## particle size distribution of grinding mill products

Size analyses of mineral products are usually made by screening with a set of sieves having mean apertures arranged in the Tyler scale, which is a geometric progression with a constant ratio equal to the square root of two. For this reason it is convenient to represent particle size as a logarithmic function of the particle diameter

The variable, x, may be defined as the logarithm to the base s of the ratio of the particle diameter to the reference diameter, d. The inverse of the transformation given in equation 1 is given by equation 2:

where yi is the weight fraction in the size interval between xi-l and xi; N is the total number of size intervals. If the reference diameter is equal to the maximum sieve aperture represented in the screen analysis and this is taken as the axis for the first moment, xo is zero and the first moment is the value of x corresponding to the mean particle size:

These higher moments contain the essential information concerning the form of the distribution function; that is, they are parameters that measure various aspects of the way in which the weight fraction is distributed about the mean. The second moment, or variance, measures the width of the distribution; the third moment measures the skewness or asymmetry; the fourth moment measures the kurtosis or peakedness; etc. Obviously, higher moments are increasingly dependent on values of the weight fraction remote from the mean which represent only a small fraction of the sample. For this reason, moments beyond the fourth are usually not considered. The third and fourth moments are usually represented as dimensionless ratios with respect to the second moment:

Moment analysis is a well-established method for analyzing and characterizing statistical distribution. Quoting from one of the standard treatises in mathematical statistics: For all ordinary purposes, therefore, a knowledge of the moments, when they exist, is equivalent to a knowledge of the distribution function: equivalent, that is, in the sense that it should be possible theoretically to exhibit all the properties of the distribution in terms of the moments.The procedures that have been developed for computation of moments and deriving distribution functions that best fit the data are fully discussed in a book by Elderton which has recently been published in a new edition. The application of moment analyses to particle size distribution data is discussed in a previous paper by the author.

When uniform sized samples of cryptocrystalline quartz were crushed by impact in a drop-weight machine, the reduction in size, as measured by the change in the mean value of x, was found to follow the relationship in equations 8 and 9

Equation 8 is an empirical relationship derived by multiple regression analyses of various combinations of independent variables from the data on the impact crushing experiments. The logarithms of E, the impact energy per unit weight of mineral, and d0, the initial mean particle diameter, gave the greatest decrease in variance between the observed and expected values. The threshold energy, Eo, which is the energy corresponding to zero size reduction, varies approximately inversely as the diameter of the particle being crushed.

Analyses of other data obtained by Hukki, using a double pendulum apparatus for impact crushing, gave slightly larger values for the coefficients in equations 8 and 9 and the exponent of do was slightly larger than one. If the exponent of do is assumed to be unity, the equation derived for the combined data from the Hukki and Bureau of Mines experiments gives the equations 10 and 11

These equations represent a best fit of data from 39 experiments on impact crushing of quartz particles ranging in size from .156 to 5.75 cm in diameter. The standard error for x is .387 as compared with 1.192 for the observed data.

The form of the distribution curve was found to be primarily a function of the size reduction. The distribution function of the initial uniform sample is a delta function. As soon as any size reduction occurs, there is a transfer of material to a wide range of sizes thereby increasing the dispersion. The change in distribution is asymetric since crushing transfers material only to the finer sizes, and so the skewness, as measured by 1, initially has a very large value. In other words, as a uniform sample is progressively decreased in size, there is a progressive increase in dispersion as measured by 2 and a decrease in skewness as measured by 1.

These changes are shown in Figure 1 which depicts the changes in distribution that occur in size distribution when quartz samples are subjected to increasing impact energy. As size reduction proceeds, the effect of the initial uniform distribution is gradually obliterated and both ant! tend to approach steady values. The variance tends to approach a steady value of 18 which corresponds to a standard deviation of 4.25 or a little more than four intervals in the Tyler sieve scale. The skewness levels off at a value between 1.6 and 1.8.

The manner in which energy is employed in a ball mill to effect size reduction is much more complex than in simple impact crushing. The application of energy in a drop-weight machine is similar to that occurring in a stamp mill, in which a considerable size reduction is produced by a single impact of relatively high energy input. In a tumbling mill, such as a ball mill, the size reduction occurs by repeated application of kinetic energy of less intensity by impact and rolling action of a large number of crushing media, in certain circumstances an appreciable amount of size reduction may occur by abrasion.

A series of studies has been made at the University of California in Berkeley on the grinding of dolomite in a ball mill. These tests were made in a specially designed mill in which the input energy can be measured.

A series of impact crushing experiments was made on a sample of the same dolomite used by Berlioz. The results of those experiments are summarized in Table 1. The results of these experiments permit comparison of the size reduction by impact and by ball mill grinding for the same energy input. Differences in the behavior of the particle size distribution may also be observed.

A comparison of the size reduction by impact and ball mill grinding is shown in figure 2. The ball mill experiments show the size reduction versus energy input for 7 series of experiments in which the weight of dolomite was 660, 1320, 1980, 2640, 3300, 3960, and 5420, respectively. The charge was composed of equal portions of -7 +8 and -8 +10 size fractions. The ball load was 455 stainless steel balls one inch in diameter, having a total weight of 30 kg. The particle size distribution was observed after 20, 40, 60, 80, 100, 150, 200 and 300 ball mill revolutions. The energy input was calculated from the net torque (corrected for the torque for the empty mill) and the number of revolutions.

For light ball loads the ball mill is less efficient as might be expected. As the load exceeds 1980 gm it has no influence on size reduction. The relationship between size reduction and input for optimum grinding in the ball mill is represented by the solid curve. The corresponding relationship for crushing by impact is represented by broken lines.

It is evident, as might be expected, that for moderate size reduction (x<5) the input energy is utilized much more effectively in the drop-weight machine than in the ball mill. For example, reduction to one-half of the original particle size requires about 4-3 kg cm per gm by impact as compared to 18.5 kg-cm by ball mill; that is, the energy requirements are less than 25 percent as much for impact crushing as for ball milling. For great size reduction the application of energy by a single impact becomes less effective so that there would be no advantage in applying more impact energy than would be required for about a four-fold reduction of particle diameter; this is equivalent to x = k. The energy required would be about percent that required for an equivalent size reduction in the ball mill.

The change in particle size distribution during ball milling is shown by the graphs of the variance and skewness in Figure 3. Here again, the points representing variance for those experiments with light ball mill loads show a noticeable divergence from those for optimum loading. The skewness is not affected by the load.

The differences in distribution between the products from the ball mill and those from impact crushing can be observed by comparing the solid lines, which represent the ball mill products, with the broken lines, which represent the products from the drop-weight experiments. The greatest difference is in the variance. The products from the ball mill show considerably greater dispersion over the size range and this dispersion develops sooner as the mean size is decreased. On the other hand, the skewness decreases earliest for impact crushing. The differences in size distribution are essentially difference in degree of dispersion and asymmetry. Actually there is a surprising similarity in the evolution of the distribution from uniform particle size to a typical skewed bell-shaped curve as the size is reduced whether by the ball mill or impact crushing.

The comminution process may be represented mathematically in terms of a set of state variables that define the mean size and size distribution of the feed to and product from a grinding machine. Three variables are required to specify the particulate state of the mineral undergoing size reduction; these variables define quantitatively the mean particle size, degree of dispersion and asymmetry of the distribution. Such a set of variables may be calculated by moment analysis of the screen analysis of the aggregate being studied. Conversely, if the mean, variance, and skewness of a size distribution are known, a gamma distribution function may be derived that will approximate the weight fraction in any given size range.

When a sized fraction of particles is subjected to comminution, the size distribution undergoes progressive change characterized by a steady increase in the variance or second moment and an abrupt decrease in skewness or third moment. As size reduction proceeds further, both the variance and skewness approach steady values and the form of the distribution becomes more and more stable.

## ball milling - an overview | sciencedirect topics

Ball milling is often used not only for grinding powders but also for oxides or nanocomposite synthesis and/or structure/phase composition optimization [14,41]. Mechanical activation by ball milling is known to increase the material reactivity and uniformity of spatial distribution of elements [63]. Thus, postsynthesis processing of the materials by ball milling can help with the problem of minor admixture forming during cooling under air after high-temperature sintering due to phase instability.

Ball milling technique, using mechanical alloying and mechanical milling approaches were proposed to the word wide in the 8th decade of the last century for preparing a wide spectrum of powder materials and their alloys. In fact, ball milling process is not new and dates back to more than 150 years. It has been used in size comminutions of ore, mineral dressing, preparing talc powders and many other applications. It might be interesting for us to have a look at the history and development of ball milling and the corresponding products. The photo shows the STEM-BF image of a Cu-based alloy nanoparticle prepared by mechanical alloying (After El-Eskandarany, unpublished work, 2014).

Ball milling, a shear-force dominant process where the particle size goes on reducing by impact and attrition mainly consists of metallic balls (generally Zirconia (ZrO2) or steel balls), acting as grinding media and rotating shell to create centrifugal force. In this process, graphite (precursor) was breakdown by randomly striking with grinding media in the rotating shell to create shear and compression force which helps to overcome the weak Vander Waal's interaction between the graphite layers and results in their splintering. Fig. 4A schematic illustrates ball milling process for graphene preparation. Initially, because of large size of graphite, compressive force dominates and as the graphite gets fragmented, shear force cleaves graphite to produce graphene. However, excessive compression force may damage the crystalline properties of graphene and hence needs to be minimized by controlling the milling parameters e.g. milling duration, milling revolution per minute (rpm), ball-to-graphite/powder ratio (B/P), initial graphite weight, ball diameter. High quality graphene can be achieved under low milling speed; though it will increase the processing time which is highly undesirable for large scale production.

Fig. 4. (A) Schematic illustration of graphene preparation via ball milling. SEM images of bulk graphite (B), GSs/E-H (C) GSs/K (D); (E) and (F) are the respective TEM images; (G) Raman spectra of bulk graphite versus GSs exfoliated via wet milling in E-H and K.

Milling of graphite layers can be instigated in two states: (i) dry ball milling (DBM) and (ii) wet ball milling (WBM). WBM process requires surfactant/solvent such as N,N Dimethylformamide (DMF) [22], N-methylpyrrolidone (NMP) [26], deionized (DI) water [27], potassium acetate [28], 2-ethylhexanol (E-H) [29] and kerosene (K) [29] etc. and is comparatively simpler as compared with DBM. Fig. 4BD show the scanning electron microscopy (SEM) images of bulk graphite, graphene sheets (GSs) prepared in E-H (GSs/E-H) and K (GSs/K), respectively; the corresponding transmission electron microscopy (TEM) images and the Raman spectra are shown in Fig. 4EG, respectively [29].

Compared to this, DBM requires several milling agents e.g. sodium chloride (NaCl) [30], Melamine (Na2SO4) [31,32] etc., along with the metal balls to reduce the stress induced in graphite microstructures, and hence require additional purification for exfoliant's removal. Na2SO4 can be easily washed away by hot water [19] while ammonia-borane (NH3BH3), another exfoliant used to weaken the Vander Waal's bonding between graphite layers can be using ethanol [33]. Table 1 list few ball milling processes carried out using various milling agent (in case of DBM) and solvents (WBM) under different milling conditions.

Ball milling of graphite with appropriate stabilizers is another mode of exfoliation in liquid phase.21 Graphite is ground under high sheer rates with millimeter-sized metal balls causing exfoliation to graphene (Fig. 2.5), under wet or dry conditions. For instance, this method can be employed to produce nearly 50g of graphene in the absence of any oxidant.22 Graphite (50g) was ground in the ball mill with oxalic acid (20g) in this method for 20 hours, but, the separation of unexfoliated fraction was not discussed.22 Similarly, solvent-free graphite exfoliations were carried out under dry milling conditions using KOH,23 ammonia borane,24 and so on. The list of graphite exfoliations performed using ball milling is given in Table 2.2. However, the metallic impurities from the machinery used for ball milling are a major disadvantage of this method for certain applications.25

Reactive ball-milling (RBM) technique has been considered as a powerful tool for fabrication of metallic nitrides and hydrides via room temperature ball milling. The flowchart shows the mechanism of gas-solid reaction through RBM that was proposed by El-Eskandarany. In his model, the starting metallic powders are subjected to dramatic shear and impact forces that are generated by the ball-milling media. The powders are, therefore, disintegrated into smaller particles, and very clean or fresh oxygen-free active surfaces of the powders are created. The reactive milling atmosphere (nitrogen or hydrogen gases) was gettered and absorbed completely by the first atomically clean surfaces of the metallic ball-milled powders to react in a same manner as a gas-solid reaction owing to the mechanically induced reactive milling.

Ball milling is a grinding method that grinds nanotubes into extremely fine powders. During the ball milling process, the collision between the tiny rigid balls in a concealed container will generate localized high pressure. Usually, ceramic, flint pebbles and stainless steel are used.25 In order to further improve the quality of dispersion and introduce functional groups onto the nanotube surface, selected chemicals can be included in the container during the process. The factors that affect the quality of dispersion include the milling time, rotational speed, size of balls and balls/ nanotube amount ratio. Under certain processing conditions, the particles can be ground to as small as 100nm. This process has been employed to transform carbon nanotubes into smaller nanoparticles, to generate highly curved or closed shell carbon nanostructures from graphite, to enhance the saturation of lithium composition in SWCNTs, to modify the morphologies of cup-stacked carbon nanotubes and to generate different carbon nanoparticles from graphitic carbon for hydrogen storage application.25 Even though ball milling is easy to operate and suitable for powder polymers or monomers, process-induced damage on the nanotubes can occur.

Ball milling is a way to exfoliate graphite using lateral force, as opposed to the Scotch Tape or sonication that mainly use normal force. Ball mills, like the three roll machine, are a common occurrence in industry, for the production of fine particles. During the ball milling process, there are two factors that contribute to the exfoliation. The main factor contributing is the shear force applied by the balls. Using only shear force, one can produce large graphene flakes. The secondary factor is the collisions that occur during milling. Harsh collisions can break these large flakes and can potentially disrupt the crystal structure resulting in a more amorphous mass. So in order to create good-quality, high-area graphene, the collisions have to be minimized.

The ball-milling process is common in grinding machines as well as in reactors where various functional materials can be created by mechanochemical synthesis. A simple milling process reduces both CO2 generation and energy consumption during materials production. Herein a novel mechanochemical approach 1-3) to produce sophisticated carbon nanomaterials is reported. It is demonstrated that unique carbon nanostructures including carbon nanotubes and carbon onions are synthesized by high-speed ball-milling of steel balls. It is considered that the gas-phase reaction takes place around the surface of steel balls under local high temperatures induced by the collision-friction energy in ball-milling process, which results in phase separated unique carbon nanomaterials.

Conventional ball milling is a traditional powder-processing technique, which is mainly used for reducing particle sizes and for the mixing of different materials. The technique is widely used in mineral, pharmaceutical, and ceramic industries, as well as scientific laboratories. The HEBM technique discussed in this chapter is a new technique developed initially for producing new metastable materials, which cannot be produced using thermal equilibrium processes, and thus is very different from conventional ball milling technique. HEBM was first reported by Benjamin [38] in the 1960s. So far, a large range of new materials has been synthesized using HEBM. For example, oxide-dispersion-strengthened alloys are synthesized using a powerful high-energy ball mill (attritor) because conventional ball mills could not provide sufficient grinding energy [38]. Intensive research in the synthesis of new metastable materials by HEBM was stimulated by the pioneering work in the amorphization of the Ni-Nb alloys conducted by Kock et al. in 1983 [39]. Since then, a wide spectrum of metastable materials has been produced, including nanocrystalline [40], nanocomposite [41], nanoporous phases [42], supersaturated solid solutions [43], and amorphous alloys [44]. These new phase transformations induced by HEBM are generally referred as mechanical alloying (MA). At the same time, it was found that at room temperature, HEBM can activate chemical reactions which are normally only possible at high temperatures [45]. This is called reactive milling or mechano-chemistry. Reactive ball milling has produced a large range of nanosized oxides [46], nitrides [47], hydrides [48], and carbide [49] particles.

The major differences between conventional ball milling and the HEBM are listed in the Table 1. The impact energy of HEBM is typically 1000 times higher than the conventional ball milling energy. The dominant events in the conventional ball milling are particle fracturing and size reductions, which correspond to, actually, only the first stage of the HEBM. A longer milling time is therefore generally required for HEBM. In addition to milling energy, the controls of milling atmosphere and temperature are crucial in order to create the desired structural changes or chemical reactions. This table shows that HEBM can cover most work normally performed by conventional ball milling, however, conventional ball milling equipment cannot be used to conduct any HEBM work.

Different types of high-energy ball mills have been developed, including the Spex vibrating mill, planetary ball mill, high-energy rotating mill, and attritors [50]. In the nanotube synthesis, two types of HEBM mills have been used: a vibrating ball mill and a rotating ball mill. The vibrating-frame grinder (Pulverisette O, Fritsch) is shown in Fig. 1a. This mill uses only one large ball (diameter of 50 mm) and the media of the ball and vial can be stainless steel or ceramic tungsten carbide (WC). The milling chamber, as illustrated in Fig. 1b, is sealed with an O-ring so that the atmosphere can be changed via a valve. The pressure is monitored with an attached gauge during milling.

where Mb is the mass of the milling ball, Vmax the maximum velocity of the vial,/the impact frequency, and Mp the mass of powder. The milling intensity is a very important parameter to MA and reactive ball milling. For example, a full amorphization of a crystalline NiZr alloy can only be achieved with a milling intensity above an intensity threshold of 510 ms2 [52]. The amorphization process during ball milling can be seen from the images of transmission electron microscopy (TEM) in Fig. 2a, which were taken from samples milled for different lengths of time. The TEM images show that the size and number of NiZr crystals decrease with increasing milling time, and a full amorphization is achieved after milling for 165 h. The corresponding diffraction patterns in Fig. 2b confirm this gradual amorphization process. However, when milling below the intensity threshold, a mixture of nanocrystalline and amorphous phases is produced. This intensity threshold depends on milling temperature and alloy composition [52].

Figure 2. (a) Dark-field TEM image of Ni10Zr7 alloy milled for 0.5, 23, 73, and 165 h in the vibrating ball mill with a milling intensity of 940 ms2. (b) Corresponding electron diffraction patterns [52].

Fig. 3 shows a rotating steel mill and a schematic representation of milling action inside the milling chamber. The mill has a rotating horizontal cell loaded with several hardened steel balls. As the cell rotates, the balls drop onto the powder that is being ground. An external magnet is placed close to the cell to increase milling energy [53]. Different milling actions and intensities can be realized by adjusting the cell rotation rate and magnet position.

The atmosphere inside the chamber can be controlled, and adequate gas has to be selected for different milling experiments. For example, during the ball milling of pure Zr powder in the atmosphere of ammonia (NH3), a series of chemical reactions occur between Zr and NH3 [54,55]. The X-ray diffraction (XRD) patterns in Fig. 4 show the following reaction sequence as a function of milling time:

The mechanism of a HEBM process is quite complicated. During the HEBM, material particles are repeatedly flattened, fractured, and welded. Every time two steel balls collide or one ball hits the chamber wall, they trap some particles between their surfaces. Such high-energy impacts severely deform the particles and create atomically fresh, new surfaces, as well as a high density of dislocations and other structural defects [44]. A high defect density induced by HEBM can accelerate the diffusion process [56]. Alternatively, the deformation and fracturing of particles causes continuous size reduction and can lead to reduction in diffusion distances. This can at least reduce the reaction temperatures significantly, even if the reactions do not occur at room temperature [57,58]. Since newly created surfaces are most often very reactive and readily oxidize in air, the HEBM has to be conducted in an inert atmosphere. It is now recognized that the HEBM, along with other non-equilibrium techniques such as rapid quenching, irradiation/ion-implantation, plasma processing, and gas deposition, can produce a series of metastable and nanostructured materials, which are usually difficult to prepare using melting or conventional powder metallurgy methods [59,60]. In the next section, detailed structural and morphological changes of graphite during HEBM will be presented.

Ball milling and ultrasonication were used to reduce the particle size and distribution. During ball milling the weight (grams) ratio of balls-to-clay particles was 100:2.5 and the milling operation was run for 24 hours. The effect of different types of balls on particle size reduction and narrowing particle size distribution was studied. The milled particles were dispersed in xylene to disaggregate the clumps. Again, ultrasonication was done on milled samples in xylene. An investigation on the amplitude (80% and 90%), pulsation rate (5 s on and 5 s off, 8 s on and 4 s off) and time (15 min, 1 h and 4 h) of the ultrasonication process was done with respect to particle size distribution and the optimum conditions in our laboratory were determined. A particle size analyzer was used to characterize the nanoparticles based on the principles of laser diffraction and morphological studies.

## analysis of particle size reduction on overall surface area and enzymatic hydrolysis yield of corn stover | springerlink

Particle size of lignocellulose materials is an important factor for enzymatic hydrolysis efficiency. In this study, corn stover was milled and sieved into different size fractions from 1.42, 0.69, 0.34, to 0.21mm, and the corresponding enzymatic hydrolysis yields were 24.69, 23.96, 25.34, and 26.97%, respectively. The results indicate that the hydrolysis yield is approximately constant with changing corn stover particle sizes in the experimental range. The overall surface area and the inner pore size measurement show that the overall specific surface area was less than 2% with the half reduction of particle size due to the greater inner pore surface area. The scanning electron microscope photographs gave direct evidence of the much greater inner pore surface area of corn stover particles. This result provided a reference when a proper size reduction of lignocellulose materials is considered in biorefining operations.

Ballesteros I, Oliva JM, Negro MJ, Manzanares P, Ballesteros M (2002) Enzymatic hydrolysis of steam exploded herbaceous agricultural waste (Brassica carinata) at different particle sizes. Process Biochem 38:187192

This research was supported by the Joint Training Program of Shanghai High School and ECUST, and the National Basic Research Program of China (2011CB707406), and the National High-Tech Program of China (2012AA022301/2014AA021901).

Li, H., Ye, C., Liu, K. et al. Analysis of particle size reduction on overall surface area and enzymatic hydrolysis yield of corn stover.
Bioprocess Biosyst Eng 38, 149154 (2015). https://doi.org/10.1007/s00449-014-1253-y

## ball mill - retsch - powerful grinding and homogenization

Ball mills are among the most variable and effective tools when it comes to size reduction of hard, brittle or fibrous materials. The variety of grinding modes, usable volumes and available grinding tool materials make ball mills the perfect match for a vast range of applications.

RETSCH is the world leading manufacturer of laboratory ball mills and offers the perfect product for each application. The High Energy Ball Mill Emax and MM 500 were developed for grinding with the highest energy input. The innovative design of both, the mills and the grinding jars, allows for continuous grinding down to the nano range in the shortest amount of time - with only minor warming effects. These ball mills are also suitable for mechano chemistry. Mixer Mills grind and homogenize small sample volumes quickly and efficiently by impact and friction. These ball mills are suitable for dry, wet and cryogenic grinding as well as for cell disruption for DNA/RNA recovery. Planetary Ball Mills meet and exceed all requirements for fast and reproducible grinding to analytical fineness. They are used for the most demanding tasks in the laboratory, from routine sample processing to colloidal grinding and advanced materials development. The drum mill is a type of ball mill suitable for the fine grinding of large feed sizes and large sample volumes.