Comparison of volumetric and section area particle compositions using the Gaudin random mineral liberation model

Transcription

1 Comparison of volumetric and section area particle compositions using the Gaudin random mineral liberation model R. L. Wiegel Mineral Processing Consultant, Lakeland, Florida, USA Abstract A well-defined geometric model for a mineral system and its liberation by size reduction is used to show the relationship between true volumetric particle composition and the composition estimates obtained by the measurement of linear intercepts or areas of simulated polished sections of particles. Based on this information, a transformation is developed between the areal measurements and the volumetric composition for this idealized mineral system. Key words: Gaudin random liberation model, Mineral liberation, Particle composition measurement Introduction There has long been a recognition that there are differences among the quantitative measurements of individual particle composition that are based on either the actual volume of the mineral grain fragments constituting a particle, or on the various mineral surface areas exposed by a polished section of a slice through a mineral particle. There have been attempts to determine how large or small these differences may be, usually based on mathematical first principles, but these have not yielded a widely supported general conclusion. In the study of mineral liberation, it is almost always the volumetric or gravimetric composition that is being sought, but only in rare cases is it possible to make measurements that directly provide this data. More frequently, it is necessary to be satisfied with composition estimates based on the exposed mineral surface areas obtained from polished sections. This paper attempts to use simulation techniques to compare these various types of measurements for an ideal, yet simple, set of locked and liberated mineral particles, as generated by the Gaudin random liberation model. It is hoped that observations and relationships that can be derived from these comparisons will lead to an improved ability to make the transformation from section areal estimates to volumetric results. Description of the Gaudin random liberation model The Gaudin random liberation model (GRLM) assumes that all mineral grains are of the same cubic size and aligned in the mineral aggregate in a Rubik cube arrangement, with a random placement of the two (or more) mineral species, designated A for waste and B for values (Gaudin, 1939; Wiegel and Li, 1967). When the mineral aggregate is broken by a randomly located but regular fracture lattice which parallels the mineral grain surfaces, the resultant particles are again of a uniform cubic size. If the particle s linear dimension (beta) is much larger than the mineral grain s linear size (alpha), each particle will be made up of many individual grains and grain fragments, and because of the random location of the various species, most of the particles will still be locked. There will be a few particles, however, which will have only contained mineral grains and grain fragments of the same species, and hence be liberated, due to the coincidental occurrence of all abutting grains in a specific region being of the same mineral. As the particle size gets smaller, or as the ratio of grain size to particle size (K) increases, the number of contained grain fragments in an individual particle decreases, until at a ratio of exactly unity, where particle size equals mineral grain size, the number of grain fragments reaches a value of eight. At this point these grain fragments occupy the eight corners of the cubic particle. As particle size gets smaller still, the types of particles which are created, as shown in Fig. 1, gradually change from those representing the junction of the grain corners (8-grain fragments) to those representing the contact at the grain edges (4-grain fragments) and those from the interface of the grain surfaces (2-grain fragments), and eventually the grain fragments which are internal to the original grains (1-grain fragment). The multi-fragment particles could be either locked or liberated depending on the composition of the adjacent grains, but the single grain fragment particle will necessarily be liberated. Because of the gross simplifications achieved by the assumptions of ideal fracture of the Rubik cube arrangement of randomly located cubic grains into the resultant cubic particles, the application of simple geometry, calculus and binary statistical concepts makes it possible to calculate the number of each type of particle created Paper number MMP Original manuscript submitted November Revised manuscript accepted for publication March Discussion of this peer-reviewed and approved paper is invited and must be submitted to SME Publications Dept. prior to August 31, Copyright 2010, Society for Mining, Metallurgy, and Exploration, Inc. February 2010 Vol. 27 No. 1 24

2 Figure 1 The four types of particles created by the Gaudin random liberation model (1, 2, 4 and 8 mineral grain fragments per particle). by the fracture lattice. These calculations are a function of volumetric composition of the mineral of interest (VB) and the ratio of mineral grain size to particle size (K). The general relationship for a specific particle size fraction, which applies over the entire grain size to particle size range, is as follows: 1/ K = t + ε (2) where I represents either waste mineral A or valued mineral B, PI is the liberated fraction of particles of either mineral A or B, VI is the volume fraction of mineral A or B in the crude ore feed size fraction, (1) K is the ratio of linear mineral grain size (alpha) to particle size (beta), t is the largest integer in 1/K and ε is the fractional remainder in 1/K In the case when particle size is smaller than or equal to mineral grain size (K 1), where most mineral liberation takes place, t = 0 and this relationship simplifies to the following: In either case the fraction of locked particles, PAB, is given by: (4) (3) Figure 2 Liberation characteristics of a 0.20 values fraction GRLM ore showing volumetric composition ranges of locked particles. Application of the Gaudin random liberation model to real mineral systems The above relationships combined with simulation techniques can be used to calculate the effect of size reduction on mineral liberation, by gradually decreasing particle size (increasing size ratio K) from one level to another level for a constant volumetric composition of values in the feed, as shown in Fig. 2. These relationships have been semi-quantitatively verified (Wiegel, 1975) for the special case where a selective laboratory ferro-magnetic separator, the Davis tube, was used to measure the liberation characteristics of disseminated magnetite ores from around the world. The use of this type of separation on narrow size fractions of crude ore permits the rejection of virtually all non-magnetic particles (PA - liberated waste) and the retention of all particles containing magnetics (PB + PAB - liberated values and locked values and waste). The comparison 25 Vol. 27 No. 1 February 2010

3 of GRLM calculations with Davis tube test results, converted to volumetric terms, for sized fractions of numerous magnetite crude ores, indicates that there is a reasonable degree of agreement with reality for binary mineral systems, despite the gross assumptions made in the GRLM derivations (Wiegel, 2002; Wiegel, 2006). In this case the binary mineral system consists of magnetite and non-magnetite grains. Liberation measurement techniques Unfortunately, there are very few situations where volumetric or gravimetric liberation data can be collected for important commercial mineral systems. It therefore usually becomes necessary to make use of the two-dimensional appearance of minerals as a measure of their state of liberation. This is determined microscopically, or by other instrumentally derived measurements on polished sections of the narrow size fractions of an ore. In many cases, the instrumentation is capable of determining with reasonable speed and great precision the identity of a mineral and the exposed section area it represents in an individual particle. By carrying out similar measurements for all minerals in an ore, the volumetric composition of each particle can be estimated from the relative section areas exposed. An additional, somewhat less sophisticated, technique has also been used, in which the narrowly sized particles in a polished section are subjected to the measurement of the length of the randomly placed linear intercept across the various exposed minerals that each particle contains, and an estimate made of the particle composition based on these intercepts. It is recognized that the information one obtains from either the sectioned area of particles or the lines scribed across sectioned particles may be considerably different from that which is obtained from volumetric or gravimetric classification of particles. This can be especially important when attempting to quantify the liberation characteristics of an ore; that is, the relationship between particle quantity, composition and size. The one-dimensional and two-dimensional approximations of the three-dimensional realities have been the subject of considerable speculation, especially regarding the magnitude of the distortion which may be introduced. When one sections or scribes a line across the section of a totally liberated particle, the composition estimate obtained indicates complete liberation, which is an exact measure of reality. However, it is also possible to obtain section areas or intercepts of particles which mistakenly indicate complete liberation, due to the section being too small to be representative, the number of grain fragments exposed being incomplete or the effect of the third dimension of grain fragment depth not being included. All of these types of errors contribute to inaccurate measures of composition throughout the locked particle composition spectrum. In the past, there have been various approaches to the problem of converting the one-dimensional linear intercept or two-dimensional section area composition estimate into the usually more useful three-dimensional volumetric composition distribution. The earliest was the use of a single empirical correction factor dependent on the ore s values composition and a locking factor (Gaudin, 1939). Computer simulation techniques were used (Jones and Horton, 1978) to compare the results of linear intercept measurements and true volumetric composition for several simple particle geometries one- and two-fragment spheres and cubes. Correction factors were developed from these simulations. A more sophisticated, mathematically based approach is the use of a particle generator program, PARGEN (Miller and Lin, 1988; Bole et al., 1993), which creates simulated particles consistent with the information available from the linear intercept or section area distribution measurements. Two parameters important to the application of this technique are the ore s values composition and dispersion density; that is, the number of mineral grains per simulated particle. One interesting point related to the description of the application of the PARGEN program is the treatment of the section or intercept-based composition distribution as not just a numerical frequency, but rather as a distribution weighted with respect to the magnitude of the individual section areas or intercept lengths. This treatment tends to minimize the effect of the composition of the smaller section particles, even though they may be large in number. Comparisons have been made among several adjustment techniques (Lin et al., 1995): 1. Large section correction consists of eliminating or filtering out all linear or areal composition estimates where the length or area is less than 90% of the maximum length or area measured, and can reduce the numerical sectioned data set by 70%; 2. Hill s fast approximation adjustment (Hill et al., 1987) applies an empirical correction to the apparently liberated sections based on previous experience with locked, single-interface capped spheres; 3. Barbery s polyhedral texture breakage correction (Barbery and Leroux, 1988; Barbery, 1991) uses parameters based on the section composition distribution; and 4. the PARGEN program adjustment developed at the University of Utah. Of these, the large section correction was found to be most useful and appears to be the simplest to apply. There was, however, considerable discussion about this paper s results, regarding the possible misapplication of the PARGEN program (King and Schneider, 1995). More recently (Latti and Adair, 2001), a comparison of two-dimensional section area composition distributions was made with the reconstructed three-dimensional particle images derived from a series of section areas of the identical particles, which were sequentially polished to remove thin slices of particle depth. The conclusion reached was that there was very little difference between these two composition distributions. The rationale reported for this conclusion was based on the fact that this somewhat limited study was made using a natural, multi-mineral system and not based on computer simulations of ideal binary mineral systems. The recognition of substantial differences between one-, two- and three-dimensional measures of particle composition by virtually all other investigators has built a solid wall of opinion counter to this isolated study s conclusion. Once the physical description of a situation is understood, the advantage of the use of particle simulation techniques is that they are in effect non-destructive of the particles, and permit the generation of relatively large numbers of results, which can be evaluated statistically. The more recent use of a microtomographic technique (Miller and Lin, 2004) for developing detailed three-dimensional mineral liberation data shows signs of overcoming many of the shortcomings of the currently used one- and two-dimensional measurements on mineral polished sections. To date, the availability of this equipment is limited and its capabilities and limitations are yet to be determined. Results of this study This project compares the results obtained if the two particle composition estimating techniques, that based on polished section mineral area and that based on mineral linear intercepts, were applied to the identical samples of GRLM particles for which true volumetric compositions were available. The cur- February 2010 Vol. 27 No. 1 26

4 Figure 3 Comparison of estimated particle compositions based on section areas or intercept lengths with true volumetric composition for binary GRLM particles. rent study was done using simulation techniques to mimic the randomly oriented sectioning of the GRLM cubic particle and to randomly scribe a line across that particle s section. In each case, the exposed mineral area or the intercepted mineral length is used to estimate the particle s composition in a binary mineral system when particle size is less than or equal to mineral grain size. The extension of this simulation approach to particles larger than the mineral grain size becomes much too complex for consideration. The situation when particle size is smaller than mineral grain size, however, is by far the more important, since it is then that mineral liberation proceeds to sufficient completion to permit economic separation and recovery operations. As described above, when particle size is less than or equal to mineral grain size, the GRLM particles contain from one to eight grain fragments, the relative frequencies of which are calculable, as are the probabilities for the various permutations of binary particle grain fragment arrangements in those eight corner locations, the four edge locations or the two side locations of a cube. In carrying out these simulations for each feed grade (VB) and particle size ratio (K) combination, the quantity-composition results were obtained for 128 to 640 random settings of the GRLM fracture lattice location with respect to the grain lattice (in x, y, z dimensions) for all 278 grain fragment arrangement permutations, in order to generate the volumetric or true composition for in excess of 35,500 particles. Each of these particles is appropriately weighted for its occurrence, given the specific feed composition (VB) of values component in the ore and a specific value of grain size to particle size ratio (K=>1, or particle size less than or equal to mineral grain size). The identical fracture lattice locations and grain fragment arrangement permutations were used for the polished section area estimates and the intercept line estimates. A single randomly oriented sectioning plane was simulated for each lattice location and grain fragment arrangement permutation, and a single line scribed across each particle s polished section was also simulated. The orientation and location of the simulated polished section plane was obtained by randomly choosing three of the six sides of the cubic particle and randomly fixing the dimensions of a point on each of these three sides. These three points then define the section plane. Measurements were made for feed grades (VB) of 0.01, 0.02, 0.05, 0.10, 0.20, and 0.50 fraction values by volume. The range of size ratios investigated was 1, 2, 4, 8, 16 and 32, with additional values using the square root of two size progressions for the 0.20 feed grade. It is worth noting that the two parameters mentioned in prior studies as important to the adjustment of section area composition distributions to volumetric composition distributions are the composition of the valuable mineral in the ore sample and the number of mineral grain fragments, or possibly different mineral interfaces, in a particle. These same parameters appear in the GRLM s mathematical description of mineral liberation. The data shown in Fig. 3 gives an indication of the typical magnitude and frequency of composition estimate errors over the entire volumetric or true composition range obtained in this study. As an example, in the set of 175 particle composition simulations shown in this chart, there were actually two volumetric observations for liberated waste and one for liberated values particles, while there were seven and seven observed in section area estimates and fifty-one and forty-nine observed in intercept length estimates. This indicates that some 56% of the linear intercept estimates of composition were either liberated waste or liberated values, while only 6% were so for the section area estimates. A further indication of the substantial difference between the linear intercept and section area composition estimates when compared to the true volumetric composition is provided by the magnitude of correlation coefficients for linear intercept versus volume of 0.65 and 0.88 for section area versus volume. These differences are sufficient to eliminate any further consideration by this author of the use of linear intercepts as a reliable indicator of particle composition when other alternatives are available. Therefore, the remainder of this paper focuses its presentation on section area composition estimates versus volumetric compositions. An additional important point regarding the magnitude of correlation coefficients is that as mineral liberation progresses, the proportion of totally liberated particles increases, and since either of the composition estimating techniques is infallible 27 Vol. 27 No. 1 February 2010

5 Figure 4 - Comparison of cumulative frequency distributions of particle composition for volume and section area data, with VB =0.20, K =1.0, 2.0, 4.0. Figure 5 Effect of section area magnitude on indicated liberation characteristics for sectioning of GRLM particles (VB = 0.20, K = 2.0) on liberated particles, the correlation coefficient will improve. In the case of the data presented here, there are so few totally liberated particles by volume that the calculated correlation coefficients represent a reasonable estimate of reality for the wide binary locked particle composition spectrum. A comparison can be made between the plot of cumulative frequency data for increasing values of composition based on section area and similar data based on composition by volume, as shown in Fig. 4. This family of curves represents the simulation of mineral liberation, as particle size is reduced (K getting larger), indicated by the relative quantity of particles of composition measured volumetrically and estimated from section area measurements. The difference between the results of the section area liberation data and the volumetric liberation data are greatest at low ratios of mineral grain size to particle size, K, and are reduced as particle size is reduced and liberation becomes more complete. This is due to the increasing number of particles which are volumetrically liberated, and for which there is no error in the section area estimate of composition. Effect of section area on composition distributions Since the random sectioning of even a closely sized fraction of particles produces a distribution of particle section areas, this introduces an additional complexity to the interpretation of mineral liberation data obtained by section area particle composition estimates, as suggested in several of the references reviewed above. The effect of the magnitude of the particle s section area on the inferred mineral liberation results could be significant, with section area tending to mimic particle size in liberation. It is therefore expected that as a particle s section area decreases, liberation would appear to be more complete. This expectation is confirmed by the simulated particle section data shown in Fig. 5. In this case, for in excess of 50,000 February 2010 Vol. 27 No. 1 28

6 Table 1 Relationship between volumetric composition range and section area composition ranges for approximately 2000 typical particles. Particle Particle Particle Particle Particle Particle Freq Comp Freq Comp Freq Comp Freq Comp Freq Comp Freq Comp Liberated 0.00 Range Range Range Range Range Volume Area Range Range Range Range Range Liberated 1.00 Volume Area simulated particle observations, the section data for an initial specific particle size fraction was segregated into several section area ranges. The cumulative quantity (number of particles) versus section composition estimate was subjected to statistical analysis to obtain these graphical results, which indicate a very substantial effect of the magnitude of section area on the inferred liberation situation. The volumetric composition for the six individual section area ranges shown in Fig. 5 averaged 0.204, with a standard deviation of and no recognizable composition versus magnitude of section area trend. Although this effect would be hidden when all particle section areas, regardless of magnitude, are included in the liberation measurement results, this introduction of significant variability to the information is reason to give consideration to a policy of eliminating the very largest and smallest particle section area range results when interpreting liberation data. This significant effect of section area on the composition estimate distribution is substantiated by the results of previous investigations, where the effect of small sections was minimized in the use of the University of Utah s PARGEN program by weighting the individual particle composition estimate by the magnitude of the section area, and by the filtering out of some 70% of the smaller section composition data when using the large section correction approach. It is reasonable to expect that each individual narrow volumetric composition range of particles will yield a reasonably reproducible but wider range of section area compositions, with a numerical average of the section compositions approximating the volumetric composition. This expectation is substantiated by the data presented in Table 1, which represents a summary of a random set of about 2000 comparative observations of a GRLM particle volumetric composition and the identical particle s section area composition. An example of the sectioning of an eight-grain fragment 29 Vol. 27 No. 1 February 2010

7 Figure 6 - Sectioning of an eight-mineral grain fragment GRLM particle with three values fragments. Figure 7 Eight-grain fragment particle, view normal to x-y plane. GRLM particle is shown in Figs. 6 and 7. In this case, the particle is composed of three fragments of the valuable mineral B and five fragments of the waste mineral A. The values are located in positions 2, 3 and 8. The fracture lattice is positioned by the x, y and z values shown in Fig. 6. These parameters define the volumetric composition of this particle to be volume fraction values, or mineral B. A randomly placed sectioning plane passes through the GRLM particle, giving the section shown in Fig. 7. This plane slices through portions of seven of the eight grain fragments and permits the estimation of the particle s composition based on the areas of these exposed surfaces. This estimation provides a value of area fraction mineral B. In addition, a randomly placed linear intercept is scribed across the particle, which provides another estimate of particle composition, fraction mineral B, based on the relative lengths of the two minerals intercepted. When evaluating the magnitude of the particle s section area, it is necessary to look at the view normal to the sectioning plane s intersection of the particle, not at that normal to the x-y plane. These two areas are directly related by a factor based on the cosine of the angle between the sectioning plane and the x-y plane. Ratio of exposed values grain perimeter to total particle exposed perimeter One other indication, in a polished section, of expected particle behavior in a separation process, and to some extent another measure of mineral composition and liberation, is the ratio of the exposed perimeter of valuable mineral grains to the total particle s exposed perimeter. This ratio is the twodimensional analogy of the ratio of the valuable mineral s exposed surface area to total particle surface area, which is thought to play an important part in determining surfacedependent separation results. This ratio increases from zero for liberated waste particles to unity for liberated valuable mineral February 2010 Vol. 27 No. 1 30

8 Figure 8 Relation between the ratio of exposed perimeter for particle sections and the true ratio of exposed surface area for GRLM particles. Figure 9 Comparison of estimated cumulative frequencies for volumetric compositions using different size ratios (VB = 0.20, K = 1,2,4). particles. The simulation techniques developed for this study have been used to generate the two-dimensional section area data for comparison with the three-dimensional volumetric data for the GRLM particles. The GRLM particle and section graphics depicted in Figs. 6 and 7 also contain information that permits calculating the values external surface ratio at , while the external perimeter ratio is A comparison of the two-dimensional and three-dimensional ratios for 175 observations over the range of compositions is shown in Fig. 8. These data are for the same particles whose composition estimates are shown in Fig. 3. The correlation coefficient for these variables is 0.84, approximately what was obtained for the 2D and 3D composition comparison. Transformation of section area to volumetric composition frequency data Using this simulation technique, it is possible to generate information that shows the statistical relationship between the volumetric composition of the GRLM particles and section area composition estimates for these same particles. This relationship suggests it should be possible to take section area data for actual ore samples and transform that into GRLM volumetric composition estimates. Since this volumetric composition estimate is usually what the mineral-processing expert would prefer to use in the recognition, evaluation and solution of processing problems, this transformation from areal composition to volumetric composition could provide a very useful function. It is of course possible that the GRLM is not a suitable approximation to the mineral system being evaluated, in which case the transformed results would be of questionable value. To demonstrate the usefulness of the results, the composition spectrum has been divided into 16 regions. A greater emphasis on the lower ranges of values composition was made to permit 31 Vol. 27 No. 1 February 2010

9 Table 2 Transformation matrices to show section area composition distribution for each volume composition range (corrected for symmetry). the eventual application of these techniques to low-grade ores. The data that was collected consisted of the GRLM volumetric composition of a particle, the number of grain fragments making up that particle and the composition estimate obtained by a simulated single random section through that particle. The compositions were then cross-grouped into the ranges listed in Table 2 for both the known volumetric value and section area estimated composition. For each volumetric composition range this provides a frequency distribution of the section area composition estimates. These frequency distributions are independent of the ratio of mineral grain to particle size and the composition of the overall sample being measured, and are symmetrical with respect to the values and waste composition ranges. By that it is meant that the section area frequency distribution for a values composition by volume would be the same as a waste composition by volume. It does turn out that there are differences in the frequency distributions depending on whether the GRLM particle sectioned is a 2-, 4- or 8-grain fragment particle. The single grain fragment particles are in fact liberated and therefore the section area composition estimate is always correct. The differences between the 2-, 4- and 8-grain fragment particle sections are measurable but not large. A slight complication in the attempt at transformation from section area composition to volumetric composition does occur, in that as liberation proceeds, the relative number of these various particle types changes. As mentioned above, at a particle size equaling the mineral grain size, all particles contain 8-grain fragments. As particle size gets smaller, the proportion of 8-grain fragment particles decreases, and the proportion of 4-grain fragment particles increases, goes through a maximum and begins to decrease as the number of 2-grain fragment and 1-grain fragment (liberated) particles increase. At this point an assumed value of 2.0 for K has been used. This results in the 1-, 2-, 4-, and 8-grain fragment particles representing 12.5%, 37.5%, 37.5% and 12.5% of the particles respectively. February 2010 Vol. 27 No. 1 32

10 The procedure used in making the section area composition frequency to volumetric composition frequency transformation is based on solving a series of 16 simultaneous linear equations, with one equation for each section area composition range. In these equations the individual frequencies of volume compositions (unknowns) are multiplied by the portion of the section area composition distribution of a specific composition range (known) and summed to be equal to the individual frequency of the sample s specific area composition range (known). This result provides 16 linear equations in 16 unknowns for solution. Following is the general form of these equations: For I = 0 to 15 where NSA (I) is the total frequency of particles with section area composition in Range I, NV (II) is the estimated total frequency of particles with volumetric composition in Range II, FK (J) is the relative quantity of particle types (1, 2, 4, 8 fragment) J = 0, 1, 2, 3; if II = I = 0 or 15 then summation is for J = 0-3, else summation is for J = 1-3 and VSA (II,I,J) is the proportion of frequency in volumetric composition range II, measured as section area composition range I, particle type J. (5) The equations defining FK (J) are FK(0) = (K-1) 3 / K 3 (6) FK(1) = 3*(K-1) 2 /K 3 (7) FK(2) = 3*(K-1)/K 3 (8) FK(3) = 1/K 3 (9) There are four 16-by-16 element matrices used in the transformation, as designated by VSA( II, I, J), one for each of the particle types (J). The values of the transformation matrices (VSA (II, I, J)) are given in Table 2 for the 2-, 4- and 8-grain fragment particles, with the matrix for the 1-fragment particles being elementary. The values of FK (J), as indicated in Equations 6 through 9 above, are a function of the ratio of mineral grain size to particle size (K). It turns out that in those transformation calculations made to date, the magnitude of the variable K has had little effect. As an example, Fig. 9 compares the use of this transformation procedure on a set of simulation data, where the original section area cumulative frequency data is compared with the actual volumetric cumulative frequency data and several versions of the estimated volumetric cumulative frequency data. To test the sensitivity of the calculations, the values of 1, 2 and 4 were assumed as the value of K, while the data was actually developed for a value of 2. The figure shows there was little effect on the transformation. The occurrence of small negative numbers in the solution of the 16 simultaneous linear equations can be avoided by the use of a linear programming approach. Summary and conclusions This paper uses simulation techniques to show comparisons between true volumetric particle composition and estimates of composition obtained from mineral areas of particle sections, exposed perimeter of mineral grains and randomly positioned linear intercepts across particle sections. The Rubik cube arrangement of the Gaudin random mineral liberation particles provides an understandable, very much idealized, yet mathematically definable representation of locked and liberated binary mineral particles. The particles contain from one to eight mineral grain fragments, depending on the amount of liberation that has taken place. The use of analytical techniques permits the simulation of the slicing of these particles with randomly oriented sectioning planes to produce the mathematical equivalent of a polished section. The areal and exposed peripheral compositions of these simulated particle sections can be calculated and compared to the true volumetric values of the original GRLM particles. This permits one to make judgments regarding the value of real sectioned particle measurements when making decisions where the volumetric or gravimetric particle composition is usually the variable of importance. Finally, a technique is presented for transforming the two-dimensional areal composition estimates of sectioned particles into three-dimensional volumetric GRLM values. This volumetric-based data is believed to be more meaningful information on which to make mineral processing evaluations and decisions. Acknowledgments The author wishes to express appreciation to Drs. Bill Whiten and Rob Morrison of the JKMRC staff of the University of Queensland, Australia, for their helpful discussion and encouragement when this project was initiated. References Barbery, G., 1991, Mineral Liberation Measurement, Simulation and Practical Use in Mineral Processing, Quebec, Canada: Editions GB. Barbery, G., and Leroux, D., 1988, Prediction of particle composition distribution after fragmentation of heterogeneous materials, International Journal of Mineral Processing, Vol. 22, pp Bole, J., Lin, C. L., and Miller, J.D., 1993, Experimental verification of the PARGEN simulator for liberation analysis, International Journal of Mineral Processing, Vol. 37, pp Gaudin, A. M., 1939, Principles of Mineral Dressing, New York, McGraw-Hill, pp Hill, G. S., Rowlands, N., and Finch, J.A., 1987, Data correction in two-dimensional liberation studies, in Process Mineralogy VII, Vassiliou, A., Hausen, D., and Carson, D., eds., Warrendale, PA, The Metallurgical Society, pp Jones, M. F., and Horton, R., 1978, Recent developments in the stereological assessment of composites (middling) particles by linear measurements, Proceedings 11th Commonwealth Mining and Metallurgical Congress, London, Institution of Mining and Metallurgy, pp King, R. P., and Schneider, C.L., 1995, Discussion of Lin, Gomez, Finch paper. In Trans. Instn. Min. Metall. (Sect. C: Mineral Process. Extr. Metall.), Vol. 106, pp Latti, D., B., and Adair, J.I., 2001, An assessment of stereological adjustment procedures, Minerals Engineering, Vol. 14, pp Lin, D., Gomez, C.O., and Finch, J.A., 1995, Comparison of stereological correction procedures for liberation measurements, Trans. Instn. Min. Metall. (Sect. C: Mineral Process. Extr. Metall.), Vol. 104, pp Miller, J. D., and Lin, C.L., 1988, Treatment of polished section data for detailed liberation analysis, International Journal of Mineral Processing, Vol. 22, pp Miller, J. D., and Lin, C.L., 2004, Three dimensional analysis of particulates in mineral processing systems by cone beam microtomography, Minerals and Metallurgical Processing, Vol. 21, pp Wiegel, R. L., and Li, K., 1967, A random model for mineral liberation by size reduction, AIME Transactions, Vol. 238, pp Wiegel, R. L., 1975, Liberation of magnetite iron formations, AIME Transactions, Vol. 258, No. 3, pp Wiegel, R. L., 2002, Size reduction/mineral liberation simulation for a magnetic taconite concentrator, Mineral and Metallurgical Processing, Vol. 19, No. 3, pp Wiegel, R. L., 2006, The rationale behind the development of one model describing the size reduction/liberation of ores, in Advances in Comminution, Komar Kawatra, S., ed., Littleton, CO, Society for Mining and Engineering, pp Vol. 27 No. 1 February 2010