203. https://doi.org/10.1016/j.mineng.2013. 09.005. Fandrich, R., Gu, Y., Burrows, D., Moeller, K., 2007. Modern SEM-based mineral liberation analysis. Int. J. Miner. Process. 84, 310–320. https://doi.org/10.1016/j.minpro. 2006.07.018. Gay, S.L., 1994. Liberation Modelling Using Particle Sections. The University of Queensland, Australia (accessed 5 July 2019). Gay, S.L., Morrison, R.D., 2006. Using two dimensional sectional distributions to infer three dimensional volumetric distributions – validation using tomography. Part. Part. Syst. Charact. 23, 246–253. https://doi.org/10.1002/ppsc.200601056. Gottlieb, P., Wilkie, G., Sutherland, D., Ho-Tun, E., Suthers, S., Perera, K., Jenkins, B., Spencer, S., Butcher, A., Rayner, J., 2000. Using quantitative electron microscopy for process mineralogy applications. J. Miner. Met. Mater. Soc. 52, 24–25. https://doi. org/10.1007/s11837-000-0126-9. Hrstka, T., Gottlieb, P., Skála, R., Breiter, K., Motl, D., 2018. Automated mineralogy and
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