Mineral Processing 345 Particle Sizing, Liberation, and Comminution Theory PDF

Mineral Processing 345 Particle sizing, liberation and comminution theory 1 Introduction In Week 4, metallurgical concepts such as grade, recovery, process efficiency, and basic mass balance considerations were discussed. The role of comminution and the importance of libe...

Mineral Processing 345 Particle sizing, liberation and comminution theory 1 Introduction In Week 4, metallurgical concepts such as grade, recovery, process efficiency, and basic mass balance considerations were discussed. The role of comminution and the importance of liberation in the performance of mineral processing plants were also explained. This week the following aspects related to comminution and liberation receive further attention: Describing the properties of material that consists of many particles, each with different properties (i.e. particle populations and property distribution functions). Analysing the grade of individual particles and estimating the degree of liberation using empirical correlations. Understanding particle breakage mechanisms, the energy requirements of comminution processes and associated empirical correlations, and how to perform mass balances on a particle size class basis (i.e. population balances). 2 Property distributions Ores consist of many particles that differ from one another in many respects: the differences that are of interest in mineral processing operations are those physical properties that influence the behaviour of a particle when subjected to treatment in any mineral processing unit operation. The two most important fundamental properties are the size of the particle and its mineralogical composition. The operation of comminution and classification are primarily dependent on the size of the particles, but the various physical properties are not independent of each other; for example, the specific gravity of a single particle is uniquely fixed once the mineralogical composition is specified. The distribution function for a particular property defines quantitatively how the values of that property are distributed among the particles in the entire population. Property distribution functions will be discussed in the context of particle sizes, but the same principles (with reference to the density distribution function, cumulative function and average property, as depicted in Equations 1 to 4) apply to other properties such as particle grade and density. Concepts related to particle size and particle size distributions, as well as statistical methods applied to describe the properties of particle mixtures, have been discussed in Particle Technology 316. The same principles apply to ore particles in the context of mineral processing. In addition, the prescribed textbook in this course has a comprehensive discussion on particle size analysis (refer to the supplementary reading references in Section 5 for more information) which will not be repeated here, but some key concepts are briefly summarised: Work carefully through the section “Sieve analysis” on pp. 91-97 of the prescribed textbook to ensure understanding of the sieve analysis method. Pay particular attention to the methodology/test procedure as well as the reporting and presentation of results in tabular and graphical form. Non-spherical particles have long and short dimensions in the various coordinate directions. A particle larger than the sieve-hole in the one dimension may fall through the sieve-hole because of other dimensions smaller than the sieve-hole. Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 1 of 18  A sieve-series is a sieve system where respective mesh sizes grow with a factor equal to the square root of 2. Thus, if a sieve has a mesh size of 75 µm, the following sieve has a rougher mesh size of 106 µm (√2 × 75). The average size of the particles above a sieve is then projected as the geometric average between the determined sieve and the following rougher sieve. It is determined by the square root of the product of the two mesh sizes. The geometric average size of particles that lie in the 75-106 µm fraction is thus 89.2 µm (√75 × 106). Material that falls through the 150 µm sieve, but remains on the 106 µm sieve is called the plus 106 µm minus 150 µm fraction (+106 – 150 µm). In the process of sieving, particles are repeatedly introduced to the sieve openings until they fall through or the process is terminated. As with all particle based separation processes, this procedure is not ideal and misplacement of the undersize in the oversize takes place. The particle size of an ore sample is often specified as a certain percentage passing a certain sieve size, for example a P80 size of 100 µm implies that 80% of the particles will pass a 100 µm sieve. A particle size distribution is separated in / specified according to the different sieve-classes. A population density function, f(l), is the differential representation of the mass fraction of the original material that occurs within a particular mesh-size interval (or is associated with a determined geometric average mesh-size) (see figure 4.7 in the prescribed textbook). The cumulative particle size distribution, F(l), represents the total mass fraction of particles smaller than a particular mesh-size in the population: it is the integral of the population density function (Equation 1), if the function is continuous, or the sum of the mass fractions of the mesh-size intervals, in the event of a discreet population density distribution (see figure 4.5 in the prescribed textbook). For the population density function f(l) for particles with geometric average particle size, l, the corresponding cumulative function, F(l), is given by: 𝑙𝑙 𝐹𝐹 (𝑙𝑙 ) = 𝑓𝑓(𝑙𝑙)𝑑𝑑𝑑𝑑 0 The value of the cumulative particle size distribution, F(l), increases monotonically from zero to one as l increases from zero to infinitely large. The average particle size (or the expected value of function f(l)) is: 𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚 𝑙𝑙 ̅ = 𝑙𝑙 ∙ 𝑓𝑓(𝑙𝑙)𝑑𝑑𝑑𝑑 , or 0 𝑙𝑙𝑚𝑚𝑚𝑚𝑚𝑚 𝑙𝑙 ̅ = 𝑙𝑙 ∙ 𝑑𝑑𝑑𝑑(𝑙𝑙) in the case of continuous functions, or 0 𝑛𝑛 𝑙𝑙 ̅ = 𝑙𝑙𝑖𝑖 ∙ ∆𝐹𝐹 (𝑙𝑙𝑖𝑖 ) where ∆𝐹𝐹 (𝑙𝑙𝑖𝑖 ) is the mass fraction material with geometric mean size 𝑙𝑙𝑖𝑖 𝑖𝑖=1 Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 2 of 18 Example 1 Analysis of an ore sample yields the following sieve analysis results: Screen 1180 850 600 425 300 212 150 106 75 53 38 0 size (µm) Mass on 6.8 1.4 2.6 6.2 13.5 27.3 40.0 39.0 16.8 11.4 6.0 33.7 screen (g) a) Plot the discrete particle size density distribution as well as the cumulative particle size distribution. b) Calculate the average particle size of the sample. c) Determine the P80 size. a) To calculate the density distribution and the cumulative size distribution, it is necessary to calculate the mass fraction of particles in each of the size classes. Start by calculating the total mass of the sample being analysed: 𝑛𝑛 𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇 = 𝑚𝑚𝑖𝑖 = 204.7 𝑔𝑔 𝑖𝑖=1 For each size class, the mass fraction material in that particular size class can be calculated as follows (results shown as column (5) in the results table): 𝑚𝑚𝑖𝑖 ∆𝐹𝐹 (𝑙𝑙𝑖𝑖 ) = 𝑚𝑚 𝑇𝑇𝑇𝑇𝑇𝑇 The discrete particle size population density distribution graph shows the mass fraction per size class (column(5)) plotted against the geometric mean of particles in the respective size classes. Note that for the largest size class, the mean particle size is simply taken as the only known sieve size; in the case of the smallest size, the arithmetic mean instead of the geometric mean of zero and the smallest sieve size is used to obtain a non-zero mean size. The cumulative size distribution (as the cumulative mass percentage of material passing size i) can be calculated from the density distribution function. Consider the largest sieve size (1180 µm): 3.3 % of the material remains on this sieve, which means that the rest of the material (i.e. 100 – 3.3 = 96.7 %) must past this sieve size. For the next sieve size (850 µm), it is known that apart from the 3.3 % of material larger than 1180 µm there is also 0.7% additional material that do not pass the 850 µm sieve; thus, in total, 4 % of material do not past the 850 µm sieve size (or 96.7 – 0.7 = 96 % does pass the 850 µm sieve). Continuing in this fashion yields the cumulative size distribution shown in column (6) of the solutions table. These values are plotted against the lower screen size for the corresponding size class to give the cumulative size distribution graph. Another approach to determine the cumulative size distribution would have been to calculate the cumulative distribution at a certain size interval as the sum of the mass fractions of material in the finer mesh-size intervals. Starting at the smallest sieve size, it is clear from the density distribution (column (5)) that 16.5% of material passes through the smallest sieve size and this also represents the cumulative percentage passing because it is the smallest sieve. Moving to the next (larger) size class, 2.9 % of the material is smaller than 53 µm but larger than 38 µm. Taking this material together with the 16.5% of material that is smaller than 38 µm yields a total (cumulative) percentage of 19.4 % of the material that is less than 53 µm. Again, continuing in this fashion will Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 3 of 18  yield the cumulative size distribution in column (6). This is probably the easiest approach because it involves addition. (1) (2) (3) (4) (5) (6) (5)/100 x (3) Upper Lower Geometric Mass on Mass % in Cumulative % Average size screen (µm) screen (µm) mean (µm) screen (g) class i passing class i calculation 1180 1180.0 6.8 3.3 96.7 39.2 1180 850 1001.5 1.4 0.7 96.0 6.8 850 600 714.1 2.6 1.3 94.7 9.1 600 425 505.0 6.2 3.0 91.7 15.3 425 300 357.1 13.5 6.6 85.1 23.5 300 212 252.2 27.3 13.3 71.8 33.6 212 150 178.3 40 19.5 52.2 34.8 150 106 126.1 39 19.1 33.2 24.0 106 75 89.2 16.8 8.2 25.0 7.3 75 53 63.0 11.4 5.6 19.4 3.5 53 38 44.9 6 2.9 16.5 1.3 38 0 19.0 33.7 16.5 3.1 204.7 201.7 Particle size density distribution 25 20 Weight % in class i 15 10 5 0 19.0 44.9 63.0 89.2 126.1 178.3 252.2 357.1 505.0 714.1 1001.5 1180.0 Mean particle size (µm) Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 4 of 18  Cumulative particle size distribution 100 90 80 Cumulative % passing 70 60 50 40 30 20 10 0 0 200 400 600 800 1000 1200 1400 Sieve size (µm) b) The average particle size is calculated using Equation 4. The calculation is shown as the last column in the results table, with the weighted average of the particle size distribution giving an average particle size of 201.7 µm. c) The P80 size is defined as the sieve size through which 80% of the material will pass. This value can be read directly off the cumulative size distribution graph as approximately 260 µm (the size corresponding to 80 % passing). Alternative, the data for the size classes directly above (85.1 % passing 300 µm) and below (71.8 % passing 212 µm) the P80 size can be used to determine the P80 size by linear interpolation: 300 − 212 𝑃𝑃80 = 212 + (80 − 71.8) = 266.3 µ𝑚𝑚 85.1 − 71.8 3 Particle grade and liberation The concept of liberation was introduced in Week 4 (Section 2 of the lecture notes). Given the dependence of metallurgical efficiency on the degree of liberation, it would be useful to determine what percentage of the total quantity of a mineral type is liberated completely, when the mineral-bearing ore particle falls within a certain sieve size class. Stated differently, it would be useful to know how far the size reduction process must proceed before a particular level of liberation is obtained. It is not the purpose of this course to determine the grain-size distribution of unliberated mineral particles in the middlings, thus only the average composition of the middlings will generally be determined. Furthermore, for the purpose of quantifying the degree of liberation in this course, the minerals in ore particles will be classified into two types only, namely: (1) the valuable minerals of interest (called mineral in further discussions), and (2) valueless minerals (called gangue). All valuable minerals are classified together as mineral, while all valueless minerals are combined in the gangue classification. The particle population is conceived as consisting of three groups, namely: (1) completely liberated particles of gangue, (2) completely liberated particles of mineral, and (3) the remainder of the particles which are all composed of mixtures of the two minerals (middlings). When describing a population of particles that have a distribution of mineral content, parameters that are essential to provide a description of the population include the average grade of mineral in the Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 5 of 18 population, the mass fraction of the population that consists of completely liberated gangue particles, and the mass fraction of the population that consists of completely liberated mineral. 3.1 Analytical techniques Traditionally, the degree of liberation was determined by means of microscopy coupled with measurements of the linear intersects of mineral grains and ore grains. These measurements could be used to further determine the volumetric relationships of mineral to gangue (and hence estimate the volumetric grade of particles). An image of a cross-section of a particle is considered, as shown in Figure 1. If the grey-shaded area (area B) represents the mineral and area A the gangue component of the particle, the particle grade can be expressed on an area basis (Figure 1(a)) or on a linear basis (Figure 1(b)): 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐵𝐵 Area grade of section through particle = 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐴𝐴 + 𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝐵𝐵 𝑏𝑏 Linear grade of line through particle section = 𝑎𝑎 + 𝑏𝑏 (a) (b) (c) Figure 1. Illustration of particle grade estimation measurements for a single particle: (a) area grade distribution, (b) linear grade distribution, and (c) effect of line probe or section position on estimation. (Redrawn from: Woollacott and Eric, 1994) From Figure 1(c) it is evident that the position of the line probe through the particle section can have a significant effect on the estimated linear grade and a wide range of linear grades for a single particle is possible. Similarly, the position (height) at which the cross-section of the particle is viewed could impact the estimated particle grades. This dependence of analysis results on the selection of the line probe or section location could lead to inaccurate particle grade estimations, which is known as stereological bias; the ore texture affects the degree of stereological bias (the bias is most significant if the mineral grain size is comparable to the particle size). Techniques to compensate for stereological bias exist, but discussion thereof falls outside the scope of this course. More advanced image processing algorithms and automated analytical techniques allow automated quantification of the average particle grades as well as the degree of liberation using systems such as the Mineral Liberation Analyser (MLA) and QEMSCAN, which were discussed in the geology section of this course. As discussed in Section 2, the grade property for a group of particles can be represented using the Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 6 of 18 same type of density distribution function, cumulative function and average property calculations previously discussed in the context of particle size. Because the tendency for liberated and nearly liberated particles to appear is greatly enhanced when the particle size is significantly smaller than the size of the mineral grains, the grade density distribution function exhibits a strong u-shape (i.e. there is a large fraction liberated mineral and a large fraction liberated gangue). On the other hand, when the particle size is distinctly larger than the sizes of the mineral grains within the ore, most particles in the population exhibit a mineral grade close to the mean value for the ore as a whole (i.e. a large fraction of material is present as middlings) and the density distribution is bell-shaped. The average particle grade (𝜆𝜆̅) can be calculated using Equation 7: 𝑛𝑛 𝜆𝜆̅ = 𝜆𝜆𝑖𝑖 ∙ ∆𝐹𝐹 (𝜆𝜆𝑖𝑖 ) where ∆𝐹𝐹 (𝜆𝜆𝑖𝑖 ) is the mass fraction material with average grade 𝜆𝜆𝑖𝑖 𝑖𝑖=1 Example 2 Analysis of an ore sample yields the following cumulative particle grade distribution: % particle 0 10 20 30 40 50 60 70 80 90 100 grade, λ Cumulative fraction with 0.2 0.5478 0.6215 0.6727 0.7137 0.7488 0.7803 0.8095 0.8375 0.8657 0.9 grade less than λ, F(λ) Calculate the average particle grade. In this example, the calculation of the average particle grade as the weighted average of the particle grade distribution using Equation 7 is analogous to the calculation of the average particle size in Example 1. In this particular example, however, the particle grade density distribution must first be calculated from the cumulative particle grade distribution. From the data provided, it is known that 20% of the material consists of liberated gangue (i.e. the particle grade is zero). Cumulatively, 54.78% of the particles has a grade of less than 10%; this implies that the difference between these two values (i.e. 54.78 – 20 = 34.78 %) represents the percentage particles with a grade between zero and 10%. Similarly, because 62.15 % of particles have a grade less than 20%, it implies that 7.73 % (62.15 – 54.78) of particles have a grade between 10 % and 20 %. Continuing in this fashion yields the particle grade distribution function shown in column (4) of the results table. The average grade per grade class is taken as arithmetic mean of the class limits, resulting in the average grade contributions and overall average grade of 28.5 % as shown in column (5). (1) (2) (3) (4) (5) = (2) x (4) Average fractional Grade range (%) F(𝜆𝜆) Fraction in range, ΔF(𝜆𝜆) 𝜆𝜆 x ΔF grade, 𝜆𝜆 0 0 0.2 0.2 0.0000 0-10 0.05 0.5478 0.3478 0.0174 10-20 0.15 0.6215 0.0737 0.0111 20-30 0.25 0.6727 0.0512 0.0128 30-40 0.35 0.7137 0.041 0.0144 40-50 0.45 0.7488 0.0351 0.0158 Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 7 of 18  50-60 0.55 0.7803 0.0315 0.0173 60-70 0.65 0.8095 0.0292 0.0190 70-80 0.75 0.8375 0.028 0.0210 80-90 0.85 0.8657 0.0282 0.0240 90-100 0.95 0.9 0.0343 0.0326 100 1 1 0.1 0.1000 1.0 0.285 3.2 Liberation models In many applications, the fraction of mineral that is completely liberated (i.e. the fraction of mineral present in particles that consist only of the specific mineral) is of primary interest. This is referred to as fractional liberation (Λ), and can be predicted as a function of particle size using liberation models such as the model proposed by Peter King. Before projecting the fractional liberation of a mineral, a few terms and the model’s background are discussed. King’s liberation model is based on the premise that an ore is a random composition of mineral and gangue grains, with clear, observable grain boundaries between proximate grains. The ore-, mineral-, and gangue grains assume any random shape and size, but each mineral grain is single phase in terms of the type of mineral considered. The model is solely designed for binary isotropic mixtures of mineral and gangue. Isotropic refers to the mineral grains having all the same properties in all directions. It is thus possible to use the binary model to evaluate the liberation of all the minerals in a multi-mineral ore, by completing a binary analysis on every respective mineral. The model predicts the liberation of minerals from ore grains by using only the linear intersects of the various minerals in the ore. The use of linear intersects means the model is not influenced by mineral grain form or intergrowth. The model also accepts that breaking occurs randomly all over grains and not along grain borders. The linear intersects can be measured microscopically by measuring and polishing ore grains inside a particular sieve size class. The grain borders between various minerals in the ore grain then become apparent, and the distribution of line intersects for a specific mineral can be determined. A great number of test lines are observed across the ore, stretching across the mineral grains and surrounding gangue matrix. A single test line for a single particle is shown in Figure 2, where the grey areas represent the mineral grains and the open areas represent the gangue in the particle. The line crosses the mineral and gangue grain borders with alternating intervals. The linear distances that cover the line segments between two grain borders, i.e. the solid line portions of the test line in Figure 2, are measured. Given the mineral grains in the ore grain are not all evenly sized, and a great number of lines are used across a large number of ore grains, a distribution of linear intercept lengths of the mineral grains is obtained. This is the population density distribution of linear mineral intercept lengths (fm(l)). The cumulative linear intercept length distribution is Fm(l). The m subscript refers to the mineral and not to the ore grain. Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 8 of 18  Figure 2. Schematic illustration of linear intercept length measurements. Note that the use of geometric average particle sizes does not imply that all the grains within a sieve interval are the same size. The ore grains themselves adopt an intersection length distribution for a given mesh-size. It is a conditional probability function of ore grains with size l that fall within the sieve interval with square geometric mesh-size D x D; this distribution is referred to as n(l|D). The cumulative linear intersect length distribution of particles with mesh-size D is N(l|D). Further, let 𝑙𝑙𝑚𝑚 be the average mineral grain intercept length and 𝑙𝑙 𝑔𝑔 the average gangue grain intercept length (where the averages from the distributions are calculated as discussed in Section 2). Accept further that the so-called Rosiwal relationship applies; this relationship states that the relationship of the mineral fractions to the gangue is equivalent to the linear intersects of mineral to gangue, or: 𝑉𝑉𝑚𝑚 𝑙𝑙𝑚𝑚 = 𝑉𝑉𝑔𝑔 𝑙𝑙 𝑔𝑔 King’s model suggests that the fractional liberation of mineral and gangue can be predicted from the dispersion of linear intersects of mineral and gangue grains and the sieve class with mesh-size D. This liberation is predicted as follows from the cumulative functions, N(l|D) and Fm(l): 𝐷𝐷 1 Λ𝑚𝑚 (𝐷𝐷) = 1 − {[1 − 𝑁𝑁(𝑙𝑙|𝐷𝐷)] ∙ [1 − 𝐹𝐹𝑚𝑚 (𝑙𝑙 )]} 𝑑𝑑𝑑𝑑 𝑙𝑙𝑚𝑚 0 If particles are assumed to be spherical, it can be shown that: 𝑙𝑙 2 𝑁𝑁(𝑙𝑙|𝐷𝐷) = 𝐷𝐷2 The population density distribution of mineral grains in an ore often takes an exponential decline, so that the fine grains appear the most liberated and then decline exponentially as the mineral grains increase in size. Thus: 1 −𝑙𝑙 𝑓𝑓𝑚𝑚 (𝑙𝑙 ) = exp 𝑙𝑙𝑚𝑚 𝑙𝑙𝑚𝑚 and 𝑙𝑙 𝐹𝐹𝑚𝑚 (𝑙𝑙 ) = 𝑓𝑓𝑚𝑚 (𝑙𝑙 )𝑑𝑑𝑑𝑑 0 so that: Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 9 of 18  −𝑙𝑙 𝐹𝐹𝑚𝑚 (𝑙𝑙 ) = 1 − exp 𝑙𝑙𝑚𝑚 By substitution of the expressions for N(l|D) and Fm(l) in Equation 9 followed by integration between the named borders, the fractional liberation of the mineral can be projected as: 2𝑙𝑙 𝑚𝑚 −𝐷𝐷 Λ𝑚𝑚 (𝐷𝐷) = 2 𝑙𝑙𝑚𝑚 − 𝑙𝑙 𝑚𝑚 + 𝐷𝐷 exp 𝐷𝐷 𝑙𝑙𝑚𝑚 The liberation across all particle sizes can be obtained by integrating Λm(D) across all mesh-sizes: 𝐷𝐷𝑚𝑚𝑚𝑚𝑚𝑚 Λ𝑚𝑚 = Λ𝑚𝑚 (𝐷𝐷) ∙ 𝑓𝑓(𝐷𝐷)𝑑𝑑𝑑𝑑 0 Where f(D)dD is the dispersion that represents the mass fraction in every particle size class. Dmax is the maximum particle size present in the population. In all cases, the integrands can be replaced by summation figures, where discreet rather than continual variables are worked with. 4 Comminution 4.1 Background Comminution is the process where the particle size of materials declines under the imposition of breaking forces. Comminution, or size reduction of particles, is an important unit process in the chemical and metallurgical industry for the following reasons: The natural sizes of particles do not lend themselves to easy processing down the line. The average particle size is too large for the client, requiring the material to be milled first. Valuable minerals or materials that occur concreted in a useless matrix must be liberated so that the mineral can be refined. The manner in which particles break when they are subjected to an external force and the way the particle size distribution will look after breaking will depend on the form, manner (and rate) with which the breaking power acts on the particle (ore/mineral/crystal/material), as well as the form, size and microstructure of the particle. The microstructure refers to the quantity of each phase present, the distribution of grains within the particle, the physical or mechanical properties of the individual minerals/phases and the occurrence of already-present defects like cracks and hairline fractures. There are many different ways in which forces can be applied that will crack or break materials. The principles of comminution are discussed on p. 109 of the prescribed textbook. In short, stress has to be applied to provide the energy required for crack propagation and particle breakage. Fracturing will occur when the local strain energy at the crack tip is sufficient to provide the surface energy of the two new surfaces produced by the fracture. Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 10 of 18 4.2 Fracture mechanisms Commercial equipment uses a combination of different methods of stress application to operationalise breaking: (1) slow compression, (2) fast compression or impact, and (3) friction. Every stress application method leads to a different daughter particle size distribution from the original mother material. Slow compression leads to breakage predominantly by a mechanism known as cleavage. Fracture by cleavage occurs when the energy applied is just sufficient to load comparatively few regions of the particle to the fracture point. Stresses are created throughout the material, and the stress level is heightened until the induced stresses are just enough to propagate through the existing cracks. Breaking only occurs at single places, resulting in the formation of only a few daughter particles; the daughter particles have a narrow particle size distribution (particles tend to all be the same size) with a relatively large average particle size. Impact (like a hammer striking a particle) leads to breakage predominantly by shatter; in this case, the applied energy is well in excess of that required for fracture. Under these conditions many areas in the particle are overloaded and the result is a comparatively large number of particles with a wide spectrum of sizes; the average particle size is significantly smaller than the original material. Breakage occurring as a results of friction between particles or between the particle and equipment surfaces is known as breakage by the abrasion mechanism. This type of breakage occurs when large shearing stresses are present at the particle’s surface layer, resulting in localised breakage at the surface area; this leads to a bi-modal particle size distribution: one peak (mode) with an average particle size that is almost as big as the original mother material, as well as a peak or mode at a very small average particle size. Figure 1 in the article by Hasan et al. (2017) (available on SunLearn) illustrates the different breakage mechanisms and resulting particle distributions. 4.3 Comminution theories Comminution theory is concerned with the relationship between energy input and the particle size produced from a given feed size. The oldest theory is that of Von Rittinger (1867), which states that the energy consumed in the size reduction is proportional to the area of new surface produced. The surface area of a known weight of particles of uniform diameter is inversely proportional to the diameter. Hence, Rittinger law equates to: 1 1 𝐸𝐸 = 𝐶𝐶𝑅𝑅 − 𝑙𝑙2 𝑙𝑙1 where: E is the work input per mass of material CR is the Rittinger law constant (determined experimentally or from literature) l1 and l2 are the feed and product particle diameters, respectively Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 11 of 18 The second theory is that of Kick (1885), who stated that the energy required to reduce the size of particles is proportional to the ratio of the initial to the final dimension. The energy required is then given by Equation 17, with the symbols defined as for Von Rittinger’s law (with CK being the Kick law constant): 𝑙𝑙1 𝐸𝐸 = 𝐶𝐶𝐾𝐾 ln 𝑙𝑙2 Bond (1952) developed a theory that the work input is proportional to the new crack tip length produced in particle breakage. In particles of similar shape, the surface area of a unit volume of material is inversely proportional to the diameter. The crack length in a unit volume is considered to be proportional to one side of that area and therefore inversely proportional to the square root of the diameter. In general terms, then: 1 1 𝐸𝐸 = 𝐶𝐶𝐵𝐵 − 𝑙𝑙2 𝑙𝑙1 where l1 and l2 are the particle sizes (in µm) before and after breakage, respectively, or the sieve sizes (in µm) through which 80% of the feed and product, respectively, passes. For practical calculations and given the wide application of the Bond’s equation, so-called Bond work indices (Wi) (equivalent to the Bond equation constant, CB) have been measured and published for a range of different materials. The Bond work index is defined as the energy (measured in kWh per short ton, where 1 short ton = 2000 lb) required to break the material from a theoretically infinite feed size to 80% passing 100 µm. In more practical terms, Equation 18 is then given as Equation 19: 1 1 𝐸𝐸 = 10𝑊𝑊𝑖𝑖 − 𝑃𝑃80 𝐹𝐹80 where: E is the work input in kWh per short ton of material Wi is the Bond work index, as defined above (values for selected minerals are provided in Table 5.1 of the prescribed textbook) P80 is the size (in µm) which 80% of the product passes F80 is the size (in µm) which 80% of the feed passes There are limitations to the use of the empirical equations described above to calculate energy requirements of comminution operations, and the accuracy of results depend on the quality of data to which the law constants have been fitted. In general, Bond’s theory applies reasonably well in the particle size range of conventional rod and ball mill grinding, while Kick’s law is reasonably accurate in the crushing particle size range of 1 cm and larger. Rittinger’s law applies in a relatively broad particle size range of`10 µm to 10 mm. Example 3 Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 12 of 18  An ore is milled from 6 mm to 0.12 mm particle diameter using a mill with a 7.5 kW motor. What reduction in throughput rate would you expect if the material was reduced to 0.08 mm instead of 0.12 mm? Assume Rittinger’s law applies. Considering the Rittinger law equation, it is clear that this problem requires a two-step approach: (1) use the initial operating conditions (feed size, product size and power) to calculate the Rittinger constant, and (2) use the Rittinger constant from part (1) to determine the power requirement at the revised product size. The initial material flow rate and the flow rate at the revised product size are denoted MF0 (t/s) and MF1 (t/s), respectively, to distinguish between the operating conditions and to calculate the reduction in material flow rate. The work per mass for the initial conditions is calculated as: 𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃𝑃 7.5 kJ 𝐸𝐸 = = t 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 𝑓𝑓𝑓𝑓𝑓𝑓𝑓𝑓 𝑀𝑀𝑀𝑀1 From Equation 16, for the initial conditions: 7.5 1 1 1 1 = 𝐶𝐶𝑅𝑅 − = 𝐶𝐶𝑅𝑅 − 𝑀𝑀𝑀𝑀1 𝑙𝑙2 𝑙𝑙1 0.12 6 7.5 0.918 kJ∙mm ∴ 𝐶𝐶𝑅𝑅 = = t 1 1 𝑀𝑀𝑀𝑀 𝑀𝑀𝑀𝑀1 − 1 0.12 6 Using the same approach for the revised product size specification, and remembering that the mill power remains constant: 7.5 1 1 0.918 1 1 = 𝐶𝐶𝑅𝑅 − = − 𝑀𝑀𝑀𝑀2 𝑙𝑙2 𝑙𝑙1 𝑀𝑀𝑀𝑀1 0.08 6 7.5𝑀𝑀𝑀𝑀1 ∴ 𝑀𝑀𝑀𝑀2 = = 0.662𝑀𝑀𝑀𝑀1 1 1 0.918 − 0.08 6 From the above result, it can be concluded that the throughput rate at the finer product size is 66.2% of the throughput rate at the larger product size. Or, stated different, reducing the product particle size from 0.12 mm to 0.08 mm for the fixed mill power will lead to a 33.8% reduction in throughput. 4.4 Population balances Using the comminution theories, it is not possible to predict the resulting particle size distribution. An empirical approach must thus be followed to predict particle size distribution, based on laboratory and plant determined measurements that include the daughter product distributions and the kinetics of milling. It is then possible to describe the breaking process quantitatively, based on analysis of particle size distributions that arise from different breaking processes under controlled circumstances. Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 13 of 18 The general technique used to model the comminution process is called population balances; this is a form of mass balancing for particulate processes that keeps track of a particulate property (size, in this specific case) as a function of the particulate space. With reference to Figure 3, consider a process where the feed material to a mill consists of uniformly sized particles (size class 1 at t0). After a certain milling time, particles from size class 1 have undergone size reduction to the extent that they move to size class 2 and smaller sizes (i.e. the particles in size class 2 at t1 appear as a result of particles breaking into size class 2 from size class 1). The number of particles that leaves size class 1 from t0 to t1 is determined by the specific rate of breakage for class 1 (or the probability of particles in size class 1 breaking per unit time); this probability/rate of breakage is known as the selection parameter for size class 1, s1. The selection function, as a function of industrial mill operating conditions, can be determined by tests under controlled conditions that are similar to the process being modelled. Of the particles that break from size class 1, the daughter particles can form in any of the smaller size classes. The manner in which the daughter particles report to the smaller size classes is specified by the breakage distribution function, b: bij specifies which fraction of the particles that do break from size j (larger particle size class) reports to size i (smaller particle size class). If, as an example and with specific reference to the breakage during the period t0 to t1 in Figure 3, b31 = 0.6, it means that 60% of the material that break out of class 1 report to size class 3. Considering the next time step from t1 to t2, the material in the mill at t1 contains particles in different size classes. For size class 1, particles can only break out of the specific size class (i.e. particles do not increase in size during milling) and the amount of material in this size class will decrease with milling time. For size classes smaller than the largest size classes and larger than the smallest size class (i.e. classes 2 and 3 in the case of Figure 3) there are two ways in which the amount of material in the size class can change: Particles can break out of the specific size class thereby reducing the amount of material in the size class. The amount of material that breaks out from a size class is specified by the selection function for that specific size class. Particles can break into the size class as a result of breakage of particles initially in larger size classes. In the case of class 3, for example, particles can enter this size as a result of breakage of particles in classes 1 and 2. The amounts of particles that enter the size class under consideration depend on the selection function and distribution function of the respective larger size classes. If the rate of particles breaking into the size class from larger size classes is greater than the rate of breakage from the particular size class, the net effect will be an increase in the amount of material in the size class, and vice versa. For the smallest size class (class 4), material can only break into the size class and the amount of material in this class will increase over time (as long as there are particles in one or more of the larger size classes). Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 14 of 18  Figure 3. Schematic illustration of size reduction and population balance concepts. (Redrawn from: Kelly and Spottiswood, 1982) Now consider an arbitrary particle size distribution of particles in a process. For any given time interval dt, a portion of particles of size class i will be broken. The breaking of particles takes place in a series of breakage events. Breaking of the original particle leads to a number of smaller particles, the proportion in each particle size class determined by the breakage function bij. These particles are also broken in turn, leading to a new distribution of particles, and so the process is repeated. If size class i is considered, which receives particles broken from larger size classes and whose particles are themselves broken further into smaller particle size classes, then a mass balance can be constructed that caters for time interval dt. The rate of mass accumulation in particle size class i is equal to the rate of formation of particles in size class i (as a result of breakage of particles in classes 1 to j, given j < i) (the first term on the right hand side of Equation 20) minus the rate of breakage of particles in size class i (the second term on the right hand side of Equation 20), i.e.: 𝑖𝑖−1 𝑑𝑑 (𝐻𝐻𝑓𝑓𝑖𝑖 ) = 𝑏𝑏𝑖𝑖𝑖𝑖 𝑠𝑠𝑗𝑗 𝐻𝐻𝑓𝑓𝑗𝑗 (𝑡𝑡) − 𝑠𝑠𝑖𝑖 𝐻𝐻𝑓𝑓𝑖𝑖 (𝑡𝑡) 𝑑𝑑𝑑𝑑 𝑗𝑗=1 where fi(t) is the mass fraction of material of size class i at time t, and H is the total mass of the material that is processed. By dividing Equation 20 by the total mass of material being processed, an expression for rate of change of the mass fraction of material in size class i is obtained: 𝑖𝑖−1 𝑑𝑑𝑓𝑓𝑖𝑖 = 𝑏𝑏𝑖𝑖𝑖𝑖 𝑠𝑠𝑗𝑗 𝑓𝑓𝑗𝑗 (𝑡𝑡) − 𝑠𝑠𝑖𝑖 𝑓𝑓𝑖𝑖 (𝑡𝑡) 𝑑𝑑𝑑𝑑 𝑗𝑗=1 Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 15 of 18 The expansion of the breaking process is then determined by the simultaneous solution of differential equations over the whole size distribution. The solution of these equations is normally achieved with a computer. Under certain assumptions, analytical solutions can be determined for certain types of comminution processes. For example, in the case of a continuous comminution process operating at steady state, integration of Equation 21 yields the following expression for the mass fraction of material in size class i in the mill outlet (product) stream, fout,i: 𝑖𝑖−1 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 = 𝑓𝑓𝑖𝑖𝑖𝑖,𝑖𝑖 + 𝜏𝜏 𝑏𝑏𝑖𝑖𝑖𝑖 𝑠𝑠𝑗𝑗 𝑓𝑓𝑗𝑗 − 𝜏𝜏𝑠𝑠𝑖𝑖 𝑓𝑓𝑖𝑖 𝑗𝑗=1 where: fin,i is the mass fraction of material in size class i in the mill inlet (feed) stream τ is the residence time of the material in the mill fi is the mass fraction of material in size class i in the mill In the case where the selection and breakage functions are defined based on the particle size distribution of the feed and the size distribution in the mill is assumed to approximate the size distribution of the feed (i.e. fi = fin,i), Equation 22 simplifies to: 𝑖𝑖−1 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 = 𝑓𝑓𝑖𝑖𝑖𝑖,𝑖𝑖 + 𝜏𝜏 𝑏𝑏𝑖𝑖𝑖𝑖 𝑠𝑠𝑗𝑗 𝑓𝑓𝑖𝑖𝑖𝑖,𝑗𝑗 − 𝜏𝜏𝑠𝑠𝑖𝑖 𝑓𝑓𝑖𝑖𝑖𝑖,𝑖𝑖 𝑗𝑗=1 On the other hand, if perfect mixing of the mill is assumed so that the size distribution of the outlet stream is assumed to be the same as the size class distribution inside the mill (i.e. fout,i = fi), rearranging Equation 22 yields: 𝑓𝑓𝑖𝑖𝑖𝑖,𝑖𝑖 + 𝜏𝜏 ∑𝑖𝑖−1 𝑗𝑗=1 𝑏𝑏𝑖𝑖𝑖𝑖 𝑠𝑠𝑗𝑗 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,𝑗𝑗 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,𝑖𝑖 = 1 + 𝜏𝜏𝑠𝑠𝑖𝑖 Example 4 The selection and breakage functions of a particular material in a ball mill are shown below: Size interval (µm) 212-150 150-106 106-75 75-53 53-37 37-0 Interval number 1 2 3 4 5 6 sj (min-1) 0.07 0.06 0.05 0.035 0.03 0 b(1,j) 0 0 0 0 0 0 b(2,j) 0.32 0 0 0 0 0 b(3,j) 0.3 0.4 0 0 0 0 b(4,j) 0.14 0.2 0.5 0 0 0 b(5,j) 0.12 0.2 0.25 0.6 0 0 b(6,j) 0.12 0.2 0.25 0.4 1 0 Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 16 of 18  Material with the size distribution shown below is ground in the mill; at the selected mill operating conditions, the ore residence time is 10 minutes. Predict the size distribution of the mill product if perfect mixing in the mill can be assumed. Interval number 1 2 3 4 5 6 Fraction in feed 0.2 0.4 0.3 0.06 0.04 0 To calculate the size distribution of the product produced by the perfectly mixed mill, start with size interval number 1. For the largest size class, particles break out of the size class but there is no particles breaking into the size class, so that Equation 24 simplifies to: 𝑓𝑓𝑖𝑖𝑖𝑖,1 0.2 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,1 = = = 0.118 1 + 𝜏𝜏𝑠𝑠1 1 + (10)(0.07) For size classes 2 to 5, breakage into and out of the size classes need to be taken into account. Using size interval 2 as an example: 𝑓𝑓𝑖𝑖𝑖𝑖,2 + 𝜏𝜏 ∑1𝑗𝑗=1 𝑏𝑏2𝑗𝑗 𝑠𝑠𝑗𝑗 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,𝑗𝑗 𝑓𝑓𝑖𝑖𝑖𝑖,2 + 𝜏𝜏𝑏𝑏21 𝑠𝑠1 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,1 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,2 = = 1 + 𝜏𝜏𝑠𝑠2 1 + 𝜏𝜏𝑠𝑠2 0.4 + (10)(0.32)(0.07)(0.118) ∴ 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,2 = = 0.266 1 + (10)(0.06) Similarly: 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,3 = 0.259 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,4 = 0.125 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,5 = 0.108 For the smallest size class, interval 6, particle only break into the size (i.e. s6 = 0) so that Equation 24 simplifies to: 5 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,6 = 𝑓𝑓𝑖𝑖𝑖𝑖,6 + 𝜏𝜏 𝑏𝑏5𝑗𝑗 𝑠𝑠𝑗𝑗 𝑓𝑓𝑜𝑜𝑜𝑜𝑜𝑜,𝑗𝑗 = 0.124 𝑗𝑗=1 5 Supplementary reading Topic Prescribed textbook reference* Comments Study carefully to understand the sieve analysis Property methodology as well as reporting of the data as “Sieve analysis”, pp. 91-97 distributions density distributions and cumulative distribution functions. Particle grade and None liberation Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 17 of 18  “Principles of comminution”, pp. 109- Read carefully to understand the theory of Fracture 110 particle breakage. mechanisms Refer to figure 1 in this article in the context of Hasan et al. (2017)** (See link below) the lecture notes. Comminution Read the discussion in the textbook for “Comminution theory”, pp. 110-111 theories additional insight. Population “Simulation of comminution processes Most of the concepts related to population balances and circuits”, pp. 112-114 balances are described in the notes. Read the discussion in the textbook for additional insight. *https://www-sciencedirect-com.ez.sun.ac.za/book/9780750644501/wills-mineral-processing-technology **https://doi.org/10.1016/j.mineng.2017.06.024 6 Acknowledgements Parts of these notes have been taken from course material prepared by Professors AJ Burger and JJ Eksteen. Notes prepared by C Dorfling Particle sizing, liberation and comminution theory Page 18 of 18